Commun. Math. Phys. 299, 1–44 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1095-7
Communications in
Mathematical Physics
Long-Time Stability of Multi-Dimensional Noncharacteristic Viscous Boundary Layers∗ Toan Nguyen, Kevin Zumbrun Department of Mathematics, Indiana University, Bloomington, IN 47402, USA. E-mail:
[email protected];
[email protected] Received: 30 July 2008 / Accepted: 17 April 2010 Published online: 30 July 2010 – © Springer-Verlag 2010
Abstract: We establish long-time stability of multi-dimensional noncharacteristic boundary layers of a class of hyperbolic–parabolic systems including the compressible Navier–Stokes equations with inflow [outflow] boundary conditions, under the assumption of strong spectral, or uniform Evans, stability. Evans stability has been verified for small-amplitude layers by Guès, Métivier, Williams, and Zumbrun. For large-amplitude layers, it may be efficiently checked numerically, as done in the one-dimensional case by Costanzino, Humpherys, Nguyen, and Zumbrun. Contents 1.
2. 3.
4.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Equations and assumptions . . . . . . . . . . . . . . . 1.2 The Evans condition and strong spectral stability . . . 1.3 Main results . . . . . . . . . . . . . . . . . . . . . . . 1.4 Discussion and open problems . . . . . . . . . . . . . Resolvent Kernel: Construction and Low-Frequency Bounds 2.1 Construction . . . . . . . . . . . . . . . . . . . . . . . 2.2 Pointwise low-frequency bounds . . . . . . . . . . . . Linearized Estimates . . . . . . . . . . . . . . . . . . . . . 3.1 Resolvent bounds . . . . . . . . . . . . . . . . . . . . 3.2 Estimates on homogeneous solution operators . . . . . 3.3 Boundary estimates . . . . . . . . . . . . . . . . . . . 3.4 Duhamel formula . . . . . . . . . . . . . . . . . . . . 3.5 Proof of linearized stability . . . . . . . . . . . . . . . Nonlinear Stability . . . . . . . . . . . . . . . . . . . . . . 4.1 Auxiliary energy estimates . . . . . . . . . . . . . . .
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∗ This work was supported in part by the National Science Foundation award number DMS-0300487.
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4.2 Proof of nonlinear stability . . . . . . . . . . . Appendix A. Physical Discussion in the Isentropic Case A.1 Existence . . . . . . . . . . . . . . . . . . . . A.2 Stability . . . . . . . . . . . . . . . . . . . . . A.3 Discussion . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction We consider a boundary layer, or stationary solution, U˜ = U¯ (x1 ),
lim U¯ (z) = U+ , U¯ (0) = U¯ 0
z→+∞
of a system of conservation laws on the quarter-space F j (U˜ )x j = (B jk (U˜ )U˜ xk )x j , x ∈ Rd+ = {x1 > 0}, t > 0, U˜ t + j
(1.1)
(1.2)
jk
U˜ , F j ∈ Rn , B jk ∈ Rn×n , with initial data U˜ (x, 0) = U˜ 0 (x) and Dirichlet type boundary conditions specified in (1.5), (1.6) below. A fundamental question connected to the physical motivations from aerodynamics is whether or not such boundary layer solutions are stable in the sense of PDE, i.e., whether or not a sufficiently small perturbation of U¯ remains close to U¯ , or converges time-asymptotically to U¯ , under the evolution of (1.2). That is the question we address here. 1.1. Equations and assumptions. We consider the general hyperbolic-parabolic system of conservation laws (1.2) in conserved variable U˜ , with 0 0 u˜ ˜ U= , B= jk jk , v˜ b1 b2 u˜ ∈ Rn−r , and v˜ ∈ Rr , where jk b2 ξ j ξk ≥ θ |ξ |2 > 0, ∀ξ ∈ Rn \{0}. σ jk
Following [MaZ4,Z3,Z4], we assume that Eqs. (1.2) can be written, alternatively, after a triangular change of coordinates as ˜ w˜ I (u) ˜ ˜ ˜ , (1.3) W := W (U ) = w˜ I I (u, ˜ v) ˜ in the quasilinear, partially symmetric hyperbolic-parabolic form ˜ A˜ 0 W˜ t + ( B˜ jk W˜ xk )x j + G, A˜ j W˜ x j = j
jk
where, defining W˜ + := W˜ (U+ ), (A1) A˜ j (W˜ + ), A˜ 0 , A˜ 111 are symmetric, A˜ 0 block diagonal, A˜ 0 ≥ θ0 > 0,
(1.4)
Stability of Multi-Dimensional Boundary Layers
3
(A2) for each ξ ∈ Rd \ {0}, no eigenvector of j ξ j A˜ j ( A˜ 0 )−1 (W˜ + ) lies in the kernel of jk ξ j ξk B˜ jk ( A˜ 0 )−1 (W˜ + ), jk 0 0 0 jk 2 ˜ ˜ ˜ (A3) B = with g( ˜ W˜ x , W˜ x ) = , b ξ j ξk ≥ θ |ξ | , and G = g˜ 0 b˜ jk O(|W˜ x |2 ). Along with the above structural assumptions, we make the following technical hypotheses: (H0)
(H1)
(H2) (H3) (H4)
F j , B jk , A˜ 0 , A˜ j , B˜ jk , W˜ (·), g(·, ˜ ·) ∈ C s+1 , with s ≥ [(d − 1)/2] + 4 in our analysis of linearized stability, and s ≥ s(d) := [(d − 1)/2] + 7 in our analysis of nonlinear stability. ˜ 11 A˜ 11 1 is either strictly positive or strictly negative, that is, either A1 ≥ θ1 > 0, 11 or A˜ 1 ≤ −θ1 < 0. (We shall call these cases the inflow case or outflow case, correspondingly.) The eigenvalues of d F 1 (U+ ) are distinct and nonzero. j The eigenvalues of j d F+ ξ j have constant multiplicity with respect to ξ ∈ Rd , ξ = 0. The set of branch points of the eigenvalues of ( A˜ 1 )−1 (iτ A˜ 0 + j =1 iξ j A˜ j )+ , τ ∈ R, ξ˜ ∈ Rd−1 is the (possibly intersecting) union of finitely many smooth curves τ = ηq+ (ξ˜ ), on which the branching eigenvalue has constant multiplicity sq (by definition ≥ 2).
Condition (H1) corresponds to hyperbolic–parabolic noncharacteristicity, while (H2) is the condition for the hyperbolicity at U+ of the associated first-order hyperbolic system obtained by dropping second-order terms. The assumptions (A1)–(A3) and (H0)–(H2) are satisfied for gas dynamics and MHD with van der Waals equation of state under inflow or outflow conditions; see discussions in [MaZ4,CHNZ,GMWZ5,GMWZ6]. Condition (H3) holds always for gas dynamics, but fails always for MHD in dimension d ≥ 2. Condition (H4) is a technical requirement of the analysis introduced in [Z2]. It is satisfied always in dimension d = 2 or for rotationally invariant systems in dimensions d ≥ 2, for which it serves only to define notation; in particular, it holds always for gas dynamics. We also assume: (B) Dirichlet boundary conditions in W˜ -coordinates: ˜ x, (w˜ I , w˜ I I )(0, x, ˜ t) = h( ˜ t) := (h˜ 1 , h˜ 2 )(x, ˜ t)
(1.5)
for the inflow case, and ˜ x, ˜ t) = h( ˜ t) w˜ I I (0, x,
(1.6)
˜ ∈ Rd . for the outflow case, with x = (x1 , x) This is sufficient for the main physical applications; the situation of more general, Neumann and mixed-type boundary conditions on the parabolic variable v can be treated as discussed in [GMWZ5,GMWZ6].
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T. Nguyen, K. Zumbrun
Example 1.1. The main example we have in mind consists of laminar solutions (ρ, u, e) (x1 , t) of the compressible Navier–Stokes equations ⎧ ∂t ρ + div(ρu) = 0 ⎪ ⎪ ⎪ ⎨ ∂t (ρu) + div(ρu t u) + ∇ p = εμ u + ε(μ + η)∇divu (1.7) ⎪ ∂t (ρ E) + div ((ρ E + p)u) = εκ T + εμdiv ((u · ∇)u) ⎪ ⎪ ⎩ + ε(μ + η)∇(u · divu), x ∈ Rd , on a half-space x1 > 0, where ρ denotes density, u ∈ Rd velocity, e specific 2 internal energy, E = e + |u|2 specific total energy, p = p(ρ, e) pressure, T = T (ρ, e) temperature, μ > 0 and |η| ≤ μ first and second coefficients of viscosity, κ > 0 the coefficient of heat conduction, and ε > 0 (typically small) the reciprocal of the Reynolds number, with no-slip suction-type boundary conditions on the velocity, u j (0, x2 , . . . , xd ) = 0, j = 1 and u 1 (0, x2 , . . . , xd ) = V (x) < 0, ˜ Under the standard assumpand prescribed temperature, T (0, x2 , . . . , xd ) = Twall (x). tions pρ , Te > 0, this can be seen to satisfy all of the hypotheses (A1)–(A3), (H0)–(H4), (B) in the outflow case (1.6); indeed these are satisfied also under much weaker van der Waals gas assumptions [MaZ4,Z3,CHNZ,GMWZ5,GMWZ6]. In particular, boundarylayer solutions are of noncharacteristic type, ¯ u, ¯ e)(x ¯ 1 /ε), with √ scaling as (ρ, u, e) = (ρ, layer thickness ∼ ε as compared to the ∼ ε thickness of the characteristic type found for an impermeable boundary. This corresponds to the situation of an airfoil with microscopic holes through which gas is pumped from the surrounding flow, the microscopic suction imposing a fixed normal velocity while the macroscopic surface imposes standard temperature conditions as in flow past a (nonporous) plate. This configuration was suggested by Prandtl and tested experimentally by G.I. Taylor as a means to reduce drag by stabilizing laminar flow; see [S,Bra]. It was implemented in the NASA F-16XL experimental aircraft program in the 1990’s with reported 25% reduction in drag at supersonic√ speeds [Bra].1 Possible mechanisms for this reduction are smaller thickness ∼ ε 0, and prescribed temperature and pressure ˜ T (0, x2 , . . . , xd ) = Twall (x),
p(0, x2 , . . . , xd ) = pwall (x) ˜
(equivalently, prescribed temperature and density). Under the standard assumptions pρ , Te > 0 on the equation of state (alternatively, van der Waals gas assumptions), this can be seen to satisfy hypotheses (A1)–(A3), (H0)–(H4), (B) in the inflow case (1.5). 1 See also NASA site http://www.dfrc.nasa.gov/Gallery/photo/F-16XL2/index.html
Stability of Multi-Dimensional Boundary Layers
5
Lemma 1.3 ([MaZ3,Z3,GMWZ5,NZ]). Given (A1)–(A3) and (H0)–(H2), a standing wave solution (1.1) of (1.2), (B) satisfies
(1.8)
(d/d x1 )k (U¯ − U+ ) ≤ Ce−θ x1 , 0 ≤ k ≤ s + 1, as x1 → +∞, s as in (H0). Moreover, a solution, if it exists, is in the inflow or strictly parabolic case unique; in the outflow case it is locally unique.
Proof. See Lemma 1.3, [NZ].
1.2. The Evans condition and strong spectral stability. The linearized equations of (1.2), (B) about U¯ are Ut = LU := (B jk Uxk )x j − (A j U )x j , (1.9) j,k
j
B jk := B jk (U¯ (x1 )), A j U := A j (U¯ (x1 ))U −(d B j1 U )U¯ (x1 ), with initial data U (0) = U0 and boundary conditions in (linearized) W˜ -coordinates of ˜ t) = h W (0, x, ˜ t) := (w I , w I I )T (0, x, for the inflow case, and w I I (0, x, ˜ t) = h for the outflow case, with x = (x1 , x) ˜ ∈ Rd , where W := (∂ W˜ /∂U )(U¯ )U . A necessary condition for linearized stability is weak spectral stability, defined as nonexistence of unstable spectra λ > 0 of the linearized operator L about the wave. As described in Sect. 2.1.1, this is equivalent to nonvanishing for all ξ˜ ∈ Rd−1 , λ > 0 of the Evans function D L (ξ˜ , λ) (defined in (2.8)), a Wronskian associated with the Fourier-transformed eigenvalue ODE. Definition 1.4. We define strong spectral stability as uniform Evans stability: |D L (ξ˜ , λ)| ≥ θ (C) > 0
(D)
for (ξ˜ , λ) on bounded subsets C ⊂ {ξ˜ ∈ Rd−1 , λ ≥ 0}\{0}. For the class of equations we consider, this is equivalent to the uniform Evans condition of [GMWZ5,GMWZ6], which includes an additional high-frequency condition that for these equations is always satisfied (see Prop. 3.8, [GMWZ5]). A fundamental result proved in [GMWZ5] is that small-amplitude noncharacteristic boundary layers are always strongly spectrally stable.2 Proposition 1.5 ([GMWZ5]). Assuming (A1)–(A3), (H0)–(H3), (B) for some fixed endstate (or compact set of endstates) U+ , boundary layers with amplitude 2 The result of [GMWZ5] applies also to more general types of boundary conditions and in some situations to systems with variable multiplicity characteristics, including, in some parameter ranges, MHD.
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T. Nguyen, K. Zumbrun
U¯ − U+ L ∞ [0,+∞] sufficiently small satisfy the strong spectral stability condition (D). As demonstrated in [SZ], stability of large-amplitude boundary layers may fail for the class of equations considered here, even in a single space dimension, so there is no such general theorem in the large-amplitude case. Stability of large-amplitude boundary-layers may be checked efficiently by numerical Evans computations as in [BDG, Br1,Br2,BrZ,HuZ,BHRZ,HLZ,CHNZ,HLyZ1,HLyZ2]. 1.3. Main results. Our main results are as follows. Theorem 1.6 (Linearized stability). Assuming (A1)–(A3), (H0)–(H4), (B), and strong spectral stability (D), we obtain asymptotic L 1 ∩ H [(d−1)/2]+5 → L p stability of (1.9) in dimension d ≥ 2, and any 2 ≤ p ≤ ∞, with rate of decay |U (t)| L 2 ≤ C(1 + t)− |U (t)| L p ≤ C(1 + t)
d−1 4
(|U0 | L 1 ∩H 3 + E 0 ),
− d2 (1−1/ p)+1/2 p
(|U0 | L 1 ∩H [(d−1)/2]+5 + E 0 ),
(1.10)
provided that the initial perturbations U0 are in L 1 ∩H 3 for p = 2, or in L 1 ∩H [(d−1)/2]+5 for p > 2, and boundary perturbations h satisfy |h(t)| L 2 ≤ E 0 (1 + t)−(d+1)/4 , x˜
|h(t)| L ∞ ≤ E 0 (1 + t)−d/2 , x˜
(1.11)
|Dh (t)| L 1 ∩H [(d−1)/2]+5 ≤ E 0 (1 + t)−d/2− , x˜
x˜
where Dh (t) := |h t | + |h x˜ | + |h x˜ x˜ |, E 0 is some positive constant, and > 0 is arbitrary small for the case d = 2 and = 0 for d ≥ 3. Theorem 1.7 (Nonlinear stability). Assuming (A1)–(A3), (H0)–(H4), (B), and strong spectral stability (D), we obtain asymptotic L 1 ∩ H s → L p ∩ H s stability of U¯ as a solution of (1.2) in dimension d ≥ 2, for s ≥ s(d) as defined in (H0), and any 2 ≤ p ≤ ∞, with rate of decay |U˜ (t) − U¯ | L p ≤ C(1 + t)− 2 (1−1/ p)+1/2 p (|U0 | L 1 ∩H s + E 0 ), d
|U˜ (t) − U¯ | H s ≤ C(1 + t)−
d−1 4
(|U0 | L 1 ∩H s + E 0 ),
(1.12)
provided that the initial perturbations U0 := U˜ 0 − U¯ are sufficiently small in L 1 ∩ H s I I (U ˜ ˜ ¯ 0 )) satisfy (1.11) and boundary perturbations h(t) := h(t)−W (U¯ 0 ) (resp. h(t)−W and Bh (t) ≤ E 0 (1 + t)−
d−1 4
,
(1.13)
with sufficiently small E 0 , where the boundary measure Bh is defined as Bh (t) := |h| H s (x) ˜ +
[(s+1)/2] i=0
for the outflow case, and similarly
|∂ti h| L 2 (x) ˜
(1.14)
Stability of Multi-Dimensional Boundary Layers
Bh (t) := |h| H s (x) ˜ +
[(s+1)/2]
7
|∂ti h 2 | L 2 (x) ˜ +
i=0
s
|∂ti h 1 | L 2 (x) ˜
(1.15)
i=0
for the inflow case. Combining Theorem 1.7 and Proposition 1.5, we obtain the following small-amplitude stability result, applying in particular to the motivating situation of Example 1.1. Corollary 1.8. Assuming (A1)–(A3), (H0)–(H4), (B) for some fixed endstate (or compact set of endstates) U+ , boundary layers with amplitude U¯ − U+ L ∞ [0,+∞] sufficiently small are linearly and nonlinearly stable in the sense of Theorems 1.6 and 1.7. Remark 1.9. The obtained rate of decay in L 2 may be recognized as that of a (d − 1)dimensional heat kernel, and the obtained rate of decay in L ∞ as that of a d-dimensional heat kernel. We believe that the sharp rate of decay in L 2 is rather that of a d-dimensional heat kernel and the sharp rate of decay in L ∞ dependent on the characteristic structure of the associated inviscid equations, as in the constant-coefficient case [HoZ1,HoZ2]. Remark 1.10. In one dimension, strong spectral stability is necessary for linearized asymptotic stability; see Theorem 1.6, [NZ]. However, in multi-dimensions, it appears likely that, as in the shock case [Z3], there are intermediate possibilities between strong and weak spectral stability for which linearized stability might hold with degraded rates of decay. In any case, the gap between the necessary weak spectral and the sufficient strong spectral stability conditions concerns only pure imaginary spectra λ = 0 on the boundary between strictly stable and unstable half-planes, so this should not interfere with investigation of physical stability regions. 1.4. Discussion and open problems. Asymptotic stability, without rates of decay, has been shown for small amplitude noncharacteristic “normal” boundary layers of the isentropic compressible Navier–Stokes equations with outflow boundary conditions and vanishing transverse velocity in [KK], using energy estimates. Corollary 1.8 recovers this existing result and extends it to the general arbitrary transverse velocity, outflow or inflow, and isentropic or nonisentropic (full compressible Navier–Stokes) case, in addition giving asymptotic rates of decay. Moreover, we treat perturbations of boundary as well as initial data, as previous time-asymptotic investigations (with the exception of direct predecessors [YZ,NZ]) do not. As discussed in Appendix A, the type of boundary layer relevant to the drag-reduction strategy discussed in Examples 1.1–1.2 is a noncharacteristic “transverse” type with constant normal velocity, complementary to the normal type considered in [KK]. The large-amplitude asymptotic stability result of Theorem 1.7 extends to multi dimensions corresponding to one-dimensional results of [YZ,NZ], reducing the problem of stability to verification of a numerically checkable Evans condition. See also the related, but technically rather different, work on the small viscosity limit in [MZ, GMWZ5,GMWZ6]. By a combination of numerical Evans function computations and asymptotic ODE estimates, spectral stability has been checked for arbitrary amplitude noncharacteristic boundary layers of the one-dimensional isentropic compressible Navier–Stokes equations in [CHNZ]. Extensions to the nonisentropic and multi-dimensional case should be possible by the methods used in [HLyZ1] and [HLyZ2] respectively to treat the related shock stability problem.
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T. Nguyen, K. Zumbrun
This (investigation of large-amplitude spectral stability) would be a very interesting direction for further investigation. In particular, note that it is large-amplitude stability that is relevant to drag-reduction at flight speeds, since the transverse relative velocity (i.e., velocity parallel to the airfoil) is zero at the wing surface and flight speed outside a thin boundary layer, so that variation across the boundary layer is substantial. We discuss this problem further in Appendix A for the model isentropic case. Our method of analysis follows the basic approach introduced in [Z2,Z3,Z4] for the study of multi-dimensional shock stability and we are able to make use of much of that analysis without modification. However, there are some new difficulties to be overcome in the boundary-layer case. The main new difficulty is that the boundary-layer case is analogous to the undercompressive shock case rather than the more favorable Lax shock case emphasized in [Z3], in that G y1 ∼ t −1/2 G as in the Lax shock case but rather G y1 ∼ (e−θ|y1 | +t −1/2 )G, θ > 0, as in the undercompressive case. This is a significant difficulty; indeed, for this reason, the undercompressive shock analysis was carried out in [Z3] only in nonphysical dimensions d ≥ 4. On the other hand, there is no translational invariance in the boundary layer problem, so no zero-eigenvalue and no pole of the resolvent kernel at the origin for the one-dimensional operator, and in this sense G is somewhat better in the boundary layer than in the shock case. Thus, the difficulty of the present problem is roughly intermediate to that of the Lax and undercompressive shock cases. Though the undercompressive shock case is still open in multi-dimensions for d ≤ 3, the slight advantage afforded by lack of pole terms allows us to close the argument in the boundary-layer case. Specifically, thanks to the absence of pole terms, we are able to get a slightly improved rate of decay in L ∞ (x1 ) norms, though our L 2 (x1 ) estimates remain the same as in the shock case. By keeping track of these improved sup norm bounds throughout the proof, we are able to close the argument without using detailed pointwise bounds as in the one-dimensional analyses of [HZ,RZ]. Other difficulties include the appearance of boundary terms in integrations by parts, which makes the auxiliary energy estimates by which we control high-frequency effects considerably more difficult in the boundary-layer than in the shock-layer case, and the treatment of boundary perturbations. In terms of the homogeneous Green function G, boundary perturbations lead by a standard duality argument to contributions consisting of integrals on the boundary of perturbations against various derivatives of G, and these are a bit too singular as time goes to zero to be absolutely integrable. Following the strategy introduced in [YZ,NZ], we instead use duality to convert these to less singular integrals over the whole space, that are absolutely integrable in time. However, we make a key improvement here over the treatment in [YZ,NZ], integrating against an exponentially decaying test function to obtain terms of exactly the same form already treated for the homogeneous problem. This is necessary for us in the multi-dimensional case, for which we have insufficient information about individual parts of the solution operator to estimate them separately as in [YZ,NZ], but makes things much more transparent also in the one-dimensional case. Among physical systems, our hypotheses appear to apply to and essentially only to the case of compressible Navier–Stokes equations with inflow or outflow boundary conditions. However, the method of analysis should apply, with suitable modifications, to more general situations such as MHD; see for example the recent results on the related small-viscosity problem in [GMWZ5,GMWZ6]. The extension to MHD is a very interesting open problem.
Stability of Multi-Dimensional Boundary Layers
9
Finally, as pointed out in Remark 1.10, the strong spectral stability condition does not appear to be necessary for asymptotic stability. It would be interesting to develop a refined stability condition similarly as was done in [SZ,Z2,Z3,Z4] for the shock case. 2. Resolvent Kernel: Construction and Low-Frequency Bounds In this section, we briefly recall the construction of resolvent kernel and then establish the pointwise low-frequency bounds on G ξ˜ ,λ , by appropriately modifying the proof in [Z3] in the boundary layer context [YZ,NZ]. 2.1. Construction. We construct a representation for the family of elliptic Green distributions G ξ˜ ,λ (x1 , y1 ), G ξ˜ ,λ (·, y1 ) := (L ξ˜ − λ)−1 δ y1 (·),
(2.1)
associated with the ordinary differential operators (L ξ˜ − λ), i.e. the resolvent kernel of the Fourier transform L ξ˜ of the linearized operator L of (1.9). To do so, we study the homogeneous eigenvalue equation (L ξ˜ − λ)U = 0, or L 0U
(B 11 U ) − (A1 U ) −i A jξjU + i B j1 ξ j U j =1
j =1
+i (B 1k ξk U ) − B jk ξ j ξk U − λU = 0, k =1
(2.2)
j,k =1
with boundary conditions (translated from those in W -coordinates) 1 0 A11 − A112 (b211 )−1 b111 ∗ U (0) ≡ , 0 b11 b11 1
(2.3)
2
where ∗ = 0 for the inflow case and is arbitrary for the outflow case. Define ˜
ξ :=
n
+j (ξ˜ ),
j=1
where +j (ξ˜ ) denote the open sets bounded on the left by the algebraic curves λ+j (ξ1 , ξ˜ ) determined by the eigenvalues of the symbols −ξ 2 B+ − iξ A+ of the limiting constantcoefficient operators L ξ˜ + w := B+ w − A+ w as ξ1 is varied along the real axis, with ξ˜ held fixed. The curves λ+j (·, ξ˜ ) comprise the ˜ essential spectrum of operators L ˜ . Let denote the set of (ξ˜ , λ) such that λ ∈ ξ . ˜
ξ+
For (ξ˜ , λ) ∈ ξ , introduce locally analytically chosen (in ξ˜ , λ) matrices 0 0 + = (φ1+ , . . . , φk+ ), 0 = (φk+1 , . . . , φn+r ),
(2.4)
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T. Nguyen, K. Zumbrun
and = (+ , 0 ),
(2.5)
whose columns span the subspaces of solutions of (2.2) that, respectively, decay at x = +∞ and satisfy the prescribed boundary conditions at x = 0, and locally analytically chosen matrices + + 0 = (ψ10 , . . . , ψk0 ), + = (ψk+1 , . . . , ψn+r )
(2.6)
= ( 0 , + ),
(2.7)
and
whose columns span complementary subspaces. The existence of such matrices is guaranteed by the general Evans function framework of [AGJ,GZ,MaZ3]; see in particular [Z3,NZ]. That dimensions sum to n + r follows by a general result of [GMWZ5]; see also [SZ]. 2.1.1. The Evans function. Following [AGJ,GZ,SZ], we define on the Evans function D L (ξ˜ , λ) := det(0 , + )|x=0 .
(2.8)
Evidently, eigenfunctions decaying at +∞ and satisfying the prescribed boundary conditions at x1 = 0 occur precisely when the subspaces span 0 and span + intersect, i.e., at zeros of the Evans function D L (ξ˜ , λ) = 0. The Evans function as constructed here is locally analytic in (ξ˜ , λ), which is all that we need for our analysis; we prescribe different versions of the Evans function as needed on different neighborhoods of . Note that includes all of {ξ˜ ∈ Rd−1 , λ ≥ 0} \ {0}, so that Definition 1.4 is well-defined and equivalent to simple nonvanishing, away from the origin (ξ˜ , λ) = (0, 0). To make sense of this definition near the origin, we must insist that the matrices j in (2.8) remain uniformly bounded, a condition that can always be achieved by limiting the neighborhood of definition. For the class of equations we consider, the Evans function may in fact be extended continuously along rays through the origin [R2,MZ,GMWZ5,GMWZ6]. 2.1.2. Basic representation formulae. Define the solution operator from y1 to x1 of ODE (L ξ˜ − λ)U = 0, denoted by F y1 →x1 , as F y1 →x1 = (x1 , λ)−1 (y1 , λ) and the projections 0y1 , +y1 on the stable manifolds at 0, +∞ as +y1 = + (y1 ) 0 −1 (y1 ), 0y1 = 0 0 (y1 ) −1 (y1 ). We define also the dual subspaces of solutions of (L ∗˜ − λ∗ )W˜ = 0. We denote ξ growing solutions + + ˜ 0 = (φ˜ 10 , . . . , φ˜ k0 ), ˜ + = (φ˜ k+1 , . . . , φ˜ n+r ),
(2.9)
Stability of Multi-Dimensional Boundary Layers
11
˜ + ) and decaying solutions ˜ := ( ˜ 0, + + ˜ + = (ψ˜ k+1 ˜ 0 = (ψ˜ 10 , . . . , ψ˜ k+ ), , . . . , ψ˜ n+r ),
˜ + ), satisfying the relations ˜ := ( ˜ 0, and ˜ ˜ ∗ S¯ ξ˜ ≡ I, 0,+ 0,+ where
⎛ −A1
˜ + i B 1ξ
˜ + i Bξ 1
˜ S¯ ξ = ⎝ −(b211 )−1 b11 I −Ir
(2.10)
(2.11)
⎞ 0 Ir ⎠ . 0
(2.12)
With these preparations, the construction of the Resolvent kernel goes exactly as in the construction performed in [ZH,MaZ3,Z3] on the whole line and [YZ,NZ] on the half line, yielding the following basic representation formulae; for a proof, see [MaZ3,NZ]. Proposition 2.1. We have the following representation: ˜ (In , 0)F y1 →x1 +y1 ( S¯ ξ )−1 (y1 )(In , 0)tr , G ξ˜ ,λ (x1 , y1 ) = ˜ −(In , 0)F y1 →x1 0y ( S¯ ξ )−1 (y1 )(In , 0)tr , 1
f or x1 > y1 , f or x1 < y1 .
(2.13)
Proposition 2.2. The resolvent kernel may alternatively be expressed as ˜ 0∗ (y1 ; λ)(In , 0)tr (In , 0)+ (x1 ; λ)M + (λ) x1 > y1 , G ξ˜ ,λ (x1 , y1 ) = 0 0 +∗ tr ˜ (y1 ; λ)(In , 0) −(In , 0) (x1 ; λ)M (λ) x1 < y1 , where ˜ ˜ −1∗ (z; λ). M(λ) := diag(M + (λ), M 0 (λ)) = −1 (z; λ)(S¯ ξ )−1 (z)
(2.14)
2.1.3. Scattering decomposition. From Propositions 2.1 and 2.2, we obtain the following scattering decomposition, generalizing the Fourier transform representation in the constant-coefficient case, from which we will obtain pointwise bounds in the low-frequency regime. ˜
Corollary 2.3. On ξ ∩ ρ(L ξ˜ ), G ξ˜ ,λ (x1 , y1 ) =
d +jk φ +j (x1 ; λ)ψ˜ k+ (y1 ; λ)∗ +
j,k
for 0 ≤ y1 ≤ x1 , and G ξ˜ ,λ (x1 , y1 ) =
j,k
φk+ (x1 ; λ)φ˜ k+ (y1 ; λ)∗
(2.15)
k
d 0jk φ +j (x1 ; λ)ψ˜ k+ (y1 ; λ)∗ −
ψk+ (x1 ; λ)ψ˜ k+ (y1 ; λ)∗ (2.16)
k
for 0 ≤ x1 ≤ y1 , where + 0 −1 + . d 0,+ jk (λ) = (I, 0)
(2.17)
12
T. Nguyen, K. Zumbrun
Proof. For 0 ≤ x1 ≤ y1 , we obtain the preliminary representation G ξ˜ ,λ (x1 , y1 ) =
d 0jk (λ)φ +j (x1 ; λ)ψ˜ k+ (y1 ; λ)∗ +
j,k
e0jk ψ +j (x1 ; λ)ψ˜ k+ (y1 ; λ)∗
jk
from which, together with duality (2.11), representation (2.13), and the fact that 0 = I − + , we have
d0 e0
+ ˜ =−
˜+
∗
A0 + −1 I − + = − + + In−k 0 + 0 + = 0 0 −Ik
0 + 0
−1
0
−1
+
+.
(2.18)
Similarly, for 0 ≤ y1 ≤ x1 , we obtain the preliminary representation G ξ˜ ,λ (x1 , y1 ) =
d +jk (λ)φ +j (x1 ; λ)ψ˜ k+ (y1 ; λ)∗ +
j,k
e+jk φ +j (x1 ; λ)φ˜ k+ (y1 ; λ)∗ ,
jk
from which, together with duality (2.11) and representation (2.13), we have
d+ e+
˜ +∗ A+ + =
+
−1 + + = (+ )−1 + 0 + 0 −1 + + = I 0 + 0 −1 + 0 I 0 0 0 + = n−k + 0 Ik 0 0 0 0
Ik . 0
(2.19)
Remark 2.4. In the constant-coefficient case, with a choice of common bases 0,+ = +,0 at 0, +∞, the above representation reduces to the simple formula N G ξ˜ ,λ (x1 , y1 ) =
−
+ ˜ +∗ j=k+1 φ j (x 1 ; λ)φ j (y1 ; λ) + ˜ +∗ j=1 ψ j (x 1 ; λ)ψ j (y1 ; λ)
k
x1 > y1 , x1 < y1 .
(2.20)
2.2. Pointwise low-frequency bounds. We obtain pointwise low-frequency bounds on the resolvent kernel G ξ˜ ,λ (x1 , y1 ) by appealing to the detailed analysis of [Z2,Z3,GMWZ1] in the viscous shock case. Restrict attention to the surface ˜
ξ := {λ : eλ = −θ1 (|ξ˜ |2 + |mλ|2 )}, for θ1 > 0 sufficiently small.
(2.21)
Stability of Multi-Dimensional Boundary Layers
13 ˜
Proposition 2.5 ([Z3]). Under the hypotheses of Theorem 1.7, for λ ∈ ξ and ρ := |(ξ˜ , λ)|, θ1 > 0, and θ > 0 sufficiently small, there hold: |G ξ˜ ,λ (x1 , y1 )| ≤ Cγ2 e−θρ
2 |x
1 −y1 |
,
(2.22)
and |∂ yβ1 G ξ˜ ,λ (x1 , y1 )| ≤ Cγ2 (ρ β + βe−θ y1 )e−θρ where γ2 := 1 +
2 |x
1 −y1 |
,
1/s j −1 ρ −1 |mλ − η+j (ξ˜ )| + ρ ,
(2.23)
(2.24)
j
and s j , η+j (ξ˜ ) are as defined in (H4). Proof. This follows by a simplified version of the analysis of [Z3], Sect. 5 in the viscous shock case, replacing − , − with 0 , 0 , omitting the refined derivative bounds of Lemmas 5.23 and 5.27 describing special properties of the Lax and overcompressive shock case (not relevant here), and setting = 0, or γ˜ ≡ 1 in definition (5.128). Here, is the multiplicity to which the Evans function vanishes at the origin, (ξ˜ , λ) = (0, 0), evidently zero under assumption (D). The key modes + , + at plus spatial infinity are the same for the boundary-layer as for the shock case. This leads to the pointwise bounds (5.37)–(5.38) given in Proposition 5.10 of [Z3] in case α = 1, γ1 ≡ 1 corresponding to the uniformly stable undercompressive shock case, but without the first O(ρ −1 ), or “pole”, terms appearing on the right-hand side, which derive from cases γ˜ ∼ ρ −1 not arising here. But, these are exactly the claimed bounds (2.22)–(2.24). We omit the (substantial) details of this computation, referring the reader to [Z3]. However, the basic idea is, starting with the scattering decomposition of Corollary 2.1.3, ˜ j, ˜ j can be approximated up to an to note, first, that the normal modes j , j , exponentially trivial coordinate change by solutions of the constant-coefficient limiting system at x → +∞ (the conjugation lemma of [MZ]) and, second, that the coefficients M jk , d jk may be well-estimated through formulae (2.14) and (2.17) using Kramer’s rule and the assumed lower bound on the Evans function |D| appearing in the denominator. This is relatively straightforward away from the branch points λ = η j (ξ˜ ) or “glancing set” of hyperbolic theory; the treatment near these points involves some delicate matrix perturbation theory applied to the limiting constant-coefficient system at x → +∞ followed by careful bookkeeping in the application of Kramer’s rule. 3. Linearized Estimates We next establish estimates on the linearized inhomogeneous problem Ut − LU = f
(3.1)
with initial data U (0) = U0 and Dirichlet boundary conditions as usual in W˜ -coordinates: W (0, x, ˜ t) := (w I , w I I )T (0, x, ˜ t) = h
(3.2)
for the inflow case, and w I I (0, x, ˜ t) = h for the outflow case, with x = (x1 , x) ˜ ∈
Rd .
(3.3)
14
T. Nguyen, K. Zumbrun
3.1. Resolvent bounds. Our first step is to estimate solutions of the resolvent equation with homogeneous boundary data hˆ ≡ 0. Proposition 3.1 (High-frequency bounds). Given (A1)–(A3), (H0)–(H2), and homogeneous boundary conditions (B), for some R, C sufficiently large and θ > 0 sufficiently small, |(L ξ˜ − λ)−1 fˆ| Hˆ 1 (x1 ) ≤ C| fˆ| Hˆ 1 (x1 ) ,
(3.4)
and |(L ξ˜ − λ)−1 fˆ| L 2 (x1 ) ≤
C | fˆ| Hˆ 1 (x1 ) , |λ|1/2
(3.5)
for all |(ξ˜ , λ)| ≥ R and eλ ≥ −θ , where fˆ is the Fourier transform of f in variable x˜ and | fˆ| Hˆ 1 (x1 ) := |(1 + |∂x1 | + |ξ˜ |) fˆ| L 2 (x1 ) . Proof. First observe that a Laplace-Fourier transformed version with respect to variables (λ, x) ˜ of the nonlinear energy estimate in Sect. 4.1 with s = 1, carried out on the linearized equations written in W -coordinates, yields (eλ + θ1 )|(1 + |ξ˜ |+|∂x1 |)W |2 ≤ C |W |2 +(1 + |ξ˜ |2 )|W || fˆ|+|∂x1 W ||∂x1 fˆ| (3.6) for some C big and θ1 > 0 sufficiently small, where |.| denotes |.| L 2 (x1 ) . Applying Young’s inequality, we obtain (eλ + θ1 )|(1 + |ξ˜ | + |∂x1 |)W |2 ≤ C|W |2 + C|(1 + |ξ˜ | + |∂x1 |) fˆ|2 .
(3.7)
On the other hand, taking the imaginary part of the L 2 inner product of U against λU = f + LU , we have also the standard estimate |mλ||U |2L 2 ≤ C|U |2H 1 + C| f |2L 2 ,
(3.8)
and thus, taking the Fourier transform in x, ˜ we obtain |mλ||W |2 ≤ C| fˆ|2 + C|(1 + |ξ˜ | + |∂x1 |)W |2 .
(3.9)
Therefore, taking θ = θ1 /2, we obtain from (3.7) and (3.9), |(1 + |λ|1/2 + |ξ˜ | + |∂x1 |)W |2 ≤ C|W |2 + C|(1 + |ξ˜ | + |∂x1 |) fˆ|2 ,
(3.10)
for any eλ ≥ −θ . Now take R sufficiently large such that |W |2 on the right-hand side of the above can be absorbed into the left-hand side, and thus, for all |(ξ˜ , λ)| ≥ R and eλ ≥ −θ , |(1 + |λ|1/2 + |ξ˜ | + |∂x1 |)W |2 ≤ C|(1 + |ξ˜ | + |∂x1 |) fˆ|2 , for some large C > 0, which gives the result.
(3.11)
Stability of Multi-Dimensional Boundary Layers
15
We next have the following: Proposition 3.2 (Mid-frequency bounds). Given (A1)–(A2), (H0)–(H2), and strong spectral stability (D), |(L ξ˜ − λ)−1 | Hˆ 1 (x1 ) ≤ C, for R −1 ≤ |(ξ˜ , λ)| ≤ R and eλ ≥ −θ,
(3.12)
for any R and C = C(R) sufficiently large and θ = θ (R) > 0 sufficiently small, where | fˆ| Hˆ 1 (x1 ) is defined as in Proposition 3.1. Proof. Immediate, by compactness of the set of frequencies under consideration together with the fact that the resolvent (λ − L ξ˜ )−1 is analytic with respect to H 1 in (ξ˜ , λ); see Proposition 4.8, [Z4]. We next obtain the following resolvent bound for low-frequency regions as a direct consequence of pointwise bounds on the resolvent kernel, obtained in Proposition 2.5. Proposition 3.3 (Low-frequency bounds). Under the hypotheses of Theorem 1.7, for ˜ λ ∈ ξ and ρ := |(ξ˜ , λ)|, θ1 sufficiently small, there holds the resolvent bound |(L ξ˜ − λ)−1 ∂xβ1 fˆ| L p (x1 ) ≤ Cγ2 ρ −2/ p ρ β | fˆ| L 1 (x1 ) + β| fˆ| L ∞ (x1 )
(3.13)
for all 2 ≤ p ≤ ∞, β = 0, 1, where γ2 is as defined in (2.24). Proof. Using the convolution inequality |g ∗ h| L p ≤ |g| L p |h| L 1 and noticing that |∂ yβ1 G ξ˜ ,λ (x1 , y1 )| ≤ Cγ2 (ρ β + βe−θ y1 )e−θρ
2 |x
1 −y1 |
,
we obtain |(L ξ˜ − λ)−1 ∂xβ1 fˆ| L p (x1 )
=
∂ yβ1 G ξ˜ ,λ (x1 , y1 ) fˆ(y1 , ξ˜ ) dy1
+ β|G ξ˜ ,λ (x1 , 0) fˆ(0, ξ˜ )| L p (x1 ) L p (x1 )
β −θ y1 −θρ 2 |x1 −y1 | ˆ
˜ ≤ Cγ2 (ρ + βe )e | f (y1 , ξ )| dy1
Lp
2 + Cγ2 β| fˆ(0, ξ˜ )||e−θρ x1 | L p (x1 ) ≤ Cγ2 ρ −2/ p ρ β | fˆ| L 1 (x1 ) + β| fˆ| L ∞ (x1 ) ,
as claimed.
Remark 3.4. The above L p bounds may alternatively be obtained directly by the argument of Sect. 12, [GMWZ1], using quite different Kreiss symmetrizer techniques, again omitting pole terms arising from vanishing of the Evans function at the origin, and also the auxiliary problem construction of Sect. 12.6 used to obtain sharpened bounds in the Lax or overcompressive shock case (not relevant here).
16
T. Nguyen, K. Zumbrun
3.2. Estimates on homogeneous solution operators. Define low- and high-frequency parts of the linearized solution operator S(t) of the linearized problem with homogeneous boundary and forcing data, f , h ≡ 0, as 1 ˜ S1 (t) := eλt+i ξ ·x˜ (L ξ˜ − λ)−1 dλd ξ˜ (3.14) (2πi)d |ξ˜ |≤r ξ˜ ∩{|λ|≤r } and S2 (t) := e Lt − S1 (t).
(3.15)
Then we obtain the following: Proposition 3.5 (Low-frequency estimate). Under the hypotheses of Theorem 1.7, for β = (β1 , β ) with β1 = 0, 1, |S1 (t)∂xβ f | L 2x ≤ C(1 + t)−(d−1)/4−|β|/2 | f | L 1x + Cβ1 (1 + t)−(d−1)/4 | f | L 1,∞ , x,x ˜ 1
|S1 (t)∂xβ f | L 2,∞ ≤ C(1 + t)−(d+1)/4−|β|/2 | f | L 1x + Cβ1 (1 + t)−(d+1)/4 | f | L 1,∞ , x,x ˜ 1
x,x ˜ 1
(3.16)
≤ C(1 + t)−d/2−|β|/2 | f | L 1x + Cβ1 (1 + t)−d/2 | f | L 1,∞ , |S1 (t)∂xβ f | L ∞ x,x ˜ x,x ˜ 1
1
where | · | L p,q denotes the norm in L p (x; ˜ L q (x1 )). x,x ˜ 1
Proof. The proof will follow closely the treatment of the shock case in [Z3]. Let u(x ˆ 1 , ξ˜ , λ) denote the solution of (L ξ˜ − λ)uˆ = fˆ, where fˆ(x1 , ξ˜ ) denotes Fourier transform of f , and 1 ˜ eλt+i ξ ·x˜ (L ξ˜ − λ)−1 fˆ(x1 , ξ˜ )dλd ξ˜ . u(x, t) := S1 (t) f = (2πi)d |ξ˜ |≤r ξ˜ ∩{|λ|≤r } Recalling the resolvent estimates in Proposition 3.3, we have |u(x ˆ 1 , ξ˜ , λ)| L p (x1 ) ≤ Cγ2 ρ −2/ p | fˆ| L 1 (x1 ) ≤ Cγ2 ρ −2/ p | f | L 1 (x) , where γ2 is as defined in (2.24). Therefore, using Parseval’s identity, Fubini’s theorem, and the triangle inequality, we may estimate |u|2L 2 (x
1
(t) = ,x) ˜
1 (2π )2d
=
1 (2π )2d
≤
1 (2π )2d
2
λt
d ξ˜ d x1 ˜ e u(x ˆ , ξ , λ)dλ 1
x1 ξ˜ ξ˜ ∩{|λ|≤r }
2 λt
˜ , λ)dλ
e u(x ˆ , ξ d ξ˜ 1
ξ˜
2 ∩{|λ|≤r } ξ˜ L (x1 )
2
eλt
d ξ˜ ˜ e | u(x ˆ , ξ , λ)| dλ 2 1 L (x1 )
ξ˜ ∩{|λ|≤r }
ξ˜
2 ≤ C| f | L 1 (x)
˜ ξ
ξ˜ ∩{|λ|≤r }
e
eλt
γ2 ρ
−1
2
dλ
d ξ˜ .
Stability of Multi-Dimensional Boundary Layers
17
˜
Specifically, parametrizing ξ by λ(ξ˜ , k) = ik − θ1 (k 2 + |ξ˜ |2 ), k ∈ R, and observing that by (2.24), γ2 ρ
−1
≤ (|k| + |ξ˜ |)−1 1 +
j
≤ (|k| + |ξ˜ |)−1 1 +
j
where := estimate
˜ ξ
1 max j s j
ξ˜ ∩{|λ|≤r }
|k−τ j (ξ˜ )| ρ
1/s j −1
|k−τ j (ξ˜ )| ρ
−1
,
(3.17)
(0 < < 1 chosen arbitrarily if there are no singularities), we
2
eeλt γ2 ρ −1 dλ
d ξ˜ ≤
2
e−θ1 (k 2 +|ξ˜ |2 )t γ2 ρ −1 dk d ξ˜
˜ R
ξ
≤
ξ˜
+
e
−2θ1 |ξ˜ |2 t
R
˜
ξ˜
j
2
−2
−θ1 k 2 t
−1
˜ |ξ | |k| dk d ξ˜
e
e−2θ1 |ξ | t |ξ˜ |−2 2
2
−θ1 k 2 t
−1
˜ × e |k − τ j (ξ )| dk d ξ˜ R
≤
ξ˜
2
2 ˜2 e−2θ1 |ξ | t |ξ˜ |−2
e−θ1 k t |k| −1 dk
d ξ˜ R
≤ Ct −(d−1)/2 . Likewise, we have |u|2 2,∞ (t) L x,x ˜ 1
1 = (2π )2d
˜
≤
1 (2π )2d
˜
≤
C| f |2L 1 (x)
ξ
ξ
ξ˜ ∩{|λ|≤r }
ξ˜ ∩{|λ|≤r }
˜ ξ
2
˜ e u(x ˆ 1 , ξ , λ)dλ
λt
L ∞ (x1 )
d ξ˜
2
eeλt |u(x ˆ 1 , ξ˜ , λ)| L ∞ (x1 ) dλ
d ξ˜
ξ˜ ∩{|λ|≤r }
e
eλt
2
γ2 dλ
d ξ˜ ,
18
T. Nguyen, K. Zumbrun
where
˜ ξ
2
2
2 ˜2
eeλt γ2 dλ
d ξ˜ ≤ e−2θ1 |ξ | t
e−θ1 k t dk
d ξ˜ ξ˜ ∩{|λ|≤r } R ξ˜ −2θ1 |ξ˜ |2 t ˜ 2−2 e |ξ | + ξ˜
j
2
2 ×
e−θ1 k t |k − τ j (ξ˜ )| −1 dk
d ξ˜ R ˜2 −(d+1)/2 ≤ Ct + C e−2θ1 |ξ | t |ξ˜ |2−2 ξ˜
2
−θ1 k 2 t
−1
× e |k| dk d ξ˜ R
≤ Ct −(d+1)/2 . Similarly, we estimate |u|
L∞ x,x ˜ 1
λt
˜ e u(x ˆ 1 , ξ , λ)dλ
d ξ˜
˜ ξ ξ˜ ∩{|λ|≤r } L ∞ (x1 ) 1 ≤ eeλt |u(x ˆ 1 , ξ˜ , λ)| L ∞ (x1 ) dλd ξ˜ (2π )d ξ˜ ξ˜ ∩{|λ|≤r } eeλt γ2 dλd ξ˜ , ≤ C| f | L 1 (x)
1 (t) ≤ (2π )d
ξ˜
ξ˜ ∩{|λ|≤r }
where as above we have 2 ˜2 eeλt γ2 dλd ξ˜ ≤ e−θ1 |ξ | t e−θ1 k t dkd ξ˜ ξ˜ ξ˜ ∩{|λ|≤r } ξ˜ R 2 ˜2 e−θ1 |ξ | t |ξ˜ |1− e−θ1 k t |k − τ j (ξ˜ )| −1 dkd ξ˜ + j
ξ˜
≤ Ct −d/2 + C
ξ˜
R
˜
e−θ1 |ξ | t |ξ˜ |1− 2
e−θ1 k t |k| −1 dkd ξ˜ 2
R
≤ Ct −d/2 . The x1 -derivative bounds follow similarly by using the resolvent bounds in Proposition 3.3 with β1 = 1. The x-derivative ˜ bounds are straightforward by the fact that β˜ ˜ ∂x˜ f = (i ξ˜ )β fˆ. Finally, each of the above integrals is bounded by C| f | L 1 (x) as the product of | f | L 1 (x) times the integral quantities γ2 ρ −1 , γ2 over a bounded domain, hence we may replace t by (1 + t) in the above estimates. Next, we obtain estimates on the high-frequency part S2 (t) of the linearized solution operator. Recall that S2 (t) = S(t) − S1 (t), where 1 ˜ L t S(t) = ei ξ ·x˜ e ξ˜ d ξ˜ (2πi)d Rd−1
Stability of Multi-Dimensional Boundary Layers
and 1 S1 (t) = (2πi)d
19
|ξ˜ |≤r
ξ˜ ∩{|λ|≤r }
˜ eλt+i ξ ·x˜ (L ξ˜ − λ)−1 dλd ξ˜ .
Then according to [Z4, Cor. 4.11], we can write −θ1 +i∞ 1 S2 (t) f = P.V. χ|ξ˜ |2 +|mλ|2 ≥θ1 +θ2 (2πi)d −θ1 −i∞ Rd−1 ˜ ˜ × ei ξ ·x+λt (λ − L ξ˜ )−1 fˆ(x1 , ξ˜ )d ξ˜ dλ.
(3.18)
Proposition 3.6 (High-frequency estimate). Given (A1)–(A2), (H0)–(H2), (D), and homogeneous boundary conditions (B), for 0 ≤ |α| ≤ s − 3, s as in (H0), |S2 (t) f | L 2x ≤ Ce−θ1 t | f | Hx3 ,
(3.19)
|∂xα S2 (t) f | L 2x ≤ Ce−θ1 t | f | H |α|+3 . x
Proof. The proof starts with the following resolvent identity, using analyticity on the resolvent set ρ(L ξ˜ ) of the resolvent (λ − L ξ˜ )−1 , for all f ∈ D(L ξ˜ ), (λ − L ξ˜ )−1 f = λ−1 (λ − L ξ˜ )−1 L ξ˜ f + λ−1 f.
(3.20)
Using this identity and (3.18), we estimate −θ1 +i∞ 1 P.V. χ|ξ˜ |2 +|mλ|2 ≥θ1 +θ2 S2 (t) f = (2πi)d −θ1 −i∞ Rd−1 ˜ ˜ ×ei ξ ·x+λt λ−1 (λ − L ξ˜ )−1 L ξ˜ fˆ(x1 , ξ˜ )d ξ˜ dλ −θ1 +i∞ 1 P.V. χ|ξ˜ |2 +|mλ|2 ≥θ1 +θ2 + (2πi)d −θ1 −i∞ Rd−1 ˜
˜ ×ei ξ ·x+λt λ−1 fˆ(x1 , ξ˜ )d ξ˜ dλ =: S1 + S2 ,
(3.21)
where, by Plancherel’s identity and Propositions 3.6 and 3.2, we have −θ1 +i∞ ≤ C |λ|−1 |eλt ||(λ − L ξ˜ )−1 L ξ˜ fˆ| L 2 (ξ˜ ,x1 ) |dλ| |S1 | L 2 (x,x ˜ 1) ≤ Ce
−θ1 −i∞ −θ1 +i∞ −θ1 t −θ1 −i∞
|λ|−3/2 (1 + |ξ˜ |)|L ξ˜ fˆ| H 1 (x1 )
L 2 (ξ˜ )
|dλ|
≤ Ce−θ1 t | f | Hx3 and |S2 | L 2x
−θ1 +i∞
−1 λt i x· ˜ ξ˜ ˆ
P.V. ˜ ˜ λ e dλ e f (x1 , ξ )d ξ
d−1 R −θ1 −i∞ L2
−θ1 +ir
1
˜ ξ˜ ˆ P.V. + λ−1 eλt dλ ei x· f (x1 , ξ˜ )d ξ˜
d−1 (2π )d
R −θ1 −ir L2
1 ≤ (2π )d
≤ Ce−θ1 t | f | L 2x ,
(3.22)
20
T. Nguyen, K. Zumbrun
by direct computations, noting that the integral in λ in the first term is identically zero. This completes the proof of the first inequality stated in the proposition. Derivative bounds follow similarly. Remark 3.7. Here, we have used the λ1/2 improvement in (3.5) over (3.4) together with modifications introduced in [KZ] to greatly simplify the original high-frequency argument given in [Z3] for the shock case. 3.3. Boundary estimates. For the purpose of studying the nonzero boundary perturbation, we need the following proposition. For h := h(x, ˜ t), define Dh (t) := (|h t | + |h x˜ | + |h x˜ x˜ |)(t), and
t h(t) :=
0
Rd−1
(3.23)
G yk B
k1
+ GA
1
(x, t − s; 0, y˜ )h( y˜ , s) d y˜ ds, (3.24)
k
where G(x, t; y) is the Green function of ∂t − L. This boundary term will appear when we write down the Duhamel formulas for the linearized and nonlinear equations (see (3.36) and (4.55)). Noting that for the outflow case, the fact that G(x, t; 0, y˜ ) ≡ 0 simplifies h to t h(t) = G y1 (x, t − s; 0, y˜ )B 11 h d y˜ ds. (3.25) 0
Rd−1
Therefore when dealing with the outflow case, instead of putting assumptions on h itself as in the inflow case, we make assumptions on B 11 h, matching with the hypotheses on W -coordinates. Proposition 3.8. Assume that h = h(x, ˜ t) satisfies |h(t)| L 2 ≤ E 0 (1 + t)−(d+1)/4 , x˜
|h(t)| L ∞ ≤ E 0 (1 + t)−d/2 , x˜ |Dh (t)| L 1 ∩H |γ |+3 ≤ E 0 (1 + t) x˜
−d/2−
x˜
(3.26) ,
for some positive constant E 0 ; here |γ | = [(d − 1)/2] + 2, and > 0 is arbitrary small for d = 2 and = 0 for d ≥ 3. For the outflow case, we replace these assumptions on h by those on B 11 h. Then we obtain |h(t)| L 2 ≤ C E 0 (1 + t)−(d−1)/4 , |h(t)| L 2,∞ ≤ C E 0 (1 + t)−(d+1)/4 , x,x ˜ 1
(3.27)
|h(t)| L ∞ ≤ C E 0 (1 + t)−d/2 , and derivative bounds |∂x h(t)| L 2,∞ ≤ C E 0 (1 + t)−(d+1)/4 , x,x ˜ 1
|∂x2˜ h(t)| L 2,∞ x,x ˜
1
for all t ≥ 0.
≤ C E 0 (1 + t)−(d+1)/4 ,
(3.28)
Stability of Multi-Dimensional Boundary Layers
21
Proof. We first recall that G(x, t − s; y) is a solution of (∂s − L y )∗ G ∗ = 0, that is, j − Gs − (G A j ) y j + G Ay j = (G yk B k j ) y j . (3.29) j
j
jk
Integrating this on Rd+ × [0, t] against g(y1 , y˜ , s) := e−y1 h( y˜ , s),
(3.30)
and integrating by parts twice, we obtain ⎛ ⎞ t ⎝ h = − G yk B k j + G A j ⎠ g y j dyds Rd+
0
−
t Rd+
0
⎛
jk
j
⎝−G s +
⎞
j G A y j ⎠ g(y, s)dyds,
j
where, recalling that
S(t) f (x) =
G(x, t; y) f (y)dy,
Rd+
we get −
t
Rd+ jk
0
t
=−
⎛ ⎝G yk B k j +
S(t − s) ⎝−
−
⎛
t 0
Rd+
t
=− 0
⎝−G s +
⎞ G A j ⎠ g y j dyds
j
⎛
0
and
(B k j gx j )xk +
jk
⎞ A j gx j ⎠ ds
j
⎞ G A y j ⎠ g(y, s)dyds j
j
⎛
S(t − s) ⎝gs +
⎞ A x j g ⎠ ds + g(x, t) − S(t)g(x, 0). j
j
Therefore combining all these estimates yields t h = g(x, t) − S(t)g0 − S(t − s)(gs − L x g(x, s))ds 0
(3.31)
with g0 (x) := g(x, 0) and L x g = − j (A j g)x j + jk (B jk gxk )x j . Now we are ready to employ estimates obtained in the previous section on the solution operator S(t) = S1 (t) + S2 (t). Noting that |g| L xp ≤ C|h| L p , x˜
22
T. Nguyen, K. Zumbrun
we estimate |h| L 2 ≤ |g| L 2 + |S1 (t)g0 | L 2 + |S2 (t)g0 | L 2 t + |S1 (t − s)(gs − Lg)| L 2 + |S2 (t − s)(gs − Lg)| L 2 ds 0
d−1
≤ |h(t)| L 2 + C(1 + t)− 4 |g0 | L 1 + Ce−ηt |g0 | H 3 x˜ t + (1 + t − s)−(d−1)/4 (|gs | + |Lg|) L 1 + e−θ(t−s) (|gs | + |Lg|) H 3 ds 0
d−1
≤ |h(t)| L 2 + C(1 + t)− 4 |h 0 | L 1 ∩H 3 x˜ x˜ x˜ t + (1 + t − s)−(d−1)/4 |Dh (s)| L 1 + e−θ(t−s) |Dh (s)| H 3 ds x˜
0
≤ C E 0 (1 + t)
− d−1 4
x˜
,
and similarly we also obtain d+1
|h| L 2,∞ ≤ |h(t)| L 2 + C(1 + t)− 4 |h 0 | L 1 ∩H 4 x˜ x˜ x˜ x,x ˜ 1 t +C (1 + t − s)−(d+1)/4 |Dh (s)| L 1 + e−θ(t−s) |Dh (s)| H 4 ds x˜
0
≤ C E 0 (1 + t)
x˜
− d+1 4
(3.32)
and d
|h| L ∞ ≤ |h(t)| L ∞ + C(1 + t)− 2 |h 0 | L 1 ∩H |γ |+3 x˜ x˜ x˜ t −d/2 +C (1 + t − s) |Dh (s)| L 1 + e−θ(t−s) |Dh (s)| H |γ |+3 ds x˜
0
≤ C E 0 (1 + t)
− d2
x˜
.
(3.33)
Similar bounds hold for derivatives. This completes the proof of the proposition.
3.4. Duhamel formula. The following integral representation formula expresses the solution of the inhomogeneous equation (3.1) in terms of the homogeneous solution operator S for f , h ≡ 0. Lemma 3.9 (Integral formulation). Solutions U of (3.1) may be expressed as t U (x, t) = S(t)U0 + S(t − s) f (·, s) + U (0, x, ˜ t),
(3.34)
0
where U (x, 0) = U0 (x), t U (0, x, ˜ t) := ( G y j B j1 +G A1 )(x, t −s; 0, y˜ )U (0, y˜ , s) d y˜ ds, 0
Rd−1
j
and G(·, t; y) = S(t)δ y (·) is the Green function of ∂t − L.
(3.35)
Stability of Multi-Dimensional Boundary Layers
23
Proof. Integrating on Rd+ the linearized equations (∂s − L y )U = f against G(x, t − s; y) and using the fact that by duality (∂s − L y )∗ G ∗ (x, t − s; y) ≡ 0, we easily obtain the lemma as in the one-dimensional case (see [YZ,NZ]), recalling that S(t) f = G(x, t; y) f (y)dy. Rd+
3.5. Proof of linearized stability. Proof of Theorem 1.6. Writing the Duhamel formula for the linearized equations ˜ t), U (x, t) = S(t)U0 + h(x,
(3.36)
with h defined in (3.24), where U (x, 0) = U0 (x) and U (0, x, ˜ t) = h(x, ˜ t), and applying estimates on low- and high-frequency operators S1 (t) and S2 (t), we obtain |U (t)| L 2 ≤ |S1 (t)U0 | L 2 + |S2 (t)U0 | L 2 + |h(t)| L 2 ≤ C(1 + t)−
d−1 4
|U0 | L 1 + Ce−ηt |U0 | H 3 + C E 0 (1 + t)−(d−1)/4
≤ C(1 + t)−
d−1 4
(|U0 | L 1 ∩H 3 + E 0 )
(3.37)
and |U (t)| L ∞ ≤ |S1 (t)U0 | L ∞ + |S2 (t)U0 | L ∞ + |h(t)| L ∞ d
≤ C(1 + t)− 2 |U0 | L 1 + C|S2 (t)U0 | H [(d−1)/2]+2 + C E 0 (1 + t)−d/2 d
≤ C(1 + t)− 2 |U0 | L 1 + Ce−ηt |U0 | H [(d−1)/2]+5 + C E 0 (1 + t)−d/2 d
≤ C(1 + t)− 2 (|U0 | L 1 ∩H [(d−1)/2]+5 + E 0 ).
(3.38)
These prove the bounds as stated in the theorem for p = 2 and p = ∞. For 2 < p < ∞, we use the interpolation inequality between L 2 and L ∞ . 4. Nonlinear Stability 4.1. Auxiliary energy estimates. For the analysis of nonlinear stability, we need the following energy estimate adapted from [MaZ4,NZ,Z4]. Define the nonlinear perturbation variables U = (u, v) by U (x, t) := U˜ (x, t) − U¯ (x1 ).
(4.1)
24
T. Nguyen, K. Zumbrun
Proposition 4.1. Under the hypotheses of Theorem 1.7, let U0 ∈ H s and U = (u, v)T be a solution of (1.2) and (4.1). Suppose that, for 0 ≤ t ≤ T , the Wx2,∞ norm of the solution U remains bounded by a sufficiently small constant ζ > 0. Then t 2 −θt 2 (4.2) e−θ(t−τ ) |U (τ )|2L 2 + |Bh (τ )|2 dτ |U (t)| H s ≤ Ce |U0 | H s + C 0
for all 0 ≤ t ≤ T , where the boundary term Bh is defined as in Theorem 1.7. Proof. Observe that a straightforward calculation shows that |U | H r ∼ |W | H r , W = W˜ − W¯ := W (U˜ ) − W (U¯ ),
(4.3)
for 0 ≤ r ≤ s, provided |U |W 2,∞ remains bounded, hence it is sufficient to prove a corresponding bound in the special variable W . We first carry out a complete proof in the more straightforward case with conditions (A1)-(A3) replaced by the following global versions, indicating afterward by a few brief remarks the changes needed to carry out the proof in the general case. (A1’) A˜ j (W˜ ), A˜ 0 , A˜ 111 are symmetric, A˜ 0 ≥ θ0 > 0, (A2’) Same as (A2), 0 w˜ I jk = B kj = 0 ˜ ˜ , B , ξ j ξk b˜ jk ≥ θ |ξ |2 , and G˜ ≡ 0. (A3’) W˜ = jk I I ˜ 0 b w˜ Substituting (4.3) into (1.4), we obtain the quasilinear perturbation equation j A 0 Wt + A j Wx j = (B jk Wxk )x j + M1 W¯ x1 + (M2 W¯ x1 )x j , j
jk
(4.4)
j
where A0 := A0 (W + W¯ ) is symmetric positive definite, A j := A j (W + W¯ ) are symmetric, 1 1 1 ¯ 1 ¯ ¯ M1 = A (W + W ) − A (W ) = d A (W + θ W )dθ W, 0 0 0 j ! 1 j1 . M2 = B j1 (W + W¯ ) − B j1 (W¯ ) = 0 ( 0 db (W¯ + θ W )dθ )W As shown in [MaZ4], we have bounds |A0 | ≤ C, |A0t | ≤ C|Wt | ≤ C(|Wx | + |wxI xI |) ≤ Cζ, |∂x A0 | + |∂x2 A0 | ≤ C(
2
|∂xk W | + |W¯ x1 |) ≤ C(ζ + |W¯ x1 |).
(4.5) (4.6)
k=1
We have the same bounds for A j , B jk , and also due to the form of M1 , M2 , |M1 |, |M2 | ≤ C(ζ + |W¯ x1 |)|W |.
(4.7)
Note that thanks to Lemma 1.3 we have the bound on the profile: |W¯ x1 | ≤ Ce−θ|x1 | , as x1 → +∞. The following results assert that hyperbolic effects can compensate for degenerate viscosity B, as revealed by the existence of a compensating matrix K .
Stability of Multi-Dimensional Boundary Layers
25
Lemma 4.2 ([KSh]). Assuming (A1’), condition (A2’) is equivalent to the following: (K1) There exist smooth skew-symmetric “compensating matrices” K (ξ ), homogeneous degree one in ξ , such that ⎞ ⎛ ξ j ξk B jk − K (ξ )(A0 )−1 ξk Ak ⎠ (W+ ) ≥ θ2 |ξ |2 > 0 (4.8) e ⎝ j,k
k
for all ξ ∈ Rd \{0}. Define α by the ODE αx1 = −sign(A111 )c∗ |W¯ x1 |α, α(0) = 1,
(4.9)
where c∗ > 0 is a large constant to be chosen later. Note that we have (αx1 /α)A111 ≤ −c∗ θ1 |W¯ x1 | =: −ω(x1 )
(4.10)
|αx1 /α| ≤ c∗ |W¯ x1 | = θ1−1 ω(x1 ).
(4.11)
and
In what follows, we shall use ·, · as the α-weighted L 2 inner product defined as f, g = α f, g L 2 (Rd+ ) and f s =
s "
∂xα f, ∂xα f
#1/2
i=0 |α|=i
as the norm in weighted H s space. Note that for any symmetric operator S, 1 S f x j , f = − Sx j f, f , j = 1, 2 1 1 S f x1 , f = − (Sx1 + (αx1 /α)S) f, f − S f, f 0 , 2 2 where ·, ·0 denotes the integration on Rd0 := {x1 = 0} × Rd−1 . Also we define f 0,s = f H s (Rd ) = 0
s " i=0 |α|=i
∂xα˜ f, ∂xα˜ f
#1/2 0
.
Note that in what follows, we shall pay attention to keeping track of c∗ . For constants independent of c∗ , we simply write them as C. Also, for simplicity, the sum symbol will sometimes be dropped where there is no confusion. We write f x = j f x j and ∂xk f = |α|=k ∂xα f .
26
T. Nguyen, K. Zumbrun
4.1.1. Zeroth order “Friedrichs-type” estimate. First, by integration by parts and estimates (4.5), (4.6), and then (4.10), we obtain for j = 1, −A j Wx j , W =
1 j A x j W, W ≤ C(ζ + |W¯ x1 |)w I , w I + Cw I I 20 2
and for j = 1, 1 1 (A1x1 + (αx1 /α)A1 )W, W + A1 W, W 0 2 2 1 1 I I ≤ (αx1 /α)A11 w , w + C(ζ + |W¯ x1 |)|W | + ω(x1 )|w I I |, |W | + Jb0 2 1 ≤ − ω(x)w I , w I + C(ζ + |W¯ x1 |)w I , w I + C(c∗ )w I I 20 + Jb0 , 2
−A1 Wx1 , W =
where Jb0 denotes the boundary term 21 A1 W, W 0 . The term |W¯ x1 |w I , w I may be easily absorbed into the first term of the right-hand side, since for c∗ sufficiently large, 1 ω(x1 )w I , w I . |W¯ x1 |w I , w I ≤ (c∗ θ1 )−1 ω(x1 )w I , w I ≤ 4C
(4.12)
Also, integration by parts yields (B jk Wxk )x j , W = −B jk Wxk , Wx j − (αx1 /α)B 1k Wxk , W − B 1k Wxk , W 0 ≤ −θ wxI I 20 + Cω(x1 )wxI I , w I I − b1k wxI kI , w I I 0 ≤ −θ wxI I 20 + C(c∗ )w I I 20 − b1k wxI kI , w I I 0 , where we used the fact that B jk Wx · W = b jk wxI I · w I I , noting that B has blockj diagonal form with the first block identical to zero. Similarly, recalling that M2 = j1 j1 B (W + W¯ ) − B (W¯ ), we have (M2 W¯ x1 )x j , W = −M2 W¯ x1 , Wx j − (αx1 /α)M21 W¯ x1 , W − M21 W¯ x1 , W 0 j
j
≤ C|W¯ x1 ||W |, |wxI I | + Cω(x1 )|W |, w I I − m 12 W¯ x1 , w I I 0 ≤ ξ wxI I 20 +C ω(x1 )w I , w I +C(c∗ )w I I 20 −m 12 W¯ x1 , w I I 0 for any small ξ, . Note that C is independent of c∗ . Therefore, for ξ = θ/2 and c∗ sufficiently large, combining all above estimates, we obtain 1 d 0 1 A W, W = A0 Wt , W + A0t W, W 2 dt 2 1 j = −A j Wx j + (B jk Wxk )x j + M1 W¯ x1 + (M2 W¯ x1 )x j , W + A0t W, W 2 1 I I II 2 I 2 ≤ − [ω(x1 )w , w + θ wx 0 ] + Cζ w 0 + C(c∗ )w I I 20 + Ib0 , 4 (4.13) where the boundary term Ib0 :=
1 1 A W, W 0 − b1k wxI kI , w I I 0 − m 12 W¯ x1 , w I I 0 2
(4.14)
Stability of Multi-Dimensional Boundary Layers
27
which, in the outflow case (thanks to the negative definiteness of A111 ), is estimated as Ib0 ≤ −
θ1 I 2 w 0,0 + C(w I I 20,0 + wxI I 0,0 w I I 0,0 ), 2
(4.15)
and similarly in the inflow case, estimated as Ib0 ≤ C(W 20,0 + wxI I 0,0 w I I 0,0 ).
(4.16)
Here we recall that · 0,s := · H s (Rd ) . 0
4.1.2. First order “Friedrichs-type” estimate. Similarly as above, we need the following key estimate, computing by the use of integration by parts, (4.12), and c∗ being sufficiently large, − Wxi , A j Wxi x j j
=
1 1 1 j Wxi , A x j Wxi + Wxi , (αx1 /α)A1 Wxi + Wxi , A1 Wxi 0 2 2 2 j
1 1 ≤ − ω(x1 )wxI , wxI + Cζ wxI 20 + Cc∗2 wxI I 20 + Wxi , A1 Wxi 0 . 4 2
(4.17)
We deal with the boundary term later. Now let us compute 1 d 0 1 A Wxi , Wxi = Wxi , (A0 Wt )xi − Wxi , A0xi Wt + A0t Wxi , Wxi . 2 dt 2
(4.18)
We control each term in turn. By (4.5) and (4.6), we first have A0t Wxi , Wxi ≤ Cζ Wx 20 and by multiplying (A0 )−1 into (4.4), |Wxi , A0xi Wt | ≤ C(ζ + |W¯ x1 |)|Wx |, (|Wx | + |wxI xI | + |W |) ≤ ξ wxI xI 20 + C(ζ + |W¯ x1 |)wxI , wxI + C(ζ + |W¯ x1 |)w I , w I + Cw I I 21 , where the term |W¯ x1 |wxI , wxI may be treated in the same way as was |W¯ x1 |w I , w I in (4.12). Using (4.4), we write the first term in the right-hand side of (4.18) as j Wxi , (A0 Wt )xi = Wxi , [−A j Wx j + (B jk Wxk )x j + M1 W¯ x1 + (M2 W¯ x1 )x j ]xi
= −Wxi , A j Wxi x j + Wxi , −A xi Wx j + (M1 W¯ x1 )xi j
j −Wxi x j , [(B jk Wxk )xi + (M2 W¯ x1 )xi ]
−(αx1 /α)Wxi , [(B 1k Wxk )xi + (M21 W¯ x1 )xi ] −Wxi , [(B 1k Wxk )xi + (M21 W¯ x1 )xi ]0 1 ≤ − ω(x1 )wxI , wxI + θ wxI xI 20 4 + C ζ w I 21 + C(c∗ )wxI I 20 + |W¯ x1 |w I , w I + Ib1 ,
28
T. Nguyen, K. Zumbrun
where Ib1 denotes the boundary terms 1 Wxi , A1 Wxi 0 − Wxi , [(B 1k Wxk )xi + (M21 W¯ x1 )xi ]0 2 1 = Wxi , A1 Wxi 0 − wxI iI , [(b1k wxI kI )xi + (m 12 W¯ x1 )xi ]0 , 2
Ib1 : =
(4.19)
and we have used (A3) for each fixed i and ξ j = (Wxi )x j to get Wxi x j , B jk Wxk xi ≥ θ Wxi x j 20 , jk
(4.20)
j
and estimates (4.17),(4.12) for w I , wxI , and Young’s inequality to obtain: Wx , −A x Wx + (M1 W¯ x1 )x ≤ C(ζ + |W¯ x1 |)|Wx |, |Wx | + |W |. j
−Wx x +(αx1 /α)Wx , (B jk Wx )x ≤ −θ wxI xI 20 +C|wxI xI |+ω(x1 )|wxI I |, (ζ +|W¯ x1 |)|wxI I | −Wx x + (αx1 /α)Wx , (M2 W¯ x1 )x ≤ C|wxI xI | + ω(x1 )|wxI I |, (ζ + |W¯ x1 |)(|Wx | + |W |). j
Putting these estimates together into (4.18), we have obtained 1 1 1 d 0 A Wx , Wx + θ wxI xI 20 + ω(x1 )wxI , wxI 2 dt 4 4 I 2 I I ¯ ≤ C ζ w 1 + |Wx1 |w , w + C(c∗ )w I I 21 + Ib1 .
(4.21)
Let us now treat the boundary term. First observe that using the parabolic equations, noting that A0 is the diagonal-block form, we can estimate ˜ t) ≤ C |wtI I | + |Wx j | + |W | (0, x, ˜ t), (4.22) (b jk wxI kI )x j (0, x, and thus for i = 1, wxI iI , [(b1k wxI kI )xi + (m 12 W¯ x1 )xi ]0 ≤ |wxI iIxi | |W | + |wxI kI | Rd0
≤C
Rd0
|W |2 + |wxI I |2 + |wxI˜ xI˜ |2
(4.23)
and for i = 1, using b1k = bk1 , (4.22), and recalling here that we always use the sum convention, 1 1k I I j1 I I II (b wxk )x1 + (b j1 wxI 1I )x j + b1k (b1k wxI kI )x1 = x1 wxk − bx j wx1 2 k ⎞ ⎛ 1 ⎝ jk I I j1 I I II (b jk wxI kI )x j ⎠ = (b wxk )x j +b1k x1 wxk −bx j wx1 − 2 j =1; k =1 ≤ C |wtI I | + |W | + |Wx j | + |wxI˜ xI˜ | . (4.24)
Stability of Multi-Dimensional Boundary Layers
29
Therefore wxI 1I , [(b1k wxI kI )x1 + (m 12 W¯ x1 )x1 ]0 I 2 |wtI I |2 + |W |2 + |wxI I |2 + |wxI˜ xI˜ |2 . ≤ |wx | + C Rd0
Rd0
For the first term in Ib1 , we consider each inflow/outflow case separately. For the outflow case, since A111 ≤ −θ1 < 0, we get A1 Wx · Wx ≤ − Therefore Ib1 ≤ −
θ1 I 2 |w | + C|wxI I |2 . 2 x
θ1 |W |2 + |wxI I |2 + |wtI I |2 + |wxI˜ xI˜ |2 . |wxI |2 + 2 Rd0 Rd0
(4.25)
Meanwhile, for the inflow case, since A111 ≥ θ1 > 0, we have |A1 Wx · Wx | ≤ C|Wx |2 . In this case, the invertibility of A111 allows us to use the hyperbolic equation to derive |wxI 1 | ≤ C(|wtI | + |wxI I | + |wxI˜ |). Therefore we get
Ib1 ≤
|W |2 + |Wt |2 + |wxI˜ |2 + |wxI I |2 + |wxI˜ xI˜ |2 .
Rd0
(4.26)
Now apply the standard Sobolev inequality |w(0)|2 ≤ Cw L 2 (R) (wx L 2 (R) + w L 2 (R) ) to control the term |wxI 1I (0)|2 in Ib1 in both cases. We get |wxI 1I |2 ≤ wxI xI 20 + CwxI I 20 . Rd0
(4.27)
(4.28)
Using this with = θ/8, (4.19), and (4.25), the estimate (4.21) reads d 0 A Wx , Wx + wxI xI 20 + ω(x1 )wxI , wxI dt ≤ C ζ w I 21 + |W¯ x1 |w I , w I + C(c∗ )w I I 21 + Ib1 , where the (new) boundary term Ib1 satisfies θ1 Ib1 ≤ − |W |2 + |wxI˜ I |2 + |wtI I |2 + |wxI˜ xI˜ |2 |wxI |2 + C 2 Rd0 Rd0 for the outflow case, and Ib1 ≤ for the inflow case.
Rd0
|W |2 + |Wt |2 + |Wx˜ |2 + |wxI˜ xI˜ |2
(4.29)
(4.30)
(4.31)
30
T. Nguyen, K. Zumbrun
4.1.3. Higher order “Friedrichs-type” estimate. For any fixed multi-index α = d (αx1 , · · · , αxd ), α1 = 0, 1, |α| = k = 2, ..., s, by computing dt A0 ∂xα W, ∂xα W and following the same spirit as the above subsection, we easily obtain d 0 α A ∂x W, ∂xα W + θ ∂xα+1 w I I 20 + ω(x1 )∂xα w I , ∂xα w I dt k−1 II 2 I 2 i I i I ¯ ≤ C C(c∗ )w k + ζ w k + |Wx1 |∂x w , ∂x w + Ibα ,
(4.32)
i=1
where ∂xα := ∂xα11 · · · ∂xαdd , ∂xα+1 :=
∂xα11 · · · ∂xαdd ∂x j , ∂xi =
j
|β|=i
∂xβ11 · · · ∂xβdd
and the boundary term Ibα satisfies Ibα
θ1 ≤− |∂ α w I |2 2 Rd0 x ⎛ ⎞ [(k+1)/2] k−1 k ⎝ +C |∂ti w I I |2 + |∂xi w I |2 + |∂xi˜ w I I |2 ⎠ Rd0
i=1
i=0
(4.33)
i=0
for the outflow case, and Ibα ≤
Rd0
⎛ ⎞ [(k+1)/2] k k ⎝ |∂ti w I |2 + |∂ti w I I |2 + |∂ i W |2 ⎠ x˜
i=0
i=1
(4.34)
i=0
for the inflow case. Now for α with α1 = 2, . . . , s we observe that the estimate (4.32) still holds. Indeed, d using integration by parts and computing dt A0 ∂xα W, ∂xα W as above leaves the boundary terms as Ibα :=
1 α ∂ W, A1 ∂xα W 0 − ∂xα w I I , ∂xα [(b1k wxI kI ) + (m 12 W¯ x1 )]0 . 2 x
(4.35)
Then we can use the parabolic equations to solve j j ¯ II ¯ wxI 1Ix1 = (b11 )−1 A02 wtI I + A2 Wx j −(b jk wxI kI )x j − b11 x1 wx1 − M1 W x1 − (m 2 W x1 )x j . Using this we can reduce the order of derivative with respect to x1 in ∂xα to one, with the same spirit as (4.23) and (4.24). Finally we use the Sobolev embedding similar to (4.28) to obtain the estimate for the normal derivative ∂x1 , and get the estimate for Ibα as claimed in (4.33) and (4.34). We recall next the following Kawashima-type estimate, presented in [Z3], to bound the term w I 2k appearing on the left-hand side of (4.32).
Stability of Multi-Dimensional Boundary Layers
31
4.1.4. “Kawashima-type” estimate. Let K (ξ ) be the skew-symmetry in (4.8). Using Plancherel’s identity and Eqs. (4.4), we compute 1 d 1 d K (∂x )∂xr W, ∂xr = i K (ξ )(iξ )r Wˆ , (iξ )r Wˆ 2 dt 2 dt = i K (ξ )(iξ )r Wˆ , (iξ )r Wˆ t j = (iξ )r Wˆ , −K (ξ )(A0+ )−1 ξ j A+ (iξ )r Wˆ j
+i K (ξ )(iξ ) Wˆ , (iξ )r Hˆ , r
where H :=
(4.36)
j (A0+ )−1 A+ − (A0 )−1 A j Wx j j
⎛ ⎞ j +(A0 )−1 ⎝ (B jk Wxk )x j + M1 W¯ x1 + (M2 W¯ x1 )x j ⎠ . jk
(4.37)
j
j By using the fact that |(A0+ )−1 A+ − (A0 )−1 A j | = O(ζ + |W¯ x1 |), we can easily obtain
∂xr H 20 ≤ Cw I I r2+2 + C
r +1 (ζ + |W¯ x1 |)∂xk w I , ∂xk w I . k=0
Meanwhile, applying (4.8) into the first term of the last line in (4.36), we get j (iξ )r Wˆ , −K (ξ )(A0+ )−1 ξ j A+ (iξ )r Wˆ j
≥ =
θ |ξ | Wˆ 20 − C|ξ |r +1 wˆ I I 20 θ ∂xr +1 w I 20 − C∂xr +1 w I I 20 . r +1
Putting these estimates together into (4.36), we have obtained the high order “Kawashima-type” estimate: d K (∂x )∂xr W, ∂xr W ≤ −θ ∂xr +1 w I 20 + Cw I I r2+2 dt r +1 +C (ζ + |W¯ x1 |)∂xi w I , ∂xi w I .
(4.38)
i=0
4.1.5. Final estimates. We are ready to conclude our result. First combining the estimate (4.29) with (4.13), we easily obtain 1 d 0 A Wx , Wx + MA0 W, W 2 dt θ 1 wxI xI 20 + ω(x1 )wxI , wxI ≤− 8 4 + C ζ w I 21 + |W¯ x1 |w I , w I + C(c∗ )w I I 21 + Ib1 M ω(x1 )w I , w I + θ wxI I 20 + C Mζ w I 20 + MC(c∗ )w I I 20 + M Ib0 . − 4
32
T. Nguyen, K. Zumbrun
By choosing M sufficiently large such that Mθ C(c∗ ), and noting that c∗ θ1 |W¯ x1 | ≤ ω(x1 ), we get 1 d 0 A Wx , Wx + MA0 W, W 2 dt ≤ − θ w I I 22 + ω(x1 )w I , w I + ω(x1 )wxI , wxI + C ζ w I 21 + C(c∗ )w I I 20 + Ib1 + M Ib0 . (4.39) We shall treat the boundary terms later. Now we use the estimate (4.38) (for r = 0) to absorb the term ∂x w I 0 into the left-hand side. Indeed, fixing c∗ large as above, adding (4.39) with (4.38) times , and choosing , ζ sufficiently small such that C(c∗ ) θ, 1 and ζ θ2 , we obtain 1 d 0 A Wx , Wx + MA0 W, W + K Wx , W 2 dt ≤ − θ w I I 22 + ω(x1 )w I , w I + ω(x1 )wxI , wxI θ 2 wxI 20 + C ζ w I 21 + C(c∗ )w I I 20 − 2 + C w I I 22 + ζ w I 20 + ω(x1 )w I , w I + ω(x1 )wxI , wxI + Ib1 + M Ib0 1 ≤ − θ w I I 22 + θ2 wxI 20 + C(c∗ )W 20 + Ib , 2 where Ib := Ib1 + M Ib0 . In view of boundary terms Ib0 and Ib1 , we treat the term Ib in each inflow/outflow case separately. Recall the inequality (4.28), wxI 1I 0,0 ≤ Cw I I 2 . Thus, using this, for the inflow case we have Ib0 ≤ C(W 20,0 +wxI I 0,0 w I I 0,0 ) ≤ C(W 20,0 +wxI˜ I 20,0 + w I I 22 ), (4.40) and for the outflow case, θ1 I 2 w 0,0 + C(w I I 20,0 + wxI I 0,0 w I I 0,0 ) 2 θ1 ≤ − w I 20,0 + C(w I I 20,0 + wxI˜ I 20,0 + w I I 22 ). 2
Ib0 ≤ −
(4.41)
Therefore these together with (4.30) and (4.31), using the good estimate of wxI xI 20 , yield θ1 |w I I |2 + |wxI˜ I |2 + |wtI I |2 + |wxI˜ xI˜ |2 (4.42) (|w I |2 + |wxI |2 ) + C Ib ≤ − 2 Rd0 Rd0 for the outflow case, and Ib1 ≤ for the inflow case.
Rd0
|W |2 + |Wt |2 + |Wx˜ |2 + |wxI˜ xI˜ |2
(4.43)
Stability of Multi-Dimensional Boundary Layers
33
Now by Cauchy-Schwarz’s inequality, |K (ξ )| ≤ C|ξ |, and positive definiteness of A0 , it is easy to see that E := A0 Wx , Wx + MA0 W, W + K (∂x )W, W ∼ W 2H 1 ∼ W 2H 1 . (4.44) α
The last equivalence is due to the fact that α is bounded above and below away from zero. Thus the above yields d E(W )(t) ≤ −θ3 E(W )(t) + C(c∗ ) W (t)2L 2 + |B1 (t)|2 , dt for some positive constant θ3 , which by the Gronwall inequality implies W (t)2H 1 ≤ Ce−θt W0 2H 1 +C(c∗ )
t 0
e−θ(t−τ ) W (τ )2L 2 +|B1 (τ )|2 dτ, (4.45)
where W (x, 0) = W0 (x) and |B1 (τ )| :=
2
Rd0
|W |2 + |Wt |2 + |Wx˜ |2 + |wxI˜ xI˜ |2
(4.46)
for the inflow case, and |B1 (τ )|2 :=
Rd0
|w I I |2 + |wxI˜ I |2 + |wtI I |2 + |wxI˜ xI˜ |2
(4.47)
for the outflow case. Similarly, by induction, we can derive the same estimates for W in H s . To do that, let us define E1 (W ) := A0 Wx , Wx + MA0 W, W + K Wx , W , Ek (W ) := A0 ∂xk W, ∂xk W + MEk−1 (W ) + K ∂xk W, ∂xk−1 W ,
k ≤ s.
Then similarly by the Cauchy-Schwarz inequality, Es (W ) ∼ W 2H s , and by induction, we obtain d Es (W )(t) ≤ −θ3 Es (W )(t) + C(c∗ )(W (t)2L 2 + |Bh (t)|2 ), dt for some positive constant θ3 , which by the Gronwall inequality yields W (t)2H s ≤ Ce−θt W0 2H s + C(c∗ )
0
t
e−θ(t−τ ) (W (τ )2L 2 + |Bh (τ )|2 )dτ, (4.48)
where W (x, 0) = W0 (x), and Bh are defined as in (1.14) and (1.15).
34
T. Nguyen, K. Zumbrun
4.1.6. The general case. Following [MaZ4,Z3], the general case that hypotheses (A1)(A3) hold can easily be covered via following simple observations. First, we may express matrix A in (4.4) as 0 O(1) , (4.49) A j (W + W¯ ) = Aˆ j + (ζ + |W¯ x1 |) O(1) O(1) where Aˆ j is a symmetric matrix obeying the same derivative bounds as described for A j , Aˆ 1 identical to A1 in the 11 block and obtained in other blocks kl by A1kl (W + W¯ ) = A1kl (W¯ ) + A1kl (W + W¯ ) − A1kl (W¯ ) = A1kl (W+ ) + O(|Wx | + |W¯ x1 |) = A1kl (W+ ) + O(ζ + |W¯ x1 |),
(4.50)
and meanwhile, Aˆ j , j = 1, obtained by A j = A j (W+ ) + O(ζ + |W¯ x1 |), similarly as in (4.50). Replacing A j by Aˆ j in the k th order Friedrichs-type bounds above, we find that the resulting error terms may be expressed as ∂xk O(ζ + |W¯ x1 |)|W |, |∂xk+1 w I I |, plus lower order terms, easily absorbed using Young’s inequality, and boundary terms k i II k I |∂x w (0)||∂x w (0)| , O i=0
resulting from the use of integration by parts as we deal with the 12-block. However these boundary terms were already treated somewhere as before. Hence we can recover the same Friedrichs-type estimates obtained above. Thus we may relax (A1 ) to (A1). Next, to relax (A3 ) to (A3), first we show that the symmetry condition B jk = B k j is not necessary. Indeed, by writing 1 1 jk (B jk + B k j )Wxk (B jk Wxk )x j = + (B − B k j )x j Wxk , 2 2 xj jk
jk
jk
we can just replace B jk by B˜ jk := 21 (B jk + B k j ), satisfying the same (A3 ), and thus still obtain the energy estimates as before, with a harmless error term (the last term in the above identity). Next notice that the term g(W˜ x ) − g(W¯ x1 ) in the perturbation equation may be Taylor expanded as 0 0 . + O(|Wx |2 ) g1 (W˜ x , W¯ x1 ) + g1 (W¯ x1 , W˜ x ) The first term, since it vanishes in the first component and since |W¯ x | decays at plus spatial infinity, yields by Young’s inequality the estimate $ I % 0 wx I I II 2 ¯ ≤ C (ζ + | W , |)w , w + w x 1 x x x 0 g1 (W˜ x , W¯ x1 ) + g1 (W¯ x1 , W˜ x ) wxI I
Stability of Multi-Dimensional Boundary Layers
35
which can be treated in the Friedrichs-type estimates. The (0, O(|Wx |2 )T nonlinear term may be treated as other source terms in the energy estimates. Specifically, the worst-case term # " 0 k k ∂x W, ∂x O(|Wx |2 ) = −∂xk+1 w I I , ∂xk−1 O(|Wx |2 ) − ∂xk w I I (0)∂xk−1 O(|Wx |2 )(0) may be bounded by ∂xk+1 w I I L 2 W W 2,∞ W H k − ∂xk w I I (0)∂xk−1 O(|Wx |2 )(0). The boundary term will contribute to energy estimates in the form (4.35) of Ibα , and thus we may use the parabolic equations to get rid of this term as we did in (4.23), (4.24). Thus, we may relax (A3 ) to (A3), completing the proof of the general case (A1) − (A3) and the proposition. 4.2. Proof of nonlinear stability. Defining the perturbation variable U := U˜ − U¯ , we obtain the nonlinear perturbation equations Ut − LU = Q j (U, Ux )x j , (4.51) j
where Q j (U, Ux ) = O(|U ||Ux | + |U |2 ), Q j (U, Ux )x j = O(|U ||Ux | + |U ||Ux x | + |Ux |2 ),
(4.52)
Q (U, Ux )x j xk = O(|U ||Ux x | + |Ux ||Ux x | + |Ux | + |U ||Ux x x |), j
2
so long as |U | remains bounded. For boundary conditions written in U -coordinates, (B) gives h = h˜ − h¯ = (W˜ (U + U¯ ) − W˜ (U¯ ))(0, x, ˜ t) ˜ ˜ ¯ = (∂ W /∂ U )(U0 )U (0, x, ˜ t) + O(|U (0, x, ˜ t)|2 )
(4.53)
in the inflow case, where (∂ W˜ /∂ U˜ )(U¯ 0 ) is constant and invertible, and h = h˜ − h¯ = (w˜ I I (U + U¯ ) − w˜ I I (U¯ ))(0, x, ˜ t) = (∂ w˜ I I /∂ U˜ )(U¯ 0 )U (0, x, ˜ t) + O(|U (0, x, ˜ t)|2 ) = m b¯1 b¯2 (U¯ 0 )U (0, x, ˜ t) + O(|U (0, x, ˜ t)|2 ) ˜ t) + O(|U (0, x, ˜ t)|2 ) = m B(U¯ 0 )U (0, x, for some invertible constant matrix m. Applying Lemma 3.9 to (4.51), we obtain t S(t − s) ∂x j Q j (U, Ux )ds + U (0, x, ˜ t), U (x, t) = S(t)U0 + 0
j
(4.54)
(4.55)
36
T. Nguyen, K. Zumbrun
where U (x, 0) = U0 (x), U (0, x, ˜ t) :=
t 0
Rd−1
(
G y j B j1 +G A1 )(x, t −s; 0, y˜ )U (0, y˜ , s) d y˜ ds, (4.56)
j
and G is the Green function of ∂t − L. Proof of Theorem 1.7. Define d−1 d (1 + s) 2 ζ (t) := sup |U (s)| L 2x (1 + s) 4 + |U (s)| L ∞ x s
+ (|U (s)| + |Ux (s)| + |∂x2˜ U (s)|) L 2,∞ (1 + s) x,x ˜ 1
d+1 4
.
(4.57)
We shall prove here that for all t ≥ 0 for which a solution exists with ζ (t) uniformly bounded by some fixed, sufficiently small constant, there holds ζ (t) ≤ C(|U0 | L 1 ∩H s + E 0 + ζ (t)2 ).
(4.58)
This bound together with continuity of ζ (t) implies that ζ (t) ≤ 2C(|U0 | L 1 ∩H s + E 0 )
(4.59)
for t ≥ 0, provided that |U0 | L 1 ∩H s + E 0 < 1/4C 2 . This would complete the proof of the bounds as claimed in the theorem, and thus give the main theorem. By standard short-time theory/local well-posedness in H s , and the standard principle of continuation, there exists a solution U ∈ H s on the open time-interval for which |U | H s remains bounded, and on this interval ζ (t) is well-defined and continuous. Now, let [0, T ) be the maximal interval on which |U | H s remains strictly bounded by some fixed, sufficiently small constant δ > 0. By Proposition 4.1, and the Sobolev embeding inequality |U |W 2,∞ ≤ C|U | H s , we have |U (t)|2H s ≤ Ce−θt |U0 |2H s + C ≤
C(|U0 |2H s
+
E 02
t 0
e−θ(t−τ ) |U (τ )|2L 2 + |Bh (τ )|2 dτ
+ ζ (t)2 )(1 + t)−(d−1)/2 ,
(4.60)
and so the solution continues so long as ζ remains small, with bound (4.59), yielding existence and the claimed bounds. Thus, it remains to prove the claim (4.58). First by (4.55), we obtain t |U (t)| L 2 ≤ |S(t)U0 | L 2 + |S1 (t − s)∂x j Q j (s)| L 2 ds 0 t + |S2 (t − s)∂x j Q j (s)| L 2 ds + |U (0, x, ˜ t)| L 2 0
≤ I1 + I2 + I3 + |U (0, x, ˜ t)| L 2 ,
(4.61)
Stability of Multi-Dimensional Boundary Layers
37
where d−1
I1 := |S(t)U0 | L 2 ≤ C(1 + t)− 4 |U0 | L 1 ∩H 3 , t I2 := |S1 (t − s)∂x j Q j (s)| L 2 ds 0 t d−1 1 d−1 ≤ C (1 + t − s)− 4 − 2 |Q j (s)| L 1 + (1 + s)− 4 |Q j (s)| L 1,∞ ds x,x ˜ 1 0 t d−1 1 d−1 − 4 −2 2 − 4 2 2 |U | 2,∞ + |Ux | 2,∞ ds ≤ C (1 + t − s) |U | H 1 + (1 + t − s) 0
t d−1 1 d−1 ≤ C(|U0 |2H s + ζ (t)2 ) (1 + t − s)− 4 − 2 (1 + s)− 2 0 d+1 − d−1 + (1 + t − s) 4 (1 + s)− 2 ds ≤ C(1 + t)−
d−1 4
L x,x ˜
1
L x,x ˜
1
(|U0 |2H s + ζ (t)2 ),
and
I3 :=
t
0
≤
0
t
|S2 (t − s)∂x j Q j (s)| L 2 ds e−θ(t−s) |∂x j Q j (s)| H 3 ds
t
≤ C
0
e−θ(t−s) (|U | L ∞ + |Ux | L ∞ )|U | H 5 ds
t
e−θ(t−s) |U |2H s ds 0 t d−1 ≤ C(|U0 |2H s + ζ (t)2 ) e−θ(t−s) (1 + s)− 2 ds ≤ C
− d−1 2
≤ C(1 + t)
0 2 (|U0 | H s
+ ζ (t)2 ).
Meanwhile, for the boundary term |U (0, x, ˜ t)| L 2 , we treat two cases separately. First for the inflow case, then by (4.53) we have |U (0, x, ˜ t)| ≤ C|h(x, ˜ t)| + O(|U (0, x, ˜ t)|2 ), and thus |U (0, x, ˜ t)| ≤ C|h(x, ˜ t)|, provided that |h| is sufficiently small. Therefore under the hypotheses on h in Theorem 1.7, Proposition 3.8 yields |U (0, ·, ·)| L 2x ≤ C E 0 (1 + t)−
d−1 4
.
Now for the outflow case, recall that in this case G(x, t; 0, y˜ ) ≡ 0. Thus (4.56) simplifies to t G y1 (x, t − s; 0, y˜ )B 11 U (0, y˜ , s) d y˜ ds. (4.62) U (0, x, ˜ t) = 0
Rd−1
To deal with this term, we shall use Proposition 3.8 as in the inflow case. In view of (4.54), |B 11 U (0, y˜ , s)| ≤ C|h( y˜ , t)| + O(|U (0, y˜ , s)|2 ),
38
T. Nguyen, K. Zumbrun
and assumptions on h are imposed as in Theorem 1.6, so that (3.26) is satisfied. To check the last term O(|U (0)|2 ), using the definition (4.57) of ζ (t), we have d
|O(|U (0, y˜ , s)|2 )| L 2 ≤ C|U | L ∞ |U | L 2,∞ ≤ Cζ 2 (t)(1 + s)− 2 − x,x ˜ 1
d+1 4
,
|O(|U (0, y˜ , s)|2 )| L ∞ ≤ C|U |2L ∞ ≤ Cζ 2 (t)(1 + s)−d , and for the term Dh with h replaced by O(|U (0, y˜ , s)|2 ), using the standard Hölder inequality to get |Dh | L 1 ≤ C(|U |2L 2,∞ + |Ux |2L 2,∞ + |Ux˜ x˜ |2L 2,∞ ) ≤ Cζ 2 (t)(1 + s)−
d+1 2
x˜
|Dh | H [(d−1)/2]+5 ≤ C|U | L ∞ |U | H s ≤ Cζ 2 (t)(1 + s)−d/2−(d−1)/4 . x˜
We remark here that Sobolev bounds (4.60) are not good enough for estimates of Dh in L 1 , requiring a decay at rate (1 + t)−d/2− for the two-dimensional case (see Proposition 3.8). This is exactly why we have to keep track of Ux˜ x˜ in the L 2,∞ norm in ζ (t) as well, to gain a bound as above for Dh . Therefore applying Proposition 3.8, we also obtain (4.62) for the outflow case. Combining these above estimates yields |U (t)| L 2 (1 + t)
d−1 4
≤ C(|U0 | L 1 ∩H s + E 0 + ζ (t)2 ).
(4.63)
Next, we estimate
t
|U (t)| L 2,∞ ≤ |S(t)U0 | L 2,∞ + |S1 (t − s)∂x j Q j (s)| L 2,∞ ds x,x ˜ 1 x,x ˜ 1 x,x ˜ 1 0 t + |S2 (t − s)∂x j Q j (s)| L 2,∞ ds + |U (0, x, ˜ t)| L 2,∞ x,x ˜ 1
0
x,x ˜ 1
≤ J1 + J2 + J3 + |U (0, x, ˜ t)| L 2,∞ ,
(4.64)
x,x ˜ 1
where d+1
J1 := |S(t)U0 | L 2,∞ ≤ C(1 + t)− 4 |U0 | L 1 ∩H 4 , x,x ˜ 1 t J2 := |S1 (t − s)∂x j Q j (s)| L 2,∞ ds x,x ˜ 1 0 t d+1 1 d+1 ≤ C (1 + t − s)− 4 − 2 |Q j (s)| L 1 + (1 + s)− 4 |Q j (s)| L 1,∞ ds x,x ˜ 1 0 t d+1 1 d+1 − 4 −2 2 − 4 2 2 |U | 2,∞ + |Ux | 2,∞ ds ≤ C (1 + t − s) |U | H 1 + (1 + t − s) 0
L x,x ˜
≤ C(|U0 |2H s + ζ (t)2 ) +(1 + t − s)− − d+1 4
≤ C(1 + t)
d+1 4
t
(1 + t − s)
0
(1 + s)−
(|U0 |2H s
d+1 2
ds
+ ζ (t) ), 2
1 − d+1 4 −2
(1 + s)
− d−1 2
1
L x,x ˜
1
Stability of Multi-Dimensional Boundary Layers
39
and (by Moser’s inequality) t J3 := |S2 (t − s)∂x j Q j (s)| L 2,∞ ds x,x ˜ 1 0 t ≤ C e−θ(t−s) |∂x j Q j (s)| H 4 ds 0 t ≤ C e−θ(t−s) |U | L ∞ |U | H 6 ds x 0 t d d−1 ≤ C(|U0 |2H s + ζ (t)2 ) e−θ(t−s) (1 + s)− 2 (1 + s)− 4 ds − d+1 4
≤ C(1 + t)
0 2 (|U0 | H s + ζ (t)2 ).
These estimates together with similar treatment for the boundary term yield |U (t)| L 2,∞ (1 + t)
d+1 4
x,x ˜ 1
≤ C(|U0 | L 1 ∩H s + E 0 + ζ (t)2 ).
(4.65)
Similarly, we have the same estimate for |Ux (t)| L 2,∞ . Indeed, we have
x,x ˜ 1
t
|Ux (t)| L 2,∞ ≤ |∂x S(t)U0 | L 2,∞ + |∂x S1 (t − s)∂x j Q j (s)| L 2,∞ ds x,x ˜ 1 x,x ˜ 1 x,x ˜ 1 0 t + |∂x S2 (t − s)∂x j Q j (s)| L 2,∞ ds + |∂x U (0, x, ˜ t)| L 2,∞ x,x ˜ 1
0
x,x ˜ 1
≤ K 1 + K 2 + K 3 + |∂x U (0, x, ˜ t)| L 2,∞ ,
(4.66)
x,x ˜ 1
where K 2 and K 3 are treated exactly in the same way as the treatment of J2 , J3 , yet in the first term of K 2 it is a bit better by a factor t −1/2 . Similar bounds hold for |Ux˜ x˜ | in L 2,∞ , noting that there are no higher derivatives in x1 involved and thus is similar to those in (4.64). Finally, we estimate the L ∞ norm of U . By Duhamel’s formula (4.55), we obtain t |U (t)| L ∞ ≤ |S(t)U0 | L ∞ + |S1 (t − s)∂x j Q j (s)| L ∞ ds 0 t + |S2 (t − s)∂x j Q j (s)| L ∞ ds + |U (0, x, ˜ t)| L ∞ 0
≤ L 1 + L 2 + L 3 + |U (0, x, ˜ t)| L ∞ ,
(4.67)
where the boundary term is treated in the same way as above, and for |γ | = [(d−1)/2]+2, d
L 1 := |S(t)U0 | L ∞ ≤ C(1 + t)− 2 |U0 | L 1 ∩H |γ |+3 , t L 2 := |S1 (t − s)∂x j Q j (s)| L ∞ ds 0 t d 1 d ≤ C (1 + t − s)− 2 − 2 |Q j (s)| L 1 + (1 + s)− 2 |Q j (s)| L 1,∞ ds x,x ˜ 1 0 t d 1 d −2−2 2 −2 2 2 |U | 2,∞ + |Ux | 2,∞ ds ≤ C (1 + t − s) |U | H 1 + (1 + t − s) 0
L x,x ˜
1
L x,x ˜
1
40
T. Nguyen, K. Zumbrun
t d 1 d−1 (1 + t − s)− 2 − 2 (1 + s)− 2 0 d d+1 + (1 + t − s)− 2 (1 + s)− 2 ds
≤ C(|U0 |2H s + ζ (t)2 )
d
≤ C(1 + t)− 2 (|U0 |2H s + ζ (t)2 ) and (again by Moser’s inequality),
t
L 3 :=
0
0
t
≤
t
≤ 0
|S2 (t − s)∂x j Q j (s)| L ∞ ds |S2 (t − s)∂x j Q j (s)| H |γ | ds e−θ(t−s) |∂x Q j (s)| H |γ |+3 ds t
e−θ(t−s) |U | L ∞ |U | H |γ |+5 ds 0 t d d−1 ≤ C(|U0 |2H s + ζ (t)2 ) e−θ(t−s) (1 + s)− 2 (1 + s)− 4 ds ≤ C
0
d
≤ C(1 + t)− 2 (|U0 |2H s + ζ (t)2 ). Therefore we have obtained d
(1 + t) 2 ≤ C(|U0 | L 1 ∩H s + E 0 + ζ (t)2 ), |U (t)| L ∞ x and thus completed the proof of claim (4.58), and the main theorem.
(4.68)
Appendix A. Physical Discussion in the Isentropic Case In this appendix, we revisit in slightly more detail the drag-reduction problem sketched in Examples 1.1–1.2, in the simplified context of the two-dimensional isentropic case. Following the notation of [GMWZ5], consider the two-dimensional isentropic compressible Navier–Stokes equations ρt + (ρu)x + (ρv) y = 0,
(A.1)
(ρu)t + (ρu )x + (ρuv) y + px = (2μ + η)u x x + μu yy + (μ + η)vx y ,
(A.2)
(ρv)t + (ρuv)x + (ρv ) y + p y = μvx x + (2μ + η)v yy + (μ + η)u yx
(A.3)
2
2
on the half-space y > 0, where ρ is density, u and v are velocities in the x and y directions, and p = p(ρ) is pressure, and μ > |η| ≥ 0 are coefficients of first (“dynamic”) and second viscosity, making the standard monotone pressure assumption p (ρ) > 0. We imagine a porous airfoil lying along the x-axis, with constant imposed normal velocity v(0) = V and zero transverse relative velocity u(0) = 0 imposed at the airfoil surface, and seek a laminar boundary-layer flow (ρ, u, v)(y) with transverse relative velocity u ∞ a short distance away the airfoil, with |V | much less than the sound speed c∞ and |u ∞ | of an order roughly comparable to c∞ .
Stability of Multi-Dimensional Boundary Layers
41
A.1. Existence. The possible boundary-layer solutions have been completely categorized in this case in Sect. 5.1 of [GMWZ5]. We here cite the relevant conclusions, referring to [GMWZ5] for the (straightforward) justifying computations. A.1.1. Outflow case (V < 0). In the outflow case, the scenario described above corresponds to case (5.15) of [GMWZ5], in which it is found that the only solutions are purely transverse flows (ρ, v) ≡ (ρ0 , V ), u(y) = u ∞ (1 − eρ0 V y/μ ),
(A.4)
varying only in the tranverse velocity u. The drag force per unit length at the airfoil, by Newton’s law of viscosity, is μu¯ y | y=0 = u ∞ ρ∞ |V |,
(A.5)
since momentum m := ρ0 V = ρ∞ V is constant throughout the layer, so that (ρ∞ , u ∞ being imposed by ambient conditions away from the wing) drag is proportional to the speed |V | of the imposed normal velocity. A.1.2. Inflow case (V > 0). Consulting again [GMWZ5] (p. 61), we find for V > 0 with specified (ρ, u, v)(0) of the orders described above, the only solutions are purely normal flows, u ≡ u(0), (ρ, v) = (ρ, v)(y),
(A.6)
varying only in the normal velocity v. Thus, it is not possible to reconcile the velocity u(0) at the airfoil with the velocity u ∞ >> c some distance away. As discussed in [MN], the expected behavior in such a case consists rather of a combination of a boundary-layer at y = 0 and one or more elementary planar shock, rarefaction, or contact waves moving away from y = 0: in this case a shear wave moving with normal fluid velocity V into the half-space, across which the transverse velocity changes from zero to u ∞ . That is, a characteristic layer analogous to the solid-boundary case detaches from the airfoil and travels outward into the flow field. In this case, one would not expect drag reduction compared to the solid-boundary case, but rather some increase. A.2. Stability. If we consider one-dimensional stability, or stability with respect to perturbations depending only on y, we find that the linearized eigenvalue equations decouple into the constant-coefficient linearized eigenvalue equations for (ρ, v) about a constant layer (ρ, v) ≡ (ρ0 , V ), and the scalar linearized eigenvalue equation λρu ¯ + mu y = μu yy
(A.7)
associated with the constant-coefficient convection-diffusion equation ρu ¯ t + mu = μu yy , m := ρ¯ v¯ ≡ ρ0 V , ρ¯ ≡ ρ0 . As the constant layer (ρ0 , V ) is stable by Corollary 1.5 or direct calculation (Fourier transform), and (A.7) is stable by direct calculation, we may thus conclude that purely transverse layers are one- dimensionally stable. Considered with respect to general perturbations, the equations do not decouple, nor do they reduce to constant-coefficient form, but to a second order system whose coefficients are quadratic polynomials in eρ0 V y . It would be very interesting to try to resolve the question of spectral stability by direct solution using this special form, or, alternatively, to perform a numerical study as done in [HLyZ2] for the multi-dimensional shock wave case.
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Remark A.1. For general laminar boundary layers (ρ, ¯ u, ¯ v)(y), ¯ the one-dimensional stability problem, now variable-coefficient, does not completely decouple, but has triangular form, breaking into a system in (ρ, v) alone and an equation in u forced by (ρ, v). Stability with respect to general perturbations, therefore, is equivalent to stability with respect to perturbations of form (ρ, 0, v) or (0, u, 0). For perturbations (ρ, u, v) = (0, u, 0), the u equation again becomes (A.7), with μ, m still constant, but ρ¯ varying in y. Taking the real part of the complex L 2 inner product of u against (A.7) gives λu2L 2 + u y 2L 2 = 0, hence for λ ≥ 0, u ≡ constant = 0. Thus, the layer is one-dimensionally stable if and only if the normal part (ρ, ¯ v) ¯ is stable with respect to perturbations (ρ, v). Stability of normal layers was studied in [CHNZ] for a γ -law gas p(ρ) = aρ γ , 1 ≤ γ ≤ 3, with the conclusion that all layers are one-dimensionally stable, independent of amplitude, in the general inflow and compressive outflow cases. Hence, we can make the same conclusion for full layers (ρ, ¯ u, ¯ v). ¯ In the present context, this includes all cases except for suction with supersonic velocity |V | > c∞ , which in the notation of [CHNZ] is of expansive outflow type, since |v| ¯ is decreasing with y, so that density ρ¯ (since m = ρ¯ v¯ ≡ constant) is increasing.
A.3. Discussion. Note that we do not achieve by subsonic boundary suction an exact laminar flow connecting the values (u, v) = (0, V ) at the wing to the values (u ∞ , 0) of the ambient flow at infinity, but rather to an intermediate value (u ∞ , V ). That is, we trade a large variation u ∞ in shear for a possibly small variation V in normal velocity, which appears now as a boundary condition for the outer, approximately Euler flow away from the boundary layer. Whether the full solution is stable appears to be a question concerning also nonstationary Euler flow. It is not clear either what is the optimal outflux velocity V . From (A.5) and the discussion just above, it appears desirable to minimize |V |, since this minimizes both drag and the imbalance between flow v∞ just outside the boundary layer and the ambient flow at infinity. On the other hand, we expect that stability becomes more delicate in the characteristic limit V → 0− , in the sense that the size of the basin of attraction of the boundary layer shrinks to zero (recall, we have ignored throughout our analysis the size of the basin of attraction, taking perturbations as small as needed without keeping track of constants). These would be quite interesting issues for further investigation. References [AGJ] [BHRZ] [Bra] [BDG] [Br1] [Br2]
Alexander, J., Gardner, R., Jones, C.: A topological invariant arising in the stability analysis of travelling waves. J. Reine Angew. Math. 410, 167–212 (1990) Barker, B., Humpherys, J., Rudd, K., Zumbrun, K.: Stability of viscous shocks in isentropic gas dynamics. Commun. Math. Phys. 281, 231–249 (2008) Braslow, A.L.: A history of suction-type laminar-flow control with emphasis on flight research. NSA History Division, Monographs in aerospace history, number 13, Washington, DC: NASA, 1999 Bridges, T.J., Derks, G., Gottwald, G.: Stability and instability of solitary waves of the fifthorder kdv equation: a numerical framework. Phys. D 172(1–4), 190–216 (2002) Brin, L.Q.: Numerical testing of the stability of viscous shock waves. PhD thesis, Indiana University, Bloomington, 1998 Brin, L.Q.: Numerical testing of the stability of viscous shock waves. Math. Comp. 70(235), 1071–1088 (2001)
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[BrZ] [CHNZ] [GZ] [GR] [GMWZ1] [GMWZ5] [GMWZ6] [HZ] [HLZ] [HLyZ1] [HLyZ2] [HoZ1] [HoZ2] [HuZ] [KK] [KNZ] [KSh] [KZ] [MaZ3] [MaZ4] [MN] [MZ] [NZ] [PW] [RZ] [R2] [R3] [S] [SZ]
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Brin, L.Q., Zumbrun, K.: Analytically varying eigenvectors and the stability of viscous shock waves. Mat. Contemp. 22, 19–32 (2002) Costanzino, N., Humpherys, J., Nguyen, T., Zumbrun, K.: Spectral stability of noncharacteristic boundary layers of isentropic navier–stokes equations. Arch. Ration. Mech. Anal. 192, 537–587 (2009) Gardner, R.A., Zumbrun, K.: The gap lemma and geometric criteria for instability of viscous shock profiles. Comm. Pure Appl. Math. 51(7), 797–855 (1998) Grenier, E., Rousset, F.: Stability of one dimensional boundary layers by using green’s functions. Comm. Pure Appl. Math. 54, 1343–1385 (2001) Guès, O., Métivier, G., Williams, M., Zumbrun, K.: Multidimensional viscous shocks i: degenerate symmetrizers and long time stability. J. Amer. Math. Soc. 18(1), 61–120 (2005) Guès, O., Métivier, G., Williams, M., Zumbrun, K.: Existence and stability of noncharacteristic boundary-layers for compressible Navier-Stokes and viscous MHD equations. Arch. Ration. Mech. Anal. http://arxiv.org/abs/0805.3333v1[math.AP] Guès, O., Métivier, G., Williams, M., Zumbrun, K.: Viscous boundary value problems for symmetric systems with variable multiplicities. J. Differ. Equ. 244, 309–387 (2008) Howard, P., Zumbrun, K.: Stability of undercompressive viscous shock waves. J. Differ. Equ. 225(1), 308–360 (2006) Humpherys, J., Lafitte, O., Zumbrun, K.: Stability of isentropic Navier-Stokes shocks in the high-Mach number limit. Commun. Math. Phys. 293(1), 1–36 (2010) Humpherys, J., Lyng, G., Zumbrun, K.: Spectral stability of ideal-gas shock layers. Arch. Ration. Mech. Anal. 194(3), 1029–1079 (2009) Humpherys, J., Lyng, G., Zumbrun, K.: Multidimensional spectral stability of large-amplitude Navier-Stokes shocks. In preparation Hoff, D., Zumbrun, K.: Multi-dimensional diffusion waves for the navier-stokes equations of compressible flow. Indiana Univ. Math. J. 44(2), 603–676 (1995) Hoff, D., Zumbrun, K.: Pointwise decay estimates for multidimensional navier-stokes diffusion waves. Z. Angew. Math. Phys. 48(4), 597–614 (1997) Humpherys, J., Zumbrun, K.: An efficient shooting algorithm for evans function calculations in large systems. Physica D 220(2), 116–126 (2006) Kagei, Y., Kawashima, S.: Stability of planar stationary solutions to the compressible navierstokes equations in the half space. Commun. Math. Phys. 266, 401–430 (2006) Kawashima, S., Nishibata, S., Zhu, P.: Asymptotic stability of the stationary solution to the compressible navier-stokes equations in the half space. Commun. Math. Phys. 240(3), 483– 500 (2003) Kawashima, S., Shizuta, Y.: Systems of equations of hyperbolic-parabolic type with applications to the discrete boltzmann equation. Hokkaido Math. J. 14(2), 249–275 (1985) Kwon, B., Zumbrun, K.: Asymptotic behavior of multidimensional scalar relaxation shocks. J. Hyperbolic Differ. Equ. 6(4), 663–708 (2009) Mascia, C., Zumbrun, K.: Pointwise green function bounds for shock profiles of systems with real viscosity. Arch. Ration. Mech. Anal. 169(3), 177–263 (2003) Mascia, C., Zumbrun, K.: Stability of large-amplitude viscous shock profiles of hyperbolicparabolic systems. Arch. Ration. Mech. Anal. 172(1), 93–131 (2004) Matsumura, A., Nishihara, K.: Large-time behaviors of solutions to an inflow problem in the half space for a one-dimensional system of compressible viscous gas. Commun. Math. Phys. 222(3), 449–474 (2001) Métivier, G., Zumbrun, K.: Viscous Boundary Layers for Noncharacteristic Nonlinear Hyperbolic Problems. Memoirs AMS 826, Providence, RI: Amer. Math. Soc., 2005 Nguyen, T., Zumbrun, K.: Long-time stability of large-amplitude noncharacteristic boundary layers for hyperbolic-parabolic systems. J. Math. Pures Appl. 92(6), 547–598 (2009) Pego, R.L., Weinstein, M.I.: Eigenvalues, and instabilities of solitary waves. Philos. Trans. Roy. Soc. London Ser. A 340(1656), 47–94 (1992) Raoofi, M., Zumbrun, K.: Stability of undercompressive viscous shock profiles of hyperbolicparabolic systems. J. Differ. Equ. 246(4), 1539–1567 (2009) Rousset, F.: Inviscid boundary conditions and stability of viscous boundary layers. Asymptot. Anal. 26(3–4), 285–306 (2001) Rousset, F.: Stability of small amplitude boundary layers for mixed hyperbolic-parabolic systems. Trans. Amer. Math. Soc. 355(7), 2991–3008 (2003) Schlichting, H.: Boundary layer theory. Translated by J. Kestin. 4th ed. McGraw-Hill Series in Mechanical Engineering. New York: McGraw-Hill Book Co., Inc., 1960 Serre, D., Zumbrun, K.: Boundary layer stability in real vanishing-viscosity limit. Commun. Math. Phys. 221(2), 267–292 (2001)
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[YZ] [Z2] [Z3] [Z4] [ZH]
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Yarahmadian, S., Zumbrun, K.: Pointwise green function bounds and long-time stability of large-amplitude noncharacteristic boundary layers. SIAM J. Math. Anal. 40(6), 2328–2350 (2009) Zumbrun, K.: Multidimensional stability of planar viscous shock waves. In: Advances in the theory of shock waves. Volume 47 of Progr. Nonlinear Differential Equations Appl., Boston, MA: Birkhäuser Boston, 2001, pp. 307–516 Zumbrun, K.: Stability of large-amplitude shock waves of compressible Navier-Stokes equations. In: Handbook of mathematical fluid dynamics. Vol. III, Amsterdam: North-Holland, 2004, pp. 311–533 (with an appendix by Helge Kristian Jenssen and Gregory Lyng) Zumbrun, K.: Planar stability criteria for viscous shock waves of systems with real viscosity. In: Hyperbolic systems of balance laws. Volume 1911 of Lecture Notes in Math., Berlin: Springer, 2007, pp. 229–326 Zumbrun, K., Howard, P.: Pointwise semigroup methods and stability of viscous shock waves. Indiana Univ. Math. J. 47(3), 741–871 (1998)
Communicated by P. Constantin
Commun. Math. Phys. 299, 45–87 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1094-8
Communications in
Mathematical Physics
Droplet Phases in Non-local Ginzburg-Landau Models with Coulomb Repulsion in Two Dimensions Cyrill B. Muratov Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102, USA. E-mail:
[email protected] Received: 14 May 2009 / Accepted: 4 May 2010 Published online: 29 July 2010 – © Springer-Verlag 2010
Abstract: We establish the behavior of the energy of minimizers of non-local Ginzburg-Landau energies with Coulomb repulsion in two space dimensions near the onset of multi-droplet patterns. Under suitable scaling of the background charge density with vanishing surface tension the non-local Ginzburg-Landau energy becomes asymptotically equivalent to a sharp interface energy with screened Coulomb interaction. Near the onset the minimizers of the sharp interface energy consist of nearly identical circular droplets of small size separated by large distances. In the limit the droplets become uniformly distributed throughout the domain. The precise asymptotic limits of the bifurcation threshold, the minimal energy, the droplet radii, and the droplet density are obtained. 1. Introduction Spatial patterns are often a result of the competition between thermodynamic forces operating on different length scales. When short-range attractive interactions are present in a system, phase separation phenomena can be observed, resulting in aggregation of particles or formation of droplets of new phase, which evolve into macroscopically large domains via coarsening or nucleation and growth (see e.g. [1]). This process, however, can be frustrated in the presence of long-range repulsive forces. As the droplets grow, the contribution of the long-range interaction may overcome the short-range forces, whereby suppressing further growth. This mechanism was identified in many energydriven pattern forming systems of different physical nature, such as various types of ferromagnetic systems, type-I superconductors, Langmuir layers, multiple polymer systems, etc., just to name a few [2–11]. Remarkably, these systems often exhibit very similar pattern formation behaviors [10,12]. One important class of systems with competing interactions are systems in which the long-range repulsive forces are of Coulomb type (for an overview, see [13,14] and references therein). The nature of the Coulombic forces may be very different from
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system to system. For example, these forces may arise when particles undergoing phase separation carry net electric charge [15–18], or they may be a consequence of entropic effects associated with chain conformations in polymer systems [19–23]. Coulomb interactions may also arise indirectly as a result of diffusion-mediated processes [4,24,25]. All this makes systems with repulsive Coulombic interactions a ubiquitous example of pattern forming systems. Studies of systems with competing short-range attractive interactions and long-range repulsive Coulomb interactions go back to the work of Ohta and Kawasaki, who proposed a non-local extension of the Ginzburg-Landau energy in the context of diblock copolymer systems [19]. Even though its validity for diblock copolymer systems may be questioned [21,26–28], the Ohta-Kawasaki model is applicable to a great number of physical problems of different origin [14]. On the other hand, mathematically the Ohta-Kawasaki model presents a paradigm of energy-driven pattern forming systems which has been receiving a growing degree of attention [9,29–37]. The Ohta-Kawasaki energy is a functional of the form [13,14,19,24,38]:
ε2 2 E[u] = |∇u| + W (u) d x 2 Ω 1 + (u(x) − u)G ¯ 0 (x, y)(u(y) − u)d ¯ x d y. 2 Ω Ω
(1.1)
Here, u : Ω → R is a scalar quantity denoting the “order parameter” in a bounded domain Ω ⊂ Rd . Different terms of the energy are as follows: the first term penalizes spatial variations of u on the scales shorter than ε, the second term, in which W is a symmetric double-well potential drives local phase separation towards the minima of W at u = ±1, and the last term is the long-range interaction, whose Coulombic nature comes from the fact that the kernel G 0 solves the Neumann problem for 1 − ΔG 0 (x, y) = δ(x − y) − , G 0 (x, y)d x = 0, (1.2) |Ω| Ω where Δ is the Laplacian in x and δ(x) is the Dirac delta-function. The parameter u¯ denotes the prescribed uniform background charge, and the overall “charge neutrality” is ensured via the constraint 1 u d x = u. ¯ (1.3) |Ω| It is important to note that the kernel G 0 solves (1.2) in the space of the same dimensionality as the order parameter u (not to be confused with the case in which the kernel solves Laplace’s equation in the space of higher spatial dimensionality, as is common in many other systems with competing interactions, see e.g. [7,16]). The parameter ε > 0 in (1.1) determines both the scale of the short-range interaction and the magnitude of the interfacial energy between the regions with different values of u when ε is sufficiently small. In fact, it is known that no patterns can form in the system if ε is sufficiently large [13,14,39]. On the other hand, when ε 1, the first term in the functional E becomes a singular perturbation, giving rise to “domain structures” (see Fig. 1), which are of particular physical interest. These patterns consist of extended regions in which u is close to one of the minima of the potential W , separated by narrow domain walls. In this situation one can reduce the energy functional appearing in (1.1)
Droplet Phases in Non-local Ginzburg-Landau Models
47
Fig. 1. A multi-droplet pattern: density plot of u in a local minimizer of E[u] with W (u) = 41 (1−u 2 )2 obtained numerically for u¯ = −0.5, ε = 0.025, and Ω = [0, 11.5) × [0, 10), with periodic boundary conditions. Dark regions correspond to u ≈ −1, and light regions correspond to u ≈ 1 (from [14])
to an expression in terms of the interfaces alone. In [13,14], such a reduction was performed for E using formal asymptotic techniques (see also [30,35,40,41]) and leads to the following reduced energy (for simplicity of notation, we choose the normalizations in such a way that the parameter ε is, in fact, the domain wall energy, see Sec. 4 for details): ε 1 E[u] = |∇u| d x + (u(x) − u)G(x, ¯ y)(u(y) − u) ¯ d x d y. (1.4) 2 Ω 2 Ω Ω Here the function u takes on values ±1 throughout Ω, and the kernel G is the screened Coulomb kernel, i.e., it solves the Neumann problem for − ΔG(x, y) + κ 2 G(x, y) = δ(x − y),
(1.5)
with some κ > 0. The constant κ has the physical meaning of the inverse of the Debye screening length [13,14]. Note that the sharp interface energy E with the unscreened Coulomb kernel (i.e. with κ = 0) was derived by Ren and Wei as the Γ -limit of the diffuse interface energy E under assumptions of weak non-local coupling (i.e., with an extra factor of ε in front of the Coulomb kernel) and u¯ ∈ (−1, 1) independent of ε, as ε → 0 [30] (see also [35,37]; note that this case is also equivalent to considering E on the domain of size O(ε1/3 )). At the same time, screening becomes important near the transition between the uniform and the patterned states which occurs near |u| ¯ = 1, the case of interest in the present paper [13,14]. Note that in the presence of screening the neutrality condition in (1.3) is relaxed. In this paper, we rigorously establish the relation between the sharp interface energy E and the diffuse interface energy E, and analyze the precise behavior of minimizers of the sharp interface energy E for ε 1 in the vicinity of the transition from the trivial minimizer to patterned states occurring near |u| ¯ = 1. We note that despite the apparent simplicity of the expression for E, the minimizers of E exhibit quite an intricate dependence on the parameters for ε 1 and |u| ¯ 1. Our analysis in this paper will be restricted to the case d = 2. While a number of our results can be extended
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to arbitrary space dimensions, our methods to obtain sharp estimates for the energy of minimizers rely critically on the properties of minimal curves in two dimensions and the logarithmic behavior of the Green’s function of the two-dimensional Laplacian near the singularity. Therefore, they cannot be readily extended to other spatial dimensionalities, and, indeed, one would expect certain important differences between these cases and the case of two space dimensions. At the same time, we will show that in the case d = 2 it is possible to obtain rather detailed information about the structure of the transition near |u| ¯ = 1 in terms of energy. Let us note that, since the case d = 1 is now well-understood [29–31,42], the remaining open case of physical interest is that of d = 3. Before turning to the analysis, let us briefly mention a perfect example of an experimental system in which the regimes studied by us could be easily realized, which is inspired by the beautiful Nobel Lecture of Prof. G. Ertl [43,44]. Consider molecules which undergo adsorption and desorption to and from a crystalline surface. On the surface, the atoms may hop around and reversibly stick to each other to form monolayer aggregates [45]. Then, within the framework of phase field models, this process may be described by the following evolution equation for the adsorbate density fraction φ [25]: φt = MΔ(W (φ) − gΔφ) + k+ (1 − φ) − k− φ,
(1.6)
where W is a double-well potential with two minima between φ = 0 and φ = 1, g is the short-range coupling constant, M is a kinetic coefficient, and k± are the adsorption and desorption rates, respectively. Note that this equation can be rewritten as ¯ φt = MΔ{W (φ) − gΔφ + kG 0 ∗ (φ − φ)},
(1.7)
where k = (k+ + k− )/M, φ¯ = k+ /(k+ + k− ), and “∗” denotes convolution in space, ¯ Upon suitwith G 0 given by (1.2), provided the spatial average of the initial data is φ. able rescaling, this is precisely the H −1 gradient flow for the energy E, i.e., we have u t = Δ(δE/δu), where u is a rescaling of φ. In particular, minimizers of E are ground states of the considered system in equilibrium in the mean-field limit. We note that the adsorption and desorption rates k± can be quite small compared to the hopping rate, resulting in very small values of ε ∼ k 1/2 . Therefore, one can achieve a very good scale separation between the interfacial thickness (atomic scales) and the size of adsorbate clusters (micro-scale) in this experimental setup. Our paper is organized as follows. In Sec. 2, we present heuristic arguments and give the statements of main results, in Sec. 3, we perform a detailed analysis of the sharp interface energy E, in Sec. 4 we establish a connection between the sharp interface energy E and the diffuse interface energy E. Finally, in Sec. 5 we conclude the proofs of the theorems. Throughout the paper, the symbols L p , H k , W k, p , C k,α , BV denote the usual function spaces, | · | denotes the d-dimensional Lebesgue measure or the (d − 1)-dimensional Hausdorff measure of a set, depending on the context, and C, c, etc., denote generic positive constants that can change from line to line. The symbols O(1) and o(1) denote, as usual, uniformly bounded and uniformly small quantities, respectively, in the limit ε → 0, etc. Finally, we will say that a statement holds for ε 1, etc., if there exists ε0 > 0 such that that statement is true for all 0 < ε ≤ ε0 . For simplicity of notation, the subscript ε is omitted for all quantities depending on ε.
Droplet Phases in Non-local Ginzburg-Landau Models
49
2. Heuristics and Main Results Let us begin our investigation by setting d = 2 and making a simplifying assumption that the domain Ω is a torus1 : Ω = [0, 1)2 . Let us also specify the domains of definition for the functionals E and E. Formally, the diffuse interface energy E[u] will be defined for all u ∈ H 1 (Ω) subject to Ω u d x = u, ¯ whereas the sharp interface energy E[u] will be defined for all u ∈ BV (Ω; {−1, 1}). The assumption that Ω is a torus, which is common in the considered class of problems, eliminates the need to deal with the boundary effects and, even more importantly, restores the translational invariance inherent in the problem on the whole of Rd (note that the choice of the size of Ω is inconsequential, the obtained energy of the minimizers scales linearly with |Ω|). As a result, the kernel of the non-local part of the energy becomes a function of x − y only. With a slight abuse of notation, in the following we will, therefore, replace G(x, y) with G(x − y) everywhere below. On heuristic grounds one would expect that the minimizers of E at ε 1 would be periodic with period R ∼ ε1/3 , whenever |u| ¯ < 1 and |u| ¯ is not too close to 1 [9,13,14,19]. A simple scaling analysis shows that in this case E ∼ ε2/3 as ε → 0 with u¯ fixed. Our first result gives a justification for this energy scaling without any assumptions about the minimizers (for statements about existence and regularity of minimizers, see the following sections). Theorem 2.1. Let W satisfy the assumptions (i)–(iv) at the beginning of Sec. 4, and let u¯ ∈ (−1, 1) be fixed. Then there exist ε0 > 0 and C > c > 0, such that cε2/3 ≤ min E, min E ≤ Cε2/3
(2.1)
for all ε ≤ ε0 . Observe also that for E this result still holds when Ω = [0, 1)d for any d, while for E it holds at least for d < 6 (see Sec. 4). We note that for u¯ = 0 and |u| ≤ 1 such a result was obtained by Choksi, using somewhat different techniques [9]. On the level of E (with κ = 0), Alberti, Choksi and Otto recently proved, among many other interesting results, a stronger statement that in the limit ε → 0, the constants in the upper and lower bounds in (2.1) can be chosen to be arbitrarily close to each other [36]. We note that the case κ = 0 and u¯ ∈ (−1, 1) fixed can be treated as the limit of energy E considered by us as κ → 0, when the constraint Ω u d x = u¯ gets automatically enforced (see (5.2)). Thus, when u¯ ∈ (−1, 1) is fixed, the energy E admits a non-trivial minimizer, whose energy scales as in (2.1) when ε 1. What about the case |u| ¯ > 1? Here, in fact, it is easy to see that the only minimizers admitted by E are the trivial ones. Consider, for example, the case u¯ < −1, the other case is equivalent by symmetry. In this case the problem admits the unique global minimizer u = −1. To see this, let us introduce the characteristic function χΩ + of the set Ω + = {u = +1} for a given u ∈ BV (Ω; {−1, 1}). Then u = 2χΩ + − 1, and by a straightforward computation 1 (2χΩ + (x) − 1 − u)G(x ¯ − y)(2χΩ + (y) − 1 − u)d ¯ xd y E[u] ≥ 2 Ω Ω (1 + u) ¯ 2 2(1 + u) ¯ ≥ − |Ω + |. (2.2) 2 2 2κ κ 1 Throughout the paper, [0, )d will always denote a d-dimensional rectangle with size and periodic boundary conditions, identified with a d-dimensional torus.
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Thus, when u¯ < −1, the second term in the last inequality in (2.2) is positive, hence, is minimized by |Ω + | = 0. But in this case u = −1 attains equality in (2.2), so u = −1 is the minimizer. Thus, when |u| ¯ > 1, non-trivial minimizers of E do not exist, and, therefore, at |u| ¯ = 1 a bifurcation occurs in the limit ε → 0. The main purpose of this paper is to investigate the transition between the trivial and the non-trivial minimizers of E and E that occurs in the neighborhood of |u| ¯ = 1 for ε 1. The energy E captures most of the difficulty associated with the considered problem. Therefore, we will spend most of our effort in this paper to the studies of E (see Sec. 3). At the same time, as we show later (see Sec. 4), the statements about the behavior of min E also extend to that of min E for ε 1 (the correspondence of minimizers of the two energies will be a subject of future study). When Ω = [0, 1)2 , the kernel G has an explicit representation G(x) =
1 K 0 (κ|x − n|), 2π 2
(2.3)
n∈Z
where K 0 is the modified Bessel function of the first kind. In particular, G > 0 and we have the following asymptotic expansion from the power series representation of K 0 [46]: G(x) = − where
1 ln(κ|x|) ¯ + O(|x|), 2π
⎛
κ¯ = 21 κ exp ⎝γ −
(2.4) ⎞
K 0 (κ|n|)⎠ ,
(2.5)
n∈Z2 \{0}
and γ ≈ 0.5772 is Euler’s constant. We also have G(x) bounded whenever |x| > δ, for any δ > 0, and (2.4) can be used to estimate derivatives of G to O(|x| ln |x|) as well. Consider the case in which the value of u¯ approaches u¯ = −1 from above, with ε 1 fixed. Clearly, for large enough deviations there exists a non-trivial minimizer. As can be seen from the arguments in the proof of Theorem 2.1, the size of the set where u = 1 on the minimizer goes to zero as u¯ → −1. Heuristically, one would, therefore, expect that in this situation the minimizer will consist of a number of isolated droplets where u = +1 of small size in the background where u = −1. Moreover, since on the scale of a droplet the interfacial energy will give a dominant contribution, these droplets are expected to be nearly circular. This motivates an introduction of the following reduced energy: E N ({ri }, {xi }) =
N 2π εri − 2π(1 + u)κ ¯ −2 ri2 − πri4 (ln κr ¯ i − 14 ) i=1
+4π 2
N −1
N
G(xi − x j )ri2 r 2j ,
(2.6)
i=1 j=i+1
which describes the energy of interaction of N well separated disk-shaped droplets of radius ri centered at xi , to the leading order. More precisely, the first term (2.6) stands for the interfacial energy of all the droplets, the second term is the energy of interactions
Droplet Phases in Non-local Ginzburg-Landau Models
51
between the droplets and the background, the third term is the self-interaction energy of each droplet, and the last term is the interaction energy of each droplet pair (for the case of a single droplet in R2 , see [14]). We can use the reduced energy in (2.6) to obtain the leading order scaling of various quantities for ε 1 by balancing different terms. From the balance of interfacial energy and the self-interaction energy, one should have ri = O(ε1/3 | ln ε|−1/3 ). Balancing this with the second term leads, in turn, to δ¯ = ε−2/3 | ln ε|−1/3 (1 + u) ¯
(2.7)
being an O(1) quantity. Similarly, balancing the last term with the first three leads to N = O(| ln ε|), and the expected behavior of min E N = O(ε4/3 | ln ε|2/3 ). One would also expect that, since the droplets repel each other, in a minimum energy configuration they would become uniformly distributed throughout Ω. Our main result proves and further quantifies this heuristic picture on the level of the sharp interface energy E. ¯ with some δ¯ > 0 fixed. Then for any σ > 0 Theorem 2.2. Let u¯ = −1 + ε2/3 | ln ε|1/3 δ, sufficiently small there exists ε0 > 0 such that for all ε ≤ ε0 : √ (i) If δ¯ < 21 3 9 κ 2 , then u = −1 is the unique global minimizer of E, with ε−4/3 | ln ε|−2/3 min E = 21 κ −2 δ¯2 . √ (ii) If δ¯ > 21 3 9 κ 2 , there exists a non-trivial minimizer of E. The minimizer is u(x) = −1 + 2
N
χΩi+ (x),
(2.8)
i=1
where χΩi+ are characteristic functions of N disjoint simply connected sets Ωi+ ⊂ Ω with boundary of class C 3 , and N = O(| ln ε|). The boundary of each set Ωi+ is O(ε2/3−σ )-close (in the Hausdorff sense) to a circle of radius ri centered at xi . Furthermore, min E = 21 ε4/3 | ln ε|2/3 κ −2 δ¯2 + E N ({ri }, {xi }) + O(ε5/3−σ ),
(2.9)
with E N = O(ε4/3 | ln ε|2/3 ), ri = O(ε1/3 | ln ε|−1/3 ), and (iii) If δ¯ >
√ 1 3 2
|xi − x j | > εσ , ∀ j = i. 9 κ 2 , in the limit ε → 0 we have ε−1/3 | ln ε|1/3ri →
√ 3 3
(2.10)
(2.11)
uniformly,
√ N 3 9 2 1 1 ¯ κ , δ(x − xi ) → √ δ− | ln ε| 2 2π 3 9 i=1
(2.12)
weakly in the sense of measures, and ε
−4/3
| ln ε|
−2/3
√ √ 3 3 9 9 2 δ¯ − κ . min E → 2 4
(2.13)
52
C. B. Muratov
Fig. 2. “Coulombic dice”: Minimizers of E N with ri = obtained using a random search algorithm
√ 3 3 ε 1/3 | ln ε|−1/3 for κ = 2 and N = 2, 3, 4, 5,
Note that √ a more detailed result on the structure of the transition occurring near δ¯ = 21 3 9 κ 2 is presented in Proposition 3.11. Let us make a few remarks related to the statements of Theorem 2.2. For small, but finite values of ε this theorem establishes an equivalence between the sharp interface energy E and the energy of N interacting droplets E N , in the sense that the minimizers of E are close to “almost” minimizers of E N , i.e., we have E N < min E N + O(ε5/3−σ ). Nevertheless, to prove closeness of minimizers of E to those of E N we also need some coercivity of the energy E N . This problem has to do with the properties of the minimizers of the pairwise interaction of the droplets, i.e. the choice of xi which minimize E N with fixed ri . This becomes a difficult problem in the case of interest, since we generally expect N 1 (for a numerical solution at a few values of N and κ = 2 see Fig. 2). It would be natural to conjecture that at small enough ε the minimizing droplets will arrange themselves into a periodic lattice close to a hexagonal (close-packed) lattice. Proving this kind of result, however, is a major challenge (see [47] for a recent proof for a certain class of pair interactions), which is one of the open questions also in many other problems, such as the problem of characterizing Abrikosov vortex lattices, for example [48]. Let us mention here a recent result by Chen and Oshita, who proved that in the case κ = 0 the hexagonal arrangement of disks is energetically the best among simple periodic lattices [49]. Yet, it is not known if the same result also holds for more general arrangements of droplets. Here we prove a weaker result that the number density of droplets becomes asymptotically uniform as ε → 0, leading also to uniform distribution of energy (compare with [36]). Moreover, we identify the precise asymptotic behavior of the minimal energy and show that the size of the minimizing droplets becomes the same as ε → 0. Lastly, we establish the asymptotic behavior of the minimal value of the diffuse interface energy E. Theorem 2.3. Let W satisfy assumptions (i)–(iv) at the beginning of Sec. 4, let u¯ = ¯ with some δ¯ > 0 fixed, and let κ be given by (4.10). Then −1 + ε2/3 | ln ε|1/3 δ, √ 3 (i) If δ¯ ≤ 21 9 κ 2 , then ε−4/3 | ln ε|−2/3 min E → 21 κ −2 δ¯2 , √ √ √ 3 3 (ii) If δ¯ > 1 3 9 κ 2 , then ε−4/3 | ln ε|−2/3 min E → 9 δ¯ − 9 κ 2 , 2
2
4
as ε → 0. Theorem 2.3 says that the energy of the minimizers of the diffuse interface energy E behaves asymptotically the same as that of the sharp interface energy E in the limit ε → 0. In particular, the transition to non-trivial minimizers occurs asymptotically at the same values of u¯ for ε 1. The proofs of Theorems 2.1–2.3 are based on a number of propositions established in Secs. 3 and 4, and are completed in Sec. 5.
Droplet Phases in Non-local Ginzburg-Landau Models
53
3. Analysis of the Sharp Interface Problem Our plan for the analysis of the sharp interface problem consists of a number of steps which we list below: ¯ 1. Introduce a suitably rescaled energy E¯ and domain Ω. 2. Establish existence and regularity of the minimizers of E¯ (subsets of Ω¯ where u = 1). 3. Establish some a priori estimates for the geometry of the minimizers of E¯ and uniform bounds on the induced long-range potential. 4. Establish that different connected components of minimizers of E¯ are separated by ¯ large distances in Ω. 5. Establish that each connected component of a minimizer of E¯ is close to a disk (hence the term “droplet”). 6. Establish equivalence between min E¯ and min E¯ N (the suitably rescaled version of E N ). 7. Improve the estimate for the separation distance between different droplets. 8. Prove uniform convergence of the rescaled droplet radii to a universal constant. 9. Prove convergence of min E¯ to a limit and convergence of the normalized droplet density in the original, unscaled domain Ω to a limit, as ε → 0. This plan is carried out in the rest of this section via a series of lemmas and propositions. 3.1. Scaling. We begin by introducing a suitable rescaling, in which the main quantities of interest become O(1) quantities in the limit ε → 0. Motivated by the discussion of Sec. 2, we define the rescaled energy E¯ (with the energy of the uniform state u = −1 subtracted) and a new coordinate x¯ ∈ Ω¯ = [0, ε−1/3 | ln ε|1/3 )2 , where Ω¯ is a twodimensional torus with period ε−1/3 | ln ε|1/3 : ¯ E = ε4/3 | ln ε|2/3 ( 21 κ −2 δ¯2 + E),
x=
ε1/3 x. ¯ | ln ε|1/3
(3.1)
The energy E¯ can be conveniently expressed in terms of the set Ω¯ + ⊂ Ω¯ in which u = 1: ¯ −2 |Ω¯ + | E¯ = | ln ε|−1 |∂ Ω¯ + | − 2δκ +2| ln ε|−2 G ε1/3 | ln ε|−1/3 (x¯ − y¯ ) d x¯ d y¯ . (3.2) Ω¯ +
Ω¯ +
We also need an expression for the rescaled energy E¯ N of a system of interacting droplets. With the help of (3.1), we can write the rescaling of (2.6) as N 2π ¯ −2 r¯i2 − 1 | ln ε|−1 r¯i4 ln(ε1/3 | ln ε|−1/3 κ¯ r¯i ) − 1 E¯ N = r¯i − δκ 2 4 | ln ε| i=1
+
N −1 N 4π 2 G(ε1/3 | ln ε|−1/3 (x¯i − x¯ j ))¯ri2 r¯ 2j , | ln ε|2 i=1 j=i+1
where r¯i and x¯i are the radii and the centers of the droplets, respectively.
(3.3)
54
C. B. Muratov
3.2. Properties of minimizers. Let us begin with the statement of a result on the existence and regularity of minimizers of E¯ (or, equivalently, of E), which is obtained by straightforwardly adapting the results of [50] for sets of prescribed mean curvature. Proposition 3.1. There exists a set Ω¯ + of finite perimeter which minimizes E¯ in (3.2). The boundary ∂ Ω¯ + of this set is a curve of class C 1,α for some α ∈ (0, 1). In view of this, in the following we will always assume that minimizers Ω¯ + of E¯ are closed sets. We also note that −1 G(ε1/3 | ln ε|−1/3 (x¯ − y¯ )) d y¯ (3.4) v(x) ¯ = | ln ε| Ω¯ +
¯ with any p > 1, and, hence, in C 1,α (Ω) ¯ for any α ∈ (0, 1). Indeed, v¯ is in W 2, p (Ω), solves the equation − Δv + ε2/3 | ln ε|−2/3 κ 2 v = | ln ε|−1 χΩ¯ + ,
(3.5)
¯ and so the result follows by stanwhere χΩ¯ + is the characteristic function of Ω¯ + , in Ω, dard elliptic regularity theory [51]. As a consequence, we have a higher regularity for the boundary of the minimizer Ω¯ + of E¯ [52]: Corollary 3.1. The boundary ∂ Ω¯ + of a minimizer Ω¯ + of E¯ is of class C 3,α . Note that this regularity result also holds more generally for local minimizers of E¯ in dimensions d ≤ 7 [50] (see also [35,53,54]), hence, in particular, the expressions for the first and second variation of E¯ in d ≤ 3 obtained in [14] are justified (see also [55] ¯ a > 0, and Ω¯ a is the set obtained for the case of arbitrary dimensions). If ρ ∈ C 1 (∂ Ω), ¯ Ω¯ a+ ) is twice by displacing ∂ Ω¯ + by aρ in the outward normal direction, then a → E( continuously differentiable at a = 0, and we have [14] (for the reader’s convenience, the computation is reproduced in Appendix C): ¯ Ω¯ a+ ) d E( ¯ −2 + 4v(x))ρ( | ln ε| = (K (x) ¯ − 2δκ ¯ x) ¯ dH1 (x), ¯ (3.6) + da a=0 ¯ ∂Ω ¯ Ω¯ a+ ) d 2 E( 2 2 | ln ε| |∇ρ( x)| ¯ = + 4ν( x) ¯ · ∇v( x) ¯ ρ ( x) ¯ dH1 (x) ¯ da 2 a=0 ∂ Ω¯ + ¯ −2 )K (x)ρ + (4v(x) ¯ − 2δκ ¯ 2 (x) ¯ dH1 (x) ¯ ∂ Ω¯ + 1 G(ε1/3 | ln ε|−1/3 (x¯ − y¯ ))ρ(x)ρ( ¯ y¯ ) dH1 (x)dH ¯ ( y¯ ), (3.7) +4| ln ε|−1 ∂ Ω¯ +
∂ Ω¯ +
where K (x) ¯ is the curvature at point x¯ ∈ ∂ Ω¯ + , with the sign convention that K > 0 if + ¯ Ω is convex, and ν(x) ¯ is the outward unit normal to ∂ Ω¯ + at that point. The associated Euler-Lagrange equation for ∂ Ω¯ + reads ¯ −2 − 4v(x), K (x) ¯ = 2δκ ¯ (3.8) which also allows to simplify the expression in (3.7) evaluated on a minimizer to ¯ Ω¯ a+ ) d 2 E( | ln ε| da 2 a=0 |∇ρ(x)| ¯ 2 + 4ν(x) = ¯ · ∇v(x) ¯ ρ 2 (x) ¯ − K 2 (x)ρ ¯ 2 (x) ¯ dH1 (x) ¯ ∂ Ω¯ + 1 G(ε1/3 | ln ε|−1/3 (x¯ − y¯ ))ρ(x)ρ( ¯ y¯ ) dH1 (x)dH ¯ ( y¯ ). (3.9) +4| ln ε|−1 ∂ Ω¯ +
∂ Ω¯ +
Droplet Phases in Non-local Ginzburg-Landau Models
55
We will use these equations later on to establish some properties of the minimizers for ε 1. Meanwhile, let us begin our analysis with some basic estimates. ¯ Then there exists C > 0 such that Lemma 3.1. Let Ω¯ + be a minimizer of E. |Ω¯ + | ≤ C| ln ε|, |∂ Ω¯ + | ≤ C| ln ε|
(3.10) (3.11)
for ε 1. Proof. First of all, by representation (2.3) we have G(x − y) ≥ c > 0 for all x, y ∈ Ω. Therefore, in view of the fact that min E¯ ≤ 0 (since E¯ = 0 if Ω¯ + = ∅), from (3.2) we have ¯ −2 |Ω¯ + | 0 ≥ | ln ε| min E¯ ≥ 2| ln ε|−1 G ε1/3 | ln ε|−1/3 (x¯ − y¯ ) d x¯ − 2δκ ≥ 2c| ln ε|
−1
Ω¯ +
Ω¯ +
¯ −2 |Ω¯ + |, |Ω¯ + |2 − 2δκ
(3.12)
which gives (3.10). On the other hand, we also have ¯ −2 |Ω¯ + |. |∂ Ω¯ + | ≤ 2δκ Therefore, from (3.10) we immediately obtain (3.11).
(3.13)
As a corollary, it follows from (3.11) that the diameter of each connected subset Ω¯ i+ of Ω¯ + is bounded by O(| ln ε|) diam(Ω¯ i+ ) ≤ C| ln ε|,
(3.14)
for some C > 0 independent of ε 1. Our next step is to show that the area of each connected component of Ω¯ + = ∅ is uniformly bounded above and below independently of ε. N Ω ¯ where Ω¯ + are the ¯ + be a non-trivial minimizer of E, Lemma 3.2. Let Ω¯ + = ∪i=1 i i + disjoint connected components of Ω¯ . Then, there exist C > c > 0 such that
c ≤ |Ω¯ i+ |, |∂ Ω¯ i+ | ≤ C,
diam(Ω¯ i+ ) ≤ C,
(3.15)
for ε 1. Proof. First, note that since by Corollary 3.1 the set ∂ Ω¯ + is of class C 3,α we have N < ∞. To see that (3.15) holds, we first write E¯ as | ln ε| E¯ =
N
¯ −2 |Ω¯ i+ | |∂ Ω¯ i+ | − 2δκ
i=1
+2| ln ε|−1 +2| ln ε|−1
Ω¯ i+
Ω¯ i+
j=i
Ω¯ i+
G(ε1/3 | ln ε|−1/3 (x¯ − y¯ )) d xd ¯ y¯ ⎞
Ω¯ +j
G(ε1/3 | ln ε|−1/3 (x¯ − y¯ )) d xd ¯ y¯ ⎠ . (3.16)
56
C. B. Muratov
Fig. 3. The graph of V0 (¯r ) from (3.18) for different values of δ¯
In view of (3.14) and (2.4), the integral in the second line in (3.16) is bounded from 1 below by 6π (1 − δ)| ln ε| |Ω¯ i+ |2 for any δ > 0, provided ε is small enough. Therefore, removing the set Ω¯ i+ from Ω¯ + will result in the change of energy Δ E¯ estimated as ¯ −2 |Ω¯ i+ | + 1 (1 − δ)|Ω¯ i+ |2 | ln ε| Δ E¯ ≤ − |∂ Ω¯ i+ | − 2δκ 3π √ + 1/2 −2 1 ¯ ≤ − 2 π |Ω¯ i | − 2δκ |Ω¯ i+ | + 3π (1 − δ)|Ω¯ i+ |2 ,
(3.17)
where in the first line we took into account that G > 0 and in the second line used the isoperimetric inequality. Then, by direct inspection (see also Fig. 3) we have Δ E¯ < 0, contradicting minimality of E¯ on Ω¯ + , unless c ≤ |Ω¯ i+ | ≤ C for some C > c > 0, independently of ε 1. Finally, the lower bound for |∂ Ω¯ i+ | follows from the isoperimetric inequality, and the upper bound is obtained by applying the previous argument to the first line in (3.17). Following the same arguments, we also immediately arrive at the following non-existence result: √ Proposition 3.2. Let δ¯ < 21 3 9 κ 2 be fixed. Then the unique minimizer of E¯ is Ω¯ + = ∅ for ε 1. Proof. Let us introduce the function Vv : [0, ∞) → R, defined as ¯ −2 )¯r 2 + 1 r¯ 4 , Vv (¯r ) = 2π r¯ + (2v − δκ 6
(3.18)
whose graph at v = 0 and several values of δ¯ is shown in Fig. 3. If Ω¯ i+ is a connected component of Ω¯ + and r¯i = ( π1 |Ω¯ i+ |)1/2 , then by the same arguments as in Lemma 3.2, the energy gained by removing Ω¯ i+ from Ω¯ + is bounded below by | ln ε|−1 (V0 (¯ri )+o(1)), as long as ε 1. Then, by direct inspection V0 (¯r ) is always positive under the assumptions of the proposition, making Ω¯ + = ∅ energetically preferred.
Droplet Phases in Non-local Ginzburg-Landau Models
57
Note that the asymptotic value of the threshold of δ¯ in Proposition 3.2 below which no non-trivial minimizers are present was computed in [14]. Another simple corollary to Proposition 3.2 is the following Lemma 3.3. Let Ω¯ + be a non-trivial minimizer of E¯ and let N be the number of disjoint connected components of Ω¯ + . Then there exists C > 0 such that N ≤ C| ln ε|,
(3.19)
for ε 1. Let us now establish a uniform bound on the potential v. Note that a version of this result is also an important component in the proofs of [36]. ¯ Then for any α ∈ (0, 1) we have Lemma 3.4. Let Ω¯ + be a non-trivial minimizer of E. 0 < v ≤ C,
||v||C 1,α (Ω) ¯ ≤ C,
(3.20)
where v is given by (3.4), for some C > 0 independent of ε 1. Proof. We start by noting that v > 0 in view of positivity of G. Let us now estimate the gradient of v. Using (2.4) and Lemmas 3.2 and 3.3, we get −1 |∇v(x)| ¯ ≤ | ln ε| |∇G(ε1/3 | ln ε|−1/3 |x¯ − y¯ |)| d y¯ + ¯ Ω −1 ≤ | ln ε| |∇G(ε1/3 | ln ε|−1/3 |x¯ − y¯ |)| d y¯ Br¯ (x) ¯ +| ln ε|−1 |∇G(ε1/3 | ln ε|−1/3 |x¯ − y¯ |)| d y¯ ≤ C(| ln ε|
−1
¯ Ω¯ + \Br¯ (x) −1
r¯ + r¯
) ≤ 2C| ln ε|−1/2 ,
(3.21)
for some C > 0, where Br¯ (x) ¯ is a disk of radius r¯ centered at x, ¯ and the last inequality is obtained by choosing r¯ = | ln ε|1/2 . Therefore, by the results of Lemma 3.2, we see that osc v(x) ¯ = o(1),
x∈ ¯ Ω¯ i+
(3.22)
for each connected component Ω¯ i+ of Ω¯ + . To see that this implies the conclusion of the lemma, suppose that, to the contrary, we have max v = M 1. Since by (3.5) the ¯ Ω¯ + , it achieves its maximum in the closure of some function v is subharmonic in Ω\ + ¯ Ωi . Therefore, in view of (3.22) we have v ≥ 21 M in Ω¯ i+ . Then, following the same arguments as in the proof of Lemma 3.2, for large enough M we can lower the energy by removing Ω¯ i+ from Ω¯ + . Finally, by [51, Theorem 9.11] we have ||v||W 2, p (B1 (x)) ¯ is the disk ¯ ≤ C, where B1 ( x) ¯ for some C > 0 and any p > 2, independently of x¯ of radius 1 centered at x¯ ∈ Ω, and ε 1. Hence, the uniform Hölder estimate on the gradient follows by Sobolev imbedding [51, Theorem 7.17]. We can also immediately conclude from (3.8) and (3.22) that the curvature of ∂ Ω¯ + is uniformly bounded both from above and below by positive constants, implying that each Ω¯ i+ is convex. Note that this result justifies the terminology “droplet” for each Ω¯ i+ which we will be using from now on.
58
C. B. Muratov
¯ Then we have Lemma 3.5. Let ∂ Ω¯ + be the boundary of a minimizer Ω¯ + of E. c ≤ K (x) ¯ ≤ C,
(3.23)
for all x¯ ∈ ∂ Ω¯ + , with some C > c > 0 independent of ε 1. In particular, when ε 1, each connected component Ω¯ i+ of Ω¯ + is convex and simply connected. Proof. The upper bound is an immediate consequence of (3.8) and positivity of v. To obtain the lower bound, let us note that by the results of Lemma 3.2, for every connected component Ω¯ i+ there exists a disk Br¯i (x¯i ), with r¯i = O(1), such that Ω¯ i+ ⊂ Br¯i (x¯i ). Therefore, translating Br¯i (x¯i ) until its boundary touches ∂ Ω¯ i+ , we obtain a point x¯i ∈ ∂ Ω¯ i+ , such that K (x¯i ) ≥ r¯i−1 ≥ 2c, for some c > 0 independent of ε 1. Now, ¯ −2 − c). At the same time, by (3.22) this implies that by (3.8) we have v(x¯i ) ≤ 21 (δκ 1 ¯ −2 v(x) ¯ ≤ 4 (2δκ − c) for all x¯ ∈ ∂ Ω¯ i+ , which, again, by (3.8) gives the statement. We now show that different connected components of Ω¯ + cannot come too close to each other when ε 1. N Ω ¯ where Ω¯ + are the ¯ + be a non-trivial minimizer of E, Lemma 3.6. Let Ω¯ + = ∪i=1 i i + disjoint connected components of Ω¯ , and let N ≥ 2. Then, there exists C > 0 such that
dist(Ω¯ i+ , Ω¯ +j ) ≥ C
∀i = j,
(3.24)
for ε 1. Proof. Let x¯i ∈ Ω¯ i+ and x¯ j ∈ Ω¯ +j be such that r = |x¯i − x¯ j | = dist(Ω¯ i+ , Ω¯ +j ) > 0. Consider the disk B centered at 21 (x¯i + x¯ j ) with radius R = 2r and the rectangle Q inscribed into B which is shown by the thick solid lines in Fig. 4. In view of the uniform bound on the curvature of ∂ Ω¯ + obtained in Lemma 3.5, the curve segments ∂ Ω¯ i+ ∩ Q and ∂ Ω¯ +j ∩ Q passing through x¯i and x¯ j , respectively, intersect ∂ Q transversally as in Fig. 4 when r 1. Furthermore, we have dist(∂ Ω¯ i+ ∩ ∂ Q + , ∂ Ω¯ +j ∩ ∂ Q + ) ≤ 2r and dist(∂ Ω¯ i+ ∩ ∂ Q − , ∂ Ω¯ +j ∩ ∂ Q − ) ≤ 2r , where ∂ Q + and ∂ Q − are the right and the left side of the boundary of the rectangle relative to the line through x¯i and x¯ j , respectively, for sufficiently small r independent of ε 1 (see Fig. 4). At the same time, we have √ |∂ Ω¯ i+ ∩ Q| + |∂ Ω¯ +j ∩ Q| ≥ 4r 3. Therefore, reconnecting the points ∂ Ω¯ i+ ∩ ∂ Q + with ∂ Ω¯ +j ∩∂ Q + , and ∂ Ω¯ i+ ∩∂ Q − with ∂ Ω¯ +j ∩∂ Q − by straight lines and including the region √ ¯ we will decrease |∂ Ω¯ + | by at least 4( 3 − 1)r . Thus, the change between them into Ω, Δ E¯ in the total energy is estimated to be √ ¯ v(x) ¯ d x¯ | ln ε|Δ E ≤ −4( 3 − 1)r + 4 Q +2| ln ε|−1 G(ε1/3 | ln ε|−1/3 (x¯ − y¯ )) d xd ¯ y¯ . (3.25) Q
Q
Finally, in view of Lemma 3.4 and (2.4), the right-hand side of (3.25) is bounded above by −C1r + C2 r 2 , with C1,2 > 0 independent of ε 1. Hence, the energy of such a rearrangement will be lower if r is sufficiently small, for all ε 1.
Droplet Phases in Non-local Ginzburg-Landau Models
(a)
59
(b)
2r r r
r 3
r
Fig. 4. Schematics of the rearrangement argument of Lemma 3.6. In (a), the set Ω¯ + is shown in gray, solid arcs show the bounds on the location of ∂ Ω¯ + , the thick solid lines show the rectangle Q. In (b), the gray region shows the rearranged Ω¯ +
As our next step, we establish that different droplets must, in fact, be sufficiently far from each other. We note that this result is a manifestation of the “repumping” instability, which does not allow two droplets to approach each other sufficiently closely. Dynamically, this instability results in the growth of one droplet at the expense of its neighbor shrinking. This instability mechanism for reaction-diffusion systems was first pointed out in [56] (see also [4]) and further studied in the context of two-dimensional periodic structures in [13,14,57]. N Ω ¯ where Ω¯ + are the ¯ + be a non-trivial minimizer of E, Lemma 3.7. Let Ω¯ + = ∪i=1 i i + ¯ disjoint connected components of Ω , and let N ≥ 2. Then there exists α > 0 such that
dist(Ω¯ i+ , Ω¯ +j ) > ε−α
∀i = j,
(3.26)
for ε 1. Proof. Consider the second variation of E¯ with respect to the perturbation, in which the boundary of each connected component Ω¯ i+ is expanded uniformly by a distance aci in the normal direction, i.e., we have ρ(x) ¯ = ci for all x¯ ∈ ∂ Ω¯ i+ . By (3.9), we have ¯ Ω¯ a+ ) d 2 E( = | ln ε|−1 Q i j ci c j , (3.27) 2 da a=0 i, j
where the coefficients Q i j of the quadratic form Q can be estimated as 2 Q ii = − |∂ Ω¯ i+ |2 + o(1), K 2 (x) ¯ dH1 (x) ¯ + (3.28) + 3π ¯ ∂ Ωi where we took into account that by (3.5) and Gauss’s theorem ∂ Ω¯ + ν(x)·∇v( ¯ x) ¯ dH1 (x) ¯ = i + −1 2/3 −2/3 −| ln ε| |Ω¯ i | + O(ε | ln ε| ) and used the expansion in (2.4) together with Lemmas 3.5, 3.4 and 3.2, for ε 1. Furthermore, since by convexity of Ω¯ i+ (see Lemma 3.5) the boundary of each Ω¯ i+ is a closed curve, by the Cauchy-Schwarz inequality we have
2 2 1 4π = K (x) ¯ dH (x) ¯ ≤ |∂ Ω¯ i+ | K 2 (x) ¯ dH1 (x). ¯ (3.29) ∂ Ω¯ i+
∂ Ω¯ i+
60
C. B. Muratov
Therefore, the diagonal elements of Q can be further estimated as Q ii ≤
2 4π 2 |∂ Ω¯ i+ |2 − + o(1). 3π |∂ Ω¯ i+ |
(3.30)
On the other hand, define αi j = | ln ε|−1 ln(dist(Ω¯ i+ , Ω¯ +j )), and suppose, to the contrary of the statement of the proposition, that αi j → 0 for some pair of indices on a sequence of ε → 0. Then, with the help of Lemma 3.6 and (2.4) we can estimate Qi j =
2 (1 − 3αi j ) |∂ Ω¯ i+ | |∂ Ω¯ +j | + o(1). 3π
(3.31)
Now, for the index pair (i, j) above let us choose ci = |∂ Ω¯ +j |, c j = −|∂ Ω¯ i+ |, and let us set ck = 0 for all other indices k. A simple calculation of the sum in (3.27) then shows that for this choice of c’s we have
¯ Ω¯ a+ ) 4|∂ Ω¯ i+ |2 |∂ Ω¯ +j |2 π3 π3 d 2 E( ≤ − | ln ε| + o(1), αi j − da 2 a=0 π |∂ Ω¯ i+ |3 |∂ Ω¯ +j |3 (3.32) where we took into account Lemma 3.2. This expression is negative as soon as αi j < 2π 3 min{|∂ Ω¯ i+ |−3 , |∂ Ω¯ +j |−3 }, which, in view of Lemma 3.2, contradicts minimality of E¯ for small enough ε.
(3.33)
Let us also point out that the proof of Lemma 3.7 gives a universal lower bound for the perimeter of the connected portions of the minimizers. Indeed, the quadratic form Q 2 has a negative eigenvalue, if 3π |∂ Ω¯ i+ |2 − 4π 2 |∂ Ω¯ i+ |−1 < 0 and ε 1, which implies the following result: N Ω ¯ where Ω¯ + are the ¯ + be a non-trivial minimizer of E, Proposition 3.3. Let Ω¯ + = ∪i=1 i i + disjoint connected components of Ω¯ . Then, for every δ > 0, √ 3 (3.34) |∂ Ω¯ i+ | ≥ π 6 − δ,
for ε 1. Note that this condition in the radially-symmetric case was obtained in [13,14,58] and is also applicable to all local minimizers (for global minimizers, a better bound will be obtained below). We also derive another quantitative estimate on v and the geometry of Ω¯ i+ that remains valid for local minimizers of low energy. N Ω ¯ where Ω¯ + are the ¯ + be a non-trivial minimizer of E, Proposition 3.4. Let Ω¯ + = ∪i=1 i i disjoint connected components of Ω¯ + . Then
0 0 and α > 0, in any disk of O(1) radius containing Ω¯ i+ for ε 1. Therefore, if x¯ ∈ ∂ Br¯i (x¯i ), by Taylor-expanding the Bessel functions [46] we have for any α < 13 , v ∗ (x) ¯ = v¯i + O(εα ),
|∇v ∗ (x)| ¯ = O(| ln ε|−1 ),
(3.42)
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C. B. Muratov
where the constant v¯i is given by v¯i = − 21 | ln ε|−1 r¯i2 ln(ε1/3 | ln ε|−1/3 κ¯ r¯i ) +π | ln ε|−1 r¯ 2j G(ε1/3 | ln ε|−1/3 κ(x¯i − x¯ j )), r¯ j = ( π1 |Ω¯ +j |)1/2 . (3.43) j=i
In the following, we will show that v(x) ¯ on ∂ Ω¯ i+ also coincides with v¯i to O(εα ), giving the balance of forces at the interface. We are now ready to state our result: N Ω ¯ where Ω¯ + are the ¯ + be a non-trivial minimizer of E, Proposition 3.5. Let Ω¯ + = ∪i=1 i i + ¯ disjoint connected components of Ω . Then there exists a constant α > 0 such that for all ε 1:
(i) For each Ω¯ i+ there exists a point x¯i ∈ Ω¯ i+ , such that Br¯i −εα (x¯i ) ⊂ Ω¯ i+ ⊂ Br¯i +εα (x¯i ),
(3.44)
¯ is the disk of radius r¯ centered at x; ¯ where r¯i = ( π1 |Ω¯ i+ |)1/2 and Br¯ (x) (ii) The values of r¯i satisfy ¯ −2 + 4v¯i = O(εα ), r¯i−1 − 2δκ
(3.45)
where v¯i are given by (3.43). Proof. Let us pick a point x¯i ∈ Ω¯ i+ , then Ω¯ i+ ⊂ B|∂ Ω¯ + | (x¯i ). Let us then replace Ω¯ i+ with i the disk of the same area centered at x¯i . By Lemmas 3.7 and 3.2, the resulting set Br¯i (x¯ ) still satisfies the bound in (3.26), and the change of energy Δ E¯ under this rearrangement can be estimated as √ | ln ε|Δ E¯ = 2 π |Ω¯ i+ |1/2 − |∂ Ω¯ i+ | + O | ln ε|−1 , (3.46) where we used (2.4), (3.41) and Lemma 3.2. Thus, the energy will decrease under this ¯ unless for some C > 0 the isoperimetric rearrangement, contradicting minimality of E, deficit of Ω¯ i+ , |∂ Ω¯ i+ | D(Ω¯ i+ ) = √ − 1 ≤ C| ln ε|−1 , 2 π |Ω¯ i+ |1/2
(3.47)
for ε 1. Choosing x¯i ∈ B|∂ Ω¯ + | (x¯i ) to minimize |Ω¯ i+ ΔBr¯i (x¯i )|, where Ω¯ i+ ΔBr¯i (x¯i ) i denotes the symmetric difference of sets Ω¯ i+ and Br¯i (x¯i ), by the results of [59] we have |Ω¯ i+ ΔBr¯i (x¯i )| ≤ C | ln ε|−1/2 , and C > 0 is a constant independent of ε 1. In fact, x¯i ∈ Ω¯ i+ , since otherwise by convexity of Ω¯ i+ we would have |Ω¯ i+ ΔBr¯i (x¯i )| ≥ 1 ¯+ 2 |Br¯i ( x¯i )|). Therefore, by Lemma 3.5 the set ∂ Ωi is uniformly close to ∂ Br¯i ( x¯i ), giving (i) to o(1). To obtain the O(εα ) bound in (i), let ρ : ∂ Br¯i (x¯i ) → R be the signed distance from a given point on ∂ Br¯i (x¯i ) to ∂ Ω¯ i+ along the outward normal to ∂ Br¯i (x¯i ). Note that by convexity of Ω¯ i+ the function ρ defines a one-to-one map between ∂ Ω¯ i+ and ∂ Br¯i (x¯i ). Furthermore, if ||ρ|| L ∞ (∂ Br¯i (x¯i )) = δ, we have ||∇ρ|| L ∞ (∂ Br¯i (x¯i )) ≤ Cδ 1/2 for some
Droplet Phases in Non-local Ginzburg-Landau Models
63
C > 0 and ε 1 in view of Lemma 3.5, and δ → 0, as ε → 0. Also, by Corollary 3.1 we have ρ ∈ C 3 (∂ Br¯i (x¯i )). Treating Ω¯ + as a perturbation of the set Ω¯ ∗ = Br¯i (x¯i ) ∪ (Ω¯ + \Ω¯ i+ ) and expanding as in Lemma C.1, we can write ¯ Ω¯ + ) − E( ¯ Ω¯ ∗ )) = | ln ε|( E( +
1 2
∂ Br¯i (x¯i )
1 + 2¯ri +
+O(δ
∂ Br¯i (x¯i )
¯ −2 + 4v ∗ (x) r¯i−1 − 2δκ ¯ ρ(x) ¯ dH1 (x) ¯
|∇ρ(x)| ¯ 2 + 4ν(x) ¯ · ∇v ∗ (x) ¯ ρ 2 (x) ¯ dH1 (x) ¯
∂ Br¯i (x¯i )
2 | ln ε|
¯ −2 )ρ 2 (x) (4v ∗ (x) ¯ − 2δκ ¯ dH1 (x) ¯
∂ Br¯i (x¯i ) ∂ Br¯i (x¯i )
1 G(ε1/3 | ln ε|−1/3 (x¯ − y¯ ) ρ(x)ρ( ¯ y¯ ) dH1 (x)dH ¯ ( y¯ )
),
2+α
(3.48)
for any α ∈ (0, 1). Moreover, in view of Lemmas 3.4 and 3.5, the error term in (3.48) is uniform in ε 1. On the other hand, since |Ω¯ i+ | = |Br¯i (x¯i )|, we have 0= =
ρ(x) ¯
∂ Br¯i (x¯i ) 0 ∂ Br¯i (x¯i )
(1 + r¯i−1r ) dr dH1 (x) ¯
ρ(x) ¯ dH1 (x) ¯ +
1 2¯ri
∂ Br¯i (x¯i )
ρ 2 (x) ¯ dH1 (x). ¯
(3.49)
Therefore, using the estimate in (3.42) we can rewrite (3.48) as ¯ Ω¯ + ) − E( ¯ Ω¯ ∗ )) = | ln ε|( E( 2 + | ln ε|
∂ Br¯i (x¯i ) ∂ Br¯i (x¯i )
+O(| ln ε|−1 ||ρ||2L 2
Br¯ (x¯i ) i
1 2
∂ Br¯i (x¯i )
|∇ρ|2 − r¯i−2 ρ 2 (x) ¯ dH1 (x) ¯
1 G(ε1/3 | ln ε|−1/3 (x¯ − y¯ ) ρ(x)ρ( ¯ y¯ ) dH1 (x)dH ¯ ( y¯ )
) + O(εα ||ρ|| L 2
Br¯ (x¯i ) i
where we took into account that δ ≤ C||ρ|| H 1 the double integral in (3.50), using ∂ Br¯
i
(x¯i )
Br¯ (x¯i ) i
) + O(δ α ||ρ||2H 1
Br¯ (x¯i ) i
(3.50)
for some C > 0. Further estimating
1 1 G(ε1/3 | ln ε|−1/3 (x¯ − y¯ )) − | ln ε| 6π
Cδ ln | ln ε| ≤ , | ln ε|
),
ρ( y¯ ) dH1 ( y¯ ) (3.51)
64
C. B. Muratov
we have ¯ Ω¯ + ) − E( ¯ Ω¯ ∗ )) | ln ε|( E( 2
1 1 −2 2 2 1 1 = |∇ρ| − r¯i ρ (x) ¯ dH (x) ¯ + ρ(x) ¯ dH (x) ¯ 2 ∂ Br¯i (x¯i ) 3π ∂ Br¯i (x¯i ) +O(εα ||ρ|| L 2
Br¯ (x¯i ) i
) + o(||ρ||2L 2
Br¯ (x¯i ) i
) + o(||ρ||2H 1
Br¯ (x¯i ) i
Now, write ρ as ρ = ρ0 + ρ1 + ρ2 , where ρ0 = x− ¯ x¯i |x− ¯ x¯i |
1 2π r¯i
).
(3.52)
¯ dH1 (x), ¯ ρ1 (x) ¯ = ∂ Br¯i (x¯i ) ρ( x) 2 ρ0 and ρ1 in L (∂ Br¯i (x¯i )). By
· b, for some vector b ∈ R2 , and ρ2 orthogonal to (3.49) we have |ρ0 | = O(||ρ||2L 2 (∂ B (x¯ )) ), which is, therefore, negligibly small comr¯i
i
pared to |b| and ||ρ2 || L 2 (∂ Br¯ (x¯i )) in all the arguments below. Then, using Poincaré’s i inequality, we find that ¯ Ω¯ + ) ≥ | ln ε| E( ¯ Ω¯ ∗ ) | ln ε| E( 1 + |∇ρ2 (x)| ¯ 2 dH1 (x) ¯ − Cεα ||ρ|| L 2 − c|b|2 , Br¯ (x¯i ) 4 ∂ Br¯i (x¯i ) i
(3.53)
for some C > 0 and 0 < c 1, whenever ε 1. This implies that ||ρ2 ||2H 1
Br¯ (x¯i ) i
≤ C εα ||ρ|| L 2
Br¯ (x¯i ) i
+ c |b|2 ,
(3.54)
for some C > 0 and 0 < c 1, for ε 1, otherwise replacing Ω¯ i+ with Br¯i (x¯i ) lowers the energy. On the other hand, we also have |b| = O(||ρ2 || H 1 ). If not, then Br¯ (x¯i ) i
∂ Ω¯ i+ will be o(|b|) close to ∂ Br¯i (x¯i + b) for ε 1. This, however, contradicts the choice of x¯i to minimize |Ω¯ i+ ΔBr¯i (x¯i )|. Therefore, we have ||ρ||2H 1
Br¯ (x¯i ) i
≤ C εα ||ρ|| H 1
Br¯ (x¯i ) i
+ c ||ρ||2H 1
Br¯ (x¯i ) i
for some C > 0 and 0 < c 1, implying ||ρ|| H 1
Br¯ (x¯i ) i
,
(3.55)
= O(εα ) and, hence, δ =
O(εα ). This gives part (i) of the statement of the proposition. Finally, to prove part (ii) of the statement, let Ω¯ a+ be obtained from Ω¯ + by expanding + ¯ Ωi by an amount a > 0, i.e., let us change ρ(x) ¯ → ρ(x) ¯ + a for every x¯ ∈ ∂ Br¯i (x¯i ). By (3.48), the change of energy can be estimated as ¯ Ω¯ a+ ) − E( ¯ Ω¯ + )) = 2πar¯i (¯r −1 − 2δκ ¯ −2 + 4v¯i ) + O(aδ) + O(a 2 ), | ln ε|( E( i
(3.56)
where we took into account (3.42). Then, since Ω¯ + is a minimizer, the right-hand side of (3.56) should vanish to O(a). Therefore, by the previous result we obtain the statement. Also, from the proof of Proposition 3.5 we obtain the following universal lower bound on |Ω¯ i+ |:
Droplet Phases in Non-local Ginzburg-Landau Models
65
N Ω ¯ where Ω¯ + are the ¯ + be a non-trivial minimizer of E, Proposition 3.6. Let Ω¯ + = ∪i=1 i i + disjoint connected components of Ω¯ . Then, for every δ > 0, √ 3 (3.57) |Ω¯ i+ | ≥ π 9 − δ,
for ε 1. Proof. Let r¯i = ( π1 |Ω¯ i+ |)1/2 and let Vvi (x¯i ) (¯ri ) be the fourth degree polynomial in r¯i √ ¯ −2 ≤ − 1 3 9 + o(1), so that defined in (3.18). First of all, observe that 2v¯i (x¯i ) − δκ 2 Vvi (x¯i ) (¯ri ) ≤ o(1) for ε 1. If not, then arguing as in Lemma 3.2, we can reduce the energy by removing Ω¯ i+ from Ω¯ + . Therefore, Vvi (x¯i ) has exactly two critical points: a strict local maximum and a strict local minimum (see Fig. 3). Furthermore, since Vvi (x¯i ) (¯ri ) ≤ o(1) and since the left-hand side of (3.45) in Proposition 3.5 is equal to d Vvi (x¯i ) (¯ri )/d r¯i , each√value of r¯i is close to the local minimum of Vvi (x¯i ) . By inspection, in this situation r¯i ≥ 3 3 − δ, for any δ > 0, provided ε is sufficiently small, hence, the claim. The results of Proposition 3.5 just obtained immediately allow to establish an asymptotic equivalence of the energy E¯ and the reduced energy E¯ N on the minimizers for ε 1. N Ω ¯ where Ω¯ + are the ¯ + be a non-trivial minimizer of E, Proposition 3.7. Let Ω¯ + = ∪i=1 i i disjoint connected components of Ω¯ + , and let r¯i and x¯i be as in Proposition 3.5. Then
min E¯ = O(1),
min E¯ = min E¯ N + O(εα ),
(3.58)
for some α > 0 independent of ε 1. Proof. The first equation in (3.58) is a direct consequence of the definition of E¯ in (3.2), according to which 0 ≤ 21 δ¯2 κ −2 + min E¯ ≤ ε−4/3 | ln ε|2/3 E[−1] = 21 δ¯2 κ −2 . The upper bound for min E¯ in the second equation follows by choosing a trial function for E¯ in the form of disks of radius r¯i centered at x¯i which minimize E¯ N and taking into consideration Lemmas 3.2 and 3.7 and (2.4). On the other hand, by Proposition 3.5(i), we have Ω¯ i+ ⊃ Br¯i −εα (x¯i ) for ε 1, hence, |∂ Ω¯ i+ | > 2π(¯ri − εα ) and |Ω¯ i+ | > π(¯ri − εα )2 . ¯ except the one involving δ, ¯ by the This controls from below all the terms of min E, corresponding terms of E¯ N . The latter, however, is controlled by the second inclusion in Proposition 3.5(i). To summarize, for 0 < ε 1 the non-trivial minimizers of E¯ have the form of well-separated nearly circular droplets. In fact, from Proposition 3.7 one should expect that the droplet-droplet interaction part of the energy, which is given by the last term in the expression (3.3) for E¯ N , should be close to the minimum for fixed droplet sizes. Proving this, however, generally requires information about coercivity of the interaction energy, which becomes difficult to establish when N 1, the asymptotic case of interest. Nevertheless, with the help of Lemma 3.4 we can prove that in the original scaling the droplets stay away from each other a distance O(εβ ) in Ω, with an arbitrary β > 0 for ε 1, i.e., that the statement of Lemma 3.7 actually holds for any α ∈ (0, 13 ), provided that ε is small enough. N Ω ¯ where Ω¯ + are the ¯ + be a non-trivial minimizer of E, Proposition 3.8. Let Ω¯ + = ∪i=1 i i disjoint connected components of Ω¯ + , and let x¯i be as in Proposition 3.5. Then, for any α ∈ (0, 13 ) we have |x¯i − x¯ j | > ε−α , for all i = j, as long as ε 1.
66
C. B. Muratov
Proof. First of all, note that by Lemma 3.7 the statement of the proposition holds for some α > 0. To prove that α could be chosen arbitrarily close to 13 , suppose that, to the contrary, there exists a sequence of ε → 0 and a pair of indices (i, j), depending on ε, such that |x¯i − x¯ j | ≤ ε−α with some 0 < α < 13 . Let us denote by I1 the set of indices of those droplets whose centers are contained in the disk B1 centered at x¯0 = 21 (x¯i + x¯ j ) with radius ε−α . By assumption we have |I1 | ≥ 2, where | · | denotes the counting measure. Also, we have |I1 | < M for some M ∈ N independent of ε 1. Indeed, by Lemmas 3.2 and 3.4, and by (2.4) we have for some c > 0, C ≥ v(x¯0 ) ≥ | ln ε|−1 G(ε1/3 | ln ε|−1/3 (x¯i − y¯ )) d y¯ ≥
1 3
¯+ k∈I1 Ωk
− α + o(1) c|I1 |,
(3.59)
for ε 1. Now, fix σ > 0 sufficiently small independently of ε, and consider a sequence of nested disks Bk of radii ε−α(1+kσ ) centered at x¯0 . By repeating the argument above, we also have |I M | ≤ M, as long as ε 1, where |Ik | is the counting measure of the set Ik of indices such that x¯l ∈ Bk for all l ∈ Ik . Therefore, in view of the fact that |I1 | > 1, we must have |Ik+1 | − |Ik | = 0 for some 1 ≤ k ≤ M − 1, implying that Bk+1 \Bk ∩ Ω¯ + = ∅. Thus, there exists a cluster of droplets, whose indices are denoted by Ik , which are within O(ε−α(1+kσ ) ) distance of x¯0 and are separated from all other droplets by O(ε−α(1+σ +kσ ) ) distance. Let us show that this contradicts the minimality of E¯ for small enough ε. Indeed, let us displace the droplets in Bk to the new locations x¯l = x¯l + λ(x¯l − x¯i ), with l ∈ Ik , which represents a dilation of Bk by a factor of 1 + λ relative to x¯i , keeping all r¯i fixed. For 0 < λ 1 the resulting change Δ E¯ of energy satisfies | ln ε|2 Δ E¯ ≤ −cλ + Cλεσ α | ln ε|,
(3.60)
for some C, c > 0 independent of ε 1, where we used Lemmas 3.1 and 3.2, and the estimate (2.4), arguing as in the derivation of (3.41). Thus, the considered rearrangement lowers the energy. ¯ As a simple corollary to this result, we actually have the following universal (δ-independent) upper bound on |Ω¯ i+ | and, hence, on r¯i : N Ω ¯ where Ω¯ + are the ¯ + be a non-trivial minimizer of E, Corollary 3.2. Let Ω¯ + = ∪i=1 i i + disjoint connected components of Ω¯ . Then, for any δ > 0, √ 2/3 |Ω¯ i+ | ≤ π 12( 2 − 1) + δ, (3.61)
when ε 1. Proof. If |Ω¯ i+ | is bigger, split Ω¯ i+ into two disks of equal area and move them apart a distance d = ε−β , with 0 < β < α < 13 . Arguing as before, the energy change Δ E¯ upon this manipulation is given by √ √ β |Ω¯ + |2 + o(1). | ln ε| Δ E¯ ≤ 2( 2 − 1) π |Ω¯ i+ |1/2 − (3.62) 2π i In view of the arbitrary closeness of β to 13 , the energy change is, therefore, negative for ε 1.
Droplet Phases in Non-local Ginzburg-Landau Models
67
Finally, we note that the argument of Proposition 3.8 still holds for local minimizers of low energy; the result can be obtained by sending λ → 0 in the proof. 3.3. Limiting behavior. We now √ investigate the limiting behavior of the minimizers of E¯ as ε → 0, with δ¯ > 21 3 9 κ 2 fixed, i.e., the situation in which minimizers are non-trivial. As the value of ε is decreased, the number of droplets in a minimizer are expected to grow. What we will show below is that in the limit ε → 0 the droplet sizes become asymptotically the same, and that the droplets become uniformly distributed throughout Ω. Let us first study the behavior of each droplet as ε → 0. We have the following result. N Ω ¯ where Ω¯ + are the ¯ + be a non-trivial minimizer of E, Proposition 3.9. Let Ω¯ + = ∪i=1 i i √ disjoint connected components of Ω¯ + , and let r¯i be as in Proposition 3.5. Then r¯i → 3 3 uniformly as ε → 0. √ Proof. First of all, by Proposition 3.6 we already know that r¯i ≥ 3 3 − δ for any δ > 0, provided that ε 1. Let us prove that the matching upper bound also holds for ε 1. Indeed, for any β ∈ (0, 13 ) let Bε−β (x¯i ) ∈ Ω¯ be the disk of radius ε−β centered at x¯i ¯ ∪ N B ε−β (x¯i ). Note that by Propodefined in Proposition 3.5, and consider Ω¯ β = Ω\ i=1 sition 3.8 the disks Bε−β (x¯i ) do not intersect for ε 1. In fact, by Proposition 3.8, for any α ∈ (β, 13 ) we have dist (Bε−β (x¯i ), Bε−β (x¯ j )) > ε−α for ε 1. Let us show that the minimum of v defined in (3.4) is attained in Ω¯ β for ε 1. Let x¯ be such that v(x) ¯ = min and let x¯i be the center of a droplet which is closest to x. ¯ Recalling the definition in (3.40) and Proposition 3.5, we can write
v(x) ¯ = vi (x) ¯ −
r¯i2 ln(ε1/3 (1 + |x¯ − x¯i |)) + o(1), 2| ln ε|
(3.63)
where we used (2.4). In particular, for any δ > 0 we have v(x) ¯ > vi (x¯i )+ 16 r¯i2 (1−3β)−δ, −β if |x¯ − x¯i | ≤ ε and ε 1, in view of (3.41), where according to Proposition 3.8, we can use α defined above, whenever ε 1. On the other hand, choosing γ ∈ (β, α) and picking any x¯ such that |x¯ − x¯i | = ε−γ , we see that for any δ > 0 we have v(x¯ ) < vi (x¯i ) + 16 r¯i2 (1 − 3γ ) + δ for ε 1. However, with δ sufficiently small this implies that v(x¯ ) < v(x) ¯ for small enough ε, contradicting minimality of v at x. ¯ √ ¯ −2 − 1 3 9 − δ, for any δ > 0, provided that Now, we demonstrate that v(x) ¯ > 21 δκ 4 ε 1. Indeed, suppose the opposite inequality holds for some δ > 0 and a sequence of ε → 0. Then, inserting a new droplet in the form of a disk of radius r¯ = O(1) centered at x¯ results in the change Δ E¯ of energy | ln ε|Δ E¯ = Vv(x) r ) + o(1), ¯ (¯
(3.64)
where V is√given by (3.18), and we used (2.4) and (3.41). Since by assumption 2v(x) ¯√− 3 ¯ −2 < 1 3 9, it is easy to verify that Vv(x) δκ attains a minimum at some r ¯ = r ¯ > 3, 0 ¯ 2 with Vv(x) r0 ) < 0. Therefore, inserting a droplet with radius r¯0 and center at x¯ would ¯ (¯ ¯ reduce energy for some ε 1, contradicting minimality of E. √ 1 ¯ −2 1 3 This, in turn, implies that vi (x¯i ) > 2 δκ − 4 9 − δ for all i. Indeed, since x¯ ∈ Ω¯ β , √ ¯ −2 − 1 3 9− 1 (1−3β)+o(1), for any x¯ ∈ ∂ Bε−β (x¯i ), from (3.63) we have vi (x¯ ) > 21 δκ 4 6π
68
C. B. Muratov
for ε 1. On the other hand, by (3.41) the same inequality holds for x¯ = x¯i . The estimate then follows in view of arbitrariness of β < 13 . √ ¯ −2 − 1 3 9 − δ when ε 1, and by Proposition 3.5(ii) the Finally, since v¯i (x¯i ) > 21 δκ 4 √ values of r¯i are√close to the minimizers of Vvi (x¯i ) (¯r ) for r¯ > 3 3 − δ, by direct inspection we have r¯i < 3 3 + δ as well, for any δ > 0 and ε 1. Let us point out that by (3.45) the uniform convergence of the droplet radii in Proposition 3.9 also implies uniform convergence of vi (x¯i ) to a space-independent constant as ε → 0:
√ 3 9 2 1 vi (x¯i ) → 2 δ¯ − κ . (3.65) 2κ 2 In fact, from the proof of Proposition 3.9 we can conclude that v stays close to the constant in (3.65) in Ω¯ β for β arbitrarily close to 13 , provided ε 1. This, in turn, implies ¯ or, equivalently, in Ω as ε → 0. that the droplets become uniformly distributed in Ω, Below we prove this fact, which also gives the leading order behavior of energy in the limit. Let us rewrite the energy E¯ N for the system of interacting droplets, using (3.18): E¯ N =
1 Vvi (x¯i ) (¯ri ) − 4π vi (x¯i )¯ri2 | ln ε| N
i=1
+
N −1 N 4π 2 G(ε1/3 | ln ε|−1/3 (x¯i − x¯ j ))¯ri2 r¯ 2j + o(1). | ln ε|2
(3.66)
i=1 j=i+1
To proceed, let us go back to the original scaling in x and introduce xi = ε1/3 | ln ε|−1/3 x¯i . Also, for any 0 < σ 1 define (our method is reminiscent of the Ewald summation technique [60]) G σ (x) =
2 1 − |x−y| 2ε2σ K 0 (κ|y − n|) dy. e 4π 2 ε2σ R2 n∈Z2
(3.67)
Here G σ is a mollified version of G, with Fourier transform Gˆ σ (q) =
Ω
1 2σ
e
iq·x
e− 2 ε |q| G σ (x) d x = 2 , κ + |q|2 2
(3.68)
and which can, e.g., be estimated as G σ (x) = G(x) + o(εσ/4 ),
|x| > εσ/2 ,
(3.69)
and G σ (x) = O(σ | ln ε|),
|x| < εσ .
(3.70)
Droplet Phases in Non-local Ginzburg-Landau Models
69
Therefore, in view of Lemmas 3.2 and 3.3 we can write E¯ N =
1 Vvi (x¯i ) (¯ri ) − 4π vi (x¯i )¯ri2 | ln ε| N
i=1
+
2π 2
N N
| ln ε|2
i=1 j=1
G σ (xi − x j )¯ri2 r¯ 2j + O(σ ).
(3.71)
Now, let us introduce the quantity 1 δ(x − xi ). | ln ε| N
ρ(x) =
(3.72)
i=1
Note that by Lemma 3.3 we have Ω ρ(x) d x = O(1). In view of Proposition 3.9 and (3.65), we can further rewrite E¯ N as √ √ N 3 9 2 1 2π 3 9 ¯ δ¯ − κ EN = Vvi (x¯i ) (¯ri ) − ρ(x)d x | ln ε| κ2 2 Ω i=1 √ 2 3 ρ(x)G σ (x − y)ρ(y) d x d y + O(σ ). (3.73) +6π 3 Ω
Ω
In fact, by Proposition 3.9 and (3.65), the first term in (3.73) goes to zero. Therefore, in terms of the Fourier coefficients, ρˆq = eiq·x ρ(x) d x, (3.74) Ω
we can write E¯ N
√ √ √ 3 9 2 2π 3 9 6π 2 3 3 2 δ¯ − κ ρˆ0 + =− 2 ρˆ0 κ 2 κ2 +6π
2
√ 3
3
1 2σ
q∈2π Z2 \{0}
e− 2 ε |q| |ρˆq |2 + O(σ ). κ 2 + |q|2 2
(3.75)
Minimizing this with respect to ρˆ0 , we obtain E¯ N
1 ≥− 2 2κ
2 √ 3 √ 9 2 3 δ¯ − κ + 6π 2 3 2
q∈2π Z2 \{0}
1 2σ
e− 2 ε |q| |ρˆq |2 + O(σ ). κ 2 + |q|2 2
(3.76)
2 √ Finally, from Lemma A.2 one can see that min E¯ N ≤ − 2κ1 2 δ¯ − 21 3 9 κ 2 + o(1) for ε 1. Hence, in view of arbitrariness of σ the constant in (3.76) is the limit of E¯ N ¯ as ε → 0. In addition, this implies that ρˆq → 0 as and, by Proposition 3.7, also of E, ε → 0 for every q = 0. Thus, we just proved
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√ Proposition 3.10. Let δ¯ > 21 3 9 κ 2 , and let ρ be defined in (3.72), with xi = ε1/3 | ln ε|−1/3 x¯i , where x¯i are as in Proposition 3.5. Then
2 √ 3 1 9 2 κ min E¯ → − 2 δ¯ − , (3.77) 2κ 2 and
√ 3 9 2 1 κ ρ→ √ δ¯ − 2 2π 3 3
(3.78)
weakly in the sense of measures, as ε → 0. We end by noting that the homogenization approach to multi-droplet patterns in a related context was first discussed in [61]. Also, let us mention that in a related class of problems existence of limiting density for the ground states of particle systems interacting via potentials like our G as the number of particles goes to infinity was proved in [62]. The difference with our result, however, is that in [62] the limit is taken at fixed positive temperature, while in our case the system’s temperature (in the usual thermodynamic sense) is strictly zero. Yet, as was pointed out in [62], the “effective” temperature of the system considered actually goes to zero as the number of particles goes to infinity, making these results closely related to ours. 3.4. Fine structure of the transition point. Finally, we briefly the appearance of √ discuss 1 3 2 ¯ non-trivial minimizers in the vicinity of the point δm = 2 9 κ (this transition point was identified in [13,14]). First, we note that by the results just obtained the transition from trivial to non-trivial minimizers appears to be quite abrupt in the limit ε → 0. In fact, in this limit one goes immediately from no droplets to infinitely many droplets upon crossing the point δ¯ = δ¯m from below. ¯ Observe that the energy E[u] (and, equivalently, E[u] − E[−1]) is a monotonically ¯ Therefore, a passage through the neighborhood of δ¯ = δ¯m decreasing function of δ. at small but finite ε will result in a monotonic increase of the number of droplets in a minimizer. This number will quickly get large as one moves away from the transition point. Therefore, in order to analyze droplet creation at the transition, we need to further zoom in on the parameter region around δ¯ = δ¯m . Let us introduce the renormalized distance to the transition (with the transition point shifted appropriately):
√ 3 9 2 | ln ε| ln | ln ε| 2 δ¯ − κ − √ τ= (3.79) κ , κ2 2 2 3 3 | ln ε| and consider the behavior of energy E¯ in the limit ε → 0 for τ = O(1). As can be easily seen, all the estimates obtained previously remain valid in this case, and minimizers are close to a collection of N disks separated by large distances, whose energy is given by E¯ N to O(εα ). We can also write the energy E¯ N in the form E¯ N =
N i=1
E¯ 1 (¯ri ) + 4π 2 | ln ε|−2
N −1
N
i=1 j=i+1
G(xi − x j )¯ri2 r¯ 2j ,
(3.80)
Droplet Phases in Non-local Ginzburg-Landau Models
where
√ 3 | ln ε| E¯ 1 (¯r ) = 2π r¯ − 21 9 r¯ 2 + 16 r¯ 4 + 13 π | ln ε|−1 ln | ln ε| (¯r 2 −
71
√ 3 9)¯r 2
−π | ln ε|−1 (¯r 2 (ln κ¯ r¯ − 41 ) + 2τ )¯r 2
(3.81)
is the energy of one disk-shaped droplet of radius r¯ . It is easy to see that in the limit ε → 0 we have E¯ 1 (¯r ) ≥ 0 for all r¯ > 0, and E¯ 1 (¯r ) = 0 √ √ 3 3 3 uniformly in a minimizer as ε → 0 with τ if and only of r¯ = 3. Therefore, r¯i → √ √ fixed. In fact, by convexity of E¯ 1 near r¯ = 3 3 we have r¯i − 3 3 = O(| ln ε|−1 ln | ln ε|) in the limit ε → 0. Therefore, we obtain (the summation is absent in the formula, if N = 1) ⎧ −1 N ⎨ N √ 3 E¯ N = 12 3 π 2 | ln ε|−2 G(xi − x j ) ⎩ i=1 j=i+1 ⎫ ⎬ N 1 1 2τ − ln κ¯ + ln 3 − + √ (3.82) + o(| ln ε|−2 ). 3 ⎭ 4π 3 4 9 From this expression it is easy to see that N = O(1) quantity. Thus, in this case the problem reduces to minimizing the pair interaction potential given by the sum in (3.82). We summarize the above discussion by stating the following result. Proposition 3.11. Let δ¯ be given by (3.79) with τ fixed. Then, there exists a strictly monotonically increasing sequence of numbers (τn ), with τn → ∞ as n → ∞, such that, provided that ε 1: √ 1 3 (i) If τ < τ1 = 24 9 (3 − 4 ln 3 − 12 ln κ), ¯ then there are no non-trivial minimizers of E. √ (ii) If τ1 < τ < τ2 , with τ2 = τ1 + 2π 3 9 min G, the minimizer of E is a single droplet. (iii) If τn < τ < τn+1 , all minimizers of E consist of precisely n droplets. The droplet n−1 n centers {xi } nearly minimize V = G(xi − x j ). i=1 j=i+1
Let us mention that local minimizers of E without screening (i.e. with κ → 0) which are close to disks of the same radius centered at the minimizers of V were constructed perturbatively in a recent work of Ren and Wei [32,33]. We note that when τ = O(1), existence of these solutions easily follows from our analysis, if one notices that in the considered regime the excess energy of a minimizing sequence controls the isoperimetric deficit of each droplet and enforces O(1) distance between them. Therefore, solutions with a prescribed number of droplets may be obtained by minimizing over all u ∈ BV (Ω; {−1, 1}), such that the support of 1 + u has a fixed number of disjoint components. In turn, by Proposition 3.11 the global minimizers of E belong to this class. Let us also mention another related recent work of Choksi and Peletier, where a version of E was considered in the regime, in which the minimizers consist of a finite number of droplets [37]. There it was shown, using the language of Γ -convergence,
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that in the limit of small volume fraction the droplets concentrate into point masses interacting via a pairwise potential, both in d = 2 and d = 3. Morover, similarly to the conclusions of Sec. 3.4, in d = 2 all limiting masses were found to be equal to each other. 4. Connection to the Diffuse Interface Energy We now turn to the study of the relationship between the sharp interface energy E and the diffuse interface energy E. Since most of our analysis here does not rely on any particular assumptions on the dimensionality of space, we will treat the general case of Ω being a d-dimensional torus: Ω = [0, 1)d . We assume that W is a symmetric double-well potential with non-degenerate minima at u = ±1, together with some natural technical assumptions: (i) W ∈ C 3 (R), W (u) = W (−u), and W ≥ 0. (ii) W (+1) = W (−1) = 0 and W (+1) = W (−1) > 0. (iii) W (|u|) is monotonically increasing for |u| ≥ 1, lim|u|→∞ W (u) = +∞, and d+2 if d > 2. |W (u)| ≤ C(1 + |u|q ), for some C > 0 and q > 1, with q < d−2 Since we are setting the surface tension to ε, we need to additionally normalize W as follows: (iv) We have
1 −1
2W (u) du = 1.
(4.1)
Note that these assumptions are satisfied for, e.g., the rescaled version of the classical 9 Ginzburg-Landau energy: W (u) = 32 (1 − u 2 )2 for d ≤ 3. Also note that this assumption is not restrictive, since it is always possible to make (4.1) hold by an appropriate rescaling. Let us begin our analysis with a few general observations. First of all, by the direct method of calculus of variations (see e.g. [63]) there exists a minimizer u ∈ H 1 (Ω) of E satisfying Ω u d x = u¯ for every ε > 0. Note that any critical point u of E, including minimizers, is a weak solution of the Euler-Lagrange equation (here and below G 0 solves (1.2) with periodic boundary conditions and has zero mean) 2 ε Δu − W (u) − v + μ = 0, v(x) = G 0 (x − y)(u(y) − u)dy, ¯ (4.2) Ω
where
μ=
Ω
W (u) d x
(4.3)
is the Lagrange multiplier. Furthermore, from the Sobolev imbedding theorem we have 2d , and hence v ∈ W 2, p (Ω) ⊂ C 0,α (Ω), for some α ∈ (0, 1), u ∈ L p (Ω) for p = d−2 if d < 6. Applying the Moser iteration technique [63, see App. B], we then find that u ∈ L p (Ω), for any p < ∞. Therefore, by standard elliptic regularity theory [51], we also have u ∈ W 2, p (Ω), so u ∈ L ∞ (Ω) and is, in fact, a classical solution of (4.2). We now show that u is uniformly bounded and that |u| cannot much exceed 1 whenever E[u] is sufficiently small, at least for d not too high.
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73
Proposition 4.1. Let d < 6 and let u be a critical point of E. Then, for every δ > 0 we have |u| < 1 + δ and |v| < δ in Ω, whenever E[u] is sufficiently small. Proof. Observe first that for every δ > 0 and E[u] small enough we have |{|u| > 1+δ}| < 1 2 . Now, suppose that the maximum value u m of |u| is greater than 1 + δ. Without a loss of generality, we may assume that u m = max u. By the preceding observation, we have μ ≤ C + 21 W (u m ), for some C > 0 independent of u m . Therefore, in view of the monotonic increase to infinity of W (u) due to hypothesis (iii) on W , we have μ ≤ 43 W (u m ) for u m sufficiently large. Now, taking into account that v ≥ −C u m for some C > 0 and large enough u m , from (4.2) we find that ε2 Δu ≥ 41 W (u m ) − C u m > 0 at the point where u = u m , in view of assumption (iii) on W , contradicting the maximality of u. Finally, to see that |u| < 1 + δ with any δ > 0, when E[u] 1, note that u → ±1 a.e. when E[u] → 0. Hence, in view of the uniform bound on u obtained earlier, we have term in the energy can be written as μ →2 0. Furthermore, since the non-local 2, 1 p (Ω) for any p < ∞, we also have |∇v| d x, and v is uniformly bounded in W 2 Ω v → 0 uniformly in Ω in this limit. Therefore, in view of positivity of W (1 + δ), we can apply the same argument as above to complete the proof of the statement. We note that while the arguments above hold for every critical point E with small energy, it is generally possible for a local minimizer of E to strongly deviate from ±1 in most of Ω: take, for instance, one-dimensional periodic solutions of (4.2) with period O(1) [4,64,65]. Of course, these critical points will have O(1) energy when ε → 0, as opposed to minimizers of E whose energy vanishes in this limit. Let us also mention that numerical evidence shows that generally max |u| > 1, even for minimizers and ε 1. We now turn to estimating the minimal energy of E from below by the minimal energy of E. For u ∈ H 1 (Ω) with u d x = u, ¯ let us separate the domain Ω into three pairwise-disjoint subdomains: δ ∪ Ω0δ , Ω = Ω+δ ∪ Ω−
(4.4)
Ω+δ = {x ∈ Ω : u(x) ≥ 1 − δ},
(4.5)
where δ Ω− Ω0δ
= {x ∈ Ω : u(x) ≤ −1 + δ},
(4.6)
= {x ∈ Ω : −1 + δ < u(x) < 1 − δ}.
(4.7)
Next, let us introduce the following three auxiliary functionals (for simplicity of notation, we will suppress the index δ in the definition of each functional): 2 ε 2 |∇u| + W (u) d x, E1 [u] = (4.8) 2 Ω0δ 1 E2 [u] = 2 (u − u 0 )2 d x 2κ Ω+δ ∪Ω−δ 1 + (u(x) − u)G ¯ 0 (x − y)(u(y) − u)d ¯ xd y, (4.9) 2 Ω Ω δ , respectively, with where u 0 (x) = ±1 whenever x ∈ Ω±
κ=√
1 , W (1)
(4.10)
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and
E3 [u] =
δ Ω+δ ∪Ω−
W (u) −
1 2 d x. (u − u ) 0 2κ 2
(4.11)
It is clear that the energy E can be estimated from below as E ≥ E1 + E2 + E3 .
(4.12)
Hence, we are going to establish a lower bound for E by considering the lower bounds for each term in the sum above. We start with the part of energy that is associated with the interfaces: δ ∪ Lemma 4.1. Let δ > 0 be sufficiently small, let u ∈ H 1 (Ω) and suppose that |Ω− δ Ω+ | > 0. Then there exists u 0 ∈ BV (Ω; {−1, 1}) such that u 0 (x) = ±1 whenever δ , and x ∈ Ω± ε E1 [u] ≥ (1 − a1 δ 2 ) |∇u 0 |d x (4.13) 2 Ω
for some a1 > 0 independent of δ and ε. δ | is zero, we can simply choose u to be constant Proof. First of all, if either |Ω+δ | or |Ω− 0 δ δ | > 0, and approximate (e.g. u 0 = −1 when |Ω+ | = 0). So, let us assume that both |Ω± u in H 1 (Ω) by a piecewise linear function u˜ with ∇ u˜ = 0 almost everywhere in Ω. Then, using the Modica-Mortola trick [66,67] and the co-area formula [68], we find 1−δ E1 [u] ˜ ≥ε 2W (u) ˜ |∇ u| ˜ dx = ε 2W (t) |{u˜ = t}| dt. (4.14) |u| 0 and all δ small enough. Now, define u˜ 0 ∈ BV (Ω; {−1, 1}) as +1, u(x) ˜ > c, (4.15) u˜ 0 (x) = −1, u(x) ˜ ≤ c. The preceding arguments imply the desired inequality for u. ˜ Passing to the limit in the approximation, we obtain the result, with u 0 = lim u˜ 0 in L 1 (Ω) upon extraction of a subsequence. Lemma 4.2. Let u and u 0 be as in Lemma 4.1, let u satisfy Ω u d x = u, ¯ and let ¯ dy ≤ δ 3 in Ω, for δ > 0 sufficiently small. |u| ≤ 1 + δ 3 and Ω G 0 (x − y)(u(y) − u) Then 1 E2 [u] ≥ (u 0 (x) − u)G(x ¯ − y)(u 0 (y) − u) ¯ d xd y − a2 δ E[u], (4.16) 2 Ω Ω 3
for some a2 ≥ 0 independent of δ and ε, whenever E[u] ≤ δ 2 (d+6) .
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75
Proof. Let us write u as follows: u = u0 + u1 + u2,
u 1 (x) = −κ
2 Ω
G 0 (x − y)(u(y) − u)dy. ¯
(4.17)
Note that by assumption, ||u 1 || L ∞ (Ω) ≤ Cδ 3 and ||u 2 || L ∞ (Ω) ≤ C, for some C > 0. Now, observe that u 1 solves ¯ − Δu 1 + κ 2 u 1 = −κ 2 (u 0 + u 2 − u),
(4.18)
and, therefore, we also have u 1 (x) = −κ 2
Ω
G(x − y)(u 0 (y) + u 2 (y) − u)dy. ¯
(4.19)
Substituting u in the form (4.17) into (4.9), we obtain E2 (u 0 + u 1 + u 2 ) = 12κ 2 (u 1 + u 2 )2 d x 1 2
δ Ω+δ ∪Ω−
(u 0 (x) + u 1 (x) + u 2 (x) − u)G(x ¯ − y)(u 0 (y) + u 2 (y) − u)d ¯ yd x 1 1 1 =− 2 u 21 d x − 2 u1u2 d x − u 2 (x)G(x − y)u 2 (y)d yd x 2κ Ω0δ κ Ω0δ 2 Ω Ω 1 1 + (u 0 (x) − u)G(x ¯ − y)(u 0 (y) − u)d ¯ yd x + 2 u2 d x 2 Ω Ω 2κ Ω+δ ∪Ω−δ 2 1 1 ≥ (u 0 (x) − u)G(x ¯ − y)(u 0 (y) − u)d ¯ yd x − 2 u2 d x 2 Ω Ω 2κ Ω0δ 1 + u 2 (x)G(x − y)(u 0 (y) − u) ¯ d yd x +
Ω
Ω0δ
1 + 2 2κ
Ω
Ω
δ Ω+δ ∪Ω−
u 22 (x) − κ 2 u 2 (x)
δ Ω+δ ∪Ω−
G(x − y)u 2 (y)dy d x.
(4.20)
In fact, the last line in (4.20) is non-negative. Indeed, writing the integral in the last line of (4.20) with the help of the Fourier Transform aˆ q of u˜ = u 2 χΩ+δ ∪Ω δ , where χΩ+δ ∪Ω δ
δ: is the characteristic function of Ω+δ ∪ Ω− eiq·x u 2 (x) d x, aˆ q =
−
(4.21)
δ Ω+δ ∪Ω−
we obtain
δ Ω+δ ∪Ω−
=
u 22 (x) − κ 2 u 2 (x)
|q|2 |aˆ q |2 ≥ 0. κ 2 + |q|2 d
q∈2π Z
−
δ Ω+δ ∪Ω−
G(x − y)u 2 (y)dy d x (4.22)
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C. B. Muratov
To estimate the remaining terms in (4.20), we note that 1 u1u2 d x + u 2 (x)G(x − y)(u 0 (y) − u) ¯ d yd x 2 δ κ Ωδ Ω0 Ω 0 = u 2 (x)G(x − y)u 2 (y) d yd x Ωδ Ω 0 ≤ |u 2 (x)|G(x − y)|u 2 (y)| d yd x Ωδ
{|u|≥1−δ 3 }
0 +
Ω0δ
{|u|
Ω˜
d 2.
Therefore, continuing the estimates in (4.23), we obtain u 2 (x)G(x − y)u 2 (y) d yd x Ωδ Ω 0 ≤ C δ 3 + δ −6/q E 1/q [u] |Ω0δ | ≤ 2Cδ 3 |Ω0δ |,
(4.25)
whenever E[u] ≤ δ 3(q+2) , where we took into account that by the assumptions of 3 3 the lemma |u 2 | ≤ |u −6 u 0 | + |u 1 | ≤3 Cδ in {|u| > 1 − δ }, and that E[u] ≥ W (u) d x ≥ cδ |{|u| < 1 − δ }| for some c > 0. {|u| 0, by choosing q = 21 (d + 2).
0
Lemma 4.3. Let u and u 0 be as in Lemma 4.1. Then E3 [u] ≥ −a3 δ E[u]
(4.27)
for some a3 ≥ 0 independent of δ and ε, for sufficiently small δ > 0. Proof. By assumption (iii) on W , we have W (u) ≥ 2κ1 2 (u − u 0 )2 whenever |u| > 1. Hence 1 W (u) − 2 (u − u 0 )2 d x E3 [u] ≥ 2κ {1−δ≤|u|≤1} ≥ −Cδ (u − u 0 )2 d x ≥ −a3 δ E[u], (4.28) {1−δ≤|u|≤1}
for some a3 ≥ 0.
Droplet Phases in Non-local Ginzburg-Landau Models
77
δ | > 0 for E[u] Combining all the results above with an observation that |Ω+δ ∪ Ω− small enough, we obtain ¯ Proposition 4.2. Let δ > 0 be sufficiently small, let u ∈ H 1 (Ω) satisfy Ω u d x = u, 3 (d+6) 3 3 2 let |u| ≤ 1 + δ and Ω G 0 (x − y)(u(y) − u) ¯ dy ≤ δ in Ω, and let E ≤ δ . Then there exists a function u 0 ∈ BV (Ω; {−1, 1}) such that E[u] ≥ (1 − δ 1/2 )E[u 0 ], with κ given by (4.10).
Importantly, the lower bound in Proposition 4.2 is sharp in the limit ε → 0 for all functions u 0 ∈ BV (Ω; {−1, 1}) obeying suitable bounds (satisfied by minimizers of E in d = 2): Proposition 4.3. Let u 0 ∈ BV (Ω; {−1, 1}), with the jump set of class C 2 , let the principal curvatures of the jump set of u 0 be bounded by ε−α for some α ∈ [0, 1), let the distance between different connected portions of the jump set be bounded by εα , and let Ω G(x − y)(u 0 (y) − u) ¯ dy ≤ δ for some δ > 0 small enough. Then there exists a function u ∈ H 1 (Ω) with Ω u d x = u, ¯ such that E[u] ≤ (1 + δ 1/2 )E[u 0 ], with κ given 1
by (4.10), whenever E[u 0 ] ≤ δ 2 (d+3) and ε 1.
Proof. For simplicity of presentation, we only give the proof in the case d = 2. With minor modifications, the proof remains valid for all d. Here we adapt the standard construction of a trial function for the local part of the Ginzburg-Landau energy. Let U (ρ) be the solution of the ordinary differential equation d 2U − W (U ) = 0, dρ 2
U (−∞) = 1, U (+∞) = −1, U (0) = 0,
(4.29)
where the last condition fixes translations. As is well-known (see e.g. [69]), this solution exists, is unique and is a strictly monotonically decreasing odd function, approaching the equilibria at ρ = ±∞ exponentially fast. Therefore, for any δ > 0 we have |U (ρ)| ≤ 1 − δ, if and only if |ρ| ≤ l, with some positive l = O(| ln δ|). Also note that by hypothesis (iv) on W,
1−δ 1 dU 2 + W (U ) dρ = 2W (s) ds = 1 + O(δ 2 ). −l 2 dρ −1+δ l
(4.30)
Now, introduce the signed distance function r (x) = ±dist(x, Ω ± ), where Ω ± = {u 0 = ±1}, whenever x ∈ Ω ∓ , and define a regularized version u ε0 of u 0 : ⎧ −1 ⎪ |r (x)| ≤ εl, ⎨U (ε r (x)), ε u 0 (x) = (1 − δ + ε−1 δ(|r (x)| − εl))u 0 (x), εl ≤ |r (x)| ≤ ε(l + 1), ⎪ ⎩ u (x), |r (x)| ≥ ε(l + 1). 0
(4.31)
Then, it is easy to see that the function u(x) = u ε0 (x) − κ 2
Ω
G(x − y)(u ε0 (y) − u)dy ¯
(4.32)
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C. B. Muratov
is in H 1 (Ω), with
u d x = u. ¯ Moreover, we have for any q > 1, ε 2 |u(x) − u 0 (x)| ≤ κ G(x − y)(u 0 (y) − u) ¯ dy Ω 2 +κ G(x − y)|u ε0 (y) − u 0 (y)| dy Ω
Ωl
≤ C(δ + |Ωl |1/q ) ≤ C (δ + | ln δ|1/q E 1/q [u 0 ]) ≤ C δ, (4.33) where we defined Ωl = {|r | ≤ ε(l + 1)}, estimated Ωl G(x − y)(u ε0 (y) − u 0 (y)) dy as in Lemma 4.2, and used the curvature bound and the assumption on E with d = 2 and q = 2. On the other hand, after a few integrations by parts, from (4.32) we also obtain |∇(u − u ε0 )|2 d x = −κ 2 (u − u ε0 )(u − u) ¯ dx Ω Ω = κ4 (u(x) − u)G ¯ 0 (x − y)(u(y) − u) ¯ d yd x ≤ 2κ 4 E[u]. (4.34) Ω
Ω
To estimate E[u], let us introduce a system of curvilinear coordinates (ρ, ξ ) consisting of the signed distance ρ to the jump set of u 0 and the projection ξ onto the jump set. By our assumptions this is possible whenever |r (x)| < εα . Therefore, for ε 1 we can write
2 ε(l+1) ∂u ε2 ∂u 2 ε2 E[u] = + (1 + ρ K )−2 + W (u) 2 ∂ρ 2 ∂ξ −ε(l+1) ∂Ω + 2 ε 1 2 |∇u| + W (u) d x × (1 + ρ K ) dH (ξ ) dρ + 2 Ω\Ωl 1 + (u(x) − u)G ¯ 0 (x − y)(u(y) − u) ¯ d yd x, (4.35) 2 Ω Ω where K = K (ξ ) is the curvature at point ξ on the jump set of u 0 . Substituting the ansatz of (4.32) into (4.35), taking into account that |∇(u − u ε0 )| ≤ C in Ωl for some C > 0 independent of ε (we have u − u ε0 uniformly bounded in W 2, p (Ω), for any p < ∞) and that ∇u = ∇(u − u 0 ) = ∇(u − u ε0 ) in Ω\Ωl , and using (4.33) and (4.34), we obtain (estimating each line in (4.35) separately) ε 1−α E[u] = 1 + O(ε | ln δ|) + O(δ| ln δ|) |∇u 0 | d x 2 Ω 1 + 2 (u − u 0 )2 d x + O(ε2 E[u]) + O(δ E[u]) 2κ Ω\Ωl 1 + (u(x) − u)G ¯ 0 (x − y)(u(y) − u) ¯ d yd x. (4.36) 2 Ω Ω Now, using the identity (u(x) − u)G ¯ 0 (x − y)(u(y) − u) ¯ d yd x + κ −2 (u − u ε0 )2 d x Ω Ω Ω ε ε = (u 0 (x) − u)G(x ¯ − y)(u 0 (y) − u) ¯ d yd x, Ω
Ω
(4.37)
Droplet Phases in Non-local Ginzburg-Landau Models
79
we can further write (4.36) as 1 (u − u ε0 )2 d x E[u] = E[u 0 ] + O(δ| ln δ| E[u 0 ]) + O(δ E[u]) − 2 2κ Ωl + (u ε0 (x) − u 0 (x))G(x − y)(u ε0 (y) − u) ¯ d yd x Ωl Ω 1 + (u ε (x) − u 0 (x))G(x − y)(u ε0 (y) − u 0 (y)) d yd x, (4.38) 2 Ωl Ωl 0 for ε 1. Finally, using the same estimates as in (4.33), we obtain (1 + O(δ))E[u] = (1 + O(δ| ln δ|))E[u 0 ] + O(δ|Ωl |) + O(|Ωl |3/2 ) = (1 + O(δ| ln δ|))E[u 0 ], from which the result follows immediately.
(4.39)
The last two propositions show asymptotic equivalence of the diffuse interface energy E with the sharp interface energy E for sufficiently well-behaved critical points and ε 1. In particular, the energies of minimizers of both E and E are asymptotically the same in the limit ε → 0 (see also [70,71] for recent related studies). It would also be natural to think that the minimizers (even local, with low energy) of E are, in some sense, close to minimizers of E when ε 1 (this will be a subject of future study). 5. Proof of the Theorems Here we complete the proofs of Theorems 2.1–2.3. Proof of Theorem 2.1. The main point of the proof is the lower bound in (2.1), since the upper bound is easily obtained by constructing a suitable trial function (as in Lemma A.1). The basic tool for the lower bound is a kind of interpolation inequality obtained in Lemma B.1. Note that the proof for E works in any space dimension. To prove the lower bound, let us denote by u a minimizer of E. Introducing aˆ q = eiq·x (u(x) − u) ¯ d x, (5.1) Ω
where q ∈ 2π Zd , we can estimate the energy of the minimizer as follows: 1 min E ≥ (u(x) − u)G(x ¯ − y)(u(y) − u) ¯ d xd y 2 Ω Ω |aˆ 0 |2 1 |aˆ q |2 ≥ = 2 2 2 q κ + |q| 2κ 2 2 1 2 1 + u¯ 2 + = 2 (u − u) ¯ d x = 2 |Ω | − , 2κ κ 2 Ω where we introduced the set Ω + = {u = +1}.
(5.2)
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In view of the upper bound in (2.1), it follows from (5.2) that |Ω + | = 21 (1 + u) ¯ + O(ε2/3 ), implying that |Ω + | is bounded away from 0 or 1 for ε 1. Hence, by the isoperimetric inequality there exists p > 0 such that P= |∇u| d x ≥ p, (5.3) Ω
whenever ε 1. Applying Lemma B.1 to u − u, ¯ we conclude that C , (5.4) P2 for some C > 0 independent of ε, for ε 1. The result then follows from an application of the Young Inequality and Propositions 4.1 and 4.2. min E ≥ ε P +
Proof of Theorem 2.2. This theorem combines a number of results proved in Sec. 3 in the original, unscaled variables. Part (i) of the theorem is the statement of Proposition 3.2. Part (ii) of the theorem is the collection of results from Lemma A.1 (taking into √ account that E[u k ] < E[−1] = 21 ε4/3 | ln ε|2/3 κ −2 δ¯2 for δ¯ > 21 3 9 κ 2 ), Corollary 3.1, Lemma 3.3, and Propositions 3.5, 3.7, and 3.8 with α = 13 − σ . Part (iii) of the theorem is contained in the statements of Propositions 3.9 and 3.10. Proof of Theorem 2.3. First of all, we have min E 1 when ε 1 and u¯ = −1 + O(ε2/3 | ln ε|1/3 ), since min E ≤ E(u) ¯ = O(ε4/3 | ln ε|2/3 ) in that case. Then, from Proposition 4.1 and Lemma 3.5 we conclude that the assumptions of Propositions 4.2 and 4.3 are satisfied for the minimizers of E. Therefore, the energies E and E are asymptotically the same √ in the considered limit, and the conclusion follows from Theorem 2.2 ¯ (the case δ¯ = 21 3 9 κ 2 is included by a monotone decrease of E¯ with δ). Acknowledgments. The author would like to acknowledge valuable discussions with M. Kiessling, H. Knüpfer, V. Moroz, M. Novaga and G. Orlandi. This work was supported, in part, by NSF via grants DMS-0718027 and DMS-0908279.
A. Upper Bound Here we construct a trial function that achieves the lower bound for the energy of the non-trivial minimizers of E. √ ¯ with δ¯ > 1 3 9 κ 2 fixed. Then there exists Lemma A.1. Let u¯ = −1 + ε2/3 | ln ε|1/3 δ, 2 u ∈ BV (Ω; {−1, 1}), such that √ 3 √ ln | ln ε| 1 9 3 E[u] = ε4/3 | ln ε|2/3 δ¯ − 9 κ2 + O , (A.1) 2 4 | ln ε| for ε 1. Proof. First, consider u 1 (x) = −1 + 2χ Br (0) (x), where χ Br (0) is the characteristic function of the disk of radius r centered at the origin. If v1 (x) = Ω G(x − y)(u 1 (y) − u) ¯ dy, then by using (2.3) we explicitly have (see (3.39)) 1 + u¯ 2 + 2 (1 − κr K 1 (κr )I0 (κ|x|)), κ2 κ 2 r I1 (κr )K 0 (κ|x + n|)), |x| ≤ r, + κ 2
v1 (x) = −
n∈Z \{0}
(A.2) (A.3)
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81
where K n and In are the modified Bessel functions of the first and second kind. Therefore, expanding the Bessel functions for r 1 [46], we can write for |x| ≤ r, 1 + u¯ r2 |x|2 − ln κr + 2γ − ln 4 − 1) − (2 κ2 2 2 2 4 K 0 (κ|x + n|) + O(r | ln r |), +r
v1 (x) = −
(A.4)
n∈Z2 \{0}
where γ ≈ 0.5772 is Euler’s constant. Substituting this expression into the definition of E, after integration we get E[u 1 ] = 2π εr + 21 (1 + u) ¯ 2 κ −2 − 2π(1 + u)κ ¯ −2 r 2 −πr 4 (ln κr + γ − ln 2 − 41 ) + πr 4
K 0 (κ|n|) + O(r 6 | ln r |). (A.5)
n∈Z2 \{0}
Now, consider a new test function u k (x) = −1 + 2
k k k1 =1 k2 =1
χ Br (e1 (k1 − 1 )+e2 (k2 − 1 )) (x), 2
2
(A.6)
consisting of k 2 disks of radius r arranged periodically in Ω (here e1 and e2 are the unit vectors along the coordinate axes). We have 2 −2 2 1 E[u k ] = 2 (1 + u) ¯ κ + π k 2εr − 2(1 + u)κ ¯ −2 r 2
−r 4 (ln κr + γ − ln 2 − 41 ) + r 4
K 0 (κk −1 |n|) + O(k 2 r 6 | ln r |).
n∈Z2 \{0}
(A.7) Approximating the sum in (A.7) by an integral: −2 −1 K 0 (κk |n|) = K 0 (κ|x|)d x + O(k −2 ln k) k n∈Z2 \{0}
R2
= 2π κ −2 + O(k −2 ln k),
(A.8)
and expanding for r 1, we can further write
¯ −2 r 2 ¯ 2 κ −2 + π k 2 2εr − 2(1 + u)κ E[u k ] = 21 (1 + u) −r 4 ln r + 2π κ −2 r 4 k 2 + O(k 2 r 4 ln k).
(A.9)
√ We now substitute r = ε1/3 | ln ε|−1/3 3 3 into the expression above. Using also the definition in (2.7), we can write √ √ 3 3 E[u k ] = ε4/3 | ln ε|2/3 21 κ −2 δ¯2 − 2π 9| ln ε|−1 κ −2 δ¯ − 21 9 κ 2 k 2 √ 3 +6π 2 3κ −2 | ln ε|−2 k 4 + O(ε4/3 | ln ε|−4/3 k 2 ln k). (A.10)
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Finally, setting
√ 3 9 2 | ln ε| k = κ + O(1), √ δ¯ − 2 2π 3 9 2
we obtain (A.1) with u = u k .
(A.11)
Let us also quote without proof a similar result concerning the upper bound for the reduced energy E N . √ ¯ with δ¯ > 1 2 9 κ 2 fixed. Then Lemma A.2. Let u¯ = −1 + ε2/3 | ln ε|1/3 δ, 2 min E N
2 √ 4/3 3 9 1 4/3 ε ln | ln ε| 2/3 ¯ 2 δ− . ≤ − 2 ε | ln ε| +O κ 2κ 2 | ln ε|1/3
(A.12)
B. Interpolation Inequality Here we present the lemma that connects the non-local part of the energy with the interfacial energy via a kind of an interpolation inequality between BV (Ω), H −1 (Ω) and L ∞ (Ω), for functions bounded away from zero. Lemma B.1. Let u ∈ BV (Ω), where Ω = [0, 1)dis a torus, and assume that m ≤ |u| ≤ M in Ω for some M ≥ m > 0. Let also Ω |∇u| d x ≥ p > 0, and let G solve (1.5) in Ω with periodic boundary conditions. Then there exists a constant C = C(d, κ/ p, m 2 /M) > 0 such that −2 u(x)G(x − y)u(y) d x d y ≥ C |∇u| d x . (B.1) Ω
Ω
Ω
Proof. First, extend u periodically to the whole of Rd . Then, introducing χδ (x) = δ −d |B1 |−1 χ (δ −1 x), where χ is the characteristic function of the unit ball B1 centered at the origin, we have 1 u(x)χδ (x − y)u(y) d y d x = u(x)u(x + δy) d y d x |B1 | Ω B1 Ω Rd 1 Mδ 2 2 |∇u(x + δt y)| dt dy d x ≥ m − Mδ |∇u| d x, (B.2) ≥m − |B1 | Ω B 1 0 Ω where the inequality is obtained by approximating u by C 1 functions and passing to the limit. Therefore, choosing −1 2M δ= |∇u| d x , (B.3) m2 Ω we obtain m2 ≤ 2
Ω
Rd
u(x)χδ (x − y)u(y) d x d y =
q
χˆ δ (q)|uˆ q |2 ,
(B.4)
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83
where we introduced the Fourier transform uˆ q of u: uˆ q = eiq·x u(x) d x,
(B.5)
Ω
with q ∈ 2π Zd . The Fourier transform χˆ δ of χδ is, in turn, explicitly given by χˆ δ (q) =
2 δ|q|
d/2
Γ
d + 1 Jd/2 (δ|q|), 2
(B.6)
where Jd/2 (x) is the Bessel function of the first kind and Γ (x) is the gamma-function. Now, applying the Cauchy-Schwarz Inequality, we obtain
|uˆ q |2 m4 2 2 2 2 ≤ χˆ δ (q)(κ + |q| )|uˆ q | 4 κ 2 + |q|2 q q ≤ sup χˆ δ2 (q)(κ 2 + |q|2 ) |uˆ q |2 q
× Taking into account that δ 2 χˆ δ2 (q)(κ 2
q
Ω
q
Ω
u(x)G(x − y)u(y) d x d y.
(B.7)
|uˆ q |2 = ||u||2L 2 (Ω) ≤ M 2 and that [46]
C1 (κ 2 m 4 M −2 p −2 + 1), |q|δ ≤ 1, + |q| ) ≤ (B.8) 2 4 −2 −2 2 2 −d−1 , |q|δ > 1, C2 (κ m M p + |q| δ )(|q|δ) 2
for some C1,2 > 0 depending only on d, we conclude that u(x)G(x − y)u(y) d x d y Cδ 2 ≤ Ω
(B.9)
Ω
for some C > 0 depending only on d, κ/ p, and m 2 /M. The result then follows immediately from (B.3). Let us also make some remarks regarding a few extensions of these arguments. First, the same estimate holds true in the case where G is the Green’s function of the Laplacian in Ω and u has zero mean. Note that in this case the constant C in (B.1) becomes independent on p. The proof easily follows by passing to the limit κ → 0 in the lemma. Another observation is that, actually, for the considered class of functions a stronger interpolation inequality involving negative Sobolev norms holds. We give only the statement of the result, the proof follows easily by modifying a few steps in the arguments above Proposition B.1. Let u be as in Lemma B.1. Then Ω
u (1 − Δ)−
d+1 2
for some C = C(d, p, m, M) > 0.
u dx ≥ C
Ω
|∇u| d x
−d−1
,
(B.10)
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C. First and Second Variation Here we present the derivation of the first and second variation of E¯ in d = 2, adapted from [14]. Lemma C.1. Let Ω¯ + ⊂ Ω¯ be a set with boundary of class C 2 and v be given by (3.4). Then, the functional E¯ is twice continuously Gâteaux-differentiable with respect to C 1 perturbations of ∂ Ω¯ + . Furthermore, the first and second Gâteaux derivatives of E¯ are given by (3.6) and (3.7). Proof. Let a > 0, let ρ ∈ C 1 (∂ Ω¯ + ), and let Ω¯ a+ be the set obtained from Ω¯ + by transporting each point of ∂ Ω¯ + by aρ in the direction of the outward normal. Note that for sufficiently small a the set ∂ Ω¯ a+ is of class C 1 , in view of regularity of ∂ Ω¯ + . Then, if ¯ Ω¯ a+ ) and E¯ = E( ¯ Ω¯ + ), from (3.2) we have explicitly E¯ a = E( ¯ = | ln ε|( E¯ a − E)
∂ Ω¯ +
(1 + a K (x)ρ( ¯ x)) ¯ 2 + a 2 |∇ρ(x)| ¯ 2 − 1 d H1 (x) ¯
+
∂ Ω¯ +
aρ(x) ¯
¯ −2 )(1 + K (x)r (4v(x¯ + r ν(x)) ¯ − 2δκ ¯ ) dr d H1 (x) ¯
0
aρ(x) ¯ aρ( y¯ ) (1 + K (x)r ¯ )(1 + K ( y¯ )r ) +2| ln ε|−1 0 ∂ Ω¯ + ∂ Ω¯ + 0 ¯ − y¯ − r ν( y¯ )) dr dr d H1 ( y¯ )d H1 (x), ¯ ×G ε 1/3 | ln ε|−1/3 (x¯ + r ν(x)
(C.1) where K (x) ¯ is curvature, ν(x) ¯ is the outward unit normal at x¯ ∈ ∂ Ω¯ + , and we rewrote the integrals in terms of the curvilinear coordinates consisting of the projection x¯ of a point x ∈ Ω¯ to ∂ Ω¯ + and signed distance r = ν(x) ¯ · (x − x), ¯ which is possible for sufficiently small a. Now, Taylor-expanding the integrands in the powers of r and integrating over r and r , after some tedious algebra we obtain that for any α ∈ (0, 1), d E¯ a a 2 d 2 E¯ a ¯ ¯ Ea = E + a + + O(a 2+α ) holds, (C.2) da a=0 2 da 2 a=0 where the derivatives are given by (3.6) and (3.7). In estimating the remainder term in ¯ and the following estimate of the terms (C.2) we took into account that v ∈ C 1,α (Ω) involving the convolution integral: aρ( y¯ ) G(ε1/3 | ln ε|−1/3 (x¯ + ν(x)r ¯ − y¯ − ν( y¯ )r )) ∂ Ω¯ + 0 −G(ε1/3 | ln ε|−1/3 (x¯ − y¯ )) dr dH1 ( y¯ ) aρ( y¯ ) |x¯ − y¯ + ν(x)r ¯ − ν( y¯ )r | 1 ≤C a + ln dr dH ( y¯ ) |x¯ − y¯ | ∂ Ω¯ + 0
aρ( y¯ ) | |ν( x)r ¯ − ν( y ¯ )r ≤ C a2 + dr dH1 ( y¯ ) |x¯ − y¯ | ∂ Ω¯ + ∩|x− ¯ y¯ |≥Ma 0 dH1 ( y¯ ) 2 ≤ C a 2 | ln a|, ≤ Ca (C.3) ∂ Ω¯ + ∩|x− ¯ y¯ |≥Ma | x¯ − y¯ |
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for a 1, where M > 0 is sufficiently large, and we used the series expansion of G [46]. Finally, for every sufficiently small C 1 -perturbation ∂ Ω¯ a+ of ∂ Ω¯ + the distance from a point x¯ ∈ ∂ Ω¯ + to ∂ Ω¯ a+ is a C 1 -function, hence the formulas obtained above apply to all such perturbations. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.
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Commun. Math. Phys. 299, 89–127 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1072-1
Communications in
Mathematical Physics
Uniqueness of Smooth Stationary Black Holes in Vacuum: Small Perturbations of the Kerr Spaces S. Alexakis1, , A. D. Ionescu2, , S. Klainerman3, 1 Massachusetts Institute of Technology, Cambridge, MA 02139, USA. E-mail:
[email protected] 2 University of Wisconsin – Madison, Madison, WI 53706, USA. E-mail:
[email protected] 3 Princeton University, Princeton, NJ 08544, USA. E-mail:
[email protected] Received: 8 July 2009 / Accepted: 14 March 2010 Published online: 8 July 2010 – © Springer-Verlag 2010
Abstract: The goal of the paper is to prove a perturbative result, concerning the uniqueness of Kerr solutions, a result which we believe will be useful in the proof of their nonlinear stability. Following the program started in Ionescu and Klainerman (Invent. Math. 175:35–102, 2009), we attempt to remove the analyticity assumption in the the well known Hawking-Carter-Robinson uniqueness result for regular stationary vacuum black holes. Unlike (Ionescu and Klainerman in Invent. Math. 175:35–102, 2009), which was based on a tensorial characterization of the Kerr solutions, due to Mars (Class. Quant. Grav. 16:2507–2523, 1999), we rely here on Hawking’s original strategy, which is to reduce the case of general stationary space-times to that of stationary and axi-symmetric spacetimes for which the Carter-Robinson uniqueness result holds. In this reduction Hawking had to appeal to analyticity. Using a variant of the geometric Carleman estimates developed in Ionescu and Klainerman (Invent. Math. 175:35–102, 2009) , in this paper we show how to bypass analyticity in the case when the stationary vacuum space-time is a small perturbation of a given Kerr solution. Our perturbation assumption is expressed as a uniform smallness condition on the Mars-Simon tensor. The starting point of our proof is the new local rigidity theorem established in Alexakis et al. (Hawking’s local rigidity theorem without analyticity. http://arxiv.org/abs/0902. 1173v1[gr-qc], 2009). Contents 1. 2.
Introduction . . . . . . . . . . . . . . . . . . 1.1 Precise assumptions and the main theorem Preliminaries . . . . . . . . . . . . . . . . . . 2.1 A system of coordinates along 0 . . . .
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The first author was partially supported by a Clay research fellowship. The second author was partially supported by a Packard Fellowship. The third author was partially supported by NSF grant DMS-0070696.
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2.2 Optical functions in a neighborhood of S0 . . . . . . 2.3 Definitions and asymptotic formulas . . . . . . . . . 3. Analysis on the Hypersurface 1 . . . . . . . . . . . . . 3.1 A Lemma of Mars . . . . . . . . . . . . . . . . . . . 3.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . 4. Properties of the Function y . . . . . . . . . . . . . . . . 4.1 Control of y in a neighborhood of S0 . . . . . . . . . 4.2 Regularity properties of the function y away from S0 4.3 T-conditional pseudo-convexity . . . . . . . . . . . 5. Construction of the Hawking Killing Vector-Field K . . . 5.1 Proof of Proposition 5.4 . . . . . . . . . . . . . . . . 5.2 Proof of Lemma 5.5 . . . . . . . . . . . . . . . . . . 6. Construction of the Rotational Killing Vector-Field Z . . 6.1 The timelike span of the two Killing fields. . . . . . . 6.2 Proof of the Main Theorem . . . . . . . . . . . . . . Appendix A. Asymptotic Identities . . . . . . . . . . . . . . .
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1. Introduction It is widely expected1 that the domains of outer communications of regular, stationary, four dimensional, vacuum black hole solutions are isometrically diffeomorphic to those of the Kerr black holes. Due to gravitational radiation, general, asymptotically flat, dynamic, solutions of the Einstein-vacuum equations ought to settle down, asymptotically, into a stationary regime. Thus the conjecture, if true, would characterize all possible asymptotic states of the general vacuum evolution. A similar scenario is supposed to hold true in the presence of matter. So far the conjecture is known to be true2 if, besides reasonable geometric and physical conditions, one assumes that the space-time metric in the domain of outer communications is real analytic. This last assumption is particularly restrictive, since there is apriori no reason that general stationary solutions of the Einstein field equations should be analytic in the ergoregion, i.e. the region where the stationary Killing vector-field becomes space-like. Hawking’s proof starts with the observation that the event horizon of a general stationary metric is non-expanding and the stationary Killing field must be tangent to it. Specializing to the future event horizon H+ , Hawking [21] (see also [26]) proved the existence of a non-vanishing vector-field K tangent to the null generators of H+ and Killing to any order along H+ . Under the assumption of real analyticity of the space-time metric one can prove, by a Cauchy-Kowalewski type argument (see [21] and the rigorous argument in [13]), that the Hawking Killing vector-field K can be extended to a neighborhood of the entire domain of outer communications. Thus, it follows, that the spacetime (M, g) is not just stationary but also axi-symmetric. To derive uniqueness, one then appeals to the theorem of Carter and Robinson which shows that the exterior region of a regular, stationary, axi-symmetric vacuum black hole must be isometrically diffeomorphic to a Kerr exterior of mass M and angular momentum a < M. The proof of this result originally obtained by Carter [7] and Robinson [31], has been strengthened and extended by many authors, notably Mazur [30], Bunting [5], Weinstein [35]; the 1 See reviews by B. Carter [9] and P. Chrusciel [12] for a history and review of the current status of the conjecture. 2 By combining results of Hawking [21], Carter [7], and Robinson [31], see also the recent work of Chrusciel-Costa [17].
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most recent and complete account, which fills in various gaps in the previous literature is the recent paper of Chrusciel and Costa [17], see also [18]. A clear and complete exposition of the ideas that come into the proof can be found in Heusler’s book, [22]. We remark that the Carter-Robinson theorem does not require analyticity. In [24] a different strategy was followed based on the tensorial characterization of the Kerr spaces, due to Mars [29] and Simon [33], and a new analytic framework based on Carleman estimates. Uniqueness of Kerr was proved for a general class of regular stationary vacuum space-times which verify a complex scalar identity along the bifurcation sphere of the horizon. Unfortunately, to eliminate this local assumption, one needs a global argument which has alluded us so far. In this paper we return to Hawking’s original strategy and show how to extend his Killing vector-field, and thus axial symmetry, from the horizon to the entire domain of outer communications, without appealing to analyticity. As noted above, once axial symmetry is extended to the entire exterior region, the Carter-Robinson theorem applies and proves that the domain of outer communications must be isometric to a Kerr exterior. Our argument, which relies on (and in fact simplifies) the Carleman estimates developed in [24] and [25], and their extensions in [1] and [2], requires a smallness assumption which is expressed, geometrically, by assuming that the Mars-Simon tensor of our stationary metric is uniformly bounded by a sufficiently small constant. Our main result is therefore perturbative; we show that any regular stationary vacuum solution which is sufficiently close to a Kerr solution K(a, m), 0 ≤ a < m must in fact coincide with it. We hope that the present theorem will play an important role in understanding the asymptotic approach to final states of non-stationary space-times which arise by the evolution of initial data close enough to the initial data of Kerr space-times. The first step of our approach has already been presented by us in [2]. There we show, under very general assumptions, how to construct the Hawking Killing vector-field in a neighborhood of a non-expanding, smooth, bifurcate horizon. The main idea, which also plays an essential role in this paper, is to turn the problem of extension into one of unique continuation, relying on Carleman estimates for systems of wave equations coupled to ordinary differential equations; see the Introduction in [2] for an informal discussion. In this paper we complete the second step of our approach, which is to extend Hawking’s vector-field to the entire domain of outer communications. For this we make the assumption that the Mars–Simon tensor S is uniformly sufficiently small along a Cauchy hypersurface in the domain of outer communications (recall that the Kerr spaces are locally characterized by the vanishing of the tensor S, see [29]). Using this smallness assumption and asymptotic flatness we are able to gain sufficient control on the geometry of the domain of outer communications. In particular, we show that the foliation given by the level sets of the function y, the real part of (1 − σ )−1 , where σ is the complex Ernst potential associated to the stationary vector-field T (see Subsect. 2.3 for definitions), is regular and satisfies a crucial T-conditional pseudo-convexity property (see Lemma 4.3). The Carleman estimates on which our extension argument is based depend on these two properties, which were previously established if S vanishes identically (see [24] and [25]). As part of our analysis we show that these properties are stable if the tensor S is sufficiently small. To summarize, the main steps of our construction are (1) A robust argument by which the problem of extension of Killing vector-fields is turned into a uniqueness problem across pseudo-convex hypersurfaces for an illposed system of covariant wave equations coupled to transport equations.
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(2) A local construction of Hawking’s Killing vector-field in a neighborhood of the bifurcate horizon. (3) A global argument by which the Hawking vector-field is extended to the entire domain of outer communications. This step, which rests on delicate control of the geometry of the domain of outer communications, requires our global smallness assumption for the Mars-Simon tensor S. The first two steps in this construction, which were accomplished in [2], are unconditional, in the sense that they do not require the smallness assumption for S. In this paper we present the third step. As mentioned before, the first key issue is to construct a regular T-conditional pseudo-convex global foliation of the domain of outer communications. This requires both local information, provided by the smallness of the tensor S, and non-local information, provided by the asymptotic flatness assumption of our space-time (Sects. 3 and 4). Then, in Sect. 5, we expand the argument described in step (1) to include extensions of Killing vector-fields across T-conditional pseudo-convex hypersurfaces (using the main Carleman estimate proved in [24]), instead of the more restrictive case of pseudo-convex hypersurfaces used in the local argument in [2]. 1.1. Precise assumptions and the main theorem. We assume that (M, g) is a smooth3 vacuum Einstein spacetime of dimension 3 + 1 and T ∈ T(M) is a smooth Killing vector-field on M. We also assume that we are given an embedded partial Cauchy surface 0 ⊆ M and a diffeomorphism 0 : E 1/2 → 0 , where Er = {x ∈ R3 : |x| > r }. We group our main assumptions4 in three categories. The first assumption is a standard asymptotic flatness assumption which, in particular, defines the asymptotic region M(end) and the domain of outer communications (exterior region) E = I − (M(end) ) ∩ I + (M(end) ). Our second assumption concerns the smoothness of the two achronal boundaries δ(I − (M(end) )) in a small neighborhood of their intersection S0 = δ(I − (M(end) ))∩ δ(I + (M(end) )). Our third assumption asserts that the Mars-Simon tensor S, whose vanishing characterizes Kerr space-times, is small. GR (Global Regularity assumption). We assume that the restriction of the diffeomorphism 0 to E R0 , for R0 sufficiently large, extends to a diffeomorphism 0 : R × E R0 → M(end) , where M(end) (asymptotic region) is an open subset of M. In local coordinates {x 0 , x i } defined by this diffeomorphism, we assume that T = ∂0 and, with r = (x 1 )2 + (x 2 )2 + (x 3 )2 , that the components of the spacetime metric verify,5 g00 = −1 +
2S j x k 2M + O6 (r −2 ), gi j = δi j + O6 (r −1 ), g0i = −i jk + O6 (r −3 ), r r3 (1.1)
for some M > 0, S 1 , S 2 , S 3 ∈ R (see [3]) such that, J = [(S 1 )2 + (S 2 )2 + (S 3 )2 ]1/2 ∈ [0, M 2 ).
(1.2)
3 M is a Hausdorff, connected, oriented, time oriented, paracompact C ∞ manifold without boundary. 4 Many of these assumptions can be justified as consequences of more primitive assumptions, see [3,4,
11,15,16,19,20,32]. For the sake of simplicity, we do not attempt to work here under general regularity assumptions. See, however, the recent paper [17] for a careful discussion. 5 We denote by O (r a ) any smooth function in M(end) which verifies |∂ i f | = O(r a−i ) for any 0 ≤ i ≤ k k i i i i with |∂ i f | = i 0 +i 1 +i 2 +i 3 =i |∂00 ∂11 ∂22 ∂33 f |.
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Let E = I − (M(end) ) ∩ I + (M(end) ), where I − (M(end) ), I + (M(end) ) denote the past and respectively future sets of M(end) . We assume that E is globally hyperbolic and 0 ∩ I − (M(end) ) = 0 ∩ I + (M(end) ) = 0 (E 1 ).
(1.3)
We assume that T does not vanish at any point of E and that every orbit of T in E is complete and intersects transversally the hypersurface 0 . SBS (Smooth Bifurcation Sphere assumption). It follows from (1.3) that δ(I − (M(end) )) ∩ 0 = δ(I + (M(end) )) ∩ 0 = S0 ,
(1.4)
where S0 = 0 ({x ∈ R3 : |x| = 1}) is an embedded 2-sphere (called the bifurcation sphere). We assume that there is a neighborhood O of S0 in M such that the sets H+ = O ∩ δ(I − (M(end) ))
and
H− = O ∩ δ(I + (M(end) ))
are smooth embedded hypersurfaces. We assume that these hypersurfaces are null, nonexpanding6 , and intersect transversally in S0 . Finally, we assume that the vector-field T is tangent to both hypersurfaces H+ and H− , and does not vanish identically on S0 . PK (Perturbation of Kerr assumption). Let σ denote the Ernst potential and S the Mars-Simon tensor, defined in an open neighborhood of 0 ∩ E in M (see Sect. 2 for precise definitions). We assume that |(1 − σ )S(T, Tα , Tβ , Tγ )| ≤ ε
on 0 ∩ E
for any α, β, γ ∈ {0, 1, 2, 3}, (1.5)
for some sufficiently small constant ε (depending only on the constant A defined in Sect. 2), where T0 is the future-directed unit vector orthogonal to 0 and T0 , T1 , T2 , T3 is an orthonormal basis along 0 . Main Theorem. Under the assumptions GR, SBS, and PK the domain of outer communications E of M is isometric to the domain of outer communications of the Kerr space-time with mass M and angular momentum J . In other words, the domain of outer communications of a stationary vacuum black hole, which satisfies suitable regularity assumptions and is sufficiently “close” to the domain of outer communications of the Kerr solution K(M, a), a < M, has to be isometric to it. This can be interpreted as a strong extension of Carter’s original theorem, see [7,8], on stationary and axi-symmetric perturbations of the Kerr spaces, in which we remove the axi-symmetry assumption and give a geometric, coordinate independent, perturbation condition. We provide below a more detailed outline of the proof of the Main Theorem. In Sect. 2 we define a system of local coordinates along our reference space-like hypersurface 0 , and define our main constant A. The small constant ε in (1.5) is to be taken sufficiently small, depending only on A. We review also the construction of two optical functions u and u in a neighborhood of the bifurcation sphere S0 , adapted to the null hypersurfaces H+ and H− , and recall the definition of the complex Ernst potential 6 A null hypersurface is said to be non-expanding if the trace of its null second fundamental form vanishes identically.
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σ and the Mars-Simon tensor S. Finally, we record some asymptotic formulas, which are proved in the Appendix. In Sect. 3 we develop the main consequences of our smallness assumption (1.5). All of our results in this section are summarized in Proposition 3.4; we prove a lower bound on |1 − σ | along 1 , as well as several approximate identities in a small neighborhood of 1 which are used in the rest of the paper. In Sect. 4 we derive several properties of the function y needed in the continuation argument in Sect. 5. We prove first that y is almost constant on S0 , as in Lemma 4.1, and increases in a controlled way in a neighborhood of S0 in 1 . Then we prove that the level sets of the function y away from S0 are regular, in a suitable sense. Finally, we prove that the function y satisfies the T-conditional pseudo-convexity property, away from S0 , see Lemma 4.3. In Sect. 5 we construct the Hawking Killing vector-field K in the domain of outer communications E. The starting point is the existence of K in a neighborhood of S0 , which was proved in [2]. We extend K to larger and larger regions, as measured by the function y, as the solution of an ordinary differential equation, see Lemma 5.3. We then prove that the resulting vector-field K is Killing (and satisfies several other bootstrap conditions) as a consequence of a uniqueness property of stationary vacuum solutions, see Proposition 5.4. The proof of this last proposition relies on Carleman estimates and the properties of the function y proved in Sect. 4. In Sect. 6 we construct a global, rotational Killing vector-field Z which commutes with T, as a linear combination of the vector-fields T and K. We also give a simple proof, specialized to our setting, that the span of the two Killing fields T, Z is timelike in E (thus the area function is nonnegative in E which is an important component of the proof of the Carter–Robinson theorem, as explained in [17]). 2. Preliminaries 2.1. A system of coordinates along 0 . Let ∂1 , ∂2 , ∂3 denote the vectors tangent to 0 , induced by the diffeomorphism 0 . Let r = 0 (Er ), where, as before, Er = {x ∈ R3 : |x| > r }. In particular, for our original spacelike hypersurface, we have 0 = 1/2 . Using (1.1) and the assumption that 0 is spacelike, it follows that there are large constants A1 and R1 ≥ R0 , such that R1 ≥ A41 , with the following properties: on 3/4 , for any X = (X 1 , X 2 , X 3 ), 3
2 A−1 1 |X | ≤
X α X β gαβ ≤ A1 |X |2
and
α,β=1
3
|g(∂α , T)| + |g(T0 , T)| ≤ A1 .
α=1
(2.1) In 0 (R × E R1 ), which we continue to denote by M(end) , T = ∂0 and (see notation in footnote 5), 6
r m+1
m=0
+
3
|∂ m (g jk − δ jk )| +
j,k=1 6
m=0
r m+3
3 i=1
6
r m+2 |∂ m (g00 + 1 − 2M/r )|
m=0
|∂ m (g0i + 2i jk S j x k r −3 )| ≤ A1 .
(2.2)
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95
of 0 ∩ E, which We construct a system of coordinates in a small neighborhood M (end) extends both the coordinate system of M in (2.2) and that of 0 . We do that with the help of a smooth vector-field T which interpolates between T and T0 . More precisely we construct T in a neighborhood of 3/4 such that T = T in 0 (R × E 2R1 ) and T = η(r/R1 )T0 + (1 − η(r/R1 ))T on 3/4 , where η : R → [0, 1] is a smooth function supported in (−∞, 2] and equal to 1 in (−∞, 1]. Using now the flow induced by T we extend the original diffeomorphism 0 : E 1/2 → 0 , to cover a full neighborhood of 1 . Thus there exists ε0 > 0 sufficiently small and a diffeomorphism 1 : which agrees with 0 on {0} × E 1−ε0 ∪ (−ε0 , ε0 ) × E 2R1 (−ε0 , ε0 ) × E 1−ε0 → M, and such that ∂0 = ∂x 0 = T . By setting ε0 small enough, we may assume that Oε0 := 1 ((−ε0 , ε0 ) × {x ∈ R3 : |x| ∈ (1 − ε0 , 1 + ε0 )}) ⊆ O, where O is the open set defined in the assumption SBS. By construction, using also (2.2) and letting ε0 sufficiently small depending on R1 , 3
|g0 j | + |g00 + 1| ≤ A1 /(R1 + r )
in M.
(2.3)
j=1
With gαβ = g(∂α , ∂β ) and T = Tα ∂α , let ⎤ ⎡ 6 3 3 ⎣ A2 = sup |∂ m gαβ ( p)| + |∂ m Tα ( p)|⎦ . m=0 p∈M
α,β=0
(2.4)
α=0
Finally, we fix A = max(R1 , A2 , ε0−1 , (M 2 − J )−1 ).
(2.5)
The constant A is our main effective constant. The constant ε in (1.5) will be fixed sufficiently small, depending only on A. To summarize, we defined a neighborhood M ε0 > 0, such that the of 0 ∩ E and a diffeomorphism 1 : (−ε0 , ε0 ) × E 1−ε0 → M, bounds (2.1), (2.2), (2.3), (2.4) hold (in coordinates induced by the diffeomorphism 1 ). 2.2. Optical functions in a neighborhood of S0 . We define two optical functions u, u in a neighborhood of S0 . We fix a smooth future-directed null pair (L , L) along S0 , satisfying g(L , L) = g( L, L) = 0,
g(L , L) = −1,
(2.6)
such that L is tangent to H+ and L is tangent to H− . In a small neighborhood of S0 , we extend L (resp. L) along the null geodesic generators of H+ (resp. H− ) by parallel transport, i.e. D L L = 0 (resp. D L L = 0). We define the function u (resp. u) along H+ (resp. H− ) by setting u = u = 0 on S0 and solving L(u) = 1 (resp. L(u) = 1). Let Su (resp. S u ) be the level surfaces of u (resp. u) along H+ (resp. H− ). We define L at every point of H+ (resp. L at every point of H− ) as the unique, future directed null vector-field orthogonal to the surface Su (resp. S u ) passing through that point and such that g(L , L) = −1. We now define the null hypersurface Hu− to be the congruence of null geodesics initiating on Su ⊂ H+ in the direction of L. Similarly we define Hu+ to be the congruence of null geodesics initiating on S u ⊂ H− in the direction of L. Both congruences are well defined in a sufficiently small neighborhood of S0 in O. The null
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hypersurfaces Hu− (resp. Hu+ ) are the level sets of a function u (resp. u) vanishing on H− (resp. H+ ). By construction L = −gμν ∂μ u∂ν ,
L = −gμν ∂μ u∂ν .
(2.7)
In particular, the functions u, u are both null optical functions, i.e. gμν ∂μ u∂ν u = g(L , L) = 0
and
gμν ∂μ u∂ν u = g( L, L) = 0.
(2.8)
To summarize, there is c0 = c0 (A) ∈ (0, ε0 ] sufficiently small and smooth optical functions u, u : Oc0 → R, where Oc0 = 1 ((−c0 , c0 ) × {x ∈ R3 : |x| ∈ (1 − c0 , 1 + c0 )}). In local coordinates induced by the diffeomorphism 1 we have7 sup
4
= C(A). (|∂ j u(x)| + |∂ j u(x)|) ≤ C
(2.9)
x∈Oc0 j=0
In addition, H+ ∩ Oc0 = { p ∈ Oc0 : u( p) = 0},
H− ∩ Oc0 = { p ∈ Oc0 : u( p) = 0}.
(2.10)
In Oc0 we define = gμν ∂μ u∂ν u = g(L , L). By construction = −1 on (H+ ∪ H− ) ∩ Oc0 (we remark, however, that is not necessarily equal to −1 in Oc0 ). By taking c0 small enough, we may assume that ∈ [−3/2, −1/2]
in Oc0 .
(2.11)
Finally, by construction, we may assume that the functions |u|, |u| are proportional to |1 − r | on the spacelike hypersurface 0 ∩ Oc0 , i.e. −1 , C] |u/(1 − r )|, |u/(1 − r )| ∈ [C
on 0 ∩ Oc0 ,
(2.12)
is a constant that depends only on A. where, as in (2.9), C 2.3. Definitions and asymptotic formulas. We recall now the definitions of the Ernst potential σ and the Mars–Simon tensor S (see [24, Sect. 4] for a longer discussion and proofs of all of the identities). In M we define the 2-form, Fαβ = Dα Tβ , and the complex valued 2-form, Fαβ = Fαβ + i ∗ F αβ = Fαβ + (i/2)∈αβ μν Fμν .
(2.13)
Let F 2 = Fαβ F αβ . We define also the Ernst 1-form σμ = 2Tα Fαμ = Dμ (−Tα Tα ) − i ∈μβγ δ Tβ Dγ Tδ .
(2.14)
j0 j1 j2 j3 7 Recall the notation |∂ j f | = j0 + j1 + j2 + j3 = j |∂0 ∂1 ∂2 ∂3 f |, where ∂0 , ∂1 , ∂2 , ∂3 are the derivatives induced by the diffeomorphism 1 . This notation will be used throughout the paper.
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It is easy to check that, in M ⎧ ⎪ ⎨Dμ σν − Dν σμ = 0; Dμ σμ = −F 2 ; ⎪ ⎩σ σ μ = g(T, T)F 2 . μ
(2.15)
= 1 ((−ε0 , ε0 ) × E 1−ε0 ) is simply connected, we Since Dμ σν = Dν σμ and the set M → C such that σμ = Dμ σ, σ = −Tα Tα , and can define the Ernst potential σ : M 0 σ → 1 at infinity along . We define the complex-valued self-dual Weyl tensor Rαβμν = Rαβμν + (i/2)∈μν ρσ Rαβρσ = Rαβμν + i ∗ R αβμν .
(2.16)
We define the tensor I ∈ T04 (M), Iαβμν = (gαμ gβν − gαν gβμ + i ∈αβμν )/4. = {p ∈ M : σ ( p) = 1}.8 We define the tensor-field Q ∈ T0 (M ), Let M 4 1 Qαβμν = (1 − σ )−1 Fαβ Fμν − F 2 Iαβμν . 3
(2.17)
(2.18)
It is easy to see that the tensor-field Q is a self-dual Weyl field, i.e. ⎧ ⎪ ⎨Qαβμν = −Qβαμν = −Qαβνμ = Qμναβ ; Qαβμν + Qαμνβ + Qανβμ = 0; ⎪ ⎩gβν Q αβμν = 0, and ∗
Q αβμν =
1 ∈μνρσ Qαβ ρσ = (−i)Qαβμν . 2
We define now the self-dual Weyl field S, called the Mars–Simon tensor, S = R + 6Q.
(2.19)
Using the Ricci identity We observe that (1 − σ )S is a smooth tensor on M. Dμ Fαβ = Tν Rνμαβ ,
(2.20)
, and proceeding as in [24, formula 4.33], we deduce that, in M Dρ [F 2 (1 − σ )−4 ] = 2(1 − σ )−4 Tν Sνργ δ F γ δ .
(2.21)
This identity will play a key role in the analysis in Sect. 3. Finally, we define the functions → R, y, z : M y + i z = (1 − σ )−1 . 8 Using the assumption PK, we will prove in Sect. 3 that ⊆ M . 1
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Simple asymptotic computations using the formula (2.2), see Appendix A, show that, for R sufficiently large depending only on A, Fαβ = O(r −2 ), Sαβγ δ = O(r −3 ), α, β, γ , δ = 0, . . . , 3.
(2.22)
More precisely, 1 − σ = 2Mr −1 + O(r −2 ),
F 2 = −4M 2 r −4 + O(r −5 )
(2.23)
and in 1 ((−ε0 , ε0 ) × E R ). In particular, 1 ((−ε0 , ε0 ) × E R ) ⊆ M − 4M 2 F 2 (1 − σ )−4 = 1 + O(r −1 )
in 1 ((−ε0 , ε0 ) × E R ).
(2.24)
In addition y=
r + O(1), 2M
z=
S1 x 1 + S2 x 2 + S3 x 3 + O(r −1 ) 2M 2 r
(2.25)
in 1 ((−ε0 , ε0 ) × E R ). Finally, z 2 + 4M 2 (y 2 + z 2 )Dμ zDμ z =
J2 + O(r −1 ) 4M 4
in 1 ((−ε0 , ε0 ) × E R ).
(2.26)
All these asymptotic identities are proved in Appendix A and will be used in Sect. 3. 3. Analysis on the Hypersurface 1 In this section we use assumption PK to prove several approximate identities on the hypersurface 1 = 0 ∩ E = 0 (E 1 ). The general idea is to prove approximate identities such as (2.24) and (2.26) first in the asymptotic region, using the asymptotic flatness assumption (2.2), and then extend them to the entire hypersurface 1 using the fact that the Mars–Simon tensor is assumed to be small. We will prove also that 1 − σ does not vanish in 1 . All of our results in this section are summarized in Proposition 3.4. We will use the notation in Sect. 2. We fix first a large constant R which depends only on our main constant A (see (2.5)). We fix r0 the smallest number in [1, R − 1] with the property that |1 − σ | ≥ R
−2
on r0 \ R ,
(3.1)
where, as before, r = 0 (Er ), Er = {x ∈ R3 : |x| > r }. Such an r0 exists if R is sufficiently large, in view of (2.23) and the continuity of 1 − σ . We will prove, among denote various constants in [1, ∞) that other things, that r0 = 1. In this section we let C may depend only on R (thus on A once R is fixed sufficiently large depending on A). The value of ε in (1.5) is assumed to be sufficiently small depending on the constants To summarize, log(A) log(R) log(C) log(ε −1 ). C. it follows We will work in the region 1 [(−ε, ε) × Er0 ]. Since |∂0 (1 − σ )| ≤ C, from (3.1) and (2.23) that |1 − σ |−1 ≤ Cr
in 1 [(−ε, ε) × Er0 ].
(3.2)
Using the assumption (1.5) and the asymptotic identities (2.22), we have min(ε, r −4 ) |(1 − σ )Tν Sνργ δ | ≤ C
in 1 [(−ε, ε) × Er0 ],
(3.3)
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in the coordinate frame ∂0 , ∂1 , ∂2 , ∂3 . Using (2.21), (2.22) and the last two inequalities, it follows that 3 min(ε, r −4 ) |∂ρ (F 2 (1 − σ )−4 )| ≤ Cr
in 1 [(−ε, ε) × Er0 ].
(3.4)
We prove now that min(r −1 , ε 1/5 ) |1 + 4M 2 F 2 (1 − σ )−4 | ≤ C
in 1 [(−ε, ε) × Er0 ].
(3.5)
Indeed, let H = 1 + 4M 2 F 2 (1 − σ )−4 . Using (2.24) and (3.4), −1 |H | ≤ Cr
4
and
3 min(ε, r −4 ) |∂ρ H | ≤ Cr
in 1 [(−ε, ε) × Er0 ].
(3.6)
ρ=0
The bound (3.5) follows from the first inequality in (3.6) at points p for which r ( p) ≥ ε −1/5 . To prove (3.5) at points p with r ( p) ≤ ε−1/5 we fix a point p ∈ 1 [(−ε, ε)× Er0 ] with r ( p ) = ε−1/5 . We integrate along a line joining the points p and p and use the −4/5 ε, which gives (3.5) second inequality in (3.6). The result is |H ( p) − H ( p )| ≤ Cε ( p )−1 = Cε 1/5 . since |H ( p )| ≤ Cr It follows from (3.4) and (3.5) that there is a smooth function G 1 : 1 [(−ε, ε) × Er0 ] → C with the properties −4M 2 F 2 = (1 − σ )4 (1 + G 1 )2 ,
|G 1 | +
3
min(r −1 , ε 1/5 ) |∂ρ G 1 | ≤ C
(3.7)
ρ=0
on 1 [(−ε, ε) × Er0 ]. In particular, using also (3.2), )−4 |F 2 | ≥ (Cr
in 1 [(−ε, ε) × Er0 ].
(3.8)
We define the smooth function P = y + i z : 1 [(−ε, ε) × Er0 ] → C, P = y + i z = (1 − σ )−1 .
(3.9)
We construct now a special null pair, similar to the principal null pair in [29, Sect. 4]. Lemma 3.1. There exists a future-directed null pair l, l, g(l, l) = −1, such that Fαβ l β = (1 + G 1 )(4M P 2 )−1lα ,
Fαβ l β = −(1 + G 1 )(4M P 2 )−1l α ,
(3.10)
in 1 [(−ε, ε) × Er0 ]. Proof of Lemma 3.1. Let Z α be complex eigenvector Fαβ Z β = λZ α with complex eigenvalue λ. Using the relation Fασ Fβ σ = (1/4)gαβ F 2 (see [24, formula 4.2]) we derive, 1 1 (1 − σ )4 (1 + G 1 )2 . λ2 = − F 2 = 4 16M 2 Thus, λ = ±(4M P 2 )−1 (1 + G 1 ). The reality of the corresponding eigenvectors l, l is a consequence of the self duality of F. They must both be null in view of the antisymmetry of F and can be normalized appropriately.
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Let e(3) = l, e(4) = l. We fix vector-fields e(1) , e(2) in 1 [(−ε, ε) × Er0 ] such that together with e(3) = l, e(4) = l they form a positively oriented null frame, i.e., g(l, e(1) ) = g(l, e(2) ) = g(l, e(1) ) = g(l, e(2) ) = g(e(1) , e(2) ) = 0, g(e(1) , e(1) ) = g(e(2) , e(2) ) =∈(1)(2)(3)(4) = 1.
(3.11)
α ∂ , μ = 1, 2, 3, 4 can be chosen such that In view of (3.8), the vector-fields e(μ) = e(μ) α 4 3 μ=1 α=0
α in 1 [(−ε.ε) × Er0 ]. |e(μ) |≤C
(3.12)
According to (3.10), (3.11) and the self duality of F, the components of F are, F(4)(1) = F(4)(2) = F(3)(1) = F(3)(2) = 0
and
F(4)(3) = iF(2)(1) = (1 + G 1 )/(4M P 2 ). (3.13)
This is equivalent to the identity, Fαβ =
1 + G1 μ ν −l . l + l l − i ∈ l l α β αβμν β α 4M P 2
(3.14)
By contracting (3.14) with 2Tα and using 2Tα Fαβ = σβ = Dβ σ we derive Dβ (y + i z) =
1 + G1 −(Tα lα )l β + (Tα l α )lβ − i ∈αβμν Tα l μl ν . 2M
In particular, if G 1 = G 1 + iG 1 , we have G 1 + G 1 1 −(Tα lα )l β + (Tα l α )lβ + ∈αβμν Tα l μl ν , 2M 2M 1 + G 1 G 1 ∈αβμν Tα l μl ν + −(Tα lα )l β + (Tα l α )lβ . Dβ z = − 2M 2M
Dβ y =
(3.15)
It follows from (3.15) and (3.7) that min(r −1 , ε 1/5 ) |D(1) y| + |D(2) y| + |D(3) z| + |D(4) z| ≤ C
in 1 [(−ε, ε) × Er0 ]. (3.16)
A direct computation using the definition of P and (2.15) shows that Dα PDα P =
Dα σ Dα σ (1 + G 1 )2 Tα Tα = − . (1 − σ )4 4M 2
(3.17)
Since −Tα Tα = σ = 1 − y/(y 2 + z 2 ) we have y (1 + G 1 )2 − (G 1 )2 1− 2 , Dα yD y − Dα zD z = 4M 2 y + z2 y (1 + G 1 )G 1 Dα yDα z = 1− 2 . 2 4M y + z2 α
α
(3.18)
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3.1. A Lemma of Mars. The following lemma is an adaptation to our situation of an important calculation which first appears in [29]. Lemma 3.2. With B = J 2 /(4M 4 ) < 1/4 we have in 1 [(−ε, ε) × Er0 ], min(r −1 , ε 1/40 ). |4M 2 (y 2 + z 2 )Dβ zDβ z + z 2 − B| ≤ C
(3.19)
Proof of Lemma 3.2. We show that the function H := 4M 2 (y 2 + z 2 )Dβ zDβ z + z 2 is almost constant by computing its derivatives with respect to our null frame (3.11). In the particular case S = 0, the constancy of H was first proved in [29] using the full Newman-Penrose formalism. A similar proof was later given in [24]. Here we give instead a straightforward proof based only on the formulas we have derived so far. We will prove that 3
1/20 |∂α H ( p)| ≤ Cε
if p ∈ 1 [(−ε, ε) × Er0 ]
and
r ( p) ≤ ε −1/40 . (3.20)
α=0
−1 in 1 [(−ε, ε)× Assuming this, the bound (3.19) follows from the bound |H −B| ≤ Cr Er0 ], see (2.26), in the same way the bound (3.5) follows from (3.4) and (2.24). We differentiate H and derive, Dα H = 8M 2 (y 2 + z 2 )Dα Dβ zDβ z + 8M 2 (yDα y + zDα z)Dβ zDβ z + 2zDα z.
(3.21)
To calculate the main term 8M 2 (y 2 + z 2 )Dα Dβ zDβ z we first calculate the second covariant derivatives of P = (y + i z), using the definition of S and (2.20), Dα Dβ P = 2(1 − σ )−3 Dα σ Dβ σ + (1 − σ )−2 Dα Dβ σ = 2(1 − σ )−3 Dα σ Dβ σ + 2(1 − σ )−2 (Fα ρ Fρβ + Tρ Tν Rναρβ ) = 2(1 − σ )−3 Dα σ Dβ σ + 2(1 − σ )−2 Fα ρ Fρβ − 12(1 − σ )−2 Tρ Tν Qναρβ + 2(1 − σ )−2 Tρ Tν Sναρβ = 2(1 − σ )−2 Fα ρ Fρβ + 2(1 − σ )−2 Tρ Tν Sναρβ − (1 − σ )−3 σα σβ + (1 − σ )−3 F 2 [gαβ (Tρ Tρ ) − Tα Tβ ] = 2P 2 Fα ρ Fρβ + P −1 [(Dρ PDρ P)gαβ − Dα PDβ P] + 2P 2 Tρ Tν Sναρβ − P 3 F 2 Tα Tβ . Thus, we have the identity Dα Dβ P = −P −1 Dα PDβ P + P −1 (Dρ PDρ P)gαβ −2P 2 Fα ρ Fβρ − P 3 F 2 Tα Tβ + 2P 2 Tρ Tν Sναρβ .
(3.22)
Since T(z) = 0 we deduce, Dα Dβ PDβ z = P −1 (Dρ PDρ P)Dα z − 2P 2 Fα ρ Fβρ Dβ z −P −1 Dα PDβ PDβ z + 2P 2 Tρ Tν Sναρβ Dβ z.
(3.23)
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Observe that Dα Dβ zDβ z = Dα Dβ PDβ z . Thus, in view of (3.23), Dα Dβ zDβ z = −2P 2 Fα ρ Fβρ Dβ z + 2P 2 Tρ Tν Sναρβ Dβ z − P −1 Dα PDβ PDβ z + P −1 (Dρ PDρ P)Dα z .
(3.24)
Now,
(y 2 + z 2 ) P −1 Dα PDβ P Dβ z = Dβ z[(y − i z)Dα (y + i z)Dβ (y + i z)] = (yDα y + zDα z)Dβ zDβ z + (yDα z −zDα y)Dβ yDβ z and,
(y 2 + z 2 ) P −1 Dρ PDρ P Dα z = Dα z[[(y − i z)Dρ (y + i z)Dρ (y + i z)] = 2yDα zDρ yDρ z − zDα z(Dρ yDρ y − Dρ zDρ z).
Therefore, back to (3.24),
(y 2 + z 2 )Dα Dβ zDβ z = (y 2 + z 2 ) −2P 2 Fα ρ Fβρ Dβ z + 2P 2 Tρ Tν Sναρβ Dβ z − yDα y Dρ zDρ z + (yDα z + zDα y) Dρ yDρ z − zDα z Dρ yDρ y.
Going back to (3.21) we derive, Dα H = 8M 2 (y 2 + z 2 ) −2P 2 Fα ρ Fβρ Dβ z + 2P 2 Tρ Tν Sναρβ Dβ z + 8M 2 zDα z(Dρ zDρ z − Dρ yDρ y) + 2zDα z + 8M 2 (yDα z + zDα y) Dρ yDρ z. Recall that we are looking to prove (3.20) at points p with r ( p) ≤ ε −1/40 . In view of (3.2) and (3.3), at such points we have 16M 2 (y 2 + z 2 )Dβ z P 2 Tρ Tν Sναρβ = O1 (ε), where, for simplicity of notation, in this lemma we let O1 (ε) denote any quantity bounded 1/20 . According to (3.7) and (3.18) we also have, by Cε y 1 1/5 1 − |Dρ yDρ z| + Dρ zDρ z − Dρ yDρ y + ≤ Cε . 4M 2 y2 + z2 Thus, using again (3.2), 2yzDα z 8M 2 zDα z Dρ zDρ z − Dρ yDρ y + 2zDa z = 2 + O1 (ε). y + z2 Consequently, 2yzD z α + O1 (ε). Dα H = −16M 2 (y 2 + z 2 ) P 2 Fα ρ Fβρ Dβ z + 2 y + z2 For (3.20) it only remains to check that, 2yzD z α = O1 (ε). − 16M 2 (y 2 + z 2 ) P 2 Fα ρ Fβρ Dβ z + 2 y + z2
(3.25)
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103
We prove this in the null frame e(1) , e(2) , e(3) , e(4) , see (3.11). Recalling (3.13) and (3.16) 1/20 for α = 3, 4. For α = 1, we easily see that both terms on the left are bounded by Cε using (3.13) and (3.7), −16M 2 (y 2 + z 2 ) P 2 F(1) (ρ) F(β)(ρ) D(β) z = −16M 2 (y 2 + z 2 ) P 2 F(1)(2) F(1)(2) D(1) z i 2yz + O1 (ε). = −16M 2 (y 2 + z 2 )D(1) z P 2 4M P 2 4M(y 2 + z 2 )2 The approximate identity (3.25) follows for α = 1. The proof of (3.25) for α = 2 is similar, which completes the proof of the lemma. 3.2. Conclusions. It follows from (3.18) and Lemma 3.2, that Dβ zDβ z =
B − z2 y2 − y + B β D yD y = + O(ε), + O(ε) (3.26) β 4M 2 (y 2 + z 2 ) 4M 2 (y 2 + z 2 )
in 1 [(−ε, ε)× Er0 ], where O(ε) denotes functions on 1 [(−ε, ε)× Er0 ] dominated by min(r −1 , ε 1/40 ). Using (3.15) we deduce that Dβ yDβ y = 1 2 (Tα lα )(Tβ l β ) + O(ε). C 2M Hence, (Tα lα )(Tβ l β ) =
y2 − y + B + O(ε) 2(y 2 + z 2 )
in 1 [(−ε, ε) × Er0 ].
(3.27)
We prove now that the value of r0 in (3.1) can be taken to be equal to 1. It view of the definition of r0 , it suffices to prove the following lemma: Lemma 3.3. Assuming R is chosen sufficiently large, we have |1 − σ | ≥ 2R
−2
on r0 \ R .
Proof of Lemma 3.3. The conclusion of the lemma is equivalent to 2
|P| ≤ R /2
on r0 \ R .
(3.28)
To prove this, we recall that we still have the flexibility to fix R sufficiently large depending on A. In view of (2.23), for (3.28) it suffices to prove that |∂α P| ≤ C(A)
on r0 ,
(3.29)
for α = 1, 2, 3 and some constant C(A) that depends only on A. The bound (3.29) follows from (2.22) and (2.23) at points p for which r ( p) ≥ R(A). Since P = (1 − σ )−1 and |1 − σ | ≥ | (1 − σ )| = |1 + Tα Tα |, the bound (3.29) also follows at points p for which r ( p) ≤ R(A) and g p (T, T) ∈ / [−3/2, −1/2]. It remains to prove the bound (3.29) at points p ∈ r0 for which r ( p) ≤ R(A) and g p (T, T) ∈ [−3/2, −1/2]. Since |1 − σ | ≤ C(A) on 1 , we have |y| + |z| ≥ C(A)−1 on 1 . It follows from (3.26) that |Dβ zDβ z| + |Dβ yDβ y| ≤ C(A)
on r0 .
(3.30)
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In addition, T(σ ) = 0, therefore Tα Dα z = Tα Dα y = 0. Since g p (T, T) ∈ [−3/2, −1/2] it follows that T p is timelike, thus the vectors Y α = Dα y and Z α = Dα z are spacelike at the point p. The elliptic bounds (2.1) show, in fact, that 3β=1 |∂α y|2 ≤ C(A)|Dβ yDβ y| and 3β=1 |∂α z|2 ≤ C(A)|Dβ zDβ z|, so (3.29) follows from (3.30). We summarize the main conclusions of our analysis so far in the following proposition: = C(A) Proposition 3.4. There is a constant C sufficiently large such that )−1 |1 − σ | ≥ (Cr
on 1−ε ,
provided that ε is sufficiently small (depending on A). Therefore the frame e(α) , α = 1, . . . , 4, the Mars–Simon tensor S, and the functions P, y, z, G 1 are well defined in 1 [(−ε, ε) × E 1−ε ]. In addition, the identities and inequalities (3.3), (3.7), (3.12), (3.13), (3.15), (3.16), (3.18), (3.22), (3.26), (3.27) hold in 1 [(−ε, ε) × E 1−ε ].
4. Properties of the Function y Our next goal is to understand the behaviour of the function y defined in (3.9) on 1 . Most of our analysis in the next section depends on having sufficiently good information on y, both in a small neighborhood of the bifurcation sphere S0 and away from this to denote various constants small neighborhood.9 In this section we use the notation C in [1, ∞) that may depend only on the main constant A. We assume implicitly that ε −1 is sufficiently large compared to all such constants C. 4.1. Control of y in a neighborhood of S0 . We analyze first the function of y in a neighborhood of the bifurcation sphere S0 . Lemma 4.1. On the bifurcation sphere S0 , |y − (1 +
3 √ 1/40 . 1 − 4B)/2| + |∂α y| ≤ Cε
(4.1)
α=0
1 = C 1 (A) 1 such that Moreover, there are constants r1 = r1 (A) > 1 and C √ 1 ε 1/40 ≥ y −(1 + 1−4B)/2 ≥ C −1 (r −1)2 − C 1 ε1/40 on 1 \r1 . 1 (r −1)2 + C C 1 (4.2) Proof of Lemma 4.1. Recall the vector-fields L , L defined in a neighborhood of S0 in Sect. 2. It is easy to prove, see for example [24, Sect. 5], that Fαβ L β = F(L , L)L α
and
Fαβ L β = F( L, L) L α
on S0 .
Since the vectors l and l constructed in Lemma 3.1 are the unique solutions of the systems of equations (Fαβ ± f gαβ )V β = 0, up to rescaling and relabeling, we may assume 9 For comparison y = r/(2M), in the Kerr space of mass M and angular momentum J , in standard Boyer–Lindquist coordinates.
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105
that L = l and L = l on S0 . Thus, we may also assume that e(1) , e(2) are tangent to S0 . Since T is tangent to S0 , L(σ ) = L β σβ = 2L β Tα Fαβ = 0
on S0 .
Similarly, L(σ ) = 0 on S0 . Using also (3.16), we conclude that e(3) (y) = e(4) (y) = 0
and
1/5 |e(1) (y)| + |e(2) (y)| ≤ Cε
on S0 .
(4.3)
The inequality on the gradient of y in (4.1) follows from (4.3). For the remaining 2/5 on S0 , thus inequality we use first (3.26). It follows from (4.3) that |Dβ yDβ y| ≤ Cε 1/40 |y 2 − y + B| ≤ Cε
on S0 .
Since B = J 2 /(4M 4 ) ∈ [0, 1/4) (see (1.2) and (2.5)), it follows that √ √ 1/40 on S0 or |y −(1− 1−4B)/2| ≤ Cε 1/40 on S0 . |y −(1 + 1 − 4B)/2| ≤ Cε (4.4) To eliminate the second alternative we start by deriving a wave equation for y. Since Dμ Dμ σ = −F 2 , Dμ σ Dμ σ = −F 2 σ (see (2.15)), and −F 2 = (1 − σ )4 (1 + G 1 )2 /(4M 2 ) we derive Dμ Dμ P = (1 − σ )−2 Dμ Dμ σ + 2(1 − σ )−3 Dμ σ Dμ σ =
(1 + G 1 )2 2P − 1 (1 − σ )(1 + σ ) = (1 + G 1 )2 . 4M 2 4M 2 P P
Thus, using (3.7), Dμ Dμ y =
2y − 1 1/5 , + E, |E| ≤ Cε 4M 2 (y 2 + z 2 )
on 1−ε .
(4.5)
We now compare y with a function y which coincides with y on H+ and verifies L(y ) = 0. We use the notation in Sect. 2. For ε1 = ε1 (A) ∈ (0, c0 ] sufficiently small we define the function y : Oε1 → R,
y = y
on H+ ∩ Oε1 ,
L(y ) = 0
in Oε1 .
(4.6)
The functions y and y are smooth on Oε1 , and, using (4.3) and the definition of y , y − y = 0
on (H+ ∪ H− ) ∩ Oε1 .
(4.7)
1/5 on S0 , for all α = 1, 2, 3, 4. In addition, using again (4.3), we infer that |e(α) (y )| ≤ Cε −1 α Using (3.15) e(a) (y ) = (2M) G 1 ∈(α)(a)(μ)(ν) T l μl ν on S0 , a = 1, 2. It follows 1/5 that |D(a) D(a) y | ≤ Cε on S0 , a = 1, 2. from (3.7) and the inequality |e(α) (y )| ≤ Cε 1/5 on Using L(y ) = 0 and |e(α) (y )| ≤ Cε , we have |D(3) D(4) y | + |D(4) D(3) y | ≤ Cε S0 . Therefore, 1/5 |g y | ≤ Cε
on S0 .
(4.8)
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S. Alexakis, A. D. Ionescu, S. Klainerman
In view of (4.7), there is ε2 = ε2 (A) ∈ (0, ε1 ) such that y − y = uu f in Oε2 , where f : Oε2 → R is smooth. Since u = u = 0 and 2Dα uDα u = −2 on S0 , it follows from (4.5) and (4.8) that f = −(1/2)Dα Dα (y − y ) =
1 − 2y 1/5 , + E , |E | ≤ Cε 8M 2 (y 2 + z 2 )
on S0 .
(4.9)
To summarize, y = y + uu f in Oε2 , where f satisfies (4.9) on S0 . We eliminate now the second alternative in (4.4). The main point is that if y is close √ to (1 − 1 − 4B)/2 < 1/2 on S0 then f is strictly positive on S0 (see (4.9)). In quanti−1 in Oε3 . Since uu ≤ 0 on tative terms, there is ε3 =√ε3 (A) ∈ (0, ε2 ) such that f ≥ C 1/40 on Oε3 (using the second alternative in 1 ∩ Oε3 and y ≤ (1 − 1 − 4B)/2 + Cε (4.4) and the construction of y ), it follows that √ −1 |uu| + Cε 1/40 on 1 ∩ Oε3 . y ≤ (1 − 1 − 4B)/2 − C √ −1 at some point in 1 (proIn particular, using (2.12), y ≤ (1 − 1 − 4B)/2 − C vided, of course, that ε is sufficiently small depending on A). Using also (2.25), it follows that the√function y : 1 → R attains its minimum at some point p ∈ 1 , and −1 . Thus T1 (y) = T2 (y) = T3 (y) = 0 at p (where y( p) ≤ (1 − 1 − 4B)/2 − C T0 , T1 , T2 , T3 is the orthonormal frame along 1 defined in assumption PK), and Dα yDα y = −(T0 (y))2 ≤ 0 at p. This however, with the identity (3.26) and the inequality y( p) ≤ √ is in contradiction, −1 . We conclude that the first alternative in (4.1) holds. (1 − 1 − 4B)/2 − C √ We prove now the second statement of the lemma: since y is close to (1+ 1−4B)/2 > −C −1 ] ε3 (A) ∈ (0, ε2 ) such that f ∈ [−C, 1/2 on S0 , it follows from √ (4.9) there is ε3 =1/40 2 −C −1 ] in Oε3 . Also |y − (1 + 1 − 4B)/2| ≤ Cε in Oε3 and uu/(r − 1) ∈ [−C, (see (2.12)). The inequalities in (4.2) follow since y = y + f uu. 4.2. Regularity properties of the function y away from S0 . We derive now the main properties of the level sets of the function y. Recall first the main inequality proved in 1 = C 1 (A) 1 such that Lemma 4.1: there are constants r1 = r1 (A) > 1 and C √ 1 ε1/40 ≥ y −(1 + 1−4B)/2 ≥ C −1 (r − 1)2 − C 1 ε 1/40 on 1 \r1 . 1 (r −1)2 + C C 1 (4.10) We define y0 = (1 +
√ −1 , 1 − 4B)/2 + C 2
(4.11)
2 (A) a sufficiently large constant depending on the constants C 1 , r1 2 = C where we fix C in (4.10) and c in Proposition 5.1. As in [24, Sect. 8], for R ∈ [y0 , ∞) we define V R = { p ∈ 1 : y( p) < R};
(4.12)
U R = the connected component of V R whose closure in 0 contains S0 . In view of (4.10),
1 \1+(4C1 C2 )−1/2 ⊆ V y0 ∩ (1 \r1 ) ⊆ V y0 +C−1 ∩ (1 \r1 ) ⊆ 1 \1+(4C1 C−1 )1/2 , 2
2
Stationary Black Holes in Vacuum
107
2 is sufficiently large and ε is sufficiently small. In particular, we deduce, provided that C 1 \1+(4C1 C2 )−1/2 ⊆ U y0 ⊆ U y0 +C−1 ⊆ 1 \1+(4C1 C−1 )1/2 . 2
2
(4.13)
For p = 1 (0, q) ∈ 1 and r ≤ ε0 we define, Br ( p) = 1 {(t, q ) ∈ (−ε0 , ε0 ) × E 1−ε0 : t 2 + |q − q |2 < r 2 } . For any set U ⊆ 1 let δ1 (U ) denote its boundary in 1 . Clearly, if p ∈ δ1 (U R ) for some R ≥ y0 then y( p) = R. We define the vector-field Y = Dα yDα in 1 [(−ε, ε) × E 1−ε ] and its projection Y along the hypersurface 1−ε , Y = Y + g(Y, T0 )T0 =
3
(Y )α ∂α .
(4.14)
α=1
The vector-field Y is smooth, tangent to the hypersurface 1−ε , and 3
on 1−ε . |(Y )α | ≤ C
α=1
In addition, Y (y) = g(Y , Y ) = g(Y, Y ) + g(Y, T0 )2 ≥ g(Y, Y ) = Dα yDα y.
(4.15)
In particular, if p ∈ δ1 (U R ) for some R ≥ y0 then y( p) = R thus, using (3.26), −1 . Therefore if p ∈ δ1 (U R ) then Y (y)( p) ≥ C {x ∈ Bδ ( p) ∩ 1 : y(x) < R} = Bδ ( p) ∩ U R ,
(4.16)
for any δ ≤ δ1 = δ1 (A) > 0. We prove now that the regions U R , R ≥ y0 , increase in a controlled way. Lemma 4.2. There is δ2 = δ2 (A) ∈ (0, δ1 ) such that, for any δ ≤ δ2 and R ∈ [y0 , ∞), ∪ p∈U R (Bδ 3 ( p) ∩ 1 ) ⊆ U R+δ 2 ⊆ ∪ p∈U R (Bδ ( p) ∩ 1 ).
(4.17)
∪ R≥y0 U R = 1
(4.18)
In addition
and UR = VR
for any R ≥ y0 .
(4.19)
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Proof of Lemma 4.2. The first inclusion in (4.17) is clear: since y is a smooth function in a neighborhood of 1 (see Proposition 3.4), it follows that y(q) < R + δ 2 for any p ∈ U R and q ∈ Bδ 3 ( p) ∩ 1 , provided that δ is sufficiently small. To prove the second inclusion, it suffices to prove that U R+δ 2 ⊆ U R ∪ [∪ p∈δ1 (U R ) (Bδ/4 ( p) ∩ 1 )],
(4.20)
for δ sufficiently small, R ≥ y0 . Assume, for contradiction, that q is a point in U R+δ 2 which does not belong to the open set in the right-hand side of (4.20). Let γ : [0, 1] → U R+δ 2 ∪ S0 be a continuous curve such that γ (0) ∈ S0 and γ (1) = q (see definition (4.12)). Let q = γ (t ), t ∈ (0, 1], denote the first point on this curve which does not belong to the open set in the right-hand side of (4.20). Clearly, q does not belong to the closure of U R in 1 , thus q belongs to the closure of the set ∪ p∈δ1 (U R ) (Bδ/4 ( p) ∩ 1 ) in 1 . Since δ1 (U R ) is a compact set (see (2.25) and (4.13)), it follows that q ∈ Bδ/2 ( p0 ) ∩ 1
for some p0 ∈ δ1 (U R ).
(4.21)
For p ∈ 1+(4C1 C2 )−1/2 and |t| ≤ δ , δ > 0 sufficiently small, let γ p (t) ⊆ 1 denote the integral curves of the vector-field Y defined in (4.14), starting at p. Using (4.15), the fact that y( p0 ) = R ≥ y0 , and (3.26), it follows that −1 Y (y) ≥ C
in Bδ ( p0 ) ∩ 1 ,
(4.22)
provided that δ is sufficiently small. With q ∈ Bδ/2 ( p0 ) being the point constructed earlier, we look at the curve γq (t), t ∈ [−δ 3/2 , δ 3/2 ]. Clearly, this curve is included in Bδ ( p0 ) ∩ 1 , assuming δ sufficiently small. Using (4.22) and the fact that y(q ) < R + δ 2 (since q ∈ U R+δ 2 ), we derive that there is a point q on the curve γq (t), t ∈ [−δ 3/2 , δ 3/2 ], such that y(q ) < R. It follows from (4.16) that q ∈ U R . Since q ∈ / UR (by construction), there is a point q = γq (t ), t ∈ [−δ 3/2 , δ 3/2 ], such that q ∈ δ1 (U R ). It follows that q ∈ Bδ/8 (q ), in contradiction with the fact that q does not belong to the set in the right-hand side of (4.20). This completes the proof of (4.20). The completeness property (4.18) follows easily from the asymptotic formula (2.25) and the fact that y is a smooth function on 1 . To prove (4.19) we notice that, in view of (4.13), it suffices to prove that V R ∩ 1+(4C1 C2 )−1/2 ⊆ U R , for any R ≥ y0 . Assume, for contradiction, that / U R0 . there is R0 > y0 and q ∈ 1+(4C1 C2 )−1/2 such that y(q) < R0 and q ∈
(4.23)
/ U R }. Since I is bounded, due to (4.18), we can take R its Let I = {R ∈ [R0 , ∞) : q ∈ least upper bound. We analyze two possibilities: q ∈ U R and q ∈ / U R . If q ∈ U R then, using (4.23), R > R0 . For δ > 0 sufficiently small (depending on R − R0 and A), it follows from (4.17) that there is R = R − δ 2 ≥ R0 + δ 1/2 and a point q ∈ U R such that |q − q | < δ. However, y(q) < R0 , see (4.23), thus y(x) < R0 + δ 1/2 ≤ R for any x ∈ Bδ (q). Since Bδ (q) ∩ U R = ∅, it follows that q ∈ U R , in contradiction with the definition of R . Finally, assume that q ∈ / U R . Then q ∈ / δ1 (U R ), in view of (4.16) and (4.23). For δ sufficiently small (smaller than the distance between q and the compact set δ1 (U R )), it follows from (4.20) that q ∈ / U R +δ 2 . This is in contradiction with the definition of R , which completes the proof of (4.19).
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4.3. T-conditional pseudo-convexity. In this subsection we prove a T-conditional pseudo-convexity property of the function y away from the bifurcation sphere S0 . This pseudo-convexity property, which was first observed in [25] in the case of the Kerr spaces and used in [24], plays a key role in the Carleman estimates and the uniqueness arguments in the next section. Qualitatively, the T-conditional pseudo-convexity property for the function y is the inequality X α X β Dα Dβ y < 0
for vectors X = 0 for which g(X, X ) = X (y) = g(T, X ) = 0.
This inequality, in fact the more quantitative version in Lemma 4.3 below, allows us to use the method of Carleman estimates to extend a Killing vector-field across increasing level sets of the function y (see Sect. 5). Geometrically, this inequality says that given a point p0 and any vector X at p0 satisfying g(X, X ) = X (y) = g(T, X ) = 0, the geodesic emanating from p0 in the direction of the vector X curves towards the region where y < y( p0 ). For comparison, the standard pseudo-convexity property of Hörmander is, qualitatively, the stronger inequality X α X β Dα Dβ y < 0
for vectors X = 0 for which g(X, X ) = X (y) = 0.
Unfortunately, this stronger inequality does not hold globally for any suitable function y, even in the case of the Schwarzschild spaces. Since U y0 = { p ∈ 1 : y( p) < y0 }, see (4.19) and the definition (4.12), it follows √ −1 in from (4.13) that y ≥ (1 + 1 − 4B)/2 + C −1 )1/2 . Using also (4.10), it 1 C 2 1+(4C 2 follows that √ −2 in 1+(8C C )−1/2 . y ≥ (1 + 1 − 4B)/2 + C 2 1 2 Lemma 4.3. Assume p ∈ 1+(8C1 C2 )−1/2 , thus y( p) ≥ (1 +
√ −2 . 1 − 4B)/2 + C 2
(4.24)
There is a constant c2 = c2 (A) > 0 and μ = μ( p) ∈ R such that X α X β (μgαβ ( p) − Dα Dβ y( p)) ≥ c2 |X |2
(4.25)
for any real vector X with the property that |X α Tα ( p)| + |X α Dα y( p)| ≤ c2 |X |.
(4.26)
is sufProof of Lemma 4.3. The bound (4.25) follows easily with μ = 1 if r ( p) ≥ C ficiently large (see (2.3) and (2.25)). Assume that r ( p) ≤ C. We shall make use of the null frame e(α) defined in Sect. 3. Using (3.22), we write X α X β Dα Dβ y = X α X β [−P −1 Dα PDβ P] + g(X, X ) [P −1 (Dρ PDρ P)] −2X α X β [P 2 Fα ρ Fβρ ] − (X α Tα )2 (P 3 F 2 ) + |X |2 O(ε), 1/40 . Since X (y) = |X |O(c2 ), where, in this proof, O(ε) denotes quantities bounded by Cε X α X β [−P −1 Dα PDβ P] =
y2
y X (z)2 + |X |2 O(c2 ), + z2
(4.27)
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2 . Using (3.17) and (3.7), where, in this proof, O(c2 ) denotes quantities bounded by Cc y 1 y −1 ρ 1− 2 + O(ε),
[P (Dρ PD P)] = 2 y + z 2 4M 2 y + z2 thus X α X β [P −1 (Dρ PDρ P)]gαβ = g(X, X )
y(y 2 + z 2 − y) + |X |2 O(ε). 4M 2 (y 2 + z 2 )2
(4.28)
To calculate X α X β [ P 2 Fα ρ Fβρ ] we recall, see (3.13) that all components of F vanish, with the exception of F34 = −F43 = − 4M1P 2 + O(ε) and F12 = −F21 = 4Mi P 2 + O(ε), a, b = 1, 2. Since F = (F) we also have, F34 = −F43 = −(4M)−1 [P −2 ] + O(ε),
F12 = −F21 = −(4M)−1 [P −2 ] + O(ε).
Therefore, X α X β Fα ρ Fβρ = 2X 3 X 4 F34 F34 + (X 1 )2 + (X 2 )2 F12 F12 + |X |2 O(ε) 1 3 4 −2 1 2 2 2 −2 2X [P = X
[P ]−i (X ) +(X ) ] +|X |2 O(ε). 16M 2 P 2 Thus, − 2X α X β [P 2 Fα ρ Fβρ ] = −X 3 X 4
y2 − z2 + |X |2 O(ε). 4M 2 (y 2 + z 2 )2
(4.29)
Therefore, denoting E(X, X ) := X α X β (μgαβ − Dα Dβ y), we write y y(y 2 + z 2 − y) − 2 X (z)2 E(X, X ) = g(X, X ) μ − 2 2 2 2 4M (y + z ) y + z2 +X 3 X 4
y2 − z2 + |X |2 O(ε) + |X |2 O(c2 ), 4M 2 (y 2 + z 2 )2
or, since g(X, X ) = −2X 3 X 4 + (X 1 )2 + (X 2 )2 , 2 2 y2 − z2 y 3 4 y(y + z − y) E(X, X ) = 2X X + −μ − 2 X (z)2 4M 2 (y 2 + z 2 )2 8M 2 (y 2 + z 2 )2 y + z2 y(y 2 + z 2 − y) 1 2 2 2 + |X |2 O(ε) + |X |2 O(c2 ) μ− + (X ) + (X ) 4M 2 (y 2 + z 2 )2 2y − 1 y 3 4 −μ − 2 X (z)2 = 2X X 2 2 2 8M (y + z ) y + z2 y(y 2 + z 2 − y) + |X |2 O(ε) + |X |2 O(c2 ). + (X 1 )2 + (X 2 )2 μ − 4M 2 (y 2 + z 2 )2 We now make use of our main identity (3.19) as well as (3.16), and derive, (D1 z)2 + (D2 z)2 = Dβ zDβ z + O(ε) =
B − z2 + O1 (ε). 4M 2 (y 2 + z 2 )
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Thus, using also D3 z = O(ε) and D4 z = O(ε), by Cauchy-Schwartz, X (z)2 ≤ (X 1 )2 + (X 2 )2 (D1 z)2 + (D2 z)2 + O(ε) ≤ (X 1 )2 + (X 2 )2
B − z2 + O(ε). 4M 2 (y 2 + z 2 )
We deduce, E(X, X ) ≥ |X |2 O(ε) + |X |2 O(c2 ) + 2X 3 X 4
+ (X ) + (X ) 1 2
2 2
2y − 1 − μ 8M 2 (y 2 + z 2 )
y(B − z 2 ) y(y 2 + z 2 − y) − , μ− 4M 2 (y 2 + z 2 )2 4M 2 (y 2 + z 2 )2
or,
2y − 1 − μ 8M 2 (y 2 + z 2 ) y(y 2 − y + B) + (X 1 )2 + (X 2 )2 . μ− 4M 2 (y 2 + z 2 )2
E(X, X ) ≥ |X |2 O(ε) + |X |2 O(c2 ) + 2X 3 X 4
(4.30)
Since X (y) = X α Dα y = |X |O(c2 ), it follows from (3.15) that X 4 (Tα lα )− X 3 (Tα l α ) = |X |O(ε) + |X |O(c2 ). On the other hand, according to (3.27), (Tα lα )(Tβ l β ) =
y2 − y + B + O(ε), 2(y 2 + z 2 )
and therefore, in view of (4.24) and B < 1/4, (Tα lα )(Tβ l β ) ≥ c = c (A) > 0 (recall We infer that r ( p) ≤ C). −1 [(X 3 )2 + (X 4 )2 ] − Cε|X 2 |X |2 . 2X 3 X 4 ≥ C |2 − Cc Thus the expression in (4.30) is bounded from below by c2 |X |2 if c2 is sufficiently small and the coefficients of ((X 1 )2 +(X 2 )2 ) and 2X 3 X 4 are both positive, for a suitable choice of μ. This holds if and only if, y(y 2 − y + B) 2y − 1 . 0 such that Oc ⊆ O , and a smooth vector-field K in O such that K = u L − u L on (H+ ∪ H− ) ∩ O , LK g = 0, [T, K] = 0, Kμ σμ = 0 in O ,
(5.1)
and g(K, K) ≤ −c(r − 1)2
on 1 ∩ O .
(5.2)
In addition, there is λ0 ∈ R such that the vector-field Z = T + λ0 K has complete periodic orbits in O . The inequality (5.2) follows from [2, Prop. 4.5] and (2.12). In this section we extend K to the exterior region E. The main result is the following: Theorem 5.2. The vector-field K constructed in Oc can be extended to a smooth vector-field in the exterior region E such that LK g = 0, [T, K] = 0, Kμ σμ = 0 in E ∪ Oc .
(5.3)
The rest of this section is concerned with the proof of Theorem 5.2. We construct the vector-field K recursively, in increasingly larger regions defined in terms of the level sets of the function y. We rely on Carleman estimates to prove, by a uniqueness argument similar to that of [2], that the extended K remains Killing at every step in the process. The initial step is, of course, that given by Proposition 5.1. Recall the definitions (4.11) and (4.12), and the identity U R = V R , see (4.19). Notice T(σ ) = Tμ σμ = 0 (see (2.14)). Therefore, using the flow t,T associated that, in M, to T and the assumption GR on the orbits of T, we can extend σ to a smooth function on E as a solution of the equation T(σ ) = 0. Then we define the smooth functions y, z : E → R by σ = y + i z, and define the connected open space-time regions, E R = { p ∈ E : y( p) < R} = ∪t∈R t,T (U R ) ⊆ E,
R ≥ y0 .
(5.4)
Clearly, E = ∪ R≥y0 E R . The main step in the proof of the theorem is the following: Main Claim. For any R ≥ y0 there is a smooth vector-field K defined in the connected open set E R , which agrees with the vector-field K defined in Proposition 5.1 in a neighborhood of U y0 in E, such that LK g = 0, [T, K] = 0, Kμ σμ = 0 in E R .
(5.5)
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The Main Claim follows for R = y0 from Proposition 5.1: we define K in a small neighborhood of U y0 in E as in Proposition 5.1 and extend it to E y0 by solving the ordinary differential equation [T, K] = 0 (recall that T does not vanish in E). The remaining identities in (5.5) hold on E y0 since they hold in a small neighborhood of U y0 in E and T is a non-vanishing Killing vector-field. Assume now that the Main Claim holds for some value R0 ≥ y0 . We would like to prove the Main Claim for some value R = R0 + δ , for some δ = δ (A, δ0 ) > 0. Where δ0 :=
inf
p∈1+(8C
−1/2 1 C2 )
|g(T, T0 )( p)| > 0.
We will use the results and the notation in Sect. 4. Recall that y, z, σ are smooth well-defined functions in E and y + i z = (1 − σ )−1 . As in the proof of Lemma 4.2 let Y α = Dα y, which is a smooth vector-field in E. Using the last identity in (5.5), Kμ Yμ = 0 in E R0 . We compute in E R0 , [K, Y ]β = Kα Dα Yβ − Y α Dα Kβ = Kα Dβ Dα y + Dα yDβ Kα = Dβ (Kα Dα y) = 0. Thus [K, Y ] = 0
in E R0 .
(5.6)
For R ≥ y0 and δ > 0 small we define δ,R = ∪ p∈δ (U ) Bδ ( p). O 1 R Clearly, for δ sufficiently small and R ≥ y0 , δ,R = { p ∈ O δ,R : y( p) < R}. ER ∩ O
(5.7)
δ,R0 , by the induction hypothesis. We would like The vector-field K is defined in E R0 ∩ O δ,R0 as the solution of an ordinary differential equation to extend it to the full open set O δ,R0 . We summarize this of the form [K, Y ] = 0, where Y is a suitable vector-field in O construction in Lemma 5.3 below. α
Lemma 5.3. There is a constant δ3 = δ3 (A) > 0, a smooth vector-field Y = Y ∂α δ3 ,R0 , 3 |Y α | ≤ δ −1 in O δ3 ,R0 , and a smooth extension of the vector-field K in O α=0 3 δ3 ,R0 ∩ E R0 ) to O δ3 ,R0 such that (originally defined in O DY Y = 0, [K, Y ] = 0, Y (y) ≥ δ3
δ3 ,R0 . in O
(5.8)
Proof of Lemma 5.3. For δ sufficiently small we define δ,R0 : y(x) = R0 }. Sδ,R0 = {x ∈ O δ,R0 , −1 in O Clearly, δ1 (U R0 ) ⊆ Sδ,R0 . Since y is a smooth function and Dα yDα y ≥ C the set Sδ,R0 is a smooth embedded hypersurface. We define Y = Y on Sδ,R0 , and extend δ ,R , δ ≤ δ by solving the geodesic equation D Y = 0. Y to an open set of the form O 0 Y δ ,R ∩ E R0 , δ ∈ (0, δ ]. Since K is tangent to We first show that [K, Y ] = 0 in O 0 Sδ,R0 , K(y) = 0, and Y = Y we deduce that [K, Y ] = 0 along Sδ,R0 . On the other hand, δ ,R ∩ E R0 (where K is Killing), we have in O 0 DY (LK Y ) = LK (DY Y ) − DLK Y Y = −DLK Y Y .
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δ ,R ∩E R0 , δ ∈ (0, δ ]. We can now extend K to O δ3 ,R0 , δ3 ≤ δ , Thus, [K, Y ] = 0 in O 0 by solving the ordinary differential equation [K, Y ] = 0. This completes the proof of the lemma. We prove now that the vector field K is indeed a Killing vector-field (and verifies δ,R0 . An argument of this type was the other identities in (5.5)) in a small open set O used in [2, Sect. 4]. For |t| sufficiently small and p0 ∈ δ1 (U R0 ) we define, in a small neighborhood of p0 , the map t,K obtained by flowing a parameter distance t along the integral curves of K. Let ∗ (g) gt = t,K
and
∗ Tt = t,K (T).
The tensor gt is a smooth Lorentz metric that satisfies the Einstein vacuum equations, and Tt is a smooth Killing vector-field for gt , in a small neighborhood of p0 and for |t| sufficiently small. In addition, since K is tangent to the hypersurface {y = R0 }, it follows from the induction hypothesis that gt = g and Tt = T in a small neighborhood of p0 intersected with E R0 . In addition, using the second identity in (5.8), with t = t,K , ∗ Y − ∗Y ∗ Y − ∗Y t−h −h d ∗ t 0 ∗ Y = lim = −t lim = −t∗ (LK Y ) = 0. h→0 h→0 dt t −h −h Thus t∗ Y = Y and we infer that Dt Y Y = 0 in a small neighborhood of p0 , for |t| sufficiently small, where Dt denotes the covariant derivative with respect to gt . The main step in proving the Main Claim is the following proposition: Proposition 5.4. Assume p0 ∈ δ1 (U R0 ), g is a smooth Lorentz metric in Bδ4 ( p0 ), δ4 ∈ (0, δ3 ], such that (Bδ4 ( p0 ), g ) is a smooth Einstein vacuum spacetime, and T is a smooth Killing vector-field for the metric g in Bδ4 ( p0 ). In addition, assume that g = g and T = T in E R0 ∩ Bδ4 ( p0 ); D Y = 0 in Bδ4 ( p0 ), Y
D
where denotes the covariant derivative induced by the metric g . Then g = g and T = T in Bδ5 ( p0 ) for some δ5 ∈ (0, δ4 ] (δ5 may depend only on δ4 , A, R0 , but not on the choice of the point p0 ). Assuming the proposition and Lemma 5.5, which we prove below, we complete now the proof of the Main Claim. It follows from Proposition 5.4 that K is a Killing vec∗ (T) = T for |t| tor-field in Bδ5 ( p0 ), for any p0 ∈ δ1 (U R0 ). In addition, since t,K sufficiently small, it follows that [T, K] = LK T = 0 in Bδ5 ( p0 ). Finally, in Bδ5 ( p0 ), g (Kμ σμ ) = Kμ Dα Dα σμ = Kμ Dμ (Dα σα ) = −LK (F 2 ) = 0, (using g K = 0, DK is antisymmetric, Dσ is symmetric, Dα σα = −F 2 , see (2.15)), and T(Kμ σμ ) = K(T(σ )) = 0. Since Kμ σμ = 0 in Bδ5 ( p0 )∩E R0 (the induction hypothesis), it follows from Lemma 5.5 below, with H = 0, that Kμ σμ = 0 in Bδ6 ( p0 ), δ6 ∈ (0, δ5 ].
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δ6 ,R0 = ∪ p ∈δ (U ) To summarize, we proved that K extends to the open set O 0 1 R0 Bδ6 ( p0 ), δ6 = δ6 (A, δ0 ) > 0, as a smooth vector, and the identities in (5.5) hold in this set. Using the inclusion (4.20), it follows that K is well defined and satisfies the identities (5.5) in a small neighborhood of U R0 +δ 2 . Thus we can extend K to the region E R0 +δ 2 , by 6 6 solving the ordinary differential equation [T, K] = 0. The Main Claim follows. 5.1. Proof of Proposition 5.4. We prove Proposition 5.4 following the same scheme as in the proof of [2, Prop. 4.3]. We first fix some smooth frames v(1) , v(2) , v(3) , v(4) = Y and v (1) , v (2) , v (3) , v (4) = Y in a small neighborhood Bδ ( p0 ), such that, for a = 1, 2, 3, 4, DY v(a) = 0
and
v(a) = v (a)
D Y v (a) = 0
in Bδ ( p0 );
in E R0 ∩ Bδ ( p0 ).
The idea of the proof is to derive ODE’s for the differences dv = v − v, d = − , dT = T − T and d F = F − F, with source terms in d R = R − R. We combine these ODE’s with an equation for g (d R) and equation for T(d R). Finally, we prove uniqueness of solutions of the resulting coupled system, see Lemma 5.5, using Carleman inequalities as in [1,2,24,25]. As in the proof of [2, Prop. 4.3], we define, for a, b, c, d = 1, . . . 4 and α, β = 0, . . . , 3, (d)(a)(b)(c) = (a)(b)(c) − (a)(b)(c) = g (v (a) , D v (c) v (b) ) − g(v(a) , Dv(c) v(b) );
(∂d)α(a)(b)(c) = ∂α [(d)(a)(b)(c) ]; (d R)(a)(b)(c)(d) = R (v (a) , v (b) , v (c) , v (d) ) − R(v(a) , v(b) , v(c) , v(d) );
(5.9)
(∂d R)α(a)(b)(c)(d) = ∂α [(d R)(a)(b)(c)(d) ]; β
β
β
(dv)(a) = v (a) − v(a) β
β
β
where v(a) = v(a) ∂β and v (a) = v (a) ∂β ;
β
(∂dv)α(a) = ∂α [(dv)(a) ]. As before, the coordinate frame ∂0 , . . . , ∂3 is induced by the diffeomorphism 1 . Let g(a)(b) = g(v(a) , v(b) ), g(a)(b) = g (v (a) , v (b) ). The identities DY v(a) = D v (a) = 0 Y
show that Y (g(a)(b) ) = Y (g(a)(b) ) = 0. Since g(a)(b) = g(a)(b) in E R0 ∩ Bδ ( p0 ) it follows that g(a)(b) = g(a)(b) := h (a)(b)
and
Y (h (a)(b) ) = 0 in Bδ ( p0 ),
(5.10)
for some constant δ = δ (A, δ0 ) ∈ (0, δ ]. Clearly, (a)(b)(4) = (a)(b)(4) = 0. We use now the definition of the Riemann curvature tensor to find a system of equations for Y [(d)(a)(b)(c) ]. We have R(a)(b)(c)(d) = g(v(a) , Dv(c) (Dv(d) v(b) ) − Dv(d) (Dv(c) v(b) ) − D[v(c) ,v(d) ] v(b) )
= g(v(a) , Dv(c) (g(m)(n) (m)(b)(d) v(n) ))−g(v(a) , Dv(d) (g(m)(n) (m)(b)(c) v(n) )) + g(m)(n) (a)(b)(n) ((m)(c)(d) − (m)(d)(c) ) = v(c) ((a)(b)(d) ) − v(d) ((a)(b)(c) ) + g(m)(n) (a)(b)(n) ((m)(c)(d) −(m)(d)(c) ) + g(a)(n) [(m)(b)(d) v(c) (g(m)(n) ) − (m)(b)(c) v(d) (g(m)(n) )] + g(m)(n) ((m)(b)(d) (a)(n)(c) − (m)(b)(c) (a)(n)(d) ).
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We set d = 4 and use (a)(b)(4) = v(4) (g(a)(b) ) = 0 and g(a)(b) = h (a)(b) ; the result is Y ((a)(b)(c) ) = −h (m)(n) (a)(b)(n) (m)(4)(c) − R(a)(b)(c)(4) . Similarly, Y ( (a)(b)(c) ) = −h (m)(n) (a)(b)(n) (m)(4)(c) − R (a)(b)(c)(4) . We subtract these two identities to derive (d)(e)( f )
Y [(d)(a)(b)(c) )] = (1) F(a)(b)(c) (d)(d)(e)( f ) − (d R)(a)(b)(c)(4)
(5.11)
for some smooth function (1) F. This can be written schematically in the form Y (d) = M∞ (d) + M∞ (d R).
(5.12)
We will use such schematic equations for simplicity of notation10 . By differentiating (5.12), we also derive Y (∂d) = M∞ (d) + M∞ (∂d) + M∞ (d R) + M∞ (∂d R).
(5.13)
With the notation in (5.9), since [v(4) , v(b) ] = −Dv(b) v(4) = − (c) (4)(b) v(c) , we have β
β
β
α α v(4) ∂α (v(b) ) − v(b) ∂α (v(4) ) = −(a)(4)(b) v(c) g(a)(c) .
Similarly, β
α
β
β
α v(4) ∂α (v (b) ) − v (b) ∂α (v(4) ) = − (a)(4)(b) v (c) g
(a)(c)
.
We subtract these two identities to conclude that, schematically, Y (dv) = M∞ (d) + M∞ (dv).
(5.14)
By differentiating (5.14) we also have Y (∂dv) = M∞ (d) + M∞ (∂d) + M∞ (dv) + M∞ (∂dv).
(5.15)
We derive now a wave equation for d R. We start from the identity (g R)(a)(b)(c)(d) − (g R )(a)(b)(c)(d) = M∞ (d R), which follows from the standard wave equations satisfied by R and R and the fact that g(m)(n) = g (m)(n) = h (m)(n) . We also have D(m) R(a)(b)(c)(d) − D (m) R (a)(b)(c)(d) = M∞ (dv) + M∞ (d) + M∞ (d R) + M∞ (∂d R). It follows from the last two equations that (m)(n)
g(m)(n) v(n) (v(m) (R(a)(b)(c)(d) )) − g v (n) (v (m) (R (a)(b)(c)(d) )) = M∞ (dv) + M∞ (d) + M∞ (∂d) + M∞ (d R) + M∞ (∂d R). 10 In general, given H = (H , . . . H ) : B ( p ) → R L we let M (H ) : B ( p ) → R L denote ∞ 1 0 0 L δ δ L All Hl , where the coefficients All are smooth on vector-valued functions of the form M∞ (H )l = l=1 Bδ ( p0 ).
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Since g(m)(n) = g (m)(n) it follows that g(m)(n) v(n) (v(m) ((d R)(a)(b)(c)(d) )) = M∞ (dv) + M∞ (∂dv) + M∞ (d) + M∞ (∂d) + M∞ (d R) + M∞ (∂d R). Thus g (d R) = M∞ (dv)+M∞ (∂dv)+M∞ (d)+M∞ (∂d)+M∞ (d R)+M∞ (∂d R). (5.16) This is our main wave equation. We collect now Eqs. (5.12), (5.13), (5.14), (5.15), and (5.16): Y (d) = Y (∂d) = Y (dv) = Y (∂dv) = g (d R) =
M∞ (d) + M∞ (d R); M∞ (d) + M∞ (∂d) + M∞ (d R) + M∞ (∂d R); M∞ (dv) + M∞ (d); (5.17) M∞ (dv) + M∞ (∂dv) + M∞ (d) + M∞ (∂d); M∞ (dv)+M∞ (∂dv)+M∞ (d)+M∞ (∂d)+M∞ (d R)+M∞ (∂d R).
This is our first main system of equations. We derive now an additional system of this type, to exploit the existence of the Killing vector-fields T and T . For a, b = 1, . . . , 4 let (dT )(a) = T (a) − T(a) = g (T , L (a) ) − g(T, L (a) );
(d F)(a)(b) = F (a)(b) − F(a)(b) = D (a) T (b) − D(a) T(b) .
(5.18)
Using the identities Dv(4) v(b) = 0 and D v(4) v (b) = 0 it follows that v(4) (T(b) ) = F(4)(b) and v(4) (T (b) ) = F(4)(b) . Thus Y (dT ) = M∞ (d F).
(5.19)
We also have, using again Dv(4) v(b) = 0, v(4) (F(a)(b) ) = D(4) F(a)(b) = g(c)(d) T(d) R(c)(4)(a)(b) = h (c)(d) T(d) R(c)(4)(a)(b) . Similarly, v(4) (F (a)(b) ) = h (c)(d) T (d) R(c)(4)(a)(b) .
Thus, in our schematic notation, Y (d F) = M∞ (dT ) + M∞ (d R). Finally, we use the identities 0 = (LT R)(a)(b)(c)(d) = T(m) D(m) R(a)(b)(c)(d) + D(a) T(m) R(m)(b)(c)(d) + D(b) T(m) R(a)(m)(c)(d) + D(c) T(m) R(a)(b)(m)(d) + D(d) T(m) R(a)(b)(c)(m) ,
(5.20)
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and 0 = (LT R )(a)(b)(c)(d) = T + D (b) T
(m)
(m)
D (m) R (a)(b)(c)(d) + D (a) T
R (a)(m)(c)(d) + D (c) T
(m)
(m)
R (m)(b)(c)(d)
R (a)(b)(m)(d) + D (d) T
(m)
R (a)(b)(c)(m) .
Thus T
(m)
D (m) R (a)(b)(c)(d) − T(m) D(m) R(a)(b)(c)(d) = M∞ (d F) + M∞ (d R),
which easily gives T(d R) = M∞ (d F) + M∞ (d R) + M∞ (d) + M∞ (dT ) + M∞ (dv). (5.21) We collect now Eqs. (5.19), (5.20), and (5.21), thus Y (dT ) = M∞ (d F); (5.22) Y (d F) = M∞ (dT ) + M∞ (d R); T(d R) = M∞ (d F) + M∞ (d) + M∞ (dT ) + M∞ (dv) + M∞ (d R). This is our second main system of differential equations. Since g = g and T = T in E R0 ∩ Bδ ( p0 ), the functions d, ∂d, dv, ∂dv, dT, d F, d R vanish in E R0 ∩ Bδ ( p0 ). Therefore, using both systems (5.17) and (5.22), the lemma is a consequence of Lemma 5.5 below. Lemma 5.5. Assume δ > 0, p0 ∈ δ1 (U R0 ) and G i , H j : Bδ ( p0 ) → R are smooth functions, i = 1, . . . , I, j = 1, . . . , J . Let G = (G 1 , . . . , G I ), H = (H1 , . . . , H J ), ∂G = (∂0 G 1 , . . . , ∂4 G I ) and assume that, in Bδ ( p0 ), ⎧ ⎪ ⎨g G = M∞ (G) + M∞ (∂G) + M∞ (H ); (5.23) T(G) = M∞ (G) + M∞ (H ); ⎪ ⎩Y (H ) = M (G) + M (∂G) + M (H ). ∞ ∞ ∞ Assume that G = 0 and H = 0 in Bδ ( p0 ) ∩ E R0 = {x ∈ Bδ ( p0 ) : y(x) < R0 }. Then δ ∈ (0, δ) sufficiently small. G = 0 and H = 0 in Bδ ( p0 ) for some Unique continuation theorems of this type in the case H = 0 were proved by two of the authors in [24] and [25], using Carleman estimates. It is not hard to adapt the proofs, using the same Carleman estimates, to the general case. The essential ingredients are the T-conditional pseudo-convexity property in Lemma 4.3 and the inequality √ −1 , see (4.24). We provide all the details below. y( p0 ) ≥ (1 + 1 − 4B)/2 + C 5.2. Proof of Lemma 5.5. We will use a Carleman estimate proved by two of the authors in [24, Sect. 3], which we recall below. We may assume that the value of δ in Lemma 5.5 α is sufficiently small. For r ≤ δ, let Br = Br ( p0 ). Notice that, if T = Tα ∂α , Y = Y ∂α in the coordinate frame induced by the diffeomorphism 1 then sup
4 3
x∈Bδ j=0 α=0
α
= C(A). (|∂ j Tα (x)| + |∂ j Y (x)|) ≤ C
(5.24)
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Definition 5.6. A family of weights h : B 10 → R+ , ∈ (0, 1 ), 1 ≤ δ will be called T-conditional pseudo-convex if for any ∈ (0, 1 ), h ( p0 ) = ,
4
j |∂ j h (x)| x∈B 10 j=1 Dα h ( p0 )Dβ h ( p0 )(Dα h Dβ h sup
≤ /1 , |T(h )( p0 )| ≤ 10 ,
(5.25)
− Dα Dβ h )( p0 ) ≥ 12 ,
(5.26)
and there is μ ∈ [−1−1 , 1−1 ] such that for all vectors X = X α ∂α at p0 , 12 [(X 1 )2 + (X 2 )2 + (X 3 )2 + (X 4 )2 ] ≤ X α X β (μgαβ − Dα Dβ h )( p0 ) + −2 (|X α Tα ( p0 )|2 + |X α Dα h (x0 )|2 ). (5.27) A function e : B 10 → R will be called a negligible perturbation if sup |∂ j e (x)| ≤ 10
for j = 0, . . . , 4.
(5.28)
x∈B 10
Our main Carleman estimate, see [24, Sect. 3], is the following: Lemma 5.7. Assume 1 ≤ δ, {h }∈(0,1 ) is a T-conditional pseudo-convex family, and e is a negligible perturbation for any ∈ (0, 1 ]. Then there is ∈ (0, 1 ) sufficiently small (depending only on 1 ) and C sufficiently large such that for any λ ≥ C and any φ ∈ C0∞ (B 10 ), λe−λ f φ L 2 + e−λ f |∂φ| L 2 ≤ Cλ−1/2 e−λ f g φ L 2 + −6 e−λ f T(φ) L 2 , (5.29) where f = ln(h + e ). We also need a Carleman inequality to exploit the last equation in (5.23). Lemma 5.8. Assume ≤ δ is sufficiently small, e is a negligible perturbation, and h : B 10 → R+ satisfies h ( p0 ) = ,
sup
2
x∈B 10 j=1
j |∂ j h (x)| ≤ 1, |Y (h )( p0 )| ≥ .
(5.30)
Then there is C sufficiently large such that for any λ ≥ C and any φ ∈ C0∞ (B 10 ), e−λ f φ L 2 ≤ 4(λ)−1 e−λ f Y (φ) L 2 ,
(5.31)
where f = ln(h + e ). This inequality was proved in [2, App. A]. See also [23, Chap. 28] for much more general Carleman inequalities under suitable pseudo-convexity conditions. To prove Lemma 5.5 we set h = y − y( p0 ) + and
e = 12 N p0 ,
(5.32)
−1 2 where = |−1 1 (x) − 1 ( p0 )| is the square of the standard euclidean norm. It is clear that e is a negligible perturbation, in the sense of (5.28), for sufficiently small. Also, it is clear that h verifies the condition (5.30), for sufficiently small, see
N p0 (x)
Lemma 5.3. We show now that there is 1 = 1 (δ) sufficiently small such that the family of weights {h }∈(0,1 ) is T-conditional pseudo-convex, in the sense of Definition 5.6.
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Condition (5.25) is clearly satisfied, since T(y) = 0. Condition 5.26 is also satisfied for −1 , see (3.26). To prove (5.27) for some sufficiently small since Dα y( p0 )Dα y( p0 ) ≥ C α α vector X we apply Lemma 4.3 if |X Tα | + |X Dα y| ≤ c2 |X |; if |X α Tα | + |X α Dα y| ≥ c2 |X | then the second term in the right-hand side of (5.27) dominates the other terms, provided that 1 is sufficiently small. It follows from the Carleman estimates in Lemmas 5.7 and 5.8 that there is = (δ, A) > 0 and a constant C = C(δ, A) ≥ 1 such that λe−λ f φ L 2 + e−λ f |∂φ| L 2 ≤ Cλ−1/2 e−λ f g φ L 2 + Ce−λ f T(φ) L 2 ; (5.33) λ1/2 e−λ f φ L 2 ≤ Cλ−1/2 e−λ f Y (φ) L 2 , for any φ ∈ C0∞ (B 10 ( p0 )) and any λ ≥ C, where f = ln(h + e ). Let η : R → [0, 1] denote a smooth function supported in [1/2, ∞) and equal to 1 in [3/4, ∞). For i = 1, . . . , I, j = 1, . . . J we define, G i = G i · 1 − η(N x0 / 20 ) = G i · η , (5.34) H j = H j · 1 − η(N x0 / 20 ) = H j · η . Clearly, G i , H j ∈ C0∞ (B 10 ( p0 )). We would like to apply the inequalities in (5.33) to the functions G i , H j , and then let λ → ∞. Using the definition (5.34), we have g G i = η g G i + 2Dα G i Dα η + G i g η ; T(G i ) = η T(G i ) + T( η )G i ; Y (H j ) = η · Y (H j ) + H j · Y ( η ). Using the Carleman inequalities (5.33), for any i = 1, . . . , I, j = 1, . . . , J we have λ · e−λ f · η G i L 2 + e−λ f · η |∂G i | L 2 ≤ Cλ−1/2 · e−λ f · η g G i L 2 + Ce−λ f · η T(G i ) L 2 −λ f α −λ f (5.35) + C e · Dα G i D η L 2 + e · G i (|g η | + |∂ η |) L 2 and η H j L 2 ≤ Cλ−1/2 e−λ f · η Y (H j ) L 2 + C λ−1/2 e−λ f · H j |∂ η | L 2 , λ1/2 e−λ f · (5.36) for any λ ≥ C and some constant C = C (A, C). Using the main identities (5.23), in B 10 ( p0 ) we estimate pointwise |g G i | ≤ M
I
I l=1
|Y (H j )| ≤ M
I l=1
|Hm |,
m=1
l=1
|T(G i )| ≤ M
J
(|∂G l | + |G l |) + M |G l | + M
J
|Hm |,
(5.37)
m=1
(|∂G l | + |G l |) + M
J m=1
|Hm |,
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for some large constant M. We add inequalities (5.35) and (5.36) over i, j. The key observation is that, in view of (5.37), the main terms in the right-hand sides of (5.35) and (5.36) can be absorbed into the left-hand sides for λ sufficiently large. Thus, for any λ sufficiently large, λ
I
e
−λ f
i=1
η G i L 2 +
I
e
−λ f
η |∂G i | L 2 + λ
1/2
i=1
≤ C
J
J
e−λ f η H j L 2
j=1
e−λ f H j |∂ η | L 2 + C
j=1
I
e−λ f Dα G i Dα η L 2
i=1
+ e−λ f G i (|g η | + |∂ η |) L 2 . η and ∂ η are supported in the set {x ∈ B 10 ( p0 ) : We obseve that the functions g η = 1 in B 100 ( p0 ). By assumption, the functions G i , |∂G i |, H j are N p0 ≥ 20 /2} and supported in {x ∈ Bδ ( p0 ) : y(x) ≥ y( p0 )}. In addition, inf
B 100 ( p0 )
e−λ f ≥ eλ/C
sup {x∈B
p 20 10 ( p0 ):N 0 ≥ /2
and y(x)≥y( p0 )}
e−λ f ,
which follows easily from the definition (5.32) We let now λ → ∞, as in [24, Sect. 8], to conclude that 1 B 100 G i = 0 and 1 B 100 H j = 0. The lemma follows. 6. Construction of the Rotational Killing Vector-Field Z In this section we extend the rotational Killing vector-field Z constructed in a small neighborhood of S0 , see Proposition 5.1, to the entire exterior region E. In E ∪ Oc we define Z = T + λ0 K, where λ0 is as in Proposition 5.1. Clearly λ0 = 0, in view of the assumption GR that T does not vanish in E, and Z does not vanish identically in E, since, by assumption SBS, T does not vanish identically on S0 . It follows from (5.3) that LZ g = 0, [T, Z] = [K, Z] = 0, Zμ σμ = 0
in E ∪ Oc .
(6.1)
As in the proof of (5.6), it follows that [Z, Y ] = 0
in E ∪ Oc .
(6.2)
In view of Proposition 5.1, there is t0 > 011 such that t0 ,Z = Id in O . Clearly s,T ( p) = s,T t0 ,Z ( p) = t0 ,Z s,T ( p)
for any
p ∈ Oc ∩ E
and
s ∈ R, (6.3)
11 Using the assumption that the orbits of T in E are complete and intersect 0 , see assumption GR, it is easy to see that any smooth vector-field V in E ∪ Oc which commutes with T and is tangent to H± ∩ Oc has complete orbits in E.
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using the commutation relation [T, Z] = 0. It follows that t0 ,Z ( p) = p for any p ∈ E y0 , recall definition (5.4). To prove this identity for any point p ∈ E, assume that t0 ,Z ( p) = p
for any p ∈ E R0 ,
for some R0 ≥ y0 . As before, it follows that t0 ,Z ( p) = p for any p ∈ E R0 . Using [Y, Z] = 0 and an identity similar to (6.3), it follows that t0 ,Z ( p) = p
for any p ∈ E R0 +δ ,
for some δ = δ (A) > 0. To summarize, we proved: Corollary 6.1. There is a nontrivial smooth vector-field Z in E ∪ Oc , tangent to H+ ∩ Oc and H− ∩ Oc , and a real number t0 > 0 such that t0 ,Z = Id, LZ g = 0, [T, Z] = 0, Zμ σμ = 0 in E. 6.1. The timelike span of the two Killing fields.. We define the area function W = −g(T, T)g(Z, Z) + g(T, Z)2 . In this subsection we show that W ≥ 0 in E. More precisely, we prove the following slightly stronger proposition: Proposition 6.2. The vector-field K constructed in Theorem 5.2 does not vanish at any point in E. In addition, at any point p ∈ E there is a timelike linear combination of the vector-fields T and K. Proof of Proposition 6.2. In view of (5.2), K does not vanish at any point in U y0 . It follows that K does not vanish at any point in E y0 , since K is constructed as the solution of [T, K] = 0 in E y0 . To prove that K does not vanish at any point p ∈ E we use the identity [K, Y ] = 0 in E, see (5.6). Let R1 = sup{R ∈ [y0 , ∞) : K does not vanish at any point in E R }. If R1 < ∞ then K has to vanish at some point p0 ∈ δ1 (U R1 ) (using the assumption that any orbit of T in E intersects 1 , and the observation that the set of points in E where K vanishes can only be a union of orbits of T). Since [K, Y ] = 0 in E, K vanishes on the integral curve γ p0 (t), |t| 1, of the vector-field Y starting at the point p0 . However, this integral curve intersects the set E R for some R < R1 , in contradiction with the definition of R1 . Thus K does not vanish at any point in E. We prove now the second part of the proposition. Let N = { p ∈ E : there is no timelike linear combination of T and K at p}. Clearly, the set N is closed in E and consists of orbits of the vector-field T. In addition, N ⊆ E\E y0 , since K itself is timelike in E y0 (see (5.2)). In view of (4.24) and (4.13) we √ −2 in E\E y0 . On the other hand g(T, T) = y/(y 2 +z 2 )−1, have y ≥ (1+ 1 − 4B)/2+ C 2 hence z 2 ≤ y − y 2 = −(y 2 − y + B) + B in N . Consequently, for some constant = C(A) C 1, √ −1 and B − z 2 ≥ C −1 in N . y ≥ (1 + 1 − 4B)/2 + C (6.4)
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Consider now the set of vector-fields T, K as well as the gradient vector-fields Y = Dα yDα , Z = Dα zDα at some point p ∈ N . Since T(σ ) = K(σ ) = 0 we have g(T, Y ) = g(T, Z ) = g(K, Y ) = g(K, Z ) = 0. In addition, using (3.26), (3.18), and (6.4), −1 , g(Y, Y ) ≥ C
−1 , g(Z , Z ) ≥ C
1/5 |g(Y, Z )| ≤ Cε
in N ,
(6.5)
= C(A). for some constant C Since the metric g is Lorentzian, it follows that the vectors T, K, Y, Z cannot be linearly independent at any point p ∈ N (if they were linearly independent then the determinant of the matrix formed by the coefficients g(T, T), g(T, K), g(K, K) would have to be negative, in contradiction with p ∈ N ). Since the triplets T, Y, Z and K, Y, Z are linearly independent it must follow that K, T are linearly dependent at points of N . Thus for any p ∈ N there is a ∈ R such that K p = aT p and g(T, T)| p ≥ 0.
(6.6)
We prove now that N = ∅. Assume that N = ∅ and let p0 denote a point in N such that y( p0 ) = inf p∈N y( p). Such a point exists since N ∩ 1 ⊆ 1+(4C1 C2 )−1/2 (see (5.2)) is compact (observe that T is timelike in R for large R). We may assume that p0 ∈ N ∩ 1 . In view of (6.6), there is a0 ∈ R such that K p0 − a0 T p0 = 0. We look at the integral curve {γ p0 (t) : |t| 1} of the vector field Y passing through p0 . Since [Y, K − a0 T] = 0 and Y p0 = 0, it follows that K = a0 T in the set γ p0 (t), |t| 1. Since Y (y) = g(Y, Y ) is strictly positive at p0 (see (6.5)), it follows that y(γ p0 (t)) < y( p0 ) if −1 , 0). Since y( p0 ) = inf p∈N y( p) (the definition of p0 ), it follows that t ∈ (−C −1 , 0)} = ∅. N ∩ {γ p0 (t) : t ∈ (−C −1 , 0)} it follows that g(T, T) < 0 in {γ p0 (t) : t ∈ Since K = a0 T in {γ p0 (t) : t ∈ (−C −1 (−C , 0)}. Using the formula g(T, T) = y/(y 2 + z 2 ) − 1, it follows that the function −1 , 0)}. Thus y − y 2 − z 2 vanishes at p0 and is strictly negative on {γ p0 (t) : t ∈ (−C Y (y − y 2 − z 2 ) ≥ 0
at p0 .
On the other hand, using (6.5) and (6.4), Dα yDα (y − y 2 − z 2 ) = (1 − 2y)Dα yDα y − 2zDα yDα z < 0
at p0 ,
provided that ε is sufficiently small. This provides a contradiction. 6.2. Proof of the Main Theorem. The proof of the Main Theorem now follows from the Carter-Robinson theorem, a complete account of which can be found in [17, Theorem 1.3]. Our situation is in fact easier than the theorem proved in [17], mainly due to our assumption on the smoothness of the bifurcation sphere (see (1.4)) and our simple proof of the fact that the span of the two Killing vector-fields is timelike, see Proposition 6.2. The proof of this fact in our setting relies, of course, on the main assumption on the smallness of tensor S.
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Appendix A. Asymptotic Identities Recall, see assumption GR, that we assumed the existence of an open subset M(end) of M which is diffeomorphic to R × ({x ∈ R3 : |x| > R}) for some R sufficiently large. In local coordinates {t, x i } defined by this diffeomorphism, we assume that T = ∂t and, with r = (x 1 )2 + (x 2 )2 + (x 3 )2 , g00 = −1 +
2S j x k 2M + O(r −2 ), gi j = δi j + O(r −1 ), g0i = −i jk + O(r −3 ), r r3 (A.1)
for some M > 0, S 1 , S 2 , S 3 ∈ R. Clearly, g00 = −1 + O(r −1 ), gi j = δi j + O(r −1 ), g0i = O(r −2 ).
(A.2)
We compute Fαβ = Dα Tβ = ∂α (g0β ) − g(∂0 , D∂α ∂β ) =
1 (∂α g0β − ∂β g0α ). 2
(A.3)
Thus, using (A.1), for j = 1, 2, 3, F0 j = −(1/2)∂ j g00 = M x j r −3 + O(r −3 ).
(A.4)
We have g01 = 2r −3 (S 3 x 2 − S 2 x 3 ) + O(r −3 ), g02 = 2r −3 (S 1 x 3 − S 3 x 1 ) + O(r −3 ),
(A.5)
g03 = 2r −3 (S 2 x 1 − S 1 x 2 ) + O(r −3 ). Thus F12 = (1/2)(∂1 g02 − ∂2 g01 ) = S 3 r −3 − 3r −5 x 3 (S 1 x 1 + S 2 x 2 + S 3 x 3 ) + O(r −4 ), F23 = (1/2)(∂2 g03 − ∂3 g02 ) = S 1 r −3 − 3r −5 x 1 (S 1 x 1 + S 2 x 2 + S 3 x 3 ) + O(r −4 ), F31 = (1/2)(∂3 g01 − ∂1 g03 ) = S 2 r −3 − 3r −5 x 2 (S 1 x 1 + S 2 x 2 + S 3 x 3 ) + O(r −4 ). (A.6) We have ∗
F αβ = (1/2) ∈αβμν Fρσ gμρ gνσ .
Thus, using (A.2), (A.4), (A.6), ∗
F 01 = F23 + O(r −4 ) = S 1 r −3 − 3r −5 x 1 (S 1 x 1 + S 2 x 2 + S 3 x 3 ) + O(r −4 ),
∗
F 02 = F31 + O(r −4 ) = S 2 r −3 − 3r −5 x 2 (S 1 x 1 + S 2 x 2 + S 3 x 3 ) + O(r −4 ),
∗
F 03 = F12 + O(r −4 ) = S 3r −3 − 3r −5 x 3 (S 1 x 1 + S 2 x 2 + S 3 x 3 ) + O(r −4 ),
∗
F 12 = −F03 + O(r −3 ) = −M x 3 r −3 + O(r −3 ),
∗
F 23 = −F01 + O(r −3 ) = −M x 1 r −3 + O(r −3 ),
∗
F 31 = −F02 + O(r −3 ) = −M x 2 r −3 + O(r −3 ).
(A.7)
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125
As a consequence, F 2 = (Fαβ + i ∗ F αβ )(F αβ + i ∗ F αβ ) = −4M 2 r −4 + O(r −5 ).
(A.8)
By definition, σμ = 2Tα (Fαμ + i ∗ F αμ ). Thus σ0 = 0; σ1 = 2M x 1r −3 + O(r −3 ) + 2i[S 1 r −3 − 3r −5 x 1 (S 1 x 1 + S 2 x 2 + S 3 x 3 ) + O(r −4 )]; σ2 = 2M x 2 r −3 + O(r −3 ) + 2i[S 2 r −3 − 3r −5 x 2 (S 1 x 1 + S 2 x 2 + S 3 x 3 ) + O(r −4 )]; σ3 = 2M x 3r −3 + O(r −3 ) + 2i[S 3 r −3 − 3r −5 x 3 (S 1 x 1 + S 2 x 2 + S 3 x 3 ) + O(r −4 )]. (A.9) Thus σ = 1 − 2Mr −1 + O(r −2 ) + i[2r −3 (S 1 x 1 + S 2 x 2 + S 3 x 3 ) + O(r −3 )]. (A.10) Thus y + i z = (1 − σ )−1 =
1 1 S x + S2 x 2 + S3 x 3 r −1 + O(1) + i + O(r ) , 2M 2M 2 r
which gives y=
r + O(1), 2M
z=
S1 x 1 + S2 x 2 + S3 x 3 + O(r −1 ). 2M 2 r
(A.11)
Thus, with J = [(S 1 )2 + (S 2 )2 + (S 3 )2 ]1/2 , Dμ zDμ z =
3 (∂ j z)2 + O(r −3 ) j=1
3 1 j −1 = [S r − x j r −3 (S 1 x 1 + S 2 x 2 + S 3 x 3 )]2 + O(r −3 ) 4M 4 j=1
1 = [J 2 r −2 − r −4 (S 1 x 1 + S 2 x 2 + S 3 x 3 )2 ] + O(r −3 ). 4M 4 It follows that 1 (S 1 x 1 + S 2 x 2 + S 3 x 3 )2 r −2 + r 2 Dμ zDμ z + O(r −1 ) 4M 4 J2 = + O(r −1 ). (A.12) 4M 4
z 2 + 4M 2 (y 2 + z 2 )Dμ zDμ z =
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References 1. Alexakis, S.: Unique continuation for the vacuum Einstein equations. Preprint (2008), http://arxiv.org/ abs/0902.1131v2[gr-qc], 2009 2. Alexakis, S., Ionescu, A.D., Klainerman, S.: Hawking’s local rigidity theorem without analyticity. Preprint (2009), http://arxiv.org/abs/0902.1173v1[gr-qc], 2009 3. Beig, R., Simon, W.: The stationary gravitational field near spatial infinity. Gen. Rel. Grav. 12, 1003–1013 (1980) 4. Beig, R., Simon, W.: On the multipole expansion for stationary space-times. Proc. Roy. Soc. London Ser. A 376, 333–341 (1981) 5. Bunting, G.L.: Proof of the Uniqueness Conjecture for Black Holes. PhD Thesis, Univ. of New England, Armidale, NSW, 1983 6. Bunting, G., Massood-ul-Alam, A.K.M.: Nonexistence of multiple black holes in asymptotically Euclidean static vacuum space-time. Gen. Rel. Grav. 19, 147–154 (1987) 7. Carter, B.: An axy-symmetric black hole has only two degrees of freedom. Phys. Rev. Lett. 26, 331–333 (1971) 8. Carter, B.: Black hole equilibrium states. In: Black holes/Les astres occlus (École d’Été Phys. Théor., Les Houches, 1972), New York: Gordon and Breach, 1973, pp. 57–214 9. Carter, B.: Has the Black Hole Equilibrium Problem Been Solved? In: The Eighth Marcel Grossmann meeting, Part A, B (Jerusalem, 1997), River Edge, NJ: World Sci. Publ., 1999, pp. 136–155 10. Christodoulou, D., Klainerman, S.: The global nonlinear stability of the Minkowski space, Princeton Math. Series 41, Princeton University Press, 1993 11. Chrusciel, P.T.: On completeness of orbits of Killing vector fields. Class. Quant. Grav. 10, 2091–2101 (1993) 12. Chrusciel, P.T.: “No Hair” Theorems-Folclore, Conjecture, Results. Diff. Geom. and Math. Phys. (J. Beem and K.L. Duggal) Cont. Math. 170, Providence, RI: AMS, 1994, pp. 23–49 13. Chrusciel, P.T.: On the rigidity of analytic black holes. Commun. Math. Phys. 189, 1–7 (1997) 14. Chrusciel, P.T., Wald, R.M.: Maximal hypersurfaces in stationary asymptotically flat spacetimes. Commun. Math. Phys. 163, 561–604 (1994) 15. Chrusciel, P.T., Wald, R.M.: On the topology of stationary black holes. Class. Quant. Grav. 11, L147– L152 (1993) 16. Chrusciel, P.T., Delay, T., Galloway, G., Howard, R.: Regularity of horizon and the area theorem. Ann. H. Poincaré 2, 109–178 (2001) 17. Chrusciel, P.T., Costa, J.L.: On uniqueness of stationary vacuum black holes. Preprint (2008), http://arxiv. org/abs/0806.0016v2[gr-qc], 2008 18. Chrusciel, P.T.: On higher dimensional black holes with abelian isometry group. J. Math. Phys. 50, 052501 (2009) 19. Friedman, J.L., Schleich, K., Witt, D.M.: Topological censorship. Phys. Rev. Lett. 71, 1846–1849 (1993) 20. Friedrich, H., Rácz, I., Wald, R.: On the rigidity theorem for spacetimes with a stationary event horizon or a compact Cauchy horizon. Commun. Math. Phys. 204, 691–707 (1999) 21. Hawking, S.W., Ellis, G.F.R.: The large scale structure of space-time, Cambridge: Cambridge Univ. Press, 1973 22. Heusler, M.: Black Hole Uniqueness Theorems, Cambridge Lect. Notes in Phys, Cambridge: Cambridge Univ. Press, 1996 23. Hörmander, L.: The analysis of linear partial differential operators IV. Fourier integral operators. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 275. Berlin: Springer-Verlag, 1985 24. Ionescu, A.D., Klainerman, S.: On the uniqueness of smooth, stationary black holes in vacuum. Invent. Math. 175, 35–102 (2009) 25. Ionescu, A.D., Klainerman, S.: Uniqueness results for ill-posed characteristic problems in curved spacetimes. Commun. Math. Phys. 285, 873–900 (2009) 26. Isenberg, J., Moncrief, V.: Symmetries of Cosmological Cauchy Horizons. Commun. Math. Phys. 89, 387–413 (1983) 27. Israel, W.: Event horizons in static vacuum space-times. Phys. Rev. Lett. 164, 1776–1779 (1967) 28. Klainerman, S., Nicolò, F.: The evolution problem in general relativity. Progress in Mathematical Physics, 25. Boston, MA: Birkhäuser Boston, Inc., 2003 29. Mars, M.: A spacetime characterization of the Kerr metric, Class. Quant. Grav. 16, 2507–2523 (1999) 30. Mazur, P.O.: Proof of Uniqueness for the Kerr-Newman Black Hole Solution. J. Phys. A: Math. Gen. 15, 3173–3180 (1982) 31. Robinson, D.C.: Uniqueness of the Kerr black hole. Phys. Rev. Lett. 34, 905–906 (1975) 32. Racz, I., Wald, R.: Extensions of space-times with Killing horizons. Class. Quant. Gr. 9, 2463–2656 (1992)
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33. Simon, W.: Characterization of the Kerr metric. Gen. Rel. Grav. 16, 465–476 (1984) 34. Sudarski, D., Wald, R.M.: Mass formulas for stationary Einstein Yang-Mills black holes and a simple proof of two staticity theorems. Phys. Rev. D47, 5209–5213 (1993) 35. Weinstein, G.: On rotating black holes in equilibrium in general relativity. Comm. Pure Appl. Math. 43, 903–948 (1990) Communicated by P.T. Chru´sciel
Commun. Math. Phys. 299, 129–161 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1069-9
Communications in
Mathematical Physics
Integrable Evolution Equations on Spaces of Tensor Densities and Their Peakon Solutions Jonatan Lenells1 , Gerard Misiołek2 , Feride Ti˘glay3,4 1 Institut für Angewandte Mathematik, Leibniz Universität Hannover, D-30167 Hannover, Germany.
E-mail:
[email protected] 2 Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA.
E-mail:
[email protected] 3 Department of Mathematics, University of New Orleans, Lake Front, New Orleans, LA 70148, USA 4 Section de Mathématiques, École Polytechnique Fédérale de Lausanne, CH–1015 Lausanne, Switzerland.
E-mail:
[email protected] Received: 17 July 2009 / Accepted: 13 February 2010 Published online: 3 July 2010 – © Springer-Verlag 2010
Abstract: We study a family of equations defined on the space of tensor densities of weight λ on the circle and introduce two integrable PDE. One of the equations turns out to be closely related to the inviscid Burgers equation while the other has not been identified in any form before. We present their Lax pair formulations and describe their bihamiltonian structures. We prove local wellposedness of the corresponding Cauchy problem and include results on blow-up as well as global existence of solutions. Moreover, we construct “peakon” and “multi-peakon” solutions for all λ = 0, 1, and “shock-peakons” for λ = 3. We argue that there is a natural geometric framework for these equations that includes other well-known integrable equations and which is based on V. Arnold’s approach to Euler equations on Lie groups. Contents 1. 2. 3. 4.
5.
6.
Introduction . . . . . . . . . . . . . . . . . . . . A Family of Equations on the Space of λ Densities Lax Pairs . . . . . . . . . . . . . . . . . . . . . . Hamiltonian Structures and Conserved Quantities 4.1 Bihamiltonian structure of μDP . . . . . . . 4.2 Hamiltonian formulation of μ-equations . . . 4.3 Orbits in Fλ . . . . . . . . . . . . . . . . . . The Cauchy Problem for the μDP Equation . . . . 5.1 Local wellposedess in Sobolev spaces . . . . 5.2 Blow-up of smooth solutions . . . . . . . . . 5.3 Global solutions . . . . . . . . . . . . . . . . 5.4 Local wellposedness for any λ . . . . . . . . Peakons . . . . . . . . . . . . . . . . . . . . . . 6.1 Traveling waves . . . . . . . . . . . . . . . . 6.2 Periodic one-peakons . . . . . . . . . . . . .
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6.3 Green’s function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Multi-peakons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Poisson structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Multi-peakons for μCH . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Shock-peakons for μDP . . . . . . . . . . . . . . . . . . . . . . . . . . 7. The μBurgers Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Burgers’ equation and the L 2 -geometry of Diff s → Diff s /S 1 . . . . . . 7.2 Lifespan of solutions, Hamiltonian structure and conserved quantities of the μBurgers equation . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Multidimensional μCH and μDP Equations . . . . . . . . . . . . . . . . . Appendix A. Proof of Theorem 6.2 . . . . . . . . . . . . . . . . . . . . . . . . . Appendix B. Proof of Theorem 6.3 . . . . . . . . . . . . . . . . . . . . . . . . . Appendix C. Conservation of H2 for μDP . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
146 146 148 149 150 151 151 152 154 155 157 159 160
1. Introduction Integrability of an infinite-dimensional dynamical system typically manifests itself in several different ways such as the existence of a Lax pair formulation or a bihamiltonian structure, the presence of an infinite family of conserved quantities or at least the ability to write down explicitly some of its solutions. In this paper we introduce and study two nonlinear partial differential equations and show that they possess all the hallmarks of integrability mentioned above. The first of these equations we shall refer to as the μBurgers (μB) equation1 −u t x x − 3u x u x x − uu x x x = 0, and the second as the μDP equation, μ(u t )−u t x x +3μ(u)u x −3u x u x x −uu x x x = 0,
(μB)
1
where μ(u) =
u d x.
(μDP)
0
Here u(t, x) is a spatially periodic real-valued function of a time variable t and a space variable x ∈ S 1 [0, 1). Both of these equations belong to a larger family that also includes the Camassa-Holm equation [CH] (see also [FF]) u t − u t x x + 3uu x − 2u x u x x − uu x x x = 0,
(CH)
the Hunter-Saxton equation [HuS] u t x x + 2u x u x x + uu x x x = 0,
(HS)
the Degasperis-Procesi equation [DP] u t − u t x x + 4uu x − 3u x u x x − uu x x x = 0,
(DP)
as well as the μ-equation which was derived recently in [KLM] (we will refer to it as the μCH equation in this paper)2 u t x x − 2μ(u)u x + 2u x u x x + uu x x x = 0,
(μCH)
all of which are known to be integrable. 1 This equation is mentioned in Remark 3.9 of [HoS] and Remark 3.2 of [Lu] as the high-frequency limit of the Degasperis-Procesi equation, see (DP) below. Our terminology will be explained in Sect. 7. 2 In [KLM] the authors refer to μCH as the μHS equation.
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131
Fig. 1. The (λ = 2) and (λ = 3) families of equations
One of the distinguishing features of the CH and DP equations which makes them attractive among integrable equations is the existence of so-called “peakon” solutions. In fact, CH and DP are the cases λ = 2 and λ = 3, respectively, of the following family of equations (Fig. 1): u t − u t x x + (λ + 1)uu x = λu x u x x + uu x x x ,
λ ∈ Z,
(1.1)
with each equation in the family admitting peakons (see [DHH]) although only λ = 2 and λ = 3 are believed to be integrable (see [DP]). One of our results will show that each equation in the corresponding μ-version of the family (1.1) given by μ(u t ) − u t x x + λμ(u)u x = λu x u x x + uu x x x ,
λ ∈ Z,
(1.2)
also admits peakon solutions. The choices λ = 2 and λ = 3 yield the μCH and μDP equations, respectively, and as with (1.1), we expect that these are the only integrable members of the family (1.2). Moreover, we will show that the μDP equation admits shock-peakon solutions of a form similar to those known for DP, see [Lu]. In Sect. 2, we present a natural setting in which all the equations above can be formally described as evolution equations on the space of tensor densities (of different weight λ) over the Lie algebra of smooth vector fields on the circle. In Sect. 3 we present Lax pairs for μDP and μB, establishing their integrability. In Sect. 4 we describe the Hamiltonian structure of the equations in (1.2). In particular, we consider the bihamiltonian structure of the μDP equation together with the associated infinite hierarchy of conservation laws. In Sect. 5 we study the periodic Cauchy problem of μDP; we prove local wellposedness in Sobolev spaces and show that while classical solutions of μDP break down for certain intial data, the equation admits global solutions for other data. In Sect. 6 we construct multi-peakon solutions of (1.2) as well as shock-peakons of μDP. In Sect. 7 we discuss the μB equation and its properties and present a geometric construction to explain its close relation to the (inviscid) Burgers equation. Finally, in Sect. 8 we consider generalizations of the μCH and μDP equations to a multidimensional setting. Left open is the question of physical significance, if any, of the two equations μDP and μB. It is possible that they may play a role in the mathematical theory of water waves (as CH, see [CH]) or find applications in the study of more complex equations (as e.g. HS, see [ST]). We do not pursue these issues here. Our approach draws heavily on [KM] and [KLM] and the present work is in some sense a continuation of those two papers. 2. A Family of Equations on the Space of λ Densities Perhaps the simplest way to introduce the equations that are the main object of our investigation is by analogy with the known cases. We shall therefore first briefly review Arnold’s approach to the Euler equations on Lie groups and then describe a more general set-up intended to capture the μDP and the μB equations.
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In a pioneering paper Arnold [A] presented a general framework within which it is possible to employ geometric and Lie theoretic techniques to study a variety of equations (ODE as well as PDE) of interest in mathematical physics (see also more recent expositions in [KM] or [KW]). Arnold’s principal examples were the equations of motion of a rigid body in R3 and the equations of ideal hydrodynamics. The formal set-up is the following. Consider a possibly infinite-dimensional Lie group G (the “configuration space” of a physical system) with Lie algebra g. Choose an inner product ·, · on g (essentially, the “kinetic energy” of the system) and, using right(or left-) translations, endow G with the associated right- (or left-) invariant Riemannian metric. The motions of the system can now be studied either through the geodesic equation defined by the metric on G (equivalently, the geodesic flow on its tangent or cotangent bundles) or else directly on the Lie algebra g using Hamiltonian reduction.3 The equation that one obtains by this procedure on g is called the Euler (or EulerArnold) equation. Using the inner product it can be reformulated as an equation on the dual algebra g∗ as follows. Let (·, ·) be the natural pairing between g and g∗ and let A : g → g∗ denote the associated inertia operator4 determined by the formula (Au, v) = u, v for any u, v ∈ g. The Euler equation on g∗ reads m t = −ad∗A−1 m m, where
ad∗
:g→
End(g∗ )
m = Au ∈ g∗ ,
(E)
is the coadjoint representation of g given by (ad∗u m, v) = −(m, [u, v])
(2.1)
for any u, v ∈ g and m ∈ g∗ . ad∗u is the infinitesimal version of the coadjoint action Ad∗ : G × g∗ → g∗ of the group G on the dual algebra g∗ . In our case G will be the group Diff(S 1 ) of orientation-preserving diffeomorphisms of the circle whose Lie algebra g is the space of smooth vector fields Te Diff(S 1 ) = vect(S 1 ). The dual vect ∗ (S 1 ) is the space of distributions on S 1 but we shall consider only its “regular part” which can be identified with the space of quadratic differentials F2 = m(x)d x 2 : m ∈ C ∞ (S 1 ) with the pairing given by 1 2 md x , v∂x = m(x)v(x) d x 0
(see e.g. Kirillov [K]). The coadjoint representation of vect(S 1 ) on the regular part of its dual space is in this case precisely the action of vect(S 1 ) on the space of quadratic differentials. By a direct calculation using (2.1) we have ad∗u∂x md x 2 = (um x + 2u x m) d x 2
(2.2)
and the Euler equation (E) on g∗ takes the form m t = −ad∗A−1 m m = −um x − 2u x m,
m = Au.
(2.3)
Equivalently, when rewritten on vect(S 1 ), Eq. (2.3) becomes Au t + 2u x Au + u Au x = 0,
(2.4)
3 We will be making use of both structures in this paper. 4 In infinite-dimensional examples A is often some positive-definite self-adjoint pseudodifferential operator
acting on the space of smooth tensor fields on a compact manifold.
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so that with an appropriate choice of the inertia operator A (which is equivalent to picking an inner product on Te Diff) ⎧ 2 ⎪ ⎨1 − ∂ x A = μ − ∂x2 ⎪ ⎩ 2 −∂x
for CH, for μCH,
(2.5)
for HS,
one recovers the CH, μCH and HS equations5 as Euler equations on vect(S 1 ); these results can be found in [M1,KM and KLM]. We now want to extend this formalism to include the DP equation as well as the two equations of principal interest in this paper: μDP and μB. Admittedly, our construction does not have the same beautiful geometric interpretation as that of Arnold’s and hence perhaps is not completely satisfactory. The main point is to replace the space of quadratic differentials with the space of all tensor densities on the circle of weight λ.6 Recall that a tensor density of weight λ ≥ 0 (respectively λ < 0) on S 1 is a section of the bundle λ T ∗ S 1 (respectively −λ T S 1 ) and set
Fλ = m(x)d x λ : m(x) ∈ C ∞ (S 1 ) . There is a well-defined action of the diffeomorphism group Diff(S 1 ) on each density module Fλ given by Fλ md x λ → m ◦ ξ (∂x ξ )λ d x λ ∈ Fλ ,
ξ ∈ Diff(S 1 ),
(2.6)
which naturally generalizes the coadjoint action Ad∗ : Diff(S 1 ) → Aut (F2 ) on the space of quadratic differentials. The infinitesimal generator of the action in (2.6) is easily calculated, L λu∂x (md x λ ) = (um x + λu x m) d x λ ,
(2.7)
and can be thought of as the Lie derivative of tensor densities. It represents the action of vect(S 1 ) on Fλ which for λ = 2 coincides with the (algebra) coadjoint action on F2 (that is L 2u∂x = ad∗u∂x ). If we think of (2.7) as defining a vector field on the space Fλ , then we can consider the equation for its flow m t = −um x − λu x m
(2.8)
in analogy with (2.3). The substitution m = Au transforms (2.8) into an equation on the space of quadratic differentials, Au t + λu x Au + u Au x = 0,
(2.9)
which is the λ-version of (2.4). 5 Note that in the case of the HS equation the inertia operator is degenerate, see [KM], Sect. 4, for details. 6 We refer to [O,OT or GR] for basic facts about the space of tensor densities.
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Setting λ = 3 and choosing suitable inertia operators A as above we obtain the DP, μDP and μB equations. More precisely, the specific choices of A are in parallel with those made for the λ = 2 family in (2.5), that is ⎧ 2 for DP, ⎪ ⎨1 − ∂x 2 A = μ − ∂x (2.10) for μDP, ⎪ ⎩ 2 −∂x for μB. More generally, letting A = μ − ∂x2 in (2.9) for any λ ∈ Z, we find the equations in (1.2). 3. Lax Pairs An elegant manifestation of complete integrability of an infinite dimensional dynamical system is the existence of a Lax pair formalism. It is often used as a tool for constructing infinite families of conserved quantities. This formalism has been quite extensively developed in recent years for the CH and DP equations, see e.g. [BSS,CM,DHH,LS]. Our next result describes the Lax pair formulations for the μDP and μBurgers equations. Theorem 3.1. The μDP and the μB equations admit the Lax pair formulations ψx x x = −λmψ, ψt = − λ1 ψx x − uψx + u x ψ,
(3.1)
where λ ∈ C is a spectral parameter, ψ(t, x) is a scalar eigenfunction and m = Au with A = μ − ∂x2 or A = −∂x2 as defined in (2.10). Proof. This is a straightforward computation which shows that the condition of compatibility (ψt )x x x = (ψx x x )t of the linear system (3.1) is equivalent to the equation μDP or μB when m is given by m = μ(u) − u x x or m = −u x x , respectively. 4. Hamiltonian Structures and Conserved Quantities In this section we describe the bihamiltonian structure of μDP (see Sect. 7 for the bihamiltonian structure of the μBurgers equation) as well as the Hamiltonian structure for the family of μ-equations (1.2) for any λ ∈ Z. In the last subsection we consider the orbits of Diff(S 1 ) in the space of tensor densities of weight λ and give a geometric interpretation of one of the conserved quantities. 4.1. Bihamiltonian structure of μDP. Recall that DP admits the bihamiltonian formulation [DHH] m t = J0 where H0 = −
9 2
δ H0 δ H2 = J2 , δm δm
m dx
and
H2 = −
1 6
u 3 d x,
Integrable Evolution Equations and Peakons
135
and the Hamiltonian operators are −1 m 1/3 ∂x m 2/3 and J2 = ∂x 4 − ∂x2 1 − ∂x2 J0 = m 2/3 ∂x m 1/3 ∂x − ∂x3 with m = (1 − ∂x2 )u. Similarly, the μDP equation admits the bihamiltonian formulation m t = J0
δ H0 δ H2 = J2 , δm δm
where now the Hamiltonian functionals H0 and H2 are 9 H0 = − 2
m dx
and
H2 = −
2 1 3 −1 μ(u) A ∂x u + u 3 d x, 2 6
(4.1)
the operators J0 and J2 are given by J0 = −m 2/3 ∂x m 1/3 ∂x−3 m 1/3 ∂x m 2/3
and
J2 = −∂x3 A = ∂x5 ,
and m = Au with A = μ − ∂x2 . The fact that J0 and J2 form a compatible bihamiltonian pair is a consequence of Theorem 2 in [HW]. It can be verified directly that H0 and H2 , as well as H1 = 21 u 2 d x are conserved in time whenever u is a solution of μDP (see Appendix C for details of this calculation in the case of H2 ). Using the standard techniques we can now construct an infinite sequence of conservation laws . . . H−1 , H0 , H1 , H2 , . . . ,
(4.2)
see Fig. 2. As in the case of the CH and DP equations the Hn ’s are nonlocal and not easy to write down explicitly for n ≥ 3, while they are readily computable recursively in terms of m and its derivatives for negative n. In fact, the first negative flow in the μDP hierarchy is μDP−1 : m t = −
2 6160m 5x − 13200mm x x m 3x + 3600m 2 m 2x m x x x 17/3 729m
− 675m 2 mm x x x x − 8m 2x x m x +27m 3 (3mm x x x x x − 50m x x m x x x ) .
Remark 4.1. One peculiarity of the μDP and D P equations is that the recursion operator J2−1 J0 maps δ Hn /δm to δ Hn+2 /δm and not to δ Hn+1 /δm. Thus, the family (4.2) is divided into two families consisting of the Hn ’s with even and odd values of n, respectively. One would like to define an operator J1 which satisfies J0 = J1 J2−1 J1 and such that J1 and J2 form a bihamiltonian pair with the corresponding Hamiltonians H1 and H2 . However, it appears difficult to find an expression for such a J1 .
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J. Lenells, G. Misiołek, F. Tı˘glay
Fig. 2. Recursion scheme for the μDP equation
4.2. Hamiltonian formulation of μ-equations. Although we expect that μCH and μDP are the only equations among the family of μ-equations in (1.2) which admit a bihamiltonian structure, we can nevertheless provide one Hamiltonian structure for any λ = 1. Indeed, if we set J0 = −
1 (m x + λm∂x )∂x−3 ((λ − 1)m x + λm∂x ) λ2
and H0 = −
λ2 λ−1
m d x,
then Eq. (1.2) is equivalent to δ H0 . δm In order to see that J0 is a Hamiltonian operator we first rewrite it in the form m t = J0
J0 = −m (λ−1)/λ ∂x m 1/λ ∂x−3 m 1/λ ∂x m (λ−1)/λ and refer to Theorem 1 of [HW].
(4.3)
(4.4)
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137
4.3. Orbits in Fλ . From Sect. 2 we know that for any value of λ a solution m of the Cauchy problem7 for Eq. (1.2) with initial data m 0 belongs to the orbit of m 0 in Fλ under the Diff(S 1 )-action defined in (2.6). It is therefore tempting to exploit this setting to study the family in (1.2) as in the case of the coadjoint action when λ = 2. We present here two results in this direction leaving a detailed investigation of the geometry of the corresponding orbits for general λ for a future work. We let L ξ (md x λ ) = m ◦ξ(∂x ξ )λ d x λ denote the action of an element ξ ∈ Diff(S 1 ) on md x λ in Fλ . The following two propositions reflect the fact that m 1/λ d x transforms as 1 a one-form. The first proposition shows the conservation of H−1 = 0 |m|1/λ d x under the flow of (1.2). Proposition 4.2. The map m d x λ → H−1 [m] =
0
1
|m|1/λ d x : Fλ → R+
is invariant under the action (2.6) of Diff(S 1 ) on Fλ . Proof. This is a straightforward change of variables 1 m ◦ ξ (∂x ξ )λ 1/λ d x = H−1 L ξ (m d x λ ) = 0
1
0
|m|1/λ ◦ ξ dξ = H−1 [m],
since ξ is a smooth orientation-preserving circle diffeomorphism and m d x λ ∈ Fλ .
In many respects the classification of orbits in Fλ resembles that of coadjoint orbits of Diff(S 1 ), cf. [GR]. In order to state the next result we denote by Mλ the set of those elements md x λ in Fλ such that m(x) = 0 for all x ∈ S 1 . Proposition 4.3. The orbit space Mλ /Diff(S 1 ) is in bijection with the set R\{0}. More precisely, the map 1 md x λ → sgn(m)(H−1 )λ : Mλ → R\{0}, H−1 = |m|1/λ d x, (4.5) 0
Diff(S 1 )-orbits
in Mλ ⊂ Fλ and induces a one-to-one corresponis constant on the dence between orbits in Mλ and R\{0}. Proof. That the map is constant on the orbits follows from Proposition 4.2. To see that it is surjective we note that for any nonzero real number a the element ad x λ is mapped to a. In order to prove injectivity we observe that the orbit through an arbitrary element md x λ of Mλ contains an element of the form ad x λ , where a is a constant. Indeed, given md x λ we define ξ ∈ Diff(S 1 ) by x 1 |m|1/λ d x. ξ(x) = H−1 0 Then
λ λ L ξ sgn(m)H−1 d x λ = sgn(m)H−1 (∂x ξ )λ d x λ = md x λ .
λ d x λ , where sgn(m)H λ ∈ R\{0} is Thus, the action of ξ −1 maps m to sgn(m)H−1 −1 constant. 7 See Sect. 5.4 for results on wellposedness of Eqs. (1.2) for any λ.
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The stabilizer in Diff(S 1 ) of a constant element ad x λ in Mλ (a ∈ R) consists exactly of the rigid rotations and is isomorphic to S 1 . It follows from Proposition 4.3 that the stabilizer of any point md x λ ∈ Mλ is conjugate to S 1 within Diff(S 1 ). In particular, each orbit in Mλ is of the form Diff(S 1 )/S 1 . 5. The Cauchy Problem for the μDP Equation In this section we turn to the periodic Cauchy problem for μDP. Rather than aiming at the strongest possible theorems we present here only basic results that display the interesting behavior of solutions. We start with local existence, uniqueness and persistence theorems for the μDP equation in Sobolev spaces. Next, we describe the breakdown of smooth solutions and prove a global existence result for the class of initial data with non-negative momentum density. At the end of this section we briefly discuss local wellposedness of the whole family (1.2) of μ-equations. In order to prove these results we will need Sobolev completions of the group of circle diffeomorphisms8 Diff(S 1 ) and its Lie algebra of smooth vector fields vect(S 1 ). We denote by H s = H s (S 1 ) the Sobolev space of periodic functions 2 s 1 2πinx 2 s v(n) < ∞ , H (S ) = v = v(n)e ˆ : v H s = n
n
where s ∈ R and the pseudodifferential operator s = (1 − ∂x2 )s/2 is defined by s v(n) = (1 + 4π 2 n 2 )s/2 v(n). ˆ In what follows we will also make use of another elliptic operator 2μ : H s (S 1 ) → H s−2 (S 1 ),
2μ v = μ(v) − vx x
(5.1)
whose inverse can be easily checked to be 1 1 x 2 x 13 x 1 − + −2 v(x) = v(x) d x + x − v(y) d yd x μ 2 2 12 0 2 0 0 x y 1 x y − v(z) dzdy + v(z) dzdyd x. (5.2) 0
0
0
0
0
5.1. Local wellposedess in Sobolev spaces. We begin with the μDP equation. Our proof follows with minor changes the approach of [M2] to the CH equation based on the original methods in [EM] developed for the Euler equations of hydrodynamics. We include it for the sake of completeness. The two above papers and [KLM] will be our main references for most of the basic facts we use in this section. We can write the Cauchy problem for μDP in the form u t + uu x + 3μ(u) ∂x −2 μ u = 0, u(0) = u 0 .
(5.3) (5.4)
Note that applying 2μ to both sides of (5.3) we obtain the equation in its original form given in the Introduction. 8 Although no longer a Lie group, Diff s (S 1 ) retains the structure of a topological group for a sufficiently high Sobolev index; see below.
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Theorem 5.1. (Local wellposedness and persistence) Assume s > 3/2. Then for any u 0 ∈ H s (T) there exists a T > 0 and a unique solution u ∈ C (−T, T ), H s ∩ C 1 (−T, T ), H s−1 of the Cauchy problem (5.3)-(5.4) which depends continuously on the initial data u 0 . Furthermore, the solution persists as long as u(t, ·)C 1 stays bounded. Our strategy will be to reformulate (5.3)-(5.4) as an initial value problem on the space of circle diffeomorphisms Diff s (S 1 ) of Sobolev class H s . It is well known that whenever s > 3/2 this space is a smooth Hilbert manifold and a topological group. We will then show that the reformulated problem can be solved on Diff s (S 1 ) by standard ODE techniques. Let u = u(t, x) be a solution of μDP with initial data u 0 . Then its associated flow t → ξ(t, x), i.e. the solution of the initial value problem9 ξ˙ (t, x) = u(t, ξ(t, x)), ξ(0, x) = x
(5.5)
is (at least for a short time) a C 1 smooth curve in the space of diffeomorphisms starting from the identity e ∈ Diff s (S 1 ). Differentiating both sides of this equation in t and using (5.3) we obtain the following initial value problem: −1 ˙ ˙ (5.6) ξ¨ = −3μ ξ˙ ◦ ξ −1 ∂x −2 μ (ξ ◦ ξ ) ◦ ξ =: −F(ξ, ξ ), ξ(0, x) = x, ξ˙ (0, x) = u 0 (x).
(5.7)
In this section we make repeated use of a technical result which we state here for convenience (for a proof refer to [M2], Appendix 1). Lemma 5.2. For s > 3/2 the composition map ξ → ω◦ξ with an H s function ω and the inversion map ξ → ξ −1 are continuous from Diff s (S 1 ) to H s (S 1 ) and from Diff s (S 1 ) to itself respectively. Moreover, ω ◦ ξ H s ≤ C(1 + ξ sH s )ω H s
(5.8)
with C depending only on inf |∂x ξ | and sup |∂x ξ |. It will also be convenient to introduce the notation Aξ = Rξ ◦ A ◦ Rξ −1 for the conjugation of an operator A on H s (S 1 ) by a diffeomorphism ξ ∈ Diff s (S 1 ). The right-hand side of (5.6) then becomes ˙ F(ξ, ξ˙ ) = −2 μ,ξ ∂x,ξ h(ξ, ξ ), where h(ξ, ω) = 3ω
1 0
(5.9)
ω ◦ ξ −1 d x.
Proof of Theorem 5.1. From Lemma 5.2 we readily see that F maps into H s (S 1 ). We aim to prove that F is Fréchet differentiable in a neighborhood of the point (e, 0) in Diff s × H s (S 1 ). To this end we compute the directional derivatives ∂ξ F(ξ,ω) and ∂ω F(ξ,ω) 9 Here “dot” indicates differentiation in t variable.
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and show that they are bounded linear operators on H s which depend continuously on ξ and ω. We use the formulas −2 −2 −1 2 (5.10) ∂ξ −2 μ,ξ (v) = −μ,ξ v ◦ ξ ∂x , μ ξ μ,ξ , (5.11) ∂ξ ∂x,ξ (v) = v ◦ ξ −1 ∂x , ∂x , ξ
and
∂ξ h (ξ,ω) (v) = 3 ω
ω ◦ ξ −1 ∂x (v ◦ ξ −1 ) d x
(5.12)
along with (5.9) to get
−1 −2 2 −1 ∂ξ F(ξ,ω) (v) = 3 −v ◦ ξ μ ∂x (ω ◦ ξ ) ω ◦ ξ −1 d x −2 −1 −1 + μ ∂x (v ◦ ξ )∂x (ω ◦ ξ ) (ω ◦ ξ −1 ) d x −1 −1 −1 − −2 ∂ (ω ◦ ξ ) (ω ◦ ξ ) ∂ (v ◦ ξ ) d x ◦ ξ, x μ x
which with the help of the identity 2 −2 μ ∂x = −1 + μ
(5.13)
can be simplified to
∂ξ F(ξ,ω) (v) = 3 v ω
2
−1
dx − v ω ◦ ξ dx + −2 ω ◦ ξ −1 d x μ,ξ ∂x,ξ v ∂x,ξ ω −1 −1 − −2 ∂ ω ω ◦ ξ ∂ (v ◦ ξ ) d x . x μ,ξ x,ξ
On the other hand we have
ω◦ξ
−1
∂ω h (ξ,ω) (v) = 3v and similarly ∂ω F(ξ,ω) (v) = −3−2 μ,ξ ∂x,ξ v
ω ◦ ξ −1 d x + 3ω
v ◦ ξ −1 d x
ω ◦ ξ −1 d x − 3−1 μ,ξ ∂x,ξ ω
v ◦ ξ −1 d x.
In order to show that v → ∂ξ F(ξ,ω) (v) is a bounded operator on H s it is sufficient to estimate the sum ∂ξ F(ξ,ω) (v) L 2 + ∂x ∂ξ F(ξ,ω) (v) ◦ ξ −1 H s−1 . Using Cauchy-Schwarz, Lemma 5.2 and the formulas above both of these terms can be bounded by a multiple of v H s ω2H s which gives the estimate ∂ξ F(ξ,ω) (v) H s ≤ Cv H s ω2H s ,
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where C depends only on inf |∂x ξ | and sup |∂x ξ |. A similar argument yields the bound for the other directional derivative ∂ω F(ξ,ω) (v) H s ≤ Cv H s ω H s . It follows that F is Gâteaux differentiable near (e, 0) and therefore we only need to establish continuity of the directional derivatives. Continuity in the ω-variable follows from the fact that the dependence of both partials on this variable is polynomial. It thus only remains to show that the norm of the difference ∂ξ F(ξ,ω) (v) − ∂ξ F(e,ω) (v)
Hs
is arbitrarily small whenever ξ is close to the identity e ∈ Diff s (S 1 ), uniformly in v and ω ∈ H s (S 1 ). Once again, it suffices to estimate the terms ∂ξ F(ξ,ω) (v) − ∂ξ F(e,ω) (v)
L2
+ ∂x ∂ξ F(ξ,ω) (v) − ∂ξ F(e,ω) (v) H s−1 .
(5.14)
We proceed as above. Using formula (5.2) and Lemma 5.2 together with the fact that for any r > 0 we have a continuous embedding H r (S 1 ) → C r −1/2 (S 1 ) with a pointwise estimate g(x) − g(y) g H r |x − y|r −1/2 , we can bound both terms in (5.14) by Cω2H s v H s
−1 ξ − e
Hs
s−3/2 −1 . + ξ − e ∞
Continuity of ξ → ∂ω F(ξ,ω) follows from a similar estimate. In this way we obtain that F is continuously differentiable near (e, 0) and applying the fundamental ODE theorem for Banach spaces (see e.g. Lang [L]) establish local existence, uniqueness and smooth dependence on the data u 0 of solutions ξ(t) and ξ˙ (t) of (5.6)-(5.7). Local wellposedness of the original Cauchy problem (5.3)-(5.4) follows now at once from u = ξ˙ ◦ ξ −1 and the fact that Diff s is a topological group whenever s > 3/2. In order to complete the proof of Theorem 5.1 we need to derive a C 1 bound on the solution u. The standard trick is to use Friedrichs’ mollifiers J with (0 < < 1) to derive the inequality 1 3 s J u ∂x s J u 2 d x − μ(u) 2 J u ∂x s −2 μ J u d x 2 0 0 s 2 uC 1 u2H s + |μ(u)| ∂x s −2 μ J u 2 J u L 2 uC 1 u H s ,
d J u2H s = − dt
1
L
where the first estimate is standard (see e.g. [T], Chap. 5) and the second follows easily with the help of (5.2) and (5.13). Passing to the limit with → 0 and using Gronwall’s inequality yields the persistence result.
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5.2. Blow-up of smooth solutions. A family of examples displaying a simple finite-time breakdown mechanism of a μDP solution is described by the following result. Theorem 5.3. Given any smooth periodic function u 0 with zero mean there exists Tc > 0 such that the corresponding solution of the μDP equation stays bounded for t < Tc and satisfies u x (t)∞ ∞ as t Tc . Proof. Let u be the solution of μDP with initial data u 0 . Differentiating the equation with respect to the space variable we obtain u t x + uu x x + u 2x + 3μ(u)∂x2 −2 μ u = 0. Since ∂x2 −2 μ = μ − 1, using conservation of the mean, we find that u t x + uu x x + u 2x = 3μ(u) (u − μ(u)) = 0
(5.15)
by the assumption on the data u 0 . If we let ξ(t) denote the flow of u as in (5.5) then ∂x ξ˙ = u x ◦ ξ ∂x ξ, and setting w = ∂x ξ˙ /∂x ξ we find that wt =
∂x ξ¨ ∂x ξ − (∂x ξ˙ )2 = (u xt + u x x u) ◦ ξ. (∂x ξ )2
With the help of these formulas we rewrite (5.15) in the form wt + w 2 = 0 and, since w(0, x) = u 0x (x), solve for w to get w(t, x) =
1 . t + (1/u 0x (x))
Furthermore, our assumptions are such that it is always possible to find a point x ∗ ∈ S 1 such that u 0x (x ∗ ) is negative. Setting Tc = −1/u 0x (x ∗ ) we conclude that the solution w must blow-up in the L ∞ norm as expected. More sophisticated break-down mechanisms for μDP can be demonstrated but we will not pursue them in this paper. 5.3. Global solutions. Our global result for μDP is obtained under a sign assumption on the initial data of the type that has been used previously in the studies of equations such as CH, μCH and DP. What follows is again a basic result. Theorem 5.4. Let s > 3. Assume that u 0 ∈ H s (S 1 ) has non-zero mean and satisfies the condition 2μ u 0 ≥ 0 (or ≤ 0). Then the Cauchy problem for μDP has a unique global solution u in C(R, H s (S 1 )) ∩ C 1 (R, H s−1 (S 1 )).
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Proof. From the local wellposedness and persistence result in Theorem 5.1 we know that the solution u is defined up to some time T > 0 and that in order to extend it we only need to show that the norm u x (t, ·)∞ remains bounded. On the one hand, given any periodic function w(x) differentiating the formula in (5.2) we readily obtain the estimate ∂x w∞ 2μ w 1 . (5.16) L
On the other hand, recall from Sect. 2 that for any solution u of μDP with initial condition u 0 ∈ H s (S 1 ) the expression m = 2μ u is a tensor density in F3 (S 1 ) and hence from the Diff s (S 1 )-action formula in (2.6) we get the following (pointwise) conservation law d 2 (μ u) ◦ ξ (∂x ξ )3 = (−u t x x −uu x x x +3μ(u)u x −3u x u x x ) ◦ ξ (∂x ξ )3 = 0, dt (5.17) where t → ξ(t) ∈ Diff s (S 1 ) is the associated flow (i.e. ξ˙ = u ◦ ξ and ξ(0) = e). Thus, for any x ∈ S 1 as long as the solution exists it must satisfy 2μ u (t, ξ(t, x)) (∂x ξ(t, x))3 = 2μ u 0 (x), and therefore by the assumption on u 0 it follows that 2μ u(t, x) ≥ 0 for any x ∈ S 1 as long as it is defined. The above inequality together with the conservation of the mean and (5.16) gives 1 2 ∂x u(t, ·)∞ μ u(t) 1 = 2μ u(t, x) d x = μ(u 0 ) < ∞, L
0
and we conclude that the solution u must persist indefinitely.
5.4. Local wellposedness for any λ. For completeness we state here a local wellposedness result for the family (1.2) of μ-equations for any λ. Using the substitution v = Av = 2μ u and the fact that, as in the case of μDP, the mean of the solution u is conserved, we can write the Cauchy problem for (1.2) in the form 3−λ 2 λμ(u)u + (5.18) u ∂ u t + uu x + −2 μ x x =0 2 u(0) = u 0 .
(5.19)
Theorem 5.5. (Local wellposedness) Let s > 3/2. For any u 0 ∈ H s (S 1 ) there exist a T > 0 and a unique solution u ∈ C (−T, T ), H s ∩ C 1 (−T, T ), H s−1 of (5.18)–(5.19) which depends continuously on the initial data u 0 . Furthermore, the solution persists as long as u(t, ·)C 1 stays bounded.
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The proof of this theorem is analogous to Theorem 5.1 and we omit it. In the previous section we proved a global existence result for μDP. Here, we formulate a similar result for any positive λ. For a fixed λ > 0 let u = u(t, x) be any smooth solution of the problem (5.18)-(5.19) and let ξ(t) ∈ Diff s (S 1 ) denote the corresponding flow of diffeomorphisms of the circle. As before, the argument rests on the pointwise conservation law obeyed by the solutions. Namely, using the formula for the action of Diff s (S 1 ) on λ-densities Fλ (S 1 ) in (2.6) we find the analogue of (5.17) to be d 2 (μ u) ◦ ξ (∂x ξ )λ = 2μ u t − uu x x x + λμ(u)u x − λu x u x x ◦ ξ (∂x ξ )λ = 0 dt so that as long as u(t, x) exists 2μ u (t, ξ(t, x)) (∂x ξ(t, x))λ = 2μ u 0 (x),
x ∈ S1,
and therefore we have Theorem 5.6. Fix λ > 0. Assume that u 0 ∈ H s (S 1 ) with s > 3 has non-zero mean and satisfies the condition 2μ u 0 ≥ 0 (or ≤ 0).
Then the Cauchy problem (5.18)-(5.19) has a unique global solution in C R, H s (S 1 ) ∩ C 1 R, H s−1 (S 1 ) . 6. Peakons The CH and DP equations famously exhibit peakon solutions, see e.g. [CH,DHH]. In this section we show that each of the μ-equations (1.2) parametrized by λ ∈ Z also admits peakons. Let us point out here that the fact that μCH admits peaked solutions went unnoticed in [KLM], so that the present discussion provides an extension of the results of that paper. We will investigate existence of traveling-wave solutions of the form u(t, x) = ϕ(x − ct). This will lead us to expressions for the one-peakon solutions. Subsequently, we will analyze the more general case of multi-peakons. 6.1. Traveling waves. We first consider Eq. (1.2) where μ(u) is replaced by a parameter ν ∈ R, i.e. − u t x x + λνu x = λu x u x x + uu x x x .
(6.1)
After analyzing traveling-wave solutions of this equation we will impose the two conditions μ(u) = ν
and
period(u) = 1
to find which of these are traveling waves of Eq. (1.2). We assume that λ = 0, 1. Substituting u(t, x) = ϕ(x − ct) into Eq. (6.1) we find cϕx x x + λνϕx = λϕx ϕx x + ϕϕx x x .
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We write this in terms of f := ϕ − c and integrate to find λν f + a =
λ−1 2 fx + f fx x , 2
(6.2)
where a in an integration constant. We multiply Eq. (6.2) by 2 f λ−2 f x and integrate the resulting equation with respect to x. The outcome is 2a λ−1 + b = f λ−1 f x2 , f λ−1 where b is another constant of integration. This equation can be written as 2ν f λ +
ϕx2 = 2ν(ϕ − c) +
b 2a + . λ − 1 (ϕ − c)λ−1
(6.3)
A solution of this equation yields the shape of a traveling wave within an interval where ϕ is smooth, and gluing such smooth wave segments together at points where ϕ = c produces the complete collection of traveling wave solutions of (6.1). The analysis proceeds along the same lines and yields similar results as for CH (or as for HS if ν = 0) cf. [Le]. Here we choose to consider the peaked solutions. For completeness we point out that the equation analogous to (6.3) when λ = 0 or λ = 1 is −2a + b(ϕ − c), λ = 0, 2 ϕx = (6.4) 2ν(ϕ − c) + 2aln|ϕ − c| + b, λ = 1. 6.2. Periodic one-peakons. The periodic peakons arise when b = 0 in (6.3).10 Indeed, setting b = 0 and replacing a by a new parameter m via a = (λ − 1)(c − m)ν in (6.3), we find ϕx2 = 2ν(ϕ − m). Solving this equation, we find the peakon solution u(t, x) = ϕ(x − ct), where ν 2 2(c − m) 2(c − m) ϕ(x) = m + x for x ∈ − , , 2 ν ν
(6.5)
and ϕ is extended periodically to the real axis. The parameters are required to satisfy (c − m)/ν ≥ 0. The period of ϕ is fixed by the condition that ϕ = c at the peak. Equation (6.5) defines a periodic peaked solution of (6.1) for any choices of the real parameters m, c, ν. In order to determine which of these solutions are solutions with period one of the μ-equation (1.2) we have to impose the conditions mean(ϕ) = ν and period(ϕ) = 1. A computation shows that mean(ϕ) = m + c−m 3 and period(ϕ) = ! 2(c−m) 2 . Therefore, u(t, x) = ϕ(x − ct), where ϕ is given by (6.5) is a period-one ν solution of Eq. (1.2) if and only if c−m c−m 2(c − m) = 1, ≥ 0, and ν = m + . 2 ν ν 3 Solving these equations we find m = 23c/26 and ν = 12c/13. This leads to the following result. 10 We henceforth assume ν = 0.
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Theorem 6.1. For any c ∈ R and λ = 0, 1, Eq. (1.2) admits the peaked period-one traveling-wave solution u(t, x) = ϕ(x − ct), where c (12x 2 + 23) 26
ϕ(x) =
(6.6)
for x ∈ − 21 , 21 and ϕ is extended periodically to the real line. We note that the one-peakon solutions of (1.2) are the same for any λ, that they travel with a speed equal to their height, and that μ(ϕ) = 12c/13. 6.3. Green’s function. For the construction of multi-peakons it is convenient to rewrite the inverse of the operator 2μ = μ − ∂x2 in terms of a Green’s function (an explicit formula for the inverse −2 μ was previously given in Eq. (5.2)). We find (−2 μ m)(x) =
1
g(x − x )m(x )d x ,
0
where the Green’s function g(x) is given by g(x) =
13 1 x(x − 1) + for x ∈ [0, 1) S 1 , 2 12
(6.7)
and is extended periodically to the real line. In other words, g(x − x ) =
|x − x | 13 (x − x )2 − + , 2 2 12
x, x ∈ [0, 1) S 1 .
Note that g(x) has the shape of a one-peakon (6.6) with c = 13/12 and peak located at x = 0. In particular, μ(g) = 1. 6.4. Multi-peakons. The construction of multi-peakons for the family of μ-equations (1.2) is similar to the corresponding construction for the family (1.1). Just like for CH and DP, the momentum m = 2μ u of an N -peakon solution is of the form m(t, x) =
N
pi (t)δ(x − q i (t)),
(6.8)
i=1
where the variables {q i , pi }1N evolve according to a finite-dimensional Hamiltonian system. Indeed, assuming that m is of the form (6.8), the corresponding u is given by u = −2 μ m =
N
pi (t)g(x − q i (t)).
i=1
We assign the value zero to the otherwise undetermined derivative g (0), so that 0, x = 0, g (x) := (6.9) 1 x − 2, 0 < x < 1.
Integrable Evolution Equations and Peakons
0.0
g (x) − g (x)
g (x)
g (x) 1.2 1.0 0.8 0.6 0.4 0.2 −0.5
147
µ
µ
1.2 1.0 0.8 0.6 0.4 0.2 0.5
1.0
1.5
2.0
x
2.5
− 0.5 0.0
0.0010 0.0005 − 0.5 − 0.0005 0.5
1.0
1.5
2.0
x
0.5
1.0
1.5
2.0
x
2.5
− 0.0010
2.5
Fig. 3. The periodic Green’s functions g(x) and gμ (x) corresponding to the operators A = 1 − ∂x2 and A = μ − ∂x2 , respectively, and their difference g(x) − gμ (x)
This definition provides a naive way to give meaning to the term −λu x m in the PDE (2.8). Substituting the multi-peakon Ansatz (6.8) into the μ-equation (1.2) then shows that pi and q i evolve according to q˙ i =
N
p j g(q i − q j ),
p˙ i = −(λ − 1)
j=1
N
pi p j g (q i − q j ).
j=1
A more careful analysis reveals that this computation is in fact legitimate and yields the following result. Note that the peakons fall outside the result of Theorem 5.5 since for a peakon u ∈ / H s (S 1 ) for s > 3/2. Theorem 6.2. The multi-peakon (6.8) satisfies the μ-equation (1.2) in the weak form (5.18) in the distributional sense if and only if {q i , pi }1N evolve according to q˙ i = u(q i ), p˙ i = −(λ − 1) pi {u x (q i )},
(6.10)
where {u x (q i )} denotes the regularized value of u x at q i defined by {u x (q i )} :=
N
p j g (q i − q j )
(6.11)
j=1
and g (x) is defined by (6.9). Proof. See Appendix A.
The formulas for the peakons of the μ-equations (1.2) along with the expressions for the previously known peakons of the CH and DP family (1.1) are summarized in Table 1. At any particular time, the multi-peakon is a sum of Green’s functions for the associated operator A, see Fig. 3. Since the inertia operator for the μ-equations (A = μ − ∂x2 ) is different from that of the family (1.1) (A = 1 − ∂x2 ), the new class of “μ-peakons” have a different form with respect to the CH and DP peakons. For a given family of equations the Green’s function is the same for all λ, but the time evolution of the positions q i and momenta pi of the peaks of course depends on λ and is for both families given by the system (6.10).
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Table 1. The periodic multi-peakon solutions of the family (1.1) (which includes CH and DP) and of the corresponding μ-family (1.2) (which includes μCH and μDP) when λ = 0, 1 Equation family (1.1) (1.2)
Green’s function (x ∈ S 1 [0, 1))
Multi-peakon u(t, x)
(x−1/2) g(x) = cosh 2sinh(1/2) 2 23 g(x) = 21 x − 21 + 24
"N
i i=1 pi (t)g(x − q (t))
"N
i i=1 pi (t)g(x − q (t))
6.5. Poisson structure. Equations (6.10) take the Hamiltonian form q˙ i = {q i , H0 },
p˙ i = { pi , H0 },
with respect to the Hamiltonian λ2 pi , H0 = − λ−1 N
i=1
and the Poisson structure λ−1 2 λ−1 pi p j G (q i − q j ), {q i , p j } = − 2 p j G (q i − q j ), λ λ 1 {q i , q j } = − 2 G(q i − q j ), λ x where G(x) = 0 g(x )d x . In order to make sense of G(q i − q j ) although G is not periodic, we restrict q i , q j to be local coordinates on S 1 satisfying say 0 < q i , q j < 1. Then −1 < q i − q j < 1 and G(q i − q j ) is well-defined. This Poisson structure can be derived from the Poisson structure corresponding to J0 in (4.3) by noting that
{ pi , p j } =
{m(x), m(y)} J0 = −
1 G(x − y)m x (x)m x (y) − λG (x − y)m x (x)m(y) λ2
+ λG (x − y)m(x)m x (y) − λ2 G (x − y)m(x)m(y) .
" Indeed, let m 0 (x) = i p0i δ(x − q0i ) be a multi-peakon of the form (6.8). Let φi : S 1 → R (resp. ψi : S 1 → R) be a function which takes the value 1 (resp. " x) in a small neighborhood of x = q0i and the value 0 elsewhere. Then, for m(x) = i pi δ(x − q i ) sufficiently close to m 0 , we obtain
S1
φi (x)m(x)d x = pi ,
S1
ψi (x)m(x)d x = q i pi .
We now find { pi , p j } from the computation { pi , p j } =
S1
S1
φi (x)φ j (y){m(x), m(y)} J0 d xd y =
λ−1 λ
2
pi p j G (q i − q j ).
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We then compute
{q pi , p j } = i
S1
=−
S1
ψi (x)φ j (y){m(x), m(y)} J0 d xd y
λ−1 pi p j G (q i − q j ) + λ2
λ−1 λ
2
q i pi p j G (q i − q j ).
Since {q i pi , p j } = q i { pi , p j }+{q i , p j } pi and we know the value of { pi , p j }, this gives the expression for {q i , p j }. Finally, the expression for {q i , q j } is obtained by calculating {q pi , q p j } = i
j
S1
S1
ψi (x)ψ j (y){m(x), m(y)} J0 d xd y
and comparing the result with the expression for {q i pi , q j p j } = q i q j { pi , p j } + q i { pi , q j } p j + q j {q i , p j } pi + {q i , q j } pi p j obtained by substituting in the expressions for { pi , p j }, { pi , q j }, and {q i , p j }.
6.6. Multi-peakons for μCH. For μCH (λ = 2) Eqs. (6.10) for {q i , pi } are canonically Hamiltonian with respect to the Hamiltonian N 1 h= pi p j g(q i − q j ), 2 i, j=1
and describe geodesic flow on T N × R N (here T N = S 1 × · · · × S 1 denotes the N -torus) with respect to the metric gi j with inverse given by g i j = g(q i − q j ). In this case the reduction to multi-peakon solutions can be elegantly understood in terms of momentum maps using the ideas of [HM]. We consider (q i , pi ) as an element of T ∗ (T N ) T N × R N . Then the map Jsing : T ∗ T N → vect(S 1 )∗ : (q i , pi ) →
N
pi δ(x − q i )
i=1
is an equivariant momentum map for the natural action of Diff(S 1 ) on T ∗ (T N ). Since momentum maps are Poisson, Jsing maps the solution curves in T ∗ (T N ) which satisfy ∗ (6.10) to solution curves in vect(S 1 )∗ which satisfy (2.3). Note that h = Jsing H is the 1 1 1 ∗ pull-back of the Hamiltonian H on vect(S ) defined by H = 2 0 umd x.
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6.6.1. Periodic two-peakon for μCH In the case of two peaks for the μCH equation (N = 2, λ = 2), the system (6.10) is given explicitly by q˙ 1 = p1 g(0) + p2 g(q 1 − q 2 ), q˙ 2 = p1 g(q 2 − q 1 ) + p2 g(0), p˙ 1 = − p˙ 2 = − p1 p2 g (q 1 − q 2 ). Letting Q = q 2 − q 1 and P = p2 − p1 and using that 13 H02 p1 p2 2 + (Q − |Q|), 24 16 2
h= we find the system
1 Q˙ = − P(Q 2 − |Q|), 2 α 1 P˙ = −2 2 Q − sgn(Q) , Q − |Q| 2 where α = 2h −
2 13 H0 12 16 .
This leads to the following equation for Q: ¨ 2 − |Q|) = (2 Q˙ 2 + α(Q 2 − |Q|)) Q − 1 sgn(Q) . Q(Q 2
The solution to this ODE can be expressed in terms of inverses of elliptic functions. 6.7. Shock-peakons for μDP. In addition to the peakon solutions the DP equation admits an even weaker class of solutions with jump discontinuities called “shock-peakons”, see [Lu]. We will show here that μDP also allows shock-peakon solutions. We will seek the solutions of the form m(t, x) =
N
pi (t)δ(x − q i (t)) + si (t)δ (x − q i (t)) .
i=1
The corresponding u is u=
N
pi g(x − q i ) + si g (x − q i ) .
(6.12)
i=1
Substituting this into the equation m t = −um x − λu x m, we find formally q˙ i = u(q i ),
p˙ i = (λ − 1) si u x x (q i ) − pi u x (q i ) , s˙i = −(λ − 2)si u x (q i ).
The following theorem states that in the case of λ = 3 (this is the case of μDP), this formal computation can be rigorously justified provided that the u x and u x x terms are appropriately regularized. The reason that such weak solutions can be made sense of in 2 the case of μDP is that the term 3−λ 2 u x in the weak formulation (5.18) is absent when λ = 3.
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Theorem 6.3. The shock-peakon (6.12) satisfies the weak form (5.3) of μDP in the distributional sense if and only if {q i , pi , si }1N evolve according to q˙ i = u(q i ), p˙ i = 2 si {u x x (q i )} − pi {u x (q i )} ,
(6.13)
s˙i = −si {u x (q i )}, where {u x (q )} = i
N
p j g (q − q ) + i
j
j=1
N
sj,
{u x x (q )} =
j=1
i
N
pj,
j=1
and g (x) is defined by (6.9). Proof. See Appendix B.
7. The μBurgers Equation In this section we discuss the μB equation and its properties. We introduced this equation in Sect. 2 as the λ = 3 analogue of the Hunter-Saxton equation. Here we take a different view and employ Riemannian geometric techniques to display its close relationship with the inviscid Burgers equation. 7.1. Burgers’ equation and the L 2 -geometry of Diff s → Diff s /S 1 . The constructions in this subsection are well known. First, recall that the inviscid Burgers equation u t + uu x = 0
(B)
is related to the geometry of the weak Riemannian metric on Diff s (S 1 ) which on the tangent space Tξ Diff s (S 1 ) at a diffeomorphism ξ is given by the L 2 inner product V, W L 2 =
1
V (x)W (x) d x,
(7.1)
0
where V, W ∈ Tξ Diff s . It is not difficult to show that this metric admits a unique Levi-Civita connection ∇ whose geodesics η(t) in Diff s (S 1 ) satisfy the equation11 ∇η˙ η˙ = η¨ = (u t + uu x ) ◦ η = 0,
(7.2)
and hence correspond to (classical) solutions of the Burgers equation. Here η(t) is simply the flow of u(t, x) so that η(t, ˙ x) = u(t, η(t, x)) and hence the second equality in (7.2) follows at once from the chain rule. Consider next the homogeneous space Diff s0 = Diff s /S 1 , where S 1 denotes the subgroup of rotations. For s > 3/2 this space is also a smooth Hilbert manifold whose 11 In fact, the geodesic equation is readily obtained from the first variation formula for the energy functional 2 dt of the L 2 metric (7.1) on Diff s (S 1 ). E(η) = 21 ab η(t) ˙ 2 L
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tangent space at the point [e] in Diff s0 (S 1 ) corresponding to the identity diffeomorphism can be identified with the periodic Sobolev functions with zero mean 1 s 1 s 1 T[e] Diff 0 = H0 (S ) = w ∈ H (S ) : w(x) d x = 0 . 0
It is useful to think of Diff s0 (S 1 ) as the space of probability densities on the circle and view the map π : Diff s → Diff s0 as a submersion given by the pull-back [ξ ] = π(ξ ) = ξ ∗ (d x), where d x is the (fixed) probability measure with density 1 (see e.g. [KY]). The fibre through ξ ∈ Diff s (S 1 ) consists of all diffeomorphisms that are obtained from ξ by a composition on the left with a rotation and thus is just a right coset of the rotation subgroup S 1 which acts on Diff s (S 1 ) by left translations. Furthermore, the L 2 metric (7.1) is preserved by this action and π becomes a Riemannian submersion with each tangent space decomposing into a horizontal and a vertical subspace Tξ Diff s = Pξ (Tξ Diff s ) ⊕ L 2 Q ξ (Tξ Diff s ) which are orthogonal with respect to (7.1). The two orthogonal projections Pξ : Tξ Diff s → Tπ(ξ ) Diff s0 and Q ξ : Tξ Diff s → R are given explicitly by the formulas 1 1 W (x) d x and Q ξ (W ) = W (x) d x. (7.3) Pξ (W ) = W − 0
0
Finally, as with any Riemannian submersion12 , note that a necessary and sufficient condition for a curve η(t) in Diff s (S 1 ) to be an L 2 geodesic satifying (7.2) (and hence correspond to a solution of Burgers’ equation) is that Pη ∇η˙ η˙ = Q η ∇η˙ η˙ = 0. We are now ready to introduce the μB equation in this set-up. 7.2. Lifespan of solutions, Hamiltonian structure and conserved quantities of the μBurgers equation. We first characterise local (in time) smooth solutions. Theorem 7.1. A smooth function u = u(t, x) is a solution of the μB equation u t x x + 3u x u x x + uu x x x = 0 if and only if the horizontal component of the acceleration of the associated flow η(t) in Diff s (S 1 ) is zero, i.e. Pη ∇η˙ η˙ = 0. In fact, given any u 0 ∈ H s (S 1 ) the flow of u has the form η(t, x) = x + t (u 0 (x) − u 0 (0)) + η(t, 0) for all sufficiently small t. Proof. Integrating the μB equation with respect to the x-variable we get 1 u 2x + uu x x d x = 0, u t x + u 2x + uu x x = 0
and integrating once again gives u t + uu x = c(t), 12 See e.g. O’Neill [ON] for details on Riemannian submersions.
(7.4)
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where c(t) is a function of the time variable only. Since the mean of a μB solution need not be preserved in time we obtain the equation13 u t + uu x = μ(u t ).
(7.5)
On the other hand, if η(t) is the flow of u in Diff s (S 1 ), that is η(t, ˙ x) = u(t, η(t, x)),
η(0, x) = x,
then from (7.2) and (7.3) we compute the horizontal component of the acceleration of η(t) in Diff s (S 1 ) to be 1 1 η¨ d x = (u t + uu x ) ◦ η − Pη ∇η˙ η˙ = η¨ − (u t + uu x ) ◦ η d x. 0
0
Using these formulas and integrating by parts we find that the equation Pη ∇η˙ η˙ = 0 is equivalent to 1 1 η¨ d x = η¨ ◦ η−1 d x (u t + uu x ) ◦ η = η¨ = 0
1
=
0
(u t + uu x ) d x =
0
1
u t d x,
(7.6)
0
which establishes the first part of the theorem. In order to prove the second statement it suffices to observe that from (7.6) we have in particular 1 η(t, ¨ x) = η¨ ◦ η−1 (t, x) d x = η(t, ¨ 0) 0
which immediately implies η(t, x) − η(t, 0) = x + t (u 0 (x) − u 0 (0)) for any 0 ≤ x ≤ 1 and any t for which η is defined.
Remark 7.2. It is easy to verify that Eq. (7.5) is also equivalent to ∇η˙ Pη η˙ = 0 which can be interpreted as saying that the horizontal component of the velocity of the flow η(t) of the μB equation is parallel transported along the flow. Remark 7.3. A similar construction can be used to derive the μB equation from the right-invariant L 2 metric on Diff s (S 1 ) defined by 1 V, W ξ = V ◦ ξ −1 (x)W ◦ ξ −1 (x) d x where V, W ∈ Tξ Diff s and ξ ∈ Diff s . 0
1 In this case the corresponding orthogonal projections are P˜ξ (W ) = W − 0 W dξ = W − Q˜ ξ (W ) and the geodesic equation in Diff s (S 1 ) reads ∇˜ η˙ η˙ = η+2 ¨ η˙ ∂x η˙ (∂x η)−1 = 0. Proceeding as in the proof of Theorem 7.1 we then get 0 = P˜η ∇˜ η˙ η˙ = (u t + 3uu x − μ(u t )) ◦ η, which after rescaling the dependent variable yields (7.5). 13 In the case when the mean μ(u) is independent of time (7.5) becomes the standard (inviscid) Burgers equation. The presence of the mean is also the reason for our terminology.
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It is therefore not surprising to find that the μB equation shares a number of properties with the Burgers equation. Corollary 7.4. Suppose that u(t, x) is a smooth solution of the μB equation and let u(0, x) = u 0 (x). (1) The following integrals are conserved by the flow of u:
1
(u − μ(u)) p d x =
0
1
(u 0 − μ(u 0 )) p d x,
p = 1, 2, 3 . . . .
0
(2) There exists Tc > 0 such that u x (t)∞ ∞ as t Tc . (3) The μB equation has a bi-Hamiltonian structure given by the Hamiltonian functionals 9 1 H0 = − m d x, H2 = − u 3 d x, 2 6 and the operators J0 = −m 2/3 ∂x m 1/3 ∂x−3 m 1/3 ∂x m 2/3 , J2 = ∂x5 . Proof. The first statement follows by direct calculation. Regarding the second, it suffices to consider the equation in (7.4) and apply the argument that was used in the proof of Theorem 5.3. The third statement follows from a straightforward computation establishing J0
δ H0 δ H2 = J2 = −um x − 3u x m, δm δm
where m = −∂x2 u and our earlier observation from Sect. 4 that J0 and J2 are compatible.
8. Multidimensional μCH and μDP Equations In this final section we briefly consider possible generalizations of the μCH and μDP equations to higher dimensions. Let X(T n ) be the space of smooth vector fields on the n-dimensional torus T n comprising the Lie algebra of Diff(T n ). Let A be a self-adjoint positive-definite operator defining an inner product on the Lie algebra. Given a vector field u ∈ X(T n ) we define the corresponding momentum density by m = Au. The EPDiff equation for geodesic flow on Diff(T n ) is given by mt + Lu m = 0,
(8.1)
where Lu m denotes the Lie derivative of the momentum one-form density m in the direction of u (see [HM]). Identifying T n Rn /Zn and letting x i , i = 1, 2, . . . , n denote standard coordinates on Rn , this can be written as ∂m i ∂u j ∂m i +uj + m + m i div(u) = 0, j ∂t ∂x j ∂ xi
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155
where m = m i d x i ⊗ d n x and u = u i ∂/∂ x i . The CH equation is the (1 + 1)-dimensional version of the EPDiff Eq. (8.1) when A = 1−. It is natural to define a multidimensional generalization of μCH as Eq. (8.1) with the operator A defined by Au = (μ − )u =
Tn
u d n x − u.
Similarly, using higher-order tensor densities we arrive at multidimensional versions of DP and μDP. Let m = m i d x i ⊗ d n x ⊗ d n x. Then Eq. (8.1) becomes ∂m i ∂u j ∂m i + m + 2m i div(u) = 0. +uj j ∂t ∂x j ∂ xi
(8.2)
In 1 + 1 dimensions this reduces to m t + m x u + 3mu x = 0. Hence, when A = 1 − , Eq. (8.2) can be viewed as a multidimensional DP equation, while if A = μ − it can be viewed as a multidimensional μDP equation. Acknowledgments. The authors thank the referee for constructive remarks. J.L. is grateful to Professor D. D. Holm for valuable discussions and suggestions. J.L. acknowledges support from a Marie Curie IntraEuropean Fellowship.
Appendix A. Proof of Theorem 6.2 In this appendix we prove Theorem 6.2, that is, we show that the multi-peakons defined in (6.8) are weak solutions in the distributional sense of the μ-equation (1.2) if and only if {q i , pi }1N evolve according to (6.10). Equation (1.2) in weak form reads14 1 3−λ g ∗ (u 2x ) = 0, u t + (u 2 )x + λμ(u)g ∗ u + 2 2
(A.1)
where g is the Green’s function defined in (6.7). This is to be satisfied in the space of distributions D (R × S 1 ), i.e.
∞
−∞
1 3−λ (g ∗ u 2x )φx d xdt = 0, uφt + u 2 φx + λμ(u)(g ∗ u)φx + 2 2 S1
for all test functions φ(t, x) ∈ Cc∞ (R × S 1 ). Let g j := g(x − q j ) and g j := g (x − q j ). We specify the value of g at x = 0 by setting g (0) := 0 as in Eq. (6.9). We will need the following identities which can be 14 Here ∗ denotes convolution: for two functions f, g : S 1 [0, 1) → R, we have ( f ∗ g)(x) =
1
0 f (x − y)g(y)dy.
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verified by direct computation: 1 13 g ∗ gj = − gj − g j , 3 12 g ∗ g j = 1 − g j ,
−(gi −
g j )(gi
−
g j )
=
2g j (q i )gi
(A.2) (A.3)
+ 2gi (q j )g j
13 13 gi + g(q j −q i )− g j , + g(q i − q j )− 12 12
(A.4)
gi g j = gi + g j + g (q i − q j )(gi − g j ) + g(q i − q j ) − 3.
(A.5)
These identities hold pointwise for all x, q i , q j ∈ S 1 (except (A.5) which holds for all x, q i ," q j ∈ S 1 unless x = q i = q j ; a fact that does not matter in what follows). Let u = i pi gi be a peakon. As explained in the appendix of [Lu], we can compute the left-hand side of (A.1) as a distribution in the variable x without having to involve test functions explicitly. This yields ut =
N
p˙ i gi − pi q˙ i gi ,
(u 2 )x =
i=1
N
2 pi p j gi g j .
i, j=1
Using (A.2) we find N λ 13 g j . pi p j g j − λμ(u)g ∗ u = − 3 12
i, j=1
Moreover, u 2x =
N
pi p j gi g j ,
i, j=1
which in view of (A.2), (A.3), and (A.5) implies that
g
∗ (u 2x )
N 1 13 i j =− pi p j (gi gi + g j g j − (gi + g j ) + 3g (q − q )(gi − g j ) . 3 12 i, j=1
Using these ingredients we can write (A.1) as N i=1
N N λ 13 g j p˙ i gi − pi q˙ i gi + pi p j gi g j − pi p j g j − 3 12 i, j=1
i, j=1
N 3−λ 13 i j − pi p j (gi gi + g j g j − (gi + g j ) + 3g (q − q )(gi − g j ) = 0. 6 12 i, j=1
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We rewrite this as ⎧ ⎛ ⎛ ⎞ ⎞ ⎫ N ⎨ ⎬ 1 13 p j ⎠ − q˙ i ⎠ gi − pi p j (gi − g j )(gi − g j ) p˙ i gi + pi ⎝ ⎝ ⎩ ⎭ 2 12 i=1
+
i= j
j
N λ−3 pi p j g (q i − q j )(gi − g j ) = 0. 2 i, j=1
Using (A.4) we find ⎧⎛ ⎞ ⎛ ⎞ ⎫ N ⎨ ⎬ 13 ⎝ p˙ i + 2 pi p j g j (q i )⎠ gi + pi ⎝ pi + p j g(q i − q j ) − q˙ i ⎠ gi ⎩ ⎭ 12 j=i
i=1
+
λ−3 2
N
j=i
pi p j g (q i − q j )(gi − g j ) = 0.
i, j=1
The last sum on the left-hand side can be written as (λ − 3)
N
pi p j g q i − q j gi ,
i, j=1
so that, using the definition (6.11) of {u x (q i )} and the fact that g(0) = 13/12, we arrive at N
( p˙ i + (λ − 1) pi {u x (q i )})gi + pi (u(q i ) − q˙ i )gi = 0.
i=1
Since gi and gi form a linearly independent set, this equation holds if and only if {q i , pi }1N evolve according to (6.10). This completes the proof of Theorem 6.2. Appendix B. Proof of Theorem 6.3 In this appendix we prove Theorem 6.3, that is, we show that the shock-peakons defined in (6.12) are weak solutions in the distributional sense of μDP if and only if {q i , pi , si }1N evolve according to (6.13). The objective of the proof is similar to the corresponding proof for DP presented in the appendix of [Lu]. However, since we consider the spatially periodic case and the Green’s functions are very different, the details of the two proofs are quite different. The μDP equation in weak form reads 1 u t + (u 2 )x + 3μ(u)g ∗ u = 0, 2
(B.1)
which is to be satisfied in the distributional sense. We use the same notation as in the proof of Theorem 6.2. Let δi (x) = δ(x − q i ) and note that gi = 1 − δi . It is enough to verify
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that (B.1) holds as a distribution in x when the functions u t , 21 (u 2 )x , and 3μ(u)g ∗ u are replaced by ut =
N
p˙ i gi − pi q˙ i gi + s˙i gi − si q˙ i (1 − δi ) ,
i=1 N 1 2 pi p j gi g j + si s j gi (1 − δ j ) + pi s j (gi g j + gi (1 − δ j )) , (u )x = 2 i, j=1
and
3μ(u)g ∗ u =
N
− pi p j
i, j=1
13 gj − g j + 3 pi s j (1 − g j ) , 12
(B.2)
respectively. In order to derive (B.2) we used the fact that g ∗ u =
N
p j (g ∗ g j ) + s j (g ∗ g j )
j=1
together with the identities (A.2) and (A.3). Putting these expressions together we find that Eq. (B.1) can be written as N
p˙ i gi − pi q˙ i gi + s˙i gi − si q˙ i (1 − δi )
i=1 N
+
pi p j gi g j + si s j gi (1 − δ j ) + pi s j (gi g j + gi (1 − δ j ))
i, j=1
+
N 13 − pi p j g j − g j + 3 pi s j (1 − g j ) = 0. 12
i, j=1
Employing (A.4) we can rewrite this as ⎧⎛ ⎞ ⎛ ⎞ N ⎨ 13 ⎝ p˙ + 2 pi p j g j (q i )⎠ gi + pi ⎝ pi + p j g(q i − q j ) − q˙ i ⎠ gi ⎩ i 12 i=1 j=i j=i ⎫ N ⎬ si s j gi + pi s j (gi g j + gi ) + s˙i gi − si q˙ i + ⎭ i, j=1 ⎛ ⎞ N N pi s j (1 − g j ) + si ⎝q˙ i − s j g j − p j g j ⎠ δi = 0. (B.3) +3 i, j=1
i=1
The identity (A.5) shows that the two terms combine to give N i, j=1
j
j
i, j
pi s j (gi g j +gi ) and 3
"
" i, j
pi s j 2(gi − g j ) + g (q i − q j )(gi − g j ) + g(q i − q j ) .
pi s j (1−g j )
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We find that (B.3) can be written as ⎛ ⎞ N ⎝ p˙ i + 2 pi p j g j (q i ) + 2 pi s j − 2si p j ⎠ gi i=1
j=i
⎛
j
j
⎛ ⎞ N ⎝ pi ⎝ + p j g j (q i ) + s j g (q i − q j ) − q˙ i ⎠ + s˙i i=1
+ si
j
+
⎛
si ⎝
⎞
p j g (q j − q i )⎠ gi
j
i=1
The term
s j − si
j N
j
⎞
p j g(q i − q j ) − q˙ i ⎠ +
j
"
i si
N
⎛ si ⎝q˙ i −
i=1
" j
s j g j −
j
⎞ p j g j ⎠ δi = 0.
j
p j g(q i − q j ) − q˙ i equals ⎛ ⎞ si ⎝ p j g(q i − q j ) + s j g j (q i ) − q˙ i ⎠ .
i
j
j
Using that u(q i ) =
p j g j (q i ) +
j
{u x (q )} = i
s j g j (q i ),
j
p j g j (q i ) +
j
sj,
{u x x (q i )} =
j
pj,
j
we arrive at the equation N
N p˙ i +2 pi {u x (q i )}−2si {u x x (q i )} gi + pi u(q i )− q˙ i + s˙i +si {u x (q i )} gi
i=1
+
i=1 N i=1
si (u(q i ) − q˙ i ) +
N
si q˙ i − u(q i ) δi = 0.
i=1
Since {1, gi , gi , δi } form a linearly independent set this equation holds if and only if {q i , pi , si }1N evolve according to (6.13). This completes the proof of Theorem 6.3. Appendix C. Conservation of H2 for μDP In this appendix we verify explicitly that the functional H2 defined in (4.1) is conserved under the flow of μDP. The following two forms of μDP are used: u t + uu x + 3μ(u)−2 μ u x = 0,
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and its derivative u t x + uu x x + u 2x + 3(μ(u))2 − 3uμ(u) = 0. −2 We also use the identities μ(−2 μ u) = μ(u) and μ u x x = −u + μ(u). We compute
1 2 d H2 −2 −2 =− 3μ(u)(μ u x )(μ u xt ) + u u t dt 2 S1 −2 2 2 3μ(u)(−2 uu u ) + u + 3μ(u) − 3uμ(u) = xx μ x μ x S1 1 . + u 2 uu x + 3μ(u)−2 μ ux 2 We use uu x x + u 2x = 21 ∂x2 (u 2 ) and find d H2 3 −2 3 −2 −2 2 2 −2 −9μ(u)2 (−2 = μ u x )μ u + 9μ(u) μ u x + μ(u)(μ ∂x (u ))μ u x dt 2 S1 1 3 3 2 −2 + u u x + u μ(u)μ u x . 2 2 Here the first two terms and the fourth term vanish by periodicity. The third term is equal to 3 3 2 2 2 2 −2 ∂ (u ) −2 μ(u) −2 μ x μ u x = μ(u)(−u + μ(u ))μ u x 2 2 whose first part cancels the fifth term and the second part vanishes by periodicity. References [A]
Arnold, V.: Sur la géometrie différentielle des groupes de lie de dimension infinie et ses application à l’hydrodynamique des fluides parfaits. Ann. Inst. Fourier (Grenoble) 16, 319–361 (1966) [BSS] Beals, R., Sattinger, D., Szmigielski, J.: Multipeakons and the classical moment problem. Adv. Math. 154, 229–257 (2000) [CH] Camassa, R., Holm, D.D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664 (1993) [CM] Constantin, A., McKean, H.P.: A shallow water equation on the circle. Comm. Pure Appl. Math. 52, 949–982 (1999) [DP] Degasperis, A., Procesi, M.: Asymptotic integrability. In: Symmetry and Perturbation Theory (Rome 1998), River Edge, NJ: World Scientific Publishers, 1999 [DHH] Degasperis, A., Holm, D.D., Hone, A.N.W.: A new integrable equation with peakon solutions. Teoret. Mat. Fiz. 133, 1463–1474 (2002) [EM] Ebin, D., Marsden, J.E.: Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. of Math. 92, 102–163 (1970) [FF] Fuchssteiner, B., Fokas, A.S.: Symplectic structures, their bäcklund transformation and hereditary symmetries. Physica D 4, 47–66 (1981) [GR] Guieu, L., Roger, C.: L’Algebre et le Groupe de Virasoro: aspects geometriques et algebriques, generalisations. Montreal: Publications CRM, 2007 [HM] Holm, D.D., Marsden, J.E.: Momentum maps and measure valued solutions (peakons, filaments, and sheets) of the Euler-Poincaré equations for the diffeomorphism group. In: The Breadth of Symplectic and Poisson Geometry, Progr. Math. 232, Boston, MA: Birkhäuser, 2005, pp. 203–235 [HW] Hone, A.N.W., Wang, J.P.: Prolongation algebras and hamiltonian operators for peakon equations. Inverse Problems 19, 129–145 (2003)
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Holm, D.D., Staley, M.F.: Wave structure and nonlinear balances in a family of evolutionary PDEs. SIAM J. Appl. Dynam. Syst. 2, 323–380 (2003) (electronic) [HuS] Hunter, J.K., Saxton, R.: Dynamics of director fields. SIAM J. Appl. Math. 51, 1498–1521 (1991) [KLM] Khesin, B., Lenells, J., Misiołek, G.: Generalized hunter-saxton equation and the geometry of the group of circle diffeomorphisms. Math. Ann. 342, 617–656 (2008) [KM] Khesin, B., Misiołek, G.: Euler equations on homogeneous spaces and virasoro orbits. Adv. Math. 176, 116–144 (2003) [KW] Khesin, B., Wendt, R.: The Geometry of Infinite-Dimensional Groups. Ergebnisse der Mathematik Vol. 51, New York: Springer, 2008 [K] Kirillov, A.: Infinite dimensional Lie groups: their orbits, invariants and representations. The geometry of moments. Lect. Notes in Math. 970, New York: Springer-Verlag, 1982, pp. 101–123 [KY] Kirillov, A., Yuriev, D.: Kähler geometry of the infinite dimensional homogeneous space m = diff + (s 1 )/rot(s 1 ). Funkt. Anal. Prilozh. 21, 35–46 (1987) [L] Lang, S.: Differential Manifolds. New York: Springer, 1972 [Le] Lenells, J.: Traveling wave solutions of the camassa-holm equation. J. Diff. Eq. 217, 393–430 (2005) [Lu] Lundmark, H.: Formation and dynamics of shock waves in the degasperis-procesi equation. J. Nonlinear Sci. 17, 169–198 (2007) [LS] Lundmark, H., Szmigielski, J.: Degasperis-procesi peakons and the discrete cubic string. Int. Math. Res. Pap. 2, 53–116 (2005) [M1] Misiołek, G.: A shallow water equation as a geodesic flow on the bott-virasoro group. J. Geom. Phys. 24, 203–208 (1998) [M2] Misiołek, G.: Classical solutions of the periodic camassa-holm equation. Geom. Funct. Anal. 12, 1080–1104 (2002) [ON] O’Neill, B.: Submersions and geodesics. Duke Math. J. 34, 363–373 (1967) [O] Ovsienko, V.: Coadjoint representation of Virasoro-type Lie algebras and differential operators on tensor-densities. In: Infinite dimensional Kähler manifolds (Oberwolfach 1995), DMV Sem. 31, Basel: Birkhäuser, 2001, pp. 231–255 [OT] Ovsienko, V., Tabachnikov, S.: Projective Differential Geometry. Cambridge: Cambridge Univ. Press, 2005 [ST] Saxton, R., Tı˘glay, F.: Global existence of some infinite energy solutions for a perfect incompressible fluid. SIAM J. Math. Anal. 40, 1499–1515 (2008) [T] Taylor, M.: Pseudodifferential Operators and Nonlinear PDE. Boston, MA: Birkhäuser, Boston, 1991 Communicated by P. Constantin
Commun. Math. Phys. 299, 163–224 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1071-2
Communications in
Mathematical Physics
Four-Dimensional Wall-Crossing via Three-Dimensional Field Theory Davide Gaiotto1 , Gregory W. Moore2 , Andrew Neitzke1 1 School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA.
E-mail:
[email protected];
[email protected] 2 NHETC and Department of Physics and Astronomy, Rutgers University,
Piscataway, NJ 08855–0849, USA. E-mail:
[email protected] Received: 2 September 2009 / Accepted: 20 December 2009 Published online: 1 July 2010 – © Springer-Verlag 2010
Abstract: We give a physical explanation of the Kontsevich-Soibelman wall-crossing formula for the BPS spectrum in Seiberg-Witten theories. In the process we give an exact description of the BPS instanton corrections to the hyperkähler metric of the moduli space of the theory on R3 × S 1 . The wall-crossing formula reduces to the statement that this metric is continuous. Our construction of the metric uses a four-dimensional analogue of the two-dimensional tt ∗ equations.
Contents 1. 2.
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Introduction and Summary . . . . . . . . . . . . . . . Preliminaries . . . . . . . . . . . . . . . . . . . . . . 2.1 d = 4, N = 2 gauge theory . . . . . . . . . . . . 2.2 The Kontsevich-Soibelman wall-crossing formula 2.3 The low energy effective theory on R3 × S 1 . . . 2.4 The semiflat geometry . . . . . . . . . . . . . . . A Twistorial Construction of Hyperkähler Metrics . . 3.1 Holomorphic data from hyperkähler manifolds . . 3.2 Twistorial construction of g . . . . . . . . . . . . 3.3 Twistorial construction of the semiflat geometry . Mutually Local Corrections . . . . . . . . . . . . . . 4.1 The exact single-particle metric . . . . . . . . . . 4.2 Hyperkähler structure . . . . . . . . . . . . . . . 4.3 The solution for Xm . . . . . . . . . . . . . . . . 4.4 Analytic properties . . . . . . . . . . . . . . . . 4.5 Differential equations . . . . . . . . . . . . . . . 4.6 Higher spin multiplets . . . . . . . . . . . . . . . 4.7 Higher rank generalization . . . . . . . . . . . .
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Mutually Non-local Corrections . . . . . . . . . . . . . . . 5.1 Defining the Riemann-Hilbert problem . . . . . . . . . 5.2 The role of the KS formula . . . . . . . . . . . . . . . 5.3 Solving the Riemann-Hilbert problem . . . . . . . . . 5.4 Constructing the symplectic form . . . . . . . . . . . . 5.5 Differential equations . . . . . . . . . . . . . . . . . . 5.6 Constructing the metric and its large R asymptotics . . 5.7 Comparison to the physical metric . . . . . . . . . . . 6. Adding Masses . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Single-particle corrections with masses . . . . . . . . . 6.2 Multiple-particle corrections with masses . . . . . . . 7. A Proof of the Wall-Crossing Formula . . . . . . . . . . . A. Verifying the KS Identity for Some SU (2) Gauge Theories B. Cauchy-Riemann Equations on M . . . . . . . . . . . . . C. Asymptotics of Integral Equations . . . . . . . . . . . . . . D. Asymptotics of Differential Equations . . . . . . . . . . . E. A Relation to the Thermodynamic Bethe Ansatz . . . . . .
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1. Introduction and Summary The main subject of this paper is a wall-crossing formula (WCF) for the degeneracies of BPS states in quantum field theories with d = 4, N = 2 supersymmetry. Our conventions and a summary of relevant definitions can be found in Sect. 2. The space Hγ ,B P S of BPS states of charge γ is the space of states in the one-particle Hilbert space, of electromagnetic charge γ , saturating the BPS bound M ≥ |Z γ (u)|. Here u denotes a point in the vector multiplet moduli space B, that is, in the Coulomb branch of the moduli space of vacua. The only available index for d = 4, N = 2 supersymmetry is the second helicity supertrace: 1 (γ ; u) := − Tr H B P S,γ (−1)2J3 (2J3 )2 , 2
(1.1)
where J3 is any generator of the rotation subgroup of the massive little group. It has been known for a long time that such indices are generally not independent of u but are only piecewise constant [1]. Indeed, (γ ; u) can jump across walls of marginal stability, where γ = γ1 + γ2 and arg Z γ1 (u) = arg Z γ2 (u). This fact played an important role in the development of Seiberg-Witten theory [2,3]. In recent years a more systematic understanding of the u-dependence of the index has begun to emerge. Formulae for the change across walls of marginal stability were given in [4] when at least one of the constituents in the decay γ → γ1 + γ2 is primitive. These primitive and semiprimitive wall-crossing formulae were derived from physical pictures based on multicentered solutions of supergravity [5,6]. However, when both constituents have nonprimitive charges, the methods of [4] are difficult to employ. Kontsevich and Soibelman [7] have proposed a remarkable wall-crossing formula for the which applies to all possible decays. We review their formula, which we sometimes refer to as the KS formula, in Sect. 2.2. On the one hand, Kontsevich and Soibelman’s “Donaldson-Thomas invariants” ˆ ; u) are not obviously the same as the (γ ; u) of interest in physics, and the tech(γ niques they use to arrive at their formula seem somewhat removed from standard physical
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considerations. On the other hand, their WCF involves striking new concepts compared to the formulation of the semiprimitive wall-crossing formulae of [4]. In particular, the WCF is expressed in terms of a certain product of symplectomorphisms of a torus (see ˆ ; u), and hence a priori depends on u. The (2.18) below) which depends on the (γ statement of the WCF is that this product is, in fact, independent of u. That in turn ˆ ; u). This development raises the question of the determines the u-dependence of (γ physical derivation and interpretation of the KS formula and holds out the promise that some essential new physical ideas are involved. This will indeed prove to be the case. In this paper we give a physical interpretation and proof of the KS formula in the case of d = 4, N = 2 field theories. The generalization to supergravity is an interesting and important problem for future work. Here is a sketch of the main ideas and the basic strategy. We consider the gauge theory on the space R3 × S 1 , where S 1 has radius R. At low energies this theory is described by a d = 3 sigma model with hyperkähler target space (M, g). This sigma model receives corrections from BPS instantons, in which the world-line of a BPS particle of the d = 4 theory is wrapped around S 1 . Expanding the metric g at large R, one can therefore read off the degeneracies (γ ; u) of the BPS particles. This immediately raises a puzzle: we know that the (γ ; u) are discontinuous, but g should be continuous! The continuity of the metric is based on the physical principle (which was crucial in [2,3]) that the only singularities in the low energy effective field theory Lagrangian arise from the appearance at special moduli of massless particles (which should not have been integrated out in the effective theory). Physically the resolution of this puzzle is similar to one recently discussed in [8]. The exact metric g is indeed smooth, but it receives corrections from multi-particle as well as single-particle states. The disappearance of a 1-instanton contribution when a particle decays is compensated by a discontinuity in the multi-instanton contribution from its decay products. Similarly, disappearing n-instanton contributions are compensated by discontinuities in the m-instanton contributions for m > n. To put this more precisely, n (n) the n-instanton corrections have the form i=1 (γi ; u)F(γi ) (u), where the sum runs over all n-tuples {γi } of charges, and F (n) are essentially universal functions of R and the Z γi . Upon crossing the wall each (γ ; u) has a discontinuity proportional to a sum of products of (γ j ; u) with γ j = γ . At the same time, the functions F (n) have discontinuities proportional to the functions F (n ) with n < n. We will see that the Kontsevich-Soibelman wall-crossing formula expresses the consistency of this tower of cancellations. The main technical hurdle in understanding the WCF is thus to give an efficient description of the corrections to g coming from the BPS instantons. A hyperkähler metric is a complicated object and it is hard to make progress by studying, say, the corrections to its components; nor is there generally a simple additive object like the Kähler potential available. To overcome this problem we borrow some ideas from twistor theory. Recall that a hyperkähler manifold is complex-symplectic with respect to a whole CP1 worth of complex structures. The basic idea is that studying g is equivalent to studying the holomorphic Darboux coordinates on M, provided that we consider all of these complex structures at once. In the main body of this paper, we assume that the Kontsevich-Soibelman wallcrossing formula holds for (γ ; u). Under this assumption we construct the metric on M, by giving a canonical set of functions Xγ (u, θ ; ζ ) on M × C× , indexed by an electromagnetic charge γ . Here (u, θ ) specifies a point of M and the parameter ζ
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labels the complex structures on M. Each Xγ is piecewise holomorphic in ζ ; the effect of the BPS instantons is to create discontinuities in the Xγ (u, θ ; ζ ), along rays in the ζ -plane. These discontinuities are identified with the symplectomorphisms introduced by Kontsevich and Soibelman. In this approach the continuity of the metric is a consequence of the WCF. In the final section, we run the argument in reverse: using general principles of supersymmetric gauge theory, we deduce properties of the metric g which are sufficient to prove the WCF. Summary. We begin in Sect. 2 with a review of the Seiberg-Witten solution of d = 4, N = 2 gauge theories and the Kontsevich-Soibelman wall-crossing formula. We then discuss the formulation of the theory on R3 × S 1 . It is a sigma model into a manifold M, which is topologically the Seiberg-Witten torus fibration over the d = 4 moduli space B, equipped with a hyperkähler metric g. This metric depends on the radius R of S 1 . As R → ∞ it approaches a simple form, which can be obtained by naive dimensional reduction of the d = 4 theory; we call this simple metric g sf (for “semiflat”). In Sect. 3 we explain our “twistorial” construction of hyperkähler metrics: given a collection of functions Xγ (u, θ ; ζ ) on M, varying holomorphically with ζ ∈ C× and obeying certain additional conditions, there is a hyperkähler metric for which Xγ (u, θ ; ζ ) are holomorphic Darboux coordinates. In particular, we give the functions Xγsf (u, θ ; ζ ) corresponding to the semiflat metric g sf . With this background in place we are ready to consider the instanton corrections. We begin this study in Sect. 4 with the simple case of a U (1) gauge theory coupled to a single matter hypermultiplet of electric charge q. In this theory the corrected metric g is known exactly [9,10]. We explain how to obtain this corrected metric by including instanton corrections which modify the functions Xγsf (u, θ ; ζ ) to new ones Xγ (u, θ ; ζ ). In this construction we already see the building blocks of the Kontsevich-Soibelman formula appear: our Xγ (u, θ ; ζ ) naturally come out with discontinuities in the ζ -plane, which are precisely the elementary Kontsevich-Soibelman symplectomorphisms corresponding to the electric charges ±q. We then turn in Sect. 5 to the more interesting case where we have multiple kinds of BPS instanton corrections, coming from mutually non-local BPS particles in d = 4. In this case we find a natural ansatz for the Xγ (u, θ ; ζ ): essentially we just require that each BPS particle independently contributes a discontinuity like the one we found for a single particle. This discontinuity is most naturally located along a ray in the ζ -plane determined by the phase of the central charge of the BPS particle. The KontsevichSoibelman factors for mutually non-local particles do not commute; but this generically presents no problem since these particles have non-aligned central charges, and hence their discontinuities appear on distinct rays in the ζ -plane. The separation between rays disappears exactly at the walls of marginal stability; here the discontinuities pile up into products of Kontsevich-Soibelman factors. The WCF is the statement that this product is the same as we approach the wall from either side. This requirement is essential for us: it implies that the metric we construct from the Xγ (ζ ) is continuous. More precisely, to determine the Xγ (u, θ ; ζ ) we specify both their discontinuities in the ζ -plane and also their asymptotics as ζ → 0, ∞. In other words, we formulate an infinite-dimensional “Riemann-Hilbert problem” whose solution is the Xγ (u, θ ; ζ ). We do not construct its solution exactly; rather we follow a strategy closely analogous to that employed by Cecotti and Vafa, who encountered a similar (but finite-dimensional) Riemann-Hilbert problem in the study of d = 2 theories with N = (2, 2) supersymmetry [11]. A variation of their arguments allows us to show that the solution to our problem
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exists, at least for sufficiently large R. Indeed, in the large R limit the desired Xγ can be obtained by successive approximations, where the zeroth approximation is just Xγsf , and the n th approximation incorporates multi-instanton effects up to n instantons. Having constructed the functions Xγ (u, θ ; ζ ) and hence the metric g, we check that g has various properties which are expected on general field theory grounds; it passes all of these tests and we therefore argue that it should be the correct physical metric on M, generalizing a similar argument in [12]. As we mentioned above, our construction of the Xγ (u, θ ; ζ ) bears a striking similarity to constructions which appeared in the d = 2 case [11]. In that case a wall-crossing formula for the degeneracies of BPS domain walls was proven using the flat “tt ∗ connection” in the bundle of vacua of the d = 2 theory. Two components of this connection give differential equations expressing the R-symmetry and scale invariance of the d = 2 theory. Our construction can similarly be phrased in terms of a flat connection A over B × CP1 × R+ , in the infinite-dimensional bundle of real-analytic functions on the torus fibers Mu of M. The Riemann-Hilbert construction guarantees the existence of this A. Each Xγ defines a flat section. In particular, this flatness gives a pair of differential equations for the ζ and R dependence of Xγ , which have their physical origin in the anomalous R-symmetry and scale invariance of the d = 4 theory. As we describe in Sect. 7, this tt ∗ -like flat connection can be discovered using only general principles of supersymmetric gauge theory. Moreover, its mere existence is strong enough to justify our ansatz for the metric a priori. In particular, the wall-crossing formula, which appeared as a consistency requirement working within that ansatz, can be understood as the existence of an “isomonodromic deformation” constructed from A. This gives a physical proof of the wall-crossing formula. For convenience, in most of this paper we use a simple form of the wall-crossing formula which does not include information about flavor symmetries, and correspondingly we set all flavor masses to zero. In Sect. 6 we explain how to restore the flavor charge and mass information. We include several appendices with additional details. In Appendix A we explain a direct verification that the wall-crossing formula gives the correct BPS degeneracies in the case of the pure SU (2) theory. In Appendix B we describe the Cauchy-Riemann equations on M, in a way that makes contact with our construction of the hyperkähler metric and with the tt ∗ equations of [13]. In Appendix C we give the asymptotic analysis necessary for extracting the large-R corrections to the metric from our Riemann-Hilbert problem. In Appendix D we discuss some details of how to extract the differential equations from the solution of the Riemann-Hilbert problem. Finally, Appendix E explains a curious relation of one of our main results, Eq. (5.13), with the Thermodynamic Bethe Ansatz. There is much more to be said about this connection, but we leave that for another occasion. Several subsubsections of the paper are devoted to global issues which are related to a subtle but important sign in the KS formula. On a first reading it would be reasonable to skip this discussion. Readers who choose this course should allow themselves to confuse in the main text. T and T˜ , as well as M and M, Discussion. Let us remark on a few particularly interesting points. • Physically, our construction of the metric on M amounts to a rule for “integrating out” mutually non-local particles in d = 4. This problem a priori appears to be difficult because one cannot find any duality frame in which all of the particles are
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electrically charged, so it is difficult to write a Lagrangian which includes all of the relevant fields. Here we have circumvented that difficulty. Our construction of the metric uses its twistorial description. The most natural physical context in which the twistor space occurs is projective superspace [14–16], in which the parameter ζ is a bosonic superspace coordinate. The fact that the corrections to g come only from BPS instantons, and that they are localized at specific rays in the ζ -plane, should have a natural explanation in the projective superspace language. One of the inspirations for the Kontsevich-Soibelman WCF was their earlier work [17], in which they gave a construction of the sheaf of holomorphic functions on a K3 surface, by “correcting” the sheaf of functions on the semiflat K3. The corrections were formulated in terms of products of symplectomorphisms similar to those which appear in the wall-crossing formula. This construction is closely related to ours, with K3 replaced by M. The key new ingredient in our work is to consider all complex structures at once, thus introducing the parameter ζ ∈ CP1 ; having done so, we can formulate the crucial Riemann-Hilbert problem. This idea might also be useful in the original K3 context. The multi-instanton expansion of g is given as a sum of basic building blocks weighted by products of the BPS degeneracies (γ ; u). These basic building blocks have intricate discontinuities at the walls of marginal stability, which conspire with the jumps of (γ ; u) to make g continuous in u. All this is reminiscent of recent work of Joyce on wall-crossing [18]. Moreover, Joyce’s work was interpreted by Bridgeland and Toledano Laredo in [19] in terms of isomonodromic deformation of a connection on CP1 , which somewhat resembles the one we consider here, but has a slightly different form: it has an irregular singularity only at t = 0, while ours has them both at ζ = 0 and ζ = ∞. There is an interesting scaling limit of our connection, R → 0 and ζ → 0 with ζ /R = t fixed, which brings it into the form of the one in [19] (albeit with a different structure group). This limit retains the information about the BPS degeneracies and their wall-crossing. It would be interesting to see whether there is any sense in which it relates our connection to the one in [19]. In our discussion we studied structures defined over the vector multiplet moduli space B. However, both Kontsevich-Soibelman and Joyce formulate their invariants over a larger space, the space of “Bridgeland stability conditions” [20]. We do not understand the meaning of our constructions when extended to this larger space. The wall-crossing formula as formulated by Kontsevich-Soibelman makes sense not only for N = 2 field theories but also for supergravity, and indeed this was the main focus of [4]. The moduli space M of the theory on R3 × S 1 is then a quaternionicKähler manifold rather than hyperkähler. Nevertheless, most of our considerations seem to make sense in that context, with appropriate modifications. For example, Hitchin’s theorem is replaced by LeBrun’s theorem characterizing the twistor space of a quaternionic-Kähler manifold in terms of holomorphic contact structures. In particular, there is still a natural notion of a “holomorphic” function Xγ (x, ζ ) (namely, a holomorphic function on the twistor space of M), and the quaternionic-Kähler analogue of Xγsf has been worked out in [21]. We expect that the instanton-corrected metric g on M can be obtained by a method parallel to the one employed in this paper: formulate a Riemann-Hilbert problem for Xγ , using Xγsf to fix the asymptotics, and the Kontsevich-Soibelman factors to fix the discontinuities. One important difficulty to overcome is that in gravity the degeneracies (γ ; u) grow very quickly with γ ; this makes the convergence of the iterative solution for Xγ less obvious in
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this case. As in the hyperkähler case, the WCF should arise as a consistency condition ensuring that g is smooth. • The analogy between the hyperkähler geometry of the fibration M → B and the tt ∗ geometry of [13,1,11] is striking: the two structures are very similar although one has to do with field theories in d = 4, the other in d = 2. Is there a direct relation between the two? One possibility is to relate them just by compactification, e.g. on S 2 . Different values of the U (1) fluxes on S 2 would then correspond to different vacua of the d = 2 theory, and BPS states of the d = 4 theory could be identified with domain walls interpolating between these vacua in d = 2.1 Related ideas have appeared in the literature before — in particular see [22–24]. See also [25,26] for a slightly different link between BPS spectra in d = 2 and d = 4. • Infinitesimal deformations of a class of hyperkähler manifolds which include the semiflat geometry have been recently studied in [27]. It would be interesting to describe the leading correction to the semiflat geometry in their language. Our Eq. (4.33) resembles their Eq. (3.38), with an appropriate choice of H and contours of integration. 2. Preliminaries 2.1. d = 4, N = 2 gauge theory. We consider a gauge theory in d = 4 with N = 2 supersymmetry, gauge group G of rank r , and a characteristic (complex) mass scale . Seiberg-Witten theory (initiated in [2,3], and reviewed more generally in e.g. [28–30]) gives a rather complete description of the behavior of such a gauge theory on its Coulomb branch at energies μ , as follows. The Coulomb branch is a complex manifold B of complex dimension r , parameterized by the vacuum expectation values of the vector multiplet scalars. We denote a generic coordinate system on B as (u 1 , . . . , u r ). At each point u ∈ B the gauge group is broken to a maximal torus U (1)r . There is a lattice u Z2r of electric and magnetic charges, equipped with an integral-valued symplectic pairing , . This lattice is the fiber of a local system over B. That is, there is a fibration of lattices with fiber u over u ∈ B, with nontrivial monodromy around the singular loci in B, of complex codimension 1, where some BPS particles become massless. We sometimes write “γ ∈ ” informally, meaning that γ is a local section of . There is a vector Z (u) ∈ u∗ ⊗Z C of “periods,” which varies holomorphically with u. For any γ ∈ we define the central charge Z γ (u) by Z γ (u) = Z (u) · γ .
(2.1)
Z (u) plays a fundamental role in the description both of the massless and the massive sectors. We begin with the massless part. Locally on B one can choose a splitting = m ⊕ e into Lagrangian sublattices of “magnetic” and “electric” charges respectively. m and e are then dual to one another using the pairing on . Such a splitting is called an electric-magnetic duality frame. Concretely we may choose a basis {α1 , . . . , αr } for m and {β 1 , . . . , β r } for e such that
α I , α J = 0, β I , β J = 0, α I , β J = δ IJ 1 This picture has been advocated to us by Cumrun Vafa.
(2.2)
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with I, J = 1, . . . , r . After choosing such a frame, we obtain a system of “special coordinates” a I on B, which are nothing but the electric central charges, i.e. Z β I = a I.
(2.3)
The magnetic central charges are then holomorphic functions of the a I . They are determined in terms of a single function F(a I ) (depending on the chosen frame), the N = 2 prepotential: Z αI =
∂F . ∂a I
(2.4)
This implies in particular that Z is not arbitrary: from the symmetry of mixed partial derivatives one obtains
d Z , d Z = 0.
(2.5)
On the left side of (2.5) we are using the antisymmetric pairing , and also the antisymmetric wedge product of 1-forms on B; the combined pairing is symmetric, so this condition is not vacuous. Indeed, (2.5) says that around u, B can be locally identified with a complex Lagrangian submanifold of u∗ ⊗Z C. The prepotential completely determines the two-derivative effective Lagrangian, written in terms of the electric vector multiplets. To write this Lagrangian we introduce the symmetric matrix τ defined by τ I J (a) =
∂ 2F , ∂a I ∂a J
(2.6)
and then adopt a notation that suppresses the gauge index, e.g. τ |da|2 for τ I J da I ∧d a¯ J . Then the bosonic part of the Lagrangian is Re τ Im τ −|da|2 − F 2 + L(4) = F ∧ F. (2.7) 4π 4π The central charges Z γ are also of fundamental importance for the massive spectrum. Indeed, the mass of any 1-particle state with charge γ obeys M ≥ |Z γ |
(2.8)
with equality if and only if the state is BPS. BPS states belong to massive short multiplets of the super Poincare symmetry; under the little group SU (2) the states at rest in such a multiplet transform as [ j] ⊗ ([1/2] + 2[0]).
(2.9)
Choosing j = 0 gives the massive hypermultiplet, while j = 21 is the massive vector multiplet. There is a standard index which “counts” the short multiplets, namely the second helicity supertrace (γ ; u). This supertrace receives the contribution +1 for each massive hypermultiplet of charge γ in the spectrum of the theory at u ∈ B, and similarly −2 for each massive vector multiplet. (γ ; u) is invariant under any deformation of the theory in which the 1-particle states do not mix with the continuum of multiparticle states. From (2.8) it follows that such mixing is very restricted; a BPS particle can decay
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only into other BPS particles, and then only if their central charges all have the same phase. Hence (γ ; u) is locally constant in u, away from the walls of marginal stability in B. These walls of marginal stability are of real codimension 1 and are defined, for a pair of linearly independent charges γ , γ , to be the locus of u ∈ B, where Z γ and Z γ are nonzero and have the same phase. Understanding the jumping behavior of (γ ; u) as u crosses a wall of marginal stability is one of the main motivations of this paper. We turn to it next.
2.2. The Kontsevich-Soibelman wall-crossing formula. In this section we review the Kontsevich-Soibelman wall-crossing formula. As originally proposed in [7] this formula determines the jumping behavior of “generalized Donaldson-Thomas invariants” ˆ ; u). As we will see below, if we identify the Donaldson-Thomas invariants with (γ ˆ ; u) = (γ ; u), then the wall-crossing formula gives the the helicity supertraces, (γ physically expected answer in several nontrivial examples: in particular, it reproduces the “primitive wall-crossing formula” of [4], as well as the wall-crossing behavior of the BPS spectrum of Seiberg-Witten theory with gauge group SU (2). A technical point: for the KS formula to make sense, the (γ ; u) are not allowed to be completely arbitrary. Introducing a positive definite norm on , one must require that there exists some K > 0 such that |Z γ | >K γ
(2.10)
ˆ ; u) = 0. Throughout this paper we will assume that this property, for all γ such that (γ called the “Support Property,” holds. The Kontsevich-Soibelman algebra. The wall-crossing formula is given in terms of a Lie algebra defined by generators eγ , with γ ∈ , and a basic commutation relation [eγ1 , eγ2 ] = (−1) γ1 ,γ2 γ1 , γ2 eγ1 +γ2 .
(2.11)
In this paper it will be important to realize this abstract Lie algebra as an algebra of complex symplectomorphisms of a complexified torus. Modulo a subtlety which will appear at the end of this section, this torus is the fiber T˜u of the local system T˜ := ∗ ⊗Z C× . Any γ ∈ gives a corresponding function X γ on T˜u , with X γ X γ = X γ +γ . Upon choosing a basis {γ 1 , . . . , γ 2r } for , we can choose X i := X γ i as coordinates for ˜ T˜u . The symplectic pairing on ∗ gives a holomorphic symplectic form T on T˜u : if i j i j = γ , γ , and i j is its inverse, ˜
T =
1 d Xi dX j i j i ∧ . 2 X Xj
(2.12)
We would like to identify eγ with the infinitesimal symplectomorphism of T˜u generated by the Hamiltonian X γ . This almost gives the algebra (2.11), but misses the extra sign (−1) γ1 ,γ2 . This sign will be important below in comparing to wall-crossing formulas known from physics; in that context it is related to the fact that the fermion number of a bound state of two particles of charges γ1 , γ2 is shifted by γ1 , γ2 .
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Over a local patch of B, we can absorb this sign by introducing a “quadratic refinement” of the Z2 -valued quadratic form (−1) γ1 ,γ2 : this means a σ : → Z2 obeying σ (γ1 )σ (γ2 ) = (−1) γ1 ,γ2 σ (γ1 + γ2 ).
(2.13)
One way to get such a σ is to choose a local electric-magnetic duality frame ∼ = e ⊕ m , e ,γ m e m
γ . Notice that write γ = γ + γ , and set σ (γ ) = (−1) σ (γ1 )σ (γ2 ) = σ (γ1 + γ2 )(−1) γ1 ,γ2
m + γ e ,γ m 2 1
e
= σ (γ1 + γ2 )(−1) γ1 ,γ2
(2.14)
as needed. At any rate, having chosen any σ (γ ), we could identify eγ with the symplectomorphism generated by the Hamiltonian σ (γ )X γ . Any two refinements σ , σ obey σ (γ )σ (γ ) = (−1)c(σ,σ )·γ for some fixed c(σ, σ ) ∈ ∗ /2 ∗ . The Hamiltonians σ (γ )X γ and σ (γ )X γ associated to these two refinements are related by the automorphism of T˜u which sends X γ → (−1)c(σ,σ )·γ X γ . The wall-crossing formula. Now we are ready to formulate the wall-crossing formula. Its basic building block is the group element Kγ := exp
∞ 1 enγ . n2
(2.15)
n=1
Under our identification, Kγ becomes a symplectomorphism acting on T˜u , given by Kγ : X γ → X γ (1 − σ (γ )X γ ) γ
,γ
.
(2.16)
Associate to each BPS particle of charge γ a ray in the complex ζ -plane, determined by the central charge, γ := {ζ : Z γ (u)/ζ ∈ R− }.
(2.17)
As we vary u ∈ B these rays rotate in the ζ -plane. The cyclic ordering of the rays changes only when u reaches a wall of marginal stability. At such a wall a set of BPS rays γ come together, corresponding to a set of charges γ for which Z γ become aligned. At a generic point on the wall of marginal stability, this set of charges can be parameterized as2 {nγ1 + mγ2 : m, n > 0}, for some primitive vectors γ1 , γ2 with Z γ1 /Z γ2 ∈ R+ . Now associate the group element Kγ to each BPS particle of charge γ , and form the product over states which become aligned at the wall: A :=
γ =nγ1 +mγ2 m>0, n>0
Kγ(γ ;u) ,
(2.18)
where the ordering of the factors corresponds to clockwise ordering of the rays γ . We can consider this product for u on either side of the wall. As u crosses the wall, the order of the factors is reversed, and the (γ ; u) jump. The statement of the wall-crossing formula is that the whole product A is unchanged. 2 To establish the existence of these γ , γ we need to use the Support Property: otherwise one can easily 1 2 imagine situations in which the aligned Z γ accumulate near the origin.
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This condition is strong enough to determine the (γ ; u + ) from the (γ ; u − ), where u ± are points infinitesimally displaced on opposite sides of the wall. To understand how to do this in practice we first have to deal with an important subtlety: since the spectrum of BPS states is typically infinite, the product (2.18) generally involves infinitely many factors. Following [7], we can understand it as follows. The product only involves the generators enγ1 +mγ2 , where m, n > 0. The Lie algebra they generate can be consistently truncated by fixing some integer L and then setting enγ1 +mγ2 = 0 whenever n + m > L. (That is, the Lie algebra is filtered by Lie subalgebras with n + m > L, and we can take quotients by subalgebras with successively larger values of L.) After such a truncation (2.18) involves only finitely many nontrivial terms; the infinite product can be understood as the limit of these truncated products as L → ∞. In a similar spirit consider the power expansion of the symplectomorphism A, A : X γ → (1 + cγm,n (2.19) X nγ1 +mγ2 )X γ , m>0,n>0
and truncate it to n + m ≤ L. We can compute this expansion on one side of the wall of marginal stability, and then recursively identify the (γ ; u) on the other side of the wall. Concretely, first set L = 1; then (γ1 ; u) and (γ2 ; u) are fixed by the requirement 0,1 that they correctly reproduce cγ1,0 and cγ . Next set L = 2 and consider the expansion −(γ ;u)
−(γ ;u)
of AKγ1 1 Kγ2 2 to extract the next set of degeneracies. This iteration can be continued in a straightforward way to determine all of the (nγ1 + mγ2 ; u). What is far from obvious — but conjectured in [7] — is that the (γ ; u) computed in this way are integers! Some examples. In the above interpretation of the Kontsevich-Soibelman formula we identified their generalized Donaldson-Thomas invariants with the physically defined (γ ; u). To motivate this identification, we now describe a few examples. As explained above, at a generic point on a wall of marginal stability the symplectomorphisms which enter the WCF are generated by a two-dimensional lattice of charges, γ = ( p, q) ∈ Z2 with canonical symplectic form ( p, q), ( p , q ) = pq − qp . We write correspondingly X 1,0 = x, X 0,1 = y. The symplectomorphisms K p,q are then determined by their action on x and y, which is explicitly
q
− p K p,q : (x, y) → 1 − (−1) pq x p y q x, 1 − (−1) pq x p y q y . (2.20) Consider a wall of marginal stability where the central charges for a single BPS particle of primitive charge (1, 0) and a single particle of primitive charge (0, 1) come together. Kontsevich and Soibelman notice a beautiful “pentagon identity”: K1,0 K0,1 = K0,1 K1,1 K1,0 .
(2.21)
Hence the WCF predicts that crossing the wall, only one extra particle will be created, a dyonic bound state of one electrically charged particle and one magnetically charged particle. Indeed the “primitive wall-crossing formula” from supergravity (which is also valid in field theory) [4] predicts that this pair of particles will form a single bound state in a hypermultiplet representation. It also predicts that a single particle of charge (1, 0) cannot be bound to more than one particle of charge (0, 1). It is quite hard to count more general bound states of several particles of different type. Their absence is already a non-trivial prediction of the KS wall-crossing formula.
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A further comparison with the primitive wall-crossing formula helps us understand the role of the sign in the commutation relation (2.11) of the eγ . Consider the product Kγ1 Kγ2 and try to rewrite it as a product in the opposite direction, (i.e. with the slopes of Z γi increasing instead of decreasing) of the form Kγ2 · · · Kγ1 . Suppose γ1 , γ2 are primitive and consider the subalgebra generated by enγ1 +mγ2 quotiented by that with n ≥ 2, m ≥ 2. The result is a Heisenberg algebra. The KS formula in the truncated Heisenberg group reads (γ +γ2 ;u + )
1 1 ;u + ) K Kγ(γ γ1 +γ2 1
(γ2 ;u − )
2 ;u + ) = K Kγ(γ γ2 2
(γ +γ2 ;u − )
Kγ1 +γ12
(γ1 ;u − )
Kγ1
,
(2.22)
where u ± are points infinitesimally displaced on either side of the wall. Now, at a generic point on the wall of marginal stability we have (γi ; u + ) = (γi ; u − ) for i = 1, 2. Moreover, Kγ1 +γ2 is central in the Heisenberg group, and therefore, computing the group commutator we reproduce the corollary of the primitive wall-crossing formula: = (−1) γ1 ,γ2 −1 γ1 , γ2 (γ1 ; u)(γ2 ; u).
(2.23)
A more elaborate version of this argument allows one to extract the semiprimitive wallcrossing formula of [4] from the KS formula.3 The example in (2.21) is exceptional in that both sides involve a finite number of terms. More typically one encounters infinite products. A second beautiful example presented by Kontsevich and Soibelman is the following: −2 2 2 2 2 2 4 2 2 2 K1,0 · · · K3,2 . (2.24) K0,1 = K0,1 K1,2 K2,3 · · · K1,1 K2,2 K2,1 K1,0 We give an instructive proof of this identity in Appendix A. By a change of basis we obtain a physically very interesting formula, −2 2 2 2 2 2 4 2 2 2 K1,−1 · · · K3,−1 , (2.25) K0,1 = K0,1 K1,1 K2,1 · · · K1,0 K2,0 K2,−1 K1,−1 which captures the spectrum of an SU (2) Seiberg-Witten theory with two massless flavors (more precisely, hypermultiplets transforming in the vector representation of an S O(4) = SU (2) A × SU (2) B flavor symmetry) as described in [31].4 On the right side we see the full weak coupling spectrum: one W boson of charge (2, 0) (which contributes −2 to the helicity supertrace), the four hypermultiplets of charge (1, 0), and a set of dyons of charge (n, ±1), with multiplicity 2. (In fact these dyons are in doublets of SU (2) A or SU (2) B , depending on the parity of n.) On the left side we see the strong coupling spectrum: a single monopole with multiplicity 2 (a doublet of SU (2) A ) and a single dyon with multiplicity 2 (a doublet of SU (2) B .) The small change of variables y → −y 2 converts the product formula (2.25) into
−2 K2,−1 K0,1 = K0,1 K2,1 K4,1 · · · K2,0 · · · K6,−1 K4,−1 K2,−1 . (2.26) This formula captures the wall-crossing behavior of the pure SU (2) Seiberg-Witten theory.5 The left side includes the two particles present at strong coupling [32]: a monopole of charge (0, 1) and a dyon of charge (2, −1). The right side includes the infinite spectrum of dyons at weak coupling, together with the W boson contribution −2 K2,0 . 3 This was shown in unpublished work with Wu-yen Chuang. 4 The relation of the identity (2.25) to Seiberg-Witten theory was first suggested by Frederik Denef. The
precise relation of (2.25) to the N f = 2 theory was worked out in collaboration with Wu-yen Chuang. 5 The close resemblance between (2.25) and (2.26) arises because the Seiberg-Witten curve for the N = 2 f theory with zero masses is a double cover of that for the N f = 0 theory.
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Adding flavor information. The product (2.25) describes the BPS spectrum of SU (2) Seiberg-Witten theory with N f = 2, but does not carry information about the flavor charges of the BPS particles. We now describe a conjectural variant of the KS formula which includes the information about flavor charges. (We will see the physical motivation for this formula in Sect. 6.) Introduce a new lattice of flavor charges f , and a new parameter log μ ∈ ( f )∗ ⊗Z C× . Then generalize the X γ to new functions labeled by (γ , γ f ) ∈ ⊕ f :6 letting a run over a basis for f , f f (X i )γi (μa )γa = X γ (μa )γa . (2.27) X γ ,γ f := i
a
a
Define refined symplectomorphisms carrying flavor information: Kγ ,γ f : X γ → X γ (1 − σ (γ )X γ ,γ f ) γ
,γ
.
(2.28) f
The central charge now depends on the masses m a , Z γ ,γ f (u) = Z γ (u) + γa m a , and determines new walls of marginal stability. (μa are functions of the m a . See Sect. 6 below.) We introduce a product analogous to (2.18), A :=
(γ ,γ f )=nγ1 +mγ2 m>0, n>0
(γ ,γ f ;u)
Kγ ,γ f
.
(2.29)
The extended WCF states the continuity of A across the walls. We derive a refined version of the infinite product (2.25), including the flavor charges, in Appendix A; combining this with the extended WCF we obtain the correct wall-crossing for the SU (2) theory with N f = 2. Global issues. So far in this section we have worked over a local patch in B, and chosen a fixed quadratic refinement σ in order to identify the Kontsevich-Soibelman algebra with an algebra of symplectomorphisms of the complexified torus T˜u , a fiber of the local system T˜ . It is impossible in general to choose such a refinement globally over B, because of the monodromies of the local system . Hence it is not true globally that the Kontsevich-Soibelman algebra is the algebra of symplectomorphisms acting on T˜ . However, by an appropriate twisting of T˜ we can define a closely related complexified torus fibration T , on which the Kontsevich-Soibelman algebra does act. T is defined so that a local choice of quadratic refinement gives an identification T T˜ , and given two different refinements σ, σ , the corresponding identifications differ by ˜ the map X γ → (−1)c(σ,σ )·γ X γ on T˜ . The fiberwise symplectic form T induces a corresponding fiberwise symplectic form T on T . We can construct a twisted fibration T with the above properties as follows. Let R denote the local system over B whose local sections are refinements σ . R is a torsor for ∗ /2 ∗ , and T is the associated fibration, (2.30) T := T˜ × R / (X γ , σ ) ∼ ((−1)c(σ,σ )·γ X γ , σ ) . 6 Strictly speaking, the full local system ˆ of charges does not split into ⊕ f globally; we really have an extension 0 → f → ˆ → → 0. However, we can always split this extension locally, and this is sufficient for our purposes.
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2.3. The low energy effective theory on R3 × S 1 . Our goal is to explain the Kontsevich-Soibelman WCF as a statement about the gauge theory on R3 × S 1 , with S 1 of radius R. We study the theory at an energy scale μ which is low compared to all other scales, i.e., μ and also μ 1/R. At this energy the theory looks effectively three-dimensional. In this section we describe some of its basic properties. In the limit of large radius, R 1/, we can determine the three-dimensional dynamics using the infrared Lagrangian (2.7). The dynamical degrees of freedom are just the x 4 -independent modes of the four-dimensional fields. These include of course the scalars a I . In addition, from the gauge sector we get the “electric” Wilson lines θeI :=
S1
A4I d x 4 ,
(2.31)
as well as another set of periodic scalars θm,I obtained by dualizing the d = 3 gauge fields AαI d x α . We will often think of these as “magnetic” Wilson lines, θm,I :=
S1
(A D,4 ) I d x 4 .
(2.32)
We can define θm,I either by working in a formulation treating the gauge fields as selfdual, or by working at fixed magnetic quantum numbers P I and introducing θm,I as their Fourier duals. All these periodic scalars coordinatize a 2r -torus Mu at any fixed u ∈ B. Letting u vary we obtain a torus fibration M. The fiber Mu degenerates over the singular loci in B. The low energy theory on R3 is a sigma model with target space M. More precisely, θ = (θeI , θm,I ) is an element in the fiber of a local system of 2r -tori := ∗ ⊗Z (R/2π Z). For any γ ∈ , we get an angular coordinate on M denoted M θγ := γ · θ . M is not exactly the same as M; there is a global twisting which we glossed over above, and which we discuss at the end of this section. In sum, the three-dimensional theory is a sigma model into a Riemannian manifold M of real dimension 4r , which is topologically a 2r -torus fibration over B. The theory enjoys N = 4 supersymmetry (8 real supercharges), which implies that the metric on M is hyperkähler. This metric is the main object of study in this paper. It was studied previously in [12], where in particular the R → 0 limit for pure SU (2) gauge theory was identified as the Atiyah-Hitchin manifold. In this paper we are more interested in the opposite limit R → ∞, because in this limit one can read off the imprint of the full BPS spectrum of the theory in d = 4. In the next section we begin by considering the leading behavior in this limit. Global issues. In the description above we were slightly naive about the precise definition of the Wilson lines. Our description is adequate over a local patch in B, but as we will see in Sect. 4, it cannot be quite correct globally. Indeed, in order for the metric on M to be smooth, we will see that the monodromies around paths in B must generally be accompanied by shifts of the Wilson lines by π . This contradicts our naive description, comes with a distinguished zero section. since the torus fibration M We propose that the correct global picture is as follows: at any fixed u ∈ B, the Wilson u , but not canonically isomorphic. lines live in a torus Mu which is isomorphic to M u upon choosing a refinement σ of the quadratic One obtains an isomorphism Mu M
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form (−1) γ1 ,γ2 on u .7 Such a refinement generally exists only locally, so the fibrations are globally different. Given two local refinements σ , σ the corresponding M and M differ by the shift θ → θ + π c(σ, σ ) acting on M. two local isomorphisms M M Of course, this discussion is closely parallel to the relation between the torus fibrations T and T˜ which we described at the end of Sect. 2.2. 2.4. The semiflat geometry. The leading behavior of the metric on M in the R → ∞ limit is governed by the d = 3 effective action obtained by simply truncating (2.7) to its x 4 -independent sector. This gives
R R 1 2 dθ L(3) = (Im τ ) − |da|2 − F (3) ∧ F (3) − 2 2 8π 2 R e
1 (2.33) dθe ∧ F (3) . + (Re τ ) 2π Then dualizing the d = 3 gauge field A I to a scalar θm,I gives after a little rearranging R 1 (3) (Im τ )−1 |dθm − τ dθe |2 . Ldual = − (Im τ )|da|2 − 2 8π 2 R
(2.34)
This is the Lagrangian for a sigma model into M, with metric locally given by g sf = R(Im τ )|da|2 +
1 (Im τ )−1 |dz|2 , 4π 2 R
(2.35)
where we introduced dz I = dθm,I − τ I J dθeJ .
(2.36)
(While this notation is very convenient, we should emphasize that the form “dz I ” is not closed on the whole M: it is only closed when restricted to each torus fiber Mu .) We call g sf the “semiflat” metric on M, because in this metric the torus fibers are flat. The expression (2.35) reflects the fact that g sf is Kähler, with respect to a complex structure on M for which da I and dz I are a basis for 1,0 . In this complex structure M is the Seiberg-Witten fibration by compact complex tori over B. (We contrast this with other complex structures on M which we will meet momentarily, in which the tori Mu are not complex submanifolds.) The fibers Mu all have volume r 1 vol (Mu ) = . R
(2.37)
The expression (2.35) is valid only locally, since it uses a choice of duality frame. Nevertheless the expressions in different frames glue together into a smooth metric, everywhere except over the singular loci of B, where g sf has a singularity. Such a singularity would be unexpected from the point of view of effective field theory; we will see that it is resolved by BPS instanton corrections in the exact quantum-corrected metric g. 7 Such quadratic refinements frequently appear in the precise formulations of self-dual gauge theories [33–36]. It seems likely that the origin of σ here can be explained in this way.
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3. A Twistorial Construction of Hyperkähler Metrics In this section we review some general facts about hyperkähler geometry, and then explain the basic idea underlying our description of g.
3.1. Holomorphic data from hyperkähler manifolds. We first recall some holomorphic data attached to any hyperkähler manifold. By definition, a hyperkähler manifold (M, g) is Kähler with respect to a triplet of complex structures J, obeying the relations J1 J2 = J3 ,
J2 J3 = J1 ,
J3 J1 = J2 ,
Jα2 = −1.
(3.1)
Let ωα denote the three corresponding Kähler forms. In fact, any hyperkähler (M, g) is Kähler with respect to a more general complex structure, namely a α Jα with 3α=1 aα2 = 1, with corresponding Kähler form a α ωα . So we have a whole S 2 worth of complex structures. One of the key insights of the twistor approach is that it is useful to consider this S 2 as the Riemann sphere, labeled by a complex parameter ζ . So we write the general complex structure and corresponding Kähler form as i(−ζ + ζ¯ )J1 − (ζ + ζ¯ )J2 + (1 − |ζ |2 )J3 , 1 + |ζ |2 i(−ζ + ζ¯ )ω1 − (ζ + ζ¯ )ω2 + (1 − |ζ |2 )ω3 = . 1 + |ζ |2
J (ζ ) =
(3.2)
ω(ζ )
(3.3)
We also organize the Kähler forms into a second combination, (ζ ) = −
i i ω+ + ω3 − ζ ω− , 2ζ 2
(3.4)
where we introduced the notation ω± = ω1 ± iω2 .
(3.5)
The essential property of (ζ ) is that for any fixed ζ ∈ CP1 it is a holomorphic symplectic form on M in complex structure J (ζ ) . (To make sense of this statement for ζ = 0, ∞ we have to rescale (ζ ) by ζ , 1/ζ respectively. Globally one could say that (ζ ) is twisted by the line bundle O(2) over CP1 .) 3.2. Twistorial construction of g. Now we describe the method of determining g from holomorphic data on M, which will be used in the rest of this paper. First we specify our assumptions. Recall that M is topologically a torus fibration over B. For any choice of local patch in B, quadratic refinement, and local section γ of the charge lattice , we assume given a locally defined C× -valued function Xγ (u, θ ; ζ ) of (u, θ ) ∈ M and ζ ∈ C× , with the following properties: • The Xγ are multiplicative, Xγ Xγ = Xγ +γ .
(3.6)
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• The Xγ obey a reality condition, Xγ (ζ ) = X−γ (−1/ζ¯ ). • All Xγ are solutions to a single set of differential equations, of the form
∂ 1 (−1) (0) X = + Au i X , A ∂u i ζ ui ∂ (0) (1) X = A i¯ + ζ A i¯ X , ¯i u¯ u¯ ∂ u¯
(3.7)
(3.8) (3.9)
, A(n) are complex vertical vector fields on the torus fiber where the operators A(n) i¯ ui u¯
Mu , with the Au(−1) linearly independent at every point, and similarly A(1) . (To motii u¯ i¯ vate these equations, note that in Appendix B we show that the Cauchy-Riemann equations on (M, g) have this form.) • For each fixed x ∈ M, Xγ (x; ζ ) is holomorphic in ζ on a dense subset of C× . (In our application below, Xγ (x; ζ ) will be holomorphic away from a countable union of lines.) To state our last three assumptions on the functions Xγ we first define (ζ ) :=
dXγ j dXγ i 1 i j ∧ , 2 8π R Xγ i Xγ j
(3.10)
where by d we mean the fiberwise differential, i.e. we treat ζ as a fixed parameter. We assume: • (ζ ) is globally defined (in particular the (ζ ) defined over different local patches of B agree with one another) and holomorphic in ζ ∈ C× . (Note that this does not imply that the Xγ are holomorphic in ζ ; in our application they will be only piecewise holomorphic.) • (ζ ) is nondegenerate in the appropriate sense for a holomorphic symplectic form, i.e. ker (ζ ) is a 2r -dimensional subspace of the 4r -dimensional TC M. • (ζ ) has only a simple pole as ζ → 0 or ζ → ∞. In the rest of this section we explain how to define a hyperkähler metric g on M, such that Xγ (ζ ) are holomorphic functions in complex structure J (ζ ) , and (ζ ) is the holomorphic symplectic form as in (3.4). We consider the manifold Z := M × CP1 . It has the following properties: 1. Z is a complex manifold. At any (x, ζ ) the 2r equations (3.8), (3.9) define a half-dimensional subspace of TC M (if ζ = 0 or ζ = ∞ this is still true after rescaling one of the equations by a factor ζ ). The direct sum of this subspace and the one generated by ∂/∂ ζ¯ is a half-dimensional subspace of TC Z. We define T 0,1 Z to be this subspace. This a priori defines only an almost complex structure on Z. However, the existence of the functions Xγ guarantees that this almost complex structure is actually integrable. (Of course, the Xγ are not everywhere holomorphic in ζ ; but they are holomorphic on a dense set, which is enough to guarantee the vanishing of the Nijenhuis tensor. It follows in particular that there exist complex coordinates on Z even around ζ = 0 or ζ = ∞.) 2. Z is a holomorphic fibration over CP1 . The projection is simply p(x, ζ ) = ζ .
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3. There is a holomorphic section of 2Z /CP1 ⊗ O(2), giving a holomorphic symplectic
form on each fiber p −1 (ζ ). This is the globally defined (ζ ). 4. There is a family of holomorphic sections s : CP1 → Z, each with normal bundle N O(1)⊕2r . Indeed, for each x ∈ M, we can define a section sx : CP1 → Z by sx (ζ ) = (x, ζ ). To see that this is a holomorphic section, note first that it is holomorphic at least away from ζ = 0, ∞, just because the local complex coordinates Xγ i (x, ζ ) of Z are holomorphic in ζ at fixed x; but it extends continuously to ζ = 0, ∞, so it must be holomorphic there as well by the Riemann removable singularity theorem. To show that the normal bundle N (sx ) O(1)⊕2r , first note that there is a 1-1 correspondence between holomorphic sections of N ∗ (sx ) and holomorphic functions on the first infinitesimal neighborhood of sx which vanish on sx . But such functions are determined by their first-order Taylor expansion around x, i.e. they correspond to holomorphic sections of the trivial bundle p ∗ ((TC∗ )x M) which annihilate the subbundle B ⊂ p ∗ ((TC )x M) defined by Eqs. (3.8), (3.9). Dualizing, we have N (sx ) p ∗ ((TC )x M)/B. On the other hand (3.8), (3.9) give 2r trivializing sections of B ⊗ O(1). So we conclude that N (sx ) ⊗ O(−1) is trivial. 5. There is an antiholomorphic involution σ : Z → Z, which covers the antipodal map on CP1 , and preserves in the sense that σ ∗ = . This involution is just σ (x, ζ ) = (x, −1/ζ¯ ). Using the reality condition (3.7) we can check that it is antiholomorphic and preserves .
These are the characteristic properties of the twistor space of a hyperkähler manifold as described in [37,38]. In particular, using the recipe of [37,38], one can reconstruct a hyperkähler metric g on M from Z. We can describe g concretely: note that from (ζ )r +1 = 0 it follows that ω+r ∧ ω3 = 0, which implies that the real 2-form ω3 is of type (1, 1) in complex structure J3 . Therefore we can use J3 and ω3 to build a Kähler metric g on M. This g coincides with the hyperkähler metric guaranteed by the twistor construction. In the following sections we will use this approach.
3.3. Twistorial construction of the semiflat geometry. The foregoing description of hyperkähler metrics is particularly convenient in the case of the semiflat metric g sf which we introduced in Sect. 2.4. As above, we work over a local patch in B, and make a local choice of quadratic refinement. Then for any γ ∈ we write the locally defined function8 Xγsf (ζ ) := exp π Rζ −1 Z γ + iθγ + π Rζ Z¯ γ .
(3.11)
These functions obey “Cauchy-Riemann equations” of the form (3.8), (3.9), where (−1)
Au i
= −iπ R
∂Z ∂ ∂ Z¯ ∂ (1) , A i¯ = −iπ R ¯ · , · i u¯ ∂u ∂θ ∂ u¯ i ∂θ (0)
Au i = 0, A
(0) u¯ i¯
= 0,
(3.12) (3.13)
8 This formula was first obtained in joint work with Boris Pioline, and is essentially the rigid limit of a formula in [21] for the quaternionic-Kähler case. It provided an important clue to discovering the constructions described in this paper.
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and Z stands for the vector of periods. Then sf (ζ ) is sf
sf
dXγ i dXγ j 1 (ζ ) := ∧ (3.14) i j 8π 2 R Xγsfi Xγsfj
1 1 i ¯ ¯
d Z , dθ + π R d Z , d Z −
dθ, dθ +iζ d Z , dθ . = 4π ζ 2π R (3.15) sf
(Note that the vanishing condition (2.5) ensures that sf (ζ ) has no terms of order ζ −2 or ζ 2 .) sf (ζ ) and Xγsf (ζ ) obey the necessary conditions for the construction we described in Sect. 3.2, so they are the holomorphic symplectic form and complex coordinates for some hyperkähler metric on M. As we now check, this metric is simply g sf as desired. First note that comparing the leading terms in (3.4) and (3.15) gives ω+sf = −
1
d Z , dθ . 2π
(3.16)
From ω+sf we can determine complex structure J3sf : indeed, after choosing an electricmagnetic duality frame, we can rewrite (3.16) as ω+sf =
1 da I ∧ dz I . 2π
(3.17)
This makes manifest that M in complex structure J3sf is just the Seiberg-Witten fibration by complex tori. This is the complex structure we already described in Sect. 2.4. Similarly, comparing the ζ -independent terms in (3.4) and (3.15) gives ω3sf =
1 R
d Z , d Z¯ −
dθ, dθ , 4 8π 2 R
which we can rewrite as
i 1 sf I J −1 I J ω3 = R(Im τ ) I J da ∧ d a¯ + ((Im τ ) ) dz I ∧ d z¯ J . 2 4π 2 R
(3.18)
(3.19)
Comparing this with (2.35) we see that g sf is indeed Kähler for complex structure J3sf and Kähler form ω3sf , and hence it is the hyperkähler metric guaranteed by the twistor construction starting from sf (ζ ). In this section we have seen that the semiflat metric on M and its hyperkähler structure can be constructed from the functions Xγsf defined in (3.11). These functions are of fundamental importance for what follows. 4. Mutually Local Corrections If we considered only the naive dimensional reduction of the massless sector, then the semiflat metric g sf would be the end of the story. However, the theory in d = 4 also contains massive BPS particles. The metric receives corrections from “instanton” configurations in which one or more of these massive particles go around S 1 . These corrections will be weighted by a factor of at least e−2π R|Z | , because of the bound M ≥ |Z | on the energy of states in the d = 4 theory.
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In this section we study these corrections in examples in which all of the BPS particles are mutually local. This is much more tractable than the general situation, because we can choose a duality frame in which these particles are all electrically charged, and hence we can work completely within an effective Lagrangian description. For most of the section we specialize further to the free U (1) gauge theory coupled to a single charged hypermultiplet. In addition to being the simplest example, this theory is physically relevant because it describes the physics near a generic singularity in B, where one BPS particle becomes much lighter than the others. 4.1. The exact single-particle metric. We consider a U (1) gauge theory on R3 × S 1 , coupled to a single hypermultiplet of charge q > 0 (along with its CPT conjugate of charge −q). The metric we will describe has been considered previously in [9,10]. The moduli space B of the d = 4 theory is coordinatized by the vector multiplet scalar a ∈ C. More precisely, B is only an open patch in C, because the d = 4 theory is not asymptotically free: there is a cutoff at |a| ∼ ||. As we explained in Sect. 2.3, the moduli space M of the d = 3 theory is a 2-torus fibration over B. The torus fibers Ma are coordinatized (temporarily ignoring the subtlety about quadratic refinements) by the electric Wilson line θe and the magnetic Wilson line θm , both with periodicity 2π . The semiflat metric g sf has an action of U (1)2 by isometries, because shifts of θe and θm are exact symmetries. The electrically charged hypermultiplet couples to θe , and hence breaks the isometry which shifts it. However, there are no magnetically charged BPS states in the theory, so shifts of θm are still exact isometries. The corrected metric g is therefore of Gibbons-Hawking form. For comparison with the Gibbons-Hawking ansatz we introduce a vector x by a = x 1 + i x 2 , θe = 2π Rx 3 .
(4.1)
θm is a local coordinate on a U (1) bundle over the open subset of R2 × S 1 parameterized by x. The metric is g = V ( x )−1
dθm + A( x) 2π
2 + V ( x )d x2 ,
(4.2)
where V is a positive harmonic function, to be calculated below, and A is a U (1) connection with curvature F = d V.
(4.3)
This is a slight generalization of the standard Gibbons-Hawking ansatz, in which one takes x to lie in (an open subset of) R3 . (We can first work over a suitable subset of R3 and then divide by a Z-action on the total space which shifts x 3 .) In the standard ansatz all A obeying (4.3) are gauge equivalent and so define the same metric. In our case this is not quite true: there is one additional gauge invariant degree of freedom associated to the holonomy around S 1 . This choice is related to the choice of a θ angle in d = 4. V ( x ) in our case can be calculated by integrating out the charged hypermultiplet at one loop. Reference [10] asserts a nonrenormalization theorem which implies that the computation is exact. The resulting V is a harmonic function with q singularities in
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R2 × S 1 . The periodicity in θe arises because one sums over the Kaluza-Klein momenta of the charged hypermultiplet on S 1 : ⎛ ⎞ ∞ 1 q2 R ⎝ − κn ⎠ . (4.4) V = 4π n=−∞ θe 2 2 2 2 q R |a| + (q + n) 2π
Here κn is a regularization constant introduced to make the sum converge. Poisson resummation of (4.4) shows that V = V sf + V inst , with V
sf
V inst
a a¯ q2 R , log + log =− ¯ 4π q 2 R inqθe = e K 0 (2π R|nqa|). 2π
(4.5)
(4.6) (4.7)
n=0
Here is an ultraviolet cutoff related to the choice of κn .9 To specify the metric fully we must also give A( x ) obeying (4.3): A = Asf + Ainst , where
a a¯ log − log dθe , ¯
q 2 R da d a¯ =− (sgn n)einqθe |a|K 1 (2π R|nqa|). − 4π a a¯
Asf = Ainst
iq 2 8π 2
(4.8)
(4.9) (4.10)
n=0
At large R the leading terms in V and A are V sf and Asf . Keeping only these terms, g becomes the semiflat metric with τ=
a q2 log . 2πi
(4.11)
This is the running coupling which comes from integrating out the hypermultiplet in d = 4. The subleading terms V inst , Ainst yield corrections to the semiflat metric. They have the form of an instanton expansion as we expected, because of the asymptotic behavior π −x e for x → +∞. They also break the translation invariance along θe as K ν (x) ∼ 2x expected. Finally, they improve the singular behavior. Recall that in g sf there is a singularity at a = 0. From (4.4) we see that the only possible singularities of g occur at a = 0, qθe = 2π n. Studying the metric near these points we find that there is an Aq−1 conical singularity at each one. So the singularity in g sf is replaced by q higher-codimension singularities in g. In the simplest case q = 1, the singularity is completely smoothed. 9 For example, if we choose κ n ˜ exp[−2 ∞ m=1 K 0 (2π m||)].
˜ 2 + n 2 )−1/2 , then we can choose = (q R)−1 ˜ = (||
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Global issues. There is a subtle issue regarding the global definition of the coordinate θm . We have chosen a gauge which is convenient for discussing the periodicity in θe . However, the presence of the logarithm in Asf signals that this gauge is singular at a = 0. q2 dθe upon continuation around the origin a → e2πi a. This Moreover Asf shifts by − 2π shift must be compensated by a gauge transformation θm → θm + q 2 θe + C. To fix C we make a gauge transformation to a new coordinate θm :
a a¯ i log − log (q 2 θe + C). θm = θm + ¯ 4π
(4.12)
(4.13)
The transformed θm is single-valued as a goes around the origin. The gauge transformed Asf is
da d a¯ i (A )sf = − − (q 2 θe + C). (4.14) 4π a a¯ Now we focus on the behavior at qθe = π . Here we have Ainst = 0, so the exact gauge field is just given by (A )sf . On the other hand, once the instanton corrections are included, there is no singularity either of the metric or of the U (1) bundle at this point (recall that the only singularities occur at a = 0, qθe = 2π n.) Since moreover θm is single-valued, it follows that (A )sf cannot have a singularity here even if we go to a = 0 (or more precisely the only allowed singularity is a quantized Dirac string), which implies C = −qπ + 2π k
(4.15)
for some integer k. So we conclude that as we go around a = 0 the angular coordinates shift by θe → θe , θm → θm + q 2 θe − qπ.
(4.16a) (4.16b)
The shift by q 2 θe is as expected from the monodromy of the torus fibration. The shift by −qπ is more surprising, but fits into our discussion in the end of Sect. 2.3, where we proposed that the Wilson lines are well defined only after choosing a local quadratic refinement σ . So far in this section we have chosen the “standard” refinement σ (γe , γm ) = (−1)γe γm . The monodromy shifts γe → γe + q 2 γm , and hence replaces σ 2 2 by σ (γe , γm ) = (−1)q γm σ (γe , γm ). This change of refinement is compensated by the shift of θm by −qπ . 4.2. Hyperkähler structure. Next we want to describe M as a hyperkähler manifold. The hyperkähler structure of any Gibbons-Hawking metric is determined by the triplet of symplectic forms
1 dθm ωα = d x α ∧ + A( x ) + αβγ V d x β ∧ d x γ . (4.17) 2π 2
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The holomorphic symplectic form (3.4) is then (ζ ) =
1 ξm ∧ ξe , 4π 2 R
(4.18)
where
1 ξm = idθm + 2πi A( da − ζ d a¯ , x ) + πi V ζ
1 ξe = idθe + π R da + ζ d a¯ . ζ
(4.19) (4.20)
In particular it follows that ξe and ξm are of type (1, 0). Moreover, ξe can be written as ξe =
dXe , Xe
(4.21)
where
a Xe = exp π R + iθe + π Rζ a¯ . ζ
(4.22)
So Xe is a holomorphic function on M in complex structure J (ζ ) . Notice that it coincides with the semiflat coordinate Xγsf given in (3.11), if we choose γ to be the unit electric charge, since in that case Z γ = a and θγ = θe . In other words, the “electric” complex coordinate is unaffected by the instanton corrections due to the electrically charged particle, Xe = Xesf .
(4.23)
To finish describing the complex geometry of M one should construct a second “magnetic” complex coordinate Xm , such that (ζ ) = −
1 dXe dXm ∧ . 4π 2 R Xe Xm
(4.24)
Such a Xm is necessarily of the form ¯ e ,ζ ) Xm = eiθm +(a,a,θ .
(4.25)
The most obvious way of constructing Xm would be to write out the Cauchy-Riemann equations on M and look for a particular solution of the form (4.25). In the next section we follow a different approach: we give a particular solution for Xm directly, in a form which will be especially convenient for what follows, and then rather than checking the Cauchy-Riemann equations we check (4.24) directly.
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4.3. The solution for Xm . Now we specialize to our M. In this case we have (ζ ) = sf (ζ ) + inst (ζ ), where
(4.26)
1 sf sf 1 (ζ ) = − 2 ξe ∧ idθm + 2πi A + πi V da − ζ d a¯ , 4π R ζ
1 inst inst inst 1 da − ζ d a¯ . (ζ ) = − 2 ξe ∧ 2πi A + πi V 4π R ζ sf
If we neglect the instanton corrections, the desired magnetic coordinate is
a a¯ Rq 2 ζ Rq 2 Xmsf (ζ ) = exp −i a log − a + iθm + i a¯ log − a¯ . ¯ 2ζ 2
(4.27) (4.28)
(4.29)
This coincides with the expression (3.11) for the holomorphic coordinate Xγsf in the semiflat geometry, if we choose γ to be the unit magnetic charge, with Z γ = and θγ = θm . A direct computation verifies that
dXmsf sf sf 1 da − ζ d a ¯ = idθ + 2πi A + πi V m Xmsf ζ
2 a a¯ dXe iq log − log − , ¯ 4π Xe
q2 a 2πi (a log −a)
(4.30)
and hence in particular sf (ζ ) = −
1 dXe dXmsf ∧ sf , 4π 2 R Xe Xm
(4.31)
as expected. Notice that Xmsf has a nontrivial monodromy around a = 0: the monodromies of log a and log a¯ combine with the monodromy of eiθm given in (4.16b) to give q2
Xmsf → (−1)q Xe Xmsf .
(4.32)
Next we include the instanton corrections. As we will demonstrate below, we can give the desired Xm obeying (4.24) by the integral formula dζ ζ + ζ iq sf log[1 − Xe (ζ )q ] Xm = Xm exp 4π + ζ ζ − ζ dζ ζ + ζ iq −q log[1 − Xe (ζ ) ] , (4.33) − 4π − ζ ζ − ζ where we choose the contours ± to be any paths connecting 0 to ∞ which lie in the two half-planes a U± = ζ : ±Re < 0 . (4.34) ζ
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The two integral contributions in (4.33) come respectively from instanton corrections of positive and negative winding around S 1 . In the rest of this section we verify that (4.33) is indeed correct. This amounts to verifying −
1 dXe dXm ∧ = (ζ ). 4π 2 R Xe Xm
(4.35)
dXm dXmsf = + I+ + I− , Xm Xmsf
(4.36)
From (4.33) we have
where iq 2 I± = − 4π
±
dζ ζ + ζ ζ ζ − ζ
Xe (ζ )±q dXe (ζ ) . 1 − Xe (ζ )±q Xe (ζ )
(4.37)
(Here we used the fact that the integrals in (4.33) depend on a, a, ¯ θm , θe only through Xe (ζ ), and are absolutely convergent, so we are free to bring the differential d inside.) Combining (4.35), (4.36), and (4.31), we see that the integrals I± need to give the instanton part inst (ζ ) on the right side of (4.35), i.e. we need
1 dXe (ζ ) dXe (ζ ) ∧ (I+ + I− ) = ∧ 2πi Ainst + πi V inst da − ζ d a¯ . (4.38) Xe (ζ ) Xe (ζ ) ζ To check this we first note that
dXe (ζ ) Xe (ζ )±q dζ ζ +ζ dXe (ζ ) dXe (ζ ) iq 2 ∧ I± = − ∧ Xe (ζ ) 4π ± ζ ζ −ζ Xe (ζ ) Xe (ζ ) 1−Xe (ζ )±q (4.39) and the two-form which appears here can be rewritten, ζ + ζ dXe (ζ ) dXe (ζ ) ζ + ζ dXe (ζ ) dXe (ζ ) dXe (ζ ) ∧ = ∧ − (4.40) ζ − ζ Xe (ζ ) Xe (ζ ) ζ − ζ Xe (ζ ) Xe (ζ ) Xe (ζ )
1 1 dXe (ζ ) ∧ da −(ζ +ζ )d a¯ , = −π R + Xe (ζ ) ζ ζ (4.41) using the explicit form (4.22) of Xe . Hence the left side of (4.38) becomes
1 1 dζ Xe (ζ )q iq 2 R dXe (ζ ) ∧ da − (ζ + + ζ )d a ¯ 4 Xe (ζ ) ζ ζ 1 − Xe (ζ )q + ζ
1 1 dζ Xe (ζ )−q . (4.42) da − (ζ + + + ζ )d a ¯ ζ ζ 1 − Xe (ζ )−q − ζ Now we are ready to evaluate the integrals. It is convenient first to deform each of the contours ± to a canonical choice lying exactly in the middle of U± , i.e. to choose a ± = ζ : ± ∈ R− . (4.43) ζ
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We first consider the terms which multiply ζ or ζ1 . Expanding the geometric series we obtain: dζ dζ Xe (ζ )q a = exp π Rqn + iqnθe + π Rqnζ a¯ (4.44) q ζ ζ + ζ 1 − Xe (ζ ) n>0 + 2eiqnθe K 0 (2π Rq|na|). (4.45) = n>0
The integral over − in (4.42) gives a similar sum over n < 0. Altogether we find that the terms which multiply ζ or ζ1 in (4.42) equal ⎛ ⎞
2 1 dXe ⎝ iq R da − ζ d a¯ ∧ eiqnθe K 0 (2π Rq|na|)⎠ Xe 2 ζ n=0
dXe 1 = da − ζ d a¯ . (4.46) ∧ iπ V inst Xe ζ For the remaining terms in (4.42) we use similarly dζ dζ Xe (ζ )q a ζ = ζ exp π Rqn +iqnθe +π Rqnζ a¯ (4.47) 1 − Xe (ζ )q ζ ζ + ζ n>0 + |a| 2 eiqnθe K 1 (2π Rq|na|) (4.48) =− a¯ n>0
and
+
dζ 1 dζ 1 Xe (ζ )q a = exp π Rqn +iqnθ +π Rqnζ a ¯ (4.49) e ζ ζ 1 − Xe (ζ )q ζ ζ ζ n>0 + |a| 2 eiqnθe K 1 (2π Rq|na|). (4.50) =− a n>0
Combining these with their counterparts from the integral over − (which come with an extra minus sign), we see that these terms in (4.42) equal ⎛ ⎞
da dXe ⎝ iq 2 R iqnθe d a¯ − ∧ − e (sgn n)|a|K 1 (2π Rq|na|)⎠ Xe 2 a a¯ n=0
=
dXe ∧ 2πi Ainst . Xe
(4.51)
So finally, summing (4.46) and (4.51), we obtain (4.38) as desired: differentiating the contour integrals in Xm has correctly produced the instanton corrections V inst and Ainst . This finishes the check that Xm is the desired “magnetic” complex coordinate on M. Remark. Xm is closely related to the so-called “Q function” in the theory of quantum integrable systems.10 We feel this is not a coincidence and points to some deep relation to integrable field theories. This feeling is reinforced by the fact that the crucial Eq. (5.13) below is a form of the Thermodynamic Bethe Ansatz, as explained in Appendix E. 10 We thank S. Lukyanov for sharing his notes on these functions with us.
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4.4. Analytic properties. We now consider the analytic behavior of the pair (Xm , Xe ) on the ζ -plane. For Xe the story is simple: it is analytic for ζ ∈ C× , with essential singularities at ζ = 0, ∞. For Xmsf the same is true, but for the full Xm the story is more intricate: the integrals in (4.33) are analytic in ζ only away from the contours ± . As ζ crosses either of these contours, the pole in the integrand crosses the path of integration. Therefore our expression for Xm defines a piecewise analytic function, with the discontinuity determined by the residue of the pole. Introduce the notation (Xm )++ , (Xm )− + for the limit of Xm as ζ approaches + in the clockwise or counterclockwise direction respectively, and similar notation for − . The discontinuity is then given by −q (Xm )++ = (Xm )− + (1 − Xe ) ,
(4.52a)
(Xm )+− =
(4.52b)
q
−q q (Xm )− − (1 − Xe ) .
These discontinuities will play a crucial role for us below: indeed we will identify them with Kontsevich-Soibelman symplectomorphisms, as follows. We consider the pair of complex functions (Xm , Xe ) as giving a map X : Ma → Ta
(4.53)
from the real 2-torus Ma coordinatized by (θm , θe ) to a complexified 2-torus Ta coordinatized by (X m , X e ). The map X varies as a function of ζ (and a, a, ¯ R). In Sect. 2.2 we introduced the Kontsevich-Soibelman factors Kγ as symplectomorphisms of Ta . Our discontinuities (4.52) say that at the ray ± , X + and X − differ by composition with K0,±q . An interesting phenomenon has occurred here. Consider the monodromy of Xm in the a-plane around a = 0. This monodromy receives two contributions: the monodromy of Xmsf given in (4.32) and the contributions from (4.52). These two contributions actually cancel one another! This fact is essentially related to the fact that the singularity of the semiflat metric at a = 0 has been smoothed out. On the other hand, if we analytically continue Xm around ζ = 0 it does not come back to itself. This monodromy does not create any problems. In particular, (ζ ) does behave well near ζ = 0: it just has a simple pole, as one expects from the discussion in Sect. 3.1. Now let us consider the asymptotics of Xe , Xm as ζ → 0, ∞. The asymptotics of Xe can be trivially read off from (4.22), exp π R ζa + iθe as ζ → 0, Xe ∼ (4.54) exp [π Rζ a¯ + iθe ] as ζ → ∞. For Xm the asymptotics are more interesting. As ζ → 0, ∞ the integrand of (4.33) simplifies: the rational function just reduces to ±1. Then expanding the logarithm and evaluating the integral gives ⎧ Rq 2 q 1 ⎪ ⎨exp −i 2ζ (a log(a/) − a) + iθm + 2πi s=0 s eisqθe K 0 (2π Rq|sa|) as ζ → 0, Xm ∼ ⎪ q 1 isqθe ⎩exp i ζ Rq 2 (a¯ log(a/ ¯ − a) ¯ ) ¯ + iθm − 2πi K 0 (2π Rq|sa|) as ζ → ∞. s=0 s e 2
(4.55) These asymptotics hold for all phases of ζ . The discontinuities (4.52) along ± do not lead to discontinuities in the asymptotics, because the jump is exponentially close to 1
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as ζ → 0, ∞ along ± : along + we have Xe → 0 exponentially fast, and along − , Xe−1 → 0 exponentially fast. On the other hand, we could also have defined a different function Xm , by analytically continuing Xm across + clockwise. It follows from (4.52a) that on the clockwise side of + we have Xm = Xm (1 − Xe )q . q
(4.56)
Suppose now that we analytically continue Xm further, clockwise to the boundary of U+ and then across into U− . In U− , Xe is exponentially large as ζ → 0. So from (4.56) it follows that the ζ → 0 asymptotics of Xm and Xm are different; in particular, Xm does not obey (4.55). Thus, the asymptotics of the analytic continuation of the function Xm is not the analytic continuation of the asymptotics. This is the hallmark of Stokes’ phenomenon. Altogether, we have been led to consider a map X : Ma → Ta , which depends holomorphically on ζ , and exhibits Stokes phenomena at ζ → 0, ∞, with Stokes factors given by composition with the Kontsevich-Soibelman symplectomorphisms acting on Ta . The crucial idea of this paper is that this picture is valid for general gauge theories, not just the abelian theory we considered here; indeed, as we will see in Sect. 5, it automatically incorporates multi-instanton effects from mutually non-local particles, and gives the exact metric on M. 4.5. Differential equations. Above we saw that the hyperkähler geometry of M is naturally described in terms of a map X which exhibits Stokes phenomena. Stokes phenomena typically arise in the theory of linear ordinary differential equations with irregular singular points. Indeed, in our case there is such a differential equation ζ ∂ζ X = A ζ X ,
(4.57)
with an irregular singularity. In this section we identify this equation. In fact, at the same time we will find a companion equation, governing the dependence on the radius of S 1 , R∂ R X = A R X .
(4.58)
Equations of the form (4.57), (4.58) are commonly encountered for finite-dimensional matrices Aζ , A R , X . Then Aζ and A R act on X by matrix multiplication from the left, and the Stokes factors act from the right, so in particular the two actions commute. In our case the solution X is a map Ma → Ta . The Stokes factors act as diffeomorphisms of Ta . Aζ and A R act as infinitesimal diffeomorphisms of Ma , i.e. as differential operators in (θm , θe ). These two actions commute with one another because they act on different spaces. Now what is the origin of the desired equations? They should be related to some symmetries of (M, g). At first glance (M, g) would appear to have a U (1) symmetry which just maps a → eiθ a. Such a symmetry would have an obvious physical origin: it would come from a U (1) R symmetry of the theory in d = 4. However, we know that this symmetry is actually anomalous once we include the matter hypermultiplet. Indeed, Asf from (4.9) contains the factor log(a/), which is invariant only under a simultaneous rotation of a and . This simultaneous rotation hence leaves the metric invariant. It does not preserve the hyperkähler forms ω, but rather rotates ω1 and ω2 into one another; hence it leaves (ζ ) invariant if combined with the action ζ → eiθ ζ . By inspection,
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both Xe and Xm are invariant under this combined rotation of a, and ζ , which leads to a differential equation:
¯ ¯ − a∂a + a∂ ζ ∂ζ X = −∂ + ∂ (4.59) ¯ a¯ X . Similarly the anomalous scale invariance of the d = 4 theory leads to a symmetry which rescales R, a and :
¯ ¯ + a∂a + a∂ R∂ R X = ∂ + ∂ (4.60) ¯ a¯ X . These equations are not yet of the desired form (4.57), (4.58) since they still involve ¯ a, a). derivatives with respect to the parameters (, , ¯ So let us consider the dependence on these parameters. The dependence of X on (a, a) ¯ is completely determined in terms of the dependence on (θe , θm ), by the requirement that (Xe , Xm ) are holomorphic in complex structure J (ζ ) . Indeed, using the basis (4.19), (4.20) for (T ∗ )1,0 M, we see that the Cauchy-Riemann equations on M are simply ∂a X = Aa X , ∂a¯ X = Aa¯ X ,
(4.61) (4.62)
where the connection form A is defined by 1 −iπ R∂θe + π(V + 2πi R Aθe )∂θm + 2π Aa ∂θm , ζ Aa¯ = 2π Aa¯ ∂θm − ζ iπ R∂θe + π(V − 2πi R Aθe )∂θm .
Aa =
(4.63) (4.64)
¯ dependence. First note that Xe is simply indeWe can similarly dispose of the (, ) 2 ¯ So writing ¯ For Xm we have ∂ Xm = i Rq a Xm , and similarly for . pendent of (, ). ∂ 2ζ A =
q 2 Ra ζ q 2 R a¯ ∂θm , A¯ = ∂θm , 2ζ 2
(4.65)
we have ∂ X = A X , ¯ ¯ X = A¯ X . ∂
(4.66) (4.67)
We can now recast Eqs. (4.59), (4.60) in the desired form (4.57), (4.58), with ¯ ¯ , Aζ = −aAa + aA ¯ a¯ − A + A ¯ ¯ . A R = aAa + aA ¯ a¯ + A + A
(4.68) (4.69)
Now we come to the crucial point: Aζ as given in (4.68) depends on ζ in a very simple way — it has only simple poles at ζ = 0, ∞: Aζ =
1 (−1) (0) (1) A + Aζ + ζ Aζ . ζ ζ
(4.70)
Equation (4.57) thus defines a meromorphic connection on CP1 , with two irregular singularities of rank 1. This motivates the appearance of Stokes phenomena, which we saw explicitly in the previous section.
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A family of differential equations very similar to (4.57), (4.58), (4.61), (4.62), defining the “tt ∗ connection,” appeared in [11,39] in the context of the analysis and classification of N = (2, 2) field theories in d = 2. The similarity is not just formal. In particular, the interpretation of their equations for the ζ and R dependence was also in terms of U (1) R symmetry and scale transformations of the underlying field theory. A crucial point of their analysis is a direct relation between the large R asymptotics of the connection A, the explicit form of the Stokes factors, and the degeneracies of BPS states in the d = 2 theory. There is a similar relation in our problem as well. Indeed this relation is the key to understanding the Kontsevich-Soibelman wall-crossing formula. A look at [11,39] also suggests a very useful technical tool for making further progress: we should convert the differential equations into a Riemann-Hilbert problem for X , defined directly in terms of the Stokes data and asymptotics as ζ → 0, ∞. Using this tool we can immediately write down the generalization to multiple mutually non-local BPS instantons. We move to that problem in Sect. 5. 4.6. Higher spin multiplets. So far we have considered in some detail the corrections to g which come from integrating out a single electrically charged hypermultiplet. One can ask similarly about the corrections due to a single electrically charged higher spin multiplet — for example the vector multiplet containing the massive W boson. In principle these corrections could be determined by a careful one-loop computation in three dimensions. Instead we exploit a trick: we consider the massive vector multiplet of an N = 4 supersymmetric theory. Decomposing under N = 2 supersymmetry this multiplet contains two hypermultiplets and one vector multiplet. On the other hand, because of the higher supersymmetry in the N = 4 theory, one expects that the metric on M will not receive any instanton corrections. The reason is that according to standard nonrenormalization theorems the Higgs branch is uncorrected [40], but the nonanomalous R-symmetry mixes the Higgs and Coulomb branches. It follows that the corrections from the N = 2 vector multiplet must precisely cancel those from the two N = 2 hypermultiplets. In other words, at least as far as these two N = 2 multiplets are concerned, the corrections are weighted by the helicity supertrace (γ ; u). More generally we may consider integrating out N = 2 multiplets with arbitrary spin. Let a j denote the weight multiplying the instanton correction from the j th N = 2 multiplet ( j = 0 for the hypermultiplet, j = 1 for the vector, …), normalized to a0 = 1. We saw above that a1 = −2. Moreover, from the fact that the contribution from any multiplet of N = 4 supersymmetry should vanish, we get a j+2 + 2a j+1 + a j = 0. This determines a j = (−1) j ( j + 1), so indeed the instanton corrections are weighted by the second helicity supertrace. 4.7. Higher rank generalization. All of our discussion can be easily generalized to the case of a rank r abelian gauge theory coupled to a set of electrically charged hypermul(s) tiplets. Let the charges be q I , where I = 1, . . . , r runs over the electric gauge fields, and s labels the set of hypermultiplets. There is a 4r -dimensional generalization of the Gibbons-Hawking ansatz, with base (R3 )r and a fiber (S 1 )r . We use coordinates (x α I ) = x I for the base, θm,I for the fiber, and write
dθm,I dθm,J + A I ( + A J ( g = [V ( x )−1 ] I J x) x ) + V ( x ) I J d x I d x J , (4.71) 2π 2π
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where A and V are related by differential equations stating that
1 dθm,I ωα = d x α I ∧ + A I + VI J αβγ d x β I ∧ d x γ J 2π 2
(4.72)
is a closed 2-form for α = 1, 2, 3. The 1-loop integral gives the natural result ⎛ ⎞ ∞ q (s) q (s) R 1 I J ⎝ VI J = Imτ I0J + − κn ⎠ . (4.73) 4π θeK (s) (s) K 2 2 s n=−∞ |q K Ra | + (q K 2π + n) Poisson resummation of (4.4) shows that VI J = VIsfJ + VIinst J ,
(4.74)
with VIsfJ
=
VIinst J =
Imτ I0J
−
q (s) q (s) R I
s (s) (s) qI qJ R
s
2π
J
4π
!
(s)
(s)
q a¯ K q aK + log K log K ¯
(s) K θe
einq K
(s)
K 0 (2π R|nq I a I |).
" ,
(4.75)
(4.76)
n=0
Also A I = AsfI + Ainst I ,
(4.77)
where " (s) (s) ! (s) (s) q K a¯ K qK a K dθeJ iq I q J sf 0 + − log dθeJ , A I = Reτ I J (4.78) log 2 ¯ 2π 8π s " (s) (s) ! (s) I qI qJ R da J d a¯ J (s) (s) inst AI = − − (s) (sgn n)einq I θe |q K a K |K 1 (2π R|nq K a K |). (s) 4π q K a K q K a¯ K n=0
(4.79) At large R the leading terms in V and A are V sf and Asf . Keeping only these terms, g becomes the semiflat metric with τ I J = τ I0J +
q (s) q (s) I
s
J
2πi
(s)
log
qK a K .
(4.80)
This is the coupling which comes from integrating out the hypermultiplets in d = 4. The holomorphic symplectic form is (ζ ) = −
1 ξ I ∧ ξm,I , 4π 2 R e
(4.81)
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where
da I + ζ d a¯ I , ζ J
da J + 2πi A I + iπ VI J − ζ d a¯ . ζ
ξeI = idθeI + π R ξm,I = idθm,I
As before, the electric coordinates agree with their semiflat approximation, aI + iθeI + π Rζ a¯ I . XeI = exp π R ζ The semiflat approximation to the magnetic ones is # ! " (s) K q (s) (s) q a π R sf I Xm,I q K a K log K = exp τ I0J a J + + iθm,I ζ 2πi e s ! "$ (s) q (s) (s) q K a¯ K 0 J K I q a¯ log + π Rζ τ I J a¯ + . ¯ 2πi K e s The full magnetic coordinates are given by the integral formula # iq (s) dζ ζ + ζ (s) sf I Xm,I = Xm,I exp XeJ (ζ )q J ] log[1 − 4π s+ ζ ζ − ζ s J $ (s) (s) iq I dζ ζ + ζ J −q J − log[1 − Xe (ζ ) ] , 4π s− ζ ζ − ζ
(4.82) (4.83)
(4.84)
(4.85)
(4.86)
J
where s± are any paths connecting 0 to ∞ which lie in the two half-planes % (s) a K qK s 0, n>0
Kγ(γ ;u) .
(5.10)
Assuming that limu→u w X from this side exists, it is the solution to a Riemann-Hilbert problem in which the discontinuity across is A (while the discontinuities along all other BPS rays are specified as before). On the other hand, we could also consider limu→u w X from the other side of the wall. For the two limits to agree, it is necessary and sufficient that they are solutions of the same Riemann-Hilbert problem: so this requires that A computed by (5.10) is the same on both sides of the wall. As we reviewed in Sect. 2.2, this is precisely the content of the KS wall-crossing formula!12 We conclude that, assuming the BPS degeneracies obey the KS formula, a solution X of the Riemann-Hilbert problem is continuous as a function of u and ζ , except at the BPS rays. Moreover, the discontinuity across the BPS ray is given by a symplectomorphism. 5.3. Solving the Riemann-Hilbert problem. Having formulated the Riemann-Hilbert problem, we would like to see that it has a solution, and understand its large-R behavior. Unlike the simple cases we considered in Sect. 4, for which all of the S commute with one another, here we cannot write an explicit integral formula for the desired X ; we have to proceed more indirectly. We exploit the fact that the problem has a structure very similar to one considered in [39,11]. Indeed our problem is an infinite-dimensional version of the one considered there. In [11] the Riemann-Hilbert problem is re-expressed as an integral equation for an analog of ϒ(ζ ). For large enough R, this equation describes as a small correction of the identity matrix. It can therefore be solved iteratively, which proves the existence of a solution for large enough R, and also gives an explicit formula for the leading corrections to the zeroth-order approximation = 1. These leading corrections are expressed directly in terms of the discontinuity factors. This is exactly the sort of information we would like to find about our map X . One direct approach would be to write down an infinite dimensional analogue of the integral equation in [11]. This approach is directly applicable only to a linear Riemann-Hilbert problem, so one would have to pass to the linear problem mentioned at the end of the previous subsection. The solution of the integral equation would then give a linear map between the function spaces; as we have described, this linear map would be X ∗ for u → T˜u . some map X : M 12 The fact that the product in (5.10) is counterclockwise, while it was clockwise in Sect. 2.2, comes from our unusual convention on composition of maps in Sect. 5.
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One minor issue is that if we follow precisely the prescription of [11] we will get a solution obeying the boundary condition ϒ0 = 1. For our construction we need a different choice of boundary condition, namely (5.4), which has the advantage of being compatible with the reality condition Xγ (ζ ) = X−γ (−1/ζ¯ ).13 Fortunately, it is straightforward to write a variant of the integral equation which takes into account this different choice of boundary condition, by a slight modification of the integral kernel. This strategy seems good enough to prove the existence of a solution, but it has an important drawback: the intermediate steps of the iterative solution need not be of the form X ∗ for any X . It is useful to have a realization of the problem where each step in u → T˜u . This is possible if we write the the approximation scheme is itself a map M following integral equation, using the abelian group structure on T˜u : # $ Xγ (ζ ) 1 dζ ζ + ζ sf log Xγ (ζ ) = Xγ (ζ ) exp . (5.11) 4πi (X S )γ (ζ ) ζ ζ −ζ
Here the sum runs over BPS rays . Any solution of (5.11) obeys the discontinuity conditions (5.7). Moreover, our choice of integral kernel ensures that the solution will also obey the reality condition (5.4). Hence a solution of (5.11) is a solution of the Riemann-Hilbert problem.14 Using the explicit form of the Kontsevich-Soibelman factors from (2.16), we have (X S )γ = Xγ (1 − σ (γ )Xγ )(γ ;u) γ ,γ (5.12) γ ∈( u )
(with ( u ) defined in (5.5)). Plug this into (5.11) to get the final integral equation for X: Xγ (ζ ) = Xγsf (ζ ) exp ⎡ ⎤ ζ + ζ 1 dζ × ⎣− log(1 − σ (γ )Xγ (ζ ))⎦ . (γ ; u) γ , γ 4πi γ ζ ζ − ζ γ
(5.13) As we have mentioned, Eq. (5.13) is a form of the Thermodynamic Bethe Ansatz. See Appendix E. In Appendix C we argue that (5.13) has a solution for sufficiently large R, and describe its expansion as R → ∞ for u away from the walls. The first nontrivial approximation is Xγ (ζ ) ∼ Xγsf (ζ ) exp ⎡ ⎤ ζ + ζ 1 dζ × ⎣− log(1 − σ (γ )Xγsf (ζ ))⎦ , (γ ; u) γ , γ 4πi γ ζ ζ − ζ γ
(5.14) 13 In this section we write X = X (ζ ) explicitly, thinking of X as a map which varies with ζ , and hence u . suppress the dependence on the coordinates θ of M 14 Note that although the Riemann-Hilbert problem is invariant under diffeomorphisms of M u , Eq. (5.11) is not; its solution is unique, not unique up to diffeomorphism.
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and is essentially a linear superposition of the 1-instanton corrections that we found in the abelian theory. Higher-order corrections involve multilinears in the (γ ; u), and have an R dependence which identifies them as multi-instanton contributions. Our arguments in Appendix C are closely related to ones given in [11] in the finitedimensional tt ∗ context. In fact, our approach leads to a simplification of the asymptotic analysis even in the finite-dimensional case; hence in Appendix C we re-analyze that case as well. Global issues. By solving the Riemann-Hilbert problem, we have obtained a map X : u → T˜u depending on the choice of the local quadratic refinement σ (γ ). This choice M affects the Riemann-Hilbert problem through the definition of the discontinuities Kγ . However, the solution X depends on σ in a simple way. Recall that for any two refine ments σ, σ there is some c(σ, σ ) ∈ u∗ /2 u∗ such that σ (γ )σ (γ ) = (−1)γ ·c(σ,σ ) . Given a solution X [σ ] of (5.11) with refinement σ , there is a corresponding solution X [σ ] with refinement σ ,
Xγ[σ ] (u, θ ; ζ ) = (−1)γ ·c(σ,σ ) Xγ[σ ] (u, θ + cπ ; ζ ).
(5.15)
u Mu and also T˜u Tu , we It follows that if we use the refinement to identify M obtain X : Mu → Tu which is independent of the choice of refinement.
5.4. Constructing the symplectic form. So far, we have solved the Riemann-Hilbert problem to give a map X : Mu → Tu , obeying the asymptotic conditions (5.3), the jump conditions (5.7), and the reality condition (5.9). Now letting u vary we obtain a map X : M → T . We then construct a complex 2-form (ζ ) on M by pullback of the canonical fiberwise symplectic form on T , (ζ ) =
dXγ j dXγ i 1 1 X ∗ T = i j ∧ . 2 2 4π R 8π R Xγ i Xγ j
(5.16)
A few properties of (ζ ) follow directly from (5.16): • Although X is only piecewise analytic in ζ , (ζ ) is honestly analytic (because the discontinuities Sγ are symplectomorphisms, i.e. they preserve T ). • Using (5.9), we have (−1/ζ¯ ) = (ζ ). • As ζ → 0, ∞ we can determine the behavior of (ζ ) using the asymptotics (5.3) of X and the explicit form (3.15) of sf (ζ ). We find that (ζ ) has a simple pole in each case, with residue Resζ =0 (ζ ) =
i ∗ i ∗ ϒ0 d Z , dθ , Resζ =∞ (ζ ) = − ϒ∞
d Z¯ , dθ . (5.17) 8π 8π
• Using lim R→∞ X = X sf , it follows that (ζ ) is nondegenerate (in the holomorphic sense) for large enough R. These properties will be important in our construction of the hyperkähler metric.
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5.5. Differential equations. Our Riemann-Hilbert problem has been formulated in terms of discontinuity factors which are universal (locally independent of all parameters of the gauge theory), together with asymptotics given by the functions Xγsf , which depend on the parameters only in a very simple way. In this section, following a standard recipe, we show that this implies that the solution X obeys a family of differential equations. As we will see, the physical meaning of these equations is rather transparent. One group expresses the fact that the functions X (ζ ) which solve the Riemann-Hilbert problem are holomorphic on M in complex structure J (ζ ) . These equations are essential for the construction of the hyperkähler metric. Another pair describe the renormalization group flow and a U (1) R -symmetry action. These are important for relating the metric to the KS wall-crossing formula. A very similar family of equations were crucial in the story of “tt ∗ geometry” which appeared in the context of massive N = (2, 2)2-dimensional theories [11,1,13,41]. We begin by recalling that the solution X of our Riemann-Hilbert problem over CP1 is only sectionally analytic; it has jumps of the form X → X S along various rays ⊂ CP1 . So consider instead15 Aζ := ζ ∂ζ X X −1 .
(5.19)
The discontinuities of X along the BPS rays cancel out in Aζ , which is therefore honestly analytic in ζ , except possibly for ζ = 0, ∞ where X becomes singular. So we can think of X as a solution of an ordinary differential equation in ζ , ζ ∂ζ X = Aζ X .
(5.20)
We can describe this equation rather concretely, using our asymptotic information about X . Note first that X sf obeys an equation of the same form. To write it we first introduce two vector fields on Mu , A(−1),sf := iπ Z · ∂θ , A(1),sf := iπ Z¯ · ∂θ . ζ ζ
(5.21)
sf ζ ∂ζ X sf = Asf ζ X ,
(5.22)
Then we have
where Asf ζ =
1 (−1),sf (1),sf A + ζ Aζ . ζ ζ
(5.23)
The important point is that the ζ dependence of Asf ζ is very simple: just a simple pole at each of ζ = 0, ∞. We can convert this information to information about Aζ , since we 15 This is the standard notation, but in our context it is somewhat mnemonic, so here is a longer description. The infinitesimal variation of the map X by applying ζ ∂ζ gives a vector field on T , which we call ζ ∂ζ X . We then pull this back using X to get the vector field Aζ on Mu . We write this pullback operation as X −1 , and because of our non-standard convention for composition, this X −1 appears on the right rather than the left; this makes our equation agree with the usual form for Riemann-Hilbert problems, and in fact this agreement is the reason we use the non-standard convention in Sect. 5. In local coordinates one would write #
$i ∂ ∂X −1 ∂X i Aζ = . (5.18) ∂ζ ∂θ ∂θ j j
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know from (5.3) that ϒ = X (X sf )−1 remains finite at both ζ = 0, ∞. This shows that Aζ also has only a simple pole at ζ = 0, ∞, and even determines the residue, Aζ =
1 (−1) (0) (1) + Aζ + ζ Aζ , A ζ ζ
(5.24)
where (1),sf −1 ϒ0−1 , A(1) ϒ∞ . Aζ(−1) = ϒ0 A(−1),sf ζ ζ = ϒ∞ Aζ
(5.25)
So we see that (5.20) defines a flat connection ζ ∂ζ − Aζ over CP1 , valued in the infinite-dimensional algebra of vector fields on Mu , with rank-1 irregular singularities at ζ = 0, ∞. X is a flat section for this connection. So our solution to the Riemann-Hilbert problem leads directly to the construction of a flat connection over CP1 . In fact, this is a standard maneuver in the theory of ordinary differential equations. The connection we obtained has irregular singularities at ζ = 0 and ζ = ∞, and hence it exhibits Stokes’ phenomenon. One of the virtues of the Riemann-Hilbert construction is that it is easy to determine the Stokes factors: they are simply the discontinuities S which entered the Riemann-Hilbert problem. The above discussion has an important extension. We have not just a single RiemannHilbert problem but a whole family of them, varying with additional parameters. These parameters include the coordinates u i on B, as well as the scale , the radius R of S 1 , and perhaps some bare gauge couplings τ 0 . (For the moment we do not introduce mass parameters; but see Sect. 6 below.) We introduce the generic notation t n to encompass all of these parameters. Importantly, the discontinuities S which define the Riemann-Hilbert problem do not depend on any of the t n . Hence just as we did above for the ζ dependence, we consider An := ∂t n X X −1 .
(5.26)
As before, the discontinuities of X cancel out, so An is analytic in ζ away from ζ = 0, ∞. Also as before, we can control the behavior near these singularities by first checking the sf sf −1 n behavior of Asf n := ∂t n X (X ) . For all of our t we have Asf n =
1 (−1),sf A + ζ A(1),sf n ζ n
(±1),sf
for some simple vector fields An as before, we obtain
An =
(5.27)
; then using the fact that ϒ is finite as ζ → 0, ∞
1 (−1) (1) A + A(0) n + ζ An , ζ n
(5.28)
where (1),sf −1 ϒ0−1 , A(1) ϒ∞ . An(−1) = ϒ0 A(−1),sf n n = ϒ∞ An
(5.29)
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Also including (5.20), the full set of equations we obtain is
1 (−1) (0) A + Au j X , ζ uj (0) (1) A j¯ + ζ A j¯ X , u¯ u¯
1 (−1) (0) A + A X , ζ (0) (1) A¯ + ζ A¯ X ,
1 (−1) (0) (1) A + AR + ζ AR X , ζ R
1 (−1) (0) (1) X. A + Aζ + ζ Aζ ζ ζ ∂u j X = ∂u¯ j¯ X = ∂ X = ¯ ¯ X = ∂ R∂ R X = ζ ∂ζ X =
(5.30) (5.31) (5.32) (5.33) (5.34) (5.35)
One also gets the extra relations (−1)
AR
(−1)
= −Aζ
(1)
(1)
, A R = Aζ ,
(5.36)
from the fact that X sf is annihilated by ζ ∂ζ + R∂ R as ζ → 0, and by ζ ∂ζ − R∂ R as ζ → ∞. We have finished constructing our equations. In Appendix D we discuss how to write them more concretely given the asymptotic expansion of X around ζ = 0. We conclude this section with a few remarks: • Since the symplectic form (ζ ) was constructed from X , (5.35), (5.34) trivially imply equations for the ζ and R dependence of (ζ ), of the form
1 LA(−1) + LA(0) + ζ LA(1) , ζ ζ ζ ζ
1 LA(−1) + LA(0) + ζ LA(1) (R ). R∂ R (R ) = R R R ζ
ζ ∂ζ =
(5.37) (5.38)
Recalling from (3.4) that (ζ ) = − 2ζi ω+ + ω3 − 2i ζ ω− , these equations can be expanded in powers of ζ to derive some interesting differential equations for the hyperkahler forms ω. • Recall that the solution X of the Riemann-Hilbert problem was ambiguous up to a transformation X → bX , with b any diffeomorphism of Mu . This ambiguity leads to ζ -independent gauge transformations of the connection A. There are several par(0) ticularly convenient gauges. One is a gauge in which A R = 0. It follows from (5.38) that in this gauge the restriction of Rω3 to each Mu is independent of R (and hence equals its R → ∞ limit, namely − 8π1 2 dθ, dθ .) It would be interesting to know whether this gauge is the one chosen by our integral equation (5.11). If we allow b to be a complexified diffeomorphism, then at least formally we can also pick a gauge in which ϒ0 = 1, so A(−1) = A(−1),sf ; this is an analogue of the “topological gauge” of [11] (dually ϒ∞ = 1 would be an “antitopological gauge”).
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• Two linear combinations of our equations have a simple physical meaning: they express the invariance under overall changes of scale and R-symmetry transformations. To see this first note that a I ∂a I + ∂ Z γ = Z γ (5.39) for all γ ∈ . It follows that ¯ ¯ X sf = 0, R∂ R − a I ∂a I − a¯ I ∂a¯ I − ∂ − ∂ ¯ ¯ X sf = 0. ζ ∂ζ + a I ∂a I − a¯ I ∂a¯ I + ∂ − ∂
(5.40) (5.41)
These equations can be interpreted as the (anomalous) scale and R-symmetry invariance of the semiflat geometry. They imply relations among the Asf n (just by replacing (±1) ∂ → A) which in turn give relations among An : we find that ¯ ¯ X = X , (5.42) R∂ R − a I ∂a I − a¯ I ∂a¯ I − ∂ − ∂ ¯ ¯ X = X , ζ ∂ζ + a I ∂a I − a¯ I ∂a¯ I + ∂ − ∂ (5.43) where , are ζ -independent vector fields on Mu . We can set = 0, = 0 by a gauge transformation. Indeed, our integral equation automatically picks the appropriate gauge: the recursive solution we give in Appendix C for large R satisfies (5.42), (5.43) term-by-term with = = 0. So there is a sense in which the scale and R-symmetry invariance survive the instanton corrections. • The compatibility between (5.34) and (5.35), together with the relations (5.36), implies a set of nonlinear differential equations for the R dependence of the quadru(±1) (0) (0) ple (Aζ , Aζ , A R ). These equations are a deformation of the Nahm equations, as we explain in Appendix D; the large R expansion of this quadruple can be produced directly by solving them iteratively. They are a possible tool for studying the behavior of our construction at small R. A generic solution of the Nahm equations would become singular at a finite value of R, and we expect that the same is true for our problem. Nevertheless, we expect that the particular solutions which we have described here, determined by the BPS degeneracies in N = 2, d = 4 field theories, actually are regular for all values of R. It is possible that this gives an interesting constraint on the possible BPS spectra and IR prepotentials of N = 2 theories. A very similar strategy was employed in [11] to constrain the properties of d = 2 theories. • Our discussion in this section gives a new perspective on the role of the wall-crossing formula. The collection of Eqs. (5.30)–(5.35) describe a flat connection over CP1 × P, where P is the parameter space coordinatized by the t n . This flat connection can be viewed equivalently as an isomonodromic family of connections over CP1 , with irregular singularities of rank 1 at ζ = 0, ∞. At each t ∈ P the Stokes data of the connection on CP1 are given by the Kontsevich-Soibelman factors. Using the parallel transport along P, one shows that the Stokes data at the irregular singularities are “invariant” in an appropriate sense. To be precise: choosing any convex sector V in the ζ -plane, the product AV =
⊂V
S
(5.44)
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is invariant, under any variation of t ∈ P for which no Stokes line enters or leaves V. Applying this statement to variations of u, we recover the wall-crossing formula. 5.6. Constructing the metric and its large R asymptotics. So far we have constructed a family of functions Xγ (ζ ) on M, and the corresponding holomorphic symplectic form (ζ ). As we have discussed in Sect. 3.2, given (ζ ) with the properties listed in Sect. 5.4, and Xγ obeying “Cauchy-Riemann” equations of the form (5.30), (5.31), there exists a corresponding hyperkähler metric g on M. This is our construction of g. Given the exact functions Xγ (ζ ) solving (5.13), we can write g in closed form as follows. Use the expansion of the kernel in (5.13) for |ζ /ζ | < 1 to obtain an asymptotic expansion for ζ → 0, log Xγ =
1 γ γ γ F + F0 + ζ F1 + O(ζ 2 ), ζ −1
(5.45)
where γ
F−1 = π R Z γ ,
dζ 1 (γ ; u) γ , γ log(1 − σ (γ )Xγ (ζ )), (5.46) 4πi ζ γ γ dζ 1 γ ¯ F1 = π R Z γ − (γ ; u) γ , γ log(1 − σ (γ )Xγ (ζ )). 2 2πi γ ζ γ
F0 = iθγ −
γ
Then, substituting into we extract i γi γj i j d F−1 ∧ d F0 , 2π 2 R 1 γi γj γi γj 2d F . ω3 = ∧ d F + d F ∧ d F i j 1 −1 0 0 8π 2 R
ω+ =
(5.47)
From these symplectic forms it is straightforward to obtain g. Now let us consider the behavior of g for large R. In Sect. 5.3 we have discussed the large R asymptotics of the Xγ , including the first BPS instanton correction, given in (5.14). Now we translate this into the correction to (ζ ). We begin by computing the dX correction to Xγγ : dXγsf dXγ = + Iγ + · · · , Xγ Xγsf
(5.48)
where sf sf dζ ζ + ζ dXγ (ζ ) σ (γ )Xγ (ζ ) 1 (γ ; u) γ , γ . (5.49) Iγ = sf sf 4πi γ ζ ζ − ζ Xγ (ζ ) 1 − σ (γ )Xγ (ζ ) γ Note that Iγ is exponentially suppressed as R → ∞ as promised, since on we have Xγsf → 0 exponentially as R → ∞. The ellipsis in (5.48) indicates the multi-instanton
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205
corrections, which are even more suppressed. The leading correction to (ζ ) therefore d X sf
arises from the wedge product between X sfγ and Iγ . γ To describe the correction more explicitly, it is convenient to consider each γ separately, and adopt a symplectic basis {γ 1 , . . . , γ 2r } in which γ = qγ γ 1 . Then the integral in (5.49) becomes essentially identical to the integral (4.37), which gave the instanton corrections to Xm in Sect. 4.3. Evaluating the corresponding correction to (ζ ) just as we did there, we obtain (ζ ) = sf (ζ ) + γinst (5.50) (ζ ) + · · · , γ ∈
where γinst (ζ )
sf 1 dXγ (ζ ) 1 inst 1 inst Aγ + Vγ , (5.51) daγ − ζ d a¯ γ = −(γ ; u) 2 4π R Xγsf (ζ ) 2 ζ
with (cf. (4.7), (4.10)) Vγinst = Ainst γ
Rqγ2 2π
=−
σ (nγ )einθγ K 0 (2π R|n Z γ |),
n>0 ! 2 Rqγ d Z γ
4π
Zγ
d Z¯ γ − Z¯ γ
"
σ (nγ )einθγ |Z γ |K 1 (2π R|n Z γ |).
(5.52)
(5.53)
n>0
From here one may expand in ζ to extract the leading corrections to ω+ , ω3 and hence obtain the leading correction to g. 5.7. Comparison to the physical metric. Having constructed a hyperkähler metric g on M for large enough R, we now summarize some of its properties: 1. g is continuous, 2. g approaches the semiflat metric g sf if all BPS particles have |Z | → ∞, 3. g is smooth except for specific physically expected singularities, located over the singular loci in B,
r 4. g has vol (Mu ) = R1 , 5. (M, g) in complex structure J3 can be identified with the Seiberg-Witten torus fibration in its standard complex structure, and after this identification, the holomorphic 1 symplectic form is ω+ = − 4π
d Z , dθ . All of these properties agree with what is expected for the physical metric on R3 × S 1 as described in [12]. The simplest consistent picture is therefore that the metric we have constructed is indeed the physical one. (In the rank 1 case it was suggested in [12] that these properties indeed determine the metric, by a non-compact analogue of Yau’s theorem. It is plausible that there could be a similar theorem more generally.) In the rest of this section we establish these properties from our construction: 1. The continuity of g follows from the wall-crossing formula, as we have explained. 2. We need only look at the form of the corrections (5.51): they are all exponentially suppressed in R|Z γ |, and hence vanish exponentially fast if all |Z γ | → ∞.
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3. In any limit where R|Z γ | → ∞ for all γ , the instanton contributions are exponentially suppressed and g approaches g sf . This is enough to establish the smoothness of g at large R, except near a singular locus where some BPS particles with charges γi become massless (Z γi = 0 and (γi ; u) = 0). To understand the behavior near these points, we consider a scaling limit where R → ∞ holding R Z γi finite. One can approximate the Riemann-Hilbert problem in this limit by one in which we keep only the BPS rays γi , dropping all the others. Indeed all other discontinuities involve factors of the form (1 − σ (γ )Xγ ), which become exponentially close to 1 in this scaling limit. In the simplest case where only a single Z γ = 0, we can always choose a duality frame such that γ is an electric charge. By shifting some of the angles θ by π , we can also arrange that the refinement σ is of the standard form σ = (−1)γe ·γm for this frame. Then we are in the situation we studied in Sect. 4, where we found a hyperkähler metric which is smooth except for a periodic array of q Aq−1 singularities. This agrees with the expectation from effective field theory in d = 3: a singularity occurs at the point where one of the Kaluza-Klein tower of charge-q hypermultiplets becomes massless. In addition to the physical singularities we have examined, where a set of mutually local BPS particles become massless, there can also be superconformal points, where mutually nonlocal particles simultaneously become massless [42,43]. We have not analyzed these singularities, although we expect them to be interesting, and we expect the quadratic refinement to play an important role in their * analysis. 4. Since Mu is a complex torus with respect to J3 , its volume is just r1! Mu ω3r . On the other hand, using (3.4) and the fact that ω± restrict to zero on Mu by (5.17), this is r 1 1 1 T r r (ζ ) = ( ) = , (5.54) vol (Mu ) = r ! Mu (4π 2 R)r r ! X (Mu ) R as desired. 5. Complex structure J3 can be determined from ω1 and ω2 , just by J3 = ω1−1 ω2 . But this information in turn is given by the residue of (ζ ) at ζ = 0; recall from (3.4) that ω+ = ω1 + iω2 is given by ω+ = 2i Resζ =0 (ζ ).
(5.55)
Our asymptotic condition (5.3) on (ζ ) precisely ensures that this is related to the residue of sf (ζ ): indeed we just have ω+ = ϒ0∗ ω+sf .
(5.56)
It follows that (M, J3 ) can be identified with (M, J3sf ) just by acting with the fiberwise diffeomorphism ϒ0 . As we explained in Sect. 3.3, the complex structure J3sf on M is just that of the Seiberg-Witten torus fibration. Moreover, under this identification ω+ is identified with ω+sf given in (3.16). 6. Adding Masses In this section we briefly indicate how the results of the previous sections should be modified to include nontrivial mass parameters.
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207
6.1. Single-particle corrections with masses. There is a simple variant of the U (1) theory considered in Sect. 4: we can consider the U (1) theory with several electrically charged hypermultiplets, of charges qi . A theory with more than one species of particle will involve flavor charges, and depend non-trivially on mass parameters. The mass parameters in four dimensions are complex numbers m i . Upon compactification to three dimensions an extra real periodic mass parameter m i3 appears, which is essentially a Wilson line for the flavor symmetry. We write ψi := 2π Rm i3 , with period 2π . The mass parameters enter the corrected metric in a very simple fashion: each particle gives an additive contribution to V and A similar to the one we met before, ⎛ ⎞ ∞ q2 R 1 i ⎝ V = − κn ⎠ . (6.1) 4π n=−∞ R 2 |q a + m|2 + (q θe + ψi + n)2 i
i
2π
2π
The coordinate Xe is unchanged: a Xe = exp π R + iθe + π Rζ a¯ . ζ It is also useful to introduce a similar combination of the mass parameters: mi + iψi + π Rζ m¯ i . μi := exp π R ζ
(6.2)
(6.3)
The semiflat Xm receives contributions from integrating out all of the particles in d = 4: Rqi qi a + m i sf iθm (qi a + m i ) log exp −i Xm (ζ ) = e 2ζ e i
ζ Rqi qi a¯ + m¯ i . (6.4) +i (qi a¯ + m¯ i ) log ¯ 2 e The monodromy of Xmsf (ζ ) around qi a + m i = 0 is q2
Xmsf → (−μi )qi Xe i Xmsf .
(6.5)
The full coordinate similarly receives instanton contributions from all of the particles, # iqi dζ ζ + ζ sf log[1 − μi Xe (ζ )qi ] exp Xm = Xm 4π i+ ζ ζ − ζ i $ iqi dζ ζ + ζ −1 −qi log[1 − μi Xe (ζ ) ] , − (6.6) 4π i− ζ ζ − ζ where we choose the contours i± to be any paths in the ζ -plane connecting 0 to ∞ which lie in the two half-planes qi a + m i (−1) j+i δqi j Mi j . ⎩ ⎭
(3.3)
j=i
Let 1 F be the characteristic function of a set F. Then Mii σi Mii σi Mii σi E =E 1ii + E 1 c , (3.4) det(t A + δ Q) det(t A + δ Q) det(t A + δ Q) ii since the boundary of ii has measure 0.1 1 This follows from the fact that the pushforward of the probability measure by Q (the probability density) 2 is absolutely continuous with respect to the Lebesgue measure on Cn and the set
⎧ ⎪ ⎨ ⎪ ⎩
Q ∈ Cn
2
2 ⎫ ⎪ d ⎬ : Q = (qi j )1≤i, j≤n , |(tσi + δqii )Mii |2 = (−1) j+i δqi j Mi j , ⎪ ⎭ j=i
has Lebesgue measure 0.
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T. J. Christiansen, M. Zworski
Now, E
Mii σi 1 det(t A + δ Q) ii
⎛ = E⎝ ⎛ = E⎝
Mii σi (tσi + δqii )Mii σi (tσi + δqii )
∞
d
j+i δq M ij ij j=i (−1)
1+
(tσi + δqii )Mii
d −
k=0
j+i δq M ij ij j=i (−1)
(tσi + δqii )Mii
⎞
−1
k
1ii ⎠ ⎞ 1ii ⎠ .
We recall that the set ii is chosen so that the infinite sum converges. The set ii is invariant under the mapping qi1 , . . . , qi,i−1 , qi,i+1 , . . . , qi,d → eiϕ qi1 , . . . , eiϕ qi,i−1 , eiϕ qi,i+1 , . . . , eiϕ qi,d (3.5) for any real number ϕ. Since Mi j ’s are independent of qi j , dj=i (−1) j+i δqi j Mi j is homogeneous of degree 1 under this same mapping and (tσi + δqii )Mii is independent of qi j for j = i, we find that σi Mii σi E 1ii = E 1ii . det(t A + δ Q) (tσi + δqii ) We do a similar computation for the second term of (3.4): Mii σi 1 c E det(t A + δ Q) ii ⎛ ⎞ −1 M σ + δq )M (tσ ii i i ii ii = E ⎝ d 1 c ⎠ 1 + d ii j+i δq M j+i δq M (−1) (−1) i j i j i j i j j=i j=i ⎛ ⎞ k ∞ M σ + δq )M (tσ ii i i ii ii = E ⎝ d 1 c ⎠ = 0, − d ii j+i δq M j+i δq M (−1) (−1) i j i j i j i j j=i j=i k=0 using, as before, the invariance properties of ii and the homogeneity of d (−1) j+i δqi j Mi j . j=i
Thus we have E(tr(t A + δ Q)−1 A) =
d E i=1
Now, 1 E
0
σi 1 (tσi + δqii ) ii
dt ≤
σi 1 (tσi + δqii ) ii
.
(3.6)
1 σi σi /δ dt = dt E |tσi + δqii | |tσi /δ + qii | 0 0 σi /δ 1 1 σi /δ = E g(s)ds, ds = |s + qii | π 0 0 1
E
where g is the function defined in Lemma 3.2. Using this, (3.6), and the results of Lemma 3.2 proves the proposition.
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321
Lemma 3.3. Let F, G be d × d matrices, with F invertible, and let β = F −1 . Then
2 E tr (F + δ Q)−1 G = tr F −1 G 1 + O(d 2 e−1/4(δβd) ) 1 2 +O
G d 4 e−1/4(dβδ) . δ The implicit constant in the error term is independent of F and G. Proof. We first note that if we replace F by its singular value decomposition, F = U SV ∗ , then
E tr (F + δ Q)−1 G = E tr (S + δ Q)−1 (U ∗ GV ) and
tr F −1 G = tr S −1 U ∗ GV .
Thus we may assume that F is a diagonal matrix. Our proof then resembles the proof of Proposition 3.1. Let χ ∈ L ∞ (R+ ) be the characteristic function of (−∞, 1/2], and, if A = (ai j ), let A sup = supi j |ai j |. We write
E tr (F + δ Q)−1 G = E tr (F + δ Q)−1 G χ (d Q sup δβ)
+ E tr (F + δ Q)−1 G (1 − χ (d Q sup δβ)) . (3.7) For the first term,
E tr (F + δ Q)−1 G χ (d Q sup δβ) ∞ −1 −1 j (−δ Q F ) G χ (d Q sup δβ) . = E tr F 0
Using the fact that the cut-off χ (d Q sup δβ) is invariant under rotations of the qi j and that the qi j are complex and independent, we find
E tr (F + δ Q)−1 G χ (d Q sup δβ) = tr F −1 G μ(Q : Q sup < 1/2δβd)
2 = tr F −1 G (1 + O(d 2 e−1/4(δβd) )). (3.8) Now we consider the remaining term of (3.7). In a way similar to the proof of Proposition 3.1, we denote the diagonal entries of F by f ii = σi , and by Mi j the (i, j) minor of F + δ Q. If G = (gi j ), we have
E tr (F + δ Q)−1 G (1 − χ (d Q sup δβ)) ⎞ ⎛ (−1)i+ j M ji g ji (1 − χ (d Q sup δβ))⎠ . = E⎝ det(F + δ Q) i, j
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T. J. Christiansen, M. Zworski
Just as in the proof of Proposition 3.1, to compute Mii gii (1 − χ (d Q sup δβ)) E det(F + δ Q) we write det(F + δ Q) = (σi + δqii )Mii +
(−1) j+i δqi j Mi j j=i
and define ii as in (3.3). Proceeding almost exactly as in the proof of Proposition 3.1, using that both ii and the support of (1 − χ (d Q sup δβ)) are invariant under the mapping (3.5), we get that Mii gii gii E (1 − χ (d Q sup δβ)) = E 1 (1 − χ (d Q sup δβ)) . det(F + δ Q) (σi + δqii ) ii But
gii E ≤ C G d 2 e−1/4(dδβ)2 . 1 (1 − χ (d Q
δβ)) sup ii (σi + δqii ) δ
To compute
(−1)i+ j M ji g ji E (1 − χ (d Q sup δβ)) det(F + δ Q) when i = j, we write det(F + δ Q) = δq ji M ji (−1)i+ j + (σi + δqii )Mii +
(−1)k+i δqki Mki
(3.9)
k=i, j
and define ji
⎧ ⎫ ⎬ ⎨ def d2 k+i = q ∈ C : |δq ji M ji | > (σi + δqii )Mii + (−1) δqki Mki . ⎩ ⎭ k=i, j
Following the proof of Proposition 3.1 but treating the term δq ji M ji as the distinguished one in the expansion of the determinant (3.9) and using the invariance of ji under rotations of q ji , we find that (−1)i+ j M ji g ji (1 − χ (d Q sup δβ)) E det(F + δ Q) (−1)i+ j M ji g ji =E 1 c (1 − χ (d Q sup δβ)) . (σi + δqii )Mii + k=i, j (−1)k+i δqki Mki ji Since on the support of 1 c , ji
1 k+i (σi + δqii )Mii + (−1) δqki Mki ) , |M ji | ≤ δ|q ji | k=i, j
Probabilistic Weyl Laws for Quantized Tori
we find
323
(−1)i+ j M ji g ji
G 2 −1/4(dδβ)2 . (1 − χ (d Q sup δβ)) ≤ C d e E det(F + δ Q) δ
Our proof of Proposition 4.1 in the next section will use Proposition 3.5. To prove this proposition we will need several preliminary results. The first lemma below follows from well-known facts about eigenvalues of complex Gaussian ensemble. We give a direct proof suggested to us by Mark Rudelson: Lemma 3.4. Let A = (a1 , . . . , ad ), with ai ∈ Cd . Then, with the notation of (3.1), | det A|−1 dL(A) < ∞.
A H S ≤1
Proof. We begin by introducing some more notation. For p ≤ d, p ∈ N, v ∈ Cd , denote by P p v projection onto the subspace spanned (over the complex numbers) by a1 , . . . , a p . This of course depends on a1 , . . . , a p , but we omit this in our notation for simplicity. Using the Gram-Schmidt process, we can, if A is invertible (as it is off a set of measure 0), write the matrix A = U R, with U a unitary matrix and R being upper triangular. The diagonal entries of R are then given by a1 and (1 − P p−1 )a p , p = 2, . . . , d. Thus | det A| = a1
(1 − P1 )a2
(1 − P2 )a3 · · · (1 − Pd−1 )ad . Note that
a1
(1 − P1 )a2
(1 − P2 )a3 · · · (1 − Pd−2 )ad−1
is independent of ad , that is, independent of a1d , a2d , . . . , add . Therefore | det A|−1 dL(A)
A H S ≤1 dL(ad )dL(ad−1 ) . . . dL(a1 ) 1 =
(1 − Pd−1 )ad
A ≤1 a1
(1 − P1 )a2 · · · (1 − Pd−2 )ad−1
HS dL(ad−1 ) . . . dL(a1 ) dL(ad ) . ≤ ···
a1 ≤1
ad ≤1 (1 − Pd−1 )ad a1
(1 − P1 )a2 · · · (1 − Pd−2 )ad−1
The value of ad ≤1 1/ (1 − Pd−1 )ad dL(ad ) depends only on d and the rank of the space spanned by a1 , . . . , ad−1 . We find 1/ (1 − Pd−1 )ad is locally integrable over R2d Cd , because ad ∈ Cd and the space spanned by a1 , . . . , ad−1 has complex dimension at most d − 1.Therefore 1 dL(ad ) ≤ C < ∞. (3.10)
(1 − P d−1 )ad
ad ≤1 Here the constant C can be chosen independent of a1 , . . . , ad−1 , as the maximum of the integral in (3.10) occurs when a1 , . . . , ad−1 span a d − 1 dimensional vector space. The proof follows by iterating the above argument.
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Proposition 3.5. Let A(s, t) be a d × d matrix depending smoothly on (s, t) ∈ U ⊂ C2 . Let Q denote a d × d random matrix, with each entry an independent complex N (0, 1) random variable. Then for δ > 0, (s, t) ∈ U , E(tr((A(s, t) + δ Q)−1 ∂t A) is smooth on U , and
∂s E tr((A(s, t) + δ Q)−1 ∂t A) = ∂t E tr((A(s, t) + δ Q)−1 ∂s A) . This proposition has the following corollary. Corollary 3.6. Let M, B, be d × d matrices independent of s and t. Then 1 1
−1 E tr (s B + M + δ Q) B ds = E tr (B + t M + δ Q)−1 M dt 0
1
−
−1 E tr (t M + δ Q) M dt +
0
0 1
E tr (s B + δ Q)−1 B ds.
0
Proof. Using the previous proposition, this follows from the Fundamental Theorem of Calculus: 1 1
E tr (s B + M + δ Q)−1 B ds − E tr (s B + δ Q)−1 B ds 0
=
1
=
0
1
0
=
1
∂t
0
0
1
∂s
E tr (s B + t M + δ Q)−1 B dsdt
E tr (s B + t M + δ Q)−1 M dtds
0
1
−1 E tr (B + t M + δ Q) M dt −
0
1
E tr (t M + δ Q)−1 M dt.
0
Proposition 3.5 follows from the subsequent two lemmas. Lemma 3.7. Let A(s, t), B(s, t) be d × d matrices depending smoothly on (s, t) ∈ U ⊂ C2 . With Q a random matrix as in Proposition 3.5 and δ > 0,
E tr (A(s, t) + δ Q)−1 B(s, t) ∈ C ∞ (U ). Proof. We prove the lemma by writing the expected value as an integral:
2 E tr((A + δ Q)−1 B) = tr((A + δ Q)−1 B)e− Q H S dL(Q) 1 2 = tr((δ Q)−1 B)e− Q− δ A H S dL(Q). d−1 /δ, where the ˜ | tr((δ Q)−1 B)| ˜ ˜ ≤ C| det Q|−1 B
Q
Now, for a d × d matrix B, constant C depends on d. Moreover, 1
Q j+k − Q− 1 A 2 2 j j HS . δ |∂s ∂tk e− Q− δ A H S | ≤ C j,k,d (
∂s ∂tk A ) e δ2 j ≤ j,k ≤k
1
Since, using Lemma 3.4 | det Q|−1 (1+ Q )m e− Q− δ A H S dL(Q) < ∞, for any finite m, the smoothness of A and B proves the lemma. 2
Probabilistic Weyl Laws for Quantized Tori
325
If M is an invertible matrix depending smoothly on s and t, then ∂t det M and ∂s tr(M −1 Mt ) = ∂t tr(M −1 Ms ). (3.11) det M The lemma below shows that something similar is true when taking expected values, even though the matrices under consideration are not invertible for some values of the random variable. tr(M −1 ∂t M) =
Lemma 3.8. Let A(s, t) be a d × d matrix depending smoothly on (s, t) ∈ U ⊂ C2 , and Q a random matrix as in Proposition 3.5. Then for δ > 0,
∂s E tr((A + δ Q)−1 ∂t A) = ∂t E tr((A + δ Q)−1 ∂s A) . Proof. Let χ ∈ C ∞ (R) satisfy χ (x) = 1 for |x| < /2 and χ (x) = 0 for |x| > . Then
∂s E tr((A + δ Q)−1 ∂t A) = ∂s E χ (det(A + δ Q)) tr((A + δ Q)−1 ∂t A)
+ ∂s E (1 − χ (det(A + δ Q))) tr (A + δ Q)−1 ∂t A . (3.12) Now
∂s E (1 − χ (det(A + δ Q))) tr (A + δ Q)−1 ∂t A
2 = (1 − χ (det(A + δ Q))) ∂s tr (A + δ Q)−1 ∂t A e− Q H S dL(Q)
2 − χ (det(A + δ Q)) (∂s det(A + δ Q)) tr (A + δ Q)−1 ∂t A e− Q H S dL(Q),
where we can freely interchange differentiation and integration since the integrand is smooth and it and its derivatives are integrable. But using (3.11), we get
∂s E (1 − χ (det(A + δ Q))) tr (A + δ Q)−1 ∂t A
2 = (1 − χ (det(A + δ Q))) ∂t tr (A + δ Q)−1 ∂s A e− Q H S dL(Q)
2 − χ (det(A + δ Q))∂t det(A + δ Q) tr (A + δ Q)−1 ∂s A e− Q H S dL(Q)
= ∂t E (1 − χ (det(A + δ Q))) tr (A + δ Q)−1 ∂s A . On the other hand, the first term on the right in (3.12) satisfies
lim ∂s E χ (det(A + δ Q))(tr((A + δ Q)−1 ∂t A)
↓0 2 = lim ∂s χ (det(A + δ Q))(tr((A + δ Q)−1 ∂t A)e− Q H S dL(Q)
↓0 1 2 = lim ∂s χ (det(δ Q))(tr((δ Q)−1 ∂t A)e− Q− δ A H S dL(Q) = 0,
↓0
1
since (tr((δ Q)−1 ∂t A)e− Q− δ A H S and its s derivative are both in L 1 , using Lemma 3.4. 2
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4. Reduction to a Deterministic Problem In this section we will show how to reduce the random problem to a deterministic one. That will be done using the singular value decomposition of the matrix f N . Let A be a square matrix, and let U SV ∗ be a singular value decomposition for A. We make the following simple observation: for ψ ∈ Cc∞ (R, R) equal to 1 on [−1, 1], (A + αψ(A A∗ /α 2 )U V ∗ )−1 = O(1/α) : 2 −→ 2 ,
(4.1)
which becomes totally transparent by writing ψ(A A∗ /α 2 )U V ∗ = U ψ((S/α)2 )V ∗ . The random problem is reduced to a deterministic one by using an operator of the form (4.1). Proposition 4.1. For a smooth curve γ define def I N (γ ) = E tr( f N + δ Q N − z)−1 dz, γ
(4.2)
where Q N is a complex N n × N n matrix, with entries independent N (0, 1) random variables. Let f N = U N S N VN∗ be a singular value decomposition of f N , and let ψ ∈ Cc∞ (R; [0, 1]) be equal to 1 on [−1, 1]. If 0 ∈ γ , |γ | < α/4,
δ α,
(4.3)
then I N (γ ) =
γ
=
E tr( f N + αψ( f N f N∗ /α 2 )U N VN∗ + δ Q N (ω) − z)−1 dz + E 1 tr( f N + αψ( f N f N∗ /α 2 )U N VN∗ − z)−1 dz + E 2 ,
(4.4)
N 4n −α 2 /4(3N n δ)2 , + e E 1 , E 2 = O d log δ δ
(4.5)
γ
where
α
and d = rank 1supp ψ f N f N∗ /α 2 . The proof of this proposition will use the following lemma. Lemma 4.2. Let f N , U N , S N , VN , ψ, δ, d, and α be as in the statement of Proposition 4.1. Let χ ∈ L ∞ (R) be the characteristic function for the support of ψ. Then, if |z| ≤ α/4,
1 0
E tr ( f N + sαχ ( f N f N∗ /α 2 )U N VN∗ − z + δ Q N )−1 αχ ( f N f N∗ /α 2 )U N VN∗ ds
satisfies the bound (4.5).
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327
˜ Proof. First suppose that for a m × m matrix A, A˜ A˜ B˜ and A˜ −1 = ˜ 11 A˜ = ˜ 11 ˜ 12 A21 A22 B21
B˜ 12 B˜ 22
(4.6)
with A˜ 11 , B˜ 11 d × d matrices and A˜ 22 , B˜ 22 (m − d) × (m − d) matrices. Then if A˜ 22 is invertible, we have the Schur complement formula,
−1 ˜ 21 B˜ 11 = A˜ 11 − A˜ 12 A˜ −1 , (4.7) A 22 see [18] for a review of some of its applications in spectral theory. We note, using ψ(A A∗ /α 2 )U V ∗ = U ψ((S/α)2 )V ∗ and the unitarity of U N , VN ,
E tr ( f N + sαχ ( f N f N∗ /α 2 )U N VN∗ − z + δ Q N )−1 αχ ( f N f N∗ /α 2 )U N VN∗
= E tr (S N + sαχ (S N S N∗ /α 2 ) − U N∗ zVN + δ Q N )−1 αχ (S N S N∗ /α 2 ) . (4.8) The main idea of the proof will be to effectively reduce the dimension of the matrices we work with, from N n to d. We can assume that U N , VN , S N are chosen so that the diagonal elements σ1 , . . . , σ N n of S N satisfy σ1 ≤ σ2 · · · ≤ σ N n . Let J denote projection onto the range of χ (S N2 /α 2 ), which is the same as projection off of the kernel of χ (S N2 /α 2 ). Then Id 0 , J = 0 0 and αχ (S N2 /α 2 ) takes the form
α Id 0
0 . 0
We also write S N + sαχ (S N S N∗ /α 2 ) − U N∗ zVN = and
QN =
Q 11 Q 21
Q 12 Q 22
sα Id + A11 A21
A12 A22
,
,
where A11 , Q 11 are d ×d-dimensional matrices, and A22 , Q 22 are (N n −d)×(N n −d)dimensional. Since S N is diagonal and |z| ≤ α/4, we have A12 ≤ α/4, A21 ≤ α/4. Using this notation, we have that A22 is invertible, with norm at most 4/3α. Now restrict Q N to the set with δ Q N − J Q N J sup ≤ α N −n /4.
(4.9)
Note that this poses no restriction on Q 11 . For such Q N , A22 + δ Q 22 is invertible, with norm at most 2/α. Restricting to this set of Q N and using (4.7), we find
tr (S N + sαχ (S N S N∗ /α 2 ) − U N∗ zVN + δ Q N )−1 αχ (S N S N∗ /α 2 )
= tr d α(sα Id + Md + δ Q 11 )−1 ,
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T. J. Christiansen, M. Zworski
where we use the notation tr d to emphasize we are taking the trace of a d × d matrix, and where Md = A11 − (A12 + δ Q 12 )(A22 + δ Q 22 )−1 (A21 + δ Q 21 ) is a d × d matrix depending on Q 12 , Q 21 , and Q 22 , but not on Q 11 . Since A11 =
J (S N − zU N∗ VN )J ≤ Cα and A12 ≤ α/4, A21 ≤ α/4, we have Md ≤ Cα, for a new constant C independent of N , δ, and Q N satisfying (4.9). Next we take the expected value in the Q 11 variables only: 1 2 E Q 11 (F(Q N )) = F(Q N )e− Q 11 H S dL(Q 11 ). 2 2 d π Q 11 ∈Cd Still requiring Q N to satisfy (4.9), which is not a restriction on Q 11 , and using Corollary 3.6, we get 1
E Q 11 α tr d (Md + sα Id + δ Q 11 )−1 ds
0
=
1 0
−
1
E Q 11 tr d (t Md + α Id + δ Q 11 )−1 Md dt + E Q 11 tr d (sα Id + δ Q 11 )−1 α ds
1 0
E Q 11 tr d (t Md + δ Q 11 )−1 Md
0
dt.
Recalling that Md ≤ Cα we see from Proposition 3.1 that the second and third terms on the right are O(d log(α/δ)), if α/δ > e. Moreover,
Md − J S N J ≤
α , 2
and S N ≥ 0. Therefore, for 0 ≤ t ≤ 1, α Id + t Md is invertible, with the inverse having norm at most 3/α. Thus from Lemma 3.3 we see that 1 4
d −α 2 /4(3dδ)2 −1 = O(d) + O . e tr (t M dt E + α I + δ Q ) M Q 11 d d d 11 d δ 0 The implicit constants in both cases are independent of Q −J QJ satisfying (4.9). Thus we get 1 0
2 /α 2 ) + δ Q − zU ∗ V )−1 αχ (S 2 /α 2 ) 1 E tr (S N + sαχ (S N {δ Q−J Q J sup ≤ N N N
2 2 = O(d log(α/δ)) + O d 4 δ −1 e−α /4(d3δ) ,
α 4N n
} ds
(4.10)
where for a set E, 1 E is the characteristic function of E. Exactly as in the proof of Lemma 3.3, we can show that
E tr (S N + sαχ (S N2 α 2 ) + δ Q N − zU N∗ VN )−1 αχ (S N2 α 2 ) 1{δ Q−J Q J sup >α/(4N n )}
2 n 2 = O N 4n δ −1 e−α /(4N δ) . Using (4.8), (4.10), and (4.11), we prove the lemma.
(4.11)
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329
We now use Lemma 4.2 in a preliminary step towards proving Proposition 4.1. Lemma 4.3. Let f N , U N , S N , VN , ψ, δ, d, α, I N (γ ), and γ be as in the statement of Proposition 4.1, and set χ = 1supp ψ . Then I N (γ ) = E tr( f N + αχ ( f N f N∗ /α 2 )U N VN∗ + δ Q N − z)−1 dz γ
4n α
N −α 2 /4(3N n δ)2 . +O e + O d log δ δ Proof. The proof uses the same type of argument as Corollary 3.6. Using the Fundamental Theorem of Calculus,
γ
E tr( f N + δ Q N − z)−1 dz −
1
=− 0
=
γ
∂s
∂z 0
1
γ
γ
E tr( f N + αχ ( f N f N∗ /α 2 )U N VN∗ + δ Q N − z)−1 dz
E tr( f N + sαχ ( f N f N∗ /α 2 )U N VN∗ + δ Q N − z)−1 dzds
E tr ( f N + sαχ ( f N f N∗ /α 2 )U N VN∗ + δ Q N − z)−1 αχ ( f N f N∗ /α 2 )U N VN∗ dsdz,
where we use Proposition 3.5. The right-hand side is
1 ∓ E tr ( f N +sαχ ( f N f N∗ /α 2 )U N VN∗ + δ Q N − z ± )−1 αχ ( f N f N∗ /α 2 )U N VN∗ ds, 0
±
where z ± are the endpoints of γ . Then using Lemma 4.2 finishes the proof.
We are now able to give a straightforward proof of Proposition 4.1. Proof of Proposition 4.1. We begin by noting that, with χ = 1supp ψ ,
( f N + αχ ( f N f N∗ /α 2 )U N VN∗ − z)−1 = O(1/α) and
( f N + αψ( f N f N∗ /α 2 )U N VN∗ − z)−1 = O(1/α) when |z| ≤ α/4. Moreover, the rank of χ ( f N f N∗ /α 2 ) is d and the rank of ψ( f N f N∗ /α 2 ) is at most d, and both operators have norm at most 1. Then
∗ /α 2 )U V ∗ − z)−1 − tr( f + αψ( f f ∗ /α 2 )U V ∗ − z)−1 dz tr( f + αχ ( f f N N N N N N N N N N γ
= α tr ( f N + αχ ( f N f N∗ /α 2 )U N VN∗ − z)−1 χ ( f N f N∗ /α 2 ) − ψ( f N f N∗ /α 2 ) U N VN∗ γ
×( f N + αψ( f N f N∗ /α 2 )U N VN∗ − z)−1 d|z| Cd ≤ dz = O(d). γ α
Thus, applying Lemmas 4.3 and 3.3 proves the proposition.
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5. Proof of Theorem The proof of the theorem will be deduced from the following local result: Proposition 5.1. Under the assumption of the main theorem, let γ ⊂ ∂ be a connected segment of length α α 1 1 < |γ | ≤ , h = , α = hρ , 0 < ρ < 2C C 2π N 2 − and let I N (γ ) be as defined by (4.2). Then for exp −h < δ < h p0 , we have I N (γ ) = N n
γ
T2n
(5.1)
( f (w) − z)−1 dL(w)dz + O(|γ |h −n+ρ(κ−1)−2 ) + O(|γ |h −n+1−2ρ ),
(5.2) where we note that (1.2) with κ > 1 implies that ( f (w) − z)−1 ∈ L 1 (T2n ) so that the first term on the right-hand side makes sense. Assuming the proposition we easily give the Proof of Theorem. We divide ∂ into J = C /α disjoint segments γ j , |γ j | ≤ α/C. Proposition 5.1 implies that E tr
∂
= Nn
( f N + δ Q N − z)−1 dz ∂ T2n
=
J
I N (γ j )
j=1
( f (w) − z)−1 dL(w)dz + O(h −n+ρ(κ−1)−2 ) + O(h −n+1−2ρ ).
We now choose ρ = 1/(κ +1), to optimize the error, that is to arrange, ρ(κ −1) = 1−2ρ. That means that the error is O(N n−β ) for any β < 1 − 2ρ = (κ − 1)/(κ + 1). Hence 1 Eω |Spec ( f N + N − p Q N (ω)) ∩ | = E tr( f N + N − p Q N (ω) − z)−1 dz 2πi ∂ dL(w) 1 n dz + O(N n−β ) N = 2πi ∂ T2n f (w) − z 1 dz n dL(w)+O(N n−β ) =N 2n 2πi f (w) − z T ∂ = N n volT2n ( f −1 ()) + O(N n−β ),
which is the statement of the theorem.
Proof of Proposition 5.1. Without loss of generality we can assume that 0 ∈ γ . From Proposition 4.1 we already know that I N (γ ) can be approximated by a deterministic expression def I N (γ ) = tr( f N + αψ( f N f N∗ /α 2 )U N VN∗ − z)−1 dz, (5.3) γ
Probabilistic Weyl Laws for Quantized Tori
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with, if α/δ N n " 0
α
n I N (γ ) − I˜N (γ ) = O e−co α/N δ + d log , δ
for some c0 > 0, where d is the rank of ψ( f N f N∗ /α 2 ). We choose α as in (5.1), α = h ρ , where h=
1 1 , 0 n + 1/2, I N (γ ) − I˜N (γ ) = O(h −n− +κρ ) + exp(−c0 h n− p0 +ρ ) = O(|γ |h −n+(κ−1)ρ− ). Thus we will prove (5.2) by showing that tr( f N + αψ( f N f N∗ /α 2 )U N VN∗
− z)
−1
=N
n
( f (w) − z)−1 dL(w)
T2n −n+1−2ρ
+ O(h
) + O(h −n+ρ(κ−1) ).
(5.4)
We first show that it is enough to consider z = 0. In fact, let U N (z)S N (z)VN (z)∗ be the singular value decomposition of f N − z, and put B N (z, w) = ( f N − w + αψ(( f N − z)( f N − z)∗ /α 2 )U N (z)VN∗ (z))−1 . def
Then tr (B N (z, z) − B N (0, z)) =
α tr B N (0, z) ψ( f N f N∗ /α 2 )U N VN∗ − ψ(( f N − z)( f N − z)∗ /α 2 )U N (z)VN∗ (z) B N (z, z) .
Since rank ψ(( f N − z)( f N − z)∗ /α 2 ) = O(h −n+κρ ) for z ∈ γ , and B(z, w) = O2 →2 (1/α) for |z − w| ≤ α/C , we obtain tr (B N (z, z) − B N (0, z)) = O(h −n+ρ(κ−1) ), which can be absorbed in the error on the right-hand side of (5.4). Thus we only need to prove (5.4) with the left-hand side replaced by B(z, z) and we can simply take z = 0. In other words we now want to prove tr( f N + αψ( f N f N∗ /α 2 )U N VN∗ )−1 dL(w) + O(h −n+1−2ρ ) + O(h −n+ρ(κ−1) ). = Nn T2n f (w)
(5.5)
The difficulty lies in the fact that the operators f N + αψ( f N f N∗ /α 2 )U N VN∗ do not seem to have a nice microlocal characterization. We are helped by the following identity: if ψ˜ ∈ Cc∞ (R, [0, 1]) is equal to 1 on the support of ψ then ˜ f N∗ f N /α 2 ))( f N + αψ( f N f N∗ /α 2 )U N VN∗ )−1 (1 − ψ( ˜ f N∗ f N /α 2 )) f N∗ ( f N f N∗ + α 2 ψ( f N f N∗ /α 2 ))−1 . = (1 − ψ(
(5.6)
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T. J. Christiansen, M. Zworski
This is a consequence of an identity from linear algebra: Lemma 5.2. Let A be a matrix and U SV ∗ be its singular value decomposition. If ψ, ψ˜ ∈ Cc∞ (R; [0, 1]), ψ is equal to 1 on [−1, 1], and ψ˜ is equal to 1 on the support of ψ, then ˜ ∗ A))(A + ψ(A A∗ )U V ∗ )−1 = (1 − ψ(A ˜ ∗ A))A∗ (A A∗ + ψ(A A∗ ))−1 . (5.7) (1 − ψ(A Proof. We first note that ˜ ∗ A) = V ψ(S ˜ 2 )V ∗ , A∗ A = V S 2 V ∗ , ψ(A ˜ and similarly ψ(A A∗ ) = U ψ(S 2 )U ∗ . Since S is a diagonal matrix, and (1 − ψ)ψ ≡ 0, we get ˜ ∗ A))(A + ψ(A A∗ )U V ∗ )−1 = V (1 − ψ(S ˜ 2 ))V ∗ V (S + ψ(S 2 ))−1 U ∗ (1 − ψ(A ˜ 2 ))(S + ψ(S 2 ))−1 U ∗ = V (1 − ψ(S ˜ 2 ))S(S 2 + ψ(S 2 ))−1 U ∗ = V (1 − ψ(S
˜ 2 ))V ∗ V SU ∗ U (S 2 + ψ(S 2 ))−1 U ∗ = V (1 − ψ(S ˜ ∗ A))A∗ (A A∗ + ψ(A A∗ ))−1 , = (1 − ψ(A
concluding the proof.
The identity (5.6) follows from (5.7) by putting A = f N /α, U = U N , and V = VN . Using this we will find a new expression for the left-hand side of (5.5) so that the identification with the right hand side will follow from a suitable semiclassical operator calculus. Lemma 5.3. We have the following approximation for the left hand side of (5.5): tr( f N + αψ( f N f N∗ /α 2 )U N VN∗ )−1 = tr f N∗ ( f N f N∗ + α 2 ψ( f N f N∗ /α 2 ))−1 + O(h −n+ρ(κ−1) ).
(5.8)
Proof. We use (5.6) and first note that 1 − ψ˜ can be removed from the left hand side since ˜ f N∗ f N /α 2 )( f N + αψ( f N f N∗ /α 2 )U N VN∗ )−1 tr ψ(
= O (rank ψ˜ f N f N∗ /α 2 ) ( f N + αψ( f N f N∗ /α 2 )U N VN∗ )−1 = O h −n+ρ(κ−1) . (5.9) The same argument works for the right-hand side once we observe that f N∗ ( f N f N∗ + α 2 ψ( f N f N∗ /α 2 ))−1 = O2 →2 (1/α), and this follows from using the singular value decomposition since for non-negative diagonal matrices S N (S N2 + α 2 ψ(S N2 /α 2 ))−1 ≤ 1/α.
Probabilistic Weyl Laws for Quantized Tori
333
In view of (5.5) and the lemma we have to prove tr f N∗ ( f N f N∗ + α 2 ψ( f N f N∗ /α 2 ))−1 dL(w) + O(h −n+1−2ρ ) + O(h −n+(κ−1)ρ ), = Nn T2n f (w)
(5.10)
but that follows from the calculus developed in Sect. 2. In fact, with the α-order function m(w, α) = α 2 + | f (w)|2 , given in Lemma 2.6, f N f N∗ + α 2 ψ( f N f N∗ /α 2 ) = TN , T ∈ S(m, α), T = T0 + h 1−2ρ T1 , T0 (w) = | f (w)|2 + α 2 ψ(| f (w)|2 /α 2 ), T1 ∈ S(m, α), where we also applied Lemma 2.8. We also have T0 ≥ m/2 and hence √
1/T0 ∈ S(1/m, α), 1/T ∈ S(1/m, α).
Since f ∈ S( m, α), we conclude that
√ f N∗ ( f N f N∗ + α 2 ψ( f N f N∗ /α 2 ))−1 = PN , P ∈ S(1/ m, α), √ f¯(w) P = P0 + h 1−2ρ P1 , P1 ∈ S(1/ m), P0 (w) = . | f (w)|2 + α 2 ψ(| f (w)|2 /α 2 )
We now apply Lemma 2.5 and obtain (with k " n) tr f N∗ ( f N f N∗ + α 2 ψ( f N f N∗ /α 2 ))−1 = Nn P(w)dL(w) + O(N −k+n ) sup |∂ β P|dL T2n
|β|≤k
= N
n
T2n
P0 (w)dL(w) + O(h
−n+(1−2ρ)
+h
−n+k(1−ρ)
)
T2n
m(w, α)−1/2 dL(w).
We have m(w, α)−1/2 ≤ | f (w)|−1 and (1.2) at z = 0 with κ > 1 implies that | f (w)|−1 is integrable (κ = 1 would mean that | f (w)|−1 is in weak L 1 ): ∞ ∞ | f (w)|−1 dL(w) = L({| f (w)| < t})t −2 dt = O(min(t κ , 1))t −2 dt < ∞. T2n
0
It remains to show that T2n def
0
|P0 (w) − f (w)−1 |dL(w) = O(h ρ(κ−1) ).
(5.11)
Putting ϕ(x) = ψ(x 2 ), we rewrite the left hand side above as ∞ −α 2 ϕ(t/α) dt L({| f (w)| < t})∂t t (t 2 + α 2 ϕ(t/α) 0 ∞ −α 2 (t/α)ϕ (t/α) = dt L({| f (w)| < t}) 2 2 t (t + α 2 ϕ(t/α)) 0 ∞ α 2 ϕ(t/α)(3t 2 + α 2 ϕ(t/α) + α 2 (t/α)ϕ (t/α)) + L({| f (w)| < t}) dt t 2 (t 2 + α 2 ϕ(t/α))2 0 2α t κ−2 dt = C α κ−1 , ≤ C 0
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which is (5.11). Since we have now established (5.10) this also completes the proof of Proposition 5.1. Acknowledgements. We would like to thank Edward Bierstone and Pierre Milman for helpful discussions of Łojasiewicz inequalities, Mark Rudelson for suggestions concerning random matrices, and Stéphane Nonnenmacher and Michael VanValkenburgh for comments on early versions of the paper. The authors gratefully acknowledge the partial support by an MU research leave, and NSF grants DMS 0500267, DMS 0654436. The first author thanks the Mathematics Department of U.C. Berkeley for its hospitality in spring 2009. Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
References 1. Bierstone, E., Milman, P.D.: Semianalytic and subanalytic sets. Publ. IHÉS 67, 5–42 (1988) 2. Bordeaux Montrieux, W.: Personal communication 3. Bordeaux Montrieux, W., Sjöstrand, J.: Almost sure Weyl asymptotics for non-self-adjoint elliptic operators on compact manifolds. http://arXiv.org/abs/0903.2937v2[math.sp] 4. Borthwick, D.: Introduction to Kähler quantization. In: First Summer school in analysis and mathematical physics, Cuernavaca, Mexico. Contemporary Mathematics series 260, Providence, RI: Amer. Math. Soc., 2000, pp. 91–132 5. Borthwick, D., Uribe, A.: On the pseudospectra of Berezin-Toeplitz operators. Meth. Appl. Anal. 10, 31–65 (2003) 6. Bouzouina, A., De Bièvre, S.: Equipartition of the eigenfunctions of quantized ergodic maps on the torus. Commun. Math. Phys. 178, 83–105 (1996) 7. Chapman, S.J., Trefethen, L.N.: Wave packet pseudomodes of twisted Toeplitz matrices. Comm. Pure Appl. Math. 57, 1233–1264 (2004) 8. Dencker, N., Sjöstrand, J., Zworski, M.: Pseudospectra of semi-classical (pseudo)differential operators. Comm. Pure Appl. Math. 57, 384–415 (2004) 9. Dimassi, M., Sjöstrand, J.: Spectral Asymptotics in the semi-classical limit. Cambridge: Cambridge University Press, 1999 10. Evans, L.C., Zworski, M.: Lectures on Semiclassical Analysis, http://math.berkeley.edu/~zworski/ semiclassical.pdf 11. Forrester, P., Snaith, N., Verbaarschot, V.: Introduction Review to Special Issue on Random Matrix Theory. J. Phys. A: Math. Gen. 36, R1–R10 (2003) 12. Hager, M.: Unpublished, 2007 13. Hager, M., Sjöstrand, J.: Eigenvalue asymptotics for randomly perturbed non-selfadjoint operators. Math. Ann. 342(1), 177–243 (2008) 14. Mehta, N.: Random Matrices. 3rd Edition, Amsterdam: Elsevier, 2004 15. Nonnenmacher, S., Zworski, M.: Distribution of resonances for open quantum maps. Commun. Math. Phys. 269, 311–365 (2007) 16. Schenck, E.: Weyl laws for partially open quantum maps. Ann. H. Poincaré 10, 714–747 (2009) 17. Sjöstrand, J.: Eigenvalue distribution for non-self-adjoint operators on compact manifolds with small multiplicative random perturbations. http://arXiv.org/abs/0809.4182v1[math.SP] 18. Sjöstrand, J., Zworski, M.: Elementary linear algebra for advanced spectral problems. Ann. Inst. Fourier (Grenoble) 57, 2095–2141 (2007) 19. Sjöstrand, J., Zworski, M.: Fractal upper bounds on the density of semiclassical resonances. Duke Math. J. 137, 381–459 (2007) 20. Zworski, M.: Numerical linear algebra and solvability of partial differential equations. Commun. Math. Phys. 229, 293–307 (2002) Communicated by S. Zelditch
Commun. Math. Phys. 299, 335–363 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1082-z
Communications in
Mathematical Physics
Hydrodynamic Limit for an Evolutional Model of Two-Dimensional Young Diagrams Tadahisa Funaki , Makiko Sasada Graduate School of Mathematical Sciences, The University of Tokyo, Komaba, Tokyo 153-8914, Japan. E-mail:
[email protected];
[email protected] Received: 30 September 2009 / Accepted: 17 March 2010 Published online: 26 June 2010 – © Springer-Verlag 2010
Abstract: We construct dynamics of two-dimensional Young diagrams, which are naturally associated with their grandcanonical ensembles, by allowing the creation and annihilation of unit squares located at the boundary of the diagrams. The grandcanonical ensembles, which were introduced by Vershik [17], are uniform measures under conditioning on their size (or equivalently, area). We then show that, as the averaged size of the diagrams diverges, the corresponding height variable converges to a solution of a certain non-linear partial differential equation under a proper hydrodynamic scaling. Furthermore, the stationary solution of the limit equation is identified with the so-called Vershik curve. We discuss both uniform and restricted uniform statistics for the Young diagrams.
1. Introduction The asymptotic shapes of two-dimensional random Young diagrams with large size were studied by Vershik [17] under several types of statistics including the uniform and restricted uniform statistics, which were also called the Bose and Fermi statistics, respectively. To each partition p = { p1 ≥ p2 ≥ · · · ≥ p j ≥ 1} of a positive integer j j n by positive integers { pi }i=1 (i.e., n = i=1 pi ), a Young diagram is associated by piling up j sticks of height 1 and side-length pi , more precisely, the height function of the Young diagram is defined by ψ p (u) =
j
1{u< pi } , u ≥ 0.
i=1 Supported in part by the JSPS Grants (A) 18204007 and 21654021. JSPS Research Fellow and supported by the JSPS Grant 21-3656.
(1.1)
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The closure of the interior of its ordinate set is called the Young diagram of the partition p. Note that, in most literatures, the figures of Young diagrams are upside-down compared with the graph defined by (1.1). n assigns an equal For each fixed n, the uniform statistics (U-statistics in short) μU probability to each of the possible partitions p of n, i.e., to the Young diagrams of area n. The restricted uniform statistics (RU-statistics in short) μnR also assigns an equal probability, but restricting to the distinct partitions satisfying q = {q1 > q2 > · · · > q j ≥ 1}. ε and These probabilities are called canonical ensembles. Grandcanonical ensembles μU ε μ R with parameter 0 < ε < 1 are defined by superposing the canonical ensembles in a similar manner known in statistical physics, see (2.2) and (2.9) below. Vershik [17] 2 2 proved that, under the canonical U- and RU-statistics μUN and μ NR (with n = N 2 ), the law of large numbers holds as N → ∞ for the scaled height variable ψ˜ pN (u) :=
1 ψ p (N u), u ≥ 0, N
(1.2)
of the Young diagrams ψ p (u) with size (i.e., area) N 2 and for ψ˜ qN (u) defined similarly, and the limit shapes ψU and ψ R are given by 1 1 log 1 − e−αu and ψ R (u) = log 1 + e−βu , u ≥ 0, (1.3) α β √ √ with α = π/ 6 and β = π/ 12, respectively. These results can be extended to the corε and με , if the averaged size of the diagrams responding grandcanonical ensembles μU R 2 is N under these measures. Such types of results are usually called the equivalence of ensembles in the context of statistical physics. The corresponding central limit theorem and large deviation principle (under canonical ensembles) were shown by Pittel [14] and Dembo et. al. [5], respectively. All these results are at the static level. The purpose of this paper is to study and extend these results from a dynamical point of view. We will see that, to the grandcanonical U- and RU-statistics, one can associate a weakly asymmetric zero-range process pt respectively a weakly asymmetric simple exclusion process qt on a set of positive integers with a stochastic reservoir at the boundary site {0} in both processes as natural time evolutions of the Young diagrams, or more precisely, those of the gradients of their height functions. Then, under the diffusive scaling in space and time and choosing the parameter ε = ε(N ) of the grandcanonical ensembles such that the averaged size of the Young diagrams is N 2 , we will derive the hydrodynamic equations in the limit and show that the Vershik curves defined by (1.3) are actually unique stationary solutions to the limiting non-linear partial differential equations in both cases. In Sect. 2, after defining the ensembles and the corresponding dynamics, we formulate our main theorems, see Theorems 2.1 and 2.2. In Sect. 3, we study the asymptotic behaviors of ε(N ) as N → ∞. The weakly asymmetric zero-range process pt with a stochastic reservoir at the boundary {0} can be transformed into the weakly asymmetric simple exclusion process η¯ t on Z without any boundary condition. In Sect. 4, we study such transformations and also those for the limit equations, and give the proof of the main theorem for the U-case (i.e. the case corresponding to the U-statistics). The hydrodynamic limit for η¯ t is indeed already known [9], and we apply this result for η¯ t . The idea of transforming pt into η¯ t , which is indeed known in the study of particle systems, is useful to avoid the difficulty in treating singularities at the boundary u = 0, which appear in the limit of ψ˜ pN (u). The main theorem for the RU-case (i.e. the case ψU (u) = −
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corresponding to the RU-statistics) is proved in Sect. 5. Our method is to apply the HopfCole transformation for the microscopic process qt , which was originally introduced by Gärtner [9]. This transformation linearizes the leading term in the time evolution qt even at the microscopic level so that one can avoid to show the one-block and two blocks’ estimates, which are usually required in the procedure establishing the hydrodynamic limit. The only task left is to study the boundary behavior of the transformed process, but a rather simple argument leads to the desired ergodic property of our process at the boundary, see Lemma 5.7 below. The corresponding dynamic fluctuations will be discussed in a separate paper [8]. Our dynamics can be interpreted as evolutional models of (non-increasing) interfaces which separate ±-phases in a zero-temperature two-dimensional Ising model defined on a first quadrant, see Spohn [16] and Remark 2.2 below. A randomly growing Young diagram was studied by Johansson [10,11] in relation to random matrices. See [7, Sect. 16.4] for a quick review of some related results. 2. Two-Dimensional Young Diagrams and Main Results In this section, for U- and RU-statistics individually, we define the grandcanonical and canonical ensembles, introduce the corresponding dynamics and then formulate the main results concerning the space-time scaling limits for them. Throughout the paper, we will use the following notation: Z+ = {0, 1, 2, . . .}, N = {1, 2, 3, . . .}, R+ = [0, ∞) and R◦+ = (0, ∞). 2.1. U-statistics. For each n ∈ N, we denote by Pn the set of all partitions of n into positive integers, that is, the set of all p = ( pi )i∈N satisfying p1 ≥ p2 ≥ · · · ≥ pi ≥ · · ·, pi ∈ Z+ and i∈N pi = n. For n = 0, we define P0 = {0}, where 0 is a sequence such that pi = 0 for all i ∈ N. We consider p as an infinite sequence by adding infinitely many 0’s rather than a finite sequence as in Sect. 1. This will be convenient from the point of view of the corresponding particle system. The union of Pn is denoted by P: P = ∪n∈Z+ Pn . The sum of pi ’s in p ∈ P is described as n( p): n( p) = i∈N pi , and called the size or area of p. For p ∈ P, we assign a right continuous non-increasing step-function ψ p on R+ called the height function as follows: 1{u< pi } , u ∈ R+ . (2.1) ψ p (u) = i∈N
∞
In particular, we always have 0 ψ p (u)du = n( p). ε be the probability measure on P determined by For 0 < ε < 1, let μU 1 εn( p) , p ∈ P, (2.2) Z U (ε) ∞ k −1 = n where Z U (ε) = ∞ k=1 (1 − ε ) n=0 p(n)ε , p(n) = Pn is the normalizing ε has the property με | ( p) = μn ( p), p ∈ P, where με | constant. The measure μU U U Pn U Pn ε on P and μn is the uniform probability stands for the conditional probability of μU n U ε and μn play similar roles to the grandcanonical and measure on Pn . The measures μU U canonical ensembles in statistical physics, respectively. ε ( p) = μU
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ε Now, we construct dynamics of two-dimensional Young diagrams, which have μU ε as their invariant measures. Let pt ≡ pt = ( pi (t))i∈N be the Markov process on P defined by means of the infinitesimal generator L ε,U acting on functions f : P → R as ε1{ pi−1 > pi } { f ( pi,+ )− f ( p)}+1{ pi>pi+1 } { f ( pi,− ) − f ( p)} , L ε,U f ( p) = i∈N
(2.3) where pi,± = ( pi,± j ) j∈N ∈ P are defined by
pi,± j
=
pj if j = i, pi ± 1 if j = i.
(2.4)
In (2.3), we regard p0 = ∞. Note that n( pt ) and n( pt ) := {i ∈ N; pi (t) ≥ 1} change ε in time but always stay finite. It is easy to under such dynamics see that μU is invariant ε ( p) = 0 for a sufficiently for every 0 < ε < 1 by showing that p∈P L ε,U f ( p)μU wide class of functions f . We introduce an equivalent system of particles on N, namely we think of pi (t) as the position of the i th particle. The total number of particles n( pt ) on the region N changes only through the creation and annihilation of particles at the boundary site {0}. In fact, the first part in the sum (2.3) with i = n( p) + 1 represents that a new particle is provided from the boundary site {0} to the site {1} (in N) with rate ε, while the second part with i = n( p) indicates that a particle at {1} jumps to {0} and disappears with rate 1. In other words, a stochastic reservoir is located at the boundary site {0} of N. For a probability measure ν on P and N ≥ 1, we denote by PνN the distribution on the path space D(R+ , P) of the process pt ≡ ptN with generator N 2 L ε(N ),U , which is accelerated by the factor N 2 and the initial measure ν. Here, ε(N ) is defined by the relation: E με(N ) [n( p)] = N 2 .
(2.5)
U
Let X U be the function space defined by X U := {ψ : R◦+ → R◦+ ; ψ ∈ C 1 , ψ < 0, lim ψ(u) = ∞, lim ψ(u) = 0}, u↓0
u↑∞
where ψ = dψ/du. With these notations our first main theorem is stated as follows. Recall that the scaled height variable ψ˜ pN (u) is defined by (1.2) for p ∈ P. Theorem 2.1. Let (ν N ) N ≥1 be a sequence of probability measures on P such that ∞ ∞ N N ˜ lim ν [| f (u)ψ p (u)du − f (u)ψ0 (u)du| > δ] = 0 (2.6) N →∞
0
0
holds for every δ > 0, f ∈ C0 (R◦+ ) and some function ψ0 ∈ X U , where C0 (R◦+ ) is the class of all functions f ∈ C(R◦+ ) having compact supports in R◦+ . Then, for every t > 0, ∞ ∞ lim P NN [| f (u)ψ˜ pN (u)du − f (u)ψ(t, u)du| > δ] = 0 (2.7) N →∞
ν
0
t
0
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holds for every δ > 0 and f ∈ C0 (R◦+ ), where ψ(t, u) is the unique classical solution (in the space X U ) of the non-linear partial differential equation (PDE): ⎧ ∂u ψ ∂u ψ ⎪ ⎪ ∂ +α ψ = ∂ , ⎪ u ⎨ t 1 − ∂u ψ 1 − ∂u ψ (2.8) ⎪ ψ(0, ·) = ψ0 (·), ⎪ ⎪ ⎩ ψ(t, ·) ∈ X U , t ≥ 0, √ where ∂t ψ = ∂ψ/∂t, ∂u ψ = ∂ψ/∂u and α = π/ 6. Remark 2.1. The function ψU defined in (1.3) is the unique stationary solution in the class X U of Eq. (2.8). The curve in the first quadrant of x y-plane determined by the equation y = ψU (x) is called the Vershik curve (in U-statistics). In this way, the derivation of the Vershik curve is understandable from the dynamical point of view. Remark 2.2. Spohn discussed in [16, App. A] two-dimensional interfacial dynamics, motivated by the zero-temperature Ising model, under the periodic boundary condition with symmetric jump rates and derived the non-linear PDE (2.8) with α = 0 under the hydrodynamic scaling limit. He studied interfaces having graphical representations as in our setting, but not necessarily being monotone. See [2, Sect. 4], [3,4] for further studies. Shlosman [15] discussed the similarity between the approach from the Young diagrams and the Wulff problem in the Ising model. Aldous and Diaconis [1] used an idea of the hydrodynamic limit to give a “soft” proof for the asymptotic behavior of the length of the longest increasing subsequence of random permutations. Remark 2.3. The large deviation rate function I (ψ) under the canonical ensemble of n is described in Theorem 1.2 of [5]. We can compute its functional derivU-statistics μU ative and find that it is given by the formula: δI ψ (u) + αψ (u)(1 − ψ (u)) = . δψ(u) ψ (u)(1 − ψ (u)) On the other hand, the right hand side of our hydrodynamic equation (2.8) is equal to ψ (u) + αψ (u)(1 − ψ (u)) . (1 − ψ (u))2 These formulas have similarity but are not exactly the same. Recall that we discuss the dynamics associated with the grandcanonical ensemble. The dynamics for the canonical ensemble involve much complexity. 2.2. RU-statistics. Denote by Qn the set of all partitions of n ∈ N into distinct positive integers, that is, the set of all q = (qi )i∈N ∈ Pn satisfying qi > qi+1 if qi > 0. The union of Qn is denoted by Q: Q = ∪n∈Z+ Qn , where Q0 = P0 . Let n(q) be the sum of qi ’s in q ∈ Q. The function ψq on R+ is assigned to q ∈ Q by the relation (2.1). For 0 < ε < 1, let μεR be the probability measure on Q determined by μεR (q) =
1 εn(q) , q ∈ Q, Z R (ε)
(2.9)
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∞ k n where Z R (ε) = ∞ k=1 (1 + ε ) = n=0 p = (n)ε , p = (n) = Qn is the normalizing constant, see [17]. The conditional measure μεR |Qn of μεR on Qn coincides with the uniform probability measure μnR on Qn . The measures μεR and μnR are the grandcanonical and canonical ensembles in the RU-statistics, respectively. Now, we construct the dynamics associated with μεR . Let qt ≡ qtε = (qi (t))i∈N be the Markov process on Q with the infinitesimal generator L ε,R acting on functions f : Q → R as ε1{qi−1>qi +1} { f (q i,+ )− f (q)}+1{qi>qi+1+1 or qi =1} { f (q i,− ) − f (q)} , L ε,R f (q) = i∈N
(2.10) where q i,± ∈ Q are defined by the formula (2.4) and we regard q0 = ∞. It is easy to see that μεR is invariant under such dynamics. Similarly to the U-case, we think of qi (t) as the position of the i th particle. The model defined by the generator (2.10) involves a stochastic reservoir at {0}. The only difference is that the creation of a new particle at {1} is allowed if this site is vacant, since, under the situation that qi = 0, the transition from q to q i,+ ∈ Q occurs only when qi−1 ≥ 2. For a probability measure ν on Q and N ≥ 1, we denote by QνN the distribution on the path space D(R+ , Q) of the process qt ≡ qtN with generator N 2 L ε(N ),R and the initial measure ν. Here, ε(N ) is defined by the relation: E με(N ) [n( p)] = N 2 .
(2.11)
R
Let X R be the function space defined by X R := {ψ : R+ → R+ ; ψ ∈ C 1 , −1 ≤ ψ ≤ 0, ψ (0) = −1/2, lim ψ(u) = 0}. u↑∞
Our second main theorem is stated as follows. The scaled height variable ψ˜ qN (u) is defined by (1.2) for q ∈ Q. Theorem 2.2. Let (ν N ) N ≥1 be a sequence of probability measures on Q such that ∞ ∞ N N ˜ f (u)ψq (u)du − f (u)ψ0 (u)du| > δ] = 0 (2.12) lim ν [| N →∞
0
0
C0 (R◦+ )
and some function ψ0 ∈ X R . Then, for every t > 0, holds for every δ > 0, f ∈ ∞ ∞ lim QνNN [| f (u)ψ˜ qNt (u)du − f (u)ψ(t, u)du| > δ] = 0 (2.13) N →∞
0
0
holds for every δ > 0 and f ∈ C0 (R◦+ ), where ψ(t, u) is the unique classical solution (in the space X R ) of the non-linear partial differential equation: ⎧ 2 ⎪ ⎨ ∂t ψ = ∂u ψ + β ∂u ψ(1 + ∂u ψ), (2.14) ψ(0, ·) = ψ0 (·), ⎪ ⎩ ψ(t, ·) ∈ X R , t ≥ 0, √ and β = π/ 12.
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Remark 2.4. The function ψ R defined in (1.3) is the unique stationary solution in the class X R of Eq. (2.14). The curve determined by the equation y = ψ R (x) is called the Vershik curve (in RU-statistics). Remark 2.5. The convergence (2.7) in Theorem 2.1 holds in a stronger sense: lim PνNN [ sup
N →∞
u∈[u 0 ,u 1 ]
|ψ˜ pNt (u) − ψ(t, u)| > δ] = 0
for every δ > 0 and 0 < u 0 < u 1 , and similar for (2.13) in Theorem 2.2. In fact, one can argue as in Step 1 of the proof of Theorem 2.1 below noting the monotonicity of ψ˜ pN and ψ˜ qN and the C 1 -property of ψ(t, ·) as functions of u. t
t
3. Asymptotic Behaviors of ε(N) Before giving the proof of our main theorems, we study in this section the asymptotic behaviors of ε(N ) defined by (2.5) and (2.11) in U- and RU-statistics, respectively, as N → ∞. 3.1. U-statistics. Let ε(N ) be defined by the relation (2.5). Lemma 3.1. We have that ε(N ) = 1 − α/N + O(log N /N 2 ) as N → ∞. Proof. First, we calculate the expected value of the size n( p) of p ∈ P under the ε . In fact, probability measure μU E μUε [n( p)] = =
1 n( p)εn( p) = ε (log Z U (ε)) Z U (ε) p ∞ ∞ ∞ ∞ kεk εm mk = kε = . k 1−ε (1 − εm )2 k=1
k=1 m=1
m=1
k 2 The last equality follows from the simple identity ∞ k=1 kz = z/(1−z) for 0 ≤ z < 1. However, the inequality of arithmetic and geometric means and some simple estimations prove 1 1 ε(−m+1)/2 ≤ , ≤ m 1 + ε + ε2 + · · · + εm−1 m and thus, recalling α 2 = π 2 /6 and ε < 1, we have that ∞ ∞ 1 εm ε 1 α2 ε [n( p)] ≤ ≤ E < . μ U (1 − ε)2 m2 (1 − ε)2 m2 (1 − ε)2 m=1
m=1
Therefore, by (2.5), we have for ε = ε(N ), 0 < α 2 − (1 − ε)2 N 2 ≤ α 2 −
∞ ∞ εm 1 − εm = . m2 m2
m=1
m=1
(3.1)
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Since the right hand side tends to 0 as ε ↑ 1 (or as N → ∞), we see that (1 − ε)N tends to α as N → ∞ which implies that ε ≡ ε(N ) = 1 − α/N + o(1/N ). To derive a more precise estimate for the error term, we will show that the right-hand side of (3.1) admits a bound: ∞ 1 − εm C log N , ≤ m2 N
(3.2)
m=1
with some C > 0. Indeed, once this is shown, the proof of the lemma is concluded. To prove (3.2), noting that the function f ε (x) := (1 − ε x )/x 2 , x > 0, is non-increasing, we have that ∞ ∞ ∞ 1 − εm 1 − e−y ≤ f (1) + f (x)d x = f (1) − log ε dy ε ε ε m2 y2 1 − log ε m=1
≤ (1 − ε) − log ε (− log(− log ε)) − log ε, where the last inequality follows by dividing the integral over [− log ε, ∞) into the sum of those over [− log ε, 1] and [1, ∞) and then by estimating the integrands by 1/y and 1/y 2 , respectively. This implies (3.2) by recalling ε = 1 − α/N + o(1/N ). Remark 3.1. A rude version of Hardy-Ramanujan’s formula: p(n) = Pn ∼ e as n → ∞ implies that ∞ e n E μUε [n( p)] ∼ n=1
√
√ 2/3π n−(log ε−1 )n
Z U (ε)
√
√ 2/3π n
.
√ √ Since the function f (x) := 2/3π x − (log ε−1 )x, x > 0 attains its maximal value √ 2 at x(ε) = π/( 6 log ε−1 ) , we see that E μUε [n( p)] behaves as x(ε) as ε ↑ 1 and this shows that ε(N ) ∼ 1 − α/N as N → ∞. 3.2. RU-statistics. Let ε(N ) be defined by the relation (2.11). Lemma 3.2. We have that ε(N ) = 1 − β/N + O(log N /N 2 ) as N → ∞. Proof. First, we calculate the expected value of n( p) under μεR : E μεR [n( p)] = =
1 n( p)εn( p) = ε (log Z R (ε)) Z R (ε) p ∞ ∞ ∞ kεk = k(−1)m−1 εmk . 1 + εk k=1 m=1
k=1
Thus, similarly to the U-case, we have that E μεR [n( p)] =
∞ m=1
(−1)m−1
εm = o (ε) − e (ε), (1 − εm )2
(3.3)
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where o (ε) and e (ε) are the sums taken over odd and even numbers, respectively, i.e.
o (ε) :=
∞ m=1
∞ ε2m−1 ε2m ,
(ε) := . e (1 − ε2m−1 )2 (1 − ε2m )2 m=1
Note that one can change the order of the sum in (3.3) since the series converges absolutely. Now recalling β 2 = α 2 /2, by (2.11), we have for ε = ε(N ) β 2 − (1 − ε)2 N 2 = α 2 − (1 − ε)2 ( o (ε) + e (ε)) − α 2 − 4(1 − ε)2 e (ε) /2. However, since e (ε) = o (ε2 ) + e (ε2 ) and (1 + ε)2 ≤ 4, we have that |β 2 − (1 − ε)2 N 2 | ≤ α 2 − (1 − ε)2 ( o (ε) + e (ε)) + α 2 − (1 − ε2 )2 ( o (ε2 ) + e (ε2 )) /2 ≤ {(1 − ε) − log ε (− log(− log ε)) − log ε} + (1 − ε2 ) − 2 log ε (− log(−2 log ε)) − 2 log ε /2. The second inequality is shown in the proof of Lemma 3.1. This first implies that ε ≡ ε(N ) = 1 − β/N + o(1) and then completes the proof of the lemma as in the last part of the proof of Lemma 3.1. The precise error estimates in Lemmas 3.1 and 3.2 are only needed in [8], see Remark 5.1 below.
4. Proof of Theorem 2.1 This section gives the proof of Theorem 2.1 for the U-case. In the process pt , the particles are distinguished from each other and numbered from the right. However, if we are only concerned with the number of particles at each site and define ξt = (ξt (x))x∈Z+ by ξt (x) = {i; pi (t) = x} ∈ Z+ for x ∈ N and ξt (0) = ∞, then ξt becomes the weakly asymmetric zero-range process on N with the weakly asymmetric stochastic reservoir at {0}. We can think of ξt (x) as the (negative) gradient of the height function ψ pt at u = x in the sense that ξt (x) = ψ pt (x − 1) − ψ pt (x). Actually, the stochastic reservoir for pt or ξt located at {0} can be removed under a simple transformation. Indeed, we transform the process pt into another process η¯ t on Z, which is roughly defined as follows: With each p ∈ P, we associate a family of particles located at (i,√ pi ) in the x y-plane and project them perpendicularly to the line {y = −x} rescaled by 2. Or, one can say that we first rotate the x y-plane by 45 degrees √ to the left-handed direction and then project the particles to the x-axis rescaled by 2. This determines a configuration η¯ on Z. Such transformation is sometimes used in the study of particle systems. As we will see, in the RU-case, one can not find this kind of nice transformation which removes the stochastic reservoir.
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4.1. Transformation for the process pt . We introduce a transformation of our process pt on N to a weakly asymmetric simple exclusion process η¯ t on Z mentioned above. Denote by χU the state space of the transformed process: (1 − η(x)) ¯ = η(x) ¯ < ∞}. χU := {η¯ ∈ {0, 1}Z ; x≤0
x≥1
In particular, if η¯ ∈ χU , then there exist x± ∈ Z such that η(x) ¯ = 1 for all x ≤ x− and η(x) ¯ = 0 for all x ≥ x+ . For η¯ ∈ χU , we assign two functions ζη¯− and ζη¯+ on Z by the following rule: ζη¯− (x) = (1 − η(z)) ¯ and ζη¯+ (x) = η(z), ¯ x ∈ Z. (4.1) z≤x
z≥x+1
By definition, ζη¯− and ζη¯+ are monotone non-negative integer-valued functions. Now, we construct one-to-one correspondence between χU and P. For η¯ ∈ χU , we assign η¯ p η¯ = ( pi )i∈N ∈ P by the following rule: η¯
pi = ζη¯− (xi ), i ∈ N, where xi is the unique element of Z which satisfies ζη¯+ (xi −1) = i and ζη¯+ (xi ) = i −1. In other words, the family {xi }i∈N is determined by numbering the set {x ∈ Z; η(x) ¯ = 1} η¯ by i ∈ N from the right and pi = {x ≤ xi ; η(x) ¯ = 0}. We can show that the map η¯ → p η¯ is well-defined and also it is a bijection from χU to P. So we denote its inverse map by p → η¯ p . Note that the origin 0 is determined by the condition ζη¯− (0) = ζη¯+ (0) or equivalently {x ≤ 0; η(x) ¯ = 0} = {x ≥ 1; η(x) ¯ = 1}, i.e., the number of empty sites on the left to the origin is equal to that of particles on the right to the site 1. We now consider the Markov process η¯ t on χU with the generator L¯ ε,U acting on functions f : χU → R as L¯ ε,U f (η) εc+ (x, η) ¯ = ¯ + c− (x, η) ¯ { f (η¯ x,x+1 ) − f (η)}, ¯ x∈Z
where ¯ = 1{η(x)=1, , c− (x, η) ¯ = 1{η(x)=0, , c+ (x, η) ¯ η(x+1)=0} ¯ ¯ η(x+1)=1} ¯ and
⎧ ⎪ ¯ if z = x, y, ⎨η(z) η¯ x,y (z) = η(y) ¯ if z = x, ⎪ ⎩η(x) ¯ if z = y.
(4.2)
(4.3)
Note that the relation ζη¯− (0) = ζη¯+ (0) is invariant under the transition from η¯ to η¯ x,x+1 for all x ∈ Z. The following lemma is easy so that the proof is omitted. Lemma 4.1. Two processes { pt }t≥0 and { p η¯ t }t≥0 have the same distributions on the path space D(R+ , P).
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For a probability measure ν on χU and N ≥ 1, we denote by P¯ νN the distribution on the path space D(R+ , χU ) of the process η¯ tN with generator N 2 L¯ ε(N ),U and the initial measure ν, where ε(N ) is defined by (2.5). By Lemma 3.1, since ε(N ) is close to 1 for large N , we can think of the process η¯ tN as a weakly asymmetric simple exclusion process on Z. The hydrodynamic limit of such process is already known. Indeed, let YU be the function space defined by 0 ∞ (1 − ρ(v))dv = ρ(v)dv < ∞}. YU := {ρ : R → (0, 1); ρ is continuous, −∞
0
Then, for the scaled empirical measures of the process η¯ tN defined by πtN (dv) =
1 N η¯ t (x)δx/N (dv), t ≥ 0, v ∈ R, N
(4.4)
x∈Z
we have the following proposition, see Gärtner [9]: Proposition 4.2. Let (ν N ) N ≥1 be a sequence of probability measures on χU such that ∞ ∞ g(v)π0N (dv) − g(v)ρ0 (v)dv| > δ] = 0 (4.5) lim ν N [| N →∞
−∞
−∞
holds for every δ > 0, g ∈ C0 (R) and some function ρ0 ∈ YU . Then, for every t > 0, ∞ ∞ g(v)πtN (dv) − g(v)ρ(t, v)dv| > δ] = 0 (4.6) lim P¯ νNN [| N →∞
−∞
−∞
holds for every δ > 0 and g ∈ C0 (R), where ρ(t, v) is the unique classical solution of the following partial differential equation:
∂t ρ = ∂v2 ρ + α∂v (ρ(1 − ρ)) , (4.7) ρ(0, ·) = ρ0 (·). Kipnis et al. [12] also studied the hydrodynamic limit of weakly asymmetric simple exclusion processes under the periodic boundary conditions. Remark 4.1. The unique solution of (4.7) satisfies that ρ(t, ·) ∈ YU for all t > 0 if ρ0 ∈ YU . This fact (except the equality of two integrals in the definition of YU ) is seen by regarding the non-linear PDE (4.7) as a linear PDE: ∂t ρ = ∂v2 ρ + b(t, v)∂v ρ with b(t, v) = α(1−2ρ(t, v)), in which ρ(t, v) is considered to be already given, and then by (t) relying, for instance, on a probabilistic representation of ρ(t, v): ρ(t, v) = E v [ρ0 (X t )] (t) in terms√of the solution (X s ) = (X s )0≤s≤t of the stochastic differential equation: d X s = 2d Bs + b(t − s, X s )ds, 0 ≤ s ≤ t, X 0 = v for each t> 0, where Bs is the 0 one-dimensional Brownian motion. The equality of two integrals: −∞ (1−ρ(t, v))dv = ∞ 0 ρ(t, v)dv follows directly from the PDE (4.7) or by taking limits from the microscopic systems. Proposition 4.2 is formulated only for the test functions g having compact supports. We also need the following asymptotic behaviors of the tails of πtN .
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Lemma 4.3. Assume that the following two conditions (4.8) and (4.9) hold for t = 0. Then, for every t > 0, we have that ∞ lim P¯ νNN [|πtN ([0, ∞)) − ρ(t, v)dv| > δ] = 0, (4.8) N →∞
0
and lim P¯ νNN [|πˆ tN ((−∞, 0]) −
N →∞
0 −∞
(1 − ρ(t, v))dv| > δ] = 0,
(4.9)
for every δ > 0, where πˆ tN (dv) =
1 (1 − η¯ tN (x))δx/N (dv), t ≥ 0, v ∈ R. N x∈Z
Proof. We easily see that (4.6) holds for a step function g = 1[a,b] with −∞ < a < b < ∞, by approximating such g by a sequence of continuous functions gn ∈ C0 (R) noting −K that 0 ≤ η¯ t (x), ρ(t, v) ≤ 1. Moreover, Remark 4.1 implies that both −∞ (1−ρ(t, v))dv ∞ and K ρ(t, v)dv are arbitrarily small for large enough K > 0. Thus, to prove (4.8) and (4.9), it is sufficient to show that for every δ > 0 there exists K > 0 such that lim P¯ νNN [πtN ([K , ∞)) > δ] = 0,
(4.10)
lim P¯ νNN [πˆ tN ((−∞, −K ]) > δ] = 0,
(4.11)
N →∞
and N →∞
respectively. We prove (4.10) only, since the proof of (4.11) is similar. To this end, take a function ϕ1 ∈ Cb2 (R) satisfying that ϕ1 ≥ 0, ϕ1 (v) = 1 for v ≥ 1 and ϕ1 (v) = 0 for v ≤ 1/2, and set ϕ K (v) := ϕ1 (v/K ) for K > 0. Then, t N 2 L¯ ε(N ),U πsN , ϕ K ds m tN (ϕ K ) := πtN , ϕ K − π0N , ϕ K − 0
is a martingale and the following two bounds: ∞ × |supp ϕ K | ≤ ϕ1 ∞ /2K , N 2 L¯ ε(N ),U π N , ϕ K ≤ ϕ K
(4.12)
and 2 ∞ × |supp ϕ K |/N ≤ t ϕ1 2∞ /2K N , (4.13) E[m tN (ϕ K )2 ] ≤ t ϕ K hold, where π, ϕ = R ϕ(v)π(dv) and |supp ϕ| stands for the Lebesgue measure of the support of ϕ. Indeed a similar computation is made in the proof of Proposition 5.4 below. Actually, because of the difference of the generators, the first sums in (5.7) and (5.8) below should be taken over x ∈ Z rather than x ∈ N and the second terms do not appear in the present setting. In particular, we have that
d m N (ϕ)t = (ϕ((x + 1)/N ) − ϕ(x/N ))2 εc+ (x, η¯ tN ) + c− (x, η¯ tN ) , dt x∈Z
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347
≥ 0, the first sum in (5.7) is bounded from which proves (4.13). Moreover, since ϕ K above by the same sum taken ε = 1 (because ε < 1). However, since c+ (x, η) ¯ − c− (x, η) ¯ = η(x) ¯ − η(x ¯ + 1), the bound (4.12) follows by the summation by parts. Accordingly, we have
πtN ([K , ∞)) ≤ πtN , ϕ K ≤ π0N , ϕ K + t ϕ1 ∞ /2K + |m tN (ϕ K )|. Therefore, the condition (4.8) for t = 0 controls the behavior of π0N , ϕ K and proves (4.10) with the help of (4.13). Remark 4.2. (1) The condition (4.8) is equivalent to (4.6) with g = 1[0,∞) . (2) The condition (4.6) can be rewritten into an equivalent form (4.6) , which is (4.6) with πtN , ρ(t, v) replaced by πˆ tN , 1 − ρ(t, v), respectively, and for all g ∈ C0 (R). Then the condition (4.9) is equivalent to (4.6) with g = 1(−∞,0] .
4.2. Correspondence between two function spaces X U and YU . We study the relationship between two function spaces X U and YU . To each ψ ∈ X U , one can associate an ele(1) ment ρ ∈ YU in the following manner: First consider a curve Cψ = {(u, w); w = ψ(u)} (2)
in the first quadrant in the plane, and then define a new curve Cψ in the upper half plane (1)
(2)
by shifting each point (u, w) in Cψ to (u − ψ(u), w). The tilt of the curve Cψ with reversed sign defines the function ρ ∈ YU . More precisely, for ψ ∈ X U , we define the function G ψ : R◦+ → R as G ψ (u) := u − ψ(u).
(4.14)
By the definition of X U , G ψ is a monotone function and furthermore a bijection from R◦+ to R. So, there exists an inverse function of G ψ . We define a function U (ψ) : −ψ (G −1 ψ (v))
R → (0, 1) as U (ψ)(v) =
1−ψ (G −1 ψ (v))
for v ∈ R. Then, we can easily see that
U (ψ) ∈ YU . In fact, we can show the following proposition. Proposition 4.4. The map U defines a one-to-one correspondence between X U and YU . Proof. The inverse map U of U can be constructed as follows. For ρ ∈ YU , we define two functions ζρ− : R → R◦+ and ζρ+ : R → R◦+ as ζρ− (v)
:=
v
−∞
(1 − ρ(v ))dv
and
ζρ+ (v)
:=
∞ v
ρ(v )dv , v ∈ R. (4.15)
Note that these functions are macroscopic correspondences to those determined by (4.1). By the definition of YU , ζρ− and ζρ+ are continuously differentiable monotone functions. Moreover, they are bijections from R to R◦+ . So, there exists an inverse function of ζρ− . We define a function U (ρ) : R◦+ → R◦+ as U (ρ)(u) = ζρ+ (ζρ− )−1 (u) for u ∈ R◦+ . Then, we can easily see that U (ρ) ∈ X U . Furthermore, U ◦ U = id X U and U ◦ U = idYU hold, which concludes the proof.
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4.3. Proof of Theorem 2.1. Step 1. We first show that under the condition (2.6), because of the monotonicity of functions ψ˜ pN and the C 1 -property of ψ0 , the sequence (ν N ) N ≥1 actually satisfies a stronger condition that lim ν N [ sup
N →∞
u∈[u 0 ,u 1 ]
|ψ˜ pN (u) − ψ0 (u)| > δ] = 0
(4.16)
for every δ > 0, 0 < u 0 < u 1 . To prove this, suppose that it is false, namely there exists some ε > 0, δ > 0, 0 < u 0 < u 1 and subsequence (n i )i∈N of N such that ν n i [ sup
u∈[u 0 ,u 1 ]
|ψ˜ pn i (u) − ψ0 (u)| > δ] ≥ ε for all i ∈ N.
Especially, we consider only the case in which ψ˜ pn i (u) − ψ0 (u) > δ] ≥ ε for all i ∈ N, ν n i [ sup u∈[u 0 ,u 1 ]
because the opposite case can be handled in much the same way. Since ψ˜ pn i (u) is nonincreasing and supu∈K |ψ0 (u)| < ∞ for any compact set K in R0+ , for small r > 0, we have ν n i [∃u ∈ [u 0 , u 1 ] s.t. ψ˜ pn i (v) − ψ0 (v) >
δ ∀v ∈ [u − r, u]] ≥ ε for all i ∈ N. 2
Therefore, fix large m ∈ N, then we obtain that ν n i [1 ≤ ∃k ≤ [mu 1 ] s.t. ψ˜ pn i (u) − ψ0 (u) >
k k+1 δ ∀u ∈ [ , ]] ≥ ε for all i ∈ N, 2 m m
where [mu 1 ] stands for the integer part of mu 1 , and especially, for every i ∈ N, there exists 1 ≤ ki ≤ [mu 1 ] such that ν n i [ψ˜ pn i (u) − ψ0 (u) >
ki ki + 1 ε δ ∀u ∈ [ , ]] ≥ . 2 m m [mu 1 ]
Now, we can take k∗ and a subsequence (n˜i ) of (n i ) such that ν n˜i [ψ˜ pn˜i (u) − ψ0 (u) >
k∗ k∗ + 1 ε δ ∀u ∈ [ , ]] ≥ for all i ∈ N, 2 m m [mu 1 ]
which contradicts (2.6). Step 2. Next, we will show that Theorem 2.1 for the process pt (≡ ptN ) follows from Proposition 4.2 for the process η¯ tN . To this end, we first see that the conditions (4.5), (4.8) and (4.9) at t = 0 are deduced from the condition (4.16) if we define η¯ and ρ0 by η¯ = η¯ p and ρ0 = U (ψ0 ), respectively. Take g ∈ Cb1 (R) satisfying g(v) = 0 for v ≤ −K and g(v) = c for v ≥ K with some K > 0 and c ∈ R. We will show the condition (4.5) for such g; recall Remark 4.2-(1) for t = 0. For a given 0 < δ < 1, determine u 0 , u 1 > 0 in such a manner that u 0 = ψ0−1 (K + 2) ∧ 1 and u 1 = ψ0−1 (δ), respectively. Now we assume the condition sup
u∈[u 0 ,u 1 ]
|ψ˜ pN (u) − ψ0 (u)| ≤ δ,
(4.17)
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349
for ψ˜ pN . Then, under this condition, we have that ψ˜ pN (u), ψ0 (u) ≥ K + 1, u ∈ (0, u 0 ], since 0 < δ < 1 and both functions are non-increasing in u, and pi > u 0 } = N ψ˜ pN (u 0 ) ≤ N (ψ0 (u 0 ) + 1). {i; N Thus, under (4.17), we have that ∞ x 1 1 pi − i + 1 = g(v)π0N (dv) = η(x)g ¯ g N N N N −∞ x∈Z i∈N di ( p) 1 pi pi − ψ˜ pN ( ) − = g N N N N i∈N di ( p) 1 pi pi − ψ˜ pN ( ) − = g N N N N pi i∈N: N >u 0
=
1 N
i∈N:
pi N
g
p
>u 0
i
N
− ψ0 (
pi ) + R N ,δ,1 , N
(4.18)
(4.19)
(4.20)
where di ( p) := { j ≤ i − 1; p j = pi } is the discrepancy in the graph of Young diagram ψ p (u) at u = pi , and the error term R N ,δ,1 satisfies that |R N ,δ,1 | ≤ C1 δ with C1 > 0. Indeed, the second equality in (4.20) follows from the fact that {x ∈ Z; η(x) ¯ = 1} = { pi −i +1; i ∈ N}, the third from ψ˜ pN ( pi /N ) = ψ p ( pi )/N = (i −1−di ( p))/N and the fourth from (4.18) since pi /N ≤ u 0 implies that pi /N −ψ˜ pN ( pi /N ) ≤ u 0 −(K +1) ≤ −K . The term R N ,δ,1 in the last line is defined by p pi di ( p) 1 pi i p pi g − ψ˜ N ( ) − −g − ψ0 ( ) R N ,δ,1 = N N N N N N pi i∈N: N >u 0
and admits the bound: |R N ,δ,1 | ≤
g ∞ N
p
i∈N: Ni >u 0
p d ( p) i i ˜ p pi − ψ0 + ψ N N N N
≤ g ∞ · 4δ(ψ0 (u 0 ) + 1), since the first summand in the above sum is bounded by δ if u 0 ≤ pi /N ≤ u 1 under the p condition (4.17) and is bounded by 2δ if pi /N ≥ u 1 by noting that 0 ≤ ψ˜ N (u), ψ0 (u) ≤ 2δ for u ≥ u 1 which follows from the monotonicity of these functions, and its second summand is bounded by ψ˜ pN ( pi /N −) − ψ˜ pN ( pi /N ) which is further bounded by 2δ from (4.17), recalling the continuity of ψ0 ; we have also used (4.19). We can further rewrite the sum in the last term of (4.20) as p p 1 1 pi i i − ψ0 ( ) = g (g ◦ G ψ0 ) N N N N N pi i∈N
i∈N: N >u 0
=
1 N 0 i∈N
pi N
(g ◦ G ψ0 ) (u)du =
0
∞
(g ◦ G ψ0 ) (u)ψ˜ pN (u)du.
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T. Funaki, M. Sasada
Note that, since g ◦ G ψ0 (u) = g(u − ψ0 (u)) = 0 if u ∈ (0, u 0 ], we have dropped the condition pi /N > u 0 from the summand of the above sums, and, by the same reason, we can replace the region of the integral in the last line from [0, ∞) to [u 0 , ∞). Consider the error R N ,δ,2 defined by ∞ N ,δ,2 = (g ◦ G ψ0 ) (u) ψ˜ pN (u) − ψ0 (u) du, R 0
which can be bounded as |R
N ,δ,2
| ≤ 2δ
K˜ u0
|(g ◦ G ψ0 ) (u)|du = C2 δ,
where K˜ is determined in such a manner that (g ◦ G ψ0 ) (v) = 0 for v ≥ K˜ . Furthermore, by the integration by parts formula, we have that ∞ ∞ (g ◦ G ψ0 ) (u)ψ0 (u)du = − (g ◦ G ψ0 )(u)ψ0 (u)du 0 0 ∞ 1 dv g(v)ψ0 (G −1 =− ψ0 (v)) 1 − ψ0 (G −1 −∞ ψ0 (v)) ∞ ∞ = g(v)U (ψ0 )(v)dv = g(v)ρ0 (v)dv. −∞
−∞
Therefore, under the condition (4.17), we have shown that ∞ ∞ N g(v)π0 (dv) − g(v)ρ0 (v)dv ≤ (C1 + C2 )δ. −∞
−∞
This implies the condition (4.5) for π0N and g ∈ Cb1 (R) satisfying g(v) = 0 for v ≤ −K and g(v) = c for v ≥ K with some K > 0 and c ∈ R. The same condition (4.5) with π0N , ρ0 replaced by πˆ 0N , 1 − ρ0 , respectively, and g ∈ Cb1 (R) satisfying g(v) = 0 for v ≥ K and g(v) = c for v ≤ −K with some K > 0 and c ∈ R can be shown by symmetry; recall Remark 4.2-(2) for t = 0. Indeed, for each p ∈ P, we denote by pˇ = ( pˇ i )i∈N the mirror image of the Young diagram p with the axis of symmetry {y = x} in the plane, i.e. pˇ i = { j; p j ≥ i}. Similarly, we denote by ψˇ0 the mirror image of the curve ψ0 with the axis of symmetry {y = x}, i.e. ψˇ0 (u) := ψ0−1 (u). Then, the condition (4.16) with ψ˜ pN , ψ0 replaced by ψ˜ pNˇ , ψˇ 0 , respectively, is deduced
from (4.16) itself. Therefore, if we denote by πˇ 0N the scaled empirical measure of the configuration η¯ pˇ and ρˇ0 the function associated with ψˇ 0 by the one-to-one map con structed in Subsect. 4.2, namely πˇ 0N (dv) = N1 x∈Z η¯ pˇ (x)δ Nx (dv) and ρˇ0 = U (ψˇ 0 ), then we see that the condition (4.5) with π0N , ρ0 replaced by πˇ 0N , ρˇ0 , respectively, holds for every δ > 0 and g ∈ Cb1 (R) satisfying g(v) = 0 for v ≤ −K and g(v) = c for v ≥ K with some K > 0 and c ∈ R by the above mentioned argument. However, since we easily see the relations: η¯ pˇ (x) = 1 − η¯ p (−x) and ρˇ0 (u) = 1 − ρ0 (−u), the condition (4.5) with π0N , ρ0 replaced by πˆ 0N , 1 − ρ0 , respectively, is shown for g ∈ Cb1 (R) satisfying g(v) = 0 for v ≥ K and g(v) = c for v ≤ −K . Step 3. In order to complete the proof of the theorem, it is now sufficient to show that (4.6) in Proposition 4.2 together with the assertions in Lemma 4.3 implies (2.7) with ψt = U (ρt ). The non-linear equation (2.8) for ψt follows from (4.7) for ρt .
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Take f ∈ C0 (R◦+ ) and t > 0 arbitrarily and fix them throughout the rest of the proof. Then we have that − ∞ 1 ζ (x) η¯ t (x), f (u)ψ˜ pNt (u)du = F t (4.21) N N 0 x∈Z
u
where F(u) = 0 f (u )du and ζt− (x) = ζη¯−t (x) defined by (4.1). For a given δ > 0, take K > 0 such that −K ∞ (1 − ρ(t, v))dv < δ/6, ρ(t, v)dv < δ/6, (4.22) −∞
K
and the conditions (4.10) and (4.11) hold with δ replaced by δ/3, recall the proof of Lemma 4.3. Now let us prove that v ζt− (x) N − lim P N [ sup | (1 − ρ(t, v ))dv | > δ] = 0 (4.23) N →∞ ν x∈Z:|x/N −v|≤θ N −∞ holds for every 0 < θ < δ/3 and v ∈ V K ,θ := {v ∈ R; |v| ≤ K + 1, v ∈ θ Z}. In fact, since ζt− (x) is non-decreasing in x, we have that πˆ tN ((−∞, v − θ ]) ≤
ζt− (x) = πˆ tN ((−∞, x/N ]) ≤ πˆ tN ((−∞, v + θ ]) N
(4.24)
for x ∈ Z such that |x/N − v| ≤ θ . However, from (4.11) and (4.6) with g = 1[−K ,v±θ] and δ replaced by δ/3, we have that v±θ N N (1 − ρ(t, v ))dv | > 2δ/3] = 0. (4.25) lim Pν N [|πˆ t ((−∞, v ± θ ]) − N →∞
−∞
v±θ
v Moreover, since | −∞ (1 − ρ(t, v ))dv − −∞ (1 − ρ(t, v ))dv | ≤ θ, if 0 < θ < δ/3, (4.24) and (4.25) imply (4.23). Since F ∞ = f ∞ < ∞, (4.23) further shows that − v ζt (x) N F −F lim P N [ sup (1 − ρ(t, v ))dv > δ f ∞ ] = 0 N →∞ ν x∈Z:|x/N−v|≤θ N −∞ (4.26) for every v ∈ V K ,θ if 0 < θ < δ/3. We now return to the formula (4.21) and divide it as ∞ f (u)ψ˜ pNt (u)du =: I1N + I2N + I3N , 0
where I1N , I2N and I3N are defined as the sums in the right hand side of (4.21) restricted for x ≤ −K N , −K N < x < K N and x ≥ K N , respectively. For the first term I1N , since f ∈ C0 (R◦+ ), we see that f (u) = 0 so that F(u) = 0 for u ∈ [0, u 0 ] with some u 0 > 0. Therefore, choosing δ > 0 such that δ/3 < u 0 , (4.11) with δ replaced by δ/3 implies that lim PνNN [I1N = 0] = 1.
N →∞
(4.27)
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For the second term I2N , by (4.26), we can show that lim PνNN [|I2N − I˜2N | > δ] = 0,
N →∞
where, assuming K /θ ∈ Z for simplicity, kθ K /θ−1 1 F (1 − ρ(t, v ))dv I˜2N = N −∞
k=−K /θ
η¯ t (x).
kθ≤x/N δ] = 0,
N →∞
where I¯2θ =
K /θ−1
F
kθ
−∞
k=−K /θ
(1 − ρ(t, v ))dv
By letting θ ↓ 0, I¯2θ converges to K IK = F −K
I3N ,
For the third term replaced by δ/3 that
(k+1)θ
ρ(t, v )dv .
kθ
v −∞
(1 − ρ(t, v ))dv ρ(t, v)dv.
since 0 ≤ I3N ≤ F∞ πtN ([K , ∞)), we see from (4.10) with δ lim PνNN [I3N > δF∞ /3] = 0.
N →∞
These computations are now summarized into ∞ N lim Pν N [| f (u)ψ˜ pNt (u)du − I | > δ] = 0, N →∞
where
I =
0
∞ −∞
F
v
−∞
(1 − ρ(t, v ))dv
ρ(t, v)dv.
K v Note that I K coincides with −∞ F −∞ (1 − ρ(t, v ))dv ρ(t, v)dv because of (4.22) recalling that δ/3 < u 0 and the integration over [K , ∞) in v can be taken small enough if K is sufficiently large. However, by the change of variables w = ζρ−t (v) and the integration by parts, we have that ∞ ∞ ζρ− (v) dζρ+t − t I = (v)dv F ζρt (v) ρ(t, v)dv = − f (u)du · dv −∞ −∞ 0 ∞ w dv dζρ+t − −1 (ζρt ) (w) dw =− f (u)du · dv dw 0 0 ∞ w d + − −1 ζρt (ζρt ) (w) dw =− f (u)du · dw 0 ∞ ∞0 + − −1 f (u)ζρt (ζρt ) (u) du = f (u)U (ρt )(u)du. = 0
This completes the proof of Theorem 2.1.
0
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353
5. Proof of Theorem 2.2 This section gives the proof of Theorem 2.2 for the RU-case, i.e. the case corresponding to the restricted uniform statistics. Similarly to the process ξt in the U-case, we consider the particle numbers (or the gradient of the height function ψqt ) ηt = (ηt (x))x∈Z+ defined by ηt (x) = {i; qi (t) = x} ∈ {0, 1} for x ∈ N and ηt (0) = ∞. Note that only 0-1 height differences are allowed under the restriction imposed on the Young diagrams q ∈ Q. Then ηt becomes the weakly asymmetric simple exclusion process with the stochastic reservoir at {0}, which inputs particles into the region N with rate ε and absorbs them with rate 1. Contrarily to the weakly asymmetric simple exclusion process η¯ t on Z considered in the U-case, ηt determines a finite particles’ system on N. In the RU-case, one does not have a nice transformation for ηt , which removes the stochastic reservoir as in the U-case. We will apply the Hopf-Cole transformation for ηt at the microscopic level, which linearizes the leading term, and study the boundary behavior of the transformed process. 5.1. The process ηt . Denote by χ R the state space of the process ηt defined from qt : χ R := {η ∈ {0, 1}N ; η(x) < ∞}. x∈N
We have a one-to-one correspondence between χ R and Q. Indeed, for η ∈ χ R , we assign q η ∈ Q by the following rule: η qi = min{x ∈ Z+ ; η(y) ≤ i − 1}, i ∈ N. y≥x+1 η
In other words, {qi }i∈N is determined by numbering the set {x ∈ N; η(x) = 1} from η the right and, if i is larger than the cardinality of this set, we define qi = 0. We can η show that the map η → q is well-defined and also it is a bijection from χ R to Q. So we denote its inverse map by q → ηq . We now consider the Markov process ηt on χ R with the generator L¯ ε,R acting on functions f : χ R → R as L¯ ε,R f (η) = L¯ iε,R f (η) + L¯ bε,R f (η), where L¯ iε,R f (η) = and
εc+ (x, η) + c− (x, η) { f (η x,x+1 ) − f (η)} x∈N
L¯ bε,R f (η) = ε1{η(1)=0} + 1{η(1)=1} { f (η1 ) − f (η)}
are the interior and boundary terms of the generator, respectively, c+ (x, η), c− (x, η) and η x,y are defined by (4.2), (4.3) with η¯ replaced by η, respectively, and
η(z) if z = 1, 1 η (z) = 1 − η(1) if z = 1. The following lemma is easy so that the proof is omitted.
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Lemma 5.1. Two processes {qt }t≥0 and {q ηt }t≥0 have the same distributions on D(R+ , Q). ¯ νN the distribuFor a probability measure ν on χ R and N ≥ 1, we denote by Q N 2 tion on D(R+ , χ R ) of the process ηt with generator N L¯ ε(N ),R and the initial measure ν, where ε(N ) is defined by (2.11). Let us define the scaled empirical measures πtN (dv), t ≥ 0, v ∈ R◦+ of the process ηtN by the formula (4.4) with η¯ tN replaced by ηtN and the sum taken over all x ∈ N rather than x ∈ Z. The hydrodynamic limit for a boundary driven exclusion process is studied by [6]. Our model involves a weak asymmetry both in dynamics and the boundary condition, and furthermore it is defined on an infinite volume N. Note that the boundary generator L¯ bε,R is invariant under the Bernoulli measure with mean ρ ε = ε/(1 + ε). This actually determines the Dirichlet boundary condition at v = 0 in the limit equation (5.5) stated below, since ρ ε converges to 1/2 as ε = ε(N ) ↑ 1. The hydrodynamic limit for models in infinite volume was discussed by several authors including [13]. It might be possible to apply these methods to our model, but we will employ the simplest way based on the Hopf-Cole transformation. 5.2. Hopf-Cole transformation. In this subsection we introduce the microscopic HopfCole transformation for the process ηtN and formulate Theorem 5.2 on its hydrodynamic behavior. Theorem 2.2 will be shown from Theorem 5.2 in Subsect. 5.4. It is well-known that the (macroscopic) Hopf-Cole transformation: ∞ ρ(t, v)dv}, u ∈ R+ ω(t, u) = exp{β u
allows us to deduce the solution of the viscous Burgers’ equation (5.5) (at least on the whole line R) to that of the linear diffusion equation (5.3) (on R). We introduce the corresponding transformation at the microscopic level, cf. [9]. Namely, we consider N the process ζtN = (ζtN (x))x∈N defined by ζtN (x) := exp −(log ε) ∞ y=x ηt (y) with ε = ε(N ) from the process ηtN and the C(R+ )-valued process ζ˜ N (t, u), u ∈ R+ by interpolating ζ˜ N (t, u) := ζtN (N u) defined for u ∈ N/N in such a manner that ⎧ ⎫⎤ ⎡ ∞ ⎨ ⎬ ηtN (y) + 1{u≥1/N } ([N u]+1− N u)ηtN ([N u]) ⎦, ζ˜ N (t, u) := exp ⎣−(log ε) ⎩ ⎭ y=[N u]+1
for u ∈ R+ . Theorem 5.2. Let (ν N ) N ≥1 be a sequence of probability measures on χ R such that ∞ ∞ N N g(v)π0 (dv) − g(v)ρ0 (v)dv| > δ] = 0 (5.1) lim ν [| N →∞
0
0
holds for every δ > 0, g ∈ Cb (R+ ) satisfying g(v) = c for v ≥ K with ∞some K > 0 and c ∈ R, and some continuous function ρ0 : R+ → [0, 1] satisfying 0 ρ0 (v)dv < ∞. Then, for every T > 0, K > 0 and δ > 0, ¯ NN [ lim Q ν
N →∞
sup 0≤t≤T,0≤u≤K
|ζ˜ N (t, u) − ω(t, u)| > δ] = 0
(5.2)
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holds, where ω(t, u) is the unique bounded weak solution of the following linear diffusion equation: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
∂t ω = ∂u2 ω + β∂u ω, u ∈ R+ , ∞ ρ0 (v)dv}, u ∈ R+ , ω(0, u) = exp{β
u ⎪ ⎪ ⎪ 2∂ ω(t, 0) + βω(t, 0) = 0, t > 0, ⎪ u ⎪ ⎩ ω(t, ∞) = 1, t > 0.
(5.3)
Namely, for every t > 0,
∞
g(u)ω(t, u)du =
0
∞
g(u)ω(0, u)du t ∞ + g (u) − βg (u) ω(s, u)duds 0
0
(5.4)
0
holds for every g ∈ C02 (R+ ) satisfying 2g (0) − βg(0) = 0 and limu→∞ ω(t, u) = 1. The following corollary, which gives the hydrodynamic limit for ηtN , is an immediate consequence of Theorem 5.2 and will be used in [8]. Corollary 5.3. Under the same assumption as Theorem 5.2, ¯ NN [| lim Q ν
N →∞
0
∞
g(v)πtN (dv) −
∞
g(v)ρ(t, v)dv| > δ] = 0
0
holds for every t > 0, δ > 0 and g ∈ C0 (R◦+ ), where ρ(t, u) is the unique classical solution of the following partial differential equation: ⎧ ⎪ ∂t ρ = ∂v2 ρ + β∂v (ρ(1 − ρ)), v ∈ R+ , ⎨ (5.5) ρ(0, v) = ρ0 (v), v ∈ R+ , ⎪ ⎩ ρ(t, 0) = 1/2, t > 0.
5.3. Proof of Theorem 5.2. This subsection proves Theorem 5.2. 5.3.1. Uniform estimate on the total mass. We prepare a proposition which gives a N uniform estimate ∞ on the scaled total mass of ηt . For the proof, the conditions (5.1) with g ≡ 1 and 0 ρ0 (v)dv < ∞ are essential. of the Proposition 5.4. Denote by X tN the process total mass of the empirical measure 1 N N N N ◦ πt , namely X t := N x∈N ηt (x) ≡ πt (R+ ) . Then, for every T > 0, we have that %
$ lim sup
λ→∞ N ≥1
¯ NN Q ν
sup 0≤t≤T
X tN
> λ = 0.
(5.6)
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Proof. For ϕ ∈ Cb2 (R◦+ ), denote by m tN (ϕ) the martingale defined by t N N N m t (ϕ) := πt , ϕ − π0 , ϕ − N 2 L¯ ε(N ),R πsN , ϕds. 0
Then, by a simple computation, we have that N 2 L¯ ε(N ),R π N , ϕ = N (ϕ((x + 1)/N ) − ϕ(x/N )) {εc+ (x, η) − c− (x, η)} x∈N
+N ϕ(1/N ) ε1{η(1)=0} − 1{η(1)=1} , and
(5.7)
d m N (ϕ)t = (ϕ((x + 1)/N ) − ϕ(x/N ))2 εc+ (x, ηtN ) + c− (x, ηtN ) dt x∈N (5.8) +ϕ(1/N )2 ε1{η N (1)=0} + 1{η N (1)=1} , t
t
for the quadratic variation of m tN (ϕ), if the right-hand sides of these equalities converge absolutely. Now take a function ϕ ∈ Cb2 (R◦+ ) such that ϕ ≥ 0, ϕ(u) = 0 for 0 < u ≤ 1 and ϕ(u) = 1 for u ≥ 2. Then, (5.7) shows that
N 2 L¯ ε(N ),R π N , ϕ ≤ ϕ ∞ , similar to the proof of (4.12). Therefore, sup πtN , ϕ ≤ π0N , ϕ + T ϕ ∞ + sup |m tN (ϕ)|
0≤t≤T
0≤t≤T
≤
X 0N
+ T ϕ ∞ + 1 + sup m tN (ϕ)2 , 0≤t≤T
where we have estimated the martingale as |m tN (ϕ)| ≤ m tN (ϕ)2 +1. One can apply Doob’s inequality to show lim N →∞ E[sup0≤t≤T m tN (ϕ)2 ] = 0 from (5.8), which, in particular, proves sup N E[sup0≤t≤T m tN (ϕ)2 ] < ∞. Since the assumption of Theorem 5.2 (especially (5.1) with g ≡ 1 and the integrability of ρ0 ) implies that limλ→∞ sup N ν N (X 0N > λ) = 0, the conclusion of the proposition follows by the inequality: X tN ≤ 2 + πtN , ϕ. 5.3.2. Tightness of {ζ˜ N } N . Let PN be the probability distribution of ζ˜ N = {ζ˜ N (t, u)} on D([0, T ], C(R+ )), where the space C(R+ ) is endowed with the topology determined by the uniform convergence on every compact set of R+ . Lemma 5.5. The family of probability measures {PN } N ≥1 is relatively compact. Proof. To conclude the lemma, by Prokhorov’s theorem, it suffices to show the following three conditions for {PN } N ≥1 : (i) For every t ∈ [0, T ], lim sup PN [ζ˜ (t, 0) > λ] = 0. λ→∞ N ≥1
(ii) For every δ > 0 and t ∈ [0, T ], lim sup PN [ sup |ζ˜ (t, u) − ζ˜ (t, v)| > δ] = 0. γ ↓0 N ≥1
(iii) For every δ > 0 and K > 0, lim lim sup PN [ γ↓0 N →∞
|u−v|≤γ
sup
|t−s|≤γ ,0≤u≤K
|ζ˜ (t, u) − ζ˜ (s, u)| > δ] = 0.
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By the relation: ζ˜ N (t, 0) = exp{−(log ε)N X tN }, we have that ¯ NN [X tN > log λ/C], PN [ζ˜ N (t, 0) > λ] ≤ Q ν note that there exists C > 0 such that 0 < − log ε ≤ C/N for ε = ε(N ) and every N ≥ 1. Proposition 5.4 proves (i). Since ζ˜ N (t, ·) is a non-increasing function, for every 0 ≤ u < v, we have that |ζ˜ N (t, u) − ζ˜ N (t, v)| ≤ ζ˜ N (t, u) exp −(log ε)I N (t, u, v) − 1 ≤ ζ˜ N (t, 0) exp{C I N (t, u, v)/N } − 1 , (5.9) where [N v]
I N (t, u, v) :=
ηtN (y) + ([N u] + 1 − N u)ηtN ([N u])
y=[N u]+1
−([N v] + 1 − N v)ηtN ([N v]), which has a trivial bound: I N (t, u, v) ≤ N (v − u). Therefore, we have that ¯ NN [eC X tN (eCγ − 1) > δ] PN [ sup |ζ˜ N (t, u) − ζ˜ N (t, v)| > δ] ≤ Q ν |u−v|≤γ
¯ NN [X tN > log(δ/(eCγ − 1))/C]. =Q ν
(5.10)
Proposition 5.4 concludes (ii). Finally we prove (iii). By the definition of ζ˜ N (t, u) and Proposition 5.4, we only need to show that for every K > 0 and δ > 0, ¯ NN [ lim lim sup Q ν γ ↓0 N →∞
sup
|t−s|≤γ ,0≤u≤K
|
∞ ∞ 1 N 1 N ηt (x) − ηs (x)| > δ] = 0. N N x=[N u]
x=[N u]
1 N N N Noting that | N1 ∞ x=[N u] ηt (x) − πt , 1[u,∞) | ≤ N ηt ([N u]), we consider smooth functions φκ (u, ·) which approximate the function 1[u,∞) as κ ↓ 0 such that φκ (u, v) = 0 0 ≤ φκ (u, v) ≤ 1 φκ (u, v) = 1 φκ (u, ·) = φκ (u + v, · + v)
for v ≤ u − κ, for u − κ ≤ v ≤ u + κ, for v ≥ u + κ, for every u and v.
In particular, we have that |πtN , φκ (u, ·) − πtN , 1[u,∞) | ≤ κ for every u. Moreover, φκ 2,∞ := supu {φκ (u, ·)∞ + φκ (u, ·)∞ } is finite. Now, it is enough to prove that for every κ, δ > 0, ¯ NN [ lim lim sup Q ν γ ↓0 N →∞
sup
|t−s|≤γ ,κ≤u≤K
|πtN , φκ (u, ·) − πsN , φκ (u, ·)| > δ] = 0.
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However, the term πtN , φκ (u, ·) − πsN , φκ (u, ·) is rewritten as t x φκ (u, )N L¯ ε(N ),R ηrN (x)dr + m˜ tN − m˜ sN , N s x∈N
where m˜ ·N is a martingale which vanishes as N goes to 0; recall (5.8). t On the other hand, the absolute value of the integral term is bounded from above by s 2κφκ 2,∞ dr , recall (5.7). This concludes the proof of (iii) and therefore the lemma. 5.3.3. Characterization of limit points. We start with considering a class of martingales associated with {ζ˜ N } N ≥1 . Let MtN (x), x ∈ N, be the martingale defined by t N 2 L¯ ε(N ),R (ζsN (x))ds. MtN (x) := ζtN (x) − ζ0N (x) − 0
Some simple computations permit us to rewrite N 2 L¯ ε(N ),R (ζsN (x)) = N 2 εζsN (x − 1) − (ε + 1)ζsN (x) + ζsN (x + 1) , (5.11) for every x ∈ N, where we define ζsN (0) := ε−1 ζsN (2). In fact, applying the generators L¯ iε,R and L¯ bε,R for the functions f x (η) := exp{−(log ε) ∞ y=x η(y)}, x ≥ 1, we have that ε f x−1 − (ε + 1) f x + f x+1 , x ≥ 2, ¯L i f x = ε,R 0 , x = 1, and L¯ bε,R f x =
0 , x ≥ 2, 2 f 2 − (ε + 1) f 1 , x = 1.
The last expression coincides with ε f 0 − (ε + 1) f 1 + f 2 , if we define f 0 := ε−1 f 2 and this shows (5.11). However, denoting β(N ) := N (1 − ε(N )) which converges to β as N → ∞ by Lemma 3.2, the right hand side of (5.11) can be rewritten further as N 2 ζsN (x) + Nβ(N )∇ζsN (x), where ζ = (ζ (x))x∈N and ∇ζ = (∇ζ (x))x∈N are defined for ζ = (ζ (x))x∈Z+ by ζ (x) = ζ (x − 1) − 2ζ (x) + ζ (x + 1), ∇ζ (x) = ζ (x) − ζ (x − 1), respectively. Thus, for every g ∈ C02 (R+ ), taking account of ζsN (0) = ε−1 ζsN (2), we have that ∞ 1 N g(u)ζ˜ N (t, u)du = ζt (x)g(x/N ) + RtN N 0 x∈N t 1 N = ζ0 (x)g(x/N ) + b N (ζsN , g)ds + MtN (g) + RtN , (5.12) N 0 x∈N
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where b N (ζ, g) =
1 N β(N ) N g(x/N )ζ (x) − ∇ g(x/N )ζ (x) N N x∈N
+N (g(1/N )ζ (2) − g(0)ζ (1)) ,
x∈N
(5.13)
with N g(x/N ) = N 2 (g((x + 1)/N ) + g((x − 1)/N ) − 2g(x/N )) , ∇ N g(x/N ) = N (g((x + 1)/N ) − g(x/N )) , x ∈ N, and MtN (g) =
1 N x Mt (x)g( ). N N x∈N
The error term
RtN
in (5.12) is defined by ∞ 1 N RtN = g(u)ζ˜ N (t, u)du − ζt (x)g(x/N ) N 0 x∈N
and admits a bound: |RtN |
≤e
C X tN
e
C/N
1 − 1 g L 1 (R+ ) + g ∞ × |supp g| N
(5.14)
in view of (5.9) and (5.10). Therefore, Proposition 5.4 shows that RtN tends to 0 as N → ∞ in probability. The martingale term in (5.12) vanishes in the limit: Lemma 5.6. E[MtN (g)2 ] converges to 0 as N → ∞. Proof. A straightforward computation leads to the following results for the quadratic and cross-variations of MtN (x): d M N (x)t = ζtN (x)2 a N c− (x − 1, ηtN ) + b N c+ (x − 1, ηtN ) , x ≥ 2, dt d M N (1)t = ζtN (1)2 a N 1{η N (1)=0} + b N 1{η N (1)=1} , t t dt N N M (x), M (y)t = 0, 1 ≤ x = y, where a N = N 2 (1−ε)2 /ε, b N = N 2 (1−ε)2 . This implies the conclusion of the lemma. To treat the boundary term appearing in b N (ζ, g) (i.e. the third term in the right hand side of (5.13)), we need the following ergodic property of the η-process at the boundary site {1}. Note that this ergodic property holds at the single site {1} without taking any average over sites near the boundary as performed in [6]. Lemma 5.7. Under the condition (5.6) in Proposition 5.4, for every 0 ≤ T1 ≤ T2 ≤ T and δ > 0, we have that ' & T2 1 1 N N ¯ lim Q N η (1)ds − > δ = 0. N →∞ ν T2 − T1 T1 s 2
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Proof. Consider the martingale :=
m tN
X tN
−
X 0N
− 0
t
N 2 L¯ ε(N ),R (X sN )ds.
By (5.7) with ϕ ≡ 1, we see that N 2 L¯ ε(N ),R (X sN ) = N (1−2ηsN (1))−β(N )(1−ηsN (1)). However, since Lemma 3.2 implies 0 < β(N ) = N (1 − ε(N )) ≤ C for N ≥ 1, this proves that T2 N ≤ 1 X N + X N + |m N | + |m N | + C T2 . {1 − 2η (1)}ds s T2 T1 T2 T1 N T1 Thus, the lemma follows from (5.6) and the estimate: E[|m TN |2 ] ≤ T , which follows from (5.8) with ϕ ≡ 1. Once the following lemma for the boundary term in b N (ζ, g) is established, the weak form (5.4) of Eq. (5.3) is easily derived from (5.12), (5.13), (5.14) and Lemma 5.6. Thus, the proof of Theorem 5.2 is concluded by the uniqueness of the weak solutions of (5.3), which will be shown in the next subsection. Lemma 5.8. If g ∈ C02 (R+ ) satisfies the condition 2g (0) − βg(0) = 0, then we have that ' & T N N >δ =0 ¯ NN lim Q N g(1/N )ζ (2) − g(0)ζ (1) dt t t ν N →∞
0
for every δ > 0. Proof. Recalling ζtN (2) = ζtN (1)e(log ε)ηt (1) , we have that N g(1/N )ζtN (2) − g(0)ζtN (1) = ζtN (1) g (0) − βg(0)ηtN (1) + rtN , N
where the error term rtN is defined by
N rtN := N (g(1/N ) − g(0)) − g (0) + N (g(1/N ) − g(0)) e(log ε)ηt (1) − 1 N +N g(0) e(log ε)ηt (1) − 1 + βηtN (1)/N ,
and tends to 0 as N → ∞ by Lemma 3.2. Therefore, by the boundary condition for g, if we can show that ' & T N N >δ =0 ¯ NN (5.15) lim Q ζ (1) η (1) − 1/2 dt t t N →∞ ν 0
for every δ > 0, the proof of the lemma is concluded. However, as we have shown in the tightness, the process {ζ·N (1)} N ≥1 has the equi-continuity: ⎡ ⎤ ¯ NN ⎢ lim lim sup Q sup ν ⎣ γ ↓0 N →∞
|t−s|≤γ 0≤s δ ⎦ = 0,
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for every δ > 0. Therefore, if we divide the interval [0, T ] into small subintervals with length γ : T /γ ] (k+1)γ ∧T [T N N N N ζt (1) ηt (1) − 1/2 dt ≤ ζt (1) ηt (1) − 1/2 dt , 0 kγ k=0
N (1) (if γ is small enough) and we can apply Lemma ζtN (1) in the integrand is close to ζkγ (k+1)γ ∧T N 5.7 for the integral kγ ηt (1) − 1/2 dt. In other words, ζtN (1) changes slowly compared with the rapid motion of ηtN (1). This proves (5.15).
Remark 5.1. For g satisfying the same condition as Lemma 5.8, a stronger assertion: ' & T √ N N N ¯ lim Q N N N g(1/N )ζt (2) − g(0)ζt (1) dt > δ = 0 N →∞ ν 0 √ holds for every δ > 0 even by multiplying an extra factor N . Indeed, this can be seen by noting that the error estimate given in the proof of Lemma 5.7 is O(1/N ) and that in Lemma 3.2 is O(log N /N 2 ) as N → ∞. This fact will be used in [8]. 5.3.4. Uniqueness of weak solutions Here, we prove the uniqueness of the weak solutions of (5.3). The method is standard, especially because the equation is linear. We first extend the class of test functions g = g(u) in the weak form (5.4) to the family of all g = g(t, u) ∈ C01,2 ([0, T ] × R◦+ ) satisfying 2∂u g(t, 0) − βg(t, 0) = 0 for every t ∈ [0, T ], and show that ∞ ∞ g(t, u)ω(t, u)du = g(0, u)ω(0, u)du 0 0 t ∞ + ∂s g(s, u) + ∂u2 g(s, u) − β∂u g(s, u) ω(s, u)duds 0
0
(5.16) holds for every such g and t ∈ [0, T ]. Indeed, this can be done by dividing the interval [0, t] into small pieces, assuming g to be constant in s on each small interval, applying the weak form (5.4) on each such small interval and finally by passing to the limit. Secondly, since the solution ω is assumed to be bounded, we can further extend the class of g’s from functions having compact supports in [0, T ] × R+ to those having the exponentially decaying property as u → ∞ in the sense that supt∈[0,T ],u∈R+ {|g(t, u)| + |∂t g(t, u)| + |∂u g(t, u)| + |∂u2 g(t, u)|}er u < ∞ for some r > 0. Finally, let ϕ ∈ C0∞ (R+ ) be given arbitrarily and define g ≡ gϕ = g(t, u) as the solution of the backward equation: ⎧ ⎪ ∂t g + ∂u2 g − β∂u g = 0, t ∈ [0, T ), u ∈ R+ , ⎨ g(T, u) = ϕ(u), u ∈ R+ , ⎪ ⎩ 2∂u g(t, 0) − βg(t, 0) = 0, t ∈ [0, T ). Such g exists and has the exponentially decaying property. By choosing this g in (5.16) with t = T , we obtain that ∞ ∞ ϕ(u)ω(T, u)du = gϕ (0, u)ω(0, u)du, 0
0
and this concludes the proof of the uniqueness of the weak solutions of (5.3).
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5.4. Proof of Theorem 2.2. We will show that Theorem 2.2 for the process qt follows from Theorem 5.2 for the process ηt . To this end, we first see that the condition (5.1) is deduced from the condition (2.12) if we define η and ρ0 by η = ηq and ρ0 = −ψ0 , respectively. For g ∈ Cb1 (R+ ) satisfying g(v) = 0 for v ≤ 1/K and g(v) = c for v ≥ K with some K > 1 and c ∈ R, taking g as f in (2.12), we have that ∞ ∞ N N ˜ g (u)ψq (u)du − g (u)ψ0 (u)du| > δ] = 0 lim ν [| N →∞
0
0
for every δ > 0. By the definition,
∞ 0
i 1 N 1 qi 1 x q g (u)du = g( ) = g( )η (x). N N N N N 0 q
g (u)ψ˜ qN (u)du =
i∈N
i∈N
On the other hand, by the integration by parts formula, ∞ ∞ g (u)ψ0 (u)du = − g(v)ψ0 (v)dv = 0
0
x∈N
∞
g(v)ρ0 (v)dv.
0
Therefore, (5.1) is shown for functions g satisfying the above conditions. However, this can be extended to a wider class of functions g ∈ Cb (R+ ) satisfying g(v) = c for v ≥ K with some K > 1 and c ∈ R, by approximating such g by a sequence of continuous functions gn ∈ Cb1 (R+ ) satisfying gn (v) = 0 for v ≤ 1/K and g(v) = c for v ≥ K with some K > 1 and c ∈ R, noting that 0 ≤ η(x), ρ0 (v) ≤ 1. In order to complete the proof of the theorem, it is now sufficient to show that (5.2) in Theorem 5.2 implies (2.13) with ψ(t, u) = β1 log ω(t, u). The non-linear equation (2.14) for ψt follows from (5.3) for ωt . Especially, the boundary condition 2∂u ω(t, 0) + βω(t, 0) = 0 implies that ∂u ψ(t, 0) = −1/2, and ω(t, ∞) = 1 implies that ψ(t, ∞) = 0 for t > 0. Since ψ˜ qNt (u) = β1 log ζ˜ N (t, u) + o(1) with an error going to 0 in probability as N → ∞ in view of (5.9) and (5.10), noting that ω(t, u), ζ˜ N (t, u) ≥ 1, (5.2) implies that % $ N N ¯ ˜ lim Q N |ψq (u) − ψ(t, u)| > δ = 0, sup N →∞
ν
0≤t≤T,0≤u≤K
t
for every T > 0, K > 0 and δ > 0. This completes the proof of Theorem 2.2. Acknowledgement.
The authors thank H. Spohn for leading them to the problems discussed in this paper.
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5. Dembo, A., Vershik, A., Zeitouni, O.: Large deviations for integer partitions. Markov Processes Relat. Fields 6, 147–179 (2000) 6. Eyink, G., Lebowitz, J.L., Spohn, H.: Lattice gas models in contact with stochastic reservoirs: Local equilibrium and relaxation to the steady state. Commun. Math. Phys 140, 119–131 (1991) 7. Funaki, T.: Stochastic interface models, Ecole d’Eté de Probabilités de Saint-Flour XXXIII – 2003, Lect. Notes Math. 1869, Berlin: Springer, 2005, pp. 103–274 8. Funaki, T., Sasada, M., Xie, B.: Fluctuations in an evolutional model of two-dimensional Young diagrams. In preparation 9. Gärtner, J.: Convergence towards Burger’s equation and propagation of chaos for weakly asymmetric exclusion processes. Stoch. Process. Appl. 27, 233–260 (1988) 10. Johansson, K.: Shape fluctuations and random matrices. Commun. Math. Phys. 209, 437–476 (2000) 11. Johansson, K.: Random Growth and Random Matrices. European Congress of Mathematics, Vol. I (Barcelona, 2000), Progr. Math., 201, Basel: Birkhäuser, 2001, pp. 445–456 12. Kipnis, C., Olla, S., Varadhan S.R, S.: Hydrodynamics and large deviation for simple exclusion processes. Comm. Pure Appl. Math. 42, 115–137 (1989) 13. Landim, C., Mourragui, M.: Hydrodynamic limit of mean zero asymmetric zero range processes in infinite volume. Ann. Inst. Henri Poincaré (B) 33, 65–82 (1997) 14. Pittel, B.: On a likely shape of the random Ferrers diagram. Adv. Appl. Math. 18, 432–488 (1997) 15. Shlosman, S.: The Wulff construction in statistical mechanics and combinatorics. Russ. Math. Survs. 56, 709–738 (2001) 16. Spohn, H.: Interface motion in models with stochastic dynamics. J. Stat. Phys. 71, 1081–1132 (1993) 17. Vershik, A.: Statistical mechanics of combinatorial partitions and their limit shapes. Func. Anal. Appl. 30, 90–105 (1996) Communicated by H. Spohn
Commun. Math. Phys. 299, 365–408 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1083-y
Communications in
Mathematical Physics
Ad S5 Solutions of Type IIB Supergravity and Generalized Complex Geometry Maxime Gabella1 , Jerome P. Gauntlett2,3 , Eran Palti1 , James Sparks4 , Daniel Waldram2,3 1 2 3 4
Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP, U.K. Theoretical Physics Group, Blackett Laboratory, Imperial College, London SW7 2AZ, U.K. The Institute for Mathematical Sciences, Imperial College, London SW7 2PE, U.K. Mathematical Institute, University of Oxford, 24-29 St Giles’, Oxford OX1 3LB, U.K. E-mail:
[email protected] Received: 2 October 2009 / Accepted: 11 February 2010 Published online: 30 June 2010 – © Springer-Verlag 2010
Abstract: We use the formalism of generalized geometry to study the generic supersymmetric Ad S5 solutions of type IIB supergravity that are dual to N = 1 superconformal field theories (SCFTs) in d = 4. Such solutions have an associated six-dimensional generalized complex cone geometry that is an extension of Calabi-Yau cone geometry. We identify generalized vector fields dual to the dilatation and R-symmetry of the dual SCFT and show that they are generalized holomorphic on the cone. We carry out a generalized reduction of the cone to a transverse four-dimensional space and show that this is also a generalized complex geometry, which is an extension of Kähler-Einstein geometry. Remarkably, provided the five-form flux is non-vanishing, the cone is symplectic. The symplectic structure can be used to obtain Duistermaat-Heckman type integrals for the central charge of the dual SCFT and the conformal dimensions of operators dual to BPS wrapped D3-branes. We illustrate these results using the Pilch-Warner solution.
1. Introduction Supersymmetric Ad S5 × Y solutions of type IIB supergravity, where Y is a compact Riemannian manifold, are dual to supersymmetric conformal field theories (SCFTs) in d = 4 spacetime dimensions with (at least) N = 1 supersymmetry. An important special subclass is when Y is a five-dimensional Sasaki-Einstein manifold S E 5 , and the only non-trivial flux is the self-dual five-form. Recall that, by definition, the six-dimensional cone metric with base given by the S E 5 space is a Calabi-Yau cone, and that the dual SCFT arises from D3-branes located at the apex of this cone. There has been much progress in understanding the AdS/CFT correspondence in this setting. For example, there are rich sets of explicit S E 5 metrics [1–3], and there are also powerful constructions using toric geometry. Moreover, for the toric case, the corresponding dual SCFTs have been identified, e.g. [4–7].
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A key aspect of this progress has been the appreciation that the abelian R-symmetry, which all N = 1 SCFTs in d = 4 possess, contains important information about the SCFT. For example, the a central charge is fixed by the R-symmetry, as are the anomalous dimensions of (anti-)chiral primary operators [8]. It is also known that the R-symmetry can be identified via the procedure of a-maximization, which, roughly, says that the correct R-symmetry is the one that maximizes the value of a over all possible admissible R-symmetries [9]. For the solutions of type IIB supergravity with Y = S E 5 , the R-symmetry manifests itself as a canonical Killing vector ξ on S E 5 . This defines a Killing vector on the Calabi-Yau cone, also denoted by ξ , which is a real holomorphic vector field. The Calabi-Yau cone is Kähler, and hence symplectic, and Y admits a corresponding contact structure for which ξ is the Reeb vector. When Y = S E 5 the a central charge is inversely proportional to the volume of S E 5 , and in [10,11] several geometric formulae for a in terms of ξ were derived. Analogous geometric formulae for the conformal dimension of the chiral primary operator dual to a D3-brane wrapped on a supersymmetric submanifold 3 ⊂ Y were also presented. Of particular interest here are the formulae that show that using symplectic geometry these quantities can be written as Duistermaat-Heckman integrals on the cone and hence can be evaluated by localization. In addition to providing a geometrical interpretation of a-maximization, these formulae and others in [10,11] also provide practical methods for calculating quantities of physical interest without needing the full explicit Sasaki-Einstein metric (which, apart from some special classes of solution, remains out of reach). The focus of this paper is on Ad S5 × Y solutions with Y more general than S E 5 . Most known solutions are actually part of continuous families of solutions containing a Sasaki-Einstein solution and correspond to exactly marginal deformations of the corresponding SCFT. For example, starting with a toric S E 5 solution one can construct new β-deformed solutions using the techniques of [12]. There is also the “Pilch-Warner solution” explicitly constructed in [13] (based on [14]). It has been shown numerically in [15] that the Z2 orbifold of the Pilch-Warner solution is part of a continuous family of solutions that includes the Sasaki-Einstein Ad S5 × T 1,1 solution. Using the results of [16,17] this should be part of a larger family of continuous solutions that are yet to be found. Similarly, in addition to the β-deformations of the Ad S5 × S 5 solution there are additional deformations [16] that are also not yet constructed (a perturbative analysis was studied in [18]). Having a better understanding of the geometry underlying general Ad S5 × Y solutions could be useful for finding these deformed solutions but more generally could be useful in constructing new solutions that are not connected with Sasaki-Einstein geometry at all. The first detailed analysis of supersymmetric Ad S5 × Y solutions of type IIB supergravity, for general Y with all fluxes activated, was carried out in [19]. The conditions for supersymmetry boil down to a set of Killing spinor equations on Y for two spinors (when Y = S E 5 there is only one such spinor). By analysing these equations a set of necessary and sufficient conditions for supersymmetry were established. In light of the progress summarized above for the Sasaki-Einstein case, it is natural to investigate the associated geometry of the cone over Y , and that is the principal purpose of this paper. As we shall discuss in detail, the cone X admits a specific kind of generalized complex geometry. Aspects of this geometry, restricting to a class of SU(2)-structures, were first studied in [20,21]. By viewing Ad S5 × Y as a supersymmetric warped product R3,1 × X , one sees immediately that the cone admits two compatible generalized almost complex structures [22,23], or equivalently two compatible pure spinors, ± . In fact d− = 0, so that − defines an integrable generalized complex geometry, while d+ is
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related to the RR flux. The cone is thus generalized Hermitian, and it is also generalized Calabi-Yau in the sense of [24]. Here, we will identify a generalized vector field ξ on the cone that is dual to the R-symmetry and another that is dual to the dilatation symmetry of the dual SCFT and show that they are both generalized holomorphic vector fields on the cone (with respect to the integrable generalized complex structure). This precisely generalizes known results for the Sasaki-Einstein case. We also note that all supersymmetric Ad S5 × Y solutions satisfy the condition of [20] that there is an SU(2)-structure on the cone. In the Sasaki-Einstein case, one can carry out a symplectic reduction of the Calabi-Yau cone to obtain a four-dimensional transverse Kähler-Einstein space which, in general, is only locally defined. Constructing locally defined Kähler-Einstein spaces has been a profitable way to construct Sasaki-Einstein manifolds, e.g. [25]. Here we will show, using the formalism of [26,27], that for general Y there is an analogous reduction of the corresponding six-dimensional generalized Calabi-Yau cone geometry to a four-dimensional space, which again is only locally defined in general, that is generalized Hermitian. More precisely, the four-dimensional geometry admits two compatible generalized almost complex structures, one of which is integrable. We present explicit expressions for the pure spinors ± associated with the sixdimensional cone in terms of the Killing spinor bilinears presented in [19]. We shall comment upon how − , associated with the integrable generalized complex structure, contains information on the mesonic moduli space of the dual SCFT and also, briefly, on some relations connected with generalized holomorphic objects and dual BPS operators. By analysing the pure spinor + , associated with the non-integrable complex structure, and focusing on the case when the five-form flux is non-vanishing, we show that, perhaps somewhat surprisingly, the cone is symplectic. We shall see that Y is a contact manifold and that the vector part, ξv , of the generalized vector ξ , which also defines a Killing vector on Y , is the Reeb vector field associated with the contact structure. We show that the symplectic structure can be used to obtain Duistermaat-Heckman type integrals for the central charge a of the dual SCFT and also for the conformal dimensions of operators dual to wrapped BPS D3-branes. Once again these results precisely generalize those for the Sasaki-Einstein case. Some of these results were first presented in [28]; here we will provide additional details and also show how they are related to the generalized geometry on the cone. Finally, we will illustrate some of our results using the Pilch-Warner solution. The paper begins with a review of generalized geometry and it ends with three appendices containing some details about our conventions, some technical derivations, and a brief discussion of the Sasaki-Einstein case. 2. Generalized Geometry We begin by reviewing some aspects of generalized complex geometry [24], to fix conventions and notation. For further details see, for example, [29].
2.1. The generalized tangent and spinor bundles. Generalized geometry starts with the generalized tangent bundle E over a manifold X , which is a particular extension of T X by T ∗ X obtained by twisting with a gerbe. A gerbe is simply a higher degree version of a U (1) bundle with unitary connection. Just as topologically a U (1) bundle is determined by its first Chern class, the topology of a gerbe is determined by a class in
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H 3 (X, Z). To define a gerbe [30], one begins with an open cover {Ui } of X together with a set of functions gi jk : Ui ∩ U j ∩ U j → U (1) defined on triple overlaps. These are required to satisfy gi jk = g jik −1 = gik j −1 = gk ji −1 , together with the cocycle condition g jkl gikl −1 gi jl gi jk −1 = 1 on quadruple overlaps. A connective structure [30] on a gerbe is a collection of one-forms (i j) defined on double overlaps Ui ∩ U j satisfying (i j) + ( jk) + (ki) = −(2π ils2 )gi−1 jk dgi jk on triple overlaps. In turn, a curving is a collection of two-forms B(i) on Ui satisfying B( j) − B(i) = d(i j) .
(2.1)
It follows that dB( j) = dB(i) = H is a closed global three-form on X , called the curvature, and, in cohomology, (2πl1 )2 H ∈ H 3 (X, Z) (in the normalization that we shall use s in this paper). In string theory, the collection of two-forms B(i) , which we write simply as B, is the NS B-field and H is its curvature. The generalized tangent bundle E is an extension of T X by T ∗ X, π
0 −→ T ∗ X −→ E −→ T X −→ 0.
(2.2)
Locally, sections of E, which we refer to as generalized tangent vectors, may be written as V = x + λ, where x ∈ (T X ) and λ ∈ (T ∗ X ). More precisely, in going from one coordinate patch Ui to another U j the extension is defined by the connective structure (2.3) x(i) + λ(i) = x( j) + λ( j) − i x( j) d(i j) . The bundle E is in fact isomorphic to T X ⊕ T ∗ X . However, the isomorphism is not canonical but depends on a choice of splitting, defined by a two-form curving B satisfying (2.1). It follows that x + (λ − i x B) ∈ (T X ⊕ T ∗ X ).
(2.4)
Thus the definition (2.3) of E can be viewed as encoding the patching of a class of two-form curvings B. Writing d = dimR X , there is a natural O(d, d)-invariant metric ·, · on E, given by V, W = 21 (i x μ + i y λ), where V = x + λ, W = y + μ, or in two-component notation, 0 21 y . V, W = (x λ) 1 μ 0 2
(2.5)
(2.6)
This metric is invariant under O(d, d) transformations acting on the fibres of E, defining a canonical O(d, d)-structure. A general element O ∈ O(d, d) may be written in terms of d × d matrices a, b, c, and d as a b , (2.7) O= c d under which a general element V ∈ E transforms by x a b x .
→ O V = V = λ c d λ
(2.8)
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The requirement that O V , O V = V, V implies a T c + c T a = 0, b T d + d T b = 0 and a T d + c T b = 1. Note that the G L(d) action on the fibres of T X and T ∗ X embeds as a subgroup of O(d, d). Concretely it maps a 0 x V → V = , (2.9) λ 0 a −T where a ∈ GL(d). Given a two-form ω, one also has the abelian subgroup 1 0 ω e = such that V = x + λ → V = x + (λ − i x ω). ω 1
(2.10)
This is usually referred to as a B-transform. Given a bivector β one can similarly define another abelian subgroup of β-transforms 1 β eβ = such that V = x + λ → V = (x + i λ β) + λ. (2.11) 0 1 Note that the patching (2.3) corresponds to a B-transform with ω = d(i j) . Similarly, the splitting isomorphism between E and T X ⊕ T ∗ X defined by B is also a B-transform eB
E T X ⊕ T ∗ X. e−B
(2.12)
There is a natural bracket on generalized vectors known as the Courant bracket, which encodes the differentiable structure of E. It is defined as [V, W ] = [x + λ, y + μ] = [x, y]Lie + Lx μ − L y λ − 21 d i x μ − i y λ , (2.13) where [x, y]Lie is the usual Lie bracket between vectors and Lx is the Lie derivative along x. The Courant bracket is invariant under the action of diffeomorphisms and B-shifts ω that are closed, dω = 0, giving an automorphism group which is a semidirect product Diff(X ) 2closed (X ). Note, however, that in string theory only B-shifts by the curvature of a unitary line bundle on X are gauge symmetries, as opposed to shifts by arbitrary closed two-forms, leading to a smaller automorphism group. Under an infinitesimal diffeomorphism generated by a vector field x and a B-shift with ω = dλ, one has the generalized Lie derivative by V = x + λ on a generalized vector field W = y + μ, δW ≡ LV W = [x, y]Lie + (Lx μ − i y dλ).
(2.14)
This is also known as the Dorfman bracket [V, W ] D , the anti-symmetrization of which gives the Courant bracket (2.13). Note that since the metric ·, · is invariant under O(d, d) transformations its generalized Lie derivative vanishes. Given a particular choice of splitting (2.1) defined by B, the Courant bracket on E defines a Courant bracket on T X ⊕ T ∗ X , known as the twisted Courant bracket. It is given by [x + λ, y + μ] H = e B [e−B (x + λ), e−B (y + μ)] = [x + λ, y + μ] + i y i x H,
(2.15)
where by an abuse of notation we are writing x + λ and y + μ for sections of T X ⊕ T ∗ X, whereas above they were sections of E. Given the metric ·, · , one can define Spin(d, d) spinors in the usual way. Since the volume element in Cliff(d, d) squares to one, one can define two helicity spin bundles
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S± (E) as the ±1 eigenspaces, and thus take spinors to be Majorana-Weyl. A section of S± (E) on Ui can be identified with a even- or odd-degree polyform ± ∈ even/odd (X ) restricted to Ui , with the Clifford action of V ∈ (E) given by V · ± = i x ± + λ ∧ ± .
(2.16)
(V · W + W · V ) · ± = 2V, W ± ,
(2.17)
It is easy to see that
as required. Using this Clifford action the B-transform (2.10) on spinors is given by ± → eω ± ,
(2.18)
where the exponentiated action is by wedge product. The patching (2.3) of E then implies that ( j)
d(i j) (i) ± . ± =e
(2.19)
Furthermore a splitting B also induces an isomorphism between S± (E) and S± (T X ⊕ T X ∗ ), eB
S± (E) S± (T X ⊕ T ∗ X ),
(2.20)
e−B
again by the action of the exponentiated wedge product. If ± is a section of S± (E), B ≡ e B for the corresponding section of S (T X ⊕ T ∗ X ) we will sometimes write ± ± ± defined by the splitting B. The real Spin(d, d)-invariant spinor bilinear on sections of S± (E) is a top form given by the Mukai pairing ± , ± ≡ (± ∧ λ( ± ))top ,
(2.21)
where one defines the operator λ λ( m± ) ≡ (−1)Int[m/2] m± ,
(2.22)
with m the degree m form in ± . The Mukai pairing is invariant under B-transforms: eω ± , eω ± = ± , ± . For d = 6 the bilinear is anti-symmetric. The usual action of the exterior derivative on the component forms of ± is compatible with the patching (2.19) and defines an action d : S± (E) → S∓ (E),
(2.23)
while the generalized Lie derivative on spinors is given by LV ± = Lx ± + dλ ∧ ± = d(V · ± ) + V · d± . Note that given a splitting B the operator on is d H defined by
B ±
∈ S± (T X
⊕ T ∗X)
B B B d H ± ≡ e B d(e−B ± ) = (d − H ∧ ) ± ,
(2.24)
corresponding to d (2.25)
where H = dB. Furthermore one has
LV = e−B LV B − i x H ∧ B ,
where V B = e B V = x + (λ − i x B).
(2.26)
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Finally, we note that there is actually a slight subtlety in the relation between generalized spinors and polyforms. Given the embedding (2.9) in O(d, d) of the GL(d) action on the fibres of T X one actually finds that the Clifford action (2.16) implies that on Ui we can identify S± (E) with |d T ∗ X |−1/2 ⊗ even/odd T ∗ X ; that is, there is an additional factor of the determinant bundle |d T ∗ X |. (This factor is the source, for instance, of the fact that the Mukai pairing is a top form, rather than a scalar.) This bundle is trivial, so generalized spinors can indeed be written as polyforms patched by (2.19), but there is no natural isomorphism to make this identification. The simplest solution, and one which will also allow us to incorporate the dilaton in a natural way, is to extend the O(d, d) action to a conformal action O(d, d) × R+ . One can then define a family of spinor k (E) transforming with weight k under the conformal factor R+ ; that is, with bundles S± sections transforming as ± → ρ k ± , where ρ ∈ R+ . If one embeds the GL(d) action on T X in O(d, d) as in (2.9) and, in addition, makes a conformal scaling by ρ = det a −1/2 then sections of S± (E) can be directly identified with polyforms patched by (2.19). 2.2. Generalized metrics and complex structures. A generalized metric G on E is the generalized geometrical equivalent of a Riemannian metric on T X . We have seen that there is a natural O(d, d) structure on E defined by the metric ·, · (2.5). The generalized metric G defines an O(d) × O(d) substructure. It splits E = C+ ⊕ C− such that the metric ·, · gives a positive-definite metric on C+ and a negative-definite metric on C− , corresponding to the two O(d) structure groups. One can define G as a product structure on E; that is, G : E → E with G 2 = 1 and GU , GV = U, V , so that 21 (1 ± G) project onto C± . In general G has the form g −1 1 0 g −1 B 1 0 0 g −1 = G= , (2.27) −B 1 B 1 g 0 g − Bg −1 B −Bg −1 where g is a metric on X and B is a two-form. The patching of E implies B satisfies (2.1), so that B may be identified with the curving of the gerbe used in the twisting of E. Thus the generalized metric G defines a particular splitting of E. In particular, we see from (2.27) that G = e−B G 0 e B , where G 0 is a generalized metric on T X ⊕ T ∗ X defined by g. The generalized metric G naturally encodes the NS fields g and B as the coset space O(d, d)/O(d) × O(d). The dilaton φ appears when one considers the conformal group O(d, d) × R+ , used to define the generalized spinors as true polyforms. To define a O(d) × O(d) substructure in O(d, d) × R+ , in addition to G which gives the embedding in the O(d, d) factor, one must give the embedding ρ in the conformal factor d ρ ∈ R+ . Recall that under diffeomorphisms ρ transforms as a section √ of T X . Given 2φ the metric g we can define the generic embedding by ρ = e / g for some positive function e2φ , which we identify as the dilaton. Note that ρ is by definition invariant under O(d, d) and so one finds the conventional T-duality transformation of the dilaton under O(d, d). Under the generalized Lie derivative, LV G = 0 implies [31] Lx g = 0,
Lx B − dλ = 0,
(2.28)
so that Lx H = 0, where H = dB. Such a V is called a generalized Killing vector. Given G we may decompose generalized spinors in Spin(d, d) under Spin(d) × Spin(d). In fact one can go further. Using the projection π : E → T X the two Spin(d)
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groups can be identified and the generalized spinors may be decomposed as bispinors of Spin(d): ± = e−φ e−B even/odd .
(2.29)
In this expression, one first uses the Clifford map to identify the bispinors with a generalized spinor even/odd of S± (T X ⊕ T ∗ X ) ∼ = even/odd T ∗ X and then uses the splitting B to map to a spinor of S± (E). The factor of e−φ appears because the polyforms are really −1/2 sections of S± (E) transforming with weight − 21 under conformal rescalings. Explicitly, if is a bispinor, ∈ ∗ (X ) a polyform, and γ i are Spin(d) gamma matrices, the Clifford map is =
1 i ···i γ i1 ···ik k! 1 k
←→
k
=
1 i ···i dx i1 ∧ · · · ∧ dx ik . k! 1 k
(2.30)
k
The Cliff(d, d) action is realized via left and right multiplication by the gamma matrices γ i . For the chiral spinors ± the sum is over k even/odd respectively. We also note here the Fierz identity =
1 1 Tr γik ···i1 γ i1 ···ik , nd k!
(2.31)
k
where the γ i are n d × n d matrices. Finally, the generalized metric also defines an action G on generalized spinors which is the analogue of the Hodge star. It is given by ± → G ± = e−B λ(e B ± ),
(2.32)
where λ is the operator defined in (2.22) and denotes the ordinary Hodge star for the metric g. If d = 2n one can also introduce a generalized almost complex structure on E. This is a map J : E → E with J 2 = −1 and J U , J V = U, V and gives a decomposition ¯ E C = L ⊕ L,
(2.33)
where L denotes the +i eigenspace of J . Note that L is maximally isotropic: U, V = J U , J V = iU , iV = −U, V = 0. This defines a U (n, n) ⊂ O(2n, 2n) structure on E. By definition U, J V + J U , V = 0, so J can be viewed either as an element of O(2n, 2n) or of the Lie algebra o(2n, 2n). A generic J can be written locally as I P J = , (2.34) Q −I ∗ where I ∗ is the linear map on T ∗ X dual to the map I on T X , P is a bivector and Q is a two-form. If the twisting (2.3) is trivial, so E = T X ⊕ T ∗ X , there are two canonical examples of generalized almost complex structures. The first is an ordinary almost complex structure I on T X , for which1 I 0 . (2.35) J1 = 0 −I ∗ 1 Note that we have chosen the opposite sign in (2.35) compared with [29]. This is so that the +i eigenspace is identified with T (1,0) ⊕ T ∗(0,1) .
Ad S5 Solutions of Type IIB Supergravity and Generalized Complex Geometry
The second is a non-degenerate (stable) two-form ω, for which 0 ω−1 J2 = . −ω 0
373
(2.36)
If dω = 0 this corresponds to a symplectic structure. More generally, a generalized almost complex structure J is integrable if L is closed under the Courant bracket. That is, given U, V ∈ (L) then [U, V ] ∈ (L). In the above two cases (2.35), (2.36), this reduces to integrability of I and the closure of ω, respectively. A generalized almost complex structure is equivalent to (the conformal class of) a pure spinor , which simply means a chiral complex generalized spinor such that the annihilator L = {U ∈ E C : U · = 0}
(2.37)
is maximal isotropic. The sub-bundle L defined by J is then identified with L . Notice that L is invariant under conformal rescalings → f , for any function f . A generalized almost complex structure is therefore more precisely equivalent to the pure spinor line bundle generated by . Integrability of J can be expressed as the condition d = V · for some V ∈ (E). If one can find a nowhere vanishing globally defined then one has an SU (n, n) structure and if in addition d = 0 then one has a generalized Calabi-Yau structure in the sense2 of [24]. For example, in the complex ¯ (n,0) , where (n,0) is the holomorphic (n, 0)-form structure case (2.35) one has = c ¯ (n,0) appears, rather than (n,0) , is directly and c is a non-zero constant (the reason why related to the comment in footnote 1). A generalized vector V = x +λ is called (real) generalized holomorphic if LV J = 0. Equivalently, LV preserves the spinor line bundle generated by the corresponding pure spinor ; that is, LV = f for some function f . Given a splitting B, one can define the corresponding generalized complex objects on T X ⊕ T ∗ X . In particular, if J is the generalized almost complex structure for a pure spinor , then the corresponding generalized almost complex structure on T X ⊕ T ∗ X is defined in terms of the annihilator of B = eB
(2.38)
J B ≡ e B J e−B .
(2.39)
and is given by
In particular, integrability of J is equivalent to integrability of J B using the twisted Courant bracket (2.15), or equivalently d H B = V · B . Viewing J as a Lie algebra element one can define its action on generalized spinors via the Clifford action [32]. Explicitly, one has J· =
1 Q mn dx m ∧ dx n ∧ + I m n [i ∂m , dx n ∧ ] + P mn i ∂m i ∂n . 2
(2.40)
Note that for any generalized vector V one has, under the Clifford action, [J ·, V ·] = (J V )·. One can also define the operator Jh : S± (E) → S± (E), Jh ≡ e 2 π J · , 1
2 Note that a different definition is used in [29].
(2.41)
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which is the spinor space representation of J as an element of the group Spin(d, d). If n is even and the pure spinor is a section of S± (E), then Jh defines a complex structure on S∓ (E), while if n is odd it defines a complex structure on S± (E). Observe that for any generalized vector V we have the Clifford action identity Jh · V · Jh−1 · = (J V )· . Finally, a pair of generalized almost complex structures J1 and J2 are said to be compatible if [J1 , J2 ] = 0,
(2.42)
G = −J1 J2
(2.43)
and the combination
is a generalized metric. If 1 and 2 are the corresponding pure spinors, (2.42) is equivalent to J1 · 2 = J2 · 1 = 0. An example of a pair of compatible pure spinors is (2.35), (2.36), with the compatibility condition being that I ki ω jk = gi j is positive definite. Note this is ωi j = −gik I kj , this mathematics convention differing by a sign to the usual physics convention. A pair of compatible almost complex structures defines an SU (n) × SU (n) structure. A generalized Kähler structure is an SU (n) × SU (n) structure where both generalized almost complex structures are integrable, while for a generalized Hermitian structure only one need be integrable. Note that an SU (n) × SU (n) structure can equivalently be specified by a generalized metric and a pair of chiral Spin(2n) spinors. For example, for d = 6 a pair of chiral spinors η+1 , η+2 can be used to construct an SU (3) × SU (3) structure given by + = e−φ e−B η+1 η¯ +2 ,
2 − = e−φ e−B η+1 η¯ − ,
(2.44)
2 ≡ (η2 )c . This will play a central role in the following sections. Similarly, for with η− + d = 4 a pair of chiral spinors η+1 , η+2 give rise to an SU (2) × SU (2) structure specified by two compatible pure spinors, but both of them consist of sums of even forms, since now (η+2 )c is a positive chirality spinor. We will see such an SU (2) × SU (2) structure in Sect. 4. That the spinors have the same chirality is necessary for them to be compatible in four dimensions [33].
3. Ad S5 Backgrounds as Generalized Complex Geometries 3.1. Supersymmetric Ad S5 backgrounds. Our starting point is the most general class of supersymmetric Ad S5 solutions of type IIB supergravity, as studied in [19]. The ten-dimensional metric in Einstein frame is g E = e2 (g Ad S + gY ) ,
(3.1)
where gY is a Riemannian metric on the compact five-manifold Y , and is a real function on Y . The Ad S5 metric g Ad S is normalized to have unit radius, so that Ricg Ad S = −4 g Ad S .
(3.2)
The ten-dimensional string frame metric is defined to be gσ ≡ eφ/2 g E . In addition to the metric, there is the dilaton φ and NS three-form H ≡ dB in the NS sector, and the forms
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F ≡ F1 + F3 + F5 in the RR sector. The RR fluxes Fn are related to the RR potentials Cn via F1 = dC0 , F3 = dC2 − H C0 , F5 = dC4 − H ∧ C2 .
(3.3) (3.4) (3.5)
These are all taken to be forms on Y , so as to preserve the S O(4, 2) symmetry, with the exception of the self-dual five-form F5 which necessarily takes the form F5 = f vol Ad S − v (3.6) olY , where f is a constant. Here v olY denotes a volume form for (Y, gY ). It is related by Y = −vol5 to the volume form of [19], where the latter was given in terms of an vol orthonormal frame as vol5 = e12345 and used to define, for instance, the Hodge star. Y In turns out that, in the Sasaki-Einstein limit, the conventional volume form is vol rather than vol5 and so here we will use the former throughout. In particular, it is the orientation that we will use when defining integrals over Y . In [19] the conditions for a supersymmetric Ad S5 background were written in terms of two five-dimensional spinors ξ1 , ξ2 on Y , giving the system of equations reproduced here in (A.1)–(A.6) of Appendix A. Various spinor bilinears involving ξ1 and ξ2 were also introduced, and used to determine the necessary and sufficient conditions for supersymmetry. For example, it was shown that A ≡ 21 ξ¯1 ξ1 + ξ¯2 ξ2 = 1, (3.7) Z ≡ ξ¯2 ξ1 = 0. It will be useful for later in this paper to recall the definitions of the following scalar bilinears: sin ζ ≡ 21 ξ¯1 ξ1 − ξ¯2 ξ2 , (3.8) S ≡ ξ¯2c ξ1 , the one-form bilinears: K ≡ ξ¯1c β(1) ξ2 , K 3 ≡ ξ¯2 β(1) ξ1 , K 4 ≡ 21 ξ¯1 β(1) ξ1 − ξ¯2 β(1) ξ2 , K 5 ≡ 1 ξ¯1 β(1) ξ1 + ξ¯2 β(1) ξ2 ,
(3.9)
i V = − (ξ¯1 β(2) ξ1 − ξ¯2 β(2) ξ2 ), 2 W = −ξ¯2 β(2) ξ1 .
(3.10)
2
and the two-form bilinears:
Here the βm generate the Clifford algebra for gY , so {βm , βn } = 2gY mn . Equivalently, with respect to any orthonormal frame, we write βˆm with {βˆm , βˆn } = 2δmn . We have also introduced the notation β(k) ≡ k!1 βm 1 ···m k dx m 1 ∧ · · · ∧ dx m k .
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A key result of [19] is that K 5# , the vector dual to the one-form K 5 , defines a Killing vector that preserves all of the fluxes. This was identified as corresponding to the R-symmetry in the dual SCFT. Another important result was e−4 f = 4 sin ζ.
(3.11)
The Killing spinors ξ1 , ξ2 were used to introduce a canonical five-dimensional orthonormal frame in Appendix B of [19], which is convenient for certain calculations. We will refer to that paper for further details. Finally, we note that Eq. (A.22) of Appendix A may be used to obtain expressions for the two-form potentials B and C2 in terms of the bilinear W introduced in (3.10): 4 B = − e6+φ/2 Re W + b2 , f 4 6+φ/2 4 C2 = − e C0 Re W − e6−φ/2 Im W + c2 . f f
(3.12) (3.13)
Here b2 and c2 are real closed two-forms. Notice that the first term in B in (3.12) is a globally defined two-form, and thus H = dB is exact. It follows that [H ] = 0 ∈ H 3 (Y, R), although notice that b2 may be taken to be globally defined if and only if the torsion class of H is zero in H 3 (Y, Z) (which for simplicity we shall assume). Similar remarks apply to C2 (up to large gauge transformations of C0 ). 3.2. Reformulation as generalized complex geometries. The supersymmetric Ad S5 geometry described above can be simply reformulated in terms of generalized complex geometry, as in the discussion of [20,21]. The basic observation is simply that these solutions can be viewed as warped products of flat four-dimensional space with a sixdimensional manifold X , satisfying a set of supersymmetry conditions that imply the existence of a particular generalized complex geometry [22,23]. As we shall explain in more detail below, combining this structure with the existence of the Killing vector K 5# precisely generalizes the correspondence between Sasaki-Einstein geometry and Calabi-Yau cone geometry. In the following we analyze this reformulation in detail. We find in particular that all supersymmetric Ad S5 solutions necessarily satisfy the condition of [20] that there is an SU(2)-structure on X . One begins by rewriting the unit Ad S5 metric in a Poincaré patch as g Ad S =
dr 2 + r 2 gR3,1 . r2
(3.14)
Switching to the string frame, we can consider (3.1) as a special case of a warped supersymmetric R3,1 solution of the form gσ = e2 A gR3,1 + g6 ,
(3.15)
where the warp factor is given by e2 A = e2+φ/2 r 2 ,
(3.16)
and the six-dimensional metric is given by g6 = e2+φ/2
dr 2 . + g Y r2
(3.17)
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We also define the six-dimensional volume form as vol6 ≡ e12+3φ r 5 dr ∧ v olY .
(3.18)
Notice that the six-dimensional manifold X , on which g6 is a metric, is a product R+ ×Y , where r may be interpreted as a coordinate on R+ . In particular, X is non-compact. It thus follows that supersymmetric Ad S5 solutions are special cases of supersymmetric R3,1 solutions. In [23] the general conditions for an N = 1 supersymmetric R3,1 background, in the string frame metric (3.15), were written in terms of two chiral six-dimensional spinors η+1 , η+2 on X , namely the system of equations given here in (A.17)–(A.20). The relation between the two sets of Killing spinors, for Ad S5 solutions, is given by first decomposing Cliff(6) into Cliff(5) via γˆm = βˆm ⊗ σ3 , γˆ6 = 1 ⊗ σ1 ,
m = 1, . . . , 5,
(3.19)
where γˆi , i = 1, . . . , 6, generate Cliff(6) and σα , α = 1, 2, 3, denote the Pauli matrices. Changing basis to ξ1 = χ1 + iχ2 , ξ2 = χ1 − iχ2 , we then have c χ1 1 A/2 1 A/2 −χ1 η+ = e , η− = e , iχ1 iχ1c c (3.20) −χ2 χ2 2 A/2 2 A/2 , η− = e , η+ = e −iχ2 −iχ2c i ≡ (ηi )c ≡ where χic ≡ D˜ 5 χi∗ denotes 5D charge conjugation, and correspondingly η− + i ∗ D6 (η+ ) where D6 = D˜ 5 ⊗ σ2 . For further details, see Appendix A. Using the two chiral spinors η1+ , η2+ we may define the bispinors
+ ≡ η+1 ⊗ η¯ +2 ,
2 − ≡ η+1 ⊗ η¯ − .
(3.21)
Notice that, in the conventions of Appendix A, we have A6 = 1, so that η¯ ≡ η† is just the Hermitian conjugate. Via the Clifford map (2.30) the bispinors for Spin(6) in (3.21) may also be viewed as elements of ∗ (X, C). We will mainly tend to think of ± as complex differential forms of mixed degree. These are then Spin(6, 6) spinors, as explained in Sect. 2.2. In fact ± in (3.21) are both pure spinors, and also compatible. They then define an SU (3) × SU (3) structure on T X ⊕ T ∗ X . In terms of (3.21), the Killing spinor equations for a general supersymmetric R3,1 solution (i.e. not necessarily associated with an Ad S5 solution, but with vanishing fourdimensional cosmological constant) may be rewritten as [34] (see also [22]) (3.22) d H e2 A−φ − = 0, ¯+ d H e2 A−φ + = e2 A−φ d A ∧ +
1 2A 2 (|a| − |b|2 )F + i(|a|2 + |b|2 ) λ(F) . (3.23) e 16
Here recall that F = F1 + F3 + F5 is the sum of RR fields and from (2.22), λ(F) = F1 − F3 + F5 .
(3.24)
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Note that the Hodge star is with respect to the metric g6 , with positive orientation given Y . The remaining Bianchi identities and equations of motion are (cf. [34] by dr ∧ vol Eq. (4.9)–(4.10)) dH = 0 , d H F = δsource , d(e4 A−2φ H ) − e4 A Fn ∧ Fn+2 = 0, (d + H ∧ )(e4 A F) = 0. The equation of motion for F can also be written as
d e4 A e−B λ (F) = 0,
(3.25) (3.26) (3.27)
(3.28)
and follows from the supersymmetry equations. In fact, for Ad S5 solutions it was shown in [19] that supersymmetry implies all of the equations of motion and Bianchi identities. We have also introduced the spinor norms |a|2 = |η1+ |2 ,
|b|2 = |η2+ |2 ,
(3.29)
which for a supersymmetric R3,1 background must satisfy |a|2 + |b|2 = e A c+ ,
|a|2 − |b|2 = e−A c− ,
(3.30)
where c± are constants. Upon squaring and subtracting the equations one obtains 1 1 2A 2 2 |± |2 = |a|2 |b|2 = e c+ − e−2 A c− . (3.31) 8 32 As we now show, for the particular case of Ad S5 solutions the above equations simplify somewhat. In this case it is possible to fix the constant c− in (3.31) by the scaling of ± with r which, using (3.16), implies that c− = 0 and hence |η1+ | = |η2+ |. This is 1 | = |η2 | consistent with the equation Z = 0 in (3.7), since from (3.20) we see that |η± ± is equivalent to Re Z = 0. Notice that c− = 0 is also a necessary condition in order to have supersymmetric probe branes [35]. The normalization that was used in [19] implies |a|2 = |b|2 = e A and hence c+ = 2. One can actually go a little further. In [20] it was assumed that there was an SU(2)-structure on the cone. In terms of the spinors ηi+ this is equivalent to the condition that, in addition to c− = 0, one has η¯ 1+ η2+ + η¯ 2+ η1+ = 0.
(3.32)
However it is easy to see that this is equivalent to Im Z = 0, which again is required by supersymmetry on Y . Thus in fact all supersymmetric Ad S5 solutions necessarily satisfy the SU(2) condition of [20]. We now define the pure spinor B − ≡ e2 A−φ − ,
(3.33)
B = 0. The associated generalized almost complex which by (3.22) is d H closed, d H − B structure J− is then integrable with respect to the twisted Courant bracket (2.15). We also define B − ≡ e−B − = e−B e2 A−φ − ,
(3.34)
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which is closed under the usual exterior derivative: d− = 0.
(3.35)
The associated generalized almost complex structure, which we denote by J− ,3 is then integrable. Combined with the fact that the norm of − , and hence of − , is nowhere vanishing, this means, in particular, that we have a generalized Calabi-Yau manifold in the sense of [24]. We similarly define + ≡ e−B e2 A−φ + .
(3.36)
However, the corresponding generalized almost complex structure J+ is not integrable in general, its integrability being obstructed by the RR fields in (3.23). If it were integrable, we would have a generalized Kähler manifold. With these definitions we can write the supersymmetry equations for Ad S5 solutions as d− = 0, i ¯ + + e3A e−B λ (F) . d+ = d A ∧ 8
(3.37)
It is worth noting that the latter equation may also be written as d e−A Re + = 0, 1 d e A Im + = e4 A e−B λ (F) , 8
(3.38) (3.39)
and that in turn Eq. (3.39) can be written as [36] e−B F = 8J− · d e−3A Im + = 8dJ− e−3A Im + ,
(3.40)
where dJ− ≡ −[d, J− ·]. For most of the paper we will demand that F5 = 0, or equivalently f = 0. Physically this corresponds to having non-vanishing D3-brane charge. It would be interesting to know whether or not all supersymmetric Ad S5 solutions of type IIB supergravity have this property.
3.3. Canonical vector fields. In this section we examine the geometric properties of the generalized vector fields r ∂r , ξ ≡ J− (r ∂r ) and η ≡ J− (d log r ). As in the Sasaki-Einstein case, r ∂r and ξ correspond respectively to the dilatation symmetry and the R-symmetry in the dual SCFT (while η is related to a contact structure on Y , as we shall show later in Sect. 6). 3 The generalized complex structures J B and J are related by (2.39). − −
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3.3.1. Dilatation symmetry. We begin with the dilatation vector field r ∂r . It immediately follows from (3.16), (3.20) and (3.21) that Lr ∂r ± = ± ,
(3.41)
Lr ∂r ± = 3± .
(3.42)
and therefore
This follows since e2 A has scaling dimension 2 (3.16), and both the B-field and the dilaton φ are pull-backs from Y . Notice that Eq. (3.42) may also be trivially rewritten in terms of the generalized Lie derivative (2.24): Lr ∂r ± = 3± .
(3.43)
Lr ∂r J± = 0.
(3.44)
This implies that
To see this, recall that J± is defined by saying that its +i eigenspace is equal to the annihilator L ± of ± , and the latter is clearly preserved under the one-parameter family of (generalized) diffeomorphisms generated by r ∂r . It further follows that Lr ∂r G = 0, where G is the generalized metric G = −J+ J− = −J+ J− , so that r ∂r is generalized Killing. Equation (3.44) says that r ∂r is a (real) generalized holomorphic vector field for the integrable generalized complex structure J− . We shall not use this terminology for J+ , since the latter is not in general integrable. Clearly, this generalizes the SasakiEinstein result where the cone is Calabi-Yau and the dilatation vector r ∂r is holomorphic. 3.3.2. R-symmetry. We next define the generalized vectors ξ ≡ J− (r ∂r ), η ≡ J− (d log r ),
(3.45) (3.46)
which are, in general, a mixture of vectors and one-forms. Recall that the generalized almost complex structures J± are related to the generalized metric via G = −J+ J− = −J− J+ . The conical form (3.17) of the metric g6 and the fact that B has no component φ
φ
along dr implies that G d log r = e−2− 2 r ∂r , G r ∂r = e2+ 2 d log r , and hence in addition to (3.45) and (3.46) we may also write φ
ξ = e2+ 2 J+ (d log r ), η=e
φ −2− 2
(3.47)
J+ (r ∂r ).
We may split ξ and η into a vector part and a one-form part, in a fixed splitting of E, ξ = ξv + ξ f , η = ηv + η f .
(3.48) (3.49)
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By carrying out a calculation, presented in Appendix B, we may then write these as bilinears constructed from the five-dimensional Killing spinors (3.9): ξv = K 5# , ξ f = i ξv b2 , ηv = e−2−φ/2 Re K 3# , 4 η f = e4 K 4 + i ηv b2 . f
(3.50)
As discussed in Appendix B, it is the B-transform, ξ B , of the generalized vector ξ that is naturally related to the bilinears of [19]. We have obtained (3.50) by performing an inverse B-transform using the expression for the B-field given in terms of bilinears presented in (3.12). In particular, this is where the closed two-form b2 appears. Since the B-transform of b2 by an exact form is a generalized diffeomorphism, and a gauge symmetry of string theory, we see that the physical information in b2 is represented by its cohomology class in H 2 (X, R). More precisely, large gauge transformations of the B-field, which correspond to tensoring the underlying gerbe by a unitary line bundle on X , lead to the torus H 2 (X, R)/H 2 (X, Z) (with suitable normalization). Turning on the two-form b2 corresponds to giving vacuum expectation values to moduli (of the NS field B) and so is a symmetry of the supersymmetry equations. It is therefore left undetermined. In the field theory dual, the cohomology class of b2 thus corresponds to a marginal deformation. In [19] it was shown that K 5# is a Killing vector that preserved all of the fluxes, and thus K 5# was identified as being dual to the R-symmetry in the SCFT. In the generalized geometry we can show the stronger conditions that Lξ J± = 0,
(3.51)
and hence ξ is a generalized holomorphic Killing vector field. In fact it is straightforward to show Lξ − = −3i− and hence Lξ J− = 0. Indeed since d− = 0 and r ∂r − iξ ∈ L − annihilates − , using (2.24) and (3.43) we have Lξ − = d (ξ · − ) = −id (r ∂r · − ) = −iLr ∂r − = −3i− .
(3.52)
In Appendix B we show that Lξ + = 0 and hence Lξ J+ = 0. There we also show that Lξ (e−B F) = 0.
(3.53)
Thus, we have established that ξ ≡ J− (r ∂r ) is a generalized holomorphic vector field, which moreover is generalized Killing for the generalized metric G = −J− J+ , and also preserves the RR fluxes. Again, this clearly generalizes the Sasaki-Einstein result, where ξ = I (r ∂r ) is a holomorphic Killing vector field for the Calabi-Yau cone. To conclude this section we note that when f = 0 the vector field ξv = K 5# is nowhere vanishing on Y = {r = 1}. One can see this from the formula |K 5# |2 = sin2 ζ + |S|2 ,
(3.54)
and using (3.11). Thus for f = 0, ξv acts locally freely on Y and hence the orbits of ξv define a corresponding one-dimensional foliation of Y . This is again precisely as in the Sasaki-Einstein case (although in the Sasaki-Einstein case the norm of ξv is constant).
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4. Generalized Reduction of AdS5 Backgrounds Recall that in the Sasaki-Einstein case one can consider the symplectic reduction of the Calabi-Yau cone metric with respect to the R-symmetry Killing vector ξ (or equivalently a holomorphic quotient with respect to r ∂r − iξ ). Generically ξ does not define a U (1) fibration and the four-dimensional reduced space is not a manifold. Nonetheless, locally one can consider the geometry on the transversal section to the foliation formed by the orbits of ξ in the Sasaki-Einstein space. The result of the reduction is that this four-dimensional geometry is Kähler-Einstein. Thus locally one can always write the Sasaki-Einstein metric as gY = η ⊗ η + gKE ,
(4.1)
where gKE is a Kähler-Einstein metric. The existence of the generalized holomorphic vectors ξ and r ∂r in the generic case suggests one can make an analogous generalized reduction to four dimensions. In this section, we show that this is indeed the case following the theory of generalized quotients developed in [26,27]. We first review the formalism and then apply it to our particular case, showing that there is a generalized Hermitian structure on the local transversal section, giving the conditions satisfied by the corresponding reduced pure spinors. 4.1. Generalized reductions. We will follow the description of generalized quotients given in [27]4 . These include both symplectic reductions and complex quotients as special cases. One first needs to introduce the reduction data. In conventional geometry, the action of a Lie group G on M is generated infinitesimally by a set of vector fields, defined by a map from the Lie algebra ψ : g → (T M). Given a vector field x ∈ (T M), the infinitesimal action of u ∈ g is then just the Lie derivative (or in this case Lie bracket) δx = Lψ(u) x = [ψ(u), x].
(4.2)
One requires that given u, v ∈ g, one has [ψ(u), ψ(v)] = ψ([u, v]) so that Lψ(u) Lψ(v) − Lψ(v) Lψ(u) = L[ψ(u),ψ(v)] = Lψ([u,v]) ,
(4.3)
and thus there is a Lie algebra homomorphism between g and the algebra of vector fields under the Lie bracket. In generalized geometry, we have a larger group of symmetries, diffeomorphisms and B-shifts, which are generated infinitesimally by the generalized Lie derivative (2.14). Thus given an action of G on M, it is natural to consider the infinitesimal “lifted action” of G on E defined by the map ψ˜ : g → (E), such that for any V ∈ (E) and u ∈ g we have δV = Lψ(u) V, ˜
(4.4)
and under the projection π : E → T M we simply get the vector fields ψ(u), that is ˜ π ψ(u) = ψ(u).
(4.5)
4 Note that the bracket [[, ]] used in [27] is the Dorfman bracket or generalized Lie derivative [[V, W ]] = LV W and is not anti-symmetric.
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Such transformations are infinitesimal automorphisms of E that have the property that they preserve both the metric ·, · on E and the Courant bracket (2.13). If we again assume that Lψ(v) − Lψ(v) Lψ(u) = Lψ([u,v]) , Lψ(u) ˜ ˜ ˜ ˜ ˜
(4.6)
then ψ˜ defines an equivariant structure on E. (Note that this is equivalent to the Courant ˜ ˜ ˜ bracket condition [ψ(u), ψ(v)] = ψ([u, v]).) In what follows it will also be assumed that ψ˜ is isotropic, that is ˜ 1 ), ψ(u ˜ 2 ) = 0 ψ(u
(4.7)
for all u 1 , u 2 ∈ g. One can actually define a more general action on E which is a homomorphism between algebras with Courant brackets rather than Lie algebras. One starts by extending g to a larger algebra. The construction considered in [27] which is relevant for us is as follows. Let h be a vector space on which there is some representation of g. Then we can form a “Courant algebra” a = g ⊕ h with Courant bracket5 given u i ∈ g and wi ∈ h, [(u 1 , w1 ), (u 2 , w2 )] = ( [u 1 , u 2 ], 21 (u 1 · w2 − u 2 · w1 ) ),
(4.8)
h∗
where u · w is the action of g on h. Suppose in addition μ : M → is a g-equivariant map, meaning Lψ(u) μ(w) = μ(u · w) for all u ∈ g and w ∈ h. Then, given some isotropic lifted action ψ˜ of g, one can then define the extended action : g ⊕ h → (E), ˜ (u, w) → ψ(u) + dμ(w),
(4.9)
which, it is easy to show, has the property that [ (u 1 , w1 ), (u 2 , w2 )] = ([u 1 , u 2 ], 21 (u 1 · w2 − u 1 · w2 )),
(4.10)
and hence defines a homomorphism of Courant algebras as opposed to Lie algebras as in (4.3). Note that the extra factor dμ(w) in (4.9) corresponds to a trivial B-shift, . Note also that μ will play the role of a moment map. In the and thus L (u,v) = Lψ(u) ˜ conventional case of symplectic reductions one has μ : M → g∗ , whereas here h can ˜ h, μ) is known as the reduction data. be any representation space. The triple (ψ, This reduction data can then be used to define a reduced generalized tangent bundle E red . First one makes the usual assumptions about μ and the G action on M so that M red = μ−1 (0)/G is a manifold. (This requires that 0 is a regular point of μ and that the G action on μ−1 (0) is free and proper.) Then define the sub-bundle K which is the image of the bundle map a × M associated to , that is
˜ K = ψ(u) + dμ(w), u ∈ g, w ∈ h ⊆ E, (4.11) and also the orthogonal bundle K ⊥ , the fibres of which are orthogonal to K with respect to the O(d, d) metric ·, · . One can then construct the generalized tangent space on M red , K ⊥ |μ−1 (0) red E = G. (4.12) K |μ−1 (0) 5 Note we take a slightly different definition of the bracket to that in [27] in to order to match the Courant bracket (2.13) on E.
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The main results of [26,27] are then to show how various geometrical structures can be transported from E to E red . The case of particular interest to us is that of generalized Hermitian reduction. As discussed in Sect. 2.2 a generalized Hermitian manifold is a generalized complex manifold with a compatible generalized metric (or equivalently a second, compatible, generalized almost complex structure). Let J be the integrable generalized complex ˜ h, μ), recallstructure and G be the generalized metric. Given some reduction data (ψ, ing L (u,v) = Lψ(u) , the structures are G-invariant if ˜ G = Lψ(u) J = 0, Lψ(u) ˜ ˜
(4.13)
for all u ∈ g. One can also define the sub-bundles K G = G K ⊥ ∩ K ⊥,
(4.14)
which is the sub-bundle of K ⊥ the fibres of which are orthogonal to K , with respect to G, and EK = K ⊕ G K,
(4.15)
which is the G-orthogonal complement to K G . Theorem 4.4 of [27] then states Theorem 1 (Generalized Hermitian reduction [27]). Let E be a generalized tangent ˜ h, μ). Suppose E is equipped with a G-invariant space over M with reduction data (ψ, generalized Hermitian structure (J , G). If over μ−1 (0), J K G = K G , or equivalently J E K = E K , then J and G can be reduced to E red where they define a generalized Hermitian structure. Even if the group action is such that the reduced space is not a manifold, one can still define a generalized Hermitian structure on the transversal section to the foliation. 4.2. Generalized reduction from ξ . We now use the reduction formalism to show that the generalized Calabi-Yau geometry on the cone X reduces to a generalized Hermitian geometry in four dimensions. This is the analogue of the reduced Kähler-Einstein geometry in the Sasaki-Einstein case. First we note that there is a group action on the cone X generated by the vectors r ∂r and ξv . These commute and the corresponding Lie algebra is simply R ⊕ R. If the orbits of ξv form a U (1) action then together r ∂r and ξv integrate to a C∗ action, but this need not be the case. The generalized vectors r ∂r and ξ give a lifted action of R ⊕ R on E, so that, if u = (a, b) ∈ R ⊕ R, ˜ ψ(u) = ar ∂r + bξ.
(4.16)
˜ By definition we have π ψ(u) = ar ∂r + bξv . Under the Courant bracket, given the expressions (3.50) we see that [r ∂r , ξ ] = 0, and hence ˜ 2 )] = 0 = ψ([u ˜ 1 , u 2 ]) ˜ 1 ), ψ(u [ψ(u
(4.17)
for all u 1 and u 2 , as required for a lifted action. Furthermore, from (B.5) we see that ψ˜ is isotropic. We also have a generalized Hermitian structure on X given by J± . The generalized complex structure J− is integrable, and we have the compatible generalized metric
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G = −J+ J− . In Sect. 3 we showed Lr ∂r J± = Lξ J± = 0, and hence the Hermitian structure is invariant under both group actions. There are then two different ways we can view the generalized reduction, mirroring the symplectic reduction and the complex quotient in the Sasaki-Einstein case. In the first reduction, we take g = R generated by ξv , and in the reduction data we take h = g and μ = log r . This is the same moment map one takes in the symplectic reduction. In the second case we take the complex Lie algebra g = C generated by r ∂r −iξv and h = 0 so there is no moment map. The reduction is then analogous to a complex quotient. We now discuss these in turn. As in the Sasaki-Einstein case, both lead to the same reduced structure. 4.2.1. g = R reduction. In this case, the reduction data is ˜ ψ(u) = uξ,
h = R,
μ = log r.
(4.18)
We have already seen that ψ˜ is an isotropic lifted action. It is also clear that μ is ˜ h, μ) are suitable reduction data. g-equivariant since, from (B.5), i ξv dμ = 0. Thus (ψ, ˜ Furthermore, J− and G are both invariant under ψ(u). We have μ−1 (0) = Y,
(4.19)
K = {uξ + v d log r }.
(4.20)
and given u ∈ g and v ∈ h,
Using Gξ = e2+φ/2 η,
G d log r = e−2−φ/2 r ∂r
(4.21)
we have G K = {u η + v r ∂r },
(4.22)
E K ≡ K ⊕ G K = {uξ + v d log r + u η + v r ∂r }.
(4.23)
and hence
Using the definitions (3.46) we immediately see that J− E K = E K . Hence, assuming the action of ξv on Y gives a U (1) fibration, using the generalized Hermitian reduction theorem, we see that we have a generalized Hermitian structure on E red over the fourdimensional space M red = Y/U (1). More generally, we get a generalized Hermitian structure on the transversal section to the ξv orbits. 4.2.2. g = C reduction. In this case, the reduction data is ˜ ψ(u) = u(r ∂r − iξ ),
h = 0,
μ = 0.
(4.24)
Given h is trivial, we have K = {u(r ∂r − iξ )}.
(4.25)
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˜ h, μ) are As before, we have already seen that ψ˜ is an isotropic lifted action and so (ψ, ˜ suitable reduction data. J− and G are both invariant under ψ(u) and finally, using (4.21), we now have G K = {u (d log r − iη)},
(4.26)
E K = {u(r ∂r − iξ ) + u (d log r − iη)}.
(4.27)
and hence
Again we immediately see that J− E K = E K . Hence, again assuming the action of ξv on Y gives a U (1) fibration, using the generalized Hermitian reduction theorem, we see that we have a generalized Hermitian structure on E red over the four-dimensional space M red = X/C∗ = Y/U (1), or more generally, we get a generalized Hermitian structure on the transversal section to the r ∂r − iξv orbits. Note that in both cases the reduced manifold M red is the same. Furthermore, the (complexified) spaces E K , and hence the G-orthogonal complements K G , also agree. As discussed in [27], K G is a model for the reduced bundle E red . Thus the two reductions give identical generalized Hermitian structures on M red .
4.3. The reduced pure spinors. We now calculate the conditions on the reduced generalized Hermitian structure implied by supersymmetry. The reduced structure can be defined by a pair of commuting generalized almost complex structures: J˜1 which is integrable and is the reduction of J− , and a non-integrable structure J˜2 , defined such that −J˜1 J˜2 is the reduced generalized metric. Equivalently, the structures are defined as ˜ 1 and ˜ 2 . It is the differential conditions on ˜ 1 and ˜ 2 implied a pair of pure spinors by supersymmetry that we will derive. In order to construct the reduced pure spinors, first note that the reduction gives a splitting of the generalized tangent space E = EK ⊕ K G
(4.28)
such that, in general, the O(d, d) metric ·, · factors into an O( p, p) metric on K G and an O(d − p, d − p) metric on E K . Thus we can similarly decompose sections of the spinor bundles S ± (E) into spinors of Spin(d − p, d − p) × Spin( p, p). In particular, generic sections ± ∈ S ± (E) can be written as ˜ + ⊕ ∓ ⊗ ˜ −. ± = ± ⊗
(4.29)
˜ ± in S ± (K G ) which correspond to the reduced It is then the spinor components of pure spinors. For the case in hand the relevant decomposition is under Spin(2, 2) × Spin(4, 4) ⊂ Spin(6, 6). As we will see below, the reduction is such that the pure spinors defining the supersymmetric background decompose as ˜ 1, − = − ⊗ ˜ 2. + = + ⊗ ˜ 1 and ˜ 2 are both positive helicity in Spin(4, 4). Thus the reduced spinors
(4.30)
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To make this explicit we need a basis for the Spin(6, 6) gamma matrices reflecting the decomposition (4.28). We first introduce coordinates adapted to the reduction. We write the R-symmetry Killing vector as6 ξv = K 5# = ∂ψ .
(4.31)
ym
be coordinates on the transversal section to the R-symmetry foliation. This means Let that i ξv dy m = 0 and, in particular, the metric decomposes as red m gY = K 5 ⊗ K 5 + gmn dy dy n ,
(4.32)
in analogy to (4.1). The reduction structure already defines a natural basis on E K given by fˆ1 = r ∂r , fˆ2 = ξ,
f 1 = d log r, f 2 = η,
(4.33)
and satisfying f i , fˆj = 21 δ i j and f i , f j = fˆi , fˆj = 0. We can then define an orthogonal basis on K G given by eˆm = e−b2 ∂ y m − η˜ m ξ, em = dy m − ηm ξ,
(4.34)
where η˜ m = 2η, e−b2 ∂ y m and ηm = 2η, dy m = i ηv dy m . This basis again satisfies em , eˆn = 21 δ m n and em , en = eˆm , eˆn = 0. Given such a basis we can then write a generic Spin(6, 6) spinor using the standard raising and lowering operator construction. Consider the polyform (0) = e−b2 ∈ S + (E). It is easy to see that we have the Clifford actions fˆi · (0) = eˆm · (0) = 0,
(4.35)
for all i and m. Thus we can regard (0) as a ground state for the lowering operators ( fˆi , eˆm ). A generic spinor is then given by acting with the anti-commuting raising operators ( f i , em ). Acting with the em first, we see that a generic (non-chiral) spinor has the form ˜ 0 + f 1 · e−b2 ˜ 1 + f 2 · e−b2 ˜ 2 + f 1 · f 2 · e−b2 ˜ 3, = e−b2
(4.36)
˜ i are polyforms in dy m , and e−b2 ˜ i transform as a Spin(4, 4) spinor under the where Clifford action of (em , eˆm ). We can now write the supersymmetry pure spinors ± in the form (4.36). Requiring that r ∂r −iξ and d log r −iη annihilate − while r ∂r −ie2+φ/2 η and d log r −ie−2−φ/2 ξ annihilate + , one finds the only possibility is ˜ 1, − = (d log r − iη) · r 3 e−3iψ e−b2 ˜ 2, + = 1 + ie2+φ/2 d log r · η · r 3 e−b2 ˜ 1 and ˜ 2 are both even polyforms in dy m , as claimed in (4.30). We have where ˜ 1 and ˜ 2 are independent of the r and ψ introduced factors of r 3 and e−3iψ so that coordinates. In general, they are then only locally defined. 6 Note that this, more conventional, normalization of ψ differs from the corresponding coordinate in [19] by a factor of three.
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˜ 1 and ˜ 2 , reduced to the transversal section, One can then derive the conditions on implied by supersymmetry. From d− = 0 one finds ˜ 1, ˜ 1 = −3iη˜ · d
(4.37)
where η˜ is a generalized vector on the transversal section defined by ˜ η = dψ + e−b2 η.
(4.38)
˜ 1 defines an integrable generalized complex structure on the transThis is means that verse section, as expected from the reduction theorem. For the second pure spinor, the condition (3.38) on Re + is equivalent to ˜ 2 = −2e−−φ/4 Re ˜ 2, d e+φ/4 η˜ · Im (4.39) ˜ 2 = 0. d e+φ/4 Im Finally, since ir ∂r F = 0, following (4.36), we can decompose the flux as ˜ e−B F = e−b2 F˜ + η · e−b2 G.
(4.40)
The final condition (3.40) is then equivalent to ˜ ˜ 2 = − 1 G, d e−−φ/4 η˜ · Re 8
˜ ˜ 2 = − 1 e3+3φ/4 F, J˜1 d(4 + φ) · Im 8 and to
(4.41)
˜ 2 = 0, J˜1 · d e−−φ/4 η˜ · Re
(4.42)
where J˜1 is the reduction to the transverse section of the generalized complex structure ˜ 2 = 0.7 J−b2 = eb2 J− e−b2 , and we have used the compatibility relation J˜1 · The conditions (4.39) and (4.42), which do not involve the flux, can be viewed as ˜ 2 satisfying these a generalization of the usual Kähler-Einstein conditions. Given an conditions, the flux is then determined by (4.41). 5. The Pure Spinor − The closed pure spinor − is associated with the integrable generalized complex-structure J− . The latter in turn holds information regarding BPS operators in the dual field theory. In this section we explore two aspects of this duality. The first is the mesonic moduli space of the dual theory, which is known to correspond to the subspace for which the polyform − reduces to a three-form. The second is the connection between generalized holomorphic objects and dual BPS operators.
7 Note that in the language defined in Sect. 5.3 below, the condition (4.42) states that d e−−φ/4 η˜ · Re ˜2
˜ is an element of U 0 . (and hence G) ˜ J1
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5.1. The general form of − . Recall that the most general pure spinor takes the form [29] = αθ1 ∧ θ2 ∧ · · · ∧ θk ∧ e−b+iω , 0
(5.1)
where α is some complex function, θi are complex one-forms, while b and ω0 are both real two-forms. The integer k is called the type of the pure spinor, which can change along various subspaces of X . Using the definition of ± , the Fierz identity (2.31), and the results of Sect. 3.1 and Appendix A, one can find expressions for ± in terms of spinor bilinears introduced in [19]. We find that in general − is of type one with − = θ ∧ e−b− +iω− ,
(5.2)
r 3 4 e (iK + Sd log r ), 8 4e6+φ/2 ω− = K 5 ∧ Im(K 3 ) − cos 2θ¯ cos 2φ¯ Re(K 3 ) ∧ d log r , 2 ¯ f (sin 2φ) 4e6+φ/2 ¯ 2 Im(K 3 ) ∧ d log r + b2 . K b− = − ∧ Re(K ) + (cos 2 φ) 4 3 ¯ 2 f (sin 2φ)
(5.3)
where θ =−
Note that ω− and b− are not uniquely defined since we can add two-forms that vanish ¯ which appear in Appendix B of [19] when wedged with θ . Here the angles θ¯ and φ, without bars, are functions on the link Y that are related to the scalar spinor bilinears through ¯ sin ζ = cos 2θ¯ cos 2φ, ¯ |S| = − sin 2θ¯ cos 2φ.
(5.4) (5.5)
Using the results of [19], we have the important result that θ is exact8
1 4 3 θ = d − e r S ≡ d r 3 θ0 . 24
(5.6)
Alternatively, from the supersymmetry equation d− = 0 and the definite scaling dimension Lr ∂r − = 3− , we immediately obtain − = 13 d(r ∂r − ),
(5.7)
the one-form part of which reduces to (5.6). 8 The fact that θ is closed was essentially observed in [37], and it was also shown to be exact in the special cases of the Pilch-Warner and Lunin-Maldacena solutions in [20].
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5.2. Type change of − and the mesonic moduli space. The pure spinor − has the property that its type can jump from type one to type three on the locus θ = 0. This ¯ Assuming f = 0, we locus can be neatly parameterized through the angles θ¯ and φ. have from (3.11) that sin ζ is nowhere zero and then (5.4) implies that both cos 2φ¯ and cos 2θ¯ are nowhere zero. Using the expression for K in Appendix B of [19], we see that when f = 0, sin 2θ¯ = 0 ⇐⇒ θ0 = 0, ¯ sin 2θ = sin 2φ¯ = 0 ⇐⇒ θ = 0.
(5.8) (5.9)
The locus θ = 0 is thus a sublocus of θ0 = 0. Notice that, where θ = 0, − is not identically zero, as one might have naively expected from (5.2), but instead reduces to a finite, non-zero three-form. Indeed, the powers of sin 2φ¯ in the denominator of b− and ω− are cancelled by those in K , K 3 and K 4 . The locus θ = 0 is precisely where a probe pointlike D3-brane in X is supersymmetric. This follows from [37] where it was shown that the pull-back of θ to the D3-brane worldvolume is equal to the F-term of the worldvolume theory. The supersymmetric locus of such a pointlike D3-brane is naturally interpreted as the mesonic moduli space. 5.3. BPS operators and generalized holomorphic spinors. In the Sasaki-Einstein case, holomorphic functions on the Calabi-Yau cone with a definite scaling weight λ under the action of r ∂r also have a charge λ under the action of ξ . This stems from the intimate connection (via Kaluza-Klein reduction on the Sasaki-Einstein manifold) between holomorphic functions on the cone and BPS operators in the dual CFT, in fact (anti-)chiral primary operators. For general Ad S5 solutions we might expect that the holomorphic functions should be replaced by polyforms and that the BPS condition of matching charges should be with respect to the generalized Lie derivative L discussed in Sect. 2. We now derive such a result, leaving the detailed connection with Kaluza-Klein reduction on the internal space Y to future work. We first recall that a generalized almost complex structure J defines a grading on generalized spinors, or equivalently differential forms. If ∈ (S± (E)) is a pure spinor corresponding to J , one defines the canonical pure spinor line bundle UJn ⊂ S± (E) as sections of the form ϕ = f for some function f . One can then define (n−k)
UJ
= ∧k L¯ ⊗ UJn .
(5.10)
Elements of UJk have eigenvalues ik under the Lie algebra action of J given in (2.40). These bundles then give a grading of the spinor bundles S± (E). A generalized vector V ∈ (E) acting on an element of UJk gives an element of UJk+1 ⊕ UJk−1 . In particular an annihilator of acts by purely raising the level by one. If the generalized complex structure J is also integrable then the exterior derivative splits into the sum d = ∂J + ∂¯J ,
(5.11)
where C ∞ UJk
∂¯J ← − C ∞ UJk−1 . − → ∂J
(5.12)
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Consider now a spinor ψ satisfying ψ ∈ UJk − ,
Lr ∂r ψ = λψ,
(5.13)
for some k and λ. Then imposing in addition ∂¯J− ψ = 0,
(r ∂r + iξ ) · ψ = 0
implies
Lξ ψ = iLr ∂r ψ.
(5.14)
In other words, subject to the constraints (5.13), a spinor is BPS if it is generalized holomorphic and is annihilated by r ∂r + iξ . To see this result, we first write r ∂r = (1/2)(r ∂r + iξ ) + (1/2)(r ∂r − iξ ) and use (5.13) to deduce that ∂J− [(r ∂r + iξ ) · ψ] + (r ∂r + iξ ) · ∂J− ψ = 0, ∂¯J− [(∂r − iξ ) · ψ] + (r ∂r − iξ ) · ∂¯J− ψ = 0.
(5.15)
In obtaining this we used the fact that since r ∂r − iξ is an annihilator of − it raises the level of ψ and similarly r ∂r + iξ lowers the level. We then compute Lξ ψ = iLr ∂r ψ − i {d [(r ∂r + iξ ) · ψ] + (r ∂r + iξ ) · dψ} = iLr ∂r ψ − i ∂¯J− [(r ∂r + iξ ) · ψ] + (r ∂r + iξ ) · ∂¯J− ψ .
(5.16)
In a similar way, given (5.13) we also have ∂J− ψ = 0,
(r ∂r − iξ ) · ψ = 0
implies
Lξ ψ = −iLr ∂r ψ.
(5.17)
6. The Pure Spinor + 6.1. The general form of + . One can see immediately from the supersymmetry equation (3.39) that if we assume F5 = 0, which we shall do, then Im + must have a scalar component and hence + is of type 0: + = α+ e−b+ +iω+ . 0
(6.1)
Using the same procedure as in the last section, we may again express these quantities in terms of the bilinears of [19]. After defining the rescaled two-form ω = e−2 A r 4 ω+0 ,
(6.2)
i f e−A r 4 , 32 4r 2 4 e (V + K 4 ∧ d log r ) , ω=− f 4 b+ = e6+φ/2 Im K 3 ∧ d log r + b2 , f
(6.3)
we find α+ = −
where b2 appears in (3.12).
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6.2. A canonical symplectic structure. The rescaling (6.2) is motivated by the fact that ω defines a canonical symplectic structure. To see this, we first observe that Y admits a contact structure defined by the one-form σ ≡
4 4 e K4. f
(6.4)
Recall that for a one-form σ to be contact, the top-degree form σ ∧ dσ ∧ dσ must be nowhere vanishing. Using (3.19) of [19], and results in Appendix B of [19], one can easily show that σ ∧ dσ ∧ dσ =
128 8 8 v olY , olY = e v f2 sin2 ζ
(6.5)
where recall v olY = −e12345 (using the orthonormal frame in Appendix B of [19]). We then observe, using (3.19) of [19], that ω = 21 d(r 2 σ ),
(6.6)
which shows that ω is closed and non-degenerate, and hence defines a symplectic structure on the cone X = R+ × Y . Alternatively, one can see the formula (6.6) for ω directly from the supersymmetry equation (3.38) on noting that e−A + has scaling dimension 2 under r ∂r . Furthermore, again using the results of Appendix B of [19], we have 1 = ξv σ,
0 = ξv dσ,
(6.7)
which shows that ξv is also the unique “Reeb vector field” associated with the contact structure. Notice also that (6.6) implies that H = r 2 /2 is precisely the Hamiltonian function for the Hamiltonian vector field ξv , i.e. dH = −i ξv ω. It is remarkable that these features, which are well-known in the Sasaki-Einstein case, are valid for all supersymmetric Ad S5 solutions with non-vanishing five-form flux. Although we have a symplectic structure, we do not quite have a Kähler structure, as in the Calabi-Yau case, but it is quite close. Using the last equation in (3.50) and the definition (6.4) we see that η f = σ + i ηv b2 , (6.8) b b and thus e 2 η |1−form = σ . Since e 2 (d log r ) = d log r manifestly, and by definition η ≡ J− (d log r ), we have, using (2.34), σ = J−b2 (d log r ) |1−form = −(I−b2 )∗ (d log r ).
(6.9)
Note this is precisely analogous to the formula for the contact form in the Sasakian case. We then have b2 1 (6.10) dJ− r 2 = −r 2 d Q b−2 + Tr I−b2 − (I−b2 )∗ (d(r 2 )), 2 where here we recall that in general we define dJ− ≡ −[d, J− ], and we use (2.40) for the action on generalized spinors. From this it follows that b2 1 1 1 (6.11) ω = ddJ− r 2 + d(r 2 ) ∧ d Q b−2 + Tr I−b2 . 4 4 2 Thus r 2 is almost a Kähler potential, for the b2 -transformed complex structure J−b2 = eb2 J− e−b2 , except for the last term.
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6.3. The central charge as a Duistermaat-Heckman integral. Recall that in any fourdimensional CFT there are two central charges, usually called a and c, that are constant coefficients in the conformal anomaly Tμμ =
1 a 2 (Euler) . c(Weyl) − 120(4π )2 4
(6.12)
Here Tμν denotes the stress-energy tensor, and Weyl and Euler denote certain curvature invariants for the background four-dimensional metric. For SCFTs, both a and c are related to the R-symmetry [8] via a=
3 3Tr R 3 − Tr R , 32
c=
1 9Tr R 3 − 5Tr R . 32
(6.13)
Here the trace is over the fermions in the theory. For SCFTs with Ad S5 gravity duals, in fact a = c holds necessarily in the large N limit [38]. The central charge of the SCFT is then inversely proportional to the dual five-dimensional Newton constant G 5 [38], obtained here by Kaluza-Klein reduction on Y . The Newton constant, in turn, was computed in Appendix E of [19], and is given by G5 =
κ2 G 10 = 10 , V5 8π V5
(6.14)
where G 10 is the ten-dimensional Newton constant of type IIB supergravity, and we have defined Y . V5 ≡ e8 vol (6.15) Y
We may derive an alternative formula for G 5 as follows. We begin by rewriting f2 V5 = 16
Y
1 Y , vol sin2 ζ
(6.16)
where we have used the relation (3.11). Importantly, the constant f is quantized, being essentially the number of D3-branes N . Specifically, we have 1 1 N= dC = (6.17) (F5 + H ∧ C2 ) . 4 (2πls )4 gs Y (2πls )4 gs Y Using the Bianchi identity DG = −P ∧ G ∗ and the result (A.22), one derives that d(H ∧ C2 ) = −(2/ f )d[e6 Im(W ∗ ∧ G)], and so we can also write ∗ 1 2e6 F Im W N= − ∧ G . (6.18) 5 (2πls )4 gs Y f We may evaluate this expression in terms of the orthonormal basis of forms ei introduced in Appendix B of [19], and after some calculation we find f 1 Y . vol N =− (6.19) 4 (2πls ) gs Y sin2 ζ
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2 = (2π )7 l 8 g 2 leads to the result Combining these formulae and using 2κ10 s s
G5 =
8V5 . π2 f 2N2
(6.20)
Consider now the integral μ=
1 (2π )3
e−r X
2 /2
ω3 . 3!
(6.21)
This is the Duistermaat-Heckman integral for a symplectic manifold (X, ω) with Hamiltonian function H = r 2 /2, which we have shown is the Hamiltonian for the Reeb vector field ξv . Using (6.6) and (6.5) we may rewrite ω3 16 Y . = 2 e8r 5 dr ∧ vol 3! f
(6.22)
Performing the r -integral in (6.21) allows us to rewrite the five-dimensional Newton constant as πμ . (6.23) G5 = 2N 2 Since μ = 1 for the round five-sphere solution, we thus obtain the ratio GG 55 = μ. S Recalling that this is, by AdS/CFT duality, the inverse ratio of central charges [38], we deduce the key result 3 aN =4 1 1 −r 2 /2 ω = = e σ ∧ dσ ∧ dσ. (6.24) a (2π )3 X 3! (2π )3 Y Here aN =4 = N 2 /4 denotes the (large N ) central charge of N = 4 super-Yang-Mills theory. The formula (6.24) implies that the central charge depends only on the symplectic structure of the cone (X, ω) and the Reeb vector field ξv . This is perhaps surprising: one might have anticipated that the quantum numbers of quantized fluxes would appear explicitly in the central charge formula. However, recall from formulae (3.12), (3.13) that the two-form potentials B and C2 are globally defined. In particular, for example, the period of H = dB through any three-cycle in Y is zero. As discussed in [11], the Duistermaat-Heckman integral in (6.24) may be evaluated by localization. The integral localizes where ξv = 0, which is formally at the tip of the cone r = 0. Unless the differentiable and symplectic structure is smooth here (which is only the case when X ∪ {r = 0} is diffeomorphic to R6 ), one needs to equivariantly resolve the singularity in order to apply the localization formula. Notice here that since ξv preserves all the structure on the compact manifold (Y, gY , σ ), the closure of its orbits defines a U (1)s action preserving all the structure, for some s ≥ 1. Here we have used the fact that the isometry group of a compact Riemannian manifold is compact. Thus (X, ω) comes equipped with a U (1)s action. Rather than attempt to describe this in general, we focus here on the special case where the solution is toric: that is, there is a U (1)3 action on Y under which σ , and hence ω under the lift to X , is invariant. Notice that we do not necessarily require that the full supergravity solution is invariant under U (1)3 – we shall illustrate this in the next section with the Pilch-Warner solution, where σ and the metric are invariant under
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U (1)3 , but the G-flux is invariant only under a U (1)2 subgroup. For the arguments that follow, it is only σ (and hence ω) that we need to be invariant under a maximal dimension torus U (1)3 . In fact any such symplectic toric cone is also an affine toric variety. This implies that there is a (compatible) complex structure on X , and that the U (1)3 action complexifies to a holomorphic (C∗ )3 action with a dense open orbit. There is then always a symplectic toric resolution (X , ω ) of (X, ω), obtained by toric blow-up. In physics language, this is because one can realize (X, ω) as a gauged linear sigma model, and one obtains (X , ω ) by simply turning on generic Fayet-Iliopoulos parameters. One can also describe this in terms of moment maps as follows. The image of a symplectic toric cone under the moment map μ : X → R3 is a strictly convex rational polyhedral cone (see [11]). Choosing a toric resolution (X , ω ) then amounts to choosing any simplicial resolution P of this polyhedral cone. Here P is the image of μ : X → R3 . Assuming the fixed points of ξv are all isolated, the localization formula is then simply [11] 1 (2π )3
e−r X
2 /2
ω3 = 3!
3
1 p . ξ , u vertices p∈P i=1 v i
(6.25)
p
Here u i , i = 1, 2, 3, are the three edge vectors of the moment polytope P at the vertex point p, and ·, · denotes the standard Euclidean inner product on R3 (where we regard ξv as being an element of the Lie algebra R3 of U (1)3 ). The vertices of P precisely correspond to the U (1)3 fixed points of the symplectic toric resolution X = X P of X . Thus, remarkably, these results of [11] hold in general, even when there are non-trivial fluxes turned on and X is not Calabi-Yau.
6.4. The conformal dimensions of BPS branes. A supersymmetric D3-brane wrapped on 3 ⊂ Y gives rise to a BPS particle in Ad S5 . The quantum field whose excitations give rise to this particle state then couples, in the usual way in AdS/CFT, to a dual chiral primary operator O = O3 in the boundary SCFT. More precisely, there is an asymptotic expansion of near the Ad S5 boundary ∼ 0 r −4 + Ar − ,
(6.26)
where 0 acts as the source for O and = (O) is the conformal dimension of O. In [39], following [40], it was argued that the vacuum expectation value A of O in a given asymptotically Ad S5 background may be computed from e−S E , where S E is the on-shell Euclidean action of the D3-brane wrapped on 4 = R+ × 3 , where R+ is the r -direction. In particular, via the second term in (6.26) this identifies the conformal dimension = (O3 ) with the coefficient of the logarithmically divergent part of the on-shell Euclidean action of the D3-brane wrapped on 4 . We refer to Sect. 2.3 of [39] for further details. We are thus interested in the on-shell Euclidean action of a supersymmetric D3brane wrapped on 4 = R+ × 3 . The condition of supersymmetry is equivalent to a generalized calibration condition, namely Eq. (3.16) of [35]. In our notation and conventions, this calibration condition reads
|a|2 Re −i+ ∧ eF |4 = det(h + F) dx1 ∧ · · · ∧ dx4 . 8
(6.27)
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Here h is the induced (string frame) metric on 4 , and F = F − B is the gauge-invariant worldvolume gauge field, satisfying dF = −H |4 .
(6.28)
Recalling from Sect. 3.2 that |a|2 = e A , we may then substitute for + in terms of + using (3.36) and (6.1) to obtain
f A+φ f e d log r ∧ σ ∧ dσ |4 − e−3A+φ r 4 (F − b+ )2 |4 , Re −i+ ∧ eF |4 = 64 64 (6.29) where, as in (6.3), b+ = e6+φ/2
4 Im K 3 ∧ d log r + b2 . f
(6.30)
Here b2 is a closed two-form, whose gauge-invariant information is contained in its cohomology class in H 2 (X, R)/H 2 (X, Z). In writing b+ in (6.29) we have chosen a particular representative two-form for the class of b2 in H 2 (X, R)/H 2 (X, Z). Then under any gauge transformation of b+ (induced from a B-transform of + ), the worldvolume gauge field F transforms by precisely the opposite gauge transformation restricted to 4 , so that the quantity F − b+ is gauge invariant on 4 . We now choose the worldvolume gauge field F to be F = b2 |4 , so that (6.29) becomes simply
f A+φ e d log r ∧ σ ∧ dσ |4 . Re −i+ ∧ eF |4 = 64
(6.31)
(6.32)
In fact, there is a slight subtlety in (6.31). If the cohomology class of b2 /(2πls )2 |4 in H 2 (4 , R) is not integral, then the choice (6.31) is not possible as F is the curvature of a unitary line bundle. Having said this, notice H 2 (4 , R) ∼ = H 2 (3 , R), and thus in particular that if H 2 (3 , R) = 0 then every closed b2 |4 is exact, and thus may be gauge transformed to zero on 4 . Then (6.31) simply sets F = 0. For every example of a supersymmetric 3 that we are aware of, this is indeed the case. In any case, we shall assume henceforth that the choice (6.31) is possible. The calibration condition (6.27) for a D3-brane with worldvolume 4 and with gauge field (6.31) is thus f d log r ∧ σ ∧ dσ = e−φ det(h − B) dx1 ∧ · · · ∧ dx4 . 8
(6.33)
Notice the right hand side is precisely the Dirac-Born-Infeld Lagrangian, up to the D3brane tension τ3 = 1/(2π )3 ls4 gs . From (6.33), and the comments above on the scaling dimension (O(3 )) of the dual operator O(3 ), we thus deduce τ3 f σ ∧ dσ. (6.34) (O(3 )) = − 8 3 (The sign just arising from a convenient choice of orientation.) Using (6.19) and (6.5) we have
Ad S5 Solutions of Type IIB Supergravity and Generalized Complex Geometry
8(2πls )4 gs N , Y σ ∧ dσ ∧ dσ
f = − and hence
2π N 3 σ ∧ dσ (O(3 )) = . Y σ ∧ dσ ∧ dσ
397
(6.35)
(6.36)
This is our final formula for the conformal dimension of the chiral primary operator dual to a BPS D3-brane wrapped on 3 . Since we may write ω2 2 (6.37) σ ∧ dσ = e−r /2 , 2! 3 4 we again see that it depends only on the symplectic structure of (X, ω) and the Reeb vector field ξv . This again may be evaluated by localization, having appropriately resolved the tip of the cone 4 . 7. The Pilch-Warner Solution In this section we illustrate the general results derived so far with the Pilch-Warner solution [13]. (Some aspects of the generalized complex geometry of this background have already been discussed in [20].) Recall that the Pilch-Warner solution is dual to a Leigh-Strassler fixed point theory [16] which is obtained by giving a mass to one of the three chiral superfields (in N = 1 language) of N = 4 SU (N ) super-Yang-Mills theory, and following the resulting renormalization group flow to the IR fixed point theory. This latter theory is an N = 1 SU (N ) gauge theory with two adjoint fields Z a , a = 1, 2, which form a doublet under an SU (2) flavour symmetry, and a quartic superpotential. Since the superpotential has scaling dimension three, this fixes (Z a ) = 3/4, implying that the IR theory is strongly coupled. The mesonic moduli space is simply Sym N C2 . The Pilch-Warner supergravity solution [13] was rederived in [19], and we shall use some of the results from that reference also. We have Y = S 5 with non-trivial metric, 1 6 cos2 ϑ 6 sin2 2ϑ gY = (σ12 + σ22 ) + σ2 6dϑ 2 + 9 3 − cos 2ϑ (3 − cos 2ϑ)2 3 2 2 cos2 ϑ σ3 +4 dϕ + , (7.1) 3 − cos 2ϑ where 0 ≤ ϑ ≤ π2 , 0 ≤ ϕ ≤ 2π , and σi , i = 1, 2, 3, are left-invariant one-forms on SU (2) (denoted with hats in [19]). The dilaton φ and axion C0 are simply constant, while the warp factor is f (3 − cos 2ϑ). 8 There is also a non-trivial NS and RR three-form flux given by (see (A.7)) (2 f )1/2 2iϕ i sin 2ϑ G= dϕ ∧ σ3 e cos ϑ dϕ ∧ dϑ − 3/2 3 3 − cos 2ϑ 4 cos2 ϑ ∧ (σ2 − iσ1 ). − dϑ ∧ σ 3 (3 − cos 2ϑ)2 e4 =
(7.2)
(7.3)
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We introduce the Euler angles (α, β, γ ) on SU (2) (as in [19]), so that σ1 = − sin γ dα − cos γ sin αdβ, σ2 = cos γ dα − sin γ sin αdβ, σ3 = dγ − cos αdβ.
(7.4)
In terms of these coordinates, the R-symmetry vector ξv is [19] ξv =
3 ∂ϕ − 3∂γ . 2
Using the explicit formulae in [19], it is easy to show that the contact form is 2 σ = − cos 2ϑ dϕ + cos2 ϑ σ3 . 3
(7.5)
(7.6)
The solution is toric, in the sense that both σ and the metric are invariant under shifts of ϕ, β and γ . However, notice that the G-flux in (7.3) is not invariant under shifts of ϕ, thus breaking this U (1)3 symmetry to only a U (1)2 symmetry of the full supergravity solution. This is expected, since the dual field theory described above has only an SU (2) × U (1) R global symmetry. On Y = S 5 there are precisely three invariant circles under the U (1)3 action, where two of the U (1) actions degenerate, namely at {ϑ = π2 }, {ϑ = 0, α = 0}, {ϑ = 0, α = π }. A set of 2π -period coordinates on U (1)3 are 1 1 ϕ1 = ϕ, ϕ2 = − (ϕ + γ − β), ϕ3 = − (ϕ + γ + β). 2 2
(7.7)
These restrict to coordinates on the above three invariant circles, respectively. On X ∼ = R6 \0 we also have three corresponding moment maps: r2 r2 r2 sin2 ϑ, μ2 = cos2 ϑ(1 + cos α), μ3 = cos2 ϑ(1 − cos α), (7.8) 3 3 3 3 so that ω = 21 d(r 2 σ ) = i=1 dμi ∧ dϕi . It follows that the image of the moment map – the space spanned by the μi coordinates – is the cone (R≥0 )3 , where the three invariant circles map to the three generating rays u 1 = (1, 0, 0), u 2 = (0, 1, 0), u 3 = (0, 0, 1). The Reeb vector (7.5) in this basis is then computed to be μ1 =
ξ=
3 3 3 3 ∂ϕ − 3∂γ = ∂ϕ1 + ∂ϕ2 + ∂ϕ3 . 2 2 4 4
(7.9)
Since the symplectic structure is smooth at r = 0, we may evaluate (6.25) by localization without having to resolve X at r = 0. In the case at hand, we have the single fixed point at r = 0, and from (7.9) one obtains the known result aN =4 1 32 . = = aPW ξ1 ξ2 ξ3 27
(7.10)
They key point about the above calculation is that we have computed this knowing only the symplectic structure of the solution and the Reeb vector field ξv . We may similarly compute the conformal dimensions of the operators det Z a , using (6.36), by interpreting them as arising from a BPS D3-brane wrapped on the threespheres at α = 0 and α = π , respectively. It is simple to check these indeed satisfy the
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calibration condition (6.33) and are thus supersymmetric. Using (6.37) and localization at r = 0 implies that (6.37) is equal to 1/ξ1 ξ2 , 1/ξ1 ξ3 , respectively, which in both cases is 8/9. The formula (6.36) thus gives (det Z a ) = 3N /4, or equivalently (Z a ) = 3/4, which is indeed the correct result. 1 4 Next recall that the complex one-form θ = d(r 3 θ0 ), where θ0 = − 24 e S, and the mesonic moduli space should be the locus θ = 0. As discussed in Sect. 5.2, this is the locus sin 2θ¯ = sin 2φ¯ = 0. For the Pilch-Warner solution, we may easily compute √ √ 3 sin2 ϑ 1 + 3 sin4 ϑ sin 2θ¯ = − √ , cos 2φ¯ = . (7.11) 1 + sin2 ϑ 1 + 3 sin4 ϑ Thus, as discussed in [20], the mesonic moduli space S = 0 is equivalent to ϑ = 0, which is a codimension two submanifold in R6 diffeomorphic to R4 . Moreover, this is C2 in the induced complex structure, and we thus see explicit agreement with the field theory N = 1 mesonic moduli space. Finally, although the Pilch-Warner solution is generalized complex, rather than complex, we note that one can nevertheless define a natural complex structure [41]. The relation between this integrable complex structure and the generalized geometry has been discussed in [20]. Let us conclude this section by elucidating this connection. One can introduce the following complex coordinates [20] in terms of the angular variables (7.7): s1 = r 3/2 sin ϑ e−iϕ1 , s2 = r 3/4 cos ϑ cos α2 eiϕ2 , s3 = r
3/4
cos ϑ sin
α 2
e
iϕ3
(7.12)
.
This makes R6 ∼ = C3 . However, because of the minus sign in the first coordinate in (7.12), the corresponding integrable complex structure I∗ is not the unique complex structure that is compatible with the toric structure of the solution: the latter instead has complex coordinates s¯1 , s2 , s3 . Indeed, also the Reeb vector field ξv is not given by I∗ (r ∂r ). This makes the physical significance of this complex structure rather unclear. Nevertheless, one can show that I∗ does in fact come from an SU (3) structure defined by a Killing spinor. Following [20], we define ξ2 1 2 A/2 , (7.13) 2aη ˆ ∗ = η+ + iη+ = e iξ2 where by definition we require η¯ ∗ η∗ = 1. It is then convenient to define aˆ ≡ |a|e ˆ iz , 1 A 1 A 2 2 where |a| ˆ = 2 e |ξ2 | = 2 e (1−sin ζ ). We then introduce the bilinears corresponding to the SU (3) structure defined by η∗ : J∗ ≡ −iη¯ ∗ γ(2) η∗ , ∗ ≡ η¯ ∗c γ(3) η∗ .
(7.14) (7.15)
One computes that d∗ = 0, implying that the corresponding complex structure I∗ is integrable, and moreover that √ 3/2 2iz 2iα 2 f e ∗ = −e ds 1 ∧ ds 2 ∧ ds 3 , (7.16) 9e3A
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implying that (7.12) are indeed complex coordinates for this complex structure. We also compute 2 1 cos2 ϑ e2 A dϕ + J∗ = − 2 d log r ∧ σ3 + (sin 2ϑσ3 ∧ dϑ 2 r 3 (1 + sin ϑ) 3(1 + sin2 ϑ) + cos2 ϑσ1 ∧ σ2 . (7.17)
8. Conclusion In this paper we have initiated an analysis of the generalized cone geometry associated with supersymmetric Ad S5 × Y solutions of type IIB supergravity. The cone is generalized Hermitian and generalized Calabi-Yau and we have identified holomorphic generalized vector fields that are dual to the dilatation and R-symmetry of the dual SCFT. We identified a relationship between “BPS polyforms”, i.e. polyforms with equal R-charge and scaling weight, and generalized holomorphic polyforms that should be worth exploring further. In particular, we would like to make a precise connection between such objects and the spectrum of chiral operators in the SCFT via Kaluza-Klein reduction on Y . We also showed how one can carry out a generalized reduction of the six-dimensional cone to obtain a new four-dimensional transverse generalized Hermitian geometry. This generalizes the transverse Kähler-Einstein geometry in the Sasaki-Einstein case. By analogy with the Sasaki-Einstein case (e.g. [25]) this perspective could be useful for constructing new explicit solutions. We also analysed the symplectic structure on the cone geometry, which exists providing that the five-form flux is non-vanishing. It would be interesting to know whether or not this includes all solutions. We obtained Duistermaat-Heckman type integrals for the central charge of the dual SCFT and the conformal dimensions of operators dual to BPS wrapped D3-branes. These formulae precisely generalize analogous formulae that were derived in [10,11] for the Sasaki-Einstein case. Other formulae for these quantities were also presented in [10,11] and we expect that these will also have precise generalizations in terms of generalized geometry. In particular, we expect a generalized geometric interpretation of a-maximization. Acknowledgements. We would like to thank Nick Halmagyi for a useful discussion. M.G. is supported by the Berrow Foundation, J.P.G. by an EPSRC Senior Fellowship and a Royal Society Wolfson Award, E.P. by a STFC Postdoctoral Fellowship and J.F.S. by a Royal Society University Research Fellowship.
A. Conventions and 6D to 5D Map We use exactly9 the same conventions as in [19], up to some simple relabelling. Here we will explain how the results of that paper concerning the five-dimensional geometry with metric gY can be uplifted to six-dimensions. In particular, we will relate the five-dimensional Killing spinors discussed in [19] to the six-dimensional chiral spinors ηi that 9 Although it will not be relevant in this paper we point out that there is a typo in [19]: the ρ matrices a generating Cliff(4, 1) actually satisfy ρ01234 = −i.
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define the bispinors ± . We first recall the Killing spinor equations in five-dimensions, related to the geometry gY , given in [19]. There are two differential conditions i i −4 1 e (∇m − Q m )ξ1 + f − 2 βm ξ1 + e−2 G mnp β np ξ2 = 0, (A.1) 2 4 8 i i −4 1 (∇m + Q m )ξ2 − e f + 2 βm ξ2 + e−2 G ∗mnp β np ξ1 = 0, (A.2) 2 4 8 and four algebraic conditions 1 i −4 e f − 4 ξ1 = 0, (A.3) β m ∂m ξ1 − e−2 β mnp G mnp ξ2 − 48 4 1 i −4 β m ∂m ξ2 − e−2 β mnp G ∗mnp ξ1 + e f + 4 ξ2 = 0, (A.4) 48 4 1 β m Pm ξ2 + e−2 β mnp G mnp ξ1 = 0, (A.5) 24 1 β m Pm∗ ξ1 + e−2 β mnp G ∗mnp ξ2 = 0. (A.6) 24 Here10 the βm generate the Clifford algebra for gY , so {βm , βn } = 2gY mn . Equivalently, with respect to any orthonormal frame the corresponding βˆm satisfy {βˆm , βˆn } = 2δmn . We have chosen βˆ12345 = +1. In addition we have set the parameter m in [19] to be m = 1, consistent with (3.2). In the usual string theory variables we have 1 i dφ + eφ F1 , 2 2 1 Q = − eφ F1 , 2 G = −ieφ/2 F3 − e−φ/2 H, P=
(A.7)
where the RR field strengths Fn are defined by (3.3). We also note that the constant f appearing in the Killing spinor equations is related to the component of the self-dual five-form flux on Y (3.6) via Y , F5 |Y = − f vol
(A.8)
where the five-dimensional volume form is defined as v olY = −e12345 and ei is the orthonormal frame introduced in Appendix B of [19]. We now provide a map between the five-dimensional spinors and Killing spinor equations (A.1)-(A.6) to six-dimensional quantities. We begin by using the Cliff(5) gamma matrices βˆm to construct Cliff(6) gamma matrices γˆi , i = 1, . . . , 6, via γˆm = βˆm ⊗ σ3 , γˆ6 = 1 ⊗ σ1 ,
m = 1, . . . , 5, (A.9)
where σα , α = 1, 2, 3, are the Pauli matrices. These satisfy {γˆi , γˆ j } = 2δi j . The corresponding gamma matrices for the six-dimensional metric g6 will be denoted γi . We define the 6D chirality operator to be γ˜ ≡ −iγˆ123456 = 1 ⊗ σ2 .
(A.10)
10 Notice we have relabelled γ → β in [19], as in this paper we want to keep the notation γ for sixm i i dimensional gamma matrices.
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We may choose the D6 intertwiner D6 = D˜ 5 ⊗ σ2 ,
(A.11)
where D˜ 5 = D5 = C5 is the intertwiner of Cliff(5) discussed in [19], and one checks D6−1 γi D6 = −γi∗ . We also note that since in [19] the interwiner A5 = 1 we have A6 = 1 and γi† = γi . If η+ is a Weyl spinor, satisfying γ˜ η+ = η+ , then the conjugate spinor η− ≡ η+c ≡ D6 η+∗ satisfies γ˜ η− = −η− . To construct the relevant 6D spinors we first write ξ1 = χ1 + iχ2 ,
ξ2 = χ1 − iχ2 ,
(A.12)
as in [19]. Given this, the normalization for ξi chosen in [19] implies that the χi are normalized as χ¯ 1 χ1 = χ¯ 2 χ2 = 21 . We then define
η+1
=e
A/2
η+2
=e
A/2
χ1 iχ1
−χ2 −iχ2
(A.13)
,
−χ1c , =e iχ1c c χ2 2 η− , = e A/2 −iχ2c
1 η−
,
A/2
(A.14)
where recall from (3.16) that e A/2 = r 1/2 e/2+φ/8 ,
(A.15)
χ c ≡ D˜ 5 χ ∗ .
(A.16)
and also from [19] that
In the conventions of [19] we have χ¯ = χ † . After some detailed calculation one finds that the five-dimensional Killing spinor equations (A.1)–(A.6), using the five-dimensional metric gY , are equivalent to the six-dimensional Killing spinor equations, using the six-dimensional metric g6 in (3.17) and volume form (3.18), given by 1 eφ Di − Hi η+1 + Fγi η+2 = 0, (A.17) 4 8 1 A 1 e ∂ A η+1 − e A+φ Fη+2 = 0, (A.18) 2 8 1 (A.19) Dη+1 + ∂(2 A − φ) − H η+1 = 0, 4 and additional equations obtained by applying the rule: η1 ↔ η2 , F → − F † ,
H → −H.
(A.20)
In these equations we are using the notation that, e.g. Hi = 21 Hi jk γ jk ,
F = F1i γ i +
i jk 1 3! F3i jk γ
+
i jklm 1 . 5! F5i jklm γ
(A.21)
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These are precisely the same equations that were used in [34] (for zero four-dimensional cosmological constant). Finally, we record the following equation of [19]: f G, (A.22) 4 where W is the two-form bilinear defined in (3.10). Using this one can show that φ 4 6+φ/2 Re W = e2+ 2 Re K 3 , (A.23) e iK# 5 f D(e6 W ) = −e6 P ∧ W ∗ +
and furthermore that
φ 2+ 2 d e Re K 3 = i ξv H.
(A.24)
To see the latter one can derive an expression for the left hand side using, amongst other things, (3.18), (3.38) and (B.10) of [19], and an expression for the right hand side using Eq. (3.38) and (B.8) of [19]. Using these results we can deduce that L K # B = d(i K # b2 ), 5
5
L K # C2 =
d(i K # c2 ),
5
(A.25)
5
where b2 , c2 were introduced in (3.12), (3.13), respectively. B. More on the Generalized Vectors ξ and η In this appendix we derive an expression for the generalized vector ξ in terms of the bilinears introduced in [19]. We also use the results of [19] to show that Lξ J± = 0. The projections of ξ onto the vector and form parts (in a fixed trivialization of E) are denoted ξv , ξ f , respectively. It will also be convenient to introduce ξ B ≡ e B ξ whose form part is given by ξ fB = ξ f − i ξv B,
(B.1)
and we recall that ξvB = ξv . We next construct the following two generalized (1, 0)− vectors, which, by definition, are in the +i eigenspace of J− : Z 1− = r ∂r − iξ,
Z 2− = d log r − iη.
(B.2)
That is, both are in the annihilator of − . We may similarly also construct the (1, 0)+ vectors, with respect to J+ : Z 1+ = e−−φ/4 r ∂r − ie+φ/4 η, Z 2+ = e+φ/4 d log r − ie−−φ/4 ξ.
(B.3)
Together Z i± are four independent generalized vectors. We next note that since Z i± live within null isotropic subspaces we have six relations of the form, using the notation of (2.5), Z i± , Z ± j = 0.
(B.4)
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Explicitly we have i ξv ξ f = 0,
i ξv d log r = 0,
ir ∂r ξ f = 0,
i ηv η f = 0,
i ηv d log r = 0,
ir ∂r η f = 0,
ξ, η = 21 .
Since Z 1− annihilates − , using the definition (3.34) we deduce that ir ∂r − = i i ξv − + ξ fB ∧ − .
(B.5)
(B.6)
To proceed we use (3.20) to write
Since − =
2 − ≡ η+1 ⊗ η¯ − = e A χ1 χ¯ 2c ⊗ (σ3 + iσ1 ). 1 i 1 ...i n odd n n! i 1 ...i n γ
(B.7)
we have
i v − = 21 {v i γi , − },
ω ∧ − = 21 [ωi γ i , − ].
(B.8)
Hence, using the Clifford algebra decomposition (3.19) and metric (3.17) we have ir ∂r − = 21 {e+φ/4 γˆ6 , − } = 21 e A++φ/4 χ1 χ¯ 2c ⊗ {σ1 , σ3 + iσ1 } = ie A++φ/4 χ1 χ¯ 2c ⊗1. (B.9) On the other hand using (B.5) we have i ξv − + ξ fB ∧ − = 21 {e+φ/4 ξvm βm ⊗ σ3 , − } + 21 [e−−φ/4 ξ fBm β m ⊗ σ3 , − ] + m − = e+φ/4 vm β ⊗ σ3 − + e+φ/4 vm − β m ⊗ σ3 + m − = e A++φ/4 vm β (χ1 χ¯ 2c ) + vm (χ1 χ¯ 2c )β m ⊗ 1 + m − −e A++φ/4 vm β (χ1 χ¯ 2c ) − vm (χ1 χ¯ 2c )β m ⊗ σ2 ,
(B.10)
where recall that {βm , βn } = 2gY mn and we have defined ± = 21 (ξvm ± e−2−φ/2 ξ fBm ). vm
(B.11)
To satisfy (B.6) we thus require + m − β (χ1 χ¯ 2c ) = vm (χ1 χ¯ 2c )β m = 21 χ1 χ¯ 2c , vm
(B.12)
which implies + m β χ1 = 21 χ1 , vm
− m vm β χ2 = 21 χ2 ,
(B.13)
or equivalently + vm =
χ¯ 1 βm χ1 , 2χ¯ 1 χ1
− vm =
χ¯ 2 βm χ2 . 2χ¯ 2 χ2
(B.14)
Hence, given the normalizations (A.13) we deduce that, in terms of the bilinears defined in (3.9), ξv = K 5# , ξ fB = e2+φ/2 Re K 3 .
(B.15)
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A similar calculation using ir ∂r + = ie2+φ/2 i ηv − + η Bf ∧ + ,
(B.16)
leads to ηv = e−2−φ/2 Re K 3# , η Bf = K 5 .
(B.17)
Using the expression for the B-field given in (3.12) we obtain the expressions for ξ f and η f given in (3.50). In [19] it was shown that K 5 is a Killing one-form, so that its dual vector field K 5# , with respect to the metric gY on Y , is a Killing vector field. In fact K 5# generates a full symmetry of the supergravity solution, in that all bosonic fields (warp factor, dilaton, NS three-form H and RR fields) are preserved under the Lie derivative along ξv = K 5# . However, importantly, the Killing spinors ξ1 , ξ2 are not invariant under ξv . In [19] it was shown that Lξv S = −3iS,
(B.18)
where S ≡ ξ¯2c ξ1 . Notice that, since ξv preserves all of the bosonic fields, one may take the Lie derivative of the Killing spinor equations (A.1)-(A.6) for ξ1 , ξ2 along ξv , showing that {Lξv ξi } satisfy the same equations as the {ξi }. It thus follows that Lξv ξi = iμξi ,
(B.19)
where μ is a constant. Now (B.18) implies that 2μ = −3, and thus Lξv ξi = −
3i ξi . 2
(B.20)
One can also derive this last equation directly from the Killing spinor equations (A.1)– (A.6) of [19]. It thus follows that Lξv + = 0, Lξv − = −3i − .
(B.21) (B.22)
From (A.24) we have dξ fB = i ξv H and we deduce that Lξ B + = i ξv H ∧ + , Lξ B − = −3i − + i ξv H ∧ − .
(B.23)
Since (A.24) is also equivalent to dξ f = Lξv B we deduce that Lξ + = 0, Lξ − = −3i− ,
(B.24)
and hence Lξ J± = 0. It is also interesting to point out that (Lξ B − i ξv H ∧)F = 0, or equivalently, Lξ (e−B F) = 0.
(B.25)
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C. The Sasaki-Einstein Case Here we discuss the special case in which the compact five-manifold Y is SasakiEinstein. Setting G = P = Q = 0, f = 4e4 and ξ2 = 0, the Killing spinor equations (A.1)-(A.6) reduce to i ∇m ξ1 + βm ξ1 = 0. (C.1) 2 In terms of Appendix B of [19] we choose θ¯ = φ¯ = 0 and e2iα¯ = −1 (these angles had no bars on them in [19]). We then have the equalities 1 ξ¯1 β(1) ξ1 = K 5 = e1 , 2 i = ξ¯1 β(2) ξ1 = −V = e25 + e43 , 2 1 = ξ¯1 β(2) ξ1c = (e2 + ie5 ) ∧ (e4 + ie3 ), 2
η= ωKE KE
(C.2)
and dη = 2ωKE , dKE = 3iη ∧ KE .
(C.3)
Observe that 1 2 ω = −e12345 = v olY . 2! KE Next using the 5D-6D map (3.20), we obtain η∧
iη¯ +1 γ(2) η+1 = r (d log r ∧ e1 + ωKE ) ≡
(C.4)
1 ωCY , r
−iη¯ +1c γ(3) η+1 = r (d log r − ie1 )(e2 − ie5 )(e4 − ie3 ) ≡
1 ¯ CY . r2
(C.5)
It is worth noting that 1 2 1 3 ω = r 6 e123456 = r 6 d log r ∧ η ∧ ωKE , 3! CY 2!
(C.6)
where e6 = dr/r . We also find, directly from (3.33), (3.36), 1 ¯ CY , 8 i ir 3 exp 2 ωCY . + = − 8 r
− =
(C.7)
A useful check is that these expressions agree with those obtained from the general expressions obtained in Sects. 5.1 and 6.1, respectively. ˜ 1 and ˜ 2 , as defined in One can also write down the corresponding reduced structures Sect. 4.3, on the Kähler-Einstein space. One finds ˜1 =
1 3iψ ¯ KE , e 8
i ˜ 2 = − eiωKE , 8
where ψ is the coordinate, defined such that K 5# = ∂ψ , introduced in (4.31).
(C.8)
Ad S5 Solutions of Type IIB Supergravity and Generalized Complex Geometry
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References 1. Gauntlett, J.P., Martelli, D., Sparks, J., Waldram, D.: Supersymmetric Ad S5 solutions of M-theory. Class. Quant. Grav. 21, 4335 (2004) 2. Gauntlett, J.P., Martelli, D., Sparks, J., Waldram, D.: Sasaki-Einstein metrics on S2 × S3 . Adv. Theor. Math. Phys. 8, 711 (2004) 3. Cvetic, M., Lu, H., Page, D.N., Pope, C.N.: New Einstein-Sasaki spaces in five and higher dimensions. Phys. Rev. Lett. 95, 071101 (2005) 4. Benvenuti, S., Franco, S., Hanany, A., Martelli, D., Sparks, J.: An infinite family of superconformal quiver gauge theories with Sasaki-Einstein duals. JHEP 0506, 064 (2005) 5. Hanany, A., Kennaway, K.D.: Dimer models and toric diagrams. http://arXiv.org/abs/hep-th/0503149v2, 2005 6. Franco, S., Hanany, A., Kennaway, K.D., Vegh, D., Wecht, B.: Brane Dimers and Quiver Gauge Theories. JHEP 0601, 096 (2006) 7. Franco, S., Hanany, A., Martelli, D., Sparks, J., Vegh, D., Wecht, B.: Gauge theories from toric geometry and brane tilings. JHEP 0601, 128 (2006) 8. Anselmi, D., Erlich, J., Freedman, D.Z., Johansen, A.A.: Positivity constraints on anomalies in supersymmetric gauge theories. Phys. Rev. D 57, 7570 (1998) 9. Intriligator, K.A., Wecht, B.: The exact superconformal R-symmetry maximizes a. Nucl. Phys. B 667, 183 (2003) 10. Martelli, D., Sparks, J., Yau, S.T.: The geometric dual of a-maximisation for toric Sasaki-Einstein manifolds. Commun. Math. Phys. 268, 39 (2006) 11. Martelli, D., Sparks, J., Yau, S.T.: Sasaki-Einstein manifolds and volume minimisation. Commun. Math. Phys. 280, 611 (2008) 12. Lunin, O., Maldacena, J.M.: Deforming field theories with U (1)×U (1) global symmetry and their gravity duals. JHEP 0505, 033 (2005) 13. Pilch, K., Warner, N.P.: A new supersymmetric compactification of chiral IIB supergravity. Phys. Lett. B 487, 22 (2000) 14. Khavaev, A., Pilch, K., Warner, N.P.: New vacua of gauged N = 8 supergravity in five dimensions. Phys. Lett. B 487, 14 (2000) 15. Halmagyi, N., Pilch, K., Romelsberger, C., Warner, N.P.: Holographic duals of a family of N = 1 fixed points. JHEP 0608, 083 (2006) 16. Leigh, R.G., Strassler, M.J.: Exactly Marginal Operators And Duality In Four-Dimensional N = 1 Supersymmetric Gauge Theory. Nucl. Phys. B 447, 95 (1995) 17. Benvenuti, S., Hanany, A.: Conformal manifolds for the conifold and other toric field theories. JHEP 0508, 024 (2005) 18. Aharony, O., Kol, B., Yankielowicz, S.: On exactly marginal deformations of N = 4 SYM and type IIB supergravity on Ad S5 × S 5 . JHEP 0206, 039 (2002) 19. Gauntlett, J.P., Martelli, D., Sparks, J., Waldram, D.: Supersymmetric Ad S5 solutions of type IIB supergravity. Class. Quant. Grav. 23, 4693 (2006) 20. Minasian, R., Petrini, M., Zaffaroni, A.: Gravity duals to deformed SYM theories and generalized complex geometry. JHEP 0612, 055 (2006) 21. Butti, A., Forcella, D., Martucci, L., Minasian, R., Petrini, M., Zaffaroni, A.: On the geometry and the moduli space of beta-deformed quiver gauge theories. JHEP 0807, 053 (2008) 22. Grana, M., Minasian, R., Petrini, M., Tomasiello, A.: Supersymmetric backgrounds from generalized Calabi-Yau manifolds. JHEP 0408, 046 (2004) 23. Grana, M., Minasian, R., Petrini, M., Tomasiello, A.: Generalized structures of N = 1 vacua. JHEP 0511, 020 (2005) 24. Hitchin, N.: Generalized Calabi-Yau manifolds. Quart. J. Math. Oxford Ser. 54, 281–308 (2003) 25. Gauntlett, J.P., Martelli, D., Sparks, J.F., Waldram, D.: A new infinite class of Sasaki-Einstein manifolds. Adv. Theor. Math. Phys. 8, 987 (2006) 26. Bursztyn, H., Cavalcanti, G., Gualtieri, M.: Reduction of Courant algebroids and generalized complex structures. Adv. Math. 211, 726 (2007) 27. Bursztyn, H., Cavalcanti, G., Gualtieri, M.: Generalized Kahler and hyper-Kahler quotients. In: Poisson geometry in mathematics and physics, Contemp. Math., 450, pp. 61–77. American Mathematics Society, Providence (2008) 28. Gabella, M., Gauntlett, J.P., Palti, E., Sparks, J., Waldram, D.: The central charge of supersymmetric AdS5 solutions of type IIB supergravity. Phys. Rev. Lett. 103, 051601 (2009) 29. Gualtieri, M.: Generalized complex geometry. http://arXiv.org/abs/math/0703298v2[math.DG], 2007 30. Hitchin, N.: Brackets, forms and invariant functionals. Asian J. Math. 10(3), 541 (2006) 31. Grana, M., Minasian, R., Petrini, M., Waldram, D.: T-duality, Generalized Geometry and Non-Geometric Backgrounds. JHEP 0904, 075 (2009)
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32. Cavalcanti, G.R.: New aspects of the dd c -lemma. Oxford Univ., D. Phil. thesis, http://arXiv.org/abs/ math/0501406v1[math.DG], 2005 33. Gualtieri, M.: Generalized complex geometry. Oxford University DPhil thesis, http://arXiv.org/abs/math/ 0401221v1[math-DG], 2004 34. Grana, M., Minasian, R., Petrini, M., Tomasiello, A.: A scan for new N = 1 vacua on twisted tori. JHEP 0705, 031 (2007) 35. Martucci, L., Smyth, P.: Supersymmetric D-branes and calibrations on general N = 1 backgrounds. JHEP 0511, 048 (2005) 36. Tomasiello, A.: Reformulating Supersymmetry with a Generalized Dolbeault Operator. JHEP 0802, 010 (2008) 37. Martucci, L.: D-branes on general N = 1 backgrounds: Superpotentials and D-terms. JHEP 0606, 033 (2006) 38. Henningson, M., Skenderis, K.: The holographic Weyl anomaly. JHEP 9807, 023 (1998) 39. Martelli, D., Sparks, J.: Baryonic branches and resolutions of Ricci-flat Kahler cones. JHEP 0804, 067 (2008) 40. Klebanov, I.R., Murugan, A.: Gauge/Gravity Duality and Warped Resolved Conifold. JHEP 0703, 042 (2007) 41. Halmagyi, N., Pilch, K., Romelsberger, C., Warner, N.P.: The complex geometry of holographic flows of quiver gauge theories. JHEP 0609, 063 (2006) Communicated by A. Kapustin
Commun. Math. Phys. 299, 409–446 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1075-y
Communications in
Mathematical Physics
The Pentagram Map: A Discrete Integrable System Valentin Ovsienko1 , Richard Schwartz2 , Serge Tabachnikov3 1 CNRS, Institut Camille Jordan, Université Lyon 1, Villeurbanne Cedex 69622,
France. E-mail:
[email protected] 2 Department of Mathematics, Brown University, Providence,
RI 02912, USA. E-mail:
[email protected] 3 Department of Mathematics, Pennsylvania State University, University Park,
PA 16802, USA. E-mail:
[email protected] Received: 14 October 2009 / Accepted: 4 February 2010 Published online: 24 June 2010 – © Springer-Verlag 2010
Dedicated to the memory of V. Arnold Abstract: The pentagram map is a projectively natural transformation defined on (twisted) polygons. A twisted polygon is a map from Z into RP2 that is periodic modulo a projective transformation called the monodromy. We find a Poisson structure on the space of twisted polygons and show that the pentagram map relative to this Poisson structure is completely integrable. For certain families of twisted polygons, such as those we call universally convex, we translate the integrability into a statement about the quasi-periodic motion for the dynamics of the pentagram map. We also explain how the pentagram map, in the continuous limit, corresponds to the classical Boussinesq equation. The Poisson structure we attach to the pentagram map is a discrete version of the first Poisson structure associated with the Boussinesq equation. A research announcement of this work appeared in [16]. Contents 1. 2.
3.
Introduction . . . . . . . . . . . Proof of the Main Theorem . . . 2.1 Coordinates for the space . . 2.2 A formula for the map . . . . 2.3 The monodromy invariants . 2.4 Formulas for the invariants . 2.5 The Poisson bracket . . . . . 2.6 The corank of the structure . 2.7 The end of the proof . . . . . Quasi-periodic Motion . . . . . . 3.1 Universally convex polygons 3.2 The Hilbert perimeter . . . . 3.3 Compactness of the level sets 3.4 Proof of Theorem 2 . . . . .
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3.5 Hyperbolic cylinders and tight polygons . . . . . 3.6 A related theorem . . . . . . . . . . . . . . . . . Another Coordinate System in Space Pn . . . . . . . 4.1 Polygons and difference equations . . . . . . . . 4.2 Relation between the two coordinate systems . . 4.3 Two versions of the projective duality . . . . . . 4.4 Explicit formula for α . . . . . . . . . . . . . . . 4.5 Recurrent formula for β . . . . . . . . . . . . . . 4.6 Formulas for the pentagram map . . . . . . . . . 4.7 The Poisson bracket in the (a, b)-coordinates . . Monodromy Invariants in (a, b)-Coordinates . . . . . 5.1 Monodromy matrices . . . . . . . . . . . . . . . 5.2 Combinatorics of the monodromy invariants . . . 5.3 Closed polygons . . . . . . . . . . . . . . . . . . Continuous Limit: The Boussinesq Equation . . . . . 6.1 Non-degenerate curves and differential operators 6.2 Continuous limit of the pentagram map . . . . . . 6.3 The constant Poisson structure . . . . . . . . . . 6.4 Discretization . . . . . . . . . . . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction The notion of integrability is one of the oldest and most fundamental notions in mathematics. The origins of integrability lie in classical geometry and the development of the general theory is always stimulated by the study of concrete integrable systems. The purpose of this paper is to study one particular dynamical system that has a simple and natural geometric meaning and to prove its integrability. Our main tools are mostly geometric: the Poisson structure, first integrals and the corresponding Lagrangian foliation. We believe that our result opens doors for further developments involving other approaches, such as Lax representation, algebraic-geometric and complex analysis methods, Bäcklund transformations; we also expect further generalizations and relations to other fields of modern mathematics, such as cluster algebras theory. The pentagram map, T , was introduced in [19], and further studied in [20 and 21]. Originally, the map was defined for convex closed n-gons. Given such an n-gon P, the corresponding n-gon T (P) is the convex hull of the intersection points of consecutive shortest diagonals of P. Figure 1 shows the situation for a convex pentagon and a convex hexagon. One may consider the map as defined either on unlabelled polygons or on labelled polygons. Later on, we shall consider the labelled case in detail. The pentagram map already has some surprising features in the cases n = 5 and n = 6. When P is a pentagon, there is a projective transformation carrying P to T (P). This is a classical result, cf. [15]; one of us learned of this result from John Conway in 1987. When P is a hexagon, there is a projective transformation carrying P to T 2 (P). It is not clear whether this result was well-known to classical projective geometers, but it is easy enough to prove. The name pentagram map stems from the fact that the pentagon is the simplest kind of polygon for which the map is defined. Letting C n denote the space of convex n-gons modulo projective transformations, we can say that the pentagram map is periodic on C n for n = 5, 6. The pentagram map certainly is not periodic on C n for n ≥ 7. Computer experiments suggest that the
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P
P T(P)
T(P)
Fig. 1. The pentagram map defined on a pentagon and a hexagon
pentagram map on C n in general displays the kind of quasi-periodic motion one sees in completely integrable systems. Indeed, this was conjectured (somewhat loosely) in [21]. See the remarks following Theorem 1.2 in [21]. It is the purpose of this paper to establish the complete integrability conjectured in [21] and to explain the underlying quasi-periodic motion. However, rather than work with closed n-gons, we will work with what we call twisted n-gons. A twisted n-gon is a map φ : Z → RP2 such that that φ(k + n) = M ◦ φ(k); ∀k. Here M is some projective automorphism of RP2 . We call M the monodromy. For technical reasons, we require that every 3 consecutive points in the image are in general position – i.e., not collinear. When M is the identity, we recover the notion of a closed n-gon. Two twisted n-gons φ1 and φ2 are equivalent if there is some projective transformation such that ◦ φ1 = φ2 . The two monodromies satisfy M2 = M1 −1 . Let Pn denote the space of twisted n-gons modulo equivalence. Let us emphasise that the full space of twisted n-gons (rather than the geometrically natural but more restricted space of closed n-gons) is much more natural in the general context of the integrable systems theory. Indeed, in the “smooth case” it is natural to consider the full space of linear differential equations; the monodromy then plays an essential rôle in producing the invariants. This viewpoint is adopted by many authors (see [9,14] and references therein) and this is precisely our viewpoint in the discrete case. The pentagram map is generically defined on Pn . However, the lack of convexity makes it possible that the pentagram map is not defined on some particular point of Pn , or that the image of a point in Pn under the pentagram map no longer belongs to Pn . That is, we can lose the 3-in-a-row property that characterizes twisted polygons. We will put coordinates in Pn so that the pentagram map becomes a rational map. At least when n is not divisible by 3, the space Pn is diffeomorphic to R2n . When n is divisible by 3, the topology of the space is trickier, but nonetheless large open subsets of Pn in this case are still diffeomorphic to open subsets of R2n . (Since our map is only generically defined, the fine points of the global topology of Pn are not so significant.) The action of the pentagram map in Pn was studied extensively in [21]. In that paper, it was shown that for every n this map has a family of invariant functions, the so-called weighted monodromy invariants. There are exactly 2[n/2] + 2 algebraically independent
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invariants. Here [n/2] denotes the floor of n/2. When n is odd, there are two exceptional monodromy functions that are somewhat unlike the rest. When n is even, there are 4 such exceptional monodromy functions. We will recall the explicit construction of these invariants in the next section, and sketch the proofs of some of their properties. Later on in the paper, we shall give a new treatment of these invariants. Here is the main result of this paper. Theorem 1. There exists a Poisson structure on Pn having co-rank 2 when n is odd and co-rank 4 when n is even. The exceptional monodromy functions generically span the null space of the Poisson structure, and the remaining monodromy invariants Poissoncommute. Finally, the Poisson structure is invariant under the pentagram map. The exceptional monodromy functions are precisely the Casimir functions for the Poisson structure. The generic level set of the Casimir functions is a smooth symplectic manifold. Indeed, as long as we keep all the values of the Casimir functions nonzero, the corresponding level sets are smooth symplectic manifolds. The remaining monodromy invariants, when restricted to the symplectic level sets, define a singular Lagrangian foliation. Generically, the dimension of the Lagrangian leaves is precisely the same as the number of remaining monodromy invariants. This is the classical picture of Arnold-Liouville complete integrability. As usual in this setting, the complete integrability gives an invariant affine structure to every smooth leaf of the Lagrangian foliation. Relative to this structure, the pentagram map is a translation. Hence Corollary 1.1. Suppose that P is a twisted n-gon that lies on a smooth Lagrangian leaf and has a periodic orbit under the pentagram map. If P is any twisted n-gon on the same leaf, then P also has a periodic orbit with the same period, provided that the orbit of P is well-defined. Remark 1.2. In the result above, one can replace the word periodic with ε-periodic. By this we mean that we fix a Euclidean metric on the leaf and measure distances with respect to this metric. We shall not analyze the behavior of the pentagram map on C n . One of the difficulties in analyzing the space C n of closed convex polygons modulo projective transformations is that this space has positive codimension in Pn (codimension 8). We do not know in enough detail how the Lagrangian singular foliation intersects C n , and so we cannot appeal to the structure that exists on generic leaves. How the monodromy invariants behave when restricted to C n is a subtle and interesting question that we do not yet fully know how to answer (see Theorem 4 for a partial result). We hope to tackle the case of closed n-gons in a sequel paper. One geometric setting where our machine works perfectly is the case of universally convex n-gons. This is our term for a twisted n-gon whose image in RP2 is strictly convex. The monodromy of a universally convex n-gon is necessarily an element of P G L 3 (R) that lifts to a diagonalizable matrix in S L 3 (R). A universally convex polygon essentially follows along one branch of a hyperbola-like curve. Let U n denote the space of universally convex n-gons, modulo equivalence. We will prove that U n is an open subset of P n locally diffeomorphic to R2n . Further, we will see that the pentagram map is a self-diffeomorphism of U 2n . Finally, we will see that every leaf in the Lagrangian foliation intersects U n in a compact set. Combining these results with our Main Theorem and some elementary differential topology, we arrive at the following result.
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Theorem 2. Almost every point of U n lies on a smooth torus that has a T -invariant affine structure. Hence, the orbit of almost every universally convex n-gon undergoes quasi-periodic motion under the pentagram map. We will prove a variant of Theorem 2 for a different family of twisted n-gons. See Theorem 3. The general idea is that certain points of Pn can be interpreted as embedded, homologically nontrivial, locally convex polygons on projective cylinders, and suitable choices of geometric structure give us the compactness we need for the proof. Here we place our results in a context. First of all, it seems that there is some connection between our work and cluster algebras. On the one hand, the space of twisted polygons is known as an example of cluster manifold, see [6,7] and discussion in the end of this paper. This implies in particular that Pn is equipped with a canonical Poisson structure, see [10]. We do not know if the Poisson structure constructed in this paper coincides with the canonical cluster Poisson structure. On the other hand, it was shown in [21] that a certain change of coordinates brings the pentagram map rather closely in line with the octahedral recurrence, which is one of the prime examples in the theory of cluster algebras, see [11,18,22]. Second of all, there is a close connection between the pentagram map and integrable P.D.E.s. In the last part of this paper we consider the continuous limit of the pentagram map. We show that this limit is precisely the classical Boussinesq equation which is one of the best known infinite-dimensional integrable systems. Moreover, we argue that the Poisson bracket constructed in the present paper is a discrete analog of the so-called first Poisson structure of the Boussinesq equation. We remark that a connection to the Boussinesq equation was mentioned in [19], but no derivation was given. Discrete integrable systems is an actively developing subject, see, e.g., [26] and the books [5,23]. The paper [4] discusses a well-known discrete version (but with continuous time) of the Boussinesq equation; see [25] (and references therein) for a lattice version of this equation. See [9] (and references therein) for a general theory of integrable difference equations. Let us stress that the r -matrix Poisson brackets considered in [9] are analogous to the second (i.e., the Gelfand-Dickey) Poisson bracket. A geometric interpretation of all the discrete integrable systems considered in the above references is unclear. In the geometrical setting which is closer to our viewpoint, see [5] for many interesting examples. The papers [1,2] considers a discrete integrable system on the space of n-gons, different from the pentagram map. The recent paper [13] considers a discrete integrable systems in the setting of projective differential geometry; some of the formulas in this paper are close to ours. Finally, we mention [12,14] for discrete and continuous integrable systems related both to Poisson geometry and projective differential geometry on the projective line. We turn now to a description of the contents of the paper. Essentially, our plan is to make a bee-line for all our main results, quoting earlier work as much as possible. Then, once the results are all in place, we will consider the situation from another point of view, proving many of the results quoted in the beginning. One of the disadvantages of the paper [21] is that many of the calculations are ad hoc and done with the help of a computer. Even though the calculations are correct, one is not given much insight into where they come from. In this paper, we derive everything in an elementary way, using an analogy between twisted polygons and solutions to periodic ordinary differential equations. One might say that this paper is organized along the lines of first the facts, then the reasons. Accordingly, there is a certain redundancy in our treatment. For instance, we
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introduce two natural coordinate systems in Pn . In the first coordinate system, which comes from [21], most of the formulas are simpler. However, the second coordinate system, which is new, serves as a kind of engine that drives all the derivations in both coordinate systems; this coordinate system is better for computation of the monodromy too. Also, we discovered the invariant Poisson structure by thinking about the second coordinate system. In §2 we introduce the first coordinate system, describe the monodromy invariants, and establish the Main Theorem. In §3 we apply the main theorem to universally convex polygons and other families of twisted polygons. In §4 we introduce the second coordinate system. In §4 and 5 we use the second coordinate system to derive many of the results we simply quoted in §2. Finally, in §6 we use the second coordinate system to derive the continuous limit of the pentagram map. 2. Proof of the Main Theorem 2.1. Coordinates for the space. In this section, we introduce our first coordinate system on the space of twisted polygons. As we mentioned in the Introduction, a twisted n-gon is a map φ : Z → RP2 such that φ(n + k) = M ◦ φ(k)
(2.1)
for some projective transformation M and all k. We let vi = φ(i). Thus, the vertices of our twisted polygon are naturally . . . vi−1 , vi , vi+1 , . . .. Our standing assumption is that vi−1 , vi , vi+1 are in general position for all i, but sometimes this assumption alone will not be sufficient for our constructions. The cross ratio is the most basic invariant in projective geometry. Given four points t1 , t2 , t3 , t4 ∈ RP1 , the cross-ratio [t1 , t2 , t3 , t4 ] is their unique projective invariant. The explicit formula is as follows. Choose an arbitrary affine parameter, then [t1 , t2 , t3 , t4 ] =
(t1 − t2 ) (t3 − t4 ) . (t1 − t3 ) (t2 − t4 )
(2.2)
This expression is independent of the choice of the affine parameter, and is invariant under the action of PGL(2, R) on RP1 . Remark 2.1. Many authors define the cross ratio as the multiplicative inverse of the formula in Eq. 2.2. Our definition, while perhaps less common, better suits our purposes. The cross-ratio was used in [21] to define a coordinate system on the space of twisted n-gons. As the reader will see from the definition, the construction requires somewhat more than 3 points in a row to be in general position. Thus, these coordinates are not entirely defined on our space Pn . However, they are generically defined on our space, and this is sufficient for all our purposes. The construction is as follows, see Fig. 2. We associate to every vertex vi two numbers: xi = vi−2 , vi−1 , ((vi−2 , vi−1 ) ∩ (vi , vi+1 )) , ((vi−2 , vi−1 ) ∩ (vi+1 , vi+2 )) , (2.3) yi = [((vi−2 , vi−1 ) ∩ (vi+1 , vi+2 )), ((vi−1 , vi ) ∩ (vi+1 , vi+2 )), vi+1 , vi+2 ], called the left and right corner cross-ratios. We often call our coordinates the corner invariants.
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v
vi+2
i−2
v
vi+1
i−1
vi
Fig. 2. Points involved in the definition of the invariants
Clearly, the construction is PGL(3, R)-invariant and, in particular, xi+n = xi and yi+n = yi . We therefore obtain a (local) coordinate system that is generically defined on the space Pn . In [21], §4.2, we show how to reconstruct a twisted n-gon from its sequence of invariants. The reconstruction is only canonical up to projective equivalence. Thus, an attempt to reconstruct φ from x1 , y1 , . . . , perhaps would lead to an unequal but equivalent twisted polygon. This does not bother us. The following lemma is nearly obvious. Lemma 2.2. At generic points, the space Pn is locally diffeomorphic to R2n . Proof. We can perturb our sequence x1 , y1 , . . . in any way we like to get a new sequence x1 , y1 , . . . . If the perturbation is small, we can reconstruct a new twisted n-gon φ that is near φ in the following sense. There is a projective transformation such that n-consecutive vertices of (φ ) are close to the corresponding n consecutive vertices of φ. In fact, if we normalize so that a certain quadruple of consecutive points of (φ ) match the corresponding points of φ, then the remaining points vary smoothly and algebraically with the coordinates. The map (x1 , y2 , . . . , xn , yn ) → [φ ] (the class of φ ) gives the local diffeomorphism.
Remark 2.3. (i) Later on in the paper, we will introduce new coordinates on all of Pn and show, with these new coordinates, that Pn is globally diffeomorphic to R2n when n is not divisible by 3. (ii) The actual lettering we use here to define our coordinates is different from the lettering used in [21]. Here is the correspondense: . . . p1 , q 2 , p3 , q 4 . . .
⇐⇒
. . . , x1 , y1 , x2 , y2 , . . . .
2.2. A formula for the map. In this section, we express the pentagram map in the coordinates we have introduced in the previous section. To save words later, we say now that we will work with generic elements of Pn , so that all constructions are well-defined. Let φ ∈ Pn . Consider the image, T (φ), of φ under the pentagram map. One difficulty in making this definition is that there are two natural choices for labelling T (φ), the left
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left
right
3
3 4
2 3
4
2 2
4 5
1
5
3 4
1
5
Fig. 3. Left and right labelling schemes
choice and the right choice. These choices are shown in Fig. 3. In the picture, the black dots represent the vertices of φ and the white dots represent the vertices of T (φ). The labelling continues in the obvious way. If one considers the square of the pentagram map, the difficulty in making this choice goes away. However, for most of our calculations it is convenient for us to arbitrarily choose right over left and consider the pentagram map itself and not the square of the map. Henceforth, we make this choice. Lemma 2.4. Suppose the coordinates for φ are x1 , y1 , . . . then the coordinates for T (φ) are 1 − xi−1 yi−1 1 − xi+2 yi+2 T ∗ xi = xi , T ∗ yi = yi+1 , (2.4) 1 − xi+1 yi+1 1 − xi yi where T ∗ is the standard pull-back of the (coordinate) functions by the map T . In [21], Eq. 7, we express the squared pentagram map as the product of two involutions on R2n , and give coordinates. From this equation one can deduce the formula in Lemma 2.4 for the pentagram map itself. Alternatively, later in the paper we will give a self-contained proof of Lemma 2.4. Lemma 2.4 has two corollaries, which we mention here. These corollaries are almost immediate from the formula. First, there is an interesting scaling symmetry of the pentagram map. We have a rescaling operation on R2n , given by the expression Rt :
(x1 , y1 , . . . , xn , yn ) → (t x1 , t −1 y1 , . . . , t xn , t −1 yn ).
(2.5)
Corollary 2.5. The pentagram map commutes with the rescaling operation. Second, the formula for the pentagram map exhibits rather quickly some invariants of the pentagram map. When n is odd, define On =
n
xi ;
En =
i=1
When n is even, define On/2 =
i even
xi +
n
yi .
(2.6)
i=1
xi ,
E n/2 =
i odd
The products in this last equation run from 1 to n.
i even
yi +
i odd
yi .
(2.7)
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Corollary 2.6. When n is odd, the functions On and E n are invariant under the pentagram map. When n is even, the functions On/2 and E n/2 are also invariant under the pentagram map. These functions are precisely the exceptional invariants we mentioned in the Introduction. They turn out to be the Casimirs for our Poisson structure. 2.3. The monodromy invariants. In this section we introduce the invariants of the pentagram map that arise in Theorem 1. The invariants of the pentagram map were defined and studied in [21]. In this section we recall the original definition. Later on in the paper, we shall take a different point of view and give self-contained derivations of everything we say here. As above, let φ be a twisted n-gon with invariants x1 , y1 , . . .. Let M be the monodromy of φ. We lift M to an element of G L 3 (R). By slightly abusing notation, we also denote this matrix by M. The two quantities 1 =
trace3 (M) trace3 (M −1 ) ; 2 = ; det(M) det(M −1 )
(2.8)
enjoy 3 properties. • 1 and 2 are independent of the lift of M. • 1 and 2 only depend on the conjugacy class of M. • 1 and 2 are rational functions in the corner invariants. We define 1 = On2 E n 1 ; 2 = On E n2 2 .
(2.9)
1 and 2 are polynomials in the corner invariants. Since the penIn [21] it is shown that tagram map preserves the monodromy, and On and E n are invariants, the two functions 1 and 2 are also invariants. We say that a polynomial in the corner invariants has weight k if we have the following equation: Rt∗ (P) = t k P.
(2.10)
Here Rt∗ denotes the natural operation on polynomials defined by the rescaling operation (2.5). For instance, On has weight n and E n has weight −n. In [21] it is shown that 1 =
[n/2] k=1
2 = Ok ;
[n/2]
Ek ,
(2.11)
k=1
where Ok has weight k and E k has weight −k. Since the pentagram map commutes 1 and 2 , it also preserves their “weighted with the rescaling operation and preserves homogeneous parts”. That is, the functions O1 , E 1 , O2 , E 2 , . . . are also invariants of the pentagram map. These are the monodromy invariants. They are all nontrivial polynomials. Algebraic Independence: In [21], §6, it is shown that the monodromy invariants are algebraically independent provided that, in the even case, we ignore On/2 and E n/2 . We will not reproduce the proof in this paper, so here we include a brief description of the
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argument. Since we are mainly trying to give the reader a feel for the argument, we will explain a variant of the method in [21]. Let f 1 , . . . , f k be the complete list of invariants we have described above. Here k = 2[n/2] + 2. If our functions were not algebraically independent, then the gradients ∇ f 1 , . . . , ∇ f k would never be linearly independent. To rule this out, we just have to establish the linear independence at a single point. One can check this at the point (1, ω, . . . , ω2n ), where ω is a (4n)th root of unity. The actual method in [21] is similar to this, but uses a trick to make the calculation easier. Given the formulas for the invariants we present below, this calculation is really just a matter of combinatorics. Perhaps an easier calculation can be made for the point (0, 1, . . . , 1), which also seems to work for all n. 2.4. Formulas for the invariants. In this section, we recall the explicit formulas for the monodromy invariants given in [21]. Later on in the paper, we will give a self-contained derivation of the formulas. From the point of view of our main theorems, we do not need to know the formulas, but only their algebraic independence and Lemma 2.8 below. We introduce the monomials X i := xi yi xi+1 .
(2.12)
1. We call two monomials X i and X j consecutive if j ∈ {i − 2, i − 1, i, i + 1, i + 2} ; 2. we call X i and x j consecutive if j ∈ {i − 1, i, i + 1, i + 2} ; 3. we call xi and xi+1 consecutive. Let O(X, x) be a monomial obtained by the product of the monomials X i and x j , i.e., O = X i1 · · · X is x j1 · · · x jt . Such a monomial is called admissible if no two of the indices are consecutive. For every admissible monomial, we define the weight |O| and sign(O) by |O| := s + t,
sign(O) := (−1)t .
With these definitions, it turns out that n
. Ok = sign(O) O; k ∈ 1, 2, . . . , 2
(2.13)
|O|=k
The same formula works for E k , if we make all the same definitions with x and y interchanged. Example 2.7. For n = 5 one obtains the following polynomials: O1 =
5 i=1
(xi yi xi+1 − xi ) ,
O2 =
5
(xi xi+2 − xi yi xi+1 xi+3 )
i=1
together with O5 . Now we mention the needed symmetry property. Let τ be the involution on the indices: xi → x1−i τ: mod n. (2.14) yi → y−i Then τ acts on the variables, monomials and polynomials.
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Lemma 2.8. One has τ (Ok ) = Ok . Proof. τ takes an admissible partition to an admissible one and does not change the number of singletons involved.
2.5. The Poisson bracket. In this section, we introduce the Poisson bracket on Pn . Let Cn∞ denote the algebra of smooth functions on R2n . A Poisson structure on Cn∞ is a map { , } : Cn∞ × Cn∞ → Cn∞
(2.15)
that obeys the following axioms: 1. 2. 3. 4.
Antisymmetry: { f, g} = −{g, f }. Linearity: {a f 1 + f 2 , g} = a{ f 1 , g} + { f 2 , g}. Leibniz Identity: { f, g1 g2 } = g1 { f, g2 } + g2 { f, g1 }. Jacobi Identity: { f 1 , { f 2 , f 3 }} = 0.
Here denotes the cyclic sum. We define the following Poisson bracket on the coordinate functions of R2n : {xi , xi±1 } = ∓xi xi+1 ,
{yi , yi±1 } = ±yi yi+1 .
(2.16)
All other brackets not explicitly mentioned above vanish. For instance {xi , y j } = 0; ∀ i, j. Once we have the definition on the coordinate functions, we use linearity and the Liebniz rule to extend to all rational functions. Though it is not necessary for our purposes, we can extend to all smooth functions by approximation. Our formula automatically builds in the anti-symmetry. Finally, for a “homogeneous bracket” as we have defined, it is well-known (and an easy exercise) to show that the Jacobi identity holds. Henceforth we refer to the Poisson bracket as the one that we have defined above. Now we come to one of the central results in the paper. This result is our main tool for establishing the complete integrability and the quasi-periodic motion. Lemma 2.9. The Poisson bracket is invariant with respect to the pentagram map. Proof. Let T ∗ denote the action of the pentagram map on rational functions. One has to prove that for any two functions f and g one has {T ∗ ( f ), T ∗ (g)} = { f, g} and of course it suffices to check this fact for the coordinate functions. We will use the explicit formula (2.4). To simplify the formulas, we introduce the following notation: ϕi = 1−xi yi . Lemma 2.4 then reads: ϕi−1 ϕi+2 T ∗ (xi ) = xi , T ∗ (yi ) = yi+1 . ϕi+1 ϕi One easily checks that {ϕi , ϕ j } = 0 for all i, j. Next,
{xi , ϕ j } = δi, j−1 − δi, j+1 xi x j y j , {yi , ϕ j } = δi, j+1 − δi, j−1 x j yi y j . In order to check the T -invariance of the bracket, one has to check that the relations between the functions T ∗ (xi ) and T ∗ (y j ) are the same as for xi and y j . The first relation to check is: {T ∗ (xi ), T ∗ (y j )} = 0.
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Indeed,
y j+1 ϕi−1 y j+1 ϕi−1 ϕ j+2 T ∗ (xi ), T ∗ (y j ) = {xi , ϕ j+2 } − {xi , ϕ j } ϕi+1 ϕ j ϕi+1 ϕ 2j xi ϕ j+2 xi ϕi−1 ϕ j+2 −{y j+1 , ϕi−1 } + {y j+1 , ϕi+1 } ϕi+1 ϕ j ϕ2 ϕ j
xi x j+2 y j+2 y j+1 ϕi−1 i+1 = δi, j+1 − δi, j+3 ϕi+1 ϕ j
xi x j y j y j+1 ϕi−1 ϕ j+2 − δi, j−1 − δi, j+1 ϕi+1 ϕ 2j
xi−1 y j+1 yi−1 xi ϕ j+2 − δ j+1,i − δ j+1,i−2 ϕi+1 ϕ j
xi+1 y j+1 yi+1 xi ϕi−1 ϕ j+2 + δ j+1,i+2 − δ j+1,i 2 ϕ ϕi+1 j = 0,
since the first term cancels with the third and the second with the last one. One then computes {T ∗ (xi ), T ∗ (x j )} and {T ∗ (yi ), T ∗ (y j )}, the computations are similar to the above one and will be omitted.
Two functions f and g are said to Poisson commute if { f, g} = 0. Lemma 2.10. The monodromy invariants Poisson commute. Proof. Let τ by the involution on the indices defined at the end of the last section. We have τ (Ok ) = Ok by Lemma 2.8. We make the following claim: For all polynomials f (x, y) and g(x, y), one has {τ ( f ), τ (g)} = −{ f, g}. Assuming this claim, we have {Ok , Ol } = {τ (Ok ), τ (Ol )} = −{Ok , Ol }, hence the bracket is zero. The same argument works for {E k , El } and {Ok , El }. Now we prove our claim. It suffices to check the claim when f and g are monomials in variables (x, y). In this case, we have: { f, g} = C f g, where C is the sum of ±1, corresponding to “interactions” between factors xi in f and x j in g (resp. yi and y j ). Whenever a factor xi in f interacts with a factor x j in g (say, when j = i + 1, and the contribution is +1), there will be an interaction of x−i in τ ( f ) and x− j in τ (g) yielding the opposite sign (in our example, − j = −i − 1, and the contribution is −1). This establishes the claim, and hence the lemma.
A function f is called a Casimir for the Poisson bracket if f Poisson commutes with all other functions. It suffices to check this condition on the coordinate functions. An easy calculation yields the following lemma. We omit the details. Lemma 2.11. The invariants in Eq. 2.7 are Casimir functions for the Poisson bracket.
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2.6. The corank of the structure. In this section, we compute the corank of our Poisson bracket on the space of twisted polygons. The corank of a Poisson bracket on a smooth manifold is the codimension of the generic symplectic leaves. These symplectic leaves can be locally described as levels Fi = const of the Casimir functions. See [27] for the details. For us, the only genericity condition we need is xi = 0,
y j = 0; ∀ i, j.
(2.17)
Our next result refers to Eq. 2.7. Lemma 2.12. The Poisson bracket has corank 2 if n is odd and corank 4 if n is even. Proof. The Poisson bracket is quadratic in coordinates (x, y). It is very easy to see that in so-called logarithmic coordinates, pi = log xi ,
qi = log yi ,
the bracket is given by a constant skew-symmetric matrix. More precisely, the bracket between the p-coordinates is given by the marix ⎞ ⎛ 0 −1 0 . . . 1 0 −1 . . . 0⎟ ⎜1 ⎟ ⎜ 1 0 . . . 0⎟ ⎜0 ⎠ ⎝. . . ... −1 0 0 ... 0 whose rank is n − 1, if n is odd and n − 2, if n is even. The bracket between the q-coordinates is given by the opposite matrix.
The following corollary is immediate from the preceding result and Lemma 2.11. Corollary 2.13. If n is odd, then the Casimir functions are of the form F (On , E n ). If n is even, then the Casimir functions are of the form F(On/2 , E n/2 , On , E n ). In both cases the generic symplectic leaves of the Poisson structure have dimension 4[(n − 1)/2]. Remark 2.14. Computing the gradients, we see that a level set of the Casimir functions is smooth as long as all the functions are nonzero. Thus, the generic level sets are smooth in quite a strong sense. 2.7. The end of the proof. In this section, we finish the proof of Theorem 1. Let us summarize the situation. First we consider the case when n is odd. On the space Pn we have a generically defined and T -invariant Poisson bracket that is invariant under the pentagram map. This bracket has co-rank 2, and the generic level set of the Casimir functions has dimension 4[n/2] = 2n − 2. On the other hand, after we exclude the two Casimirs, we have 2[n/2] = n − 1 algebraically independent invariants that Poisson commute with each other. This gives us the classical Arnold-Liouville complete integrability. In the even case, our symplectic leaves have dimension 4[(n − 1)/2] = 2n − 4. The invariants E n/2 and On/2 are also Casimirs in this case. Once we exclude these, we have 2[(n − 1)/2] = n − 2 algebraically independent invariants. Thus, we get the same complete integrability as in the odd case. This completes the proof of our Main Theorem. In the next chapter, we consider geometric situations where the Main Theorem leads to quasi-periodic dynamics of the pentagram map.
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3. Quasi-periodic Motion In this chapter, we explain some geometric situations where our Main Theorem, an essentially algebraic result, translates into quasi-periodic motion for the dynamics. The universally convex polygons furnish our main example.
3.1. Universally convex polygons. In this section, we define universally convex polygons and prove some basic results about them. We say that a matrix M ∈ S L 3 (R) is strongly diagonalizable if it has 3 distinct positive real eigenvalues. Such a matrix represents a projective transformation of RP2 . We also let M denote the action on RP2 . Acting on RP2 , the map M fixes 3 distinct points. These points, corresponding to the eigenvectors, are in general position. M stabilizes the 3 lines determined by these points, taken in pairs. The complement of the 3 lines is a union of 4 open triangles. Each open triangle is preserved by the projective action. We call these triangles the M-triangles. Let φ ∈ Pn be a twisted n-gon, with monodromy M. We call φ universally convex if • M is a strongly diagonalizable matrix. • φ(Z) is contained in one of the M-triangles. • The polygonal arc obtained by connecting consecutive vertices of φ(Z) is convex. The third condition requires more explanation. In RP2 there are two ways to connect points by line segments. We require the connection to take place entirely inside the M-triangle that contains φ(Z). This determines the method of connection uniquely. We normalize so that M preserves the line at infinity and fixes the origin in R2 . We further normalize so that the action on R2 is given by a diagonal matrix with eigenvalues 0 < a < 1 < b. This 2 × 2 diagonal matrix determines M. For convenience, we will usually work with this auxilliary 2 × 2 matrix. We slightly abuse our notation, and also refer to this 2 × 2 matrix as M. With our normalization, the M-triangles are the open quadrants in R2 . Finally, we normalize so that φ(Z) is contained in the positive open quadrant. Lemma 3.1. U n is open in Pn . Proof. Let φ be a universally convex n-gon and let φ be a small perturbation. Let M be the monodromy of φ . If the perturbation is small, then M remains strongly diagonalizable. We can conjugate so that M is normalized exactly as we have normalized M. If the perturbation is small, the first n points of φ (Z) remain in the open positive quadrant, by continuity. But then all points of φ (Z) remain in the open positive quadrant, by symmetry. This is to say that φ (Z) is contained in an M -triangle. If the perturbation is small, then φ (Z) is locally convex at some collection of n consecutive vertices. But then φ (Z) is a locally convex polygon, by symmetry. The only way that φ(Z) could fail to be convex is that it wraps around on itself. But, the invariance under the 2 × 2 hyperbolic matrix precludes this possibility. Hence φ(Z) is convex.
Lemma 3.2. U n is invariant under the pentagram map. Proof. Applying the pentagram map to φ(Z) all at once, we see that the image is again strictly convex and has the same monodromy.
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v
vi+2
i−2
v
vi+1
i−1
vi Fig. 4. Points involved in the definition of the invariants
3.2. The Hilbert perimeter. In this section, we introduce an invariant we call the Hilbert perimeter. This invariant plays a useful role in our proof, given in the next section, that the level sets of the monodromy functions in U n are compact. As a prelude to our proof, we introduce another projective invariant – a function of the Casimirs – which we call the Hilbert Perimeter. This invariant is also considered in [19], and for similar purposes. Referring to Fig. 2, we define z k = [(vi , vi−2 ), (vi , vi−1 ), (vi , vi+1 ), (vi , vi+2 )].
(3.1)
We are taking the cross ratio of the slopes of the 4 lines in Fig. 4. We now define a “new” invariant H = n
1
i=1 z i
.
(3.2)
Remark 3.3. Some readers will know that one can put a canonical metric inside any convex shape, called the Hilbert metric. In case φ is a genuine convex polygon, the quantity − log(z k ) measures the Hilbert length of the thick line segment in Fig. 4. (The reader who does not know what the Hilbert metric is can take this as a definition.) Then log(H ) is the Hilbert perimeter of T (P) with respect to the Hilbert metric on P. Hence the name. Lemma 3.4. H = 1/(On E n ). Proof. This is a local calculation, which amounts to showing that z k = xk yk . The best way to do the calculation is to normalize so that 4 of the points are the vertices of a square. We omit the details.
3.3. Compactness of the level sets. In this section, we prove that the level sets of the monodromy functions in U are compact. Let U n (M, H ) denote the subset of U n consisting of elements whose monodromy is M and whose Hilbert Perimeter is H . In this section we will prove that U n (M, H ) is compact. For ease of notation, we abbreviate this space by X . Let φ ∈ X . We normalize so that M is as in Lemma 3.1. We also normalize so that φ(0) = (1, 1). Then there are numbers (x, y) such that φ(n) = (x, y), independent of the choice of φ. We can assume that x > 1 and y < 1. The portion of φ of interest to us, namely φ({0, . . . , n − 1}), lies entirely in the rectangle R whose two opposite
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corners are (1, 1) and (x, y). Let (vi , v j ) denote the line determined by vi and v j . Here vk = φ(k). In particular, let L i = (vi , vi+1 ). Lemma 3.5. Suppose that {φk } ∈ X is a sequence that does not converge on a subsequence to another element of X . Then, passing to a subsequence we can arrange that at least one of the two situations holds: there exists some i such that 1. The angle between L i and L i+1 tends to 0 as k → ∞ whereas the angle between L i+1 and L i+2 does not; 2. The points vi and vi+1 converge to a common point as k → ∞ whereas vi+2 converges to a distinct point. Proof. Suppose that there is some minimum distance between all points of φk in the rectangle R. In this case, the angle between two consecutive segments must tend to 0 as k → ∞. However, not all angles between consecutive segments can converge to 0 because of the fixed monodromy. The first case is now easy to arrange. If there is no such minimum , then two points coalesce, on a subsequence. For the same reason as above, not all points can coalesce to the same point. The second case is now easy to arrange.
Lemma 3.6. X is compact. We will suppose we have the kind of sequence we had in the previous lemma and then derive a contradiction. In the first case above, the slopes of the lines (vi+2 , vi ) and (vi+2 , vi+1 ) converge to each other as k → ∞, but the common limit remains uniformly bounded away from the slopes of (vi+2 , vi+3 ) and (vi+2 , vi+4 ). Hence z i+2 → 0. Since z j ∈ (0, 1) for all j, we have H → ∞ in this case. This is a contradiction. To deal with the second case, we can assume that the first case cannot be arranged. That is, we can assume that there is a uniform lower bound to the angles between two consecutive lines L i and L i+1 for all indices and all k. But then the same situation as in Case 1 holds, and we get the same contradiction.
3.4. Proof of Theorem 2. In this section, we finish the proof of Theorem 2. Recall that the level sets of our Casimir functions give a (singular) foliation by symplectic leaves. Note that all corner invariants are nonzero for points in U n . Hence, our singular symplectic foliation intersects U n in leaves that are all smooth symplectic manifolds. Let k = [(n − 1)/2]. Let M be a symplectic leaf. Note that M has dimension 4k. Consider the map F = (O1 , E 1 , . . . , Ok , E k ),
(3.3)
made from our algebraically independent monodromy invariants. Here we are excluding all the Casimirs from the definition of F. Say that a point p ∈ M is regular if d F p is surjective. Call M typical if some point of M is regular. Given our algebraic independence result, and the fact that the coordinates of F are polynomials, we see that almost every symplectic leaf is typical. Lemma 3.7. If M is typical then almost every F-fiber of M is a smooth submanifold of M.
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Proof. Let S = F(M) ⊂ R2k . Note that S has positive measure since d F p is nonsingular for some p ∈ M. Let ⊂ M denote the set of points p such that d F p is not surjective. Sard’s theorem says that F( ) has measure 0. Hence, almost every fiber of M is disjoint from .
Let M be a typical symplectic leaf, and let F be a smooth fiber of F. Then F has dimension 2k. Combining our Main Theorem with the standard facts about Arnold-Liouville complete integrability (e.g., [3]), we see that the monodromy invariants give a canonical affine structure to F. The pentagram map T preserves both F, and is a translation relative to this affine structure. Any pre-compact orbit in F exhibits quasi-periodic motion. Now, T also preserves the monodromy. But then each T -orbit in F is contained in one of our spaces U n (H, M). Hence, the orbit is precompact. Hence, the orbit undergoes quasi-periodic motion. Since this argument works for almost every F-fiber of almost every symplectic leaf in U n , we see that almost every orbit in U n undergoes quasiperiodic motion under the pentagram map. This completes the proof of Theorem 2. Remark 3.8. We can say a bit more. For almost every choice of monodromy M, the intersection F(M) = F ∩ U n (H, M)
(3.4)
is a smooth compact submanifold and inherits an invariant affine structure from F. In this situation, the restriction of T to F(M) is a translation in the affine structure. 3.5. Hyperbolic cylinders and tight polygons. In this section, we put Theorem 2 in a somewhat broader context. The material in this section is a prelude to our proof, given in the next section, of a variant of Theorem 2. Before we sketch variants of Theorem 2, we think about these polygons in a different way. A projective cylinder is a topological cylinder that has coordinate charts into RP2 such that the transition functions are restrictions of projective transformations. This is a classical example of a geometric structure. See [24 or 17] for details. Example 3.9. Suppose that M acts on R2 as a nontrivial diagonal matrix having eigenvalues 0 < a < 1 < b. Let Q denote the open positive quadrant. Then Q/M is a projective cylinder. We call Q/M a hyperbolic cylinder. Let Q/M be a hyperbolic cylinder. Call a polygon on Q/M tight if it has the following 3 properties: • It is embedded; • It is locally convex; • It is homologically nontrivial. Any universally convex polygon gives rise to a tight polygon on Q/M, where M is the monodromy normalized in the standard way. The converse is also true. Moreover, two tight polygons on Q/M give rise to equivalent universally convex polygons iff some locally projective diffeomorphism of Q/M carries one polygon to the other. We call such maps automorphisms of the cylinder, for short. Thus, we can think of the pentagram map as giving an iteration on the space of tight polygons on a hyperbolic cylinder. There are 3 properties that give rise to our result about periodic motion.
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θ (0,0) Fig. 5. The cylinder (θ, d)
1. The image of a tight polygon under the pentagram map is another well-defined tight polygon. 2. The space of tight polygons on a hyperbolic cylinder, modulo the projective automorphism group, is compact. 3. The strongly diagonalizable elements are open in S L 3 (R). The third condition guarantees that the set of all tight polygons on all hyperbolic cylinders is an open subset of the set of all twisted polygons. 3.6. A related theorem. In this section we prove a variant of Theorem 2 for a different family of twisted polygons. We start with a sector of angle θ in the plane, as shown in Fig. 5, and glue the top edge to the bottom edge by a similarity S that has dilation factor d. We omit the origin from the sector. The quotient is the projective cylinder we call (θ, d). When d = 1 we have a Euclidean cone surface. When θ = 2π we have the punctured plane. We consider the case when θ is small and d is close to 1. In this case, (θ, d) admits tight polygons for any n. (It is easiest to think about the case when n is large.) When developed out in the plane, these tight polygons follow along logarithmic spirals. Let S(θ, d) denote the subset of R2 consisting of pairs (θ , d ), where 0 < θ < θ ; 1 < d < d. Define (d, 1) =
(θ , d ).
(3.5)
(3.6)
(θ ,d )∈S(θ,d)
(θ, d) is the space of polygons that are more tightly coiled than One might say that
those on (θ, d). Theorem 3. Suppose that θ > 0 is sufficiently close to 0 and d > 1 is sufficiently close to (θ, d) lies on a smooth torus that has a T -invariant affine 1. Then almost every point of
(θ, d) undergoes quasi-periodic structure. Hence, the orbit of almost every point of
motion. Proof. Our proof amounts to verifying the three properties above for the points in our =
(θ, d). space. We fix (θ, d) and let
1. Let P be a tight polygon on (θ , d ). If θ is sufficiently small and d is sufficiently close to 1, then each vertex v of P is much closer to its neighbors than it is to the
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origin. For this reason, the pentagram map acts on, and preserves, the set of tight polygons on (θ , d ). The same goes for the inverse of the pentagram map. Hence is a T -invariant subset of Pn .
2. Let Z (θ , d , α) denote the space of tight polygons on (θ , d ) having Hilbert perimeter α. We consider these tight polygons equivalent if there is a similarity of (θ , d ) that carries one to the other. A proof very much like the compactness argument given in [19], for closed polygons, shows that Z (θ , d , α) is compact for θ near 0 and d near 1 and α arbitrary. Hence, the level sets of the Casimir functions in compact sets. intersect
3. The similarity S is the monodromy for our tight polygons. S lifts to an element of S L 3 (R) that has one real eigenvalue and two complex conjugate eigenvalues. Small is open in Pn . perturbations of S have the same property. Hence,
We have assembled all the ingredients necessary for the proof of Theorem 2. The same argument as above now establishes the result.
Remark 3.10. The first property crucially uses the fact that θ is small. Consider the case θ = 2π . It can certainly happen that P contains the origin in its hull but T (P) does not. We do not know the exact bounds on θ and d necessary for this construction. 4. Another Coordinate System in Space Pn 4.1. Polygons and difference equations. Consider two arbitrary n-periodic sequences (ai ), (bi ) with ai , bi ∈ R and i ∈ Z, such that ai+n = ai , bi+n = bi . Assume that n = 3 m. This will be our standing assumption whenever we work with the (a, b)coordinates; its meaning will become clear shortly. We shall associate to these sequences a difference equation of the form Vi+3 = ai Vi+2 + bi Vi+1 + Vi ,
(4.1)
for all i. A solution V = (Vi ) is a sequence of numbers Vi ∈ R satisfying (4.1). Recall a well-known fact that the space of solutions of (4.1) is 3-dimensional (any solution is determined by the initial conditions (V0 , V1 , V2 )). We will often understand Vi as vectors in R3 . The n-periodicity then implies that there exists a matrix M ∈ SL(3, R) called the monodromy matrix, such that Vi+n = M Vi . Proposition 4.1. If n is not divisible by 3 then the space Pn is isomorphic to the space of the equations (4.1). Proof. First note that since PGL(3, R) ∼ = SL(3, R), every M ∈ PGL(3, R) corresponds to a unique element of SL(3, R) that (abusing the notations) we also denote by M. A. Let (vi ), i ∈ Z be a sequence of points vi ∈ RP2 in general position with monodromy M. Consider first an arbitrary lift of the points vi to vectors V˜i ∈ R3 with the condition V˜i+n = M(V˜i ). The general position property implies that det(V˜i , V˜i+1 , V˜i+2 ) = 0 for all i. The vector V˜i+3 is then a linear combination of the linearly independent vectors V˜i+2 , V˜i+1 , V˜i , that is, V˜i+3 = ai V˜i+2 + bi V˜i+1 + ci V˜i , for some n-periodic sequences (ai ), (bi ), (ci ). We wish to rescale: Vi = ti V˜i , so that det(Vi , Vi+1 , Vi+2 ) = 1
(4.2)
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for all i. Condition (4.2) is equivalent to ci ≡ 1. One obtains the following system of equations in (t1 , . . . , tn ): ti ti+1 ti+2 = 1/ det(V˜i , V˜i+1 , V˜i+2 ), tn−1 tn t1 = 1/ det(V˜n−1 , V˜n , V˜1 ), tn t1 t2 = 1/ det(V˜n , V˜1 , V˜2 ).
i = 1, . . . , n − 2,
This system has a unique solution if n is not divisible by 3. This means that any generic twisted n-gon in RP2 has a unique lift to R3 satisfying (4.2). We proved that a twisted n-gon defines Eq. (4.1) with n-periodic ai , bi . Furthermore, if (vi ) and (vi ), i ∈ Z are two projectively equivalent twisted n-gons, then they correspond to the same Eq. (4.1). Indeed, there exists A ∈ SL(3, R) such that A(vi ) = vi for all i. One has, for the (unique) lift: Vi = A(Vi ). The sequence (Vi ) then obviously satisfies the same Eq. (4.1) as (Vi ). B. Conversely, let (Vi ) be a sequence of vectors Vi ∈ R3 satisfying (4.1). Then every three consecutive points satisfy (4.2) and, in particular, are linearly independent. Therefore, the projection (vi ) to RP2 satisfies the general position condition. Moreover, since the sequences (ai ), (bi ) are n-periodic, (vi ) satisfies vi+n = M(vi ). It follows that every Eq. (4.1) defines a generic twisted n-gon. A choice of initial conditions (V0 , V1 , V2 ) fixes a twisted polygon, a different choice yields a projectively equivalent one.
Proposition 4.1 readily implies the next result. Corollary 4.2. If n is not divisible by 3 then Pn = R2n . We call the lift (Vi ) of the sequence (vi ) satisfying Eq. (4.1) with n-periodic (ai , bi ) canonical. Remark 4.3. The isomorphism between the space Pn and the space of difference equations (4.1) (for n = 3m) goes back to the classical ideas of projective differential geometry. This is a discrete version of the well-known isomorphism between the space of smooth non-degenerate curves in RP2 and the space of linear differential equations, see [17] and references therein and Sect. 6.1. The “arithmetic restriction” n = 3m is quite remarkable. Equations (4.1) and their analogs were already used in [9] in the context of integrable systems; in the RP1 -case these equations were recently considered in [14] to study the discrete versions of the Korteweg - de Vries equation. It is notable that an analogous arithmetic assumption n = 2m is made in this paper as well. Remark 4.4. Let us now comment on what happens if n is divisible by 3. A certain modification of Proposition 4.1 holds in this case as well. Given a twisted n-gon (vi ) with monodromy M, lift points v0 and v1 arbitrarily as vectors V0 , V1 ∈ R3 , and then continue lifting consecutive points so that the determinant condition (4.2) holds. This implies that Eq. (4.1) holds as well. One has: M(Vi ) = ti Vi+n
(4.3)
for non-zero reals ti , and (4.2) implies that ti ti+1 ti+2 = 1 for all i ∈ Z. It follows that the sequence ti is 3-periodic; let us write t1+3 j = α, t2+3 j = β, t3 j = 1/(αβ). Applying the monodromy linear map M to (4.1) and using (4.3), we conclude that an+i =
ti+2 ti+1 ai , bn+i = bi , ti ti
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that is, 1 α a3 j+1 , an+3 j+2 = a3 j+2 , 2 α β β β 1 = b3 j+1 , bn+3 j+2 = b3 j+2 . α αβ 2
an+3 j = αβ 2 a3 j , an+3 j+1 = bn+3 j = α βb3 j , bn+3 j+1 2
(4.4)
We are still free to rescale V0 and V1 . This defines an action of the group R∗ × R∗ : V0 → uV0 , V1 → vV1 , u = 0, v = 0. The action of the group R∗ × R∗ on the coefficients (ai , bi ) is as follows: v 1 a3 j+1 , a3 j+2 → 2 a3 j+2 , u uv 1 → uv 2 b3 j+1 , b3 j+2 → 2 b3 j+2 . u v
a3 j → u 2 v a3 j , a3 j+1 → b3 j
u → b3 j , b3 j+1 v
(4.5)
When n = 3m, according to (4.4), this action makes it possible to normalize all ti to 1 which makes the lift canonical. However, if n = 3m then the R∗ × R∗ -action on ti is trivial, and the pair (α, β) ∈ R∗ × R∗ is a projective invariant of the twisted polygon. One concludes that Pn is the orbit space [{(a0 , . . . , an−1 , b0 , . . . , bn−1 )}/(R∗ × R∗ )] × (R∗ × R∗ ) with respect to R∗ × R∗ -action (4.5). This statement replaces Proposition 4.1 in the case of n = 3m. It would be interesting to understand the geometric meaning of the “obstruction” (α, β). If the obstruction is trivial, that is, if α = β = 1, then there exists a 2-parameter family of canonical lifts, but if the obstruction is non-trivial then no canonical lift exists. 4.2. Relation between the two coordinate systems. We now have two coordinate systems, (xi , yi ) and (ai , bi ). Assuming that n is not divisible by 3, let us calculate the relations between the two systems. Lemma 4.5. One has: xi =
ai−2 , bi−2 bi−1
yi = −
bi−1 . ai−2 ai−1
(4.6)
Proof. Given four vectors a, b, c, d in R3 , the intersection line of the planes Span(a, b) and Span(c, d) is spanned by the vector (a × b) × (c × d). Note that the volume element equips R3 with the bilinear vector product: R3 × R3 → R3 . Using the identity (a × b) × (b × c) = det(a, b, c) b, and the recurrence (4.1), let us compute lifts of the quadruple of points (vi−1 , vi , (vi−1 , vi ) ∩ (vi+1 , vi+2 ), (vi−1 , vi ) ∩ (vi+2 , vi+3 ))
(4.7)
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involved in the left corner cross-ratio. One has Vi−1 = Vi+2 − ai−1 Vi+1 − bi−1 Vi . Furthermore, it is easy to obtain the lift of the intersection points involved in the left corner cross-ratio. For instance, (vi−1 , vi ) ∩ (vi+1 , vi+2 ) is (Vi−1 × Vi ) × (Vi+1 × Vi+2 ) = ((Vi+2 − ai−1 Vi+1 − bi−1 Vi ) × Vi ) × (Vi+1 × Vi+2 ) = Vi+2 − ai−1 Vi+1 . One finally obtains the following four vectors in R3 : (Vi+2 −ai−1 Vi+1 −bi−1 , Vi , Vi , Vi+2 −ai−1 Vi+1 , bi Vi+2 −ai−1 Vi −ai−1 bi Vi+1 ). Similarly, for the points involved in the right corner cross-ratio (ai Vi+2 + bi , Vi+1 + Vi , Vi+2 , bi Vi+1 + Vi , bi Vi+2 − ai−1 Vi − ai−1 bi Vi+1 ) . Next, given four coplanar vectors a, b, c, d in R3 such that c = λ1 a + λ2 b,
d = μ1 a + μ2 b,
where λ1 , λ2 , μ1 , μ2 are arbitrary constants, the cross-ratio of the lines spanned by these vectors is given by [a, b, c, d] =
λ2 μ1 − λ1 μ2 . λ2 μ1
Applying this formula to the two corner cross-ratios yields the result.
Formula (4.6) implies the following relations: xi yi = −
1 1 , xi+1 yi = − , ai−1 bi−2 ai−2 bi
ai xi yi−1 = , ai−3 xi+1 yi+1
bi xi−1 yi−1 = , bi−3 xi+1 yi (4.8)
that will be of use later. Remark 4.6. If n is a multiple of 3 then the coefficients ai and bi are not well defined and they are not n-periodic anymore; however, according to formulas (4.4) and (4.5), the right hand sides of formulas (4.6) are still well defined and are n-periodic. 4.3. Two versions of the projective duality. We now wish to express the pentagram map T in the (a, b)-coordinates. We shall see that T is the composition of two involutions each of which is a kind of projective duality. 2 The notion 2of projective duality in RP is based on the fact that 2the dual projecis the space of one-dimensional subspaces of RP which is again tive plane RP equivalent to RP2 . Projective duality applies to smooth curves: it associates to a curve γ (t) ⊂ RP2 the 1-parameter family of its tangent lines. In the discrete case, there are different ways to define projectively dual polygons. We choose two simple versions. Definition 4.7. Given a sequence of points vi ∈ RP2 , we define two sequences α(vi ) ∈ (RP2 ) and β(vi ) ∈ (RP2 ) as follows:
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Fig. 6. Projective dual for smooth curves and polygons
Fig. 7. Iteration of the duality maps: α 2 (vi ) = α(vi ) ∩ α(vi+1 ), β 2 (vi ) = β(vi−1 ) ∩ β(vi+1 ) and (α ◦ β) (vi ) = β(vi ) ∩ β(vi+1 )
1. α(vi ) is the line (vi , vi+1 ), 2. β(vi ) is the line (vi−1 , vi+1 ), see Fig. 6. Clearly, α and β commute with the natural PGL(3, R)-action and therefore are welldefined on the space Pn . The composition of α and β is precisely the pentagram map T . Lemma 4.8. One has α 2 = τ,
β 2 = Id,
α ◦ β = T,
(4.9)
where τ is the cyclic permutation: τ (vi ) = vi+1 .
(4.10)
Proof. The composition of the maps α and β, with themselves and with each other, associates to the corresponding lines (viewed as points of (RP2 ) ) their intersections, see Fig. 7.
The map (4.10) defines the natural action of the group Z on Pn . All the geometric and algebraic structures we consider are invariant with respect to this action. 4.4. Explicit formula for α. It is easy to calculate the explicit formula of the map α in terms of the coordinates (ai , b j ). As usual, we assume n = 3 m. Lemma 4.9. Given a twisted n-gon with monodromy (vi ), i ∈ Z represented by a difference equation (4.1), the n-gon (α(vi )), i ∈ Z is represented by the Eq. (4.1) with coefficients α ∗ (ai ) = −bi+1 , where, as usual,
a∗
α ∗ (bi ) = −ai ,
stands for the pull-back of the coordinate functions.
(4.11)
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Proof. Consider the canonical lift (Vi ) to R3 . Let Ui = Vi × Vi+1 ∈ (R3 ) . This is obviously a lift of the sequence (α(vi )) to (R3 ) . We claim that (Ui ) is, in fact, a canonical lift. Indeed, Ui is a lift of u i since Vi × Vi+1 is orthogonal to Vi and to Vi+1 . Next, using the identity (4.7) one has det(Ui ×Ui+1 , Ui+1 ×Ui+2 , Ui+2 ×Ui+3 ) = [(Ui ×Ui+1 )×(Ui+1 ×Ui+2 )] · (Ui+2 ×Ui+3 ) = Ui+1 · (Ui+2 × Ui+3 ) = det(Ui+1 , Ui+2 , Ui+3 ) = 1. It follows that the sequence Ui ∈ R3 satisfies the equation Ui+3 = α ∗ (ai ) Ui+2 + α ∗ (bi ) Ui+1 + Ui with some α ∗ (ai ) and α ∗ (bi ). Let us show that these coefficients are, indeed, given by (4.11). For all i, one has Ui+1 · Vi = 1,
Ui · Vi+2 = 1,
Ui+3 · Vi+3 = 0.
Using (4.1), the last identity leads to: α ∗ (bi ) Ui+1 · Vi + ai Ui · Vi+2 = 0. Hence α ∗ (bi ) = −ai . The first identity in (4.11) follows from formula (4.9). Indeed, one has α ∗ (α ∗ (ai )) = ai+1 and α ∗ (α ∗ (bi )) = bi+1 , and we are done.
4.5. Recurrent formula for β. The explicit formula for the map β is more complicated, and we shall give a recurrent expression. Lemma 4.10. Given an n-gon (vi ), i ∈ Z represented by a difference equation (4.1), the n-gon (β(vi )), i ∈ Z is represented by the Eq. (4.1) with coefficients β ∗ (ai ) = −
λi bi−1 , λi+2
β ∗ (bi ) = −
λi+3 ai+1 , λi+1
(4.12)
where the coefficients λi are uniquelly defined by λi λi+1 λi+2 = −
1 1 + bi−1 ai
(4.13)
for all i. Proof. The lift of the map β to R3 takes Vi to Wi = λi Vi−1 × Vi+1 , where the coefficients λi are chosen in such a way that det(Wi , Wi+1 , Wi+2 ) = 1 for all i. The sequence Wi ∈ R3 satisfies the equation Wi+3 = β ∗ (ai ) Wi+2 + β ∗ (bi ) Wi+1 + Wi . To find β ∗ (ai ) and β ∗ (bi ), one substitutes Wi = λi Vi−1 × Vi+1 , and then, using (4.1), expresses each V as a linear combination of Vi , Vi+1 , Vi+2 . The above equation is then equivalent to the following one: ∗
β (ai ) λi+2 + bi−1 λi Vi × Vi+1
+ ai+1 λi+3 + β ∗ (bi ) λi+1 Vi × Vi+2
+ (1 + bi ai+1 ) λi+3 + β ∗ (ai ) ai λi+2 − λi Vi+1 × Vi+2 = 0.
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Since the three terms are linearly independent, one obtains three relations. The first two equations lead to (4.12) while the last one gives the recurrence λi+3 = λi
1 + ai bi−1 . 1 + ai+1 bi
On the other hand, one has λi λi+1 λi+2 det (Vi−1 × Vi+1 , Vi × Vi+2 , Vi+1 × Vi+3 ) = 1. Once again, expressing each V as a linear combination of Vi , Vi+1 , Vi+2 , yields λi λi+1 λi+2 (1 + ai bi−1 ) = −1,
and one obtains (4.13).
4.6. Formulas for the pentagram map. We can now describe the pentagram map in terms of (a, b)-coordinates and to deduce formulas (2.4). Proposition 4.11. (i) One has: T ∗ (xi ) = xi
1 − xi−1 yi−1 , 1 − xi+1 yi+1
T ∗ (yi ) = yi+1
1 − xi+2 yi+2 . 1 − xi yi
(ii) Assume that n = 3m + 1 or n = 3m + 2; in both cases, T ∗ (ai ) = ai+2
m 1 + ai+3k+2 bi+3k+1 , 1 + ai−3k+2 bi−3k+1
T ∗ (bi ) = bi−1
k=1
m 1 + ai−3k−2 bi−3k−1 . 1 + ai+3k−2 bi+3k−1
k=1
(4.14) Proof. According to Lemma 4.8, T = α ◦ β. Combining Lemmas 4.9 and 4.10, one obtains the expression: T ∗ (ai ) =
λi+4 ai+2 , λi+2
T ∗ (bi ) =
λi bi−1 , λi+2
where λi are as in (4.13). Equation (4.6) then gives T ∗ (xi ) = =
T ∗ (ai−2 ) T ∗ (bi−2 ) T ∗ (bi−1 )
=
λi+2 ai λi λi+1 λi λi−2 bi−3 λi−1 bi−2
1 + bi−3 ai−2 ai bi−2 bi−3 1 + bi−1 ai
=
ai−3 1 + bi−3 ai−2 ai bi−2 bi−3 1 + bi−1 ai ai−3
= xi−1
1− 1−
1 xi−1 yi−1 1 xi+1 yi+1
xi yi−1 1 − xi−1 yi−1 = xi , xi+1 yi+1 1 − xi+1 yi+1
and similarly for yi . We thus proved formula (2.4). To prove (4.14), one now uses (4.8).
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+ Fig. 8. The Poisson bracket for n=5 and n = 7
4.7. The Poisson bracket in the (a, b)-coordinates. The explicit formula of the Poisson bracket in the (a, b)-coordinates is more complicated than (2.16). Recall that n is not a multiple of 3 so that we assume n = 3m + 1 or n = 3m + 2. In both cases the Poisson bracket is given by the same formula. Proposition 4.12. The Poisson bracket (2.16) can be rewritten as follows: {ai , a j } =
m
δi, j+3k − δi, j−3k ai a j , k=1
{ai , b j } = 0, {bi , b j } =
(4.15)
m
δi, j−3k − δi, j+3k bi b j . k=1
Proof. One checks using (4.5) that the brackets between the coordinate functions (xi , y j ) coincide with (2.16).
Example 4.13. a) For n = 4, the bracket is
{ai , a j } = δi, j+1 − δi, j−1 ai a j (and with opposite sign for b), the other terms vanish. b) For n = 5, the non-zero terms are:
{ai , a j } = δi, j+2 − δi, j−2 ai a j , corresponding to the “pentagram” in Fig. 8. c) For n = 7, one has:
{ai , a j } = δi, j+1 − δi, j−1 − δi, j+3 + δi, j−3 ai a j . d) For n = 8, the result is
{ai , a j } = δi, j+2 − δi, j−2 − δi, j+3 + δi, j−3 ai a j .
5. Monodromy Invariants in (a, b)-Coordinates The (a, b)-coordinates are especially well adapted to the computation of the monodromy matrix and the monodromy invariants. Such a computation provides an alternative deduction of the invariants (2.11), independent of [21].
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5.1. Monodromy matrices. Consider the 3 × ∞ matrix M constructed recurrently as follows: the columns C0 , C1 , C2 , . . . satisfy the relation Ci+3 = ai Ci+2 + bi Ci+1 + Ci ,
(5.1)
and the initial 3 × 3 matrix (C0 , C1 , C2 ) is unit. The matrix M contains the monodromy matrices of twisted n-gons for all n; namely, the following result holds. Lemma 5.1. The 3×3 minor Mn = (Cn , Cn+1 , Cn+2 ) represents the monodromy matrix of twisted n-gons considered as a polynomial function in a0 , . . . , an−1 , b0 , . . . , bn−1 . Proof. The recurrence (5.1) coincides with (4.1), see Sect. 4.1. It follows that Mn represents the monodromy of twisted n-gons in the basis C0 , C1 , C2 .
Let
⎛
⎞ 0 0 1 Nj = ⎝1 0 bj ⎠ . 0 1 aj
The recurrence (5.1) implies the following statement. Lemma 5.2. One has: Mn = N0 N1 . . . Nn−1 . In particular, det Mn = 1. To illustrate, the beginning of the matrix M is as follows: ⎛ ⎞ a1 a2 + b2 ... 1 0 0 1 a1 ⎝ 0 1 0 b0 b0 a1 + 1 b0 a1 a2 + a2 + b0 b2 ...⎠. 0 0 1 a0 a0 a1 + b1 a0 a1 a2 + b1 a2 + a0 b1 + 1 . . . The dihedral symmetry σ , that reverses the orientation of a polygon, replaces the monodromy matrices by their inverses and acts as follows: σ : ai → −b−i ,
bi → −a−i ;
this follows from rewriting Eq. (4.1) as Vi = −bi Vi+1 − ai Vi+2 + Vi+3 , or from Lemma 4.5.1 Consider the rescaling 1-parameter group ϕτ : ai → eτ ai ,
bi → e−τ bi .
It follows from Lemma 4.5 that the action on the corner invariants is as follows: xi → e3τ xi ,
yi → e−3τ yi .
Thus our rescaling is essentially the same as the one in (2.5) with t = e3τ . The trace of Mn is a polynomial Fn (a0 , . . . , an−1 , b0 , . . . , bn−1 ). Denote its homogeneous components in s := eτ by I j , j = 0, . . . , [n/2]; these are the monodromy invariants. One has Fn = I j . The s-weight of I j is given by the formula: w( j) = 3 j − k if n = 2k,
and w( j) = 3 j − k + 1 if n = 2k + 1 (5.2)
1 Since all the sums we are dealing with are cyclic, we slightly abuse the notation and ignore a cyclic shift in the definition of σ in the (a, b)-coordinates.
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(this will follow from Proposition 5.3 in the next section). For example, M4 is the matrix ⎛ ⎞ a1 a1 a2 +b2 a1 a2 a3 +a3 b2 +a1 b3 +1 ⎝ a1 b0 +1 ⎠ a1 a2 b0 +a2 +b0 b2 a1 a2 a3 b0 +a2 a3 +a3 b0 b2 +a1 b0 b3 +b3 +b0 a0 a1 +b1 a0 a1 a2 +a2 b1 +a0 b2 +1 a0 a1 a2 a3 +a2 a3 b1 +a0 a3 b2 +a0 a1 b3 +a0 +a3 +b1 b3 and I0 = b0 b2 +b1 b3 , I1 = a0 +a1 +a2 +a3 +b0 a1 a2 +b1 a2 a3 +b2 a3 a0 +b3 a0 a1 , I2 = a0 a1 a2 a3 .
Likewise, for n = 5, (b0 + b0 b2 a3 ), I0 =
I1 =
(a0 a1 + b0 a1 a2 a3 ),
I 2 = a0 a1 a2 a3 a4 ,
where the sums are cyclic over the indices 0, . . . , 4. One also has the second set of monodromy invariants J0 , . . . , Jk constructed from the inverse monodromy matrix, that is, applying the dihedral involution σ to I0 , . . . , Ik . 5.2. Combinatorics of the monodromy invariants. We now describe the polynomials Ii , Ji and their relation to the monodromy invariants E k , Ok . Label the vertices of an oriented regular n-gon by 0, 1, . . . , n − 1. Consider marking of the vertices by the symbols a, b and ∗ subject to the rule: each marking should be coded by a cyclic word W in symbols 1, 2, 3, where 1 = a, 2 = ∗ b, 3 = ∗ ∗ ∗. Call such markings admissible. If p, q, r are the occurrences of 1, 2, 3 in W then p + 2q + 3r = n; define the weight of W as p − q. Given a marking as above, take the product of the respective variables ai or bi that occur at vertex i; if a vertex is marked by ∗ then it contributes 1 to the product. Denote by T j the sum of these products over all markings of weight j. Then A := Tk is the product of all ai ; let B be the product of all bi ; here k = [n/2]. Proposition 5.3. The monodromy invariants I j coincide with the polynomials T j . One has: Ej =
Ik− j B for j = 1, . . . , k, and E n = (−1)n 2 . A A
J j are described similarly by the rule 1 = b, 2 = a ∗, 3 = ∗ ∗ ∗, and are similarly related to O j : O j = (−1)n+ j
Jk− j A for j = 1, . . . , k, and On = 2 . B B
Proof. First, we claim that the trace Fn is invariant under cyclic permutations of the indices 0, 1, . . . , n − 1. Indeed, impose the n-periodicity condition: ai+n = ai , bi+n = bi . Let Vi be as (4.1). The matrix Mn takes (V0 , V1 , V2 ) to (Vn , Vn+1 , Vn+2 ). Then the matrix (V1 , V2 , V3 ) → (Vn+1 , Vn+2 , Vn+3 ) is conjugated to Mn and hence has the same trace. This trace is Fn (a1 , b1 , . . . , an , bn ), and due to n-periodicity, this equals Fn (a1 , b1 , . . . , a0 , b0 ). Thus Fn is cyclically invariant. Now we argue inductively on n. Assume that we know that I j = T j for j = n − 2, n − 1, n. Consider Fn+1 . Given an admissible labeling of n − 2, n − 1 or n-gon, one may insert ∗ ∗ ∗, ∗ b or a between any two consecutive vertices, respectively, and obtain
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an admissible labeling of n + 1-gon. All admissible labeling are thus obtained, possibly, in many different ways. We claim that Fn+1 contains the cyclic sums corresponding to all admissible labeling. Indeed, consider an admissible cyclic sum in Fn−2 corresponding to a labeled n − 2-gon L. This is a cyclic sum of monomials in a0 , . . . , bn−3 ; these monomials are located in the matrix M on the diagonal of its minor Mn−2 . By recurrence (5.1), the same monomials will appear on the diagonal of Mn+1 , but now they must contribute to a cyclic sum of variables a0 , . . . , bn . These sums correspond to the labelings of n + 1-gon that are obtained from L by inserting ∗ ∗ ∗ between two consecutive vertices. Likewise, consider a term in Fn−1 , a cyclic sum of monomials in a0 , . . . , bn−2 corresponding to a labeled n − 1-gon L. By (5.1), these monomials are to be multiplied by bn−2 , bn−1 or bn (depending on whether they appear in the first, second or third row of M) and moved 2 units right in the matrix M, after which they contribute to the cyclic sums in Fn+1 . As before, the respective sums correspond to the labelings of n + 1-gon obtained from L by inserting ∗ b between two consecutive vertices. Similarly one deals with a contribution to Fn+1 from Fn : this time, one inserts symbol a. Our next claim is that each admissible term appears in Fn+1 exactly once. Suppose not. Using cyclicity, assume this is a monomial an P (or, similarly, bn P). Where could a monomial an P come from? Only from the bottom position of the column Cn+2 (once again, according to recurrence (5.1)). But then the monomial P appears at least twice in this position, hence in Fn , which contradicts our induction assumption. This completes the proof that I j = T j . Now let us prove that E j = Ik− j /A. Consider E j as a function of x, y and switch to the (a, b)-coordinates using Lemma 4.5: x1 =
a−1 , b−1 b0
y1 = −
b0 a−1 a0
and its cyclic permutations. Then y0 x1 y1 =
1 a−2 a−1 a0
and the cyclic permutations. An admissible monomial in E j then contributes the factor −bi /(ai−1 ai ) for each singleton yi+1 and 1/(ai−2 ai−1 ai ) for each triple yi xi+1 yi+1 . Admissibility implies that no index appears twice. Clear denominators by multiplying by A, the product of all a’s. Then, for each singleton yi+1 , we get the factor −bi and empty space ∗ at the previous position i − 1, because there was ai−1 in the denominator and, for each triple yi xi+1 yi+1 , we get empty spaces ∗ ∗ ∗ at positions i − 2, i − 1, i. All other, “free”, positions are filled with a’s. In other words, the rule 1 = a, 2 = ∗ b, 3 = ∗ ∗ ∗ applies. The signs are correct as well, and the result follows. Finally, E n is the product of all yi+1 , that is, of the terms −bi /(ai−1 ai ). This product equals (−1)n B/A2 .
Remark 5.4. Unlike the invariants Ok , E k , there are no signs involved: all the terms in polynomials Ii are positive. Similarly to Remark 4.6, the next lemma shows that one can use Proposition 5.3 even if n is a multiple of 3. In particular, this will be useful in Theorem 4 in the next section. Lemma 5.5. If n is a multiple of 3 then the polynomials I j , J j of variables a0 , . . . , bn−1 are invariant under the action of the group R∗ × R∗ given in (4.5).
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Proof. Recall that, by Lemma 5.2, Mn = N0 N1 . . . Nn−1 , where ⎛ ⎞ 0 0 1 Nj = ⎝1 0 bj ⎠ . 0 1 aj The action of R∗ × R∗ on the matrices N j depends on j mod 3 and is given by the next formulas: ⎞ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ v ⎛ 00 1 10 0 00 1 00 1 u 0 0 ⎝ 1 0 b0 ⎠ → ⎝ 1 0 uv b0 ⎠ = ⎝ 0 uv 0 ⎠ ⎝ 1 0 b0 ⎠ ⎝ 0 12 0 ⎠ , u v 0 1 a0 0 1 a0 0 1 u 2 v a0 0 0 u2v 0 0 1 ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞⎛ 1 00 1 1 0 0 0 0 00 1 00 1 2 uv ⎝ 1 0 b1 ⎠ → ⎝ 1 0 uv 2 b1 ⎠ = ⎝ 0 uv 2 0 ⎠ ⎝ 1 0 b1 ⎠ ⎝ 0 u 0 ⎠ , v v v 0 1 a1 0 1 a1 0 0 u 0 1 u a1 0 01 ⎛ ⎞ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞⎛ 2 00 1 1 0 0 00 1 00 1 u v 0 0 1 1 ⎝ 1 0 b2 ⎠ → ⎝ 1 0 2 b2 ⎠ = ⎝ 0 2 0 ⎠ ⎝ 1 0 b2 ⎠ ⎝ 0 uv 2 0 ⎠ . u v u v 0 1 a2 0 1 a2 0 0 1 0 1 uv1 2 a2 0 0 uv1 2 Note that ⎛v
⎛ 1 ⎞⎛ ⎞⎛ ⎞ ⎞ 1 0 0 0 0 1 0 0 2 0 0 uv v ⎝ 0 12 0 ⎠ ⎝ 0 uv 2 0 ⎠ = E, ⎝ 0 u 0 ⎠ ⎝ 0 12 0 ⎠ = 1 E, u v u v v u uv 2 0 0 uv 0 0 uv1 2 0 01 0 0 1 u
and
⎛
⎞⎛ ⎞ 1 0 0 u2v 0 0 ⎝ 0 uv 2 0 ⎠ ⎝ 0 uv 0 ⎠ = u 2 v E, 0 0 u2v 0 0 1
where E is the unit matrix. Therefore the R∗ × R∗ -action on Mn ⎞ ⎛ 2 ⎛ u v 0 1 ⎝ 1 0u 0 ⎠ 0 v 0 N0 N1 . . . Nn−1 ⎝ 0 uv 2 Mn → 2 u v 0 0 u2v 0 0
is as follows: ⎞ 0 0 ⎠ ∼ Mn , 1
where ∼ means “is conjugated to”. It follows that the trace of Mn , as a polynomial in a0 , . . . , bn−1 , is R∗ × R∗ -invariant, and so are all its homogeneous components.
5.3. Closed polygons. A closed n-gon (as opposed to a merely twisted one) is characI j = 3 (and, of course, terized by the condition that Mn = I d. This implies that J j = 3 as well). There are other linear relations on the monodromy invariants which we discovered in computer experiments. All combined, we found five identities. Theorem 4. For a closed n-gon, one has: k j=0
Ij =
k j=0
J j = 3,
k j=0
w( j)I j =
k
w( j)J j = 0,
j=0
where k = [n/2] and w( j) are the weights (5.2).
k j=0
w( j)2 (I j − J j ) = 0,
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Proof. The monodromy M ∈ S L(3, R) is a matrix-valued polynomial function of ai , bi , and M(τ ) = M ◦ ϕτ , where ϕτ is the scaling action ai → eτ ai , bi → e−τ bi . The characterization of Cn is M(0) = I d. Let eλ1 , eλ2 , eλ3 be the eigenvalues of M(τ ) considered as functions of ai , bi , τ . Then λi = 0 for τ = 0 and each i = 1, 2, 3, and λ1 + λ 2 + λ 3 = 0
(5.3)
identically. The eigenvalues of M −1 are similar with negative λs as exponents. The definition of I (a, b) and J (a, b) implies: eτ w( j) I j , e−λ1 + e−λ2 + e−λ3 = e−τ w( j) J j , (5.4) eλ1 + eλ2 + eλ3 = where w( j) are the weights. Setting τ = 0 in (5.4) we obtain the obvious relations Ij = J j = 3. Differentiating (5.4) with respect to τ and setting τ = 0, we get 3
λi (0) =
w( j)I j =
w( j)J j .
i=1
By (5.3), the left hand side is zero, and we obtain two other relations stated in the theorem. Differentiate (5.4) twice and set τ = 0: 3
λi (0) + λi (0)2 =
w( j)2 I j ,
i=1
3
−λi (0) + λi (0)2 =
w( j)2 J j .
i=1
Subtract and use the fact that in τ twice) to obtain:
3
i=1 λi (0)
= 0 (as follows from (5.3) by differentiating
w( j)2 (I j − J j ) = 0.
This is the fifth relation of the theorem.
(5.5)
Example 5.6. In the cases n = 4 and n = 5, it is easy to solve the equation Mn = I d. For n = 4, the solution is a single point a0 = a1 = a2 = a3 = 1,
b0 = b1 = b2 = b3 = −1,
and then I0 = 2, I1 = 0, I2 = 1. For n = 5, one has a 2-parameter set of solutions with free parameters x, y: a0 = x, a1 = y, a2 = −
1+x 1+y , a3 = −(1 − x y), a4 = − , 1 − xy 1 − xy
and bi = −ai+2 . Hence I0 = J0 = 2 − z, I1 = J1 = 1 + 2z, I2 = J2 = −z with z =
x y(1 + x)(1 + y) . 1 − xy
Remark 5.7. C n has codimension 8 in P n , and we have the five relations of Theorem 4. We conjecture that there are no other relations between the monodromy invariants that hold identically on Cn .
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6. Continuous Limit: The Boussinesq Equation Since the theory of infinite-dimensional integrable systems on functional spaces is much more developed than the theory of discrete integrable systems, it is important to investigate the n → ∞ “continuous limit” of the pentagram map. It turns out that the continuous limit of T is the classical Boussinesq equation. This is quite remarkable since the Boussinesq equation and its discrete analogs are thoroughly studied but, to the best of our knowledge, their geometrical interpretation remained unknown. We will also show that the Poisson bracket (2.16) can be viewed as a discrete version of the well-known first Poisson structure of the Boussinesq equation. 6.1. Non-degenerate curves and differential operators. We understand the continuous limit of a twisted n-gon as a smooth parametrized curve γ : R → RP2 with monodromy: γ (x + 1) = M(γ (x)),
(6.1)
for all x ∈ R, where M ∈ P S L(3, R) is fixed. The assumption that every three consecutive points are in general position corresponds to the assumption that the vectors γ (x) and γ (x) are linearly independent for all x ∈ R. A curve γ satisfying these conditions is usually called non-degenerate. As in the discrete case, we consider classes of projectively equivalent curves. The continuous analog of the space Pn , is then the space, C, of parametrized non-degenerate curves in RP2 up to projective transformations. The space C is very well known in classical projective differential geometry, see, e.g., [17] and references therein. Proposition 6.1. There exists a one-to-one correspondence between C and the space of linear differential operators on R: 3 d d + v(x), (6.2) A= + u(x) dx dx where u and v are smooth periodic functions. This statement is classical, but we give here a sketch of the proof. Proof. A non-degenerate curve γ (x) in RP2 has a unique lift to R3 , that we denote by (x), satisfying the condition that the determinant of the vectors (x), (x), (x) (the Wronskian) equals 1 for every x: (x) (x) (x) = 1. (6.3) The vector (x) is a linear combination of (x), (x), (x) and the condition (6.3) is equivalent to the fact that this combination does not depend on (x). One then obtains: (x) + u(x) (x) + v(x) (x) = 0. Two curves in RP2 correspond to the same operator if and only if they are projectively equivalent. Conversely, every differential operator (6.2) defines a curve in RP2 . Indeed, the space of solutions of the differential equation A f = 0 is 3-dimensional. At any point x ∈ R, one considers the 2-dimensional subspace of solutions vanishing at x. This defines a curve in the projectivization of the space dual to the space of solutions.
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Fig. 9. Evolution of a non-degenerate curve
Remark 6.2. It will be convenient to rewrite the above differential operator as a sum of a skew-symmetric operator and a (zero-order) symmetric operator: A=
d dx
3
d d 1 u(x) + u(x) + w(x), + 2 dx dx
(6.4)
where w(x) = v(x) − u 2(x) . The pair of functions (u, w) is understood as the continuous analog of the coordinates (ai , bi ). 6.2. Continuous limit of the pentagram map. We are now defining a continuous analog of the map T . The construction is as follows. Given a non-degenerate curve γ (x), at each point x we draw a small chord: (γ (x − ε), γ (x + ε)) and obtain a new curve, γε (x), as the envelop of these chords, see Fig. 9. The differential operator (6.4) corresponding to γε (x) contains new functions (u ε , wε ). We will show that u ε = u + ε2 u + O(ε3 ),
wε = w + ε 2 w + O(ε3 )
and calculate u, w explicitly. We then assume that the functions u(x) and w(x) depend on an additional parameter t (the “time”) and define an evolution equation: u˙ = u,
w˙ = w
that we understand as a vector field on the space of functions (and therefore of operators (6.4)). Here and below u˙ and w˙ are the partial derivatives in t, the partial derivatives in x will be denoted by . Theorem 5. The continuous limit of the pentagram map T is the following equation: u˙ = w , w˙ = −
u u u − . 3 12
(6.5)
Proof. The (lifted) curve ε ⊂ R3 , corresponding to γε (x) satisfies the following conditions: |(x + ε), (x − ε), ε (x)| = 0, ε (x), (x + ε) − (x − ε), (x) = 0. ε
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We assume that the curve ε (x) is of the form: ε = γ + ε A + ε2 B + (ε3 ), where A and B are some vector-valued functions. The above conditions easily imply that A is proportional to , while B satisfies: (x), (x), B (x) = 0, 1 2 (x), (x), (x) + (x), (x), B(x) = 0. It follows that B = 21 + g, where g is a function. We proved that the curve ε (x) is of the form:
2 ε = 1 + ε f + ε2 g + ε2 + (ε3 ), where f and g are some functions. It remains to find f and g and the corresponding differential operator. To this end one has to use the normalization condition (6.3). Lemma 6.3. The condition (6.3) implies f (x) ≡ 0 and g(x) = Proof. A straightforward computation.
u(x) 3 .
One has finally the following expression for the lifted curve: ε2 ε2 + (ε3 ). ε = 1 + u + 3 2
(6.6)
We are ready to find the new functions u ε , vε such that ε (x) + u ε (x) ε (x) + vε (x) ε (x) = 0. After a straightforward calculation, using the additional formula (V ) = −(2u + v) − (u + 2v − u 2 ) − (v − uv) , one gets directly from (6.6): u , u ε = u + ε2 v − 2 The result follows.
vε = v + ε2
v uu u − − 2 3 3
.
Remark 6.4. Equation (6.5) is equivalent to the following differential equation: 2 u u (I V ) − , u¨ = − 6 12 which is nothing else but the classical Boussinesq equation. Remark 6.5. It is not hard to compute that the continuous limit of the scaling symmetry is given by the formula: u(x) → u(x), w(x) → w(x) + t, where t is a constant.
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Remark 6.6. The fact that the continuous limit of the pentagram map is the Boussinesq equation is discovered in [19] (not much details are provided). The computation in [19] is made in an affine chart R2 ⊂ RP2 . In this lift, different from the canonical one (characterized by constant Wronskian), one obtains the curve flow ˙ = −
1 [ , ] 1 + , 3 [ , ] 2
where [., .] is the cross-product; this is equivalent to Eq. (6.6) (we omit a rather tedious verification of this equivalence). 6.3. The constant Poisson structure. Eq. (6.5) is integrable. In particular, it is Hamiltonian with respect to (two) Poisson structures on the space of functions (u, w). We describe here the simplest Poisson structure usually called the first Poisson structure of the Boussinesq equation. Consider the space of functionals of the form H (u, w) = h(u, u , . . . , w, w , . . .) d x, S1
where h is a polynomial. The variational derivatives, δu H and δw H , are the smooth functions on S 1 given by the Euler-Lagrange formula, e.g., ∂h ∂h ∂h − δu H = + − +··· , ∂u ∂u ∂u and similarly for δw H . Definition 6.7. The constant Poisson structure on the space of functionals is defined by
{G, H } = δu G (δw H ) + δw G (δu H ) d x. (6.7) S1
Note that the “functional coordinates” (u(x), w(x)) play the role of Darboux coordinates. Another way to define the above Poisson structure is as follows. Given a functional H as above, the Hamiltonian vector field with Hamiltonian H is given by u˙ = (δw H ) ,
(6.8)
w˙ = (δu H ) . The following statement is well known, see, e.g., [8]. Proposition 6.8. The function
H=
S1
w2 u3 uu − − 2 18 24
is the Hamiltonian function for Eq. (6.5). Proof. Straightforward from (6.8).
dx
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This statement has many consequences. For instance, the following functions are the first integrals of (6.5): H1 = u d x, H2 = w d x, H3 = uw d x. S1
S1
S1
Note that the functions H1 and H2 are precisely the Casimir functions of the structure (6.8). 6.4. Discretization. Let us now test the inverse procedure of discretization. Being more “heuristic”, this procedure will nevertheless be helpful for understanding the discrete limit of the classical Poisson structure of the Boussinesq equation. Given a non-degenerate curve γ (x), fix an arbitrary point vi := γ (x) and, for a small ε, set vi+1 := γ (x + ε), etc. We then have: vi = γ (x),
vi+1 = γ (x + ε)
Lifting γ (x) and (vi ) to looking for a and b in
R3
vi+2 = γ (x + 2 ε)
vi+3 = γ (x + 3 ε).
so that the difference equation (4.1) is satisfied, we are
(x + 3 ε) = a(x, ε) (x + 2 ε) + b(x, ε) (x + ε) + (x), as functions of x depending on ε, where ε is small. Lemma 6.9. Representing a(x) and b(x) as a series in ε: a(x, ε) = a0 (x) + ε a1 (x) + · · · ,
b(x, ε) = b0 (x) + ε b1 (x) + · · · ,
one gets for the first four terms: a0 = 3,
b0 = −3,
a1 = 0,
b1 = 0,
a2 = −u(x),
b2 = u(x),
a3 = − 47 u (x) −
1 2
w(x),
b3 =
5 4
u (x) −
(6.9) 1 2
w(x).
Proof. Straightforward Taylor expansion of the above expression for (x + 3 ε).
The Poisson structure (2.16) can be viewed as a discrete analog of the structure (6.7) and this is, in fact, the way we guessed it. Indeed, one has the following two observations: (1) Formula (6.9) shows that the functions log a and log b are approximated by linear combinations of u and w and therefore (4.15), a discrete analog of the bracket (6.7), should be a constant bracket in the coordinates (log a, log b). (2) Consider the following functionals (linear in the (a, b)-coordinates): F f (a, b) = f (x) a(x, ε) d x, G f (a, b) = f (x) b(x, ε) d x. S1
S1
Using (6.7) and (6.9), one obtains: {F f , Fg } = ε5 f g d x + O(ε6 ), {F f , G g } = O(ε6 ), 1 S 5 f g d x + O(ε6 ). {G f , G g } = −ε S1
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One immediately derives the following general form of the “discretized” Poisson bracket on the space Pn : {ai , a j } = Pi, j ai a j ,
{ai , b j } = 0,
{bi , b j } = −Pi, j bi b j ,
where Pi, j are some constants. Furthermore, one assumes: Pi+k, j+k = Pi, j by cyclic invariance. One then can check that (4.15) is the only Poisson bracket of the above form preserved by the map T . 7. Discussion We finish this paper with questions and conjectures. Second Poisson structure. The Poisson structure (2.16) is a discretization of (6.7) known as the first Poisson structure of the Boussinesq equation. We believe that there exists another, second Poisson structure, compatible with the Poisson structure (2.16), that allows to obtain the monodromy invariants (and thus to prove integrability of T ) via the standard bi-Hamiltonian procedure. Note that the Poisson structure usually considered in the discrete case, cf. [9], is a discrete version of the second Adler-Gelfand-Dickey bracket. We conjecture that one can adapt this bracket to the case of the pentagram map. Integrable systems on cluster manifolds. The space Pn is closely related to cluster manifolds, cf. [7]. The Poisson bracket (2.16) is also similar to the canonical Poisson bracket on a cluster manifold, cf. [10]. Example 5.6 provides a relation of the (a, b)-coordinates to the cluster coordinates. A twisted pentagon is closed if and only if a0 =
a3 + 1 , a1
a2 =
a1 + a3 + 1 , a1 a3
a4 =
a1 + 1 , a3
and bi = −ai+2 . Note that ai+5 = ai . This formula coincides with formula of coordinate exchanges for the cluster manifold of type A2 , see [7]. The 2-dimensional submanifold of P5 with M = Id is therefore a cluster manifold; the coordinates (a1 , a3 ), etc. are the cluster variables. It would be interesting to compare the coordinate systems on Pn naturally arising from our projective geometrical approach with the canonical cluster coordinates and check whether the Poisson bracket constructed in this paper coincides with the canonical cluster Poisson bracket. We think that analogs of the pentagram map may exist for a larger class of cluster manifolds.2 Polygons inscribed into conics. We observed in computer experiments that if a twisted polygon is inscribed into a conic then one has: E k = Ok for all k; the same holds for polygons circumscribed about conics. As of now, this is an open conjecture. Working on this conjecture, we discovered, by computer experiments, a variety of new configuration theorems of projective geometry involving polygons inscribed into conics; these results will be published separately. Let us also mention that twisted n-gons inscribed into a conic constitute a coisotropic submanifold of the Poisson manifold Pn . Dynamical consequences of this observations will be studied elsewhere.3 2 See the recent preprint arXiv:1005.0598. 3 See the recent preprints arXiv:0910.1952, arXiv:1004.4311.
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Acknowledgements. We are endebted to A. Bobenko, V. Fock, I. Marshall, S. Parmentier, M. Semenov-TianShanski and Yu. Suris for stimulating discussions. V. O. and S. T. are grateful to the Research in Teams program at BIRS. S. T. is also grateful to Max-Planck-Institut in Bonn for its hospitality. R. S. and S. T. were partially supported by NSF grants, DMS-0604426 and DMS-0555803, respectively.
References 1. Adler, V.E.: Cuttings of polygons. Funct. Anal. Appl. 27, 141–143 (1993) 2. Adler, V.E.: Integrable deformations of a polygon. Phys. D 87, 52–57 (1995) 3. Arnold, V.I.: Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics, 60. New York: Springer-Verlag, 1989 4. Belov, A., Chaltikian, K.: Lattice analogues of W -algebras and classical integrable equations. Phys. Lett. B 309, 268–274 (1993) 5. Bobenko, A., Suris, Yu.: Discrete Differential Geometry: Integrable Structure. Providence, RI: Amer. Math. Soc., 2008 6. Fock, V., Goncharov, A.: Moduli spaces of convex projective structures on surfaces. Adv. Math. 208, 249–273 (2007) 7. Fomin, S., Zelevinsky, A.: Cluster algebras I: foundations. J. Amer. Math. Soc. 12, 497–529 (2002) 8. Falqui, G., Magri, F., Tondo, G.: Reduction of bi-Hamiltonian systems and the separation of variables: an example from the Boussinesq hierarchy. Theoret. and Math. Phys. 122, 176–192 (2000) 9. Frenkel, E., Reshetikhin, N., Semenov-Tian-Shansky, M.: Drinfeld-Sokolov reduction for difference operators and deformations of W -algebras. I. The case of Virasoro algebra. Commun. Math. Phys. 192, 605–629 (1998) 10. Gekhtman, M., Shapiro, M., Vainshtein, A.: Cluster algebras and Poisson geometry. Mosc. Math. J. 3, 899– 934 (2003) 11. Henriques, A.: A periodicity theorem for the octahedron recurrence. J. Alg. Combin. 26, 1–26 (2007) 12. Hoffmann, T., Kutz, N.: Discrete curves in CP1 and the Toda lattice. Stud. Appl. Math. 113, 31–55 (2004) 13. Konopelchenko, B.G., Schief, W.K.: Menelaus’ theorem, Clifford configurations and inversive geometry of the Schwarzian KP hierarchy. J. Phys. A: Math. Gen. 35, 6125–6144 (2002) 14. Marshall, I., Semenov-Tian-Shansky, M.: Poisson groups and differential Galois theory of Schroedinger Equation on the circle. Commun. Math. Phys 284, 537–552 (2008) 15. Motzkin, Th.: The pentagon in the projective plane, with a comment on Napiers rule. Bull. Amer. Math. Soc. 52, 985–989 (1945) 16. Ovsienko, V., Schwartz, R., Tabachnikov, S.: Quasiperiodic motion for the pentagram map. Electron. Res. Announc. Math. Sci. 16, 1–8 (2009) 17. Ovsienko, V., Tabachnikov, S.: Projective Differential Geometry old and New, from Schwarzian Derivative to the Cohomology of Diffeomorphism Groups. Cambridge Tracts in Mathematics, 165, Cambridge: Cambridge University Press, 2005 18. Robbins, D., Rumsey, H.: Determinants and alternating sign matrices. Adv. Math. 62, 169–184 (1986) 19. Schwartz, R.: The pentagram map. Experiment. Math. 1, 71–81 (1992) 20. Schwartz, R.: The pentagram map is recurrent. Experiment. Math. 10, 519–528 (2001) 21. Schwartz, R.: Discrete monodromy, pentagrams, and the method of condensation. J. of Fixed Point Theory and Appl. 3, 379–409 (2008) 22. Speyer, D.: Perfect matchings and the octahedron recurrence. J. Alg. Combin. 25, 309–348 (2007) 23. Suris, Yu.: The Problem of Integrable Discretization: Hamiltonian Approach, Progress in Mathematics, 219, Basel: Birkhauser Verlag, 2003 24. Thurston, W.: Three-dimensional Geometry and Topology. Vol. 1, Princeton Mathematical Series, 35, Princeton, NJ: Princeton University Press, 1997 25. Tongas, A., Nijhoff, F.: The Boussinesq integrable system: compatible lattice and continuum structures. Glasg. Math. J. 47, 205–219 (2005) 26. Veselov, A.: Integrable mappings. Russ. Math. Surv. 46(5), 1–51 (1991) 27. Weinstein, A.: The local structure of Poisson manifolds. J. Diff. Geom. 18, 523–557 (1983) Communicated by N.A. Nekrasov
Commun. Math. Phys. 299, 447–467 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1050-7
Communications in
Mathematical Physics
The Fundamental Solution and Strichartz Estimates for the Schrödinger Equation on Flat Euclidean Cones G. Austin Ford Department of Mathematics, Northwestern University, 2033 Sheriden Road, Evanston, IL 60208, USA. E-mail:
[email protected] Received: 22 October 2009 / Accepted: 18 January 2010 Published online: 29 April 2010 – © Springer-Verlag 2010
Abstract: We study the Schrödinger equation on a flat euclidean cone R+ × S1ρ of cross-sectional radius ρ > 0, developing asymptotics for the fundamental solution both in the regime near the cone point and at radial infinity. These asymptotic expansions remain uniform while approaching the intersection of the “geometric front,” the part of the solution coming from formal application of the method of images, and the “diffractive front” emerging from the cone tip. As an application, we prove Strichartz estimates for the Schrödinger propagator on this class of cones. 0. Introduction In this paper, we study the initial value problem for the Schrödinger equation, (Dt − ) u(t, r, θ ) = 0 u(0, r, θ ) = u 0 (r, θ ), def
(1) def
on a flat euclidean cone. This is an incomplete manifold C(S1ρ ) = R+ ×S1ρ , where S1ρ = R 2πρZ is the circle of radius ρ > 0, equipped with the metric g(r, θ ) = dr 2 + r 2 dθ 2 , and the Laplacian is taken to be the Friedrichs extension of g ∞ 1 . Specifically, Cc C(Sρ )
we are interested in the behavior of the fundamental solution eit δ(r0 ,θ0 ) to (1). We begin by using Cheeger’s functional calculus for cones, developed first in [Che79], to show that the Schrödinger propagator on C(S1ρ ) has the series representation K eit (r1 , θ1 , r2 , θ2 ) ⎫ r 2 +r 2 ⎧ ⎬ exp 14it 2 ⎨ ∞ r r j r1 r2 1 2 +2 cos =− i j/ρ J j/ρ , J0 (θ1 − θ2 ) ⎭ 4πiρt ⎩ 2t 2t ρ j=1
(2)
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G. A. Ford
where Jν (x) is the Bessel function of order ν. Employing an integral representation for Jν (x) due to Schläfli, we then show that the quantity in braces in (2) can be represented by the loop integral π 1 x 1 2 + η + i log[v] v− cot exp S(x, η) = 4π C 2 v 2ρ π − η + i log[v] dv , (3) + cot 2 2ρ v def
def
where x = r12tr2 and η = θ1 − θ2 . Due to the fact that the amplitude of S(x, η) has poles which move with η and collide with the stationary points v = ±i of the phase 21 v − v1 , the standard techniques of asymptotic analysis will not produce an expansion of (3) which is uniform in η. However, by modifying a version of the method of steepest descent due to van der Waerden [vdW51], we are able to produce a uniform asymptotic expansion of S(x, η) in decreasing powers of x as x −→ ∞, and this leads us to an expansion for K eit in decreasing powers of r12tr2 as this quantity approaches infinity. Namely, we show K eit (r1 , θ1 , r2 , θ2 ) ∞ r r − 1 r r − 2k+1 1 r1 r2 2 2 1 2 1 2 α α α −→ ∞ ∼ + D− 2k+1 G0 + G− 1 as t 2t 2t 2t 2 2 α=±1
k=0
(4) in the regime away from poles of (3) coinciding with the stationary points of the phase, i.e. θ1 −θ2 ≡ −π , 0, or π (mod 2πρ). These functions Gαs and Dαs are piecewise-smooth and bounded in all variables, and their precise definitions can be found in Sect. 4. We also provide asymptotics as r12tr2 −→ 0, showing r r σ r12 + r22 1 1 2 K eit (r1 , θ1 , r2 , θ2 ) = − exp 1+O 4πiρt 4it 2t
(5)
def in this regime, where σ = min 2, ρ1 . This only makes use of elementary estimates for Bessel functions. This asymptotic expansion (4) shows that the Schrödinger kernel K eit is separated −s α into two parts, the “geometric” factors r12tr2 Gs analogous to those that would arise −s α Ds terms arising from the formal application of the method of images, and the r12tr2 from a “diffractive effect” emerging from the cone tip. The diffractive terms have noticeably better decay vis-à-vis r12tr2 , being of order − 21 in this variable, whereas the geometric terms as a whole are of order 0. This is analogous to the classical results for the wave equation of Sommerfeld [Som96] and Friedlander [Fri58] in the presence of obstacles and later work of Cheeger and Taylor in the setting of product cones [CT82a,CT82b] and of Melrose and Wunsch for manifolds with cone points [MW04]. In each case, they show the diffractive front is 21 degree “weaker,” i.e. more regular in an appropriate sense. It is also morally consistent with the parametrix construction of Hassell and Wunsch [HW05] for the Schrödinger equation on scattering manifolds, where they show the leading order part of the propagator is given by the sojourn relation.
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449
As an application of our asymptotic expansion, we prove the Strichartz estimates U(t) f (r, θ ) L p L q (r dr dθ) f L 2 (r dr dθ) , t U(−s)F(s, r, θ ) ds F(t, r, θ ) p q , 2 L t L (r dr dθ) L (r dr dθ) U(t − s)F(s, r, θ ) ds F(t, r, θ ) p˜ q˜ p q L t L (r dr dθ)
(6) (7) (8)
L t L (r dr dθ)
s 0, which we will write as S1ρ = R 2πρZ. Equipping it with the metric h(θ ) = dθ 2 inherited from R, the Laplace operator on S1ρ is h = −∂θ2 , and its eigenvalues and eigenfunctions are i jθ j 1 , where j = 0, 1, 2, . . .. (15) μj = , ± exp (θ ) = ϕ± √ j ρ ρ 2πρ Note that the positive eigenvalues μ j > 0 have multiplicity 2, whereas μ0 = 0 has multiplicity 1. Moving to the cone C(S1ρ ) with metric g = dr 2 + r 2 dθ 2 , the associated Laplacian g is g = −∂r2 −
1 1 ∂r − 2 ∂θ2 , r r
(16)
which we see to be the standard Laplacian for R2 written in polar coordinates. In the following, we will write for the Friedrichs extension of this Laplace operator on C(S1ρ ), understanding that the metric dependence is implicit. Consider the solution operator for the Schrödinger equation (1), U(t) = eit .
(17)
Using Cheeger’s formulae (12) and (13) for the Schwartz kernel of functions of the Laplacian, we see that ∞ ij 1 ˜ | j| exp K eit (r1 , θ1 , r2 , θ2 ) = K eit r1 , r2 , (θ1 − θ2 ) , 2πρ ρ ρ j=−∞
(18)
Schrödinger Equation on Flat Euclidean Cones
where K˜ eit (r1 , r2 , ν) =
451
∞ 0
2
eitλ Jν (λr1 ) Jν (λr2 ) λ dλ.
(19)
Letting t = is, we obtain an expression for the heat kernel e−s . In particular, applying Weber’s second exponential integral [Wat95, §13.31(1)] to the radial coefficient K˜ e−s gives us the expression r 2 +r 2 exp − 14s 2 r r 1 2 Iν , (20) K˜ e−s (r1 , r2 , ν) = 2s 2s where Iν (x) is the modified Bessel function of the first kind, ∞ x ν+2 j 1 def Iν (x) = . (21) j! (ν + j + 1) 2 j=0
Analytic continuation in s and setting s = −it then returns an expression for the Schrödinger propagator, r 2 +r 2 i ν+1 exp 14it 2 r r 1 2 K˜ eit (r1 , r2 , ν) = , (22) Jν 2t 2t ν where we use the fact that Iν (i x) = i Jν (x). Substituting this into (18) and combining ij the exponentials exp ρ (θ1 − θ2 ) and exp − iρj (θ1 − θ2 ) for positive j, we obtain the following proposition. Proposition 2.1. The Schrödinger propagator eit on C(S1ρ ) has Schwartz kernel K eit (r1 , θ1 , r2 , θ2 ) ⎫ r12 +r22 ⎧ ⎬ exp 4it ⎨ ∞ j r1 r2 r1 r2 +2 cos =− i j/ρ J j/ρ . (23) J0 (θ1 − θ2 ) ⎭ 4πiρt ⎩ 2t 2t ρ j=1
3. An Integral Representation for the Fundamental Solution The next step in our analysis is to transform the expression (23) for the Schwartz kernel of the propagator into one more amenable to calculation. Before we start, we simplify the calculation by introducing the dummy variables x and η, defined to be def r1 r2 def x = and η = θ1 − θ2 ; (24) 2t these are the arguments of the Bessel functions and the cosines respectively. We also introduce the name S(x, η) for the quantity in braces in (23), i.e. ∞ j def i j/ρ J j/ρ (x) cos S(x, η) = J0 (x) + 2 η . (25) ρ j=1
This function S(x, η) will be the primary target for our analysis, and its asymptotics will provide asymptotics of K eit .
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Lemma 3.1. The function S(x, η) has a loop integral representation π + η + i log[v] 1 1 x v− cot 2 S(x, η) = exp 4π C 2 v 2ρ π − η + i log[v] dv , + cot 2 2ρ v
(26)
where C is a contour starting at −∞, encircling the unit circle in a counterclockwise direction, and returning to −∞. Proof. Consider the Schläfli loop integral representation1 for the Bessel function Jν (x) [Wat95, §6.2(2)], (0+) 1 dv 1 x Jν (x) = v− exp . (27) 2πi −∞ 2 v v ν+1 Substituting this formula for the Bessel functions in the definition of S(x, η) (25) and exchanging the summation and integration, we have the expression ⎫ ⎧ ⎨ (0+) ∞ i j/ρ cos j η ⎬ ρ 1 1 x dv v− . (28) S(x, η) = exp 1+2 ⎭ v ⎩ 2πi −∞ 2 v v j/ρ j=1
This exchange is justifiable by taking the contour to be sufficiently far away from the origin; choosing a contour so that |v| > 1 + ε for some ε > 0 will ensure the resulting integral is absolutely convergent. Under these same conditions, we can take advantage of the fact that the quantity in braces in (28) is a sum of two geometric series: ⎧ ⎨ (0+) ∞ i j π2 + η + i log[v] 1 1 x S(x, η) = v− exp exp 1+ ⎩ 2πi −∞ 2 v ρ j=1 ⎫ ∞ i j π2 − η + i log[v] ⎬ dv + exp ⎭ v ρ j=1 ⎧ i π ⎨ (0+) + η + i log[v] exp ρ 2 1 1 x v− = exp 1+ ⎩ 2πi −∞ 2 v 1 − exp i π + η + i log[v] i π
⎫ ⎬ dv exp ρ 2 − η + i log[v] . + 1 − exp ρi π2 − η + i log[v] ⎭ v
ρ
2
(29)
Here, the logarithm is chosen to have its branch along the nonpositive real axis so as not to interfere with the integration contour. Now, we note the equality α 1 exp[iα] i + = cot . (30) 2 1 − exp[iα] 2 2 Substituting this into the above gives us the desired form.
1 The notation “ (0+) ” signifies that the contour begins at −∞, wraps around the origin with positive −∞ (counterclockwise) orientation, and returns to −∞.
Schrödinger Equation on Flat Euclidean Cones
453
Remark 3.2. Deser and Jackiw use a similar method in [DJ88], though they apply it to another of Schläfli’s integral representations [Wat95, §6.2(3)]. Our choice has the merit of producing a simpler expression for the exponential phase in S(x, η), facilitating its asymptotic development in what follows. Meromorphic continuation in v of the integrand in (26) shows that it is holomorphic away from the logarithmic branch along the nonpositive real axis and a finite number of poles. These poles all lie on the unit circle and are of the form eiϕ for ϕ in the set Pρ (±η) of “pole phases,” ! def π Pρ (±η) = ± η + 2πρk; k ∈ Z ∩ [−π, π ). (31) 2 The sign of η here denotes to which summand of the amplitude the pole belongs, and the intersection with [−π, π ) restricts the poles to lying on a single sheet of the universal cover of the punctured plane. This observation allows us to deform our contour C as we wish. 4. Asymptotics of the Fundamental Solution We will now calculate the asymptotics of S(x, η) as x −→ 0 and x −→ ∞ for general cross-sectional radius ρ. These will in turn give us the asymptotics of the fundamental solution eit δ(r0 ,θ0 ) of (1) which we will use to prove Strichartz estimates in Sect. 5. We begin by addressing the x −→ 0 regime in the following proposition. Proposition 4.1. The Schwartz kernel of eit has leading order asymptotics K eit (r1 , θ1 , r2 , θ2 ) r r σ r12 + r22 1 1 2 exp 1+O =− 4πiρt 4it 2t
as
r1 r2 −→ 0, 2t
(32)
def where σ = min 2, ρ1 . Proof. We begin with the bound [Wat95, §3.31(1)] x ν 1 |Jν (x)| , (ν + 1) 2
(33)
valid for Bessel functions with x real and ν > − 21 . For |x| < 2, this estimate implies ∞ j def |S(x, η) − 1| = (J0 (x) − 1) + 2 η i j/ρ J j/ρ (x) cos ρ j=1 ∞ x j/ρ ∞ (−1) j x 2 j 1 + 2 2 ( j!)2 2 j +1 j=1
j=1
∞ 2 j x j=1
4
+
2 1 ρ
+1
∞ j=1
ρ
x j/ρ 2
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G. A. Ford
=
x 1/ρ x2 2 + . 4 − x 2 1 + 1 21/ρ − x 1/ρ ρ
This proves the desired asymptotics.
(34)
To handle the x −→ ∞ regime, we proceed by applying a modified version of the method of steepest descent2 developed in van der Waerden’s article [vdW51]. Our approach will differ from van der Waerden’s in that our poles move with changing η. This produces a spurious singularity if we follow [vdW51] to the letter, however a straightforward modification prevents this kind of degeneration. Before diving into the calculation, we define π 1 x def 1 2 + αη + i log[v] dv Sα (x, η) = v− cot , (35) exp 4π C 2 v 2ρ v where α = ±1. Thus S(x, η) = S+ (x, η) + S− (x, η). 4.1. Van der Waerden’s change of variables. We introduce into the integral (35) the change of variables 1 1 u(v) = v− , (36) 2 v taking the phase of Sα (x, η) as our new base variable. This map u : C −→ C is a branched double cover of the complex plane with branch points at i and −i. The two sheets of this cover are the images of the reverse change of variables maps 1 2 v± (u) = u ± u 2 + 1 , (37) namely v− (C) = {v ∈ C; Re[v] < 0} ∪ {ib ∈ iR; |b| 1} , v+ (C) = {v ∈ C; Re[v] > 0} ∪ {ib ∈ iR; |b| 1} ,
(38)
as shown in Fig. 1(a). Here we take the principal branch of the square root, requiring 1 2 0. Our original variable v is therefore a multi-valued function of u whose Re z branches are given by v± . Since one part of C lies on the v− -sheet and the other on the v+ -sheet, the image contour u(C) crosses the branch cuts emanating from u = ±i; see Fig. 1(b) for an illustration. We shall write C± for the part of the contour C lying in the sheet v± (C). Expressing Sα (x, η) in terms of u produces the equation ⎡ 2 1 ⎤ π 2 + αη + i log u + σ u + 1 ⎥ ⎢2 du 1 ⎥ Sα (x, η) = e xu cot ⎢ 1 , ⎦ 2 ⎣ 4π u(C) 2ρ σ u +1 2 (39) 2 For the details of the standard method of steepest descent, see Olver’s book [Olv97] or any book on asymptotics or special functions.
Schrödinger Equation on Flat Euclidean Cones
(a)
455
(b)
Fig. 1. The change of variables v u and its mapping properties (quadrants of the plane denoted by bold roman numerals, unit circle by thin dashed line, logarithmic cut by thick gray dashes, and branch cuts by thin gray dashes)
where σ = σ (u) is ±1 on u(C± ) and serves to correct the branch of the square root. We remark that the poles of the integrand, located initially at eiϕ in the v-plane for ϕ iϕ in Pρ (αη), move to the points u e = i sin(ϕ) in the segment of the imaginary axis between −i and i in the u-plane. 4.2. Contour deformations and local uniformizations. For the remainder of this section, we will assume that none of the poles u = i sin(ϕ) coincide with the branch points at u = ±i, i.e. αη ≡ −π or 0 (mod 2πρ).
(40)
As we shall see, this assumption puts us in the regime where the “geometric front,” which is the part of the fundamental solution that arises from formal application of the method of images, and the “diffractive front” emanating from the cone tip do not interact. We now deform the contour u(C). This starts by separating u(C) into u(C− ) and u(C+ ) (see Fig. 2), the parts of u(C) on which the integrand is single-valued; in what follows, we always ensure that after the deformations the endpoints of these contours match. Changing u(C− ) or u(C+ ) will vary the contributions of the individual pieces to the contour integral, but the integral over the entire contour u(C) = u(C− ) ∪ u(C+ ) will remain the same. We replace the u(C− ) contour with one consisting of straight horizontal lines running ± from (negative) infinity to the branch points, shown in Fig. 3(a), and we denote by C− the piece of u(C− ) containing u = ±i. Turning to u(C+ ), we exchange it for a collection of horizontal and vertical lines together with small loops around the poles of the integrand, displayed in Fig. 3(b). (Note that the endpoints of u(C− ) and this transitional u(C+ ) match.) The final deformation comes from pulling these vertical components of u(C+ ) out to (negative) infinity, allowable due to the exponential decay of the integrand of Sα (x, η) for x > 0 and αη ≡ −π or 0 (mod 2πρ). We label the keyhole contour surrounding the pole at u = i sin(ϕ) by Cϕ ,
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G. A. Ford
(a)
(b)
Fig. 2. The contour u(C) separated into u(C− ) and u(C+ ) and example poles of the integrand (ρ = √1 , 21
η = 2π 9 )
and we call the purely horizontal pieces C+± depending on which branch point u = ±i they contain. The resulting contour is shown in Fig. 3(c). The contribution to Sα (x, η) coming from the Cϕ pieces of u(C+ ) reduces to a residue calculation, for the contributions coming from integration to and from negative infinity along the horizontal components sum to zero. Noting that σ = +1 along these contours, a simple calculation shows the residue of the integrand of Sα (x, η) at one of these poles is
Res
⎧ ⎪ ⎪ ⎨ e xu
u=i sin(ϕ) ⎪ ⎪ 4π
⎩
=
⎡ ⎢ cot ⎢ ⎣
π 2
⎫ 1 ⎤ ⎪ ⎪ + αη + i log u + u 2 + 1 2 ⎬ ⎥ 1 ⎥ 1 ⎪ ⎦ 2 2ρ u +1 2 ⎪ ⎭
ρ exp[i x sin(ϕ)] . 2πi
(41)
An application of the residue theorem then gives the following. Lemma 4.2. The contribution to Sα (x, η) of one of the pole-enclosing contours Cϕ is
1 4π
Cϕ
⎡ ⎢ e xu cot ⎢ ⎣
π 2
1 ⎤ + αη + i log u + σ u 2 + 1 2 ⎥ ⎥ ⎦ 2ρ
du 1 σ u2 + 1 2
= ρ exp[i x sin(ϕ)] .
(42) β
± . Treating the two contours C± together, We now work with the horizontal pieces C± where β = ±1 is the sign of the crossed branch point, we write
Schrödinger Equation on Flat Euclidean Cones
457
(b)
(a)
(c) Fig. 3. Example poles of the integrand and associated deformation of the contour u(C) (ρ = √1 , η = 2π 9 ) 21
1 4π
⎡
β
⎢ e xu cot ⎢ ⎣
β
C− ∪C+
=−
−
β 4π
β 4π
1 ⎤ + αη + i log u + σ u 2 + 1 2 ⎥ du ⎥ 1 ⎦ 2 2ρ σ u +1 2 ⎡
βi
−∞+βi
π 2
⎢ e xu cot ⎢ ⎣ ⎡
−∞+βi βi
⎢ e xu cot ⎢ ⎣
π 2
π 2
1 ⎤ + αη + i log u − u 2 + 1 2 ⎥ du ⎥ 1 ⎦ 2ρ − u2 + 1 2 2 1 ⎤ 2 + αη + i log u + u + 1 ⎥ du ⎥ 1 . ⎦ 2ρ u2 + 1 2
(43)
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G. A. Ford
We make the change of variables def
1
s(u) = σ (u) (βi − u) 2
(44)
along these contours, again taking the principal branch of the square root and correcting with σ where necessary. In the language of Riemann surfaces, this new variable s is a local uniformizer at the branch point u = βi, which is to say it unravels the doubling action of the map u(v) at that point. The inverse is given by u(s) = βi − s 2 .
(45)
Under this change, (43) becomes ⎡ 1 ⎤ π 2 − |s| s 2 − 2βi 2 + αη + i log βi − s 0 ⎥ ⎢2 β 2 ⎥ − e x βi−s cot ⎢ ⎦ ⎣ 2π −∞ 2ρ ×
+
s ds 1 |s| s 2 − 2βi 2
β 2π
×
∞
ex
βi−s 2
⎡
0
⎢ cot ⎢ ⎣
π 2
2 1 ⎤ 2 2 + αη + i log βi − s + |s| s − 2βi ⎥ ⎥ ⎦ 2ρ
s ds 1 , |s| s 2 − 2βi 2
(46)
which we rewrite as β e xβi 2π
⎡ ∞
−∞
⎢ 2 e−xs cot ⎢ ⎣
ds × 1 . s 2 − 2βi 2
π 2
2 1 ⎤ 2 2 + αη + i log βi − s + s s − 2βi ⎥ ⎥ ⎦ 2ρ (47)
The multiplicative factor of β appears here to correct for the direction of integration along the real axis, which varies with the branch point at which we localize; see Fig. 4. Remark 4.3. Pausing for a moment to consider the case where ρ = N1 for N a positive integer, we can see how our analysis reduces to what one expects from the method of images. Consider the α = −1 term in (46). We can factor out a negative sign in the cotangent to obtain 1 N π 2 2 2 − η + i log βi − s + s s − 2βi cot 2 2 1 π N 2 . (48) − + η + i log −βi + s 2 + s s 2 − 2βi = − cot 2 2
Schrödinger Equation on Flat Euclidean Cones
(a)
459
(b) Fig. 4. The s-planes and location of poles of the integrand
Using the fact that the cotangent is π -periodic, we have 1 N π 2 − cot + Nπ − + η + i log −βi + s 2 + s s 2 − 2βi 2 2 1 N π 2 2 2 + η + i log[−1] + i log −βi + s + s s − 2βi = − cot 2 2 1 N π 2 2 2 . (49) + η + i log βi − s − s s − 2βi = − cot 2 2 We now substitute this into (46) and make the change of variables s −s: 1 β e xβi ∞ −xs 2 N π 2 2 2 − η + i log βi − s + s s − 2βi e cot 2π 2 2 −∞ ds × 1 s 2 − 2βi 2 1 β e xβi ∞ −xs 2 N π 2 + η + i log βi − s 2 + s s 2 − 2βi =− e cot 2π 2 2 −∞ ds (50) × 1 . s 2 − 2βi 2 ± of the contour sum Thus, when we sum (46) over α and β, the horizontal parts C± to zero, and we are left with only the contributions from the Cϕ contours calculated previously. Hence, S(x, η) is a sum of the terms from Lemma 4.2, and the Schrödinger propagagtor is ⎤ ⎡ 2 + r 2 − 2r r cos θ − θ − 2π j N −1 r 1 2 1 2 1 2 N 1 ⎦. K eit (r1 , θ1 , r2 , θ2 ) = − exp⎣ 4πit 4it j=0
(51) This is precisely what one obtains from the method of images.
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G. A. Ford
4.3. Preliminary asymptotics. At this stage, we can obtain a preliminary asymptotic expansion for the fundamental solution in our regime. We start with the integral (46), which is amenable to the usual method of saddle points [Olv97]. Its application generates the expansion ⎡ 2 1 ⎤ π 2 2 + αη + i log βi − s + s s − 2βi ⎥ ⎢2 ds β e xβi ∞ −xs 2 ⎥ e cot ⎢ 1 ⎦ 2 ⎣ 2π 2ρ −∞ s − 2βi 2 β eβ (x+ 4 )i π
=
1
(2π ) 2
3 (1 − β) π2 + αη − 1 x 2 + O x− 2 cot 2ρ
as x −→ ∞.
(52)
To obtain an expansion for Sα (x, η), we sum over β = ±1 and add to the result the contributions coming from the Cϕ contours: ρ exp[i x sin(ϕ)] Sα (x, η) = π π ϕ∈Pρ (αη)∩(− 2 , 2 ) 1 αη (x+ π )i αη + π −(x+ π )i 4 e 4 − cot e + (2π x)− 2 cot 2ρ 2ρ 3 −2 as x −→ ∞. (53) +O x def
def
Lastly, we sum over α = ±1, substitute the definitions of x = r12tr2 and η = θ1 − θ2 , r12 +r22 1 and multiply by the leading factor − 4πρit exp 4it from (23) to obtain the leading order asymptotics of the Schrödinger kernel. Proposition 4.4. Uniformly away from θ1 −θ2 ≡ −π , 0, and π (mod 2πρ), the Schrödinger propagator has asymptotics K eit (r1 , θ1 , r2 , θ2 ) ⎧ 1⎨ 1 = − t ⎩ 4πi
r12 + r22 − 2r1r2 cos(θ1 − θ2 ± 2πρ j) exp 4it −π 0,
(62)
and making the change s −s gives us a formula for Im[a] < 0:
∞ −∞
2 1 e−xs 2 ds = −iπ e−xa erfc i x 2 a s−a
if Im[a] < 0.
(63)
Noting that 1 π 1 sgn Im σϕ (βi − i sin(ϕ)) 2 = sgn Im σϕ eβ 4 i (1 − β sin(ϕ)) 2 = σϕ β
(64)
and applying the above formulae shows aϕ(α,β)
∞
e−xs
2
ds 1 s − σϕ (βi − i sin(ϕ)) 2 π 1 1 σϕ ρ exp[i x sin(ϕ)] erfc e−β 4 i x 2 (1 − β sin(ϕ)) 2 . =− 2 −∞
Summing over ϕ in Pρ (αη) proves the lemma.
(65)
Recall that the complementary error function erfc[z] is entire with everywhere convergent Taylor series: 2 ∞ 2 e−z 2k z 2k+1 erfc[z] = 1 − √ , (2k + 1)!! π
(66)
k=0
and it has the asymptotic expansion 2 ∞ e−z (−1)k (2k)! erfc[z] ∼ √ k! (2z)2k πz
as z −→ ∞ in |arg(z)|
0 independent of ε such that for 0 ≤ ε ≤ ε0 , 3
sup {ε 2
0≤t≤ε−m
+ +
{1 + |v|2 }β+1 FRε (t) 3 ∞ + ε 2 ∇φ εR ∞ } √ ωM
sup {ε5 ∇x,v (
0≤t≤ε−m
{1 + |v|2 }β FRε (t) )∞ } + ε5 ∇x ∇x φ εR ∞ } √ ωM
FRε (t) sup { √ + ∇φ εR } ω 0≤t≤ε−m
{1 + |v|2 }β+1 FRε (0) {1 + |v|2 }β FRε (0) ∞ + ε5 ∇x,v ( )∞ √ √ ωM ωM F ε (0) + ∇φ εR (0) + 1} + √R ω(0) 3
≤ C{ε 2
for all 0 < m ≤
1 2k−3 2 2k−2 .
Remark 1.2. While we get a uniform in ε estimate for the L 2 norm of the remainders, for the weighted W 1,∞ norms, we only obtain a uniform estimate of ε5 FRε and ε5 ∇x φ εR , which is why we need higher order expansion k ≥ 6 in (1.3). With these higher order Hilbert expansions, our uniform estimates lead to the Euler-Poisson limit 2k−3 sup0≤t≤ε−m ||F ε − ω|| = O(ε) for all 0 < m ≤ 21 2k−2 . Our result provides a rare example such that the Hilbert expansion is valid for all time. In the absence of the electrostatic interaction, it is well-known [2] that a similar Hilbert expansion is only valid local in time, before shock formations in the pure compressible Euler flow, for example, see [10]. Recently, such a validity has been established near a shock [16]. By a classical result [11], it is well-known that even for arbitrary small perturbations of a motionless steady state, singularity does form in finite time for the Euler system for a compressible fluid. In contrast, the validity time in the Euler-Poisson 1 2k−3 limit is ε− 2 2k−2 for irrotational flow, which implies global in-time convergence from the Vlasov-Poisson-Boltzmann to the Euler-Poisson system (1.2). The key difference, in the presence of electrostatic interaction, is that small irrotational flows exist forever without any shock formation for the Euler-Poisson (1.2), see [4]. Such a surprising result is due to an extra dispersive effect in the presence of a self-consistent electric field, which is characterized by so-called ‘plasma frequency’ in the physics literature. This leads to a ‘Klein-Gordon effect’ which enhances the linear decay rate and destroys the possible shock formation. Our method of proof relies on a recent L 2 − L ∞ approach to study the Euler limit of the Boltzmann equation [8,9]. The improvement over Caflisch’s classical paper is that now the positivity of the initial datum can be guaranteed. The main idea of our approach
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is to use the natrual L 2 energy estimate as the first step. The most difficult term in the energy estimate is √ 1 {∂t + v · ∇x + ∇x φ0 · ∇v } ω ε (∂t + v · ∇x )θ0 f ε − θ0 f , √ 2 ω which involves the cubic power of |v|, and it is hard to control by an L 2 type of norm only. We introduce a new weighted L ∞ space to control such a term. The second step is to estimate such a weighted L ∞ norm along the trajectory, based on the L 2 estimate in the first step, but with a singular negative power of ε. Such a simple interplay between L 2 and L ∞ norms fails to yield a closed estimate in our study, unlike the compressible Euler limit [9]. The new analytical difficulty to overcome in the present work is to obtain a delicate pointwise estimate of the distribution function in the presence of a curved trajectory caused by the self-consistent electric field. It turns out that, due to the Poisson coupling, we need to further estimate the W 1,∞ norm along the trajectory to close our estimate. This requires a higher expansion (1.3) to compensate the more singular power of ε for the W 1,∞ estimate. In order to obtain the uniform estimate over the time scale 1 2k−3 of ε− 2 2k−2 , we must carefully analyze temporal growth of the coefficients F j . Recently, there have been quite a few mathematical studies of the Vlasov-PoissonBoltzmann system (1.1) for ε = 1. Among others, global solutions near a Maxwellian were constructed [5] in a periodic box. In [14,15], global solutions near a Maxwellian were constructed for the whole space. In [12], a self-consistent magnetic effect was also included. We remark that in this paper, the assumption of a hard-sphere collision is made for analytical reasons. From a physical point of view, the collisions between electrons are better described by the Landau-Fokker-Planck kernel. Take ε = 1. In the absence of the self-consistent eletric field, the global solutions have been constructed near Maxwellians for the Landau-Fokker-Planck collision kernel [7]. Unfortunately, it has remained an outstanding open problem to construct global solutions near Maxwellians for the Vlasov-Poisson-Landau-Fokker-Planck system, due to the severe singularity for large velocities in the Landau-Fokker-Planck collision kernel. On the other hand, the relativistic Landau-Fokker-Planck kernel is much better behaved analytically, and global soltuions have been constructed in [13]. Our paper is organized as follows. In Sect. 2, we construct the coefficients Fi for the Hilbert expansion (1.3), starting with the global smooth irrotational solution to the Euler-Poisson system (1.2) constructed in [4]. In particular, we study carefully the growth in time t for Fi . In Sect. 3, we use the L 2 energy estimate for the remainder FRε around the local Maxwellian F0 (1.8), in terms of the weighted L ∞ norm of h ε . Section 4 is a study of the curved trajcetory. Section 5 is the main techinical part of the paper, in which L ∞ and W 1,∞ norms of h ε are estimated along the curved trajectories in terms of the L 2 energy to close the whole argument. Section 6 is a direct proof of our main theorem based on the L 2 − L ∞ estimates. Throughout this paper, we use C to denote possibly different constants but independent of t and ε.
2. Coefficients of the Hilbert Expansion In this section, we discuss the existence and regularity of Fi , inherited from F0 = ω as defined in (1.8). Write √Fωi as the sum of macroscopic and microscopic parts as follows:
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for each i ≥ 1, Fi Fi Fi √ = P( √ ) + {I − P}( √ ) ω ω ω ⎧ ⎫
3 ⎨ ρ ρ0 j ρ0 θi ⎬ Fi i ≡ √ χ0 + u · χj + χ4 + {I − P}( √ ). ⎩ ρ0 θ0 i 6 θ0 ⎭ ω
(2.1)
j=1
Fi ’s will be constructed inductively as follows: Lemma 2.1. For each given nonnegative integer k, assume Fk ’s are found. Then the microscopic part of F√k+1 is determined through the equation for Fk in (1.5): ω Fk+1 {I − P}( √ ) ω {∂t + v · ∇x }Fk + i+ j=k ∇x φi · ∇v F j − i+ j=k+1 Q(Fi , F j ) i, j≥0 i, j≥1 ). = L −1 (− √ ω For the macroscopic part, ρk+1 , u k+1 , θk+1 satisfy the following: ∂t ρk+1 + ∇ · (ρ0 u k+1 + ρk+1 u 0 ) = 0, ρ0 {∂t u k+1 + (u k+1 · ∇)u 0 + (u 0 · ∇)u k+1 − ∇φk+1 } −
ρk+1 ∇(ρ0 θ0 ) ρ0
ρ0 θk+1 + 3θ0 ρk+1 ) = fk , (2.2) 3 2 ρ0 {∂t θk+1 + (θk+1 ∇ · u 0 + 3θ0 ∇ · u k+1 ) + u 0 · ∇θk+1 + 3u k+1 · ∇θ0 } = gk , 3 φk+1 = ρk+1 , + ∇(
where f k = −∂ j
j
{(v i − u i0 )(v j − u 0 ) − δi j
|v − u 0 |2 }Fk+1 dv + ρ j ∇x φi , 3 i+ j=k+1 i, j≥1
j j gk = −∂i { (v i − u i0 )(|v − u 0 |2 − 5θ0 )Fk+1 dv + 2u 0 {(v i − u i0 )(v j − u 0 ) − δi j
|v − u 0 |2 (ρ0 u j + ρ j u 0 )∇x φi . }Fk+1 dv} − 2u 0 · f k + 3 i+ j=k+1 i, j≥1
Here we use the subscript k for forcing terms f and g in order to emphasize that the right hand sides depend only on Fi ’s and ∇x φi ’s for 0 ≤ i ≤ k. Proof of Lemma 2.1. We shall only derive the equations for F1 . From the coefficient of ε0 in (1.5), the microscopic part of F1 should be
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F1 {∂t + v · ∇x + ∇x φ0 · ∇v }ω {I − P}( √ ) = L −1 (− ). √ ω ω
(2.3)
Since L −1 preserves decay in v [2], √ F1 |{I − P}( √ )| ≤ (∂ρ0 ∞ + ∂u 0 ∞ + ∂θ0 ∞ + ∇φ0 ∞ )(1 + |v|3 ) ω, ω
(2.4)
where ∂ is either ∂t or ∇x . For macroscopic variables ρ1 , u 1 , θ1 of F1 in (2.1), note that (v − u 0 )F1 dv = ρ0 u 1 , v F1 dv = ρ0 u 1 + ρ1 u 0 , F1 dv = ρ1 , |v − u 0 |2 F1 dv = (|v − u 0 |2 − 3θ0 )F1 dv + 3θ0 ρ1 = ρ0 θ1 + 3θ0 ρ1 , 2 |v| F1 dv = |v − u 0 + u 0 |2 F1 dv = ρ0 θ1 + 3θ0 ρ1 + ρ1 |u 0 |2 + 2ρ0 u 0 · u 1 , |v − u 0 |2 j i j }F1 dv v v F1 dv = {(v i − u i0 )(v j − u 0 ) − δi j 3 ρ0 θ1 + 3θ0 ρ1 j j j , + ρ0 u i0 u 1 + ρ0 u 0 u i1 + u i0 u 0 ρ1 + δi j 3 i 2 v |v| F1 dv = (v i − u i0 )|v|2 F1 dv + (ρ0 θ1 + 3θ0 ρ1 + ρ1 |u 0 |2 + 2ρ0 u 0 · u 1 )u i0 j j = (v i − u i0 )(|v − u 0 |2 − 5θ0 )F1 dv + 2u 0 {(v i − u i0 )(v j − u 0 ) |v − u 0 |2 2 }F1 dv + (5θ0 + |u 0 |2 )ρ0 u i1 + (ρ0 θ1 + 3θ0 ρ1 )u i0 3 3 + (ρ0 θ1 + 3θ0 ρ1 + ρ1 |u 0 |2 + 2ρ0 u 0 · u 1 )u i0 . − δi j
Project the equation for F1 in (1.5) onto 1, v, |v|2 to get equations of ρ1 , u 1 , θ1 with forcing terms as follows: ∂t ρ1 + ∇ · (ρ0 u 1 + ρ1 u 0 ) = 0, ρ0 θ1 + 3θ0 ρ1 j j j ) ∂t (ρ0 u i1 + ρ1 u i0 ) + ∂ j (ρ0 u i0 u 1 + ρ0 u 0 u i1 + ρ1 u i0 u 0 + δi j 3 |v − u 0 |2 j − ρ1 ∂i φ0 − ρ0 ∂i φ1 = −∂ j {(v i − u i0 )(v j − u 0 ) − δi j }F1 dv, 3 ∂t (ρ0 θ1 + 3θ0 ρ1 + ρ1 |u 0 |2 + 2ρ0 u 0 · u 1 ) 2 + ∂i {(5θ0 + |u 0 |2 )ρ0 u i1 + (ρ0 θ1 + 3θ0 ρ1 )u i0 + (ρ0 θ1 + 3θ0 ρ1 + ρ1 |u 0 |2 3 + 2ρ0 u 0 · u 1 )u i0 } − 2∇x φ0 · (ρ0 u 1 + ρ1 u 0 ) − 2∇x φ1 · (ρ0 u 0 ) j j = −∂i { (v i − u i0 )(|v − u 0 |2 − 5θ0 )F1 dv + 2u 0 {(v i − u i0 )(v j − u 0 ) − δi j
|v − u 0 |2 }F1 dv}, 3 φ1 = ρ1 .
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By using the equations for ρ0 , u 0 and ρ1 , the equation for u 1 can be reduced to ρ1 ρ0 {∂t u 1 + (u 1 · ∇)u 0 + (u 0 · ∇)u 1 − ∇φ1 } − ∇(ρ0 θ0 ) ρ0 ρ0 θ1 + 3θ0 ρ1 +∇( ) = ∂ j (μ(θ0 )∂ j u i0 ), 3 where μ(θ0 ) ≡ θ0 Bi j Lω −1 (Bi j ω)dv > 0. Here j
(v i − u i0 )(v j − u 0 ) |v − u 0 |2 Bi j = − δi j , Lω g ≡ −{Q(ω, g) + Q(g, ω)}, θ0 3θ0 and for the last term we have used the coefficient of ε0 in (1.5): − Bi j F1 dv = ∂ j u i0 Bi j Lω −1 (Bi j ω)dv. Letting Ai = we obtain
(v i − u i0 ) |v − u 0 |2 5 ( − ), √ 2θ0 2 θ0
−
Ai F1 dv = ∂i θ0
and define
κ(θ0 ) ≡ 2θ0
Ai Lω −1 (Ai ω)dv,
Ai Lω −1 (Ai ω)dv > 0.
Similarly, the equation for θ1 can be reduced to 2 ρ0 {∂t θ1 + (θ1 ∇ · u 0 + 3θ0 ∇ · u 1 ) + u 0 · ∇θ1 + 3u 1 · ∇θ0 } 3 = ∇ · (κ(θ0 )∇θ0 ) + 2μ(θ0 )|∇u 0 |2 . We rewrite the fluid equations for the first order coefficients ρ1 , u 1 , θ1 , φ1 : ∂t ρ1 + ∇ · (ρ0 u 1 + ρ1 u 0 ) = 0, ρ0 {∂t u 1 + (u 1 · ∇)u 0 + (u 0 · ∇)u 1 − ∇φ1 } −
ρ1 ∇(ρ0 θ0 ) ρ0
ρ0 θ1 + 3θ0 ρ1 ) = ∂ j (μ(θ0 )∂ j u i0 ), 3 2 ρ0 {∂t θ1 + (θ1 ∇ · u 0 + 3θ0 ∇ · u 1 ) + u 0 · ∇θ1 + 3u 1 · ∇θ0 } 3 = ∇ · (κ(θ0 )∇θ0 ) + 2μ(θ0 )|∇u 0 |2 , + ∇(
φ1 = ρ1 .
(2.5)
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Here, μ(θ0 ) and κ(θ0 ) represent the viscosity and heat conductivity coefficients respectively. This is reminiscent of the derivation of compressible Navier-Stokes equations from the Boltzmann equation. We refer to [1] for more details. This completes the proof for F1 and higher expansion coefficients Fk can be found in the same way. Since ρ1 , u 1 , θ1 , φ1 solve linear equations with coefficients and forcing terms coming from the smooth functions ρ0 , u 0 , θ0 , the initial value problem for (2.5) is well-posed in the Sobolev spaces, and moreover, we will show in Lemma 2.2 that |F1 (t, x, v)| ≤ C(1 + |v|3 )ω, where C only depends on the regularity of ρ0 , u 0 , θ0 , φ0 and the given initial data ρ1 (0), u 1 (0), θ1 (0). From (2.1) and (2.3), we also deduce that |∇v F1 (t, x, v)| ≤ C(1 + |v|4 )ω and |∇x F1 (t, x, v)| ≤ C(1 + |v|5 )ω. Recall [4] that the Euler-Poisson system (1.2) for a 53 − law perfect gas admits smooth small global solutions ρ0 , u 0 , ∇φ0 with the following pointwise uniform-in-time decay: ρ0 − ρW s,∞ + u 0 W s,∞ + ∇φ0 W s,∞ ≤
C (1 + t) p
(2.6)
for any 1 < p < 3/2 and for each s ≥ 0. In the next lemma, we show that the corresponding Hilbert expansion coefficients Fi cannot grow arbitrarily in time. Lemma 2.2. Let smooth global solutions ρ0 , u 0 , ∇φ0 to the Euler-Poisson system (1.2) 2
be given and let θ0 = Kρ03 . For each k ≥ 0, let ρk+1 (0, x), u k+1 (0, x), θk+1 (0, x) ∈ H s , s ≥ 0 be given initial data to (2.2). Then the linear system (2.2) is well-posed in H s , and furthermore, there exists a constant C > 0 depending only on the initial data (independent of t) such that for each t, |Fi | ≤ C(1 + t)i−1 (1 + |v|3i )ω, |∇x φi | ≤ C(1 + t)i−1 , |∇v Fi | ≤ C(1 + t)i−1 (1 + |v|3i+1 )ω, |∇x Fi | ≤ C(1 + t)i−1 (1 + |v|3i+2 )ω, (2.7) |∇v ∇v Fi | ≤ C(1 + t)i−1 (1 + |v|3i+2 )ω, |∇x ∇v Fi | ≤ C(1 + t)i−1 (1 + |v|3i+3 )ω. Proof. The well-posedness easily follows from the linear theory, for instance see [3]. Here we provide the a priori estimates for (2.7). The proof relies on the induction on i. We first prove for F1 . Write the linear system (2.5) as a symmetric hyperbolic system with the corresponding symmetrizer A0 : A0 {∂t U − V } +
3
Ai ∂i U + BU = F,
(2.8)
i=1
where U, V, A0 , and Ai ’s are given as follows: ⎛ ⎞ ⎛ ⎛ ⎞ (θ0 )2 0 0 ρ1 t t (ρ0 )2 θ0 I U ≡ ⎝ (u 1 ) ⎠ , V ≡ ⎝ (∇φ1 ) ⎠ , A0 ≡ ⎝ 0 θ1 0 0 0 ⎞ ⎛ i 2 2 ρ0 (θ0 ) ei 0 (θ0 ) u 0 ⎜ ρ (θ )2 (e )t (ρ )2 θ u i I (ρ0 )2 θ0 (e )t ⎟ Ai ≡ ⎝ 0 0 i 0 0 0 i ⎠. 3 0
(ρ0 )2 θ0 3 ei
(ρ0 )2 u i0 6
⎞ 0 0 ⎠,
(ρ0 )2 6
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479
(·)t denotes the transpose of row vectors, ei ’s for i = 1, 2, 3 are the standard unit (row) base vectors in R3 , and I is the 3 × 3 identity matrix. B and F, which consist of ρ0 , u 0 , θ0 , and first derivatives of ρ0 , u 0 , θ0 , can be easily written down. In particular, C 3 we have BW s,∞ + FW s,∞ ≤ (1+t) p for any 1 < p < 2 and any s ≥ 0. Note that (2.8) together with φ1 = ρ1 is strictly hyperbolic and thus we can apply the standard energy method of the linear symmetric hyperbolic system to (2.8) to obtain the following energy inequality: for each s ≥ 0, d C C {U 2H s + V 2H s } + {U H s + V H s }. {U 2H s + V 2H s } ≤ dt (1 + t) p (1 + t) p (2.9) Hence, we obtain U H s + V H s ≤ C and therefore, from (2.1) and (2.3), the inequality (2.7) for i = 1 follows. Now suppose (2.7) holds for 1 ≤ i ≤ n. For i = n + 1, we is bounded first note that from the coefficient of εn in (1.5), the microscopic part of F√n+1 ω by √ Fn+1 |{I − P}( √ )| ≤ C(1 + t)n (1 + |v|3(n+1) ) ω ω by the induction hypothesis. For the macroscopic part, we project the equation for Fn+1 in (1.5) onto 1, v, |v|2 as in the F1 case to obtain fluid equations for ρn+1 , u n+1 , θn+1 , ∇φn+1 . See (2.2). Since the structure of the left hand side of (2.2) is the same as in ρ1 , u 1 , θ1 , ∇φ1 , one can write the equations for ρn+1 , u n+1 , θn+1 , ∇φn+1 as the linear symmetric hyper bolic system. The difference is that there are extra terms coming from i+ j=n+1 ∇x φi · i, j≥1
∇v F j . From the induction hypothesis, one can get the following corresponding inequality for U, V as in (2.9): d C {U 2H s + V 2H s } ≤ {U 2H s + V 2H s } + C(1 + t)n {U H s + V H s }. dt (1 + t) p By the Gronwall inequality, we obtain U H s + V H s ≤ C(1 + t)n+1 , and this verifies (2.7) for i = n + 1. 3. L 2 Estimates for Remainder FRε In this section, we perform the L 2 energy estimates of remainders f ε = Here is the main result of this section.
Fε √R ω
and ∇x φ εR .
Proposition 3.1. Recall (1.10) and (1.11). There exists a constant C independent of t, ε such that for each t and ε, d δ0 { θ0 f ε 2 + ∇φ εR 2 } + θ M {I − P} f ε 2ν dt 2ε ≤ C{ε2 h ε ∞ f ε + εk−1 h ε ∞ f ε 2 + εk h ε ∞ f ε ∇φ εR } C + { f ε 2 + ∇φ εR 2 } + CI1 {ε f ε 2 + ε∇φ εR 2 } + CI2 εk−1 f ε , (1 + t) p
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where I1 and I2 are given as follows: I1 =
2k−1
[ε(1 + t)]i−1 + (
i=1
2k−1
[ε(1 + t)]i−1 )2 ; I2 =
i=1
εi+ j−2k (1 + t)i+ j−2 .
2k≤i+ j≤4k−2
(3.1) Proof. First we write the equation for f ε from (1.6): v − u0 √ 1 ω · ∇x φ εR + L f ε θ0 ε √ 2k−1 {∂t + v · ∇x + ∇x φ0 · ∇v } ω ε Fi =− εi−1 { ( √ , f ε ) f + εk−1 ( f ε , f ε ) + √ ω ω
∂t f ε + v · ∇x f ε + ∇x φ0 · ∇v f ε −
i=1
Fi + ( f ε , √ )} ω − εk ∇x φ εR · ∇v f ε + εk ∇x φ εR ·
2k−1 v − u0 ε i ∇v Fi f − ε {∇x φi · ∇v f ε + ∇x φ εR · √ } 2θ0 ω i=1
+
2k−1
εi ∇x φi ·
i=1
v − u0 ε f + εk−1 A, 2θ0
2 ε 0 where A = √Aω . Note that ∇v ω = − v−u θ0 ω. Take the L inner product with θ0 f on both sides to get
√ 1 d δ0 ε 2 θ0 f − ( (v − u 0 ) ω f ε dv) · ∇x φ εR d x + inf θ0 ||{I − P} f ε ||2ν 2 dt ε t,x √ {∂t + v · ∇x + ∇x φ0 · ∇v } ω ε ε 1 ≤ (∂t + v · ∇x )θ0 f ε , f ε − θ0 f ,f √ 2 ω + εk−1 θ0 ( f ε , f ε ), f ε + θ0
2k−1 i=1
+ εk ∇x φ εR ·
Fi Fi εi−1 { ( √ , f ε ) + ( f ε , √ )}, f ε ω ω
2k−1 v − u0 ε ε ∇v Fi f , f − θ0 εi ∇x φ εR · √ , f ε 2 ω i=1
2k−1
+
i=1
εi ∇x φi ·
v − u0 ε ε f , f + εk−1 θ0 A, f ε . 2
√ From φ εR = R3 f ε ωdv and (1.6), we obtain −∂t φ εR = −
R3
∂t FRε dv =
√ v · ∇x ( ω f ε )dv.
(3.2)
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481
Take the L 2 inner product with φ εR on both sides to get √ 1 d ∇φ εR 2 = −∂t φ εR · φ εR d x = v · ∇x ( ω f ε )φ εR dvd x 2 dt √ = − ( v ω f ε dv) · ∇φ εR d x.
(3.3)
Combining (3.2) and (3.3), we obtain √ ε 1 d δ0 ε 2 ε 2 ε 2 { θ0 f + ∇φ R } + θ M {I − P} f ν ≤ − u 0 ω f dv · ∇x φ εR d x 2 dt 2ε √ {∂t + v · ∇x + ∇x φ0 · ∇v } ω ε ε 1 ε ε f ,f + (∂t + v · ∇x )θ0 f , f − θ0 √ 2 ω + εk−1 θ0 ( f ε , f ε ), f ε + θ0
2k−1 i=1
+ εk ∇x φ εR ·
Fi Fi εi−1 { ( √ , f ε ) + ( f ε , √ )}, f ε ω ω
2k−1 v − u0 ε ε ∇v Fi f , f − θ0 εi ∇x φ εR · √ , f ε 2 ω i=1
2k−1
+
εi ∇x φi ·
i=1
v − u0 ε ε f , f + εk−1 θ0 A, f ε . 2
(3.4)
The first term in the right hand side is controlled by √ ε − u0 ω f dv · ∇x φ εR d x = − u 0 x φ εR · ∇x φ εR d x = − u i0 ∂ j ∂ j φ εR ∂i φ εR d x 1 i ε ε = ∂ j u 0 ∂ j φ R ∂i φ R d x − ∂i u i0 |∂ j φ εR |2 d x 2 C ≤ ∇φ εR 2 from (2.6) . (1 + t) p √
φ ·∇ } √x 0 v The key difficult term 21 (∂t + v · ∇x )θ0 − θ0 {∂t +v·∇x +∇ ω
{1 + |v|2 }3/2
fε
ω
is a cubic polynomial in
{1 + |v|2 }−2 h ε ,
v, and since ≤ for β ≥ 7/2 in (1.11), the second line in (3.4) can be estimated as follows: from the uniform bounds for ρ0 , u 0 , θ0 , ∇φ0 again in (2.6), √ 1 {∂t + v · ∇x + ∇x φ0 · ∇v } ω ε ε ε ε (∂t + v · ∇x )θ0 f , f − θ0 f ,f √ 2 ω + = |v|≥ √κε
|v|≤ √κε
≤ C{∂ρ0 + ∂u 0 + ∂θ0 + ∇φ0 } × {1 + |v|2 }3/2 f ε 1|v|≥ √κ ∞ × f ε ε
+C{∂ρ0 ∞ + ∂u 0 ∞ + ∂θ0 ∞ + ∇φ0 ∞ } × {1 + |v|2 }3/4 f ε 1|v|≤ √κ 2 ε
C ≤ Cκ ε h ∞ f + {{1 + |v|2 }3/4 P f ε 1|v|≤ √κ 2 ε (1 + t) p 2
ε
ε
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+{1 + |v|2 }3/4 {I − P} f ε 1|v|≤ √κ 2 } ε
≤ Cκ ε2 h ε ∞ f ε +
C κ2 ε 2 {I − P} f ε 2ν }. { f + (1 + t) p ε
By applying Lemma 2.3 in [5] and (1.11), the third line in (3.4) can be estimated as follows: εk−1 θ0 ( f ε , f ε ), f ε ≤ Cεk−1 ν f ε ∞ f ε 2 ≤ Cεk−1 h ε ∞ f ε 2 . From collision symmetry, we get θ0
2k−1 i=1
= θ0
Fi Fi εi−1 { ( √ , f ε ) + ( f ε , √ )}, f ε ω ω 2k−1 i=1
2k−1
Fi Fi εi−1 { ( √ , f ε ) + ( f ε , √ )}, {I − P} f ε ω ω
Fi (1 + |v|) √ dv∞ f ε {I − P} f ε ν ω i=1 2k−1 2 ε κ2 F i ε 2 ≤ εi−1 θ0 v √ dv∞ f + {I − P} f ε 2ν κ2 ε ω ≤
εi−1 θ0
i=1
≤ Cκ (
2k−1
[ε(1 + t)]i−1 )2 ε f ε 2 +
i=1
Next we estimate εk ∇x φ εR · ε
k
∇x φ εR
v−u 0 2
κ2 {I − P} f ε 2ν from (2.7). ε
f ε , f ε :
v − u0 ε ε f , f ≤ εk ∇φ εR ( · 2
1
| |v − u 0 || f ε |2 dv|2 d x) 2 √ 1 ωM ε 2 = εk ∇φ εR · ( | |v − u 0 | f ε √ h dv| d x) 2 w(v) ω 1 |v − u 0 |2 ω M k dv) 2 ∞ h ε ∞ ∇φ εR · f ε ≤ ε ( |w(v)|2 ω ≤ Cεk h ε ∞ ∇φ εR · f ε .
From (2.7), −θ0
2k−1 i=1
2k−1 1 ∇v Fi |∇v Fi |2 dv) 2 ∞ ∇φ εR · f ε εi ∇x φ εR · √ , f ε ≤ εi θ0 ( ω ω i=1
≤ C(
2k−1 i=1
[ε(1 + t)]i−1 ){ε∇φ εR 2 + ε f ε 2 }.
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483
Next, 2k−1
i=1
v − u0 ε ε εi ∇x φi · f ,f ≤ 2
2k−1
2 εi ∇x φi ∞
i=1
≤ Cκ (
2k−1
ε κ2 ε 2 f + {I − P} f ε 2ν κ2 ε
εi (1 + t)i−1 )2 ε f ε 2 +
i=1
Lastly, by recalling (1.7) and from (2.7), εk−1 θ0 A, f ε ≤ C(
κ2 {I − P} f ε 2ν . ε
εi+ j−2k (1 + t)i+ j−2 ) εk−1 f ε .
2k≤i+ j≤4k−2
We choose κ sufficiently small to absorb κε {I − P} f ε 2ν terms into the dissipation in the left hand side of (3.2). Together with Lemma 2.2, we complete the proof of our proposition. 2
4. Characteristics In this section, we study the curved trajectory for the Vlasov-Poisson-Boltzmann system (1.1). For any function φ ∈ L ∞ ([0, T ]; C 2,α ), we define the characteristics [X (τ ; t, x, v), V (τ ; t, x, v)] passing though (t, x, v) such that d X (τ ; t, x, v) = V (τ ; t, x, v), X (t; t, x, v) = x, dτ d V (τ ; t, x, v) = ∇x φ ε (τ, X (τ ; t, x, v)), V (t; t, x, v) = v. dτ Lemma 4.1. Recall (1.11). Assume 0 ≤ T ≤
1 ε
(4.1)
and
sup εk ||h ε (τ )||W 1,∞ ≤
√
0≤τ ≤T
ε.
(4.2)
Then we have sup {∂x X (t)∞ + ∂v V (t)∞ } ≤ C,
(4.3)
0≤t≤T
where C is independent of t, ε. Moreover, there exists 0 < T0 ≤ T such that for 0 ≤ τ ≤ t ≤ T0 , ∂ X (τ ) 1 3 3 |t − τ | ≤ det (4.4) ≤ 2|t − τ | , 2 ∂v |∂v X (τ )| ≤ 2|t − τ |, (4.5) ∂ V (τ ) 1 ∂ X (τ ) 1 (4.6) ≤ det ≤ 2, 2 ≤ det ≤ 2, 2 ∂v ∂x sup 0≤τ ≤T0 ,x0 ∈R3 ,|v|≤N
≤ C N for N ≥ 1.
1/2 (|∂xv X (τ ; t, x, v)| + |∂vv X (τ ; t, x, v)| )d x 2
|x−x0 |≤C N
2
(4.7)
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Y. Guo, J. Jang
√ω M ε 2k−1 i Proof. Since φ ε = (ω + i=1 ε Fi )dv + εk w h dv − ρ¯ and by assumption (4.2), we obtain that for any 0 < α < 1, for 0 ≤ T ≤ 1ε , ∇x φ εR C 1,α ≤ Ch ε W 1,∞ and ∇x φ ε C 1,α ≤ C + CεI1 + Cεk h ε W 1,∞ ≤ C. (4.8) Noting that these characteristics are uniquely determined under the Lipschitz continuity condition of ∇x φ ε , we have for ∂ = ∂x or ∂v , d 2 ∂ X (τ ; t, x, v) = ∇x x φ ε (τ, X (τ ; t, x, v))∂ X. dτ 2
(4.9)
By integrating in time, from the H¨older continuity of ∇x φ ε as in (4.8), one can deduce that for each 1 ≤ i, j ≤ 3,
∂X j ∂X j ∂V j ∂V j ∞ + ∞ + ∞ + ∞ ≤ C(1 + εI1 ) + Cεk h ε W 1,∞ ∂vi ∂ xi ∂vi ∂ xi
(4.10)
so that (4.3) follows. In order to see the Jacobian of the change of variables : v → X (τ ), consider Taylor expansion of ∂ X∂v(τ ) in τ around t: ∂ X (t) d ∂ X (τ ) (τ − t)2 d 2 ∂ X (τ¯ ) ∂ X (τ ) = + (τ − t) |τ =t + ∂v ∂v dτ ∂v 2 dτ 2 ∂v 2 2 (τ − t) d ∂ X (τ¯ ) = (τ − t)I + 2 dτ 2 ∂v for τ ≤ τ¯ ≤ t. The Jacobian matrix ∂ X∂v(τ ) is given by
∂ X (τ ) ∂v ⎛ ⎜ ⎜ =⎜ ⎝
(4.11)
⎛
⎞ ∂v1 X 1 (τ ) ∂v2 X 1 (τ ) ∂v3 X 1 (τ ) 2 2 2 = ⎝ ∂v1 X (τ ) ∂v2 X (τ ) ∂v3 X (τ ) ⎠ ∂v1 X 3 (τ ) ∂v2 X 3 (τ ) ∂v3 X 3 (τ )
τ −t +
(τ −t)2 d 2 ∂ X 1 (τ 1 ) 2 dτ 2 v1
(τ −t)2 d 2 ∂ X 2 (τ 4 ) 2 dτ 2 v1 (τ −t)2 d 2 ∂ X 3 (τ 7 ) 2 dτ 2 v1
(τ −t)2 d 2 ∂ X 1 (τ 2 ) 2 dτ 2 v2
τ −t +
(τ −t)2 d 2 ∂ X 2 (τ 5 ) 2 dτ 2 v2
(τ −t)2 d 2 ∂ X 3 (τ 8 ) 2 dτ 2 v2
(τ −t)2 d 2 ∂ X 1 (τ 3 ) 2 dτ 2 v3 (τ −t)2 d 2 ∂ X 2 (τ 6 ) 2 dτ 2 v3
τ −t +
⎞ ⎟ ⎟ ⎟. ⎠
(τ −t)2 d 2 ∂ X 3 (τ 9 ) 2 dτ 2 v3
We claim that if T0 is sufficiently small, the determinant of the Jacobian is bounded from below and above by |t − τ |3 . Note that from (4.9), 2 d ∂ X (τ¯ ) ∂ = ∇x φ ε (τ , X (τ ; t, x, v)) ≤ |∇x ∇x φ ε || ∂ X (τ¯ ) |. (4.12) dτ 2 ∂v ∂v ∂v Now ∇xε φC 1,α ≤ C(1 + εk h ε W 1,∞ ) + CεI1 ≤ C for t ≤ 1ε , and thus by (4.10), we can choose T0 sufficiently small so that (τ − t) d 2 ∂ X (τ¯ ) C T0 1 2 dτ 2 ∂v ≤ 2 ≤ 8 ,
Global Hilbert Expansion for the Vlasov-Poisson-Boltzmann System
485
and in turn ∂ X (τ ) 3|t − τ |3 |t − τ |3 ≤ det . ≤ 2 ∂v 2 We then deduce both (4.4) and (4.5). On the other hand, consider the Taylor expansion of
∂ X (τ ) ∂x
in τ around t:
∂ X (τ ) (τ − t)2 d 2 ∂ X (τ¯ ) ∂ X (t) d ∂ X (τ ) = + (τ − t) |τ =t + ∂x ∂x dτ ∂ x 2 dτ 2 ∂ x 2 2 (τ − t) d ∂ X (τ¯ ) =I+ 2 dτ 2 ∂ x for τ ≤ τ¯ ≤ t. Note that from (4.9), we have 2 d ∂ X (τ¯ ) ∂ = ∇x φ ε (τ , X (τ ; t, x, v)) ≤ |∇x ∇x φ ε || ∂ X (τ¯ ) |, dτ 2 ∂ x ∂ x ∂x and thus (4.6) is valid for
∂ X (τ ) ∂x
and for T0 small. We also have for τ ≤ τ¯ ≤ t,
∂ V (τ ) ∂ V (t) d ∂ V (τ¯ ) = + (τ − t) ∂v ∂v dτ ∂v ∂ ∂ X (τ¯ ) ∇x φ ε (τ¯ ) (τ − t). =I+ ∂x ∂v By (4.5), (4.6) is true for But from (4.9),
∂ V (τ ) ∂v
for T0 sufficiently small.
d2 ∂∂v X (τ ) = ∂{∇x x φ ε (τ, X (τ ; t, x, v))∂v X } dτ 2 = ∇x3 φ ε (τ, X (τ ; t, x, v)){∂v X }{∂ X } + ∇x2 φ ε (τ, X (τ ; t, x, v)){∂∂v X (τ )}. We thus conclude that by integrating twice in time: ∂∂v X (τ ) L 2 (|x−x0 |≤N ) ≤
T02 ∇x,v X 2∞ 2 +
sup ∇x3 φ ε (τ, X (τ ; t, x, v)) L 2 (|x−x0 |≤N )
0≤τ ≤T0
T02 ∇x2 φ ε ∞ 2
sup ∂∂v X (τ ) L 2 (|x−x0 |≤N ) .
0≤τ ≤T0
We note that for |v| ≤ N , from the characteristic equation, |X (τ ; t, x, v) − x0 | ≤ |X (τ ; t, x, v) − x| + |x − x0 | ≤ |v|(t − τ ) t s1 |∇x φ ε (s)|dsds1 + N + τ
τ
≤ T0 N + C T02 + N ≤ C N ,
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Y. Guo, J. Jang
for T0 ≤ 1 sufficiently small and N ≥ 1. From boundedness of ∇x2 φ ε ∞ and (4.6), we make a change of variables x → X (τ ; t, x, v) in ∇x3 φ ε and absorb the last term to the left hand side to get sup 0≤τ ≤T0 ,x0 ∈R3 ,|v|≤N
≤C ≤C ≤C
∂∂v X (τ ) L 2 (|x−x0 |≤N )
sup 0≤τ ≤T0 ,x0 ∈R3 ,|v|≤N
sup 0≤τ ≤T0 ,x0 ∈R3 ,|v|≤N
sup 0≤τ ≤T0 ,x0
∈R3 ,|v|≤N
∇x3 φ ε (τ, X (τ ; t, x, v)) L 2 (|X (τ ;t,x,v)−x0 |≤C N )
∇x3 φ ε (τ ) L 2 (|X (τ )−x0 |≤C N ) det
d X (τ ; t, x, v) −1/2 dx
∇x3 φ ε (τ ) L 2 (|X (τ )−x0 |≤C N ) ,
for T0 small. To control ∇x3 φ ε (τ ) L 2 (|X (τ )−x0 |≤C N ) , we make use of the Poisson equa √ω M ε 2k−1 i tion φ ε = (ω + i=1 ε Fi )dv + εk ¯ Note w h dv − ρ. ∂x φ ε =
(∂x ω +
2k−1
εi ∂x Fi )dv + εk
i=1
√ ωM ∂x h ε dv. w
Letting χ be a smooth cutoff function of |x − x0 | ≤ C N + 1, we have ε
∂x {χ φ } = χ (∂x ω +
2k−1 i=1
√ ωM ∂x h ε dv+ ε ∂x Fi )dv+ε χ w i
k
∂ α χ ∂ β φε .
|α+β|=3, |β|≤2
It thus follows that, from the assumption (4.2), and the fact ∂x ω, ∂x Fi ∈ L 2 , we conclude ∇x3 φ ε (τ, x) L 2 (|x−x0 |≤C N ) ≤ C + C N 3/2 εk h ε W 1,∞ + Cφ ε C 2 N 3/2 ≤ C N 3/2 . We then complete the proof of (4.7). 5. W 1,∞ Estimates for Remainder FRε In this section we establish the W 1,∞ estimate for h ε with suitable factors of ε. To be more precise, we will show that for sufficiently small ε, ε3/2 h ε ∞ and ε5 ∇x,v h ε ∞ are bounded by f ε and initial data. Recall I1 and I2 in (3.1). We now turn to the main estimates of h ε . As the first preparation, we define √ √ 1 LMg = −√ {Q(ω, ω M g) + Q( ω M g, ω)} = {ν(ω) + K M }g ωM
(5.1)
Global Hilbert Expansion for the Vlasov-Poisson-Boltzmann System
as in [2]. Letting K M,w g ≡ wK M
g w
487
, from (1.6) and (1.11), we obtain
ν(ω) ε 1 h + K M,w h ε ε ε √ √ √ 2k−1 ε ε k−1 hε ωM w ε w h ωM h ωM , )+ ) = √ Q( εi−1 √ {Q(Fi , ωM w w ωM w i=1 √ √ hε ωM ωM ε w ε + Q( ∇v ( , Fi )} − ∇x φ · √ )h w ωM w
∂t h ε + v · ∇x h ε + ∇x φ ε · ∇v h ε +
w w − ∇x φ εR · √ ∇v (ω + εi Fi ) + εk−1 √ A, (5.2) ωM ωM i=1 2k−1 n ε k 1,∞ estimates where ∇φ ε = n=0 ε ∇φn + ε ∇φ R . Our main task is to derive W ε of h : 2k−1
Proposition 5.1. Let 0 < T ≤ 1ε be given and the electric fields ∇φ εR and ∇φ ε satisfy the estimates (4.8). For all ε sufficiently small, there exists a constant C > 0 independent of T and ε such that sup {ε3/2 h ε (s)∞ } ≤ C{ε3/2 h 0 ∞ + sup f ε (s) + ε(2k+1)/2 },
0≤s≤T
(5.3)
0≤s≤T
as well as sup {ε3/2 (1 + |v|)h ε (s)∞ + ε5 ∇x,v h ε (s)∞ }
0≤s≤T
≤ C{ε3/2 (1 + |v|)h 0 ∞ + ε5 ∇x,v h 0 ∞ + sup f ε (s) + ε(2k+1)/2 }. (5.4) 0≤s≤T
The proof of the proposition relies on the following two lemmas: Lemma 5.2. Assume (4.2) is valid. There exists a T0 > 0 such that 0 ≤ T0 ≤ T ≤ and for all ε sufficiently small, sup {ε3/2 ||h ε (s)||∞ } ≤ C{||ε3/2 h 0 ||∞ + sup || f ε (s)|| + ε(2k+1)/2 },
0≤s≤T0
1 ε
(5.5)
0≤s≤T
and moreover, ε3/2 h ε (T0 )∞ ≤
1 3/2 ε h 0 ∞ + C{ sup f ε (s) + ε(2k+1)/2 }. 2 0≤s≤T
(5.6)
Lemma 5.3. For T0 > 0 obtained in Lemma 5.2, there exists a sufficiently small ε0 > 0 such that for all ε ≤ ε0 , sup {ε5 Dx h ε (s)∞ + ε5 Dv h ε (s)∞ } ≤ C{ε5 (1 + |v|)h 0 ∞ + ε5 Dx h 0 ∞
0≤s≤T0
+ ε5 Dv h 0 ∞ + ε3/2 h 0 ∞ + ε1/2 sup f ε (s) + εk+1 },
(5.7)
0≤s≤T
and moreover, 1 5 {ε (1 + |v|)h 0 ∞ + ε5 Dx h 0 ∞ 2 + ε5 Dv h 0 ∞ + ε3/2 h 0 ∞ } + C{ε1/2 sup f ε (s) + εk+1 }. (5.8)
ε5 Dx h ε (T0 )∞ + ε5 Dv h ε (T0 )∞ ≤
0≤s≤T
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Y. Guo, J. Jang
Once we establish Lemma 5.2 and 5.3, by bootstrapping the time interval into the given time T , we can readily conclude Proposition 5.1. We also remark that in light of the estimate in Proposition 5.1, the assumption (4.2) will be automatically satisfied by a continuity argument. Proof of Proposition 5.1. If t ≤ T0 , the conclusion directly follows from Lemma 5.2 and Lemma 5.3. Assume that T0 ≤ t ≤ T . Then there exists a positive integer n so that t = nT0 + τ, where 0 ≤ τ ≤ T0 . Apply (5.6) in Lemma 5.2 repeatedly to get for each n, 1 3/2 ε h({n − 1}T0 + τ )∞ + C{ sup f ε (s) + ε(2k+1)/2 } 2 0≤s≤T 1 C ≤ ε3/2 h({n − 2}T0 + τ )∞ + { + C}{ sup f ε (s) + ε(2k+1)/2 } 4 2 0≤s≤T ≤ ··· 1 ≤ n ε3/2 h(τ )∞ + 2C{ sup f ε (s) + ε(2k+1)/2 }, 2 0≤s≤T
ε3/2 h(t)∞ ≤
since 1+ 21 + 41 + · · · + 21n ≤ 2 for each n. From (5.5), the estimate (5.3) follows. Similarly, one can deduce (5.4) from the above two lemmas. In the following two subsections, we prove the above two lemmas. d h(s, X (s; t, x, v), V (s; t, x, v)) = ∂t h + 5.1. L ∞ bound : Proof of Lemma 5.2. Since ds dX dV ∇x h · ds + ∇v h · ds , the solution to the following transport equation:
∂t h ε + v · ∇x h ε + ∇x φ ε · ∇v h ε +
ν(ω) ε h =0 ε
t can be written as h ε (t, x, v) = exp{− 1ε 0 ν(τ )dτ }h ε (0, X (0; t, x, v), V (0; t, x, v)). Thus for any (t, x, v), integrating along the backward trajectory (4.1), by Duhamel’s Principle, the solution h ε (t, x, v) of the original nonlinear equation (5.2) can be written as follows: 1 t h ε (t, x, v) = exp{− ν(τ )dτ }h ε (0, X (0; t, x, v), V (0; t, x, v)) ε 0 t 1 t 1 K M,w h ε (s, X (s), V (s))ds − exp{− ν(τ )dτ } ε s ε 0 k−1 √ √ t t 1 ε w hε ωM hε ωM + , ) (s, X (s), V (s))ds exp{− ν(τ )dτ } √ Q( ε s ωM w w 0 2k−1 √ t hε ωM 1 t i−1 w + ) (s, X (s), V (s))ds exp{− ν(τ )dτ } ε √ Q(Fi , ε s ωM w 0 i=1 2k−1 √ t hε ωM 1 t i−1 w + , Fi ) (s, X (s), V (s))ds exp{− ν(τ )dτ } ε √ Q( ε s ωM w 0 i=1 √ t ωM ε 1 t w ε − )h (s, X (s), V (s))ds exp{− ν(τ )dτ } ∇x φ · √ ∇v ( ε s ωM w 0
Global Hilbert Expansion for the Vlasov-Poisson-Boltzmann System
489
2k−1 w i − ν(τ )dτ } ·√ ∇v (ω + ε Fi ) (s, X (s), V (s))ds ωM 0 s i=1 t w 1 t + exp{− ν(τ )dτ } εk−1 √ A (s, X (s), V (s))ds. (5.9) ε s ωM 0
t
1 exp{− ε
t
∇x φ εR
We will prove only (5.6). The estimate of (5.5) can be obtained in the same way by ν0 s ε rather than e 2ε directly estimating h h ε ∞ in (5.22) as done in [8,9]. ∞ √ √ hε ωM hε ωM w Since | √ω Q( w , w )| ≤ Cν(ω)h ε 2∞ from Lemma 10 in [6], and since M
ν(ω)
t 0
|v − u|ωdv (1 + |v|)ρ0 (t, x) ν M (v), ν(ω) ≥ 2ν0 > 0,
1 exp{− ε
≤ Cεe
−
t
ν(ω)(τ )dτ }ν(ω)e
−
ν0 s ε
ds ≤ Ce
−
ν0 t ε
s
ν0 t ε
t
exp{− 0
ν M (t − s) }ν M ds ε
,
the third line in (5.9) is bounded by t ν0 t ν0 s 1 t ν(ω)(τ )dτ }ν(ω)h ε (s)2∞ ds ≤ Cεk e− ε sup {e 2ε h ε (s)∞ }2 . Cεk−1 exp{− ε s 0≤s≤t 0 From Lemma 10 in [6] again, 2k−1
ε
i=1
i−1
√ √ hε ωM hε ωM w ) + Q( , Fi )} {Q(Fi , √ ωM w w
2k−1 w i−1 ≤ ν M h ε ∞ √ ε Fi ∞ , ωM i=1
so that the fourth and fifth lines in (5.9) are bounded by t ν0 t ν0 s ν M (t − s) }ν M h ε (s)∞ ds ≤ CεI1 e− 2ε sup {e 2ε h ε (s)∞ }. exp{− ε 0≤s≤t 0 Since | √w ∇v ( ω M
√ ωM w )|
≤ C(1 + |v|) and | √w ∇v (ω + ω M
2k−1 i=1
εi Fi )| ≤ C + εCI1 , ν0 t
the sixth and seventh lines in (5.9), from (4.8), are bounded by (C + εCI1 )εe− 2ε ν0 s sup0≤s≤t {e 2ε h ε (s)∞ }. The last line in (5.9) is clearly bounded by CI2 εk . We shall mainly concentrate on the second term on the right-hand side of ( 5.9). Let l M (v, v ) be the corresponding kernel associated with K M in [2]; we have |l M (v, v )| ≤ C{|v − v | +
1 ||v|2 − |v |2 |2 } exp{−c|v − v |2 − c }. |v − v | |v − v |2
Since ν(ω) ν M , we bound the second term by 1 t 1 t exp{− ν(τ )dτ } |l M,w (V (s), v )h ε (s, X (s), v )|dv ds, ε 0 ε s R3
(5.10)
(5.11)
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Y. Guo, J. Jang
where l M,w (v, ˜ v) = (5.11). Denoting
w(v) ˜ ˜ v ). w(v ) l M (v,
We now use (5.9) again to evaluate K M,w h ε in
[X (s1 ), V (s1 )] ≡ [X (s1 ; s, X (s; t, x, v), v ), V (s1 ; s, X (s; t, x, v), v )], we can further bound (5.11) by 1 ε
t
exp{− 0
+
1 ε2
t
1 ε
t
ν(τ )dτ −
s
s
exp{− 0
0
1 ε
t s
1 ε
s
ν(τ )dτ }
0
ν(τ )dτ −
1 ε
s
R3
|l M,w (V (s), v ) h ε (0, X (0), V (0))|dv ds
ν(τ )dτ }
s1
R3 ×R 3
|l M,w (V (s), v ) l M,w (V (s1 ), v )
· h ε (s1 , X (s1 ), v )|dv dv ds1 ds 1 t s 1 t 1 s + exp{− ν(τ )dτ − ν(τ )dτ } |l M,w (V (s), v ) ε 0 0 ε s ε s1 R3 k−1 √ √ hε ωM hε ωM ε w · √ Q( , ) (s1 , X (s1 ), V (s1 ))dv ds1 ds ωM w w 1 t s 1 t 1 s + exp{− ν(τ )dτ − ν(τ )dτ } |l M,w (V (s), v ) ε 0 0 ε s ε s1 R3 2k−1 √ √ hε ωM hε ωM i−1 w ε √ {Q(Fi , · ) + Q( , Fi )} (s1 , X (s1 ), V (s1 ))dv ds1 ds ωM w w i=1 1 t 1 s 1 t s exp{− ν(τ )dτ − ν(τ )dτ } |l M,w (V (s), v ) + ε 0 0 ε s ε s1 R3 √ 2k−1 ωM ε w w ε ε i · ∇x φ · √ ∇v ( ∇v (ω + ε Fi ) (s1 , X (s1 ), V (s1 ))dv ds1 ds )h + ∇x φ R · √ ωM w ωM i=1 1 t s 1 t 1 s + exp{− ν(τ )dτ − ν(τ )dτ } |l M,w (V (s), v ) ε 0 0 ε s ε s1 R3 w · εk−1 √ A (s1 , X (s1 ), V (s1 ))dv ds1 ds. (5.12) ωM
˜ v )|dv < +∞ from Lemma 7 in [6], and by the previous estiSince supv˜ R3 |l M,w (v, mates, there is an upper bound except for the second term as ν0 t ν0 s t − ν0 t ε e ε h (0)∞ + εk e− ε sup {e 2ε h ε (s)∞ }2 ε 0≤s≤t ν0 t
ν0 s
+ (1 + I1 )εe− 2ε sup {e 2ε h ε (s)∞ } + I2 εk 0≤s≤t
up to a constant multiple. We now concentrate on the second term in (5.12) and we follow the same spirit of the proof of Theorem 20 in [6]. From (4.8) and (4.1), fix N > 0 large enough so that for T0 small N ≥ sup |V (s) − v|. 2 0≤t≤T0 , 0≤s≤T0 Note that by Lemma 7 in [6] (the Grad estimate), |l M,w (V (s), v ) l M,w (V (s1 ), v )|dv dv ≤
C . 1 + |V (s)|
(5.13)
Global Hilbert Expansion for the Vlasov-Poisson-Boltzmann System
491
We divide into four cases according to the size of v, v , v and for each case, an upper bound of the second term in (5.12) will be obtained. Case 1. |v| ≥ N . In this case, since |V (s)| ≥ N2 , (5.13) implies that C |l M,w (V (s), v )l M,w (V (s1 ), v )|dv dv ≤ , N and thus we have the following bound t s C ν M (t − s) ν M (s − s1 ) exp{− } exp{− }h ε (s1 )∞ ds1 ds ε2 N 0 0 ε ε ν0 s C ν0 t ≤ e− 2ε sup {e 2ε h ε (s)∞ }. N 0≤s≤t Case 2. |v| ≤ N , |v | ≥ 2N , or |v | ≤ 2N , |v | ≥ 3N . Observe that |V (s) − v | ≥ |v − v| − |V (s) − v| ≥ |v | − |v| − |V (s) − v|, |V (s1 ) − v | ≥ |v − v | − |V (s1 ) − v | ≥ |v | − |v | − |V (s1 ) − v |, thus we have either |V (s)−v | ≥ N2 or |V (s1 )−v | ≥ are valid correspondingly for η > 0: η
N 2 , and either one of the following η
2
|l M,w (V (s), v )| ≤ e− 8 N |l M,w (V (s), v )e 8 |V (s)−v | |, 2
(5.14) η 2 η 2 |l M,w (V (s1 ), v )| ≤ e− 8 N |l M,w (V (s1 ), v )e 8 |V (s1 )−v | |. η 2 From Lemma 7 in [6], both |l M,w (V (s), v )e 8 |V (s)−v | |dv and |l M,w (V (s1 ), v ) η 2 e 8 |V (s1 )−v | |dv are still finite for sufficiently small η > 0. We use (5.14) to combine the cases of |V (s) − v | ≥ N2 or |V (s1 ) − v | ≥ N2 to get the following bound t s + 0
≤
|v|≤N ,|v |≥2N , |v |≤2N ,|v |≥3N t s Cη − η N 2 ν M (s − s1 ) ν M (t − s) } exp{− }h ε (s1 )∞ ds1 ds e 8 exp{− 2 ε ε ε 0 0 0
η
ν0 t
ν0 s
≤ Cη e− 8 N e− 2ε sup {e 2ε h ε (s)∞ }. 2
(5.15)
0≤s≤t
Case 3a. |v| ≤ N , |v | ≤ 2N , |v | ≤ 3N . This is the last remaining case because if |v | > 2N , it is included in Case 2; while if |v | > 3N , either |v | ≤ 2N or |v | ≥ 2N are also included in Case 2. We further assume that s − s1 ≤ εκ, for κ > 0 small. We bound the second term in (5.12) by ν M (s − s1 ) ν M (t − s) CN t s } exp{− }h ε (s1 )∞ ds1 ds exp{− 2 ε ε ε 0 s−κε t s ν0 t ν0 t 1 1 ν M (t − s) exp{− ≤ C N e− 2ε sup {e 2ε h ε (s)∞ } }ds ds1 ε 0 2ε 0≤s≤t s−εκ ε ν0 t
ν0 t
≤ κC N e− 2ε sup {e 2ε h ε (s)∞ }. 0≤s≤t
(5.16)
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Case 3b. |v| ≤ N , |v | ≤ 2N , |v | ≤ 3N , and s − s1 ≥ εκ. We now can bound the second term in (5.12) by s−κε ν (t−s) ν (s−s ) C t − Mε − M ε 1 e e |l M,w (V (s), v ) l M,w (V (s1 ), v ) h ε (s1 , X (s1 ), v )|, ε2 0 B 0 where B = {|v | ≤ 2N , |v | ≤ 3N }, By (5.10), l M,w (v, v ) has possible integrable 1 singularity of |v−v | , we can choose l N (v, v ) smooth with compact support such that 1 (5.17) sup |l N ( p, v ) − l M,w ( p, v )|dv ≤ . N | p|≤3N |v |≤3N Splitting l M,w (V (s), v ) l M,w (V (s1 ), v ) = {l M,w (V (s), v ) − l N (V (s), v )}l M,w (V (s1 ), v ) + {l M,w (V (s1 ), v ) − l N (V (s1 ), v )}l N (V (s), v ) + l N (V (s), v ) l N (V (s1 ), v ), we can use such an approximation (5.17) to bound the above s1 , s integration by C ε sup h (s)∞ · sup |l M,w (V (s1 ), v )|dv + sup |l N (V (s), v )|dv N 0≤s≤t |v|≤N |v |≤2N ν M (t−s) ν M (s−s1 ) C t s−κε e− ε e− ε + 2 ε 0 0 B × |l N (V (s), v ) l N (V (s1 ), v ) h ε (s1 , X (s1 ), v )|dv dv ds1 ds. (5.18) Introduce a new variable y = X (s1 ) = X (s1 ; s, X (s; t, x, v), v )
(5.19)
|y − X (s)| = |X (s1 ) − X (s)| ≤ C(s − s1 ).
(5.20)
such that
We now apply Lemma 4.1 to X (s1 ; s, X (s; t, x, v), v ) with x = X (s; t, x, v), τ = s1 , t = s. By (4.4), we can choose small but fixed T0 > 0 such that for s − s1 ≥ κε, |
dy κ 3 ε3 . | ≥ dv 2
(5.21)
Since l N (V (s), v ) l N (V (s1 ), v ) is bounded, we first integrate over v to get CN |h ε (s1 , X (s1 ), v )|dv |v |≤2N
≤ CN ≤
ε
|v |≤2N
CN κ 3/2 ε3/2
CN ≤ 3/2 3/2 κ ε
2
1 (X (s1 ))|h (s1 , X (s1 ), v )| dv
|y−X (s)|≤C(s−s1 )N
R3
1/2
|h ε (s1 , y, v )|2 dy
|h ε (s1 , y, v )|2 dy
1/2 .
1/2
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By (1.11) and (1.10), we then further control the last term in (5.18) by: C N ,κ ε 7/2 ≤ ≤
T0
s−κε
0
0 3/2 C N ,κ T0 ε 7/2
0
e− T0
ν(v)(T0 −s) ε
s−κε
e−
e−
ν(v )(s−s1 ) ε
|v |≤3N
ν(v)(T0 −s) ε
e−
ν(v )(s−s1 ) ε
R3
|h ε (s, y, v )|2 dy
|v |≤3N
0
R3
1/2
dv ds1 ds
| f ε (s, y, v )|2 dydv
1/2 ds1 ds
C N ,κ,T0 sup f ε (s). ε 3/2 0≤s≤T
In summary, we have established, for any κ > 0 and large enough N > 0, ν0 s
sup {e 2ε h ε (s)∞ }
0≤s≤T0
ν0 s ν0 s s Cκ ≤ C sup {(1+ )e− 2ε }h 0 ∞ +{C(1+I1 (T0 ))ε + Cκ + } sup {e 2ε h ε (s)∞ } ε N 0≤s≤T0 0≤s≤T0 ν0 s ν T ν0 T0 C 0 0 N ,κ + Cεk sup {e 2ε h ε (s)∞ }2 + 3/2 e 2ε sup f ε (s) + CI2 (T0 )e 2ε εk . ε 0≤s≤T0 0≤s≤T
(5.22) ν0 s − 2ε
Note that (1+ εs )e is unformly bounded in s and ε and I1 (T0 ) and I2 (T0 ) are unformly bounded in ε. For sufficiently small ε > 0, first choosing κ small, then N sufficiently large so that {C(1 + I1 (T0 ))ε + Cκ + CNκ } < 21 , we obtain, in light of assumption (4.2), ν0 s
sup {e 2ε h ε (s)∞ } ≤ Ch 0 ∞ +
0≤s≤T0
ν0 T0 C N ,κ ν0 T0 ε 2ε 2ε ε k . e sup f (s) + CI (T )e 2 0 ε3/2 0≤s≤T
Letting s = T0 in the above and multiplying by ε3/2 e− small ε, ε3/2 h ε (T0 )∞ ≤
ν0 T0 2ε
, we obtain for sufficiently
1 3/2 ε h 0 ∞ + C sup f ε (s) + Cε(2k+3)/2 . 2 0≤s≤T
5.2. W 1,∞ bound : Proof of Lemma 5.3. We will prove only (5.8). The estimate (5.7) can be done in the same way. Let Dx be any x derivative. We now take Dx of Eq. (5.2) to get ∂t (Dx h ε ) + v · ∇x (Dx h ε ) + ∇x φ ε · ∇v (Dx h ε ) +
ν(ω) Dx h ε ε
Dx ν(ω) ε 1 h − Dx (K M,w h ε ) ε ε √ √ √ 2k−1 ε ε k−1 h ωM h ωM hε ωM ε w i−1 w + √ Dx (Q( ε √ {Dx (Q(Fi , , )) + ) ωM w w ωM w i=1 √ √ hε ωM ωM ε w ε , Fi ))} − Dx (∇x φ · √ )h ) ∇v ( + Q( w ωM w
= −∇x (Dx φ ε ) · ∇v h ε −
2k−1 w w − Dx (∇x φ εR · √ ∇v (ω + εi Fi )) + εk−1 √ (Dx A). ωM ωM i=1
(5.23)
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Thus the solution Dx h ε of Eq. (5.23) can be expressed as follows: 1 t Dx h ε (t, x, v) = exp{− ν(τ )dτ }Dx h ε (0, X (0; t, x, v), V (0; t, x, v)) ε 0 t 1 t − exp{− ν(τ )dτ } ∇x (Dx φ ε ) · ∇v h ε (s, X (s), V (s))ds ε s 0 t 1 t Dx ν(ω) ε − exp{− ν(τ )dτ } h (s, X (s), V (s))ds ε s ε 0 t t 1 1 exp{− ν(τ )dτ } − Dx (K M,w h ε ) (s, X (s), V (s))ds ε s ε 0 k−1 √ √ t hε ωM hε ωM 1 t ε w + exp{− ν(τ )dτ } √ Dx (Q( , )) (s, X (s), V (s))ds ε s ωM w w 0 2k−1 √ t t hε ωM 1 i−1 w + exp{− ν(τ )dτ } ε √ Dx (Q(Fi , )) (s, X (s), V (s))ds ε s ωM w 0 i=1 2k−1 √ t hε ωM 1 t w + exp{− ν(τ )dτ } εi−1 √ Dx (Q( , Fi )) (s, X (s), V (s))ds ε s ωM w 0 i=1 √ t t ωM ε 1 w − exp{− ν(τ )dτ } Dx (∇x φ ε · √ ∇v ( )h ) (s, X (s), V (s))ds ε s ωM w 0 t t 2k−1 1 w ε i − exp{− ν(τ )dτ } Dx (∇x φ R · √ ∇v (ω + ε Fi )) (s, X (s), V (s))ds ε s ωM 0 i=1 t 1 t w + exp{− ν(τ )dτ } ε k−1 √ (Dx A) (s, X (s), V (s))ds. (5.24) ε s ωM 0
Note that since ω is a local Maxwellian depending on t, x, and v, the right hand side contains not only Dh ε terms but also h ε terms coming from commutators. In addition, there is a ∇v h ε term coming from forcing, which we will estimate afterwards. The terms involving Dx h ε can be estimated similarly as done in the h ε ∞ estimate. The terms from commutators are lower order, but they carry extra weight 1 + |v|2 ; they will be either controlled by L ∞ norm of (1 + |v|)h ε or absorbed by the stronger exponential decay factor ω. We will estimate line by line as in the previous section. It is easy to see that the second line in (5.24) is bounded by ν0 t
ν0 s
εe− 2ε {CI1 ε + C(1 + εk )h ε W 1,∞ } sup {e 2ε ∇v h ε ∞ }, 0≤s≤t
where we have used the elliptic regularity (4.8). Since |Dx ν(ω)| ≤ Cν(ω), the third line is bounded by ν0 t
ν0 s
Ce− 2ε sup {e 2ε h ε ∞ }. 0≤s≤t
√ hε ω
√ hε ωM
In order to estimate the fifth line, first write the term Dx (Q( w M , w √ √ √ √ Dx h ε ω M h ε ω M hε ωM hε ωM , ) + Q( , ) (Dx Q)( w w√ w w √ h ε ω M Dx h ε ω M + Q( , ), w w
)) as
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495
where Dx Q is a commutator which consists of the terms √ that are √given rise to when √ hε ωM hε ωM hε ωM the derivative hits other than w . Note that |(Dx Q)( w , w )(v)| ≤ C(1 + |v|2 )|Q(
√ √ hε ωM hε ωM w , w )(v)|.
By Lemma 10 in [6], √ h ωM hε ωM w |√ Dx (Q( , ))| ≤ Cν(ω){(1 + |v|)h ε 2∞ +h ε ∞ Dx h ε ∞ }, ωM w w ε√
and hence the fifth line in (5.24) is bounded by Cεk e−
ν0 t ε
ν0 s
ν0 s
ν0 s
sup {(e 2ε (1 + |v|)h ε ∞ )2 + (e 2ε h ε ∞ )(e 2ε Dx h ε ∞ )}.
0≤s≤t
Commutators in the sixth and seventh lines also have the extra weight (1 + |v|2 ), but this weight can be absorbed into the exponential decay of Fi ’s. Thus the sixth and seventh lines in (5.24) are bounded by ν0 t
ν0 s
CI1 εe− 2ε sup {e 2ε h ε W 1,∞ }. 0≤s≤t
Similarly, one can deduce that the eighth through tenth lines are bounded by ν0 t
ν0 s
ε(C + CI1 ε)e− 2ε sup {e 2ε h ε W 1,∞ } + Cεk . 0≤s≤t
We shall concentrate on the fourth line in (5.24). Write Dx (K M,w h ε ) as ε ε Dx (K M,w h )(v) = (Dx l M,w )(v, v )h (v )dv + l M,w (v, v )(Dx h ε )(v )dv , where l M,w is the corresponding kernel associated with K M,w . Note that |(Dx l M,w )(v, v )| ≤ C(1 + |v|)(1 + |v − v |) l M,w (v, v )(1 + |v |) ≤ Cν M (1 + |v − v |) l M,w (v, v )(1 + |v |) due to the dependence of l M,w on the local Maxwellian ω. Thus we can bound the fourth line in (5.24) by 1 t 1 t exp{− ν(τ )dτ }ν M |(1+|V (s)−v |)l M,w (V (s), v )(1+|v |)h ε (s, X (s), v )|dv ds ε 0 ε s R3 1 t 1 t exp{− ν(τ )dτ } l M,w (V (s), v )(Dx h ε )(s, X (s), v )dv ds ≡ (I ) + (I I ). + ε 0 ε s R3
Letting ! h ε ≡ (1+|v|)h ε , ! h ε satisfies Eq. (5.2) with a different weight w1 ≡ (1+|v|)w(v). We now use the Duhamel equation (5.9) for ! h ε with the weight w1 to evaluate (I ). Recall ∞ ε (5.12). Note that the L estimates of h do not depend on the strength of the weight, and also both l M,w and l M,w1 inherit Grad estimates (5.10). Thus we can follow the previous estimates to obtain the bound for (I ), ν0 t ν0 s t ν0 t ε (I ) ≤ C e− ε ! h (0)∞ + Cεk e− ε sup {e 2ε ! h ε (s)∞ }2 ε 0≤s≤t ν0 s Cκ − ν0 t C N ,κ + (C(1+I1 )ε+Cκ + )e 2ε sup {e 2ε ! h ε (s)∞ }+ 3/2 sup f ε (s)+CI2 εk . N ε 0≤s≤t 0≤s≤T0
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And we use Eq. (5.24) to evaluate (I I ). The estimation is again exactly the same as the L ∞ bound except the very last part where f ε comes up. Here we will only present this last case: recall the Case 3b. in the previous section and see (5.18), C t s−κε 1 t 1 s exp{− ν(τ )dτ − ν(τ )dτ }l N (V (s), v ) l N (V (s1 ), v ) ε2 0 0 ε s ε s1 B × Dx h ε (s1 , X (s1 ), v )dv dv ds1 ds.
As before in (5.19), we introduce a new variable y = X (s1 ) = X (s1 ; s, X (s; t, x, v), v ) such that |y − X (s)| = |X (s1 ) − X (s)| ≤ C(s − s1 ). We now apply Lemma 4.1 to X (s1 ; s, X (s; t, x, v), v ) with x = X (s; t, x, v), τ = s1 , t = s. Make a change of variables from v to y and integrate by parts: t s−κε 1 1 t 1 s exp{− ν(τ )dτ − ν(τ )dτ }l N (V (s), v ) l N (V (s1 ), v ) ε s ε s1 ε2 0 0 B dv × Dx h ε (s1 , y, v )| |dydv ds1 ds dy t s−κε 1 t 1 s 1 exp{− ν(τ )dτ − ν(τ )dτ }Dx (l N (V (s), v ) l N (V (s1 ), v )) ≤− 2 ε s ε s1 ε 0 0 Bˆ
dv |dydv ds1 ds dy t s−κε 1 t 1 s 1 exp{− ν(τ )dτ − ν(τ )dτ }l N (V (s), v ) l N (V (s1 ), v ) − 2 ε s ε s1 ε 0 0 Bˆ dv |)dydv ds1 ds × h ε (s1 , y, v )Dx (| dy C N ,κ (5.25) + 3 h ε ∞ (boundary contribution), ε × h ε (s1 , y, v )|
where Bˆ = {|y − X (s)| ≤ C(s − s1 )N , |v | ≤ 3N }. For the first term in the right hand side, since Dx (l N (V (s), v ) l N (V (s1 ), v )) is bounded, and by (5.21), following the same argument in the L ∞ bound, one can deduce that it is bounded by C N ,κ sup f ε (s). ε3 0≤s≤T For the second term, we need to estimate Dx ( dv dy ). First note that dv 1 dy 1 ) = − Dx (det Dx (det ), ) = Dx ( dy dy dy dv 2 det dv {det dv } where 1 1 4 ≤ ≤ by (4.4) in Lemma 4.1. dy 4(s1 − s)6 (s − s)6 1 {det dv }2 Since
dy dv
⎛
⎞ ∂v1 X 1 (s1 ) ∂v2 X 1 (s1 ) ∂v3 X 1 (s1 ) ⎜ ⎟ = ⎝ ∂v1 X 2 (s1 ) ∂v2 X 2 (s1 ) ∂v3 X 2 (s1 ) ⎠ , ∂v1 X 3 (s1 ) ∂v2 X 3 (s1 ) ∂v3 X 3 (s1 )
Global Hilbert Expansion for the Vlasov-Poisson-Boltzmann System
497
by the product rule and (4.5) in Lemma 4.1, we get |Dx (det
dy )| ≤ C|∂v X (s1 )|2 |∂x ∂v X (s1 )| ≤ C(s1 − s)2 |∂x ∂v X (s1 )|. dv
Thus, by s1 − s ≥ εκ in case 3b |Dx (det
C|∂x ∂v X (s1 )| 1 dv dy C|∂x ∂v X (s1 )| |Dx [det ]| ≤ )| = ≤ . 4 dy dv (s1 − s) κ 4 ε4 {det dy }2 dv
Therefore, we obtain dy 2 Cκ |) L 2 ( B) ≤ 4 ∂x ∂v X (s1 )||2L 2 ( B) ˆ ˆ dv ε Cκ |∂x ∂v X (s1 ; s, X (s, x, v), v )|2 d{X (s1 ; s, X (s, x, v), v )}dv dv ds1 ds ε4 Bˆ ∂ X (s1 ; s, X (s, x, v), v )} Cκ 2 |∂ ∂ X (s ; s, z, v )| det dzdv dv ds1 ds x v 1 ε4 Bˆ ∂ X (s, x, v) Cκ,T0 ,N |∂x ∂v X (s1 ; s, z, v )|2 dz ε4 |z−x|≤T0 N Cκ,T0 ,N , ε4
Dx (| = = ≤ ≤
where we have used (4.6) and (4.7). Hence, by Cauchy-Schwarz’s inequality, the second integral in (5.25) is bounded by C N ,κ,T0 sup f ε (s). ε4 0≤s≤T In summary, for any x derivative Dx , we have shown that for any κ > 0 and large enough N > 0, ν0 s
sup {e 2ε Dx h ε (s)∞ } ≤ C{(1 + |v|)h ε (0)∞ + Dx h(0)∞ }
0≤s≤T0
ν0 s
+ Cε sup {e 2ε ∇v h ε (s)∞ } + 0≤s≤T0
+ {Cε + Cκ +
ν0 s C sup {e 2ε h ε (s)∞ } ε3 0≤s≤T0
ν0 s ν0 s Cκ } sup {e 2ε Dx h ε (s)∞ + e 2ε (1 + |v|)h ε (s)∞ } N 0≤s≤T0 ν0 s
ν0 s
+ Cεk sup {(e 2ε Dx h ε (s)∞ )2 + (e 2ε (1 + |v|)h ε (s)∞ )2 } 0≤s≤T0
ν0 T0 C N ,κ ν0 T0 + 4 e 2ε sup f ε (s) + Ce 2ε εk−1 . ε 0≤s≤T
(5.26)
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As a final step, it now remains to estimate ∇v h ε ∞ that appears in (5.26) – the estimate of Dx h ε ∞ . Let Dv be any v derivative. Take Dv of Eq. (5.2) to get ∂t (Dv h ε ) + v · ∇x (Dv h ε ) + ∇x φ ε · ∇v (Dv h ε ) +
ν(ω) Dv h ε ε
Dv ν(ω) ε 1 h − Dv (K M,w h ε ) ε ε √ √ √ 2k−1 ε k−1 hε ωM ε w h ωM hε ωM w i−1 + Dv ( √ Q( ε Dv ( √ {Q(Fi , , ))+ ) ωM w w ωM w i=1 √ √ hε ωM ωM ε w , Fi )}) − ∇x φ ε · Dv ( √ )h ) ∇v ( + Q( w ωM w
= −Dx h ε −
2k−1 w w − ∇x φ εR · Dv ( √ ∇v (ω + εi Fi )) + εk−1 Dv ( √ A), ωM ωM
(5.27)
i=1
where Dx is a spatial derivative obtained from Dv (v) · ∇x . By Duhamel principle, the solution Dv h ε of Eq. (5.27) can be expressed as follows: 1 t Dv h ε (t, x, v) = exp{− ν(τ )dτ }Dv h ε (0, X (0; t, x, v), V (0; t, x, v)) ε 0 t 1 t exp{− ν(τ )dτ }(Dx h ε )(s, X (s), V (s))ds − ε s 0 t 1 t Dv ν(ω) ε h (s, X (s), V (s))ds exp{− ν(τ )dτ } − ε s ε 0 t t 1 1 − Dv (K M,w h ε ) (s, X (s), V (s))ds exp{− ν(τ )dτ } ε s ε 0 √ √ t t hε ωM hε ωM εk−1 w 1 , ) (s, X (s), V (s))ds + exp{− ν(τ )dτ }Dv √ Q( ε s ωM w w 0 ⎞ ⎛ t 2k−1 ε √ω h w 1 t M ⎠ ) (s, X (s), V (s))ds + exp{− ν(τ )dτ }Dv ⎝ εi−1 √ Q(Fi , ε s ωM w 0 i=1 ⎞ ⎛ t 2k−1 ε √ω h w 1 t M , Fi )⎠ (s, X (s), V (s))ds + exp{− ν(τ )dτ }Dv ⎝ εi−1 √ Q( ε s ωM w 0 i=1 √ t ωM ε 1 t w )h ) (s, X (s), V (s))ds exp{− ν(τ )dτ } ∇x φ ε · Dv ( √ ∇v ( − ε s ωM w 0 ⎛ ⎞ t 2k−1 1 t w exp{− ν(τ )dτ } ⎝∇x φ εR · Dv ( √ ∇v (ω + εi Fi ))⎠ (s, X (s), V (s))ds − ε s ωM 0 i=1 t 1 t w exp{− ν(τ )dτ } εk−1 Dv ( √ A) (s, X (s), V (s))ds. (5.28) + ε s ωM 0
As in the spatial derivative (Dx h ε ) case, the right-hand side contains not only Dv h ε terms but also h ε terms coming from commutators. But this time the terms from commutators carry the weight 1 + |v| at most since they are from v derivatives. The estimates will be almost the same as in the spatial derivative case, so we won’t present every detail. We
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499
rather give some brief explanations. For instance, since |Dv ν(ω)| ≤ C, the third term ν0 t ν0 s in the right hand side of (5.28) is bounded by Ce− 2ε sup0≤s≤t {e 2ε h ε ∞ }, and since √ hε ω
Q( w M , |Dv ( √w ωM line is bounded by Cεk e−
ν0 t ε
√ hε ωM w ))|
≤ Cν(ω)h ε ∞ {(1 + |v|)h ε ∞ + Dv h ε ∞ }, the fifth
ν0 s
ν0 s
ν0 s
sup {(e 2ε h ε ∞ )(e 2ε (1 + |v|)h ε ∞ + e 2ε Dv h ε ∞ )}.
0≤s≤t
Other terms except the fourth line can be estimated in the same way as before. For the intriguing term in the fourth line, we need to control 1ε Dv (K M,w h ε ) (s, X (s), V (s)). First note for T0 sufficiently small, by (4.6), ∂ V∂v(s) is non-sigular. We therefore can write Dv (K M,w h ε ) (s, X (s), V (s)) # " ∂ V (s) −1 = Dv(s) (K M,w h ε )(s, X (s), V (s)). ∂v But for Dv(s) (K M,w h ε )(s, X (s), V (s)), we can employ Lemma 2.2 in [5] so that Dv(s) (K M,w h ε )(s, X (s), V (s)) 1 2 h ε )(s, X (s), V (s)) + (K M,w ∂v h ε )(s, X (s), V (s)) = (K M,w 1 2 in which the kernels in both K M,w and K M,w satisfy the Grad estimate (5.10). We then 1 2 can repeat the same procedure to K M,w and K M,w . We use integration by parts in v so
that we do not need to take derivatives for the determinant of ( dv dy ) which is independent 2 ε of v for (K M,w ∂v h )(s, X (s), V (s))). Therefore, we have established the following W 1,∞ estimates: ν0 s
sup {e 2ε ∇x,v h ε (s)∞ } ≤ C{(1 + |v|)h ε (0)∞ + ∇x,v h ε (0)∞ }
0≤s≤T0
+ {Cε + Cκ +
ν0 s ν0 s Cκ } sup {e 2ε ∇x,v h ε (s)∞ + e 2ε (1 + |v|)h ε (s)∞ } N 0≤s≤T0 ν0 s
ν0 s
+ Cεk sup {(e 2ε ∇x,v h ε (s)∞ )2 + (e 2ε (1 + |v|)h ε (s)∞ )2 } 0≤s≤T0
ν0 s ν0 T0 C C N ,κ,T0 ν0 T0 + 3 sup {e 2ε h ε (s)∞ } + e 2ε sup f ε (s) + Ce 2ε εk−1 . 4 ε 0≤s≤T0 ε 0≤s≤T
(5.29)
From Lemma 5.2, we have the estimates of sup0≤s≤T0 ε3/2 h ε (s)∞ . Due to the singular term, the first term in the fourth line in (5.29), we first multiply both sides by ε5 and combine this estimate with L ∞ bound of h˜ ε ≡ (1 + |v|)h ε in Lemma 5.2, and choose κ small, N large to deduce that for sufficiently small ε, ε5 ∇x,v h ε (T0 )∞ ≤
1 5 {ε (1 + |v|)h ε (0)∞ + ε5 ∇x,v h(0)∞ + ε3/2 h(0)∞ } 2 + C{ε1/2 sup f ε (s) + εk+1 }. 0≤s≤T
We thus conclude the proof of (5.8), and the proof of (5.7) can be carried out similarly ν0 s without using the factor e 2ε .
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6. Proof of Theorem 1.1 Proof of Thoerem 1.1. Combining Proposition 3.1 and Proposition 5.1, we deduce d c0 { f ε 2 + ∇φ εR 2 } + {I − P} f ε 2ν dt ε $ % √ 3 2k+1 ε ≤ C ε ε 2 h 0 ∞ + sup f + ε 2 [ f ε + εk−3 f ε 2 + εk−2 f ε ∇φ εR ] 0≤s≤T
1 + I1 ε){ f ε 2 + ∇φ εR 2 } + CI2 εk−1 f ε . + C( (1 + t) p
(6.1)
Gronwall inequality yields f ε (t) + ∇φ εR (t) + 1 ≤ ( f ε (0) + ∇φ εR (0) + 1) t √ 3 × exp{ C{ ε(ε 2 h 0 ∞ + sup f ε ) + (1 + s)− p + I1 ε + I2 εk−1 }ds} 0
≤ ( f
ε
0≤s≤T
(0) + ∇φ εR (0) + 1)
√ 3 × exp{C + Ct ε(ε 2 h 0 ∞ + sup f ε ) + CI1 tε + CI2 tεk−1 }, 0≤s≤T
where we have used 1 2k−3 2 2k−2
(
c > π. cπ 2 Then for any 1 ≤ k < 4 3 there exists a subinterval J1 = [a1 , b1 ] ⊂ [a, b], such that if F is the first return map to region (i), then for any s ∈ J1 , we have
3 kI kI 2 kI ≤ I2 (F(z(s))) ≤ +c , 4 4 4 3π k I + , ϕ(F(z(a1 ))) = − 2 4 3π ϕ(F(z(b1 ))) = − . 2 Proof. We assume the rotating direction is clockwise. At the point z(a) we have π √ ϕ = + I, 2 I1 − I2 = 3x 2 . √ From the definition of I2 , we know that x = 1 + I2 cos ϕ.
506
B. Zhong
3 2/3 It follows that I1 = 4I + O I 2 , and I1 (z(b)) = I2 (z(b)) = I. Since k < cπ 4 there is an interval [a− , b− ] ⊂ [a, b], such that 3
I1 (a− ) = k I + cI 2 , I1 (b− ) = k I, k ≥ 1,
and
k I ≤ I (s) ≤ k I + cI 3/2
for s ∈ [a− , b− ].
Let V1 (z(s)) denote the rotation speed under outer billiard map T1 starting from point z(s). Then
3 √ V1 (z(a− )) − V1 (z(b− )) = 2 k I + cI 3/2 − 2 k I + O I 2 c = √ I + O(I 3/2 ). k We assume that after n steps the point z(a− ) will be mapped into the third quadrant, i.e.
3 π √ π − − k I + O I 2 ≤ θn ≤ − . 2 2 From formula (1) we know that
3 √ θn = θ0 − 2n k I + O I 2 ,
3 π √ π + k I + O I 2 ≥ θ0 ≥ . 2 2 So π π √ ≤n ≤ √ +1 2 kI 2 kI n((V1 (z(a− ))) − V1 (z(b− ))) > If we assume that c > following relation:
4 π
and 1 ≤ k
2 k I . √ Now consider region(ii): ϕ ∈ − π2 , − π2 + k I . From formula (2) we know that √ 1 + I1 cos ϕ and k I ≤ I1 ≤ k I + cI 3/2 . I1 − I2 = 3x , x = 2 √ Plugging ϕ = − π2 + k I into the above formulas one gets 2
3 kI kI − O(I 2 ) ≤ I2 (ϕ) ≤ +O I2 . 4 4 We want the action I to decrease, so k < 4, i.e. c
0 if p is sufficiently close to γ then one has 1+ n (n) I ≤ I. 4
508
B. Zhong
Lemma 2. In the above model problem, when an orbit of outer billiards starts√from
√ (I1 , θ0 ) satisfying θ0 ∈ − π2 , π2 goes into singularity zone − π2 − I1 , − π2 + I1 , then I1 − I2 = π
3 I1 − 31 I1 + 31 2 I1 + O(I12 ), 4
π ∈ [0, 1). When it starts from (I2 , ϕ0 ) satisfying ϕ0 ∈ − 3π 2 ,−2 √ √
3π goes into singularity zone − 3π 2 − I2 , − 2 + I2 , then where 1 =
+θ0 2√
2 I1
+
1 2
I1 − I2 = 3I2 − 12I2 2 + 12I2 22 + O(I22 ), 3π +ϕ0 1 2√ where 2 = 2 I + 2 ∈ [0, 1), I1 , I2 are as before and the rotation direction is 2
clockwise. Proof. Let’s prove the first part; we √ assume after n steps √ the initial point will be mapped into singularity zone, i.e. − π2 − I1 ≤ θn ≤ − π2 + I1 . From formula (1) we know that 3/2 θn = θ0 − 2n I1 + O(I1 ). (3) It follows that π 2
π + θ0 + θ0 1 1 √ − ≤ n ≤ 2√ + , 2 2 2 I1 2 I1
i.e.
π
+ θ0 1 n= √ + 2 2 I1 π π + θ + θ0 1 1 0 = 2√ + − 2√ + , 2 2 2 I1 2 I1 2
where [ ]and{} denote the integer part and fractional part respectively. Taking the n into formula (3), one gets π/2 + θ0 1 π 3/2 + O(I1 ), + θn = − − I1 + 2 I1 √ 2 2 2 I1 π +θ where 0 ≤ 1 := 22√ I 0 + 21 < 1. 1 From the definition of I1 and I2 , we know that I1 − I2 = 3x 2 , √ 1 + I1 cos θn . x= 2 So I1 − I2 =
3 I1 − 31 I1 + 31 2 I1 + O(I12 ), 4
Diffusion Speed in Piecewise Smooth Billiards
509
i.e. I2 =
I1 + 31 I1 − 31 2 I1 + O(I12 ). 4
Almost in the same way one can prove the second conclusion of the lemma. The only difference is that we need to use formula (2) to evaluate the number of steps. The next result follows directly from Lemma 2. Corollary 1.
1 (4+)n
I ≤ I (n) ≤ (4 + )n I .
Proof. From the lemma we have 3 I − 31 I + 31 2 I + O(I 2 ), 4 I (1) − I2 = 3I2 − 12I2 2 + 12I2 22 + O(I22 ), π 3π +θ +ϕ where 1 = 2 √ 0 + 21 ∈ [0, 1); 2 = 22√ I 0 + 21 ∈ [0, 1). From the proof of I − I2 =
2 I
2
Lemma 2 it is easy to see that ϕ0 = θn . Simple computation shows that 4I + O(I 2 ) ≤ I (1) ≤ 4I + O(I 2 ). Inductively one gets 41 I (n−1) + O((I (n−1) )2 ) ≤ I (n) ≤ 4I (n−1) + 1 (n) ≤ (4 + )n I . O((I (n−1) )2 ), i.e. for any small enough > 0 we have (4+) n I ≤ I Theorem 2. Let (I0 , θ0 ) be the initial point, then a. There exists an orbit whose action variable under first return map F satisfies 1+ n 1 (n) I ≤ I ≤ I0 , 0 (4 + )n 4 where (I (n) , θ (n) ) = F n (I0 , θ0 ). b. If we denote by T the standard outer billiard map, and let (Im , θm ) be an orbit starting from initial point (I0 , θ0 ) i.e. (Im , θm ) = T m (I0 , θ0 ), then we have lim inf Im m 2 ≥ m→∞
9 2 π . 16
c. There exists an orbit such that lim sup Im m 2 ≤ 4π 2 . m→∞
Proof. Part a is a direct consequence of Remark 1 and Corollary 1. Below we will prove part b. Fix 0 sufficiently small. We consider two cases: (1) Im ≥ 0 , then Im m 2 ≥ 0 m 2 ; (2) There exist M > 0 such that for any m > M the following inequality holds ∞ such that the orbit is in the sinIm < 0 . Consider consecutive numbers {m i }i=0
π gularity zone 1: θ ∈ 2iπ − 2 − Im i , 2iπ − π2 + Im i at time m i and for any i we have Im i > Im i+1 . Let m ˜ i be the first
the orbit enters time after m i such that into singularity zone 2: θ ∈ 2iπ + π2 − Im˜ i , 2iπ + π2 + Im˜ i .
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B. Zhong
Denote Im 0 = I M+1 . We know that the action variable can decrease only in singularity zone 1 and the action variable can increase only in region 2. By Lemma 2 and ∞ , {b }∞ , {c }∞ and {c˜ }∞ such that Corollary 1 there exist sequences {ai }i=1 i i=1 i i=1 i i=1 1 ≤ ai < 1, 1 ≤ bi < 4 + , 4+ −1 ≤ ci ≤ 1, −1 ≤ c˜i ≤ 1, Im i = Im i−1 ai , Im˜ i = bi Im i , θm˜ i = 2 jπ + θm i + π + ci Im i for any j ∈ Z+ , θm i+1 = 2 jπ + θm˜ i + π + c˜i Im˜ i for any j ∈ Z+ . By the definition of the sequence m k we have Im ≥ Im k = Im 0
k
ak .
i=0
By Lemma 2 and formula (4) π + ci Im i m˜ i − m i ≥ , and 2 Im i π + c˜i bi Im i m i+1 − m˜ i ≥ . 2 bi Im i Therefore for any m k ≤ m < m˜ k we have π + c1 Im 0 π + c˜1 b1 Im 0 π + c2 Im 1 π + c˜2 b2 Im 1 m ≥ mk ≥ + + + 2 Im 0 2 b1 Im 0 2 Im 1 2 b2 Im 1 π + ck Im k−1 π + c˜k bk Im k−1 +··· + + +M 2 Im k−1 2 bk Im k−1 π = 2 Im 0
k 1 π 1 1 1+ √ + ··· + 1+ √ + ci 2 bk b1 2 Im k−1 i=1
1 c˜i + M 2 k
+
i=1
π π 1 + ··· + ≥ 1+ √ +M −k 2 Im k−1 2 Im 0 4+ 1 1 π 1 1 + √ + ··· + √ 1+ √ + M − k. = a1 a1 · · · ak−1 2 Im 0 4+
(4)
Diffusion Speed in Piecewise Smooth Billiards
Hence Im m ≥ 2
Im k m 2k
≥
k
ai Im 0
π 2 Im 0
511
1+ √
1
4+ 2 1 1 × 1 + √ + ··· + √ +M −k a1 a1 · · · ak−1 ⎛ ⎞2 k √ √ π 1 ( ak + · · · + a1 · · · ak ) + ak Im 0 (M − k)⎠ 1+ √ =⎝ 2 4+ i=1 2 (M − k) Im 0 1 1 π 1 + 1+ √ ≥ . + ··· + √ √ 2 4+ 4+ ( 4 + )k (4 + )k i=1
Thus lim inf Im m 2 ≥ m→∞
9 2 π . 16
This is exactly the conclusion of part b. To prove part c, let’s consider the orbit of part a. For any integer m k ≤ m < m k+1 , 1+ k 1 I0 and we have (4+) k I0 ≤ Im ≤ 4 2π 2π m < m k+1 ≤ √ + · · · + 2 I0 2 Im k π = √ (1 + · · · + (4 + )k ). I0 Thus lim sup Im m 2 ≤ 4π 2 . m→∞
≥ Remark 2. If we wanted to show only that there exists a number c such that lim inf Im c, we could estimate m k by the time of the last rotation. However we believe that the above proof gives the optimal estimate. m2
3. Outer Billiards: General Case In this section we consider a convex curve γ which has curvature jump at one point. Similar arguments apply for curves with several discontinuity points. From [T1] we know that in a small neighborhood of γ , outer billiard map has following form: s1 = s + 2r −
2K 2 r + O(r 3 ), 3K
(5) 2K 2 r + O(r 3 ), 3K where s is the length parameter on γ with the direction of γ measured counter-clockwise, r is the distance from a point outside of γ to the tangency point on γ and K (s) is the curvature function of γ . r1 = r −
512
B. Zhong 1
Lemma 3. Let I = r K 3 , l =
! 1 s 2 0
1
K (t) 3 dt, then (5) takes the following form ([T1]):
I1 = I + O(I 3 ),
(6)
l1 = l + I + O(I 2 ). Proof. Set I = r φ(s), then I1 − I = r1 φ(s1 ) − r φ(s) 2K (s) 2 2K (s) 2 3 3 r + O(r ) φ s + 2r − r + O(r ) − r φ(s) = r− 3K (s) 3K (s) 2K (s) 2 = 2φ (s)r 2 − r φ(s) + O(r 3 ). 3K (s) Letting φ (s) −
1 K (s) φ(s) = 0, i.e. φ = K (s) 3 , 3K (s)
we obtain the first formula of (6). Set l = l(s). It follows that l1 − l = l(s1 ) − l(s) 2K (s) 2 3 r + O(r ) − l(s) = l s + 2r − 3K (s) = l (s)2r + O(r 2 ). 1
Substituting r = I K (s)− 3 into the above equation and setting l(s) = get the second formula of (6).
! 1 s 2 0
1
K (t) 3 dt we
The lemma shows that the motion in new coordinates is almost rigid. To simplify the notation below, denote " 1 1 |γ | C1 = l(|γ |) = K (t) 3 dt, 2 0 where |γ | denotes the whole arc-length of γ in the (r, s) coordinate system. Now, we assume that at a point p (with angle coordinate l∗ ) the curvature of γ has a jump. Denote K 1 = liml →l∗+ K (l) and K 2 = liml →l∗− K (l). We assume that 0 < K 2 < K 1 < ∞. Theorem 3. Denote (I0 , l0 ) as the initial point, then a. There exist orbits whose action variable under first return map F satisfies the following relation n n 3 K2 K2 + 3 (n) ≤ I ≤ I0 I0 K1 + K1 b. If we denote by (Im , lm ) the image of the initial point under standard outer billiard map T i.e. (Im , lm ) = T m (I0 , l0 ), then 1
lim inf Im m ≥ C1 m→∞
K 23 1
1
K 13 − K 23
Diffusion Speed in Piecewise Smooth Billiards
513
Fig. 1. Two parts of γ are approximated by circles
c. There exists an orbit such that lim sup Im m ≤ C1 m→∞
K1 K2
1 3
Remark 3. Since Im is proportional to the distance between the point and the tangency point, the maximal distance between the point and γ is of order Im2 . Hence the the orbit can accumulate to the boundary with the speed 1/m 2 as suggested by the model problem. Proof. Pr oo f o f par t a. To prove the left side of the inequality of part a, let’s consider the problem in the x y-plane; let p be the origin of the x y-plane (see Fig. 1). In the left neighborhood of p we name γ by γ1 and in the right neighborhood of p name γ by γ2 . Pick any point q in the singularity zone (1) l ∈ [l∗ − I0 , l∗ + I0 ] with coordinates (x, y), draw two tangent lines to γ1 and γ2 respectively, denote the distance between tangency points and q by r1 and r2 . In the neighborhood of p we use circles with curvature K 1 (K 2 ) at the point p to approximate γ1 (r esp γ2 ) and denote the distance between their tangency points and q by r1 and r2 . Then 1 2 = x + y− K 12 1 2 r2 + 2 = x 2 + y − K2 r1 + 2
So r1 − r2 = −2y 2
2
1 K1 1 K2
2 , 2
1 1 − K1 K2
(7) .
.
However, for any point in the singularity zone we have y ≤ 0 and so our assumption that K 2 < K 1 implies r1 ≤ r2 .
514
B. Zhong
Since in the x y-plane any curve with curvature K at the origin could be written in the following form in a small neighborhood of origin: y=
K 2 x + L x 3, 2
we have ri = ri + O((ri )2 ) for i=1,2. Therefore r1 + O(r12 ) ≤ r2 .
(8)
Suppose I0 is given by the formula of γ1 , i.e. 1
I0 = r1 K 13 . Thus −1
r 1 = I0 K 1 3 . The action with respect to γ2 still makes sense in the singularity zone (1), 1
I2 = r2 K 23 . Plugging the relation (8) into the above formula we obtain I2 ≥ I0
K2 K1
1 3
+ O(r12 ) = I0
K2 K1
1 3
+ O(I02 ).
From the normal form formula (6) we have I
(1)
= I2 +
O(I23 )
≥ I0
K2 K1
1 3
+ O(I02 ).
(9)
Inductively we get O(I
(n−1) 2
)+ I
(n−1)
K2 K1
1 3
≤ I (n) ,
i.e. I0
K2 K1 +
n
3
≤ I (n) .
To prove the right side of the inequality, denote z(t) = (r (t), s(t)) a curve defined in the region [l∗ , l∗ + I0 ], with a ≤ t ≤ b, such that l(z(a)) = l∗ + I0 , l(z(b)) = l∗ , | I1 (z(t)) − I0 |≤ cI02 , for any t ∈ [a, b], where I0 is the action variable of the initial point.
Diffusion Speed in Piecewise Smooth Billiards
515
Obviously the action of γ2 still makes sense in this region. From the proof of the left side of the inequality of part a we know that for any t ∈ [a, b], 1 K2 3 I0 + O(I02 ) ≤ I2 (z(t)) ≤ I0 + O(I02 ), K1 so there exists a subinterval [a− , b− ] ⊂ [a, b] such that for some constant c > 0, 1 K2 3 I0 + cI02 , I2 (z(a− )) = K1 1 K2 3 I2 (z(b− )) = I0 . K1 Assume that after n times z(a− ) will be mapped into # 1 $ K2 3 −1 l I0 C1 + l∗ , C1 + l∗ + ; K1 this implies C1 ≤ n I2 (z(a− )). So n≥
C1 C1 ≥ 1 I2 (z(a− )) K2 3 K1
. I0
Let V (z(s)) be the rotation speed under map T of the point z(s), then n(V (z(a− )) − V (z(b− ))) ≥ cI02 n. If we take c >
2 C1
K2 K1
1 3
then
n(V (z(a− )) − V (z(b− ))) > 2 and so
I1 (T n (z(a− ))) = I1 (T n (z(b− ))) =
K2 K1 K2 K1
K2 K1
1 3
I0 ,
1 3
I0 + cI02 + O(I03 ),
1 3
I0 + O(I03 ).
Therefore there exists a subinterval [a1 , b1 ] ⊂ [a− , b− ] ⊂ [a, b] such that 1 1 2 K2 3 K2 3 K2 3 2 I0 ≤ I (1) = I1 (T n (z(t))) ≤ I0 +c I0 for any t ∈ [a1 , b1 ], K1 K1 K1 1 K2 3 l(F(z(a1 ))) = l∗ + I0 , K1 l(F(z(b1 ))) = l∗ .
516
B. Zhong
Inductively one can easily show that there exist subintervals: · · · ⊂ [an , bn ] ⊂ · · · ⊂ [a1 , b1 ] ⊂ [a, b] such that for any point a∗ ∈ ∞ n=1 [an , bn ] the orbit of z(a∗ ) converges to γ , moreover, denoting by F the first return map into zone (1) we have n K2 3 2 (n) I ≤ I0 + cI (n−1) , K1 i.e. for any small enough > 0 we get n K2 + 3 (n) I ≤ I0 . K1 This completes the proof of part a. pr oo f o f par t b. This proof is quite similar to the proof of part of b in Theorem 2. Fix 0 sufficiently small; we consider two cases: (1) For any M > 0, Im ≥ 0 holds for any m > M, then Im m 2 ≥ 0 m 2 . (2) There exist M > 0 such that for any m > M the following inequality holds: ∞ such that the orbit is in the singuIm < 0 . Consider consecutive numbers {m i }i=0 larity zone: [iC1 + l0 − Im i , iC1 + l0 + Im i ] at the time m i and such that Im i > Im i+1 . ∞ and {c }∞ such that From the proof of part a there exist sequences {ai }i=1 i i=1
K2 K1 +
1 3
≤ ai < 1, and
I m 1 = a1 I m 0 ,
Im k =
k
− 1 ≤ ci ≤ 1,
for any integer i,
ak I m 0 ,
i=1
lm i+1 = jC1 + lm i + ci Im i , for j=1, 2, 3, 4, · · · . %k %k+1 ak Im 0 ≤ Im < i=1 ak Im 0 , and Therefore for any m k ≤ m < m k+1 we have i=1 C1 + ck Im k−1 C1 + c1 Im 0 C1 + c2 Im 1 + + ··· + +M m ≥ mk ≥ Im 0 Im 1 Im k−1 1 1 C1 1+ + M − k. + ··· + ≥ Im 0 a1 a1 · · · ak−1 Thus Im m ≥ Im k m k ≥ C1 (ak + · · · + a1 · · · ak + (a1 · · · ak )(M − k)) 1 k k 3 3 3 K2 K2 K2 ≥ C1 + ··· + . + (M − k) K1 + K1 + K1 + Hence 1
lim inf Im m ≥ C1 m→∞
K 23 1
1
K 13 − K 23
.
Diffusion Speed in Piecewise Smooth Billiards
517
Pr oo f o f par t c. Consider the orbit constructed in part a. For any integer m k ≤ m < m k+1 , we have
K2 K1 +
k
m < m k+1
3
I0 ≤ Im ≤
K2 + K1
k
3
I0 and
C1 C1 C1 ≤ + ··· + = I0 Im k I0
K1 + K2
1+
1 3
+ ··· +
K1 + K2
k 3
.
Therefore lim sup Im m ≤ C1 m→∞
K1 K2
1 3
.
Remark 4. If we assume that 0 < K 1 < K 2 < ∞ and still choose the direction of γ to be counter-clockwise, similar arguments will apply. However the formulas will be different, for example the result of part (a) will be I0
K2 K1 +
n
6
≤ I (n) ≤ I0
K2 + K1
n
6
.
The brief computation is the following. Since 0 ≤ x ≤ r1 the first formula of (7) implies that 0≤y≤
K1 2 4 r + O(r1 ). 2 1
Substituting this into the second formula of (7) we have r2 ≥
K1 K2
1 2
r1 . We have seen
above that in a small neighborhood of the origin ri = ri + O(ri 2 ) for i=1,2. Therefore r2 ≥
K1 K2
1 2
r1 + O(r12 ),
i.e. −1 I2 K 2 3
≥
K1 K2
1
− 31
3
I0 K 1
+ O(I12 ).
Combining this with the first formula of (6) we obtain I
(1)
= I2 +
O(I23 )
≥ I0
K1 K2
1 6
+ O(I02 ).
The proof of the upper bound for I (1) is almost the same as in the K 2 < K 1 case.
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B. Zhong
4. Inner Billiard [L] considers a billiard with smooth convex boundary and provides an asymptotic formula for the dynamics near the boundary. Let S be the standard inner billiard map in the domain bounded by a curve γ . Denote by s the arc-length parametrization on γ (going counter-clockwise). Let r ∈ [0, π ] be the angle between our trajectory and the tangent vector to γ at the point q. Let (s1 , r1 ) be the image of (s, r ) under map S, i.e. (s1 , r1 ) = S(s, r ). Then the aforementioned formula has the following form: s1 = s + 2ρr + 43 ρ ρr 2 + O(r 3 ) (10) r1 = r − 23 ρ r 2 + 49 ρ 2 − 23 ρ ρ r 3 + O(r 4 ), where ρ denotes the curvature radius of γ . Consider the following transformation (the idea comes from [BB]) " s 2 1 θ = C1 ρ − 3 (s )ds , wher e C1 = ! |γ | , 2 − 3 (s)ds 0 0 ρ 1 r I = C2 ρ 3 (s) sin , wher e C2 = 4C1 . 2 Then (10) becomes
θ1 = θ + I + O(I 3 ) I1 = I + O(I 4 ).
(11)
Now we assume at point p with coordinate (I0 , θ0 ) the curvature of γ has a jump. Denote ρ1 = lim ρ(s) ρ2 = lim ρ(s). θ →θ0 +
θ →θ0 −
Consider the singularity zone (2) given by the condition θ ∈ [θ0 − I0 , θ0 + I0 ]. Let F be the first return map to this zone. Lemma 4. Assume that 0 < ρ1 < ρ2 < ∞. Denote by (I0 , θ0 ) the initial point, then the action variable of all orbits under F satisfies the following relation: I0
ρ1 ρ2 +
n
6
≤I
(n)
≤ I0
ρ2 + ρ1
n
3
.
Proof. Similar to the outer billiard case, let us first consider the problem in the x y-plane; let p be the origin of the x y-plane (Fig. 2). Consider a small neighborhood of p and let γ1 be the part of γ to the left of p and γ2 be the part of γ to the right of p. Pick any point q in the singularity zone (2) with coordinates (x, y) and draw a line m with slope k passing through q. Let p1 be the intersection of m with γ1 and p2 be the intersection of m with γ2 . Denote the angle between the line m and the tangent line to γ1 (γ2 ) at point p1 ( p2 ) by r1 (r esp r2 ). In the neighborhood of p we use circle γ1 (r esp. γ2 ) with curvature radius ρ1 (r esp ρ2 ) at the point p to approximate γ1 (r esp. γ2 ) and denote the angle between m
Diffusion Speed in Piecewise Smooth Billiards
519
Fig. 2. Two parts of γ are approximated by circles
and the tangent lines by r1 and r2 , assume m intersects with γ1 at point p1 (x1 , y1 ) and intersects with γ2 at point p2 (x2 , y2 ). Then the elementary geometry gives tan r1 − k x1 = −& , 1 + k tan r1 2 ρ − x2 y1 = ρ1 −
&
1
1
ρ12 − x12 , & y2 = ρ2 − ρ22 − x22 ,
y1 − y2 k = tan α = , x1 − x2 r2 = − arctan k + arctan &
(12)
x2 ρ22 − x22
.
Set x1 = ρ1 cos s, where s is the polar angle. A direct computation shows that s = 3π 2 + cr1 with −2 ≤ c ≤ 0. Thus (12) implies ' (ρ2 − ρ1 )(2c + c2 ) 2 r2 = 1 + r1 + O((r1 ) ). ρ2 Since in the x y-plane any curve with curvature K = ρ1 at the point of origin could be written in the following form in a small neighborhood of origin, y=
K 2 x + L x 3, 2
we have ri = ri + O((ri )2 ) for i = 1, 2. Therefore
' r2 =
1+
(ρ2 − ρ1 )(2c + c2 ) r1 + O((r1 )2 ). ρ2
(13)
520
B. Zhong
Since −2 ≤ c ≤ 0 we have O(r12 ) +
ρ1 ρ2
1 2
r1 ≤ r2 ≤ r1 + O(r12 ).
(14)
Let I0 be the action with respect to γ1 , i.e. 1
I0 = C2 ρ13 sin
r1 . 2
Then r1 =
2 − 13 ρ I0 + O(I02 ). C2 1
Plugging this equation into (14) we obtain 1 1 ρ1 6 ρ2 3 I0 + O(I02 ) ≤ I (1) = I2 + O(I24 ) ≤ I0 + O(I02 ). ρ2 ρ1 Inductively we get the results of the lemma.
Similar to Theorem 3 we prove the following theorem. Theorem 4. Denote (I0 , θ0 ) the initial point, then we have a. There exists an orbit whose action variable under the first return map F satisfies n n 6 ρ1 ρ1 + 6 I0 ≤ In ≤ I0 . ρ2 + ρ2 b. If we denote (Im , θm ) the image of initial point under inner map S, i.e. (Im , θm ) = S m (I0 , θ0 ), then 1
lim inf Im m ≥ m →∞
ρ16 1
1
ρ26 − ρ16
.
c. Under the map S, there exists an orbit such that 1 ρ2 6 lim sup Im m ≤ . ρ1 m →∞ Remark 5. Since Im is proportional to the angle between the billiard trajectory and the curve, the maximal distance from the trajectory to γ is of order Im2 . So again the orbit can approach γ with the speed 1/m 2 . Remark 6. Similar to Sect. 3 our argument also works in the case ρ2 < ρ1 but the formulas have to be modified. For example part c of Theorem 4 will read 1 ρ2 3 lim sup Im m ≤ . ρ1 m →∞ Acknowledgement. This work is done under the direction of Prof. Dolgopyat during my visit to the University of Maryland at College Park. I’d like to thank Prof. Dolgopyat for so many useful suggestions and careful reading of the manuscript. I also thank Prof. Kaloshin for inviting me here.
Diffusion Speed in Piecewise Smooth Billiards
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References [B] Boyland, P.: Dual billiards, twist maps and impact oscillators. Nonlinearity 9, 1411–1438 (1996) [BB] Babitch, V.M., Bouldirev, V.S.: Asymptotic Methods in Problems of Short Wave Diffraction (Russian). Moskow: Nauka, 1972 [Bu] Bunimovich, L.A.: On the ergodic properties of nowhere dispersing billiards. Commun. Math. Phys. 65, 295–312 (1979) [DF] Dolgopyat, D., Fayad, B.: Unbounded orbits for semicircular outer billiard. Ann. Henri Poincaré 10, 357–375 (2009) [Do] Douady, R.: Thèse de 3-ème cycle. Université de Paris 7, 1982 [G] Genin, D.: Hyperbolic outer billiards: a first example. Nonlinearity 19, 1403–1413 (2006) [GK] Gutkin, E., Katok, A.: Caustics for inner & outer billiards. Commun. Math. Phys. 173, 101–133 (1995) [H] Hubacher, A.: Instability of the boundary in the billiard ball problem. Commun. Math. Phy 108, 483–488 (1987) [L] Lazutkin, V.: KAM Theory and Semiclassical Approximations to Eigenfunctions. Berlin-HeidelbergNew York: Springer-Verlag, 1993 [LY] Levi, M., You, J.: Oscillatory escape in a duffing equation with a polynomial potential. J. Diff. Eqs. 140, 415–426 (1997) [M] Mather, J.: Glancing billiards. Erg. Th. & Dyn. Sys 2, 397–403 (1982) [Mo] Moser, J.: Stable and Random Motions in Dynamical Systems. Annals of Math. Studies 77 Princeton, NJ: Princeton University Press, 1973 [S] Schwartz, R.: Outer Billiards on Kites. Annals of Math, Studies, Princeton, NJ: Princeton Univ. Press, 2009 [T1] Tabachnikov, S.: On the dual billiard problem. Adv. Math. 115, 221–249 (1995) [T2] Tabachnikov, S.: Geometry and Billiards. Student Mathematical Library, Volume 30, Providence, RI: Amer. Math. Soc., 2005 [T3] Tabachnikov, S.: Asymptotic dynamics of the dual billiard transformation. J. Stat. Phys. 83, 27–38 (1996) [W] Wojtkowski, M.: Principles for the design of billiards with nonvanishing lyapunov exponents. Commun. Math. Phys. 105, 391–414 (1986) Communicated by G. Gallavotti
Commun. Math. Phys. 299, 523–560 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1085-9
Communications in
Mathematical Physics
On the Lebesgue Measure of Li-Yorke Pairs for Interval Maps Henk Bruin1, , Víctor Jiménez López2, 1 Department of Mathematics, University of Surrey, Guildford, Surrey,
GU2 7XH, UK. E-mail:
[email protected] 2 Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo,
30100 Murcia, Spain. E-mail:
[email protected] Received: 11 November 2009 / Accepted: 25 February 2010 Published online: 10 July 2010 – © Springer-Verlag 2010
Abstract: We investigate the prevalence of Li-Yorke pairs for C 2 and C 3 multimodal maps f with non-flat critical points. We show that every measurable scrambled set has zero Lebesgue measure and that all strongly wandering sets have zero Lebesgue measure, as does the set of pairs of asymptotic (but not asymptotically periodic) points. If f is topologically mixing and has no Cantor attractor, then typical (w.r.t. twodimensional Lebesgue measure) pairs are Li-Yorke; if additionally f admits an absolutely continuous invariant probability measure (acip), then typical pairs have a dense orbit for f × f . These results make use of so-called nice neighborhoods of the critical set of general multimodal maps, and hence uniformly expanding Markov induced maps, the existence of either is proved in this paper as well. For the setting where f has a Cantor attractor, we present a trichotomy explaining when the set of Li-Yorke pairs and distal pairs have positive two-dimensional Lebesgue measure. 1. Introduction In interval dynamics there are many ways to deal with the notion of asymptotic complexity (“chaos”). Probably it is pointless to try and decide which is the best of them, but in applications there are two of them which are by far the most popular, see e.g. [D, MMN]. One is topological chaos, that is, the existence of an uncountable scrambled set in the sense of the famous Li and Yorke paper [LY]. The other one is ergodic chaos, that is, the existence of an invariant probability measure absolutely continuous with HB gratefully acknowledges the support of EPSRC grant EP/F037112/1 and also the hospitality of the University of Murcia, Delft University of Technology and the Mittag-Leffler Institute in Stockholm. VJL was partially supported by MICINN (Ministerio de Ciencia e Innovacion) and FEDER (Fondo Europeo de Desarrollo Regional), grant MTM2008-03679/MTM, and Fundación Séneca (Agencia de Ciencia y Tecnología de la Región de Murcia, Programa de Generación de Conocimiento Científico de Excelencia, II PCTRM 2007-10), grant 08667/PI/08.
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respect to Lebesgue measure (acip). Neither of them is without drawbacks. (To keep this Introduction at the expository level, we have deferred most of the definitions to the subsequent sections.) There are easy conditions implying the existence of Li-Yorke chaos and its stability under small perturbations. One such condition is the existence of a periodic orbit of period not a power of two [Bl]. Nevertheless, this chaos need not be “observable”: for instance, the orbits of almost all points (in the sense of Lebesgue measure) may be attracted by this periodic orbit [Gu]. On the other hand, ergodic chaos ensures complicated dynamics for a set of points with positive measure. For instance, if a smooth multimodal map (with non-flat critical points) f has an acip, then, as a simple consequence of the zero measure of Cantor metric attractors [SV] and Proposition 9 below, there is a positive measure set of points whose orbit under f is dense in some interval. However, the converse is not true, even for the family of logistic maps [Jo,Ly2], so acips can only exist under additional conditions (for instance, hyperbolic repelling periodic points and |D f n ( f (c))| → ∞ for any critical point c of the map f [BRSS]). A map f is infinitely renormalizable if there is an infinite collection of nested cycles of periodic intervals; the intersection of these cycles is a Cantor set, called a solenoidal attractor (because the suspension over this attractor is a topological solenoid). The Feigenbaum map, or more correctly Coullet-Tresser-Feigenbaum, (see e.g. [MS, pp. 151-152]) is the best known example of this. A solenoidal attractor is Lyapunov stable and as shown in [BrJ] (see Proposition 31 below), points in the basin of such an attractor are approximately periodic: Definition 1. A point x is approximately periodic if for every ε > 0 there is a periodic point p such that lim supn→∞ | f n (x) − f n ( p)| < ε. Hence, up to small errors, almost all points eventually behave as periodic points. Remark 2. An intrinsic characterization of adding machines (namely any system in which every point is regularly recurrent) is presented in [BK]. In our setting, it applies to the solenoidal attractor itself, whereas approximate periodicity gives information on a neighborhood of the solenoidal attractor. However, a multimodal (even polynomial) map may have a dense orbit while, simultaneously, almost all orbits are attracted by a Cantor set (a so-called wild attractor) [BKNS]. As we will explain below, in such a case it is still possible but not necessary that a.e. point is approximately periodic. We see that there is a variety of smooth multimodal maps featuring a certain degree of “observable” dynamical complexity which is, however, not strong enough to be realized by an acip. It is natural to return to the Li-Yorke notion of chaos and investigate to what extent it can be used to measure this complexity. This is what we intend in the present paper. Definition 3. Let f : I = [0, 1] → I be a continuous map. A pair of points (x, y) is called: • distal if lim inf n→∞ | f n (y) − f n (x)| > 0; • asymptotic if limn→∞ | f n (y) − f n (x)| = 0; • Li-Yorke if it is neither asymptotic nor distal, that is, 0 = lim inf | f n (y) − f n (x)| < lim sup | f n (y) − f n (x)|. n→∞
n→∞
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We denote the set of distal, asymptotic and Li-Yorke pairs by Dis, Asymp and LY, respectively. A pair that is not distal (hence asymptotic or Li-Yorke) is called proximal. The set of Li-Yorke pairs with lim supn→∞ | f n (y) − f n (x)| ≥ ε is denoted as LYε . Note that f n (x) = f n (y) for all n ≥ 0 whenever (x, y) is a Li-Yorke or distal pair. Definition 4. Let f : I → I be a continuous map. • A set S ⊂ X is called scrambled if any two distinct points in S form a Li-Yorke pair. • The map f is called chaotic (in the sense of Li-Yorke) if it has an uncountable scrambled set. The above definitions can be strengthened by assuming that there is a positive lower bound of lim sup | f n (x) − f n (y)| independently of x, y: • If there is ε > 0 such that for every x = y ∈ S, lim sup | f n (x) − f n (y)| ≥ ε independent of x, y ∈ S, then S is called ε-scrambled. • If there is ε > 0 such that for every x ∈ X and every neighborhood U x, there is y ∈ U such that (x, y) ∈ LYε , then f is Li-Yorke sensitive. We emphasize that ε-scrambled is a stronger property than just scrambled. For example, [BHS, Prop. 5] shows the possibility of having scrambled sets that are not ε-scrambled for any ε > 0. Our concrete aim is to investigate the Lebesgue measure of the above sets for C 2 (or sometimes C 3 ) multimodal maps from the interval I into itself with non-flat critical 2 (I ) and C 3 (I ), respectively). The advantage of working in this points (denoted by Cnf nf setting is that there are many tools at hand to deal with measure-theoretic properties. 2 (I ) Remarkably the most important of these tools is purely topological: maps from Cnf have no wandering intervals, see [MS, Theorem A, p. 267]. As it happens, some of the results in the paper are based on a generalization of this property which is of interest in itself. Definition 5. A point x is asymptotically periodic, written x ∈ AsPer, if there is a periodic point p such that limn→∞ | f n (x) − f n ( p)| = 0. A measurable set W is strongly wandering if f n (W )∩ f m (W ) = ∅ for all n > m ≥ 0 and W contains no asymptotically periodic points. A wandering interval is just an interval which is strongly wandering. The notion of strongly wandering set was introduced by Blokh and Lyubich in [BL] (in a slightly different way). Under the assumption of negative Schwarzian derivative, the non-existence of strongly wandering sets of positive measure was proved in the unimodal case (and stated in the multimodal case without inflection points) in [BL]. Here 2 (I ), for which inflection points are now allowed. we prove it for maps from Cnf 2 (I ). Then every strongly wandering set has zero Lebesgue Theorem A. Let f ∈ Cnf measure.
Concerning the size of Li-Yorke chaos, the first natural question is whether smooth multimodal maps may have scrambled sets of positive Lebesgue measure. There is extensive literature on the subject. Examples of continuous maps possessing scrambled sets of positive or even full measure are well known: [K,S2,Mi,BH]. In fact, if f is chaotic (respectively, has a dense orbit), then it is topologically conjugate to a map having a positive (respectively, full) measure scrambled set [JS,S3] (respectively, [SS]).
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Of course, it is not possible that the whole interval is scrambled (in fact, a scrambled set cannot be residual on any subinterval of I , see [G]), although there are maps with scrambled forward invariant Cantor sets [HY]. However such maps cannot be multimodal (see Proposition 26). It is worth emphasizing that maps of type 2∞ in the Sharkovskiy ordering (that is, those having periodic points of periods all powers of 2, but no other periods) can possess scrambled sets of positive measure, but not of full measure because any Li-Yorke chaotic map of type 2∞ has a wandering interval. Indeed, if a map of type 2∞ has no wandering intervals, then all points are approximately periodic [S3]. However the following result is well known (see e.g. [BC, p. 144]): Proposition 6. If f : I → I is continuous and x, y are approximately periodic points, then (x, y) is either asymptotic or distal. Hence a scrambled set can contain at most one approximately periodic point. Finally, positive measure scrambled sets may also exist for C ∞ maps (with flat critical points) or C 1 maps with non-flat critical points, but a C 1 map cannot have a full measure scrambled set [J1,J2,BJ2]. Nevertheless, it is a widely held view that these are rather pathological examples. 2 (I ) with hyperbolic periodic points For instance, it is known that neither maps from Cnf 3 (I ) with and whose critical points satisfy the Misiurewicz condition, nor maps in Cnf negative Schwarzian derivative and having no wild attractors, may possess scrambled sets of positive measure [BJ1,BrJ]. Our next result confirms these expectations: 3 (I ), then it has no measurable scrambled sets of positive LebesTheorem B. If f ∈ Cnf gue measure.
It seems rather paradoxical that scrambled sets have zero measure even in the case when there is an acip, but this is not really so. The key point is measurability. For instance, one can easily derive from [S1] that the full logistic map f (x) = 4x(1 − x) possesses a non-measurable scrambled set with full exterior Lebesgue measure. Since the sets Dis, Asymp, LY, LYε , AsPer are all Borel, hence measurable sets [J3,BrJ], the moral is that we should measure these sets rather than scrambled sets. The idea of passing to the square I × I to study topological or measure-theoretic properties of Li-Yorke chaos is due to Lasota and was first used by Piórek [P]. To begin with, we prove that there are almost no “non-trivial” asymptotic pairs. 2 (I ), then Asymp \(AsPer × AsPer) has zero (two-dimensional) Theorem C. If f ∈ Cnf Lebesgue measure.
We can consider Theorem C as a weak form of sensitivity to initial conditions in the absence of periodic attractors. It is really quite weak: it applies in particular to the infinitely renormalizable case where almost all points are attracted by a solenoidal set, so there is no sensitivity to initial conditions in the standard Guckenheimer sense [Gu]. What Theorem C emphasizes is that there are no “privileged” routes (that is, with positive measure) to measure-theoretic attractors. The Li-Yorke property describes how chaotically pairs of points behave with respect to each other, and is hence a property of the Cartesian product (I 2 , f 2 ), for I 2 = I × I and f 2 (x, y) := ( f × f )(x, y) = ( f (x), f (y)). If (x, y) is Li-Yorke, then orb((x, y)) accumulates on, but does not converge to, the diagonal of I 2 . Hence LY is a weaker property than (x, y) having a dense orbit in I 2 . We want to find conditions ensuring that
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LY has positive (or full) mass w.r.t. two-dimensional Lebesgue measure λ2 . Recall that a map g is called topologically mixing if for every pair of non-empty open sets A, B there is N ∈ N such that f n (A) ∩ B = ∅ for all n ≥ N . According to Proposition 9, the orbit of Lebesgue a.e. point is either attracted by a periodic point or a solenoidal set, or eventually falls into an interval K such that f r (K ) = K for some r and the restriction g = f r | K is topologically mixing. As we explained earlier, approximately periodic points cannot be used to produce Li-Yorke pairs. On the other hand, all sets from Definition 3 are the same for f as for any of its iterates f r (except that the ε in LYε may change, but not its positivity). Thus we can restrict ourselves to the case where the map f itself is topologically mixing. Now, if f is topologically mixing, then either almost all points have a dense orbit, or the orbits of almost all points are attracted by finitely many pairwise disjoint wild Cantor attractors. It turns out that in the first case λ2 -a.e. pair (x, y) is Li-Yorke. 3 (I ) be a topologically mixing map having no Cantor attracTheorem D. Let f ∈ Cnf tors. Then the Cartesian product (I 2 , λ2 , f 2 ) is ergodic and for every x ∈ I there is a full measure set C x ⊂ I such that
lim inf | f n (y) − f n (x)| = 0, n→∞
lim sup | f n (y) − f n (x)| ≥ diam(I )/2, n→∞
for every y ∈ C x . In particular, LYdiam(I )/2 has full measure and f is Li-Yorke sensitive. Weaker versions of this result under additional Misiurewicz or negative Schwarzian conditions were proved earlier [BJ1,BrJ]. Ergodicity of (I 2 , λ2 , f 2 ) is parallel to Lebesgue measure λ being weak mixing, although in its standard definition, weak mixing applies to invariant measures only, see Subsect. 2.4. If f admits an acip, then we can say more: λ2 -a.e. pair (x, y) has a dense orbit in I 2 , see Corollary 28. However, it seems possible that there are cases where f admits no acip, λ2 -a.e. pair (x, y) is Li-Yorke, but has no dense orbit in I 2 . 2 (I ) is a topologically mixing map It remains to consider the case where f ∈ Cnf having a wild attractor A. Let the basin Bas(A) be the set of points whose orbit is attracted by A. We already know (Theorem C) that Asymp has zero measure. Also, if some x ∈ Bas(A) is approximately periodic, then by Proposition 31, A is conjugate to an adding machine, so all points in Bas(A) are approximately periodic. Finally, recall that every pair of approximately periodic points is either asymptotic or distal. Our result in this area classifies which behaviors can (and indeed do) occur: 2 (I ) be a topologically mixing map with a wild attractor A. Theorem E. Let f ∈ Cnf Then one of the following alternatives must occur:
(a) Lebesgue a.e. pair of points in Bas(A) is distal and every point in Bas(A) is approximately periodic; (b) Lebesgue a.e. pair of points in Bas(A) is distal and no point in Bas(A) is approximately periodic; (c) Lebesgue a.e. pair of points in Bas(A) is Li-Yorke; (d) Both Dis and LY have positive Lebesgue measure in Bas(A) × Bas(A). There are examples of polynomial unimodal maps of all above types (a)-(d) so that additionally, in cases (b)-(d), Bas(A) contains ε-scrambled sets for a fixed ε > 0 and f is Li-Yorke sensitive on Bas(A).
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2. Preliminaries 2.1. Interval maps. A continuous map f : I → I (for I = [0, 1]) is called multimodal if [0, 1] can be decomposed into finitely many subintervals on which f is (strictly) monotone. A point c is critical if f (c) = 0; the set of critical points is denoted by Crit. Critical points can be turning points (if f assumes a local extremum at c) or inflection points, and hence there can be more critical points than maximal intervals of monotonicity. A differentiable map having exactly one critical (turning) point is called unimodal. Near a turning point c, there is a largest interval [a, b] such that f (a) = f (b) and f is monotone on each of the intervals [a, c] and [c, b]. Then there is a continuous involution τc : [a, b] → [a, b] such that f (τc (x)) = f (x) and τc (x) = x for every x = c. k (I ) if f : I → I has a finite critical set Crit, is of class C k We say that f ∈ Cnf and each critical point is non-flat, i.e., for each c ∈ Crit, there exist a C k diffeomorphism ϕc with ϕc (c) = 0 and an c ∈ (1, ∞), called the critical order of c, such that f (x) = ±|ϕc (x)|c + f (c) for x close to c. It follows that if n is the smallest integer ≥ c , then D m f (c) = 0 for 1 ≤ m < n, but D n f (c) = 0. Conversely, if f is C k+1 near c and D n f (c) = 0 for some 2 ≤ n ≤ k, then c is non-flat. We can always enlarge the domain of f (without adding new critical points) and rescale such that f (∂ I ) ⊂ ∂ I
and
Crit ∩∂ I = ∅.
(1)
Observe that this operation does not change the zero or positive measure quality of the sets from Definition 3. Hence, except when f is topologically mixing, we will always assume that (1) holds. Also, we will assume without loss of generality that Crit does not contain any periodic point,
(2)
because if c is a periodic critical point, then we can modify slightly f near the orbit of c so that the resulting map has a periodic orbit containing no critical points and attracting the same points as previously attracted by the orbit of c. In a metric space, topological mixing (see its definition before Theorem D) implies that every iterate f n has a dense orbit. This excludes the case that there is a proper compact subinterval J of I such that f r (J ) ⊂ J for some r ≥ 1 (i.e., f is non-renormalizable). The map f is topologically exact (also called locally eventually onto) if for every non-degenerate interval J ⊂ I , there is n such that f n (J ) = I . For multimodal interval maps, topologically mixing, the existence of a dense orbit for f n for each iterate n ∈ N, and topologically exactness are all equivalent, see e.g. [BC, pp. 157–158]. ∞ of a point x is denoted by orb(x). 2.2. Attractors of interval maps. The orbit { f n (x)} n=0 n More the orbit of a set A is orb(A) := ∞ n=0 f (A). The ω-limit set ω(x) := generally, n n∈N Cl m≥n f (x) of x is the set of limit points of the orbit of x. If A is a subset of I , then we call Bas(A) = {x ∈ I : ω(x) ⊂ A} the basin (of attraction) of A. A periodic orbit O is called attracting if its basin contains an open set. The union of the components of Bas(O) intersecting O is called the immediate basin of O. If Bas(O) contains a neighborhood of O, then O is called a two-sided attracting periodic orbit; otherwise it is called a one-sided attracting periodic orbit. If p is a periodic point of period r and |D f r ( p)| is less than, equal to, or greater than 1, then the orbit of p is called hyperbolic attracting, parabolic, or hyperbolic repelling respectively. Of course, only hyperbolic 2 (I ), then it is well known attracting and parabolic orbits can be attracting. If f ∈ Cnf
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that all periodic orbits of sufficiently high period are hyperbolic repelling [MMS]. Fur2 (I ) have no wandering intervals. One of the consequences of this thermore, maps in Cnf is the following useful result. Below dist(A, B) denotes the distance between the sets A and B (by convention dist(A, ∅) = ∞). Proposition 7. (The non-Contraction Principle) Let f : I → I be a multimodal map without wandering intervals. Then for every ε > 0 there is δ > 0 such that if J is an interval such that |J | < δ and dist(J, p) > ε for each attracting periodic point p, then every component of every preimage f −n (J ) of J has length less than ε. Proof. This follows easily from the Contraction Principle as stated in [MS, p. 305]. The principle is a bit of a misnomer, so we added non- in our version. Definition 8. We call a closed invariant set A a (measure-theoretic) attractor if its basin has positive Lebesgue measure, and there is no proper subset A ⊂ A with the same properties. Note that A need not be an attractor in any topological sense: the basin Bas(A) need not contain a neighborhood of A, nor be of second Baire category. 2 (I ), In order to understand the nature of measure-theoretic attractors of maps from Cnf certain types of interval cycles are of particular interest. We say that a compact interval K is periodic (of period r) if K , . . . , f r −1 (K ) have disjoint interiors and f r (K ) ⊂ K . −1 i f (K ) a cycle of intervals and denote it by cyc(K ). If K 0 ⊃ We call the union ri=0 K1 ⊃ · · · is a nested sequence of periodic intervals of periods r0 < r1 < · · ·, then ∞ S = i=0 cyc(K i ) is called a solenoidal set and all points from S are called solenoidal points. If ri+1 = 2ri for every i sufficiently large, then we say that both S and its points are of Feigenbaum type. If i 0 is such that cyc(K i0 ) ∩ Crit = S ∩ Crit and all periodic points in cyc(K i0 ) are hyperbolic repelling, then we say that cyc(K i0 ) is solenoidal. Thus all solenoidal sets are of Cantor type and all cycles cyc(K i ) are solenoidal if i is sufficiently large. A compact invariant set A is Lyapunov stable if for every neighborhood U of A, there exists a neighborhood V of A such that f n (V ) ⊂ U for every n. We have the following classification of measure-theoretic attractors, see [M,Ke,BL,Ly1,MS,SV]. 2 (I ), then f has countably many attractors A, which are of the Proposition 9. If f ∈ Cnf following types:
(1) A is an attracting periodic orbit; (2) A is a cycle of intervals on which Lebesgue a.e. orbit is dense;. (3) A is a solenoidal set (the infinitely renormalizable case). Then A is Lyapunov stable and the basin Bas(A) is of second Baire category. (4) A is a minimal Cantor set, but not of the above type. In particular, A is not Lyapunov stable, and Bas(A) is of first Baire category. For Lebesgue a.e. x ∈ I , either x has a finite orbit (that is, x is eventually periodic) or ω(x) is one of the attractors above, and the number of attractors of type (2)-(4) together is no more than the number of critical points (because each of them must contain at least one critical point). More can be said: there can be countably many disjoint cycles, but all but finitely many of them must be disjoint from the basins of periodic attractors. If a cycle is solenoidal, then almost all its orbits are attracted by the solenoidal set contained in the cycle. If a cycle cyc(K ) contains a dense orbit, then one of the following two holds:
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• The whole cycle is an attractor and almost all its points are dense in cyc(K ). If K has period r this can still mean that K consists of two intervals J and J with a common boundary point such that f r (J ) = J and f r (J ) = J . In this case f 2r is topologically mixing on J . Otherwise f r is topologically mixing on K . • The cycle contains finitely many attractors of type (4) attracting the orbits of almost all points in the cycle (hence almost no point has a dense orbit in the cycle). An attractor of type (3) or (4) is called a solenoidal and a wild attractor respectively. Proposition 9 implies that the orbit of λ-a.e. x ∈ / AsPer accumulates on Crit, so every solenoidal set is in fact a solenoidal attractor. It is well known that attractors of type (1) and (3) are uniquely ergodic, and it follows from [BSS, Theorem 4] that this is also true for attractors of type (4). The existence of wild attractors has been proved for unimodal maps only if the combinatorial properties of the map are very specific, and the critical order c is sufficiently large (c 2). The prototype is the unimodal Fibonacci map [BKNS], but there are other Fibonacci-like combinatorics that allow wild attractors, see [Br3]. For multimodal maps, there are combinatorial types that allow wild attractors also if all critical orders are c = 2 [vS]. 2.3. Distortion results. In what follows we denote by |A| or λ(A) the Lebesgue measure of a measurable set A ⊂ I (also, λ2 will denote the two-dimensional Lebesgue measure). The density of a set X in J is |X ∩ J |/|J |. A point x is a (Lebesgue) density point of X if limε→0 |X ∩ (x − ε, x + ε)|/2ε = 1. Many of the arguments in this paper rely on measuring images under f n of neighborhoods U of density points of certain sets. If f n |U is diffeomorphic then the Koebe Principle (see Proposition 13) is used to estimate how densities change, but in general, U can visit several critical points in its first n iterates. In this case, we need more advanced techniques and results (relying on work in [BM1,MS,SV]), which we summarize below in Theorems 20 and 21. We call a sequence (G i )li=0 of intervals a chain if G i is a maximal interval such that f (G i ) ⊂ G i+1 , i = 0, . . . , l − 1. If (Hi )li=0 and (G i )li=0 are chains and Hi ⊂ G i for every i, then we will write (Hi )li=0 ⊂ (G i )li=0 . If x ∈ G 0 (or J is a subinterval of G 0 ), then we call (G i )li=0 , or sometimes just the interval G 0 , the pullback (chain) of G l along x, . . . , f l (x) (or along J, . . . , f l (J )). The order of a chain is the number of intervals G i , 0 ≤ i < l, intersecting Crit. 2 (I ), if G is a small interval not too close to Remark 10. Under the hypothesis f ∈ Cnf l any attracting periodic orbit, then all intervals G i , i < l, are also very small by Proposition 7. (Notice that the closure of the set of attracting periodic points only contains periodic points because the periods of attracting orbits are bounded and recall that Crit only contain non-periodic points, see assumption (2).)
Given intervals J ⊂ K , we say that J is ξ -well inside K if the components L and R of K \J satisfy |L|, |R| ≥ ξ |J |. If in addition ξ |L| ≤ |R| and ξ |R| ≤ |L|, then we say that J is ξ -well centered in K . A differentiable map without critical points f : J → R has distortion bounded by κ > 0 if sup x,y∈J
| f (x)| ≤ κ. | f (y)|
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We emphasize that if the density of a subset X of J is very close to 1, say > 1 − ε, then the density of f (X ) in f (J ) is > 1 − κε, so it is very close to 1 as well. An open subset V of R is called nice if orb(∂ V ) ∩ V = ∅. The first entry map to a nice set V is the map φV : D(V ) → V defined on the domain D(V ) = ∪n≥1 f −n (V ) by φV (x) = f r V (x), where r V = min{k > 0 : f k (x) ∈ V } is the first entry time. The maximal intervals J on which the first entry time r V is constant are called entry domains. By convention we assume that J does not intersect V : if J is a subset of a component of V with first entry time r V (J ) ≡ r , then we prefer to call J a return domain with return time r . The main reason why nice sets are “nice” is that the return and entry domains are all disjoint. Furthermore, all components in the backward orbit of a nice set are nice, and two such intervals are either nested or disjoint. 2 (I ). Then for every ξ > 0 there exists ξ = ξ (ξ, f ) > 0 such Lemma 11. Let f ∈ Cnf that if T is a component of the preimage of an interval V and U is an interval ξ -well inside V (respectively, ξ -well centered in V ), then the preimage J of U in T is ξ -well inside T (respectively, ξ -well centered in T ).
Proof. This follows easily from [BM1, Lemmas 3.2 and 3.3].
/ Crit by The Schwarzian derivative of a C 3 map f is defined for every x ∈ f (x) 3 f (x) 2 − . S f (x) = f (x) 2 f (x) A C 1 version reads: f has negative Schwarzian derivative if 1/ | f | is convex on every interval where it is defined. If f has negative Schwarzian derivative, then we can frequently estimate distortion using the Koebe Principle, see [MS, Sect. IV.1] or [BM1, “Koebe lemma”]: Proposition 12. (Koebe Principle for negative Schwarzian derivative maps) If f : G → f (G) is a diffeomorphism with negative Schwarzian derivative, H ⊂ G and f (H ) is ξ -well inside f (G), then H is ξ 3 /(2(3ξ + 2)2 )-well inside G and f |G has distortion bounded by ((1 + ξ )/ξ )2 . However, in this paper we will also use a C 2 -version of these classical results. 2 (I ), there is a function Q : (0, ∞) → Proposition 13. C 2 Koebe Principle) Given f ∈ Cnf (0, ∞) with limε→0 Q(ε) = 0 such that the following holds. Suppose that H ⊂ G are intervals such that f l |G is a diffeomorphism and f l (H ) is ξ -well-inside f l (G) for some ξ > 0. Then there exists ξ = ξ (ξ, f ) > 0 (with ξ → ∞ as ξ → ∞), and l−1 1+ξ 2 i i | f (H )| · (3) κ = exp Q( max | f (G)|) · 0≤i 0 and k ≥ 0, there are ξ = Corollary 14. Let f ∈ Cnf ξ (ξ, k, f ) > 0, σ = σ (ξ, k, f ) > 0 such that the following statement holds: Let (Hi )li=0 ⊂ (G i )li=0 be chains such that (G i ) has order at most k, G l is a small interval close enough to Crit, and the intervals Hi are pairwise disjoint. If Hl is ξ -well inside G l , then H0 is ξ -well inside G 0 . If in addition k = 0, then there is κ = κ(ξ, f ) > 0 such that f l | H0 has distortion bounded by κ.
Proof. By saying that “G l is a small interval close enough to Crit” we mean that there is ε0 = ε0 ( f ) such that |G l | < ε0 and dist(G l , c) < ε0 for some non-periodic critical point c, where ε0 is chosen so that, if ε1 satisfies Q(ε) < 1 for every ε ≤ ε1 in Proposition 13, then |G i | < ε1 for every i < l (see Remark 10). The case k = 0 is just Proposition 13. Let us give the proof for k = 1; the idea is the same for k > 1. Let G t be the interval from the chain containing the critical t t , (H )t+1 ⊂ (G )t+1 and point. Now we construct three subchains, (Hi )i=0 ⊂ (G i )i=0 i i=t i i=t (Hi )li=t+1 ⊂ (G i )li=t+1 . Applying Proposition 13 to the third chain, we find ξ1 = ξ1 (ξ, f ) and κ1 = κ1 (ξ, f ) such that Ht+1 is ξ1 -well inside G t+1 . Applying Lemma 11 to the middle chain, we find ξ2 = ξ2 (ξ1 , f ) such that Ht is ξ2 -well inside G t . Finally, applying again Proposition 13 to the first chain, we find ξ = ξ(ξ2 , f ) such that H0 is ξ -well inside G0. The Koebe property refers to distortion control in the presence of Koebe space. Slightly weaker is the Macroscopic Koebe property, which refers to the preservation of Koebe space under pullback. Hence the fact that H is ξ -well inside G in Proposition 13 is basically a Macroscopic Koebe statement. In order to use the above results, we need conditions guaranteeing the existence of Koebe space at the end of chains. The following propositions are particularly useful in this regard. 2 (I ). Then for every ξ > 0 there exists ξ = ξ (ξ, f ) > 0 Proposition 15. Let f ∈ Cnf such that if V and U are nice intervals, U is ξ -well inside V, x ∈ V and f k (x) ∈ U (with k ≥ 1 not necessarily minimal), then the pullback of U along x, . . . , f k (x) is ξ -well inside the return domain to V containing x. In particular, if U is a return domain to V which is ξ -well inside V , then all return domains to U are ξ -well inside U .
Proof. This is Theorem C(1) from [SV] (see also the remark below Theorem C(1) and the erratum to that paper). The second statement follows easily from the first one, by fixing an arbitrary return domain K to U and x ∈ K , and choosing k as the return time of K . The interval K is then the pullback of U along x, . . . , f k (x) and U is the return domain to V containing x. 2 (I ) and let x be a recurrent point of f which is neither Proposition 16. Let f ∈ Cnf periodic nor of Feigenbaum type. Then there are ξ0 = ξ0 ( f ) > 0 and an arbitrarily small nice neighborhood J of x such that the return domain to J containing x is ξ0 -well inside J . Assume in addition that x is not solenoidal, and that I0 is a nice neighborhood of x so small that it contains no periodic neighborhood of x. Let (Im )∞ m=0 be the sequence of nice intervals such that Im is the return domain to Im−1 containing x. In this case, there are infinitely many m such that Im+1 is ξ0 -well inside Im .
Proof. This is a mixture of Theorems A(1) and A’(2) from [SV].
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2.4. Notions from ergodic theory. Let X be a topological space with Borel σ -algebra B and let f : X → X be a Borel measurable map. Recall that a probability measure μ on B is called invariant (respectively, non-singular) if μ( f −1 (A)) = μ(A) (respectively, μ(A) = 0 if and only if μ( f −1 (A)) = 0) for any A ∈ B. In what follows we assume that μ is non-singular but not necessarily invariant. Definition 17. Let (X, μ, f ) be defined as above. Write X 2 := X × X, μ2 := μ × μ, f 2 := f × f and let B2 be the Borel σ -algebra in X × X . We call the system (X, μ, f ) • conservative if for every set A ∈ B with μ(A) > 0, there is n ≥ 0 such that μ(A ∩ f n (A)) > 0; • ergodic if f −1 (A) = A ∈ B implies μ(A) = 0 or 1. If f is ergodic and conservative, then μ-a.e. orbit is dense in supp(μ), see Lemma 19 below. • exact if f −n ( f n (A)) = A ∈ B for every n ≥ 0 implies μ(A) = 0 or 1; • mixing if μ is invariant and lim μ(A ∩ f −n (B)) = μ(A)μ(B)
n→∞
for all sets A, B ∈ B; • weak mixing if μ is invariant, but 1 is the only eigenvalue corresponding to a measurable eigenfunction of the operator P : L 1 (μ) → L 1 (μ) defined by Pψ = ψ ◦ f −1 . An equivalent definition of weak mixing is that the Cartesian product system (X 2 , μ2 , f 2 ) is ergodic. • For non-invariant measures, we can still speak of mildly mixing: A non-singular probability measure is mildly mixing if for every set A ∈ B of positive measure lim inf μ(A ∩ f −n (A)) > 0. n→∞
Mild mixing implies that (X 2 , μ2 , f 2 ) is ergodic. (If f is invertible, then mild mixing is equivalent to (X 2 , μ2 , f 2 ) being ergodic. In this case f also preserves a probability measure equivalent to μ, but neither of these stronger statements holds in general if f is non-invertible. See [HS] for more results.) Lemma 18. If (X, μ, f ) is exact, then (X 2 , μ2 , f 2 ) is ergodic. Proof. Assume by contradiction that (X 2 , μ2 , f 2 ) is not ergodic, so there is U ∈ B2 such that f 2−1 (U ) = U and 0 < μ2 (U ) < 1. Then there is a ∈ X such that 0 < μ(Ua ) < 1 for Ua = {y ∈ X : (a, y) ∈ U }. Let Va = X \Ua . Then f n (Ua ) ∩ f n (Va ) = ∅ for all n ≥ 0, so ( f −n ◦ f n )(Ua ) = Ua for all n, contradicting that μ is exact. Lemma 19. If X is separable and (X, μ, f ) is ergodic and conservative, then μ-a.e. x has a dense orbit in supp(μ). Proof. Let {Un }n∈N ⊂ B ∩ supp(μ) be a countable basis for the relative topology of supp(μ), i.e., Un = supp(μ) ∩ Un , where Un are those elements of a countable basis of X that intersect supp(μ). Since supp(μ) is by definition the smallest closed set of full measure, μ(Un ) > 0 for every n. −1 Set Yn := {x ∈ X : f k (x) ∈ Un infinitely often}. Then f (Yn ) = Yn , so μ(Yn ) = 0 or 1 for each n. If μ(Yn ) = 1 for each n, then Y := n Yn has full measure, and every x ∈ Y has a dense orbit in supp(μ).
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So assume that n is such that μ(Yn ) = 0. For Z =i Un \Yn , we can write Z = k≥0 Z k , where Z k = {x ∈ Un : k = max{i ≥ 0 : f (x) ∈ Un }}. Since μ(Z ) = μ(Un ) > 0, there is k such that μ(Z k ) > 0, and hence μ( f k (Z k )) > 0. But then f k (Z k ) ⊂ Un is a set of positive measure that never visits Un again, and this contradicts that μ is conservative. The following diagram summarizes the implications between these various notions of mixing. The notions on the bottom line can be defined for non-invariant measures μ, but any implication to the top line requires that μ is f -invariant and in particular conservative.
μ invariant:
mixing
⇑
μ not necessarily invariant:
exact
⇒
⇒
weak mixing
ergodic
@ @ R @ R @
@ @ (X 2 , f 2 , μ2 ) ergodic ⇑ @ R @ R @ : : PP P PP P P P
mildly mixing
⇒
1 1
ergodic
3. Inducing to Critical Neighborhoods and Strongly Wandering Sets In this section, we prove that strongly wandering sets have zero measure as a consequence of Theorem 20 below. Theorem 20 is a refinement of Theorem 1 of [CL] (where 3 (I ) is used), which in turns improves Theorem D the slightly stronger hypothesis f ∈ Cnf of [SV]. 2 (I ). Then there are positive constants ξ = ξ( f ), κ = κ( f ), Theorem 20. Let f ∈ Cnf δ = δ( f ) and, for every ε > 0 and c ∈ Crit, open intervals c ∈ Uc ⊂ Vc , with |Vc | < ε, such that the following conditions hold: (i) The set U = c∈Crit Uc is nice and Uc is ξ -well inside Vc for every c ∈ Crit. (ii) If J is an entry domain of the first entry map φ to U , say φ| J = f j | J and φ(J ) = Uc , then there is K ⊃ J such that f j | K is a diffeomorphism and f j (K ) = Vc . Moreover, φ| J has distortion bounded by κ. (iii) If c ∈ Crit, then there are an open interval Wc ⊂ Uc and kc ∈ N such that |Wc | > δ|Uc |, f kc |Wc is a diffeomorphism with distortion bounded by κ, and (a) either Wc ⊂ f kc (Wc ) ⊂ Uc (if c is of Feigenbaum type); (b) or kc = 1 and ∂ Wc ∩ D(U ) = ∅ (if c is not of Feigenbaum type).
The existence of the sets Uc ⊂ Vc from Theorem 20 suggests the construction of an induced Markov map F : U → U with all branches mapping monotonically onto a component of U . The first, very detailed, constructions of such induced maps go back to Jakobson [Ja], but the abstract statement for unimodal maps having no periodic or Cantor attractors comes from Martens’ PhD. thesis [Ma]. During the writing of this paper, we
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learned about a multimodal C 3 version by Cai and Li [CL], see Proposition 45. Their construction precludes the existence of parabolic points, hence we improve it slightly by showing the version below, where Crit denotes the set of critical points interior to metric attractors of type (2) in Proposition 9. Let E := n≥0 f −n (A), where A is the union of all attractors of type (2) in Proposition 9, that is, the union of cycles of intervals in which λ-a.e. orbit is dense. 3 (I ) and let U , V , c ∈ Crit , be defined as in Theorem 20 for Theorem 21. Let f ∈ Cnf c c ε sufficiently small and let U = c∈Crit Uc . Then for Lebesgue a.e. x ∈ E we can find k x ∈ N and intervals G x ⊃ Hx x (with either Hx = Hy or Hx ∩ Hy = ∅) such that f k x : G x → Vc is diffeomorphic for some c ∈ Crit and f k x (Hx ) = Uc . Let F : U → U be defined by F| Hx = f k x : Hx → Uc for the appropriate c ∈ Crit . Then all iterates of F are well defined λ-a.e and its branches have uniformly bounded distortion, i.e., there is κ = κ( f ) > 0 such that for every n ∈ N and every interval H on which F n | H : H ⊂ U → Uc is a diffeomorphism, the distortion of F n | H is bounded by κ.
We postpone the somewhat technical proofs of Theorems 20 and 21 to the Appendix. 2 (I ) and assume that W is a strongly wandering set Proof of Theorem A. Let f ∈ Cnf of positive measure. Let x be a density point of W . It is not restrictive to assume that x belongs to the interior of a cycle cyc(K ) which either contains a dense orbit or is of solenoidal type. We can also assume that orb(x) accumulates on Crit. Let τ > 0 be such that (x − τ, x + τ ) ⊂ cyc(K ). We can assume that the density of W in any subinterval of (x − τ, x + τ ) containing x is larger than 1 − ε0 , with ε0 > 0 so small that if d = # Crit, then 1 − δ −d κ 2d+1 ε0 > 1/2, where δ = δ( f ) and κ = κ( f ) are the numbers from Theorem 20. Finally we fix ε > 0 such that if U is defined as in Theorem 20, then all components of U and entry domains to U have length less than τ (by the non-Contraction Principle). Let J1 be either the entry domain to U containing x, say φ| J1 = f j1 | J1 , f j1 (J1 ) = Uc1 , or the component Uc1 of U containing x (then we take j1 = 0). Since J1 ⊂ (x − τ, x + τ ) ⊂ cyc(K ) and cyc(K ) is invariant, the density of W in J1 is > 1 − ε0 and Uc1 ⊂ cyc(K ). Let Wc1 be the interval from Theorem 20, part (iii). Since |Wc1 | > δ|Uc1 | and the density of f j1 (W ) in Uc1 is > 1 − κε0 , the density of f j1 (W ) in Wc1 is > 1 − δ −1 κε0 and the density of f j1 +k1 (W ) in f k1 (Wc1 ) is > 1 − δ −1 κ 2 ε0 , where k1 = kc1 is as in Theorem 20(iii). If c1 is a Feigenbaum critical point, then Wc1 ⊂ f kc1 (Wc1 ) ⊂ Uc1 , so the density of f j1 (W ) in f kc1 (Wc1 ) is also > 1 − δ −1 κε0 . Therefore the densities of both f j1 (W ) and f j1 +k1 (W ) in f kc1 (Wc1 ) are > 1/2, hence f j1 (W ) ∩ f j1 +k1 (W ) = ∅, contradicting that W is strongly wandering. Now we deal with the non-Feigenbaum case. Here k1 = 1, f |Wc1 is a diffeomorphism and ∂ Wc1 ∩ D(U ) = ∅. On the other hand, Uc1 ⊂ cyc(K ), so f (Wc1 ) ⊂ cyc(K ) as well. Moreover, the choice of cyc(K ) guarantees that the orbits of almost all its points accumulate on Crit. Hence f (Wc1 ) is the pairwise disjoint union (up to a measure zero set) of entry domains to U and components of U . Since the density of f j1 +1 (W ) in f (Wc1 ) is > 1 − δ −1 κ 2 ε0 , it is also > 1 − δ −1 κ 2 ε0 in one of these entry domains or components, call it J2 . Now we repeat the argument. Say that φ| J2 = f j2 | J2 , f j2 (J2 ) = Uc2 , or J2 is a component Uc2 of U (when we take j2 = 0). Then f j1 + j2 +1 (W ) has density > 1 − δ −2 κ 3 ε0 in Uc2 . If c2 is a critical point of Feigenbaum type, then we get a contradiction as before.
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If not, then we find an entry domain to U or a component of U , call it J3 , such that f j1 + j2 +2 (W ) has density > 1 − δ −2 κ 4 ε0 in J3 . Proceeding in this way, we find intervals J1 , J2 , . . . , Jd+1 , each of them either an entry domain to U or a component of U , say f ji (Ji ) = Uci , such that the density of f ti (W ) in Uci is > 1 − δ −(i−1) κ 2i−1 ε0 for every i = 1, 2, . . . , d + 1, with ti = j1 + · · · + ji + i − 1. Find i < i such that ci = ci . Then the densities of f ti (W ) and f ti (W ) in Uci are > 1/2, and we have the required contradiction. 4. lim sup Fullness of Lebesgue Measure The following definition was first used in Barnes [Ba]. Definition 22. The system (X, μ, f ) is called lim sup full if lim supn→∞ μ( f n (A)) = 1 for any A ∈ B with μ(A) > 0. 3 (I ) be a topologically mixing map having no Cantor attracTheorem 23. Let f ∈ Cnf tors. Then f is lim sup full with respect to Lebesgue measure. 3 (I ) (and satisfying (1)) has an invariant interval Proof. It suffices to show that if f ∈ Cnf I such that f | I is topologically mixing and has no Cantor attractors, then we have lim supn λ( f n (A)) = λ(I ) for any measurable set A ⊂ I of positive measure. Fix a critical point c interior to I and let ε > 0 be small enough so that Theorem 21 holds and the corresponding interval Uc lies in I . Let x be a density point of A. Since f has no Cantor attractors in I , there is no loss of generality in assuming ω(x) = I and (after replacing if necessary A by some of its iterates) x ∈ Uc . Since all iterates of the induced map F from Theorem 21 are well defined for λ-a.e. point in Uc , we can assume that this is the case for x. This implies that there are intervals x ∈ Hn ⊂ Uc with n Hn = {x} and integers kn such that f kn | Hn : Hn → Uc are diffeomorphisms with uniform distortion bound for some c ∈ Crit . Hence λ( f kn (A ∩ Hn )) → λ(Uc ) as n → ∞. Observe that Uc ⊂ I , so there is j such that I = f j (Uc ) (because f | I is topologically mixing). Then also λ( f kn + j (A ∩ Hn )) → λ(I ), as required. 3 (I ) be topologically mixing. Then the following statements Proposition 24. Let f ∈ Cnf are equivalent:
(i) f has an acip μ; (ii) lim inf n λ( f n (A)) > 0 for every measurable set A of positive Lebesgue measure; (iii) lim inf n λ( f n (A)) = 1 for every measurable set A of positive Lebesgue measure. In this case μ is equivalent to λ (that is, μ(A) if and only if λ(A) = 0) and λ2 is ergodic and conservative. Remark 25. Under C 3 assumptions, no transitive Cantor set can have positive measure, [SV, Theorem E(1), cf. also Remark 1], so in particular a Cantor attractor has Lebesgue measure 0, and cannot support an acip. We expect this to be true in the C 2 setting as well, but we have no proof. Proof. The implication (iii)⇒(ii) is trivial. We prove (ii)⇒(i). According to [St], the existence of an acip for f is equivalent to the existence of δ > 0 and 0 < α < 1 such that λ(A) < δ implies λ( f −n (A)) < α for every n and A. Hence, if f does not admit an acip, then there are sets {Ak } and
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integers (n k )k≥1 with n k → ∞, λ(Ak ) → 0 and λ( f −n k (Ak )) ≥ 1 − 2−k−1 . Take A = k f −n k (Ak ), then λ(A) ≥ 21 and lim inf n λ( f n (A)) = 0, contrary to condition (ii). For the implication (i)⇒(iii), assume that f admits an acip μ. Since (wild) Cantor attractors have zero Lebesgue measure, μ cannot be supported on them. Then Theorem 23 applies. We claim that μ is equivalent to λ. Assume to find a measurable set the contrary −n (A). Then μ(B) = 0 but, by A such that μ(A) = 0 < λ(A). Let B = ∞ f n=0 Theorem 23, λ(B) = 1. This is impossible. The equivalence of μ and λ (or just the absolute continuity of μ) and Theorem 23 imply that if μ(A) > 0, then lim supn μ( f n (A)) = 1. Since μ is invariant, the sequence {μ( f n (A))}n∈N is non-decreasing. Therefore limn μ( f n (A)) = 1, and also limn λ( f n (A)) = 1 by the equivalence of μ and λ. Moreover, λ is exact, hence λ2 is ergodic (Lemma 18), and μ2 is conservative (because of Poincaré recurrence), so λ2 is conservative as well (because μ2 and λ2 are equivalent). 5. Li-Yorke Pairs and Scrambled Sets In this section we prove our results on scrambled sets and the λ2 -measure of the sets Dis, Asymp and LY. Proof of Theorem B. Assume that a scrambled set S has positive measure. We can assume that all its points are attracted by the same attractor, which must be either a cycle of intervals containing a dense orbit, or a minimal Cantor set. Since f n is oneto-one on S for every n, the first possibility can be immediately discarded by Theorem 23. Hence we may assume that all points of S are attracted by a minimal Cantor set W . (In fact W must be a wild attractor, because points attracted by solenoidal sets are approximately periodic and a scrambled set can contain at most one approximately periodic point by Proposition 6.) By Theorem A there are integers n < m such that f n (S) ∩ f m (S) = ∅. Let x ∈ f n (S) ∩ f m (S). Then there is y ∈ f n (S) such that f m−n (y) = x. Since (x, y) is a Li-Yorke pair, so is ( f m−n (x), f m−n (y)) = ( f m−n (x), x). Find a sequence {lk } such that | f m−n+lk (x) − f lk (x)| → 0. We may assume that ( f lk (x))k∈N accumulates at p ∈ W . Then f m−n ( p) = p, which is impossible because W is infinite and minimal. Proposition 26. A multimodal map f has no closed invariant scrambled set (apart from a singleton set). Proof. Suppose by contradiction that the closed non-singleton S is invariant and scrambled. Then clearly it can contain at most one fixed point. Let y ∈ S be a non-fixed point. Then because (y, f (y)) is Li-Yorke, there is a sequence (n k )k∈N such that limk | f n k (y)− f n k ( f (y))| = 0 and { f n k (y)} converges. By continuity limk f n k (y) = limk f n k +1 (y) = f (limk f n k (y)), so the limit is a fixed point p ∈ S. Since (y, f (y)) is Li-Yorke, { f n (y)} does not converge to p. Since f is multimodal, there are a sequence (m k ) and a number ε > 0 such that f m k (y) → p but | f m k −1 (y) − p| > ε. By taking a subsequence, we can assume that { f m k −1 (y)}k∈N converges, say to q. But then p = limk f m k (y) = f (limk f m k (y))) = f (q), so ( p, q) is not Li-Yorke. This contradiction proves the proposition. Proof of Theorem C. Assume that λ2 (Asymp \(AsPer × AsPer)) > 0. Then there are a point x ∈ I \ AsPer and a Borel set Y ⊂ I \ AsPer of positive measure such that
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limn→∞ | f n (x) − f n (y)| = 0 for every y ∈ Y . We can assume that orb(x) accumulates on a non-periodic point u. For every n ≥ 0, let dn : Y → R be defined by dn (y) = supm≥n | f m (x) − f m (y)|. Then (dn ) is a sequence of Borel measurable maps converging pointwise to zero. According to Egorov’s Theorem we can remove from Y a small set (so that the remaining set Z has positive measure) in such a way that {dn | Z }n∈N converges uniformly to zero, that is, diam( f n (Z )) → 0 as n → ∞. Use Theorem A to find integers k > m such that f k (Z ) ∩ f m (Z ) = ∅. Recall that orb(Z ) accumulates at a point u with | f j (u) − u| = ε > 0 for j = k − m. Find δ ∈ (0, ε/4) such that | f j (v) − f j (w)| < ε/2 whenever |v − w| < 2δ. Next take l > m so that dist( f l (Z ), u) < δ and diam( f l (Z )) < δ. Then | f l (z) − u)| < 2δ < ε/2, hence | f l+ j (z) − f j (u)| < ε/2 for every z ∈ Z . Thus f l+ j (Z ) ∩ f l (Z ) = ∅, contradicting f m+ j (Z ) ∩ f m (Z ) = ∅ and l > m. Proposition 27. If (I, μ, f ) is exact, then μ2 (Dis) = 0. Proof. Assume by contradiction that μ2 (Dis) > 0. Write Disx = {y ∈ I : (x, y) is distal} and G Dis := {x : μ(Disx ) > 0}. Then by Fubini’s Theorem, μ(G Dis ) > 0. For x ∈ G Dis , take Yε := {y ∈ Disx : lim inf | f n (x) − f n (y)| ≥ ε}. n→∞
Clearly Yδ ⊂ Yε if δ > ε and f n (Yε ) ∩ f n (I \Yε ) = ∅ for all n ≥ 0. If for some ε > 0, both Yε and I \Yε have positive measure, then we have a contradiction to exactness. The remaining possibility is that there is ε = ε(x) > 0 such that Yε has full measure, whereas λ(Yδ ) = 0 for all δ > ε. Now take η > 0 such that G η := {x ∈ G Dis : ε(x) > η} has positive measure. Let R > 1/η, and take distinct points x0 , x1 , . . . , x R ∈ G η . Clearly these points can be chosen so that lim inf n | f n (xi ) − f n (x j )| > η for all i = j. Take N minimal such that | f n (xi ) − f n (x j )| > η for all i = j and n ≥ N . However, by the choice of R, there is no space in I to fit the points f N (xi ), i = 0, . . . , R, so that they have pairwise distance greater than η. This contradiction proves that μ2 (Dis) = 0. Next, for the proof of Theorem D, recall that we assumed that f has no wild attractor. Proof of Theorem D. Theorem 23 implies that f is lim sup full w.r.t. Lebesgue measure. By [Ba, Theorem A], if f : I → I is a multimodal surjective map that is lim sup full with respect to a measure μ, then f is exact. (The proof is stated for d-to-1 maps, but applies with minor changes to the “at most d-to-1” setting as well.) Hence λ is exact. By Lemma 18, λ2 is ergodic for (I 2 , f 2 ). Take x ∈ I . We prove that there is a full measure set A x such that lim supn→∞ | f n (y)− n f (x)| ≥ diam(I )/2 for every y ∈ A x . If the opposite is true, then there are a set A of positive measure and an m ∈ N such that | f n (y)− f n (x)| < diam(I )/2 for every y ∈ A and every n ≥ m. But this contradicts the conclusion of Theorem 23 applied to A, i.e., that lim supn λ( f n (A)) = 1. Similarly, we can prove that there is a full measure Bx such that lim supn→∞ | f n (y) − f n (x)| = 0 for every y ∈ Bx . Indeed, if this were false then there are m ∈ N, ε > 0 and a set B of positive measure such that | f n (y) − f n (x)| > ε for every y ∈ A and every n ≥ m. Again, this contradicts that lim supn λ( f n (B)) = 1. Hence lim inf | f n (y) − f n (x)| = 0, lim sup | f n (y) − f n (x)| ≥ diam(I )/2, n→∞
for every y ∈ Ux = A x ∩ Bx .
n→∞
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539
By Proposition 24 and Lemma 19 we immediately recover a well-known result on weak-mixing. 3 (I ) is topologically mixing and has an acip, then λ -a.e. (x, y) Corollary 28. If f ∈ Cnf 2 2 has a dense orbit in I .
Remark 29. Probably the earliest result in this direction dates back to Ledrappier who proves in [Le, Theorem 1] that a certain class of interval maps is weak Bernoulli. Keller [Ke] proved (weak-)mixing for acips μ of multimodal maps assuming negative Schwarzian derivative. He also showed that dμ/dλ is bounded away from zero on supp(μ). One can prove that μ is also mixing by means of an induced map F : U → U as in Theorem 21 which possesses an acip ν, cf. [Y]. In fact, by taking an appropriate power of F N , U decomposes into a finite number of F N -invariant parts on which ν is invariant and mixing. Pulling back ν to the original system, we recover μ and by the topologically mixing condition, μ can have only one mixing component. Conservativity of λ2 is crucial in Lemma 19, and even if λ itself is conservative for (I, f ), this does not guarantee that λ2 is conservative for the Cartesian product. It is for this reason that Proposition 27 and Theorem D are not just direct consequences of ergodicity of λ2 from Lemma 18. The following conjecture suggests conditions under which λ2 -a.e. pair is Li-Yorke, but has no dense orbit. Conjecture 30. We think that if (I, f, λ) is conservative and has no acip, but instead the induced time k x associated to the induced map in Theorem 21 is non-integrable w.r.t. Lebesgue, and in fact the tail λ({x : k x > s}) ≥ 1/ log s, then the product system (I 2 , f 2 , λ2 ) is dissipative. 6. Li-Yorke Chaos in the Presence of Cantor Attractors In this section, we concentrate on C 2 unimodal maps with Cantor attractors A = ω(c) for the unique critical point c. If f is infinitely renormalizable (i.e., type (3) in Proposition 9), then the situation regarding Li-Yorke pairs is well-known: there are none. Instead, the attractor is Lyapunov stable and conjugate to an adding machine (, g). In other words, = {(ω j ) j≥1 : 0 ≤ ω j < p j }, for some sequence ( pi )i≥1 of integers pi ≥ 2 (where p1 · · · pi are the periods of the periodic intervals) such that it is equipped with product topology and the map g of “adding 1 and carry”: 0, 0, , . . . , 0, ωk + 1, ωk+1 , ωk+2 , . . . if k = min{i : ωi < pi −1}; g(ω1 , ω2 , . . . ) = (4) 0, 0, 0, 0, . . . if ωi = pi −1 for all i ≥ 1. The following classification is due to [BrJ, Prop. 5.1]. Proposition 31. Let f : I → I be a continuous map and x ∈ I . The system (ω(x), f ) is conjugate to some adding machine if and only if x is approximately but not asymptotically periodic, see Definition 1. However, there are several constructions leading to strange adding machines, i.e., (critical) omega-limit sets that are conjugate to adding machines, but not involving periodic intervals, see [BKM,BrJ,Br4]. In [Br4] it is shown that a strange adding machine can still be an attractor, but in this case it is a wild and not a solenoidal attractor.
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The dynamics on such attractors can frequently be understood in terms of generalized adding machines as done in [BKS] and in the proof of Theorem 34 below. Such generalized adding machines are based on the sequence of cutting time, which we will define now. Let Z n (x) be the n-cylinder, i.e., maximal interval containing x on which f n is monotone. Unless x ∈ ∪m 1. In this case Sk is the period of renormalization. Proof. By definition of the kneading map, n = S Q(k+1) is the smallest positive iterate such that f n ([c, c Sk ]) c, implying statement (a). Under the assumption that Q(k) → ∞, this means that |c Sk − c| → 0 and also |c S Q(k) − c| → 0 as k → ∞. Since D Sk = [c Sk , c S Q(k) ], the non-Contraction Principle shows that |Dn | → 0 as well. Minimality of ω(c) follows as in i.e., [Br2, Prop. 2] (which proves that c is persistently recurrent) and [Br2, Lemma 8]. This proves statement (b). For statement (c), observe that D1+Sl ⊂ D1+Sk for all l ≥ k + B because of the assumption maxk {k − Q(k)} ≤ B and statement (a). It follows that Dm+Sk+B ⊂ Dm+Sk Sk+B for all m ≥ 1, and ∪n≥1+Sk Dn ⊂ ∪n=1+S Dn . By definition, ω(c) ⊂ Cl ∪n≥1+Sk Dn ⊂ k Sk+B Sk+B Cl ∪n=1+S Dn = ∪n=1+S Dn as asserted. k k Part (d) is [Br2, Proposition 1(iii)].
Special types of unimodal maps are the Feigenbaum map (Sk = 2Sk−1 ) and the Fibonacci map (Sk = Sk−1 + Sk−2 ). We call f Fibonacci-like if {k − Q(k)}k is bounded. The proof of the existence of wild attractors was first established in [BKNS] for Fibonacci maps with sufficiently large critical order . In [Br3], this result was extended to C 3 unimodal maps with negative Schwarzian derivative, sufficiently large critical order
Lebesgue Measure of Li-Yorke Pairs for Integrable Maps
541
and eventually non-decreasing kneading map Q such that lim supk k − Q(k) ≤ B. If the cutting times increase more slowly than Fibonacci-like, then no unimodal map with finite critical order can have a wild attractor. Before we continue, let us give a short exposition how the existence of the wild attractor is proved for unimodal maps. There are two approaches, both based on a random walk on a Markov graph. In [BKNS,Br3,BHa], this Markov graph of I is based on preimages of the orientation reversing fixed point p. The states of that Markov system are pairs of intervals Uk ⊂ (Z S+k (c) ∪ Z S−k (c))\(Z S+k+B (c) ∪ Z S−k+B (c)) and ∂Uk belongs to the backward orbit of p. The induced Markov map defined by G|Uk = f Sk preserves ˆ p], where f −1 ( p) = { p, p}. ˆ Viewing the dynamics of G as a the partition {Uk } of [ p, random walk, we define “random variables” χn by χn (x) = k if G n (x) ∈ Uk .
(6)
It is then shown that this process has positive drift, i.e., the expectations (measured with respect to Lebesgue measure) E(χn+1 − k|χn = k) ≥ η > 0 uniformly in n and l. The second moments E((χn+1 − k)2 |χn = k) are shown to be bounded as well. It follows that χn (x) → ∞ for λ-a.e. x and hence G n (x) → c. Since Uk ⊂ (Z S+k (c) ∪ Z S−k (c)), f Sk (Uk ) ⊂ D Sk and since |Dn | → 0 as n → ∞, this means for the original map that f n (x) → ω(c) for λ-a.e. x, so A = ω(c) is an attractor. In [BHa, Theorem 5.2], a further conclusion is drawn from the positive drift, namely a Borel-Cantelli Lemma argument shows that for λ-a.e. x ∈ Bas(A), there is k0 = k0 (x) such that such that for all k ≥ k0 , / Uk for j > k. if G m (x) ∈ Uk , then G m+ j (x) ∈
(7)
In this paper, we will use a second approach from [Br1,BKS], where the Markov system is a disjoint union Iˆ = n≥2 Dn (the Hofbauer tower) for intervals Dn defined above. Let fˆ : Iˆ → Iˆ be defined by if n is not a cutting time; Dn+1 fˆ(Dn ) = D1+Sk (D1+S Q(k) \{c1 }) if n = Sk is a cutting time. Then i ◦ fˆ = f ◦ i, where i : Iˆ → I is the inclusion map, and fˆ is continuous, except at the points c ∈ D Sk , k ≥ 1, which are mapped to c1 ∈ D1+Sk . Since these are only countable many points, this has no effect on the Lebesgue typical behavior. The collection {Dn }n≥2 is a Markov partition of Iˆ, because fˆ maps each Dn to the union of partition elements (ignoring again the point c1 ∈ D1+S Q(k) ). Starting in some Dn , the subsequence intervals visited are unique determined up to the moment we reach some D Sk , where we have a choice between D1+Sk and D1+S Q(k) . Therefore it suffices to consider the transitions from interval El := D1+Sl realized by the S Q(l+1) -th image of f , see Fig. 1. It follows that {El }l≥0 is a Markov partition for the induced map F : l≥0 El → l≥0 El ,
F| El = fˆS Q(l+1) .
One can show that if f is renormalizable of period Sk , then l≥k−1 El is a trapping region for F, see part (d) of Proposition 32. Now we need to translate the results on positive drift to the current set-up. Write χˆ n (y) = l if F n (y) ∈ El .
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Fig. 1. The transitions from D1+Sl , l ≥ 1. Backward transitions go from D1+Sl to D1+S Q(l+1)
Lemma 33. If {χn }n≥0 has positive drift as in (6) (and bounded second moments), then for λ-a.e. y ∈ l≥0 El , we have χˆ n (y) → ∞. Furthermore, there is kˆ0 = kˆ0 (y) and ˆ then χˆ n+ j (y) > kˆ for all j > C k. ˆ C > 0 such that for all kˆ ≥ kˆ0 , if χˆ n (y) = k, Proof. The positive drift of {χn } ensures that for λ-a.e. x and k ∈ N, there is n k such that G n (x) ∈ ∪ j≥k U j for all n ≥ n k . Since Uk ⊂ Z S+k (c)∪ Z S−k (c), the image f Sk (Uk ) ⊂ D Sk and f 1+Sk (Uk ) ⊂ D1+Sk ∪D1+S Q(k) = E k ∪ E Q(k) . This means that y := f 1+Sk ◦G n (x) ∈ E k ∪ E Q(k) . If consequently G n+1 (x) ∈ Ul , l > k, then f Sl (y) ∈ El ∪ E Q(l) but the move from E k or E Q(k) up to El or E Q(l) in j≥0 E j involves passages through the intermediate states E i as well, but “lower” states E i , i < k − B ≤ min j≥k Q( j) are avoided. Therefore χˆ n ( f (x)) → ∞ for all n → ∞. For the second statement, observe that the passage from E k or E Q(k) up to El or E Q(l) requires u n iterates of F for some l − k − B ≤ u n ≤ l − k + B. Suppose G m (x) ∈ Uk , / Uk for j ≥ k. Take kˆ ∈ {k, Q(k)} such that y = k ≥ k0 (x) as in (7), then G m+ j (x) ∈
1+S m k f (G (x)) ∈ E kˆ . The iterates m+1, . . . , m+k of G correspond to m+k j=m+1 u j iterates of F. Each u j ≥ 1, and if χm+ j (x) < χm+ j−1 (x), then this single iterate of G corresponds to a single iterate of F, reducing the index of the state by at most B. If χm+ j (x) χm+ j−1 (x), then one iterate of G corresponds to many iterates of F, but if some of these iterates brings y above state E k+k ˆ B , then it would take more than k steps of G to return,
kˆ hence this will not occur. Thus, if the m+ j=m+1 u j iterates of F (corresponding k iterates
m+kˆ ˆ where the last inequality of G) keep y close to state E kˆ , then j=m+1 u j ≤ 2Bk ≤ 3B k, follows because k − B ≤ kˆ ≤ k. This proves the second statement for C = 3B. The dynamics of wild attractors has been investigated in [BKS]. In that paper, various combinatorial types are presented for which (A, f ) is semi-conjugate to a monothetic group (G, g), where g : G → G is an isometry for which every orbit is dense. The best known example goes back to Lyubich and Milnor [LM]; it is the Fibonacci map and its omega-limit set ω(c) factorizes over the golden mean circle rotation. In [BKS], similar examples are shown factorizing onto tori of any dimension, and even onto a solenoid. On the other hand, [BKS] gives examples for which (A, f ) is weak mixing with respect to the unique invariant probability measure μ supported on A. In [BHa], the simplest such example is shown to be Lebesgue exact as well. Let fp(x) := x − round(x) ∈ [− 21 , 21 ) be the signed distance of x to the nearest integer. Theorem 34. Assume that a unimodal map has a wild attractor with positive drift. If there exists ρ such that the cutting times satisfy
Lebesgue Measure of Li-Yorke Pairs for Integrable Maps
k
543
k max | fp(ρ Sk )| < ∞, i≥k−B
for B = lim supk k − Q(k), then λ2 (Dis) = 1. Proof. Step 1: Construction of the factor map. An enumeration scale is a symbolic system resembling an adding machine as in (4) based on, in this case, the sequence of cutting times. Any non-negative integer n can be written in a canonical way as a sum of
cutting times: n = j e j S j , where e j :=
1 if j = max{k; Sk ≤ n − 0 otherwise.
m>k em Sm },
In particular e j = 0 if S j > n. In this way we can code the non-negative integers as zero-one sequences with a finite number of ones: n → n ∈ {0, 1}N . Let E 0 = N∪{0} be the set of such sequences, and let E be the closure of E 0 in the product topology. This results in E := {e ∈ {0, 1}N ; ei = 1 ⇒ e j = 0 for Q(i + 1) ≤ j < i}. The condition in this set follows because if ei = e Q(i+1) = 1, then this should be rewritten to ei = e Q(i+1) = 0 and ei+1 = 1. It follows immediately that for each e ∈ E and j ≥ 0, e0 S0 + e1 S1 + · · · + e j S j < S j+1 .
(8)
We denote by g the standard addition of 1 by means of “add and carry”, cf. (4). Let n be the representation of n ∈ N ∪ {0} in the enumeration scale based on {Sk }k≥0 . Obviously g(n) = n + 1. Under the condition that Q(k) →
∞, g : E → E is continuous, and is invertible on E\{0}, see [BKS,GLT]. Since k | fp(ρ Sk )| < ∞, we can define a continuous projection πρ : E → S1 by πρ (e) =
ek fp(ρ Sk ) mod 1,
k
and πρ ◦ g = Rρ ◦ πρ for the circle rotation Rρ : x → x + ρ mod 1. At the same time, the map P : E → A defined as the continuous extension of P(n) = f n (c) satisfies P ◦ g = f ◦ P. We know from [BKS] that there is semi-conjugacy π = πρ ◦ P −1 :
A → S1 such that π ◦ f = Rρ ◦ π , provided k | fp(ρ Sk )| < ∞. A more direct way to construct π : A → S1 is by setting π(x) =
ρn mod 1 if x = cn , lim j→∞ ρn j mod 1 if x ∈ A\ orb(c) and (n j ) j∈N is such that x ∈ ∩ j Dn j .
In [BHa] it was shown how to extend the map πρ ◦ P −1 to a measurable factor map π˜ : Bas(A) → S1 . Here we will give a construction of π˜ which is more closely connected to [BKS].
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(E, g) P
@ π = πρ ◦ P −1
(A, f )
π
@ ρ R @
- (S1 , Rρ )
* π˜
(Bas(A), f )
For any x with χˆ n (x) → ∞, the number bn (x) defined as bn (x) := max{ j : f j (Z j (x)) = Dn } j
exists. If x ∈ f −k (c) for some k ≥ 0, then we need to write Z n± (x) for n > k, because x is the common boundary point of two cylinder sets, and f n (x) = cn−k ∈ Dn−k for n sufficiently large. So bn (x) is well-defined in this case too. Set π˜ n (x) := − fp(ρbn (x) − nk Sk ) mod 1 = −ρ(bn (x) − n) mod 1. (9) k
If n + 1 is not a cutting time, then bn+1 (x) = bn (x) + 1; in this case π˜ n+1 (x) = π˜ n (x). If n +1 = Sk is a cutting time, f bn (x) (Z bn (x) (x)) = D Sk −1 and f bn (x)+1 (Z bn (x) (x)) = D Sk , but bn+1 (x) can be strictly larger than bn (x) + 1. In this case, however, bn+1 (x) = bn (x) + 1 + d j S Q( j) j≥k
for some non-negative integers d j . Recall from Lemma 33 that for λ-a.e. x ∈ Bas(A), there is kˆ0 = kˆ0 (x) such that for all k > kˆ0 , if F m (x) ∈ E k , then F m+ j (x) ∈ / E k for all j > Ck. (10)
This means that j≥k d j ≤ k, and therefore bn (x) − n and bn+1 (x) − (n + 1) are sequences which differ by at most Ck entries and the indices of these entries are ≥ k − B. This means that by the definition of (9), |π˜ n (x) − π˜ n+1 (x)| ≤ Ck max | fp(ρ Si )|, i≥k−B
(11)
which is summable over k by assumption. It follows that {π˜ n (x)}n is a Cauchy sequence in S1 for λ-a.e. x ∈ Bas(A), and hence π˜ (x) := limn→∞ π˜ n (x) exists. Let us complete Step 1 by showing that π˜ ◦ f = Rρ ◦ π˜ for λ-a.e. x ∈ Bas(A). Assume that x ∈ / ∪ j f − j (c), then f (Z j (x)) = Z j−1 ( f (x)) for all sufficiently large j. Therefore bn ( f (x)) = bn (x) − 1, so substituting into (9) gives π˜ n ( f (x)) = π˜ n (x) + ρ for each n. In the limit, π˜ ◦ f = Rρ ◦ π˜ .
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545
Remark 35. It can be shown that π˜ is well-defined on A and coincides with π , but since it plays no role in Theorem 34, we will omit the proof. Step 2: The measure of Dis. We will show that if π˜ (x) = π˜ (y), then (x, y) form a distal pair. This is more involved than in Proposition 38 below, because π˜ |Bas(A) is not continuous. With the exception of a set of measure zero, we can assume that x and y ˜ = π(y) ˜ and take N ∈ N satisfy (10); let k1 = max{kˆ0 (x), kˆ0 (y)}. Suppose that π(x) and η > 0 such that |π˜ n (x) − π˜ n (y)| > 2η for all n ≥ N . Take k2 ≥ k1 so large that Ck maxi≥k−B | fp(ρ Si )| < η for all k ≥ k2 and C as in Lemma 33. Sk
+B
2 We know that A ⊂ ∪n≥Sk2 Dn , but by Proposition 32, part (c), ω(c) ⊂ ∪n=1+S Dn . k2 Take ε > 0 so small that every two intervals Dn , Dn , Sk2 < n, n ≤ Sk2 +B either are at least ε apart or their intersection has length at most ε. For λ-a.e. x ∈ Bas(A), we have that χˆ n (x) → ∞. Hence f i (x) ∈ D for i sufficiently large, even though D only contains a “one-sided” open neighborhood of ω(c). (In fact it is possible to prove that the same statement is true for all x ∈ Bas(A), but this is enough for our purposes.) Suppose now by contradiction that (x, y) is not distal. Then there is i ≥ N such that f i (x), f i (y) ∈ D and | f i (x) − f i (y)| < ε. So f i (x) and f i (y) belong to the same interval Dn for some n ≥ Sk2 , and taking i larger if necessary, we can assume that n is a cutting time. By (10), bn (x) − i and bn (y) − i are sequences which differ by at most Ck entries and the indices of these entries are ≥ k − B. Similar to (11), we have
|π˜ n (x) − π˜ n (y)| ≤ Ck max{| fp(ρ Si )| : k − B ≤ i ≤ 2k − B} < η. This contradiction to the choice of η and N proves that (x, y) is distal. Therefore proximal pairs (x, y) can only exist within fibers of π˜ . Finally, if W = π˜ −1 (s) with λ(W ) > 0 for some s ∈ S1 , then since π˜ ◦ f = Rρ ◦ π˜ and Rρ is invertible, it follows that f m (W ) ∩ f n (W ) = ∅ for all 0 ≤ m < n, and this contradicts the non-existence of strongly wandering sets. Therefore each fiber has measure zero. This completes the proof. Corollary 36. If Sk = Sk−1 + Sk−d for d = 2, 3, 4, and f is a map with cutting times {Sk }k≥0 and sufficiently large critical order so that A is a wild attractor with positive drift, then λ2 (Dis) = 1. Proof. We know from [BKS] that the dynamics (A, f ) is semi-conjugate to a minimal rotation on a d − 1-dimensional torus. If fact, the characteristic equation λd = λd−1 + 1 of the recursive relation Sk = Sk−1 + Sk−d has a leading root ρ which is a Pisot number, i.e., all its algebraic conjugates lie within the unit
disk. It follows easily that fp(ρ Sk ) is exponentially small in k so that the condition k k| fp(ρ Sk )| < ∞ is obviously satisfied, and Theorem 34 shows that λ2 (Dis) = 1. By defining : Bas(A) → Td−1 as (x) = (π˜ ρ (x), . . . , π˜ ρ d−1 (x)) (which is well-defined λ-a.e.), we obtain a factor map onto (Td−1 , Rρ,...,ρ d−1 ) with Haar measure, which is the maximal automorphic factor of (Bas(A), λ, f ). Proposition 37. Under the conditions of Theorem 34 and if there is a continuous factor of (A, f ) onto a d-dimensional torus Td (with d ≥ 1), then the fiber π˜ −1 (τ ) ⊂ Bas(A) for each τ ∈ Td contains an uncountable ε-scrambled set for some ε > 0. Proof. Assume for simplicity that d = 1 and take ρ ∈ R such that π˜ ◦ f = π˜ + ρ mod 1 for λ-e.a. x ∈ A. Because a zero-dimension set cannot be mapped injectively onto a set
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Fig. 2. Different loops ending at E κ j and E κˆ j respectively. Both loops contain the same path from E κˆ j to E κ j , depicted in the middle. Backward arrows go from El to E Q(l+1)
of higher dimension, see [E], the continuity of the factor map π : A → Td implies that π cannot be injective. Therefore we can find a = aˆ ∈ A such that f (a) = f (a). ˆ By Proposition 32, part (c), we can find, for every k, integers κ and κˆ with k < κ, κˆ ≤ k + B and N ≤ min(Sκ , Sκˆ ), such that D Sκ −N a and D Sκˆ −N a. ˆ Hence we can find two sequences (κ j ) j∈N and (κˆ j ) j∈N with |κ j − κˆ j | ≤ B, but possibly κ j+1 κ j , and another sequence (N j ) j∈N such that D Sκ j −N j a and D Sκˆ −N j a. ˆ j Recall the Markov map was defined as F : l≥0 El → l≥0 El . For each j, we will create loops from E κ j to itself and from E κˆ j to itself, as indicated in Fig. 2. Both loops require the same steps under F, only arranged in a different order, hence they involve the same number s j of iterates of f . Because lim supk k − Q(k) ≤ B, both loops involve no more than 2B steps, and the width (highest vertex minus smallest vertex) is less than 2B as well. Assume that κ j < κˆ j and E κ j ⊃ E κˆ j . (The other three cases can be treated similarly.) Since {El }l≥0 is a Markov partition for F, there are intervals J j ⊂ E κ j and Jˆj ⊂ E κˆ j
such that f s j : J j → E κ j and f s j : Jˆj → E κˆ j are diffeomorphic and onto. There is a similar interval Kˆ j ⊂ E κˆ representing the path from E κˆ to E κ j , that is if the path from j
j
E κˆ j to E κ j requires t j iterates of f , then f t j : Kˆ j → E κ j is diffeomorphic and onto. Combining the two, we find intervals H j ⊂ J j such that ⎧ s ⎪ ⎨ (a) f j (H j ) = Kˆ j ⊂ E κˆ j ⊂ E κ j , (b) f s j +t j (H j ) = E κ j , and ⎪ ⎩ (c) f s j −(N j +1) (H j ) contains, or is close to, a.
Similarly, we can find Hˆ j ⊂ Jˆj such that ⎧ ˆ f s j ( Hˆ j ) = Kˆ j ⊂ E κˆ j , ⎪ ⎨ (a) ˆ f s j +t j ( Hˆ j ) = E κ j , and (b) ⎪ ⎩ (c) ˆ f s j −(N j +1) ( Hˆ j ) contains, or is close to, a. ˆ Let ⊂ {0, 1}N be an uncountable scrambled subset of the full shift. The idea is now, for each τ ∈ T and each σ ∈ , to find a point x ∈ π −1 (τ ) such that D Sκ j −N j a if σ j = 0, rj (12) f (x) ∈ D Sκˆ −N j aˆ if σ j = 1, j
where the sequence r j depends of t but not on σ . Start with some y ∈ D2 = E 0 with π˜ (y) = τ + ε, where ε will be determined later. Then, when the orbit of y under iteration of F goes from E κ j to E κˆ j , we insert one of
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the loops as in Fig. 2 according to whether σ j = 0 or 1, and the extra path from E κ j to E κˆ j . That is, when q1 is such that f q1 (y) ∈ E κ j , we insert one of the extended loops, both taking s1 + t1 iterates, and iterate r1 := q1 + s1 − (N1 + 1) brings the path close to a or a, ˆ depending on whether σ1 = 0 or 1. Then, when y visits E κ2 , the new extended loop takes s1 + t1 iterates more to reach it; call this number q2 , insert the appropriate extended loop of s2 + t2 iterates, and find that after r2 = q2 + s2 − (N2 + 1) iterates, the path will be close to a or a, ˆ etc. Due to the Markov property, there is some x ∈ E 0 whose infinite path under F is precisely
the path we have created, so x satisfies (12). Furthermore, the values bn (x) = bn (y) + j,q j +B 0. On the other hand, lim j→∞ | f q j (x) − f q j (x )| = 0, because f q j (x) and f q j (x ) both belong to E κ j or to ˆ subset of π −1 (τ ), as required. E κˆ j Therefore {x(σ ) : σ ∈ } is an |a − a|-scrambled Proposition 38. Let f be a unimodal map with kneading map Q(k) = max{k − d, 0} for d = 2, 3, 4, and let μ denote the unique invariant probability measure supported by ω(c). Then • μ2 -a.e. pair of points is distal; • if d = 2 (the Fibonacci map) then ω(c) contains no Li-Yorke pair, and the only asymptotic pairs (x, y) are such that f n (x) = f n (y) = c (only one such pair for each n ≥ 1); • if d = 3, 4 then there are uncountably many Li-Yorke pairs in ω(c). Proof. As in Corollary 36, there is a continuous map π : ω(c) → Td−1 onto the d − 1-dimensional torus and an irrational rotation R := Rρ,...,ρ d : Td → Td such that π ◦ f = R ◦ π . If π(x) = π(y) then (x, y) is distal, as in Corollary 36. It follows from [BKS] that this happens for μ2 -a.e. (x, y). Furthermore, π −1 (b) consists of at most d points a1 , . . . , ad for any b ∈ Td−1 . If there are indeed d distinct points, then there is n ≥ 0 such that f n (a1 ) = f n (a2 ) = · · · = f n (ad ) = c, cf. [Br4]. For the Fibonacci map, this accounts for all non-distal pairs. For d = 3, 4, other non-singleton fibers π −1 (b) are possible, and Li-Yorke pairs exist within such fibers. They are related to incidences in the substitution shift description, as described implicitly in [BD]. The question is whether the situation is the same for the “next” Fibonacci-like map with Q(k) = max{k − 5, 0}. In this case, the system of (ω(c), f ) with its unique probability measure is weak mixing, so there is no continuous (or even measurable) factor map
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onto a group rotation. The difference with the previous cases is that the characteristic equation of the recursive relation 0 = λ5 − λ4 − 1 = (λ2 − λ + 1)(λ3 − λ − 1) is reducible, and more decisively, its leading root is not a Pisot number. The following curiosity about the cutting times holds in this case: ⎧ ⎨ +1 if k ≡ 2, 3 mod 6; (13) Sk = Sk−2 + Sk−3 + −1 if k ≡ 5, 0 mod 6; ⎩ 0 if k ≡ 1, 4 mod 6. Note that the same algebraic curiosity holds for any characteristic equation √λ6m−1 − λ6m−2 − 1 = 0, because in each such case, λ2 − λ + 1 (with solutions λ = 1±i2 3 on the unit circle) divides the equation. As an example, the case m = 2 gives: λ11 − λ10 − 1 = (λ2 − λ + 1)(λ9 − λ7 − λ6 + λ4 + λ3 − λ − 1), and one can indeed check that
⎧ ⎨ +1 Sk = Sk−2 + Sk−3 − Sk−5 − Sk−6 + Sk−8 + Sk−9 + −1 ⎩ 0
if k ≡ 2, 3 mod 6; if k ≡ 5, 0 mod 6; if k ≡ 1, 4 mod 6.
Proposition 39. If Sk = Sk−1 + S Q(k) for Q(k) = max{0, k − 5} and f is a map with cutting times {Sk }k≥0 and a wild attractor with positive drift, then λ2 (LYε ) = 1 for some ε > 0. Proof. As pointed out before, it was shown in [Br3] that A = ω(c) is a wild attractor, and from [BHa] it follows that dynamics on the basin of the attractor is Lebesgue exact. Thus Proposition 27 implies that λ2 (LY) = 1. In fact, the construction of Proposition 37 can also be used here to show that there is ε > 0 such that λ2 -a.e. pair belongs to LYε . Example 40. Let f be a unimodal map with cutting times satisfying S0 = 1, S1 = 2, S2 = 3, S3 = 4, S4 = 6, S5 = 8, S6 = 10, S7 = 12 and Sk = Sk−1 + Sk−5 for k ≥ 8. This means that the cutting times Sk are even for k ≥ 3 and eventually are twice the numbers occurring in the example of Proposition 39. Assume also that the critical order of is so large that f has a wild attractor A. Then A decomposes into two disjoint Cantor sets A0 and A1 which are permuted by f . Note, however, that f is not renormalizable, see Proposition 32, and therefore f is topologically mixing on [c2 , c1 ]. Let B0 and B1 be disjoint neighborhoods of A0 and A1 ; for example we can take B0 = [c2 , c14 ] ∪ [c4 , c6 ] and B1 = [c3 , c15 ] ∪ [c5 , c1 ]. Every point in the basin of A will eventually be trapped in B0 ∪ B1 . But every pair (x, y) with x ∈ B0 and y ∈ B1 such that orb(x), orb(y) ⊂ B0 ∪ B1 is distal. On the other hand f 2 |A0 and f 2 |A1 behave like the example of Proposition 39, so λ2 -a.e. every pair (x, y) ∈ B0 × B0 (or (x, y) ∈ B1 × B1 ) such that orb(x), orb(y) ⊂ B0 ∪ B1 is Li-Yorke.
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Proof of Theorem E. From Theorem C we know that λ2 (Asymp) = 0. Also, if some x ∈ Bas(A) is approximately periodic, then by Proposition 31, A is conjugate to an adding machine, so all points in Bas(A) are approximately periodic. By Proposition 6, Bas(A) contains no Li-Yorke pairs. Therefore (a)-(d) are the only possibilities, and they all occur: (a) The strange adding machine case as wild attractor, see [Br4]. (b) The Fibonacci-like map with kneading map Q(k) = max{k − d, 0}, d = 2, 3, 4, see Theorem 34 and Corollary 36. (c) The Fibonacci-like map with kneading map Q(k) = max{k − 5, 0}, see Proposition 39. (d) See Example 40. Proposition 37 implies the existence of ε-scrambled sets in the fibers π˜ −1 (τ ) for all τ ∈ S1 and factor maps π˜ in case (b), and for cases (c) and (d), the ε-scrambled set is also immediate. Li-Yorke sensitivity follows as well. Remark 41. In case (a) and (b) of Theorem E, (A, μA , f ) is not weakly mixing. Instead, there is an f 2 -invariant set of positive μ2 -measure that is bounded away from the diagonal of A2 . We expect that case (c) always corresponds to the weakly mixing case, cf. Proposition 39, and in particular, μA × μA -a.e. pair (x, y) has a dense orbit in A × A, and hence is Li-Yorke. Remark 42. In case (a), points x in the basin of A have a distinct target point tx ∈ A such that dist( f n (x), f n (tx )) → 0, see Proposition 6. In case (b), such target points tx ∈ A do not exist in general, cf. the proof of Proposition 37 and also [BHa, Remark 2]. 7. Appendix In this Appendix, we prove the more technical results of the paper, divided into four parts. First we give a result on the structure of solenoidal sets of Feigenbaum type, next we prove Theorem 20, then we formulate an improved C 3 Koebe distortion lemma, and finally we prove Theorem 21. 7.1. Neighborhoods of Solenoidal Sets. Assume that S is a solenoidal set and cyc(K ) is a solenoidal cycle of period r containing S. Let Si = S ∩ f i (K ), 0 ≤ i < r . Since the critical points in cyc(K ) belong to S, the convex hulls Ji of Si form a cycle of intervals, that is, f (Ji ) = Ji+1 for every i (with this we also mean f (Jr −1 ) = J0 ). We call it the r -minimal solenoidal cycle covering S. We emphasize that the intervals Ji are pairwise disjoint (because a solenoid contains no periodic points). Moreover, if they are ordered in the real line as Ji1 < Ji2 < · · · < Jir , then there is a union of periodic orbits −1 such that Ji1 < p1 < Ji2 < p2 < · · · < pr −1 < Jir (see [MT or AJS]). P = { pi }ri=1 The existence of these periodic orbits of smaller period interlaced among the intervals Ji allows us to prove immediately that the union L i of all periodic intervals of period r containing Ji is also periodic of period r . Clearly, f (L i ) ⊂ L i+1 and f (∂ L i ) ⊂ ∂ L i+1 for every i. However, although the intervals L i have pairwise disjoint interiors, they do not form a cycle of intervals because f (L i ) = L i+1 need not hold. In fact (L i )ri=0 is the pullback chain of L r := L 0 along J0 , J1 , . . . , Jr = J0 . Let Mi = Int L i . We call −1 Mi the r -shell covering S. Then M is nice and Ji ⊂ Mi for every i. M = ri=0
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Fig. 3. The graph of f r : M0 = (a, b) → M0 with J and R and relevant points 2 (I ) and let S be a solenoidal set of Feigenbaum type. Let Proposition 43. Let f ∈ Cnf −1 i f (J ), and let T be the r -minimal solenoidal cycle covering S for some r, T = ri=0 r −1 M = i=0 Mi be the r -shell covering S, with f i (J ) ⊂ Mi for every 0 ≤ i < r . Then there is ξ = ξ( f ) > 0 such that f i (J ) is ξ -well centered in Mi for every i.
Proof. Since S is a solenoid, there is a turning point c ∈ S. There is no loss of generality in assuming that c ∈ J . Also, we can assume that r is large enough so that each interval Mi contains at most one critical point of f and there are no critical points in Cl M outside T . Hence (Mi )ri=0 is the pullback chain of Mr := M0 along J, f (J ), . . . , f r (J ) = J . Let cyc(R) be the 2r -minimal solenoidal cycle covering S, with c ∈ R. We can for example assume that [u, v] = R, f r (R) = [w, z] and c ∈ R is a local maximum. Note that J is the convex hull of R ∪ f r (R). Observe that there is a periodic point p of period r such that f i ( p) lies between f i (R) and f i+r (R) for every i. Since T is a solenoidal cycle, orb( p) is hyperbolic repelling. In fact, since f is monotone on each of the intervals connecting f i (R) and f i+r (R), it is the only r -periodic orbit in T and f r ([u, p)) = ( p, z], f r (( p, z]) = [u, p). Then (τc ( p), p) is clearly the only nice periodic interval of period 2r containing c. By [SV, Theorem A’(1)] there are an integer s > 0, a number ξ0 = ξ0 ( f ) > 0 and nice periodic intervals N m c of periods s2m , m ≥ 0, such that N m+1 is ξ0 -well inside N m for every m. We have shown in the previous paragraph that if s2m is large enough, then there is just one nice periodic interval of period s2m containing c. Thus we can assume that M0 = N m and (τc ( p), p) = N m+1 for some m, with r = s2m . Let q > p be the closest point to p satisfying f r (q) = τc ( p). Then V = ( p, q) is just the pullback of U = (τc ( p), p) along f r (R), f r +1 (R), . . . , f 2r (R) = R. Since U is ξ0 -well inside M0 , there is ξ1 = ξ1 ( f ) such that V is ξ1 -well inside M0 , due to Corollary 14. Let W = (q, b] with b the right endpoint of M0 . Then ξ0 |U | ≤ |V | + |W | and ξ1 |V | ≤ |W |, hence ξ0 ξ1 (|U | + |V |) ≤ |W |. 1 + ξ0 + ξ1 Then (τc (q), q) is well centered in M0 and so is any subinterval of (τc (q), q). In particular, J is well centered in M0 .
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To finish the proof we must show that every interval f i (J ) is well centered in the corresponding interval Mi . Note that we cannot directly use Corollary 14 because it only guarantees that f i (J ) is well inside Mi . Instead we proceed as follows. According to [SV, Lemma 2], Mr := M0 is well inside an interval G r which contains at most e = 2d+1 + 3 of the intervals f i (M0 ), 0 ≤ i < r , with d = # Crit. Therefore it also contains at most e of the intervals Mi . Now [SV, Lemma 3] implies that the pullback chain (G i )ri=0 of G r along M0 , . . . , f r (M0 ) has order bounded by 2(e + d(e + 2)) + 1. Hence, by Corollary 14, Mi is well inside G i for every i and, additionally, if some iterate f l maps diffeomorphically Mi onto Mi+l , then this diffeomorphism has bounded distortion. Thus, if f i+l (J ) is well centered in Mi+l , f i (J ) is well centered in Mi . Using now Lemma 11 and recalling that J = f r (J ) is well centered in M0 = Mr , we conclude that every interval f i (J ) is well centered in Mi as we set out to show. 7.2. Proof of Theorem 20. Later on we will apply the lemma below to the set Q = Crit 2 (I ); recall that we are assuming that f has no of critical points of our map f ∈ Cnf periodic critical points, see (2). Lemma 44. Let f : I → I be a multimodal map without wandering intervals and let Q ⊂ I \∂ I be a finite set containing no periodic points. Let x ∈ Q. Then there is an arbitrarily small nice interval J x such that orb(Q) ∩ ∂ J = ∅ and dist(orb(∂ J ), Q) > 0. n Proof. Let Q = ∞ n=−∞ f (Q). We claim that the set P = AsPer \Q is dense in I . This implies the lemma. Indeed, let ε > 0 be small. Since x is not periodic, there is no loss of generality in assuming that orb(Q) ∩ (x − ε, x + ε) does not contain any periodic point. Take points aˆ ∈ (x − ε, x) ∩ P, bˆ ∈ (x, x + ε) ∩ P, and let a < x < b be the ˆ closest to x from both sides. We emphasize that both points from Cl(orb(a) ˆ ∪ orb(b)) ˆ aˆ and b are asymptotically periodic and x is not periodic, so a and b are well defined. ˆ this is obvious because ˆ ∪ orb(b) Moreover, (a, b) is nice. Also, a ∈ / Q . If a ∈ orb(a) ˆ ˆ a, ˆ b∈ / Q . If a ∈ / orb(a) ˆ ∪ orb(b), then a must belong to a periodic orbit attracting either ˆ and again a ∈ orb(a) ˆ or orb(b), / Q because Q contains no periodic points and neither does orb(Q) ∩ (x − ε, x + ε). Similarly, b ∈ / Q . We have shown that orb(Q) ∩ {a, b} = ∅ and (orb(a) ∪ orb(b)) ∩ Q = ∅. Since a and b are asymptotically periodic and Q contains no periodic points, the property (orb(a) ∪ orb(b)) ∩ Q = ∅ implies in fact that dist(orb(a) ∪ orb(b), Q) > 0. Thus J = (a, b) is the small nice interval we are looking for. We prove that every interval K intersects P. If the intervals { f n (K )}n are pairwise disjoint, then the absence of wandering intervals for f implies that these intervals are attracted by a periodic orbit. Since f is multimodal, the set of points in K ∩ Q is countable. Thus K intersects P. Now assume that f n (K ) ∩ f m (K ) = ∅ for some n < m. Let k = m − n. Using again that f is multimodal, we get that T = Cl( r∞=0 f n+r k (K )) is a nondegenerate interval. Moreover, it is invariant for f k . If f k |T has finitely many periodic points, then all points from T are asymptotically (or eventually) periodic, see e.g. [BC, p. 127], so all points from K are asymptotically periodic as well and K ∩ P = ∅ as before. If f k |T has infinitely many periodic points, there is a family of disjoint periodic orbits {O j }∞ j=1 such that orb(K ) ∩ O j = ∅ for every j. If j is large enough, then orb(Q) ∩ O j = ∅. Let y ∈ K be a preimage of such O j . Then y ∈ / Q , which finishes the proof. The next proposition strengthens [CL, Prop. 5]. In what follows, we say that a nice set V is ξ -nice if all return domains to V are ξ -well inside the components of V containing
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them. Sometimes we say that V is uniformly nice if it is ξ -nice for some constant ξ depending only on f . We denote Z = {x ∈ I : x ∈ / orb(Crit), dist(orb(x), Crit) > 0}. Observe that f −1 (Z ) = Z and that (reasoning as in the proof of Lemma 44) Z is dense 2 (I ). We denote by Feig the critical points of Feigenbaum type. for f ∈ Cnf 2 (I ). Then there are ξ = ξ ( f ) > 0 and, for every ε > 0 Proposition 45. Let f ∈ Cnf 0 0 and c ∈ Crit \ Feig, open intervals c ∈ Vc with |Vc | < ε, such that V = c∈Crit \ Feig Vc is ξ0 -nice and ∂ V ⊂ Z .
Proof. In Cai and Li’s version of this proposition, all critical points of solenoidal type (not only those of Feigenbaum type) are excluded and no additional properties on ∂ V are obtained. Nevertheless, the proof remains very much the same. We sketch it below, emphasizing the specific points where it must be modified. First of all, we prove: Claim 1. There is ξ1 = ξ1 ( f ) such that if c ∈ Crit \ Feig, then there are arbitrarily small ξ1 -nice intervals J containing c such that ∂ J ⊂ Z . This is [CL, Cor. 3], but including also solenoidal critical points of non-Feigenbaum type. If c is recurrent, then Claim 1 follows immediately from Propositions 15 and 16. In fact, if c is not solenoidal, and we take as the starting interval I0 in Proposition 16 a small nice interval containing c with ∂ I0 ⊂ Z (which is possible by Lemma 44), then the central return domain Im to Im−1 satisfies ∂ Im ⊂ Z as well for all m. Thus if m is large enough and Im+1 is well inside Im , then, by Proposition 15, Im+1 is the uniformly nice interval we need. If c is solenoidal, then Propositions 15 and 16 again imply that there is an arbitrarily small uniformly nice interval J containing c. Observe that if J is sufficiently small, then it is contained in a solenoidal cycle very close to ω(c), which in particular implies that the orbits of its endpoints cannot accumulate on any critical point outside ω(c). Also, since J is nice and ω(c) is minimal, they cannot accumulate on ω(c) either. Thus ∂ J ⊂ Z . If c is not recurrent then the argument from [CL, Cor. 3] applies without any significant changes. Namely, let I c be a small nice interval with ∂ I ⊂ Z . We can assume that c ∈ / D(I ). Let δ be the minimal length of the components of I \{c} and take an interval (a , b ) ⊂ (c − δ/2, c + δ/2) with a , b ∈ Z . This is possible by Lemma 44. If a ∈ / D(I ), define a = a . Otherwise, let K be the return domain to I containing a and let a be the endpoint of K in (a , c). The point b is defined similarly. Then J = (a, b) is 1/2-well inside I , ∂ J ⊂ Z and ∂ J ∩ D(I ) = ∅. If x ∈ J ∩ D(J ), then the return domain L to J containing x is well inside the return domain L to I containing x by Proposition 15. Since ∂ J ∩ D(I ) = ∅, we obtain L ⊂ J . Then L is well inside J and J is uniformly nice. k Vi Claim 2. Let c1 , c2 , . . . , ck ∈ Crit and Vi ci be nice intervals such that V = i=1 is a ξ -nice set and ∂ V ⊂ Z . Then there is ξ = ξ (ξ ) > 0 such that the following hold: (1) For each 1 ≤ i ≤ k, there exist nice intervals Wi ⊃ V˜i ci such that – V˜i is ξ -well inside Wi and Wi is ξ -well inside Vi ; k – ∂ V˜i ∩ D(V ) = ∅ and ∂ Wi ∩ D(V ) = ∅. In particular, i=1 V˜i is a nice set.
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(2) For each x ∈ Vi \ Cl V˜i , there is a nice interval Jx x such that Jx is ξ -well inside Vi and Jx ∩ V˜i = ∅, ∂ Jx ∩ D(V ) = ∅. (3) The endpoints of all intervals above belong to Z . Claim 2 is exactly [CL, Lemma 3], except that we additionally request ∂ V ⊂ Z and get the extra property (3) in return. The proof requires no changes: only, instead of defining the auxiliary interval ( p , q ) = (a − ξ (b − a)/4, b + ξ (b − a)/4) for V˜i = (a, b), we choose p , q ∈ Z with p ∈ (a − ξ (b − a)/3, a − ξ (b − a)/4) and q ∈ (b + ξ (b − a)/4, b + ξ (b − a)/3). We are now in position to prove Proposition 45. This is done inductively. Let Crit \ Feig = {c1 , . . . , cm }. If m = 1, then this is just Claim 1. Assume that we have k Vi constructed intervals Vi ci with |Vi | < ε and ∂ Vi ⊂ Z , 1 ≤ i ≤ k, such that i=1 ˜ is ξk -nice for some constant ξk > 0. We will show that there are smaller intervals Vi ci k+1 with ∂ V˜i ⊂ Z , and a constant ξk+1 depending only on ξk , such that i=1 V˜i is ξk+1 -nice. The intervals V˜i , 1 ≤ i ≤ k, are those from Claim2. To define V˜k+1 and conclude k Vi ), then Cai and Li’s proof the proof, two cases must be considered. If ck+1 ∈ D( i=1 works without any changes (it uses Claim 2 in its full extension). k If ck+1 ∈ / D( i=1 Vi ), then we need to find intervals ck+1 ⊂ V˜k+1 ⊂ Vk+1 with k+1 ˜ |Vk+1 | < ε, Vk+1 well inside Vk+1 and ∂ V˜k+1 ⊂ Z , such that i=1 Vi is nice and k+1 ∂ V˜i ∈ / D( i=1 Vi ) for every 1 ≤ i ≤ k + 1. We define Vk+1 and V˜k+1 . Since V˜k+1 is well inside Vk+1 and ∂ V˜k+1 ⊂ Z , we only k+1 / D( i=1 Vi ). (Recall that the endpoints of the interneed to show that ∂ Vk+1 , ∂ V˜k+1 ∈ vals Vi , V˜i , i ≤ k, belong to Z , hence their orbits cannot visit Vk+1 if it is sufficiently small.) As in the proof of Claim 1, three possibilities arise for ck+1 . The simplest case is when V˜k+1 is the return domain to ck+1 is solenoidal. Then Vk+1 is defined as in Claim 1 and k / D( i=1 Vi ). Vk+1 containing ck+1 ; everything works because ck+1 ∈ Now assume that ck+1 is not solenoidal. Starting from a small interval (a , b ) ck+1 with a , b ∈ Z and repeating the reasoning in Case 2 of Cai and Li’s proof, we find an k Vi is nice. If ck+1 interval ck+1 ∈ (a, b) ⊂ (a , b ) such that a, b ∈ Z and (a, b) ∪ i=1 is recurrent, we take I0 = (a, b) in Proposition 16, and define accordingly the intervals k k Vi is nice for every m because ck+1 ∈ / D( i=1 Vi ). Now it suffices Im . Then Im ∪ i=1 ˜ to fix m such that Im is uniformly nice and define Vk+1 = Im and Vk+1 = Im +1 . If ck+1 is non-recurrent, then we find an interval J ck+1 , similarly as we did in Claim 1, such k Vi ) = ∅, and J is 1/2-well inside (a, b). Then that ∂ J ⊂ Z , ∂ J ∩ D((a, b) ∪ i=1 Vk+1 = (a, b) and V˜k+1 = J are adequate to our purposes. k+1 ˜ Let us finally show that with this choice, V˜ = i=1 Vi is uniformly nice. Let ˜ ˜ ˜ x ∈ Vn ∩ D(V ), say φV˜ (x) ∈ Vl . Let 0 ≤ s ≤ t be the entry time of x to V˜l and the return time of x to V˜ , respectively, and denote the return domain to V˜ and the entry domain to Vl containing x by J and K respectively, so x ∈ J ⊂ K . By Proposition 15, the pullback H of V˜l along f s (x), . . . , f t (x) is well inside Vl . Note that J is the pullback of H along x, . . . , f s (x) and K is the pullback of Vl along x, . . . , f s (x). Hence J is k+1 / D( i=1 Vi ), K is contained in V˜n , so J is well inside K by Corollary 14. Since ∂ V˜n ∈ well inside V˜n . This proves that V˜ is uniformly nice. Proof of Theorem 20. Fix ε > 0. We define the sets c ∈ Uc ⊂ Vc and Wc as follows. If c ∈ Crit \ Feig, then Vc is the component of the set V from Proposition 45 containing c.
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Write Vc = (a, b) and say, for instance, that (c, b) is the smallest component of Vc \{c}. Define the convex combinations v0 :=
c + ξ0 b 1 + ξ0
and
vi :=
vi−1 + ξ0 b , i = 1, 2, 3, 1 + ξ0
with ξ0 = ξ0 ( f )/2 and ξ0 ( f ) the constant from Proposition 45. Take v ∈ (v0 , v1 ) ∩ Z close to v0 . If v ∈ / D(V ), then set v = v ; otherwise let v be the right endpoint of the return domain to V containing v , and by having chosen v close to v0 , we obtain v0 < v ≤ v < v1 . This gives v ∈ ((v0 , v1 ) ∩ Z )\D(V ), and, similarly, there is w ∈ ((v2 , v3 ) ∩ Z )\D(V ). If c ∈ D(V ), let u denote the left endpoint of the return domain to V containing c; if c ∈ / D(V ), take u 0 = c − (v0 − c) = 2c − v0 and find as before u ∈ (u 0 , c) belonging to Z \D(V ). Finally we define Uc = (u, w) and Wc = (v, w). For critical points of Feigenbaum type we rely on Proposition 43. More precisely, for every Feigenbaum solenoidal set S, we find a minimal solenoidal cycle T = cyc( Jˆ) = s−1 i ˆ i=0 f ( J ) containing S whose constituting intervals are small enough to ensure that |Vc | < ε for the components of the corresponding set V . Then the intervals Uc are those s−1 Mi of T containing points from Crit (that is, from Feig). To from the shell M = i=0 define the sets Vc assume, after reordering, that M0 = Ms is the smallest of all intervals Mi . Since the intervals f i ( Jˆ) are well centered in the intervals Mi by Proposition 43, there is an interval G s such that Ms is well inside G s and G s intersects no other interval from the solenoidal cycle T than Jˆ = f s ( Jˆ). Pulling back G s along f ( Jˆ), . . . , f s ( Jˆ), we construct similarly intervals G i , i = 1, . . . , s, such that Mi is well inside G i and G i ∩ T = f i ( Jˆ). The intervals Vc are those intervals G i containing points from Crit. To define the sets Wc we take a boundary point q of M0 with f s (q) = q (in fact the proof of Proposition 43 shows that it is possible that f s/2 (q) = q). For every 0 ≤ i < s, let u i denote the middle point between f i (q) and the endpoint of f i ( Jˆ) closest to f i (q). If Mi contains a critical point, let L i be the interval with endpoints f i (q) and u i ; if not, let L i = Mi . Finally, let L i ⊂ L i be the largest interval with endpoint f i (q) such that f j (L i ) ⊂ L i+ j for all 0 ≤ j < s with indices taken mod s, and hence f s | L i is a diffeomorphism. Now the intervals Wc are those intervals L i contained in intervals Uc = Mi intersecting Crit. We show that (i)-(iii) in Theorem 20 hold. Clearly, the construction implies the existence of a number ξ = ξ( f ) > 0, thus not depending on ε, such that Uc is ξ -well inside Vc for every c ∈ Crit. Moreover, since the endpoints of all sets Wc , Uc , Vc , c ∈ Crit \ Feig, belong to Z \D(V ), we can assume that they do not belong to D(Vc ), c ∈ Feig, either. If c ∈ Feig, then the niceness and invariance of shells still guarantees ∂Uc ∩ D(Uc ) = ∅ for every c ∈ Crit and ∂Uc ∩ D(Vc ) = ∅for every c ∈ Crit not belonging to the same solenoidal set as c . In particular, U = c∈Crit Uc is nice. This proves Theorem 20(i). Now let J be an entry domain to U , say φ| J = f j | J and φ(J ) = Uc . Then f j | J is a diffeomorphism. We want to show that f j | J extends to a diffeomorphism f j | K : K → Vc . Assume by contradiction that this is not the case. Then there is K ⊃ J such that f j | K is a diffeomorphism, one of the endpoints a of K satisfies c = f n (a) ∈ Crit for some 1 ≤ n < j, and f j ({a} ∪ K ) ⊂ Vc . Let b ∈ K such that f n (b) ∈ ∂Uc (here we use that f n (J ) does not intersect U ). We can assume that both c and c belong to the same Feigenbaum solenoidal set S, because otherwise ∂Uc ∩ D(Vc ) = ∅, which contradicts f j (b) ∈ Vc . Let T = cyc( Jˆ) be the
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minimal solenoidal cycle for S we used earlier to construct the sets Uc , Vc . Then there is i ∈ N such that f j (J ) = Uc ⊃ f i ( Jˆ). Since f j (a) = f j−n (c ) ∈ S, there is i ∈ N such that f j (a) ∈ f i ( Jˆ). But f j |{a}∪K is a homeomorphism, so f i ( Jˆ) and f i ( Jˆ) are different. This is impossible, because by its definition Vc intersects exactly one interval from T . We have shown that if J is an entry domain to U , then f j | J : J → Uc extends to a diffeomorphism f j | K : K → Vc . Since Uc is well inside Vc , there is κ = κ( f ) such that f j | J has distortion bounded by κ by the C 2 Koebe Principle (Proposition 15). This finishes the proof of Theorem 20(ii). It remains to prove Theorem 20(iii). If c ∈ Crit \ Feig, then the definition of Wc easily implies that it is not too small compared to Uc . On the other hand, it is not too large compared to the subinterval of Uc between c and Wc , so f |Wc has bounded distortion because c is non-flat. Since ∂ Wc ∩ D(U ) = ∅, Theorem 20(iii) holds in this case. Assume now that c ∈ Feig and Uc is one of the above intervals Mi1 , with Wc = L i 1 . Put kc = s and let i 1 < i 2 < · · · < i t < i 1 +s be the indices i such that Mi contains some critical point. As shown in the proof of Proposition 43, each map f ir +1 −ir −1 | Mir +1 has bounded distortion. Also, each map f | L ir has bounded distortion because |L ir | is less than half the distance from f ir (q) to the critical point in Mir and non-flatness applies. Hence f kc |Wc = f s | L i has bounded distortion. Recall that all periodic orbits in the 1
intervals Mi are repelling, and the same is true for orb(q). Then f kc (Wc ) ⊃ Wc . Finally, let us show that Wc is not too small compared to Uc . From its definition, L i1 is not too small compared to Uc . Let A j denote the largest interval with endpoint f j (q) such that f k (A j ) ⊂ L j+k for all 0 ≤ k ≤ i j − i j−1 ; hence A1 = L i1 and At = Wc . We claim that A j+1 is not too small compared with A j . Indeed, L i j+1 is not too small compared to Mi j+1 , so a fortiori is not too small compared to f i j+1 −i1 (A j ). Since f i j+1 −i1 | A j has bounded distortion and A j+1 = A j ∩ f −(i j+1 −i1 ) (L i j+1 ), A j+1 cannot be too small compared to A j either. This concludes the proof of Theorem 20. 7.3. A C 3 Koebe Distortion Lemma. 3 (I ). Then for any ξ > 0 and k ≥ 0, there is ξ = ξ (ξ, k, f ) > 0 Lemma 46. Let f ∈ Cnf such that the following statement holds: Let (Hi )li=0 ⊂ (G i )li=0 be chains such that (G i )li=0 has order at most k and G l is a small nice interval close enough to Crit. If Hl is ξ -well inside G l , then H0 is ξ -well inside G 0 . Moreover, if k = 0, then there is κ = κ(ξ, f ) > 0 such that f l | H0 has distortion bounded by κ.
A slightly weaker version of this lemma (requiring that the intervals G i are not too close to parabolic periodic points) appears in [LS], which in turn refers to [SV, Theorem C(2)]. Our proof is based on [SV, Prop. 3] and Theorem 20. Proof of Lemma 46. Assuming that the components of U in Theorem 20 are sufficiently small, we can conclude that iterates of f one beyond those mapping into U have negative Schwarzian derivative. The idea behind obtaining negative Schwarzian derivative goes back to Kozlovski [Ko], see also [GSS], and the precise statement is as follows. 3 (I ) and U be as in Theorem 20. If the components U of U are suffiLet f ∈ Cnf c ciently small, and x is such that f n (x) ∈ U and f i (x) ∈ / Crit for every 0 ≤ i ≤ n, then the Schwarzian derivative of f n+1 at x is negative.
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“Sufficiently small” in this statement should be interpreted as that for each component Uc of U and each i ≥ 0, each component of f −i (Uc ) has length ≤ τ , where τ = τ ( f ) is taken from [SV, Prop. 3] (choosing, with the notation in [SV, Prop. 3], S = 1, N = 0 and δ the number ξ0 = ξ( f ) from Theorem 20). If Uc is sufficiently small, then this holds n by Proposition 7. To prove the statement, let (Ji )i=0 be the pullback chain of Uc along n x, f (x), . . . , f (x) ∈ Uc . Let t1 < t2 < · · · < tm be the iterates such that f t j (x) ∈ U ,
t j+1 |Ji | ≤ 1. and hence the intervals Ji , t j + 1 ≤ i ≤ t j+1 are pairwise disjoint and i=t j +1 Fix j > 1 for the moment. Then Jt j +1 is contained in an entry domain J to U , say f t j+1 −t j −1 (J ) = Uc . By Theorem 20(ii), there is K ⊃ J such that f t j+1 −t j −1 maps diffeomorphically K onto Vc , hence the pullback chain of Vc along Jt j +1 , . . . , Jt j+1 has order 0. Moreover, Jt j+1 is contained in Uc , so it is ξ0 -well inside Vc . Therefore [SV, Prop. 3] implies that f t j+1 −t j has negative Schwarzian derivative at f t j +1 (x). For j = 1 and t1 > 0 (so x ∈ / U ), f t1 +1 has negative Schwarzian derivative at x by the same reasoning. If t1 = 0 (so x ∈ U ), then f has negative Schwarzian derivative at x by non-flatness. Since compositions of maps with negative Schwarzian derivatives have negative Schwarzian derivative, the statement follows. Now we continue with the proof of the lemma. Assume first k = 0. Fix ξ > 0 and let (Hi )li=0 ⊂ (G i )li=0 be chains such that (G i )li=0 has order 0 and G l is contained in a component Uc of the set U above. If G 0 is an entry domain to G l , then the statement is just Proposition 13 because the intervals G i are pairwise disjoint (G l is nice). If not, again due to the niceness of G l , there is an interval G t ⊂ G l such that the intervals G t+1 , . . . , G l are pairwise disjoint, so we can apply Proposition 13 to the subchains (Hi )li=t+1 ⊂ (G i )li=t+1 . Also, f t+1 |G 0 has negative Schwarzian derivative by the above t+1 ⊂ (G )t+1 . statement, so we can apply Proposition 12 to the subchains (Hi )i=0 i i=0 The general case k > 0 follows easily from this one (take also Lemma 11 into account). Now we need G l to be small enough so that every component of every preimage of G l is contained in U whenever it intersects Crit (again by Proposition 7). 7.4. Proof of Theorem 21. Let x ∈ I be such that orb(x) is disjoint from ∂ I . For every n ∈ N and every 0 < ε < d( f n , ∂ I ) we construct the pullback chain of ( f n (x) − ε, f n (x) + ε) along x, f (x), . . . , f n (x). We define rnk (x) = sup{ε > 0 : order of the pullback chain of ( f n (x) − ε, f n (x) + ε) ≤ k}. If for every k we have rnk (x) → 0 as n → ∞, then we call x a super-persistent point. In 2 (I ), the ω-limit set [BM1, Theorem 2.7] (see also [BM2]) it is shown that for f ∈ Cnf of any super-persistent recurrent point of f is minimal. Proof of Theorem 21. By definition of E and type (2) attractors, λ-a.e. x ∈ E has a dense orbit in some cycle cyc(K ) and therefore cannot be super-persistently recurrent, nor map into ∂ I . We can then find a sequence n j → ∞ and N x ∈ N and δx > 0 such that the pullback chain of ( f n j (x) − δx , f n j (x) + δx ) along x, . . . , f n j (x) has order at most N x . Clearly we can take δ f (x) ≥ δx and N f (x) ≤ N x (in fact, we have equality for large j whenever x is not a critical point). Since Lebesgue measure has only finitely many ergodic components [Ly1, Theorem 2], it follows that there are a single δ > 0 and N ∈ N such that δx ≥ δ and N x ≤ N for λ-a.e. x ∈ E. Next choose ε > 0 in Theorem 20 so small that the intervals Uc ⊂ Vc , c ∈ Crit , satisfy:
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557
• if c is in the interior of the metric attractor cyc(K )r, then dist(Vc , ∂ cyc(K )) > 2δ; moreover, if d ∈ Crit \ Crit , then either d ∈ ∂ cyc(K ) or dist(d, cyc(K )) > δ; • every component of every preimage of Vc , and every image f n (Vc ), n ≤ 2r (with r the period of cyc(K )), has length less than δ. Recall that U = c∈Crit Uc , V = c∈Crit Vc . From their definition, both sets are nice. Moreover, ∂U ∩ D(V ) = ∅, where as before D(V ) = ∪n>0 f −n (V ). Since orb(x) is dense in cyc(K ), we can assume that the numbers n j are large enough so that f n j (x) ∈ cyc(K ) for every j. Let J be the entry domain to U (or the component of U ) containing f n j (x), say f m j (J ) = Uc . Two possibilities arise. If J ⊂ cyc(K ), then c ∈ Crit . Moreover, if K ⊃ J is such that f m j maps K diffeomorphically onto Vc , then K ⊂ ( f n j (x) − δ, f n j (x) + δ), so the pullback chain of Vc along x, . . . , f n j +m j (x) has order at most N . The second possibility is that J contains some point from ∂ cyc(K ), call it a. It is not possible that a belongs to a periodic orbit contained in ∂ cyc(K ), because this would imply that one of the points of this periodic orbit belongs to Crit \ Crit , which contradicts that cyc(K ) contains dense orbits. Hence there is r j ≤ 2r (with r the period of cyc(K )) such that | f n j −r j (x) − c | < δ for some c ∈ Crit . In fact, f n j −r j (x) ∈ Uc , because otherwise f n j −r j (x) would belong to an entry domain J to U contained in cyc(K ), which leads to the contradiction f r j (J ) = J ⊂ cyc(K ). Since f r j (Vc ) ⊂ ( f n j (x) − δ, f n j (x) + δ), the pullback chain of Vc along x, . . . , f n j −r j (x) has order at most N . We have proved that there is a number N = N ( f ) such that, for a.e x ∈ D, there are c ∈ Crit and a sequence s j → ∞ such that f s j (x) ∈ Uc and the pullback of Vc along x, . . . , f s j (x) has order at most N . However, we need critical order 0, not N , for this theorem, so a further argument is required. Let k x ≥ 1 be the smallest integer for which there are G x ⊃ Hx x and c ∈ Crit such that f k x maps G x diffeomorphically onto Vc and f k x (Hx ) = Uc . If x ∈ / U , ω(x) = cyc(K ) and the entry domain Hx x to U is contained in cyc(K ), then k x exists; it is the first entry time k x = rU (x). But if x ∈ U , then k x is more difficult to find. Claim 1. The set B := {x ∈ E : k x exists} has full Lebesgue measure in E. Assume by contradiction that this claim fails, and that x is a density point of E\B. j sj j sj We can find the sequence (s j ) j∈N and c ∈ Crit as above, and let (G i )i=0 and (Hi )i=0 j j be the pullback chains of G s j = Vc and Hs j = Uc along x, . . . , f s j (x), respectively. j s
j The chains (G i )i=0 have order ≤ N . Let (Wt )t∈N be an enumeration of the return domains to V within Vc \Uc and also take W0 := Uc . Recall that ∂Uc ∩ D(V ) = ∅, so (Wt )t≥0 is a family of pairwise disjoint intervals whose union has full measure in V . Since f s j |G j has at most N critical 0
values, we can arrange the enumeration of the Wt s such that no critical value of f s j |G j 0 is contained in Wt if t > N . By Proposition 45 there is ξ = ξ( f ) such that each Wt is ξ -well inside Vc . By Lemma 46, there is ζ = ζ (ξ, N , f ) such that each component N j j of G 0 ∩ f −s j ( t=0 Wt ) is ζ -well inside G 0 . This holds in particular for the at most j N M ≤ 2 N +1 − 1 components of f −s j ( t=0 Wt ) in G 0 . Let us denote these components as Yi , numbered in order of decreasing size. j
j
Claim 2. There are b = b(ζ, N ) > 0 and B j ⊂ G 0 such that λ(B j ) ≥ bλ(G 0 ) and for every y ∈ B j , there are k y ≥ s j , G y ⊃ Hy y and c(y) ∈ Crit such that f k y maps G y diffeomorphically onto Vc(y) and f k y (Hy ) = Uc(y) .
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Take b = ζ M /(M + 1)!. The components Yi are pairwise disjoint, and have ζ -collars j around them in G 0 , which may intersect other components Yi or their collars. We will j j show that λ( i Yi ) ≤ (1 − b)λ(G 0 ). If λ(Y1 ) < λ(G 0 )/(M + 1), then we get directly
λ(
j
j
j
Yi ) < λ(G 0 )M/(M + 1) = (1 − 1/(M + 1))λ(G 0 ) < (1 − b)λ(G 0 ).
i j
Thus we can assume λ(Y1 ) ≥ λ(G 0 )/(M + 1) and the ζ -collar of Y1 has two compoj j nents of length ≥ ζ λ(G 0 )/(M +1). If the second largest Y2 satisfies Y2 < ζ λ(G 0 )/((M + j 1)M), then thereis a set in the ζ -collar of Y1 of measure > ζ λ(G 0 )/((M + 1)M) which do not intersect i Yi . Hence
λ(
j
j
Yi ) < (1 − ζ /((M + 1)M))λ(G 0 ) < (1 − b)λ(G 0 )
i
again. Continuing this way, we find that at least one component Yk has a ζ -collar disjoint j j from i Yi and its size is ≥ ζ M λ(G 0 )/(M + 1)! = bλ(G 0 ). j j Now take B j := E ∩ (G 0 \ i Yi ). Then clearly λ(B j ) ≥ bλ(G 0 ), and if y ∈ B j , we s j find k y as follows. We have z := f (y) ∈ Wt for some t > N . There is rt such that f rt maps Wt diffeomorphically onto a component of V . If f rt (z) ∈ U , then k y = s j + rt . Otherwise f rt (z) ∈ V \U and in fact f rt (z) belong to another return domain to V . We continue iterating diffeomorphically until finally z falls into U , and we choose k y j j accordingly. This proves Claim 2. Obviously B j ⊂ B, so λ(B ∩ G 0 ) ≥ bλ(G 0 ) indej pendently of j. Since λ(G 0 ) → 0 as j → ∞, this contradicts that x is a density point of I \B, proving Claim 1. From Claim 1 the first part of Theorem 21 easily follows: if x, y ∈ B and k x ≤ k y , then either Hx ∩ Hy = ∅ or Hy ⊂ Hx . Now an easy maximality argument allows us to redefine these sets if necessary so that either Hx ∩ Hy = ∅ or Hy = Hx . It remains to prove the second part of Theorem 21. Since a.e. point from U belongs to B, F : U → U is well defined a.e. It is important to note that if F is defined on x, then Hx ⊂ G x ⊂ U because ∂(U ) ∩ D(V ) = ∅. Now it is immediate to show by induction on n that every branch F n | H = f j | H : H ⊂ U → Uc of F n admits a diffeomorphic extension to an interval H ⊂ G ⊂ U with f j (G) = Vc . Therefore F n | H has distortion bounded by a constant κ depending neither on n nor on H by Lemma 46. References [AJS] [BJ1] [BJ2] [BD] [Ba] [BrJ]
Alsedà, L., Jiménez López, V., Snoha, L.: All solenoids of piecewise smooth maps are period doubling. Fund. Math. 157, 121–138 (1998) Balibrea, F., Jiménez López, V.: A structure theorem for C 2 functions verifying the Misiurewicz condition. In: Proceedings of the European Conference on Iteration Theory (Lisbon, 1991), Singapore: World Sci. Publishing, 1992, pp. 12–21 Balibrea, F., Jiménez López, V.: The measure of scrambled sets: a survey. Acta Univ. M. Belii Ser. Math. No. 7, 3–11 (1999) Barge, M., Diamond, B.: Proximality in Pisot tiling spaces. Fund. Math. 194, 191–238 (2007) Barnes, J.A.: Conservative exact rational maps of the sphere. J. Math. Anal. Appl. 230, 350–374 (1999) Barrio Blaya, A., Jiménez López, V.: An almost everywhere version of Smítal’s order-chaos dichotomy for interval maps. J. Austral. Math. Soc. 85, 29–50 (2008)
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[J1]
Jiménez López, V.: C 1 weakly chaotic functions with zero topological entropy and non-flat critical points. Acta Math. Univ. Comenian. (N.S.) 60, 195–209 (1991) Jiménez López, V.: Large chaos in smooth functions of zero topological entropy. Bull. Austral. Math. Soc. 46, 271–285 (1992) Jiménez López, V.: Order and chaos for a class of piecewise linear maps. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 5, 1379–1394 (1995) Johnson, S.D.: Singular measures without restrictive intervals. Commun. Math. Phys. 110, 185–190 (1987) Kan, I.: A chaotic function possessing a scrambled set with positive Lebesgue measure. Proc. Amer. Math. Soc. 92, 45–49 (1984) Keller, G.: Exponents, attractors and Hopf decompositions for interval maps. Erg. Th. Dynam. Syst. 10, 717–744 (1990) Kozlovski, O.S.: Getting rid of the negative Schwarzian derivative condition. Ann. of Math. (2) 152, 743–762 (2000) Ledrappier, F.: Some properties of absolutely continuous invariant measures on an interval. Erg. Th. Dynam. Syst. 1, 77–93 (1981) Li, S., Shen, W.: Hausdorff dimension of Cantor attractors in one-dimensional dynamics. Invent. Math. 171, 1629–1643 (2008) Li, T., Yorke, J.A.: Period three implies chaos. Amer. Math. Monthly 82, 985–992 (1975) Lyubich, M.: Ergodic Theory for Smooth one Dimensional Dynamical Systems. Stony Brook preprint 1991/11 Lyubich, M.: Combinatorics, geometry and attractors of quasi-quadratic maps. Ann. of Math. (2) 140, 347–404 (1994) Lyubich, M., Milnor, J.: The Fibonacci unimodal map. J. Amer. Math. Soc. 6, 425–457 (1993) Majumdar, M., Mitra, T., Nishimura, K., (eds.): Optimization and Chaos. Studies in Economic Theory 11, Berlin: Springer-Verlag, 2000 Mañé, R.: Hyperbolicity, sinks and measure in one-dimensional dynamics. Commun. Math. Phys. 100, 495–524 (1985). Erratum in Comm. Math. Phys. 112, 721–724 (1987) Martens, M.: Interval Dynamics. Ph.D. Thesis, Delft University of Technology, 1990 Martens, M., de Melo, W., van Strien, S.: Julia-Fatou-Sullivan theory for real one-dimensional dynamics. Acta Math. 168, 273–318 (1992) Martens, M., Tresser, C.: Forcing of periodic orbits for interval maps and renormalization of piecewise affine maps. Proc. Amer. Math. Soc. 124, 2863–2870 (1996) de Melo, W., van Strien, S.: One Dimensional Dynamics. Ergebnisse Series 25, Berlin: SpringerVerlag, 1993 Misiurewicz, M.: Chaos almost everywhere. In: Iteration Theory and its Functional Equations (Lochau, 1984), Lecture Notes in Math. 1163, Berlin: Springer, 1985, pp. 125–130 Piórek, J.: On the generic chaos in dynamical systems. Univ. Iagel. Acta Math. 25, 293–298 (1985) Smítal, J.: A chaotic function with some extremal properties. Proc. Amer. Math. Soc. 87, 54–56 (1983) Smítal, J.: A chaotic function with a scrambled set of positive Lebesgue measure. Proc. Amer. Math. Soc. 92, 50–54 (1984) Smítal, J.: Chaotic functions with zero topological entropy. Trans. Amer. Math. Soc. 297, 269–282 (1986) Smítal, J., Štefánková, M.: Omega-chaos almost everywhere. Discrete Contin. Dyn. Syst. 9, 1323–1327 (2003) Straube, E.: On the existence of invariant, absolutely continuous measures. Commun. Math. Phys. 81, 27–30 (1981) van Strien, S.: Transitive maps which are not ergodic with respect to Lebesgue measure. Erg. Th. Dynam. Syst. 16, 833–848 (1996) van Strien, S., Vargas, E.: Real bounds, ergodicity and negative Schwarzian for multimodal maps, J. Amer. Math. Soc. 17, 749–782 (2004). Erratum in J. Amer. Math. Soc. 20, 267–268 (2007) Young, L.-S.: Recurrence times and rates of mixing. Israel. J. Math. 110, 153–188 (1999)
[J2] [J3] [Jo] [K] [Ke] [Ko] [Le] [LS] [LY] [Ly1] [Ly2] [LM] [MMN] [M] [Ma] [MMS] [MT] [MS] [Mi] [P] [S1] [S2] [S3] [SS] [St] [vS] [SV] [Y]
Communicated by G. Gallavotti
Commun. Math. Phys. 299, 561–575 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1084-x
Communications in
Mathematical Physics
Absence of Squirt Singularities for the Multi-Phase Muskat Problem Diego Córdoba1 , Francisco Gancedo2 1 Instituto de Ciencias Matematicas, CSIC, C/Serrano, 123,
28006 Madrid, Spain. E-mail:
[email protected] 2 Department of Mathematics, University of Chicago, 5734 University Ave.,
Chicago, IL 60637, USA. E-mail:
[email protected] Received: 20 November 2009 / Accepted: 13 February 2010 Published online: 10 July 2010 – © Springer-Verlag 2010
Abstract: In this paper we study the evolution of multiple fluids with different constant densities in porous media. This physical scenario is known as the Muskat and the (multiphase) Hele-Shaw problems. In this context we prove that the fluids do not develop squirt singularities. 1. Introduction We consider the dynamics of the two interphases between three incompressible and immiscible fluids in a porous medium without surface tension. The free boundaries are prescribed by the jump of densities between the fluids. Although the present paper is devoted to the evolution of two interphases in R3 , the same approach can be performed when additional phases are considered in R2 or R3 . The governing equations for the 2D incompressible porous media are identical with those modeling the dynamics in a Hele-Shaw cell (see [17]). The precise formulation of this problem is as follows [1]: the scalar density ρ = ρ(x, t) of the fluid is convected by the incompressible velocity flow u = (u 1 (x, t), u 2 (x, t), u 3 (x, t)) which satisfies Darcy’s law, i.e. ρt + u · ∇ρ = 0 ∇ ·u =0 u = −∇ p − (0, 0, ρ)
(Conservation of mass), (Incompressibility), (Darcy’s law),
(1)
where the scalar p = p(x, t) is the pressure and the acceleration due to gravity is taken equal to one to simplify the notation. Darcy’s law yields the velocity in terms of the density by means of singular integral operators as follows: 2 u(x, t) = P V K (x − y) ρ(y, t)dy − (0, 0, ρ(x)) , x ∈ R3 , (2) 3 R3
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where the kernel K is given by 1 K (x) = 4π
x1 x3 x2 x3 2x 2 − x12 − x22 3 5 ,3 5 , 3 |x| |x| |x|5
.
The integral operator is defined in the Fourier side by ξ12 + ξ22 ξ1 ξ3 ξ2 ξ3 u (ξ ) = , ,− ρ (ξ ) |ξ |2 |ξ |2 |ξ |2 that shows that the velocity and the density are at the same level in terms of regularity. The fluid is characterized by three different constant values of density ρ 1 < ρ 2 < ρ 3 : ⎧ 1 ⎨ ρ in 1 = {x3 > f (x1 , x2 , t)}, ρ(x1 , x2 , x3 , t) = ρ 2 in 2 = { f (x1 , x2 , t) > x3 > g(x1 , x2 , t)}, ⎩ 3 ρ in 3 = {g(x1 , x2 , t) > x3 }, where f (x1 , x2 , t) > g(x1 , x2 , t). The moving surfaces S f (x1 , x2 , t) = {(x1 , x2 , x3 ) ∈ R3 : x3 = f (x1 , x2 , t)}, Sg (x1 , x2 , t) = {(x1 , x2 , x3 ) ∈ R3 : x3 = g(x1 , x2 , t)}, have the property (see below in Sect. 2) that they can be parameterized as a graph for all time. Since the flow is incompressible the velocity of each interphase is continuous in the normal direction to the moving surface. Moreover, from the formulation it implies that the pressures are equal across the surfaces. The system is highly non-local in the sense that the equation for the moving surfaces involves singular integral operators and they are coupled together. Within the formulation we can recover the dynamics of a single interphase by taking ρ 1 = ρ 2 or ρ 2 = ρ 3 , which has been shown to be well-posed in the stable scenario with a maximum principle (see [7 and 8]). This case is known as the Muskat problem [16] (in 2D it is also known as the two-phase Hele-Shaw) which has been broadly studied in [4,7,8,10,11,18] and references therein. The aim of this paper is first to show that for multiple interphases, in the stable case (ρ 1 < ρ 2 < ρ 3 ), the system is well posed in a chain of Sobolev spaces. In the unstable case (ρ 1 > ρ 2 or ρ 2 > ρ 3 ) there is Rayleigh-Taylor instability [7] and the system is ill-posed. Secondly we rule out a squirt singularity in the three phase system, i.e. that both interphases can not collapse in such a way that a positive volume of fluid between the interphases gets ejected in finite time. Let’s assume that both surfaces collapse at time T in a domain D such that lim [ f (x1 , x2 , t) − g(x1 , x2 , t)] = 0 for (x1 − x1 )2 + (x2 − x 2 )2 < a 2 ,
t→T −
x2 , x3 ) ∈ D are fixed. Consider the where the constant a > 0 and the point x = ( x1 , domain (t) denoted by (t) = {(x1 , x2 , x3 ) : (x1 − x1 )2 + (x2 − x2 )2 ≤ (R(t))2 , g(x1 , x2 , t) ≤ x3 ≤ f (x1 , x2 , t)} with 0 < R(t) < a and a section of its boundary given by S(t) = {(x1 , x2 , x3 ) : (x1 − x1 )2 + (x2 − x2 )2 = (R(t))2 , g(x1 , x2 , t) ≤ x3 ≤ f (x1 , x2 , t)}.
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Then by using the divergence free vector field u it follows that d Vol (t) = [R (t) − u · ν] d (Area), dt S (t) where ν is the unit normal to S(t). If the integral choose a time t0 ∈ [0, T ) and take R(t) =
1 a − 2
T
T 0
|u| L ∞ dt is bounded then we can
|u| L ∞ dτ for t0 ≤ t < T
t
d such that 0 < R(t) < a for all t ∈ [t0 , T ). Consequently dt Vol (t) ≥ 0 for all t ∈ [t0 , T ) which prevents a collapse (squirt singularity) forming in between the moving surfaces (for more details see [5]). For a general n-dimensional definition of a squirt singularity see [6]. Therefore, the third part of the paper is devoted to show a bound of the velocity of the fluid in terms of C 1,γ norms (0 < γ < 1) of the free boundary. The estimate is based on the property that in the principal value (2) the mean of the kernels K are zero on hemispheres. This extra cancellation was used by Bertozzi and Constantin [2] for the vortex patch problem of the 2D Euler equation to prove no formation of singularities. For this system the convected vorticity takes constant values in disjoint domains and is related with the incompressible velocity by the Biot-Savart law (see [14] for more details). Let us point out that the system we are dealing with is a more singular one. Also, we quote the work [15] of Mateu, Orobitg and Verdera where this cancelation was used in quasiconformal mappings theory. Finally we would like to emphasize how the character of the kernels becomes crucial here since for analogous active scalar models we can not obtain this result. For the 2D surface quasi-geostrophic equation (SQG) [3] the kernels are odd (Riesz transforms) and for the patch problem [12] the velocity is not in L ∞ , it is in B M O (see [19] to get the definition and properties of the B M O space). Furthermore in the case of regular initial data for SQG and the system (1) the problem is also open [9]. The structure of the article is as follows. In Sect. 2 we derived the equations of both interphases that are coupled together and in Sect. 3 we show that the system is well-posed in Sobolev spaces. Finally, in Sect. 4, we give a proof of boundedness of the velocity in terms of the smoothness of the interphases.
2. The Evolution Equation of the Moving Surfaces The goal is to obtain the dynamics of a fluid that takes three different constant values of density ρ 1 , ρ 2 and ρ 3 as follows: ⎧ 1 ⎨ ρ in {x3 > f (x, t)}, (3) ρ(x1 , x2 , x3 , t) = ρ 2 in { f (x, t) > x3 > g(x, t)}, ⎩ 3 ρ in {g(x, t) > x3 }, with f (x, t) > g(x, t) for all x = (x1 , x2 ) ∈ R2 . Hence ∇ρ = (ρ 2 − ρ 1 )(∂x1 f, ∂x2 f, −1)δ(x3 − f (x, t)) +(ρ 3 − ρ 2 )(∂x1 g, ∂x2 g, −1)δ(x3 − g(x, t)),
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where δ is the Dirac distribution defined by < hδ, η >=
R2
h(x)η(x, f (x, t))d x,
for η(x1 , x2 , x3 ) a test function. For a divergence free velocity field, Darcy’s law provides u = (∂x1 −1 ∂x3 ρ, ∂x2 −1 ∂x3 ρ, −∂x1 −1 ∂x1 ρ − ∂x2 −1 ∂x2 ρ),
(4)
and therefore (y1 , y2 , ∇ f (x − y, t) · y) ρ2 − ρ1 PV u(x1 , x2 , x3 , t) = − dy 2 + (x − f (x − y, t))2 ]3/2 2 4π [|y| 3 R (y1 , y2 , ∇g(x − y, t) · y) ρ3 − ρ2 PV dy, − 2 2 3/2 2 4π R [|y| + (x 3 − g(x − y, t)) ]
(5)
for f (x, t) = x3 = g(x, t). Taking x3 → f (x, t) yields ρ2 − ρ1 (y1 , y2 , ∇ f (x − y, t) · y) u(x, f (x, t), t) = − dy PV 2 2 3/2 4π R2 [|y| + ( f (x, t) − f (x − y, t)) ] (y1 , y2 , ∇g(x − y, t) · y) ρ3 − ρ2 PV dy, − 2 2 3/2 4π R2 [|y| + ( f (x, t) − g(x − y, t)) ] without considering tangential terms. In any case the evolution of the surfaces are given by the normal velocity, the tangential terms are related with the parameterization of the free boundary [13]. Consequently (x, f (x, t))t · (−∂x1 f, −∂x2 f, 1) = u(x, f (x, t), t) · (−∂x1 f, −∂x2 f, 1), and therefore (∇ f (x, t) − ∇ f (x − y, t)) · y) ρ2 − ρ1 PV f t (x, t) = dy 2 2 3/2 2 4π R [|y| + ( f (x, t) − f (x − y, t)) ] ∇ f (x, t) · y − ∇g(x − y, t) · y ρ3 − ρ2 PV dy. + 2 2 3/2 4π R2 [|y| + ( f (x, t) − g(x − y, t)) ]
(6)
In a similar way we obtain gt (x, t) =
(∇g(x, t) − ∇g(x − y, t)) · y) ρ3 − ρ2 PV dy 2 2 3/2 4π R2 [|y| + (g(x, t) − g(x − y, t)) ] ∇g(x, t) · y − ∇ f (x − y, t) · y ρ2 − ρ1 PV dy. + 2 2 3/2 4π R2 [|y| + (g(x, t) − f (x − y, t)) ]
(7)
This coupled system describes the evolution of the moving boundaries S f (x, t) and Sg (x, t).
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3. Well-posedness in the Stable Scenario ρ 1 < ρ 2 < ρ 3 Let us define the function d( f, g)(x, y, t) by the formula d( f, g)(x, y, t) =
[|y|2
1 + ( f (x, t) − g(x − y, t))2 ]1/2
∀ x, y ∈ R2 ,
which measures the distance between the contours f and g. Therefore we consider f (x, t) approaching the value C∞ > 0 as |x| → ∞ to avoid that the surfaces collapse at infinity. The section is devoted to prove the following theorem: Theorem 3.1. Let f 0 (x) − C∞ , g0 (x) ∈ H k (R2 ) for k ≥ 4, d( f 0 , g0 ) ∈ L ∞ and ρ3 > ρ2 > ρ1 . Then there exists a time T > 0 so that there is a unique solution to (6) and (7) given by ( f (x, t), g(x, t)) where f (x, t) − C∞ , g(x, t) ∈ C 1 ([0, T ]; H k (R2 )), f (x, 0) = f 0 (x) and g(x, 0) = g0 (x). Proof. We shall show that the following estimate holds: d E(t) ≤ C(E(t) + 1) p dt
(8)
for universal constants (C, p) and the function E(t) defined by E(t) = f − C∞ H k (t) + g H k (t) + d( f, g) L ∞ (t). Applying standard energy estimate arguments permit us to conclude local existence. This is based on introducing a regularized version of the system (6) and (7) that allows to take limits satisfying uniformly the a priori bound (8). To simplify the exposition we shall consider ρ 2 − ρ 1 = ρ 3 − ρ 2 = 4π and k = 4, the rest of the cases being analogous. Some of the terms can be estimated exactly as in [7] and therefore we shall show below how to deal with the different ones. We have 1 d (∇ f (x)−∇ f (x − y)) · y f − C∞ 2L 2 (t) = ( f (x)−C∞ )P V d yd x 2 +( f (x)− f (x − y))2 ]3/2 2 2 2 dt [|y| R R y + ( f (x)−C∞ )∇ f (x)· P V 2 2 3/2 2 R2 [|y| +( f (x)−g(x − y)) ] R ∇g(x − y) · y − ( f (x) − C∞ )P V 2 2 3/2 2 2 R R [|y| + ( f (x) − g(x − y)) ] = I1 + I2 + I3 . For I1 we decompose further: I1 = J1 + J2 + J3 where (∇ f (x) − ∇ f (x − y)) · y ( f (x) − C∞ ) d yd x, J1 = 2 + ( f (x) − f (x − y))2 ]3/2 2 [|y| |y|1 |y|>1 [|y| + ( f (x) − f (x − y)) ] and
J3 = −
|y|>1
( f (x) − C∞ )P V
|y|>1
[|y|2
∇ f (x − y) · y d yd x. + ( f (x) − f (x − y))2 ]3/2
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Since
∂xi f (x) − ∂xi f (x − y) =
1 0
∇∂xi f (x + (s − 1)y) · y ds,
we get
| f (x) − C∞ ||∇ 2 f (x + (s − 1)y)| d xd y 2 −2 3/2 |y|1 R2 [|y|2 + (( f (x) − f (x − y))2 ]5/2 | f (x)− f (x − y)||y|−1 (|∇ f (x)|+|∇ f (x − y)|) 3 |y|−3 | f (x)−C∞ |2 d xd y ≤ 2 |y|>1 [1+(( f (x)− f (x − y))2 |y|−2 ]5/2 R2 ≤ C f − C∞ 2L 2 ∇ f 2L ∞ ≤ C f − C∞ 2L 2 ∇ f 2H 2 . In a similar way for J3 we have 2|y|2 − ( f (x) − f (x − y))2 ( f (x) − C∞ )( f (x − y) − C∞ ) d xd y J3 = [|y|2 + ( f (x) − f (x − y))2 ]5/2 |y|>1 R2 ( f (x)− f (x − y))∇ f (x − y) · y +3 ( f (x)−C∞ )( f (x − y)−C∞ ) d xd y 2 [|y|2 +( f (x)− f (x − y))2 ]5/2 |y|>1 R |y|2 − ( f (x)−C∞ )( f (x − y)−C∞ ) d xdσ (y) [|y|2 +( f (x)− f (x − y))2 ]3/2 |y|=1 R2 ≤ C( ∇ f L ∞ + 1) f − C∞ 2L 2 . The term I2 is estimated as follows: ( f (x) − g(x − y))(∇ f (x) − ∇g(x − y)) · y 3 I2 = ( f (x) − C∞ )2 d xd y 2 R2 R2 [|y|2 + ( f (x) − g(x − y))2 ]5/2 =3 d xd y + 3 d xd y, |y|>1 R2 |y|1 R2 3 + | f (x) − C∞ |2 (|∇ f (x)| + |∇g(x − y)|)(d( f, g)(x, y))3 d xd y 2 |y|1 R2 + |∂x41 f (x)||∇ f (x)|(|∂x41 f (x)| + |∂x41 g(x − y)|)(d( f, g)(x, y))3 d xd y ≤
|y|1 R 4 +C |∂x1 f (x)||∇ f (x)|(|∂x1 f (x)|+|∂x1 g(x − y)|)4 (d( f, g)(x, y))6 d xd y ≤
|y|1
Hence
|J7 | ≤ C
We rewrite J8 as
|y|1
|y|−1 dy ∇ 2 f L ∞ ≤ C ∇ 2 f L ∞ .
y 1 − [1 + ( f (x, t) − f (x − y, t))2 |y|−2 ]3/2 dy, |y|3 [1 + ( f (x, t) − f (x − y, t))2 |y|−2 ]3/2
and considering the function Q(α) = [1 + α 2 ]3/2 , the mean value theorem gives y −3[1 + β 2 ]1/2 β| f (x) − f (x − y)| J8 = ∇ f (x, t) · dy, 4 2 −2 3/2 |y|>1 |y| [1 + ( f (x, t) − f (x − y, t)) |y| ] where 0 ≤ β ≤ | f (x) − f (x − y)||y|−1 . Therefore |J8 | ≤ C ∇ f L ∞ f − C∞ L ∞ . In a similar manner as for J3 , integration by parts in J9 yields |J9 | ≤ C( ∇ f L ∞ + 1) f − C∞ L ∞ . For I8 we split
I8 = P V
|y|>1
dy +
|y| 0. Hence
d( f, g) L ∞ (t + h) ≤ d( f, g) L ∞ (t) exp
t+h
C(E(s) + 1) p ds .
t
The above estimate applied to the following limit: d d( f, g) L ∞ (t + h) − d( f, g) L ∞ (t) d( f, g) L ∞ (t) = lim+ h→0 dt h allows us to get finally d d( f, g) L ∞ (t) ≤ C(E(t) + 1) p . dt
(14)
Combining estimates (11), (12) and (14) we conclude that (8) holds for universal constants C and p.
For uniqueness we proceed as in [7] which leads to the desired result. 4. Bound for the Fluid Velocity: Even Kernels In this section we prove the following lemma: Lemma 4.1. Let f, g be solutions of the contour system (6) and (7). Then the velocity of the fluid satisfies the following bound: 1 1 ∞ u L ≤ C 1 + + ln(1 + ∇ f C γ + ∇g C γ ) γ γ + ln(1 + f − C∞ L ∞ + C∞ + ∇ f L 2 + g L ∞ + ∇g L 2 ) , (15) where 0 < γ < 1 and the constant C = C(ρ 1 , ρ 2 , ρ 3 ) depends on ρ 1 , ρ 2 and ρ 3 . Proof. We shall denote x ∈ R2 and x = (x, x3 ) ∈ R3 . In order to acquire the above inequality, we can split ρ as follows: ρ = ρ 1 χ+ (t) + ρ 1 χ1 (t) + ρ 2 χ2 (t) + (ρ 3 − ρ 2 )χ3 (t) + ρ 3 χ− (t) , where + (t) = {x3 > M}, 1 (t) = {M > x3 > f (x, t)},
− (t) = {x3 < −M}, 2 (t) = { f (x, t) > x3 > −M}
and 3 (t) = {g(x, t) > x3 > −M} for M = f − C∞ L ∞ (t) + C∞ + g L ∞ (t) + 1.
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From (4) we check that (∂x3 −1 ∂x1 , ∂x3 −1 ∂x2 , −(∂x1 −1 ∂x1 + ∂x2 −1 ∂x2 ))(χ± (t) ) = 0. Thus the Fourier transform yields u = T (ρ 1 χ1 (t) + ρ 2 χ2 (t) + (ρ 3 − ρ 2 )χ3 (t) ) 2 − (0, 0, ρ 1 χ1 (t) + ρ 2 χ2 (t) + (ρ 3 − ρ 2 )χ3 (t) ), 3 where
1 PV T (h)(x) = K (x − y)h(y)d y, 4π R3
and
x1 x3 x2 x3 2x 2 − x12 − x22 K (x) = 3 5 , 3 5 , 3 |x| |x| |x|5
(16)
x ∈ R3 .
Then we have to deal with ρ 1 T (χ1 (t) ),
ρ 2 T (χ2 (t) ) and
(ρ 3 − ρ 2 )T (χ3 (t) ).
We will consider T (χ1 (t) ) since the other terms are analogous. We proceed as in [2]. The key point is that the kernels in T are even. Therefore in the principal value the mean of the kernels are zero on a hemisphere. We consider the three different coordinates of T (χ1 (t) ) = (T1 , T2 , T3 ). Then for a fixed x ∈ R3 we have (x1 − y1 )(x3 − y3 ) 3 PV T1 (x) = d y. 4π |x − y|5 1 (t) We take a distance δ given by δ=
1 . 3(1 + ∇ f C γ + ∇g C γ )1/γ
Then if d(x, 1 (t)) > δ we consider 1 (t) = U 1 (t) ∪ U 2 (t) for U 1 (t) = 1 ∩ {(x1 − y1 )2 + (x2 − y2 )2 ≤ L 2 }, and U 2 (t) = 1 ∩ {(x1 − y1 )2 + (x2 − y2 )2 ≥ L 2 }, where L = 2(1 + f − C∞ L ∞ + C∞ + ∇ f L 2 + g L ∞ + ∇g L 2 ). The splitting T1 = I1 + I2 for (x1 − y1 )(x3 − y3 ) 3 PV I1 (x) = d y, 4π |x − y|5 U 1 (t) (x1 − y1 )(x3 − y3 ) 3 I2 (x) = PV dy 4π |x − y|5 U 2 (t)
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gives √
|I1 (x)| ≤ C
2L
δ
√ r −1 dr ≤ C ln( 2L/δ).
We write I2 = lim R→+∞ I2R such that (x3 − y3 ) 3 R d y, I2 = ∂y 4π U R2 (t) 1 |x − y|3 where U R2 (t) = 1 ∩ {L 2 ≤ (x1 − y1 )2 + (x2 − y2 )2 ≤ R 2 }. Then integration by parts gives I2R and therefore I2R =
3 4π − −
3 = 4π
∂U R2 (t)
3 4π 3 4π
L 0. Therefore (−∇ f (a, t), 1) · (a − x) (−∇ f (a, t), 1) · (x + r z − a)
| sin ϕ(z)| = + 2 2 1 + |∇ f (a, t)| r 1 + |∇ f (a, t)| r d (−∇ f (a, t), 1) · (x + r z − a)
≤ + r 1 + |∇ f (a, t)|2 r (a3 − f (a, t)) − (x3 + r z 3 − f (x + r z, t))
+ 1 + |∇ f (a, t)|2 r d | f (x + r z, t) − f (a, t) − ∇ f (a) · (x + r z − a)| ≤ + r r and finally d ∇ f C γ (d + r )1+γ d ∇ f C γ |x + r z − a|1+γ + ≤ + r r r r 1+γ 1+γ 1+γ γ d ∇ f C (d +r ) d + r 1+γ d ≤ + 2γ ≤ + 2γ γ r r δ r r γ 2r d . ≤ (1 + 2γ ) + r δ
| sin ϕ(z)| ≤
The above estimate and the fact that (x1 − y1 )(x3 − y3 ) dσ (y) = 0 |x − y|5 S + (x) yield for J1 the following bound: δ 2π π/2 3π 3 dr ≤ (1 + 2γ (1 + 1/γ )). |J1 (x)| ≤ χ{ϕ(z): z∈Dr (x)} | cos ϕ|dϕdθ 4π d 0 r 2 −π/2 We write J2 = K 1 + K 2 , where (x1 − y1 )(x3 − y3 ) 3 PV d y, K 1 (x) = c 1 4π |x − y|5 U (t)∩Bδ (x)
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and 3 PV K 2 (x) = 4π
U 2 (t)∩Bδc (x)
(x1 − y1 )(x3 − y3 ) d y. |x − y|5
Then one can deal as for I1 and I2 respectively to obtain the same estimates. We write the second coordinate as follows: (x3 − y3 ) 3 d y, T2 (x) = ∂ y2 PV 4π |x − y|3 1 (t) to get analogous bounds. For the third coordinate we have 2(x3 − y3 )2 − (x1 − y1 )2 − (x2 − y2 )2 1 PV T3 (x) = d y, 4π |x − y|5 1 (t) and we can proceed as before. But for the term I2R we have M (x3 − y3 ) (x3 − y3 ) 1 1 d dy3 dy I2R = ∂ y3 y = ∂ y 4π U R2 (t) |x − y|3 4π L 0, as one can take ν2 = ν0 . Lemma 1.1 (Asymptotics of ϕ(N )). Let the pair-potential v satisfy Assumption (V), then the limit ϕ(N ) ϕ(N ) = inf ∈ (−∞, 0), N →∞ N N ∈N N
ϕ˜ = lim
(1.5)
exists and is finite. The existence of the limit relies on subadditivity, the finiteness on Assumption (V) (5), and the negativeness is provided by the presence of negative interactions according to Assumption (V) (4). Remark 1.2. The point configurations that minimise the energy ϕ(N ) received attention in the literature. In [GR79,Th06] crystallisation is proved for d = 1, resp. d = 2. This is the phenomenon that the minimising particle configuration approaches, as N → ∞, a certain regular lattice which is unique up to translation and rotation. See also [AFS09] for more recent results. Physically speaking, these results are about zero temperature.
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By Lemma 1.1, the extended sequence (θκ : κ ∈ N ∪ {∞}) given by θκ =
ϕ(κ) κ ,
ϕ, ˜
if κ ∈ N, if κ = ∞,
(1.6)
is a continuous map from N ∪ {∞} to R. Now we identify the logarithmic asymptotics of the partition function Z N (β N , N ): Theorem 1.3 (Free energy). Suppose the pair-potential v satisfies Assumption (V). Let N ⊂ Rd be a centred cube and β N → ∞ such that, for some c ∈ (0, ∞), the particle density N = N /| N | satisfies N = e−cβ N . Then the free energy per particle, (c) = − lim
N →∞
1 log Z N (β N , N ), Nβ N
(1.7)
exists and is given by
(c) = inf
⎧ ⎨ ⎩
κ∈N∪{∞}
qκ : q ∈ [0, 1]N∪{∞} , qκ θκ − c κ κ∈N
κ∈N∪{∞}
⎫ ⎬ qκ = 1 . (1.8) ⎭
Remark 1.4. In the case of positive particle density at fixed positive temperature, the existence of the free energy per particle and of a close-packing phase transition when the potential is infinite in a neighbourhood of zero, is a classical fact, see e.g. [Ru99, Theorem 3.4.4]. The probability sequence q = (qκ )κ∈N∪{∞} appearing in (1.8) has an interpretation, which we informally describe now. Since the support of v is bounded, any point configuration {x1 , . . . , x N } in the integral on the right of (1.2) can be decomposed into connected components such that no particles of different components interact with each other. The quantity qκ characterises the relative frequency of components of cardinality κ among all these components, more precisely the configuration {x1 , . . . , x N } consists of N qκ /κ components of cardinality κ for each κ ∈ N. In the case κ = ∞, one should speak of components whose cardinalities tend to infinity as some function of N . Each component of cardinality κ is chosen optimally, i.e., as a minimiser of the right-hand side in the definition (1.4) of ϕ(κ). Then the term κ∈N∪{∞} qκ θκ expresses the energy
coming from such a configuration, and the term κ∈N qκ /κ describes its entropy. Now the logarithmic asymptotics of the partition function Z N (β N , N ) is determined by optimal configurations, i.e., by those configurations whose component structure follows the frequency distribution of any minimiser q of the right hand side of (1.8). By a straightforward, but technical, extension of the proof of the upper bound in (1.7), one could see that configurations with cluster frequencies different from the optimal q do not contribute to the limiting free energy, but we do not carry out details here. Neither information about the locations of the components relative to each other, nor about their shape is present in (1.8). The optimal shapes of cardinality κ are precisely those that minimize the right-hand side in the definition (1.4) of ϕ(κ), but it goes far beyond the scope of the present paper to give more specific information about them.
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Fig. 2. Example with η = 3 phase transitions and κ1 = 1, κ2 = 2, κ3 = 5
1.3. The phase transitions. Now we analyse the minimisers q on the right-hand side of (1.8). It turns out that, as the temperature parameter c decreases from infinity to zero, the minimiser q jumps between Dirac sequences on increasing component sizes, beginning with the size one. This means that, for sufficiently large c, the interparticle distance diverges, and that for smaller values of c all components of the system have the same finite size, which depends on the phase. The number of phases may be finite or infinite (depending on the interaction potential v). If it is finite then there is a phase with components of unbounded size. We interpret the phases with finite component sizes as the gaseous phases of the system. It remains open whether the phase with infinite component size is a fluid or solid phase. Let us turn to the details. Consider the sequence of points (1/κ, θκ ) with κ ∈ N∪{∞}, and extend them to the graph of a piecewise linear function [0, 1] → (−∞, 0]. Pick those of them which determine the largest convex minorant of this function. In formulas, let κ1 = 1 and, for n ∈ N, θκn − θ j θκn − θi = max , (1.9) κn+1 = max i ∈ N ∪ {∞} : i > κn and j>κn 1/κn − 1/j 1/κn − 1/i if κn = ∞. Observe that the maximum in (1.9) exists since the set { j ∈ N∪{∞} : j > κn } θ −θ j is compact and the mapping j → 1/κκnn −1/j is continuous. Hence, the sequence (κ1 , κ2 , . . . ) either terminates at κη+1 = ∞ for some η ∈ N or continues infinitely, in which case we put η = ∞. We thus have η = sup{n : κn < ∞} ∈ N ∪ {∞}. By cn =
θκn − θκn+1 , 1/κn − 1/κn+1
for 1 ≤ n < η + 1,
(1.10)
we denote the slope of the convex minorant in the n th interval. If η = ∞ define c∞ = inf n∈N cn (Fig. 2). For 1 ≤ n < η + 1, let I (n) = κn ≤ i ≤ κn+1 : (1/i, θi ) lies on the straight line passing through (1/κn , θκn ) and (1/κn+1 , θκn+1 ) .
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For κ ∈ N ∪ {∞}, we denote by q(κ) the Dirac sequence that has a one in the κ th entry and zeros everywhere else; we use the conventions 1/∞ = 0 and 1/0 = ∞. Let Q(n) be the convex hull of all q(i) with i ∈ I (n) , i.e., ⎧ ⎫ ⎨ ⎬ Q(n) = λi q(i) : λi = 1, λi ≥ 0 for all i ∈ I (n) . ⎩ (n) ⎭ (n) i∈I
i∈I
Now we can identify the free energy introduced in Theorem 1.3. It turns out that there are precisely η phase transitions, and they occur at the critical values cn with 1 ≤ n < η + 1 (with one more phase transition at c∞ in the case when η = ∞ and c∞ > 0). In particular, the function c → (c) is continuous and is linear between cn and cn−1 , with slope equal to −1/κn , which is strictly decreasing in n. Theorem 1.5 (Analysis of the variational formula). Let be as in Theorem 1.3. Then (i) The sequence (cn )1≤n 0), • for c ∈ (cn , cn−1 ), with some 2 ≤ n < η + 1, it is equal to q(κn ) , • for c ∈ (c1 , ∞) it is equal to q(κ1 ) = q(1) . (iv) If c = cn for some 1 ≤ n < η + 1, then the set of the minimisers of (1.8) is equal to Q(n) . Note that η ≥ 1, that is, there is at least one phase transition. In particular, the high temperature phase corresponding to the last case in (1.11) is always present. In this phase the relevant configurations {x1 , . . . , x N } on the right-hand side of (1.2) consist of single points, i.e., there is no interaction between any of the particles of the configuration and we are in an entropy dominated regime. The first case in (1.11) is the low-temperature phase, where the relevant configurations consist of components whose cardinalities tend to infinity as N → ∞. This case is empty if η = ∞ and cη = c∞ = 0. The second case in (1.11) is the case of intermediate temperatures; here those configurations dominate whose connected components have a finite cardinality ≥ 2. If η = 1, then the second case is empty, that is, only one-point configurations appear for c above the critical point c1 = −ϕ˜ and infinitely large ones for c below that point. This happens if and only if neither of the points (1/κ, θκ ) with 1 < κ < ∞ lies below the line connecting ˜ that is, if and only if ϕ(κ) ˜ (1, θ1 ) = (1, 0) and (0, θ∞ ) = (0, ϕ), κ−1 ≥ ϕ. We do not offer any criterion under which the value of η is determined. See Sect. 6 for an example of a potential v for which there are two phase transitions. 1.4. Outline of the proof of Theorem 1.3. As v(x) = 0 for x ≥ R, points do not interact if the distance between them is larger than R. We therefore introduce a graph structure
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609
on point configurations, connecting two points by an edge if the distance between them is at most R. A finite or countable set C ⊂ Rd is called connected if for any two elements a, b ∈ C there exist k ∈ N and a0 , . . . , ak ∈ C such that a0 = a, ak = b and |ai − ai−1 | ≤ R for i ∈ {1, . . . , k}. For a configuration x = (x1 , . . . , x N ) of points in N , we define i to be the largest subset of {1, . . . , N } containing i such that the set {x j : j ∈ i } is connected. Now the i th connected cloud is defined by
[xi ] :=
δx j ,
j∈ i
and the shifted i th connected cloud by [xi ] − xi :=
δx j −xi ,
j∈ i
where δ y denotes the Dirac measure at y ∈ Rd . If [xi ]({a}) = 0 we say that a ∈ [xi ] and denote #[xi ] = [xi ](Rd ), which is equal to the number of points in the i th connected cloud. The main object of our analysis is the empirical measure on the connected components of the graph induced by the configuration, translated such that any of its points is at the origin with equal measure, Y N(x) =
N 1 δ[xi ]−xi . N
(1.12)
i=1
Observe that the energy of the configuration may be written as VN (x) =
N
N v |xi − x j | = v |xi − x j | i=1
i, j=1 i = j
=
N i=1
j =i x j ∈[xi ]
1 v(|x − y|) = N Y N(x) , #[xi ] x,y∈[x ] x = y
i
where, for a suitable class of probability measures Y on configurations, we define (Y ) =
Y (d A)
1 v(|x − y|). # A x,y∈A x = y
) Let X be a vector of independent random variables X 1(N ) , X 2(N ) , . . . , X (N N uniformly distributed on N , and write Y N = Y N(X ) . Hence we can represent the partition function as
| N | N E N exp {−Nβ N (Y N )} , Z N β N , e−cβ N = N!
(1.13)
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A. Collevecchio, W. König, P. Mörters, N. Sidorova
where E N and P N denote the expectation and probability with respect to X . The main step, formulated as Proposition 2.2, is a large deviation principle for Y N in the weak topology with speed N log(1/ N ) and a rate function 1 . J (Y ) = 1 − Y (d A) #A By definition, this means that 1 log P N (Y N ∈ · ) ⇒ − inf J (Y ) , Y∈· N log(1/ N ) where ⇒ denotes weak convergence. Large deviation principles for particle systems with interactions were also considered in [Ge94], though in the non-dilute case which is significantly different. By our assumption N log(1/ N ) = cNβ N and hence an informal application of Varadhan’s Lemma (whose direct application is impossible by the lack of continuity of the functional ) to (1.13) implies that lim
N →∞
1 log E N exp {−Nβ N (Y N )} = − inf {(Y ) + c J (Y )} . Y Nβ N
Using (1.13) and | N | N 1 e O(N ) = exp{cNβ N } eo(Nβ N ) , = exp N log N! n
(1.14)
it is easily seen that this gives Theorem 1.3. The remainder of this paper is organised as follows. In Sect. 2, we analyse the distribution of Y N(X ) and prove a large deviation principle. In Sect. 3 we analyse ϕ(N ) and its asymptotics and prove in particular Lemma 1.1. The proofs of Theorems 1.3 and 1.5 are in Sects. 4 and 5. Finally, an example of a potential v that admits several phase transitions is described in Sect. 6. 2. Large Deviations for Y N In this section we analyse the distribution of the empirical measure Y N introduced in (1.12) asymptotically as N → ∞. In particular, we state and prove one of our main tools, the large deviation principle. In Sect. 2.1 we introduce the topological framework. The principle is formulated in Sect. 2.2 and proved in Sects. 2.3 and 2.4. 2.1. Topological framework. A measure A on Rd is called a configuration or a point
measure if A = i∈I δai for some finite or countable collection (ai : i ∈ I ) of (not necessarily distinct) points of Rd such that A(K ) < ∞ for any compact set K ⊂ Rd . For any configuration A, we denote # A = A(Rd ) and, for a ∈ Rd , we write a ∈ A if A({a}) ≥ 1. We call a point a a multiple point of a configuration A if A({a}) ≥ 2. Denote by N the space of all configurations on Rd . We equip N with the vague topology, that is, An converges to A if lim supn→∞ An (K ) ≤ A(K ) and lim inf n→∞ An (O) ≥ A(O), for any compact set K ⊂ Rd and open set O ⊂ Rd . The vague convergence is metrisable and N is a Polish space, cf. [Res87, Prop. 3.17].
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A configuration A = i∈I δai ∈ N is called connected if the set {ai : i ∈ I } is connected in the sense explained in Sect. 1.4. We denote by N R(0) the set of all configurations A ∈ N which are connected and such that 0 ∈ A. It is easy to see from [Res87, Prop. 3.13] that the set N R(0) is closed in N . Hence N R(0) is a Polish space. We equip N with the Borel σ -algebra and denote by M1 = M1 (N R(0) ) the set of all probability measures on N R(0) . We equip M1 with the weak topology and observe that it is Polish and so is N R(0) (see [Bil68, App. III]). In the sequel, it will often be more convenient to work with atomic measures in M1 that are concentrated on finitely many finite configurations without multiple points and without points at the critical distance R. Observe that the clouds [X i(N ) ] have this property with probability one; the shifted clouds have this property too, and they additionally (0) contain zero. Denote by N R, the set of all finite configurations A ∈ N R(0) which satisfy A({x}) = 1, for any x ∈ A, and |x − y| = R, for any x, y ∈ A. Let m m (0) pi δ Ai : m ∈ N, pi = 1, and pi > 0, Ai ∈ N R, for all i M1, := Y = i=1
i=1
denote the set of all convex combinations of finitely many Dirac measures configura(0) tions in N R, . Observe that Y N belongs to the space M1, with probability one but it additionally has a lot of symmetries. Namely, for each A ∈ N R(0) and a ∈ A one has Y N ({A − a}) = Y N ({A}). Hence, with probability one, Y N is an element of (0) M(0) 1, := Y ∈ M1, : Y ({A − a}) = Y ({A}) for each A ∈ N R and a ∈ A . (0) Let us denote by M(0) 1 the closure of M1, in M1 . It will be the main state space for our analysis. Finally, for ν > 0 and n ∈ N, denote (0) (0) N R,n,ν := A ∈ N R, : # A ≤ n, |x − y| > ν for any distinct x, y ∈ A . (2.1)
Our last preparation is the following continuity property. Lemma 2.1. The map A → # A from N R(0) to N ∪ {∞} is continuous in the vague topology. Proof. Let {An : n ∈ N} be a sequence in N R(0) converging to some A ∈ N R(0) . Since Rd is open we have lim inf n→∞ # An = lim inf n→∞ An (Rd ) ≥ A(Rd ) = # A. If # A = ∞ we are done. If # A < ∞ we need to prove that lim supn→∞ # An ≤ # A. If this is not the case then # An ≥ # A + 1 for infinitely many n and, for those n, An (K ) ≥ # A + 1, where K is a closed ball of radius R(# A + 1) centred at zero. Then lim supn→∞ An (K ) ≥ # A + 1 > A(Rd ) ≥ A(K ) which is a contradiction. 2.2. The large deviation principle. Now we formulate the main result of this section, the large deviation principle for the empirical measures Y N introduced in (1.12). Let L , J : M1 → [0, ∞) be defined by 1 and J (Y ) := 1 − L(Y ), (2.2) L(Y ) := Y (d A) (0) # A NR where we recall our conventions 1/∞ = 0 and 1/0 = ∞.
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Proposition 2.2 (Large deviation principle for Y N ). (i) The sequence (Y N ) N ∈N satisfies a large deviation principle on the space M(0) 1 with speed N log(1/ N ) and rate function J , that is, the following holds. For any open set O ⊂ M(0) 1 , lim inf N →∞
1 log P N (Y N ∈ O) ≥ − inf J (Y ), Y ∈O N log(1/ N )
(2.3)
and for any closed set C ⊂ M(0) 1 , lim sup N →∞
1 log P N (Y N ∈ C) ≤ − inf J (Y ). Y ∈C N log(1/ N )
(2.4)
Furthermore, J is continuous. (0) (ii) Let O ⊂ M(0) 1 be an open set and Y ∈ M1, ∩ O. Denote n(Y ) = max {# A : Y ({A}) > 0} and pick ν < ν(Y ), where ν(Y ) = min { |x − y| : x, y ∈ A for some A with Y ({A}) > 0 and x = y} . Then lim inf N →∞
1 (0) = 1 ≥ −J (Y ). log P N Y N ∈ O, Y N N R,n(Y ),ν N log(1/ N )
Proposition 2.2 shows that the distribution of Y N depends, on the exponential scale N log(1/ N ), only on the distribution of the component sizes, i.e., on the distribution of the mapping qˆ = (qˆκ )κ∈N : M1 → [0, 1]N defined by its coordinates qˆκ (Y ) = Y ({A : # A = κ}),
for κ ∈ N.
(2.5)
This can be understood as follows. Given q, the probability that the configuration {X 1 , . . . , X N } has about N qκ /κ components of size κ for each κ ∈ N, is determined by the probability to place, for any κ ∈ N, N qκ /κ points somehow into N and to control the combinatorics coming from the possible choices of the indices 1, . . . , N . The excluded-volume effect requiring that these points are sufficiently distant from each other may be ignored when deriving upper bounds and has a negligible effect because of the diluteness. Placing the remaining points in such a way that components of the required sizes arise has an effect on the scale e−O(N ) , which is negligible. This explains why the large-deviation rate function is only a function of the numbers qˆκ (Y ). We refer to (2.3) and (2.4) as to the lower bound for open sets and the upper bound for closed sets, respectively. The version of the large deviation lower bound in Proposition 2.2 (ii) is used in Sect. 4.2, where we apply Varadhan’s lemma to a functional without sufficient continuity properties. Observe that the continuity of J follows from the continuity of L, which itself immediately follows from Lemma 2.1. In Sect. 2.3 we prove (ii) and in Sect. 2.4 we prove the upper bound for closed sets. Observe that the lower bound for open sets immediately (0) follows from (ii) by the continuity of J , since M(0) 1, is dense in M1 . By Br (x) we d denote the Euclidean ball of radius r > 0 around x ∈ R .
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(0) 2.3. Proof of Proposition 2.2 (ii). Let O ⊂ M(0) 1 be an open set, and let Y ∈ M1, ∩ O. Let us write Y in the form
Y =
m
pi δ Ai ,
i=1 (0) where m ∈ N, p1 , . . . , pm > 0 add up to one, and the configurations Ai ∈ N R, are (0) pairwise distinct. The symmetries of Y as a member of the space M1, imply that any two atoms of Y that are shifts of each other appear with the same probability. This allows us to introduce an equivalence relation on the integers {1, . . . , m}: we say that i ∼ j if there is x ∈ Ai such that A j = Ai − x. It is easy to see that this is indeed an equivalence relation: i ∼ i as 0 ∈ Ai ; if i ∼ j then j ∼ i since 0 ∈ Ai and so −x ∈ A j ; if i ∼ j and j ∼ k then A j = Ai − x for some x ∈ Ai and Ak = A j − y for some y ∈ A j , which implies Ak = Ai − x − y, and it suffices to notice that x + y ∈ Ai since y =
z − x for (0) some z ∈ Ai . For each configuration A ∈ N R, we denote by b(A) = #1A a∈A a its barycentre. If i ∼ j then # Ai = # A j , Ai − b(Ai ) = A j − b(A j ) and pi = p j . Denote the set of equivalence classes by C and, for any c ∈ C, define E c = Ai − b(Ai ) and qc = pi for some i ∈ c. For technical reasons, we extend the set C by one extra element c and write C = C ∪ {c }. Finally, we denote E c = {0} and qc = 0. Hence, we may write qc δ E c −e . (2.6) Y =
c∈C
e∈E c
For any N , let kc(N ) = N qc if c ∈ C and kc(N ) = N − c∈C kc(N ) #E c , also put k (N ) =
(N ) c∈C kc . Below we prescribe certain ways to place N points x1 , . . . , x N into N such that, (0) for large N , the resulting measure Y N(x) lies in O and Y N(x) (N R,n(Y ),ν ) = 1. (N ) Roughly speaking, we pick, for any c ∈ C , precisely kc places in N at which we put slightly perturbed copies of the set E c ; let {x1 , . . . , x N } be the union of these copies. If all these places are not too close to each other, then these copies are its connected components, and Y N(x) ≈ Y . Afterwards, we give a lower bound for the total mass of the choices of these places and the choices of the perturbed copies under the N -fold product of the uniform distribution on N . Using elementary combinatorics, we show that the exponential rate of this probability is bounded from below by −J (Y ). The role of c is the following. Having taken care of the components that appear with positive probability in Y , we have distributed only c∈C kc(N ) #E c points; the remaining kc(N ) points will be placed by default at the origin. Let us turn to the details. Denote r = max { |x − y| : x, y ∈ Ai for some i} ∈ [0, ∞), ρ = max { |x − y| : x, y ∈ Ai for some i and |x − y| < R} ∈ [0, R). Observe that ρ < R since Y ∈ M(0) 1, , and that |e| ≤ r for any e ∈ E c for any c ∈ C . Let = 2r + 2R + 1 and define the grid
D N := x ∈ Zd ∩ N : dist(x, ∂ N ) > r + R
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A. Collevecchio, W. König, P. Mörters, N. Sidorova
and observe that #D N ≥ −d | N | (1 + o(1)). Let (N ) Z (N ) := z = (z c,i : c ∈ C , i ≤ kc(N ) ) ∈ (D N )k : z c,i = z c ,i for (c, i) = (c , i ) , the set of vectors of places where the perturbed copies of the E c will be located. Note that, for any z ∈ Z (N ) , for (c, i) = (c , i ), the distance between z c,i and z c ,i is larger than 2r + 2R. Observe that, as N ↑ ∞, #Z (N ) =
(N ) k
(N ) k (N ) k 1− #D N (k (N ) )2 (N ) O(1) = | N |k eo(N log(1/ N )) . exp − | N |
[#D N − j + 1] ≥ (#D N )k
j=1
≥ | N |k
(N )
e O(N )
(N )
(2.7)
For each z ∈ Z (N ) , denote (N )
(N )
σz
=
c k
z c,i + supp(E c )
c∈C i=1
and
Sz(N ) =
δx ∈ N .
(N ) x∈σz
Since all E c do not have multiple points, and due to our choice of Z (N ) , the configuration Sz(N ) consists of N distinct points of mass 1 belonging to N , which form kc(N ) connected components of size #E c for each c ∈ C . Later we will introduce a perturbation of the set σz(N ) to obtain the set {x1 , . . . , x N } mentioned in the rough explanation below (2.6). Fix 0 < ρ < min{ν(Y ) − ν, R − ρ }. Observe that the ρ/2-balls around the points of Sz(N ) lie in N and have distance larger than ν to each other. The former follows from ρ/2 + r < r + R and the latter from ρ + ν < ν(Y ) for balls centred at points belonging to the same cloud and from ρ + 2r + ν < 2r + 2R (provided by the conditions ν < R and ρ < R) for balls centred at points belonging to different clouds. Further, if y1 , y2 ∈ Sz(N ) are such that |y1 − y2 | < R, then the same is true for any y1 ∈ Bρ/2 (y1 ), y2 ∈ Bρ/2 (y2 ). Indeed, |y1 − y2 | < R implies |y1 − y2 | ≤ ρ and | y1 − y2 | < R follows from ρ + ρ < R. On the other hand, if y1 , y2 ∈ Sz(N ) lie in different clouds, then |y1 − y2 | ≥ (2r + 2R) − 2r = 2R. Since ρ < R we obtain | y1 − y2 | ≥ 2R − ρ > R for any y1 ∈ Bρ/2 (y1 ), y2 ∈ Bρ/2 (y2 ) and also y1 and y2 lie in different clouds. Finally, (0) the case | y1 − y2 | = R is impossible since Y is concentrated on configurations in N R, .
For any A = i∈I δxi ∈ N and any ε > 0, we denote A(ε) = δ xi : | xi − xi | < ε for all i ∈ I . i∈I
The preceding arguments imply that any configuration S ∈ Sz(N ) (ρ/2), like Sz(N ) itself, consists of N points which form kc(N ) clouds of size #E c for each c ∈ C . In particular, S does not contain clouds of size larger than n(Y ) since #E c ≤ n(Y ) for all c. Moreover, S has no multiple points, and any two distinct points of S have distance larger than ν to each other. Finally, any two points of S belonging to the same cloud have distance smaller than R to each other. We write {x1 , . . . , x N } for the support of S. This is the set mentioned in the rough explanation below (2.6). Recall that the measure Y N(x) is defined (0) in (1.12). Summarising, we have that Y N(x) is concentrated on the set N R,n(Y ),ν .
Dilute Particle System with Lennard-Jones Potential
615
Observe that Y N(x) can be written as (N )
(x)
YN
kc 1 = δ E c,i − e(e,c,i) , N ∗ c∈C i=1 e∈E c
where E c,i ∈ E c (ρ/2) and e(e, c, i) ∈ E c,i is uniquely determined by the condition | e(e, c, i) − e| < ρ/2. Let us show that Y N(x) ∈ O for any support x = {x1 , . . . , x N } of an S ∈ Sz(N ) (ρ/2) with z ∈ Z (N ) . There is an open set UY ⊂ O of the form UY =
n
∈ M(0) : Y 1
(d A) F j (A) − Y
Y (d A) F j (A) < η ,
j=1
where η > 0, n ∈ N and F1 , . . . , Fn : N R(0) → R are continuous and bounded. Let M be such that |F j (A)| ≤ M for all A and all j ∈ {1, 2, . . . , n}. Denote q = min{qc : c ∈ C}. We have, for N ≥ 1/q,
(x)
Y N (d A) F j (A) −
Y (d A) F j (A)
(N )
kc 1 = F j E c,i − e(e, c, i) − qc F j (E c − e) N ∗ c∈C i=1 e∈E c
c∈C e∈E
(N )
kc 1 qc ≤ F j E c,i − e(e, c, i) − (N ) F j (E c − e) N kc c∈C e∈E i=1 c
(N ) kc
+
1 F j (E c ,i − e(0, c , i)) . N i=1
Observe that E c,i − e(e, c, i) ∈ (E c − e)(ρ) = Al (ρ) for some l ≤ m. Since the functions F1 , . . . , Fn are continuous and since the configurations A1 , . . . , Am are finite, there is ε > 0 such that A ∈ Ai (ε) implies |F j (A) − F j (Ai )| < η/2 for all i, j. We further require that ρ < ε, then we have η |F j E c,i − e(e, c, i) − F j (E c − e)| < 2 for all j, c, i, e. Further, notice that 1 qc qc 1 qc 1 1 − (N ) = − ≤ − ≤ . N N qc N N qc − 1 N N (N q − 1) kc
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Using the boundedness of all F j by M we obtain
Y N(x) (d A) F j (A) − kc ! 1 (N )
≤
c∈C e∈E c i=1
Y (d A) F j (A) F j E c,i − e(e, c, i) − F j (E c − e)
N
1 qc − (N ) N kc
+
" F j (E c − e)
+
kc(N ) M N
(N ) " kc ! k (N ) M k (N ) M 1 η M η M ≤ + + c = + + c . N 2 Nq − 1 N 2 Nq − 1 N
c∈C e∈E c i=1
Observe that kc(N ) = N −
kc(N ) #E c = N −
c∈C k
N qc #E c ≤
c∈C
#E c ,
(2.8)
c∈C
(N ) M
which implies that c N → 0, since the right hand side of (2.8) does not depend on N . M → 0 as well, there is N0 such that for all N ≥ N0 , Since N q−1
(x)
Y N (d A) F j (A) −
Y (d A) F j (A) < η
for all j ≤ n. This proves that, for N ≥ N0 , Y N(x) ∈ O for any S ∈ Sz(N ) (ρ/2), z ∈ Z (N ) . ) Recall that X 1(N ) , . . . , X (N N under P N are independent and chosen uniformly from the set N . For N ≥ N0 , we have ⎛ ⎞ (0) )⎠ ⎝ (N , (2.9) P N Y N(X ) ∈ O, Y N(X ) N R,n(Y z ),ν = 1 ≥ P N z∈Z (N )
where (N )
z :=
N i=1
(N )
δ X (N ) ∈ Sz (ρ/2) . i
) For a fixed z, (N z is the event that each of N fixed non-intersecting balls of radius ρ/2 ) contains exactly one of the points X 1(N ) , . . . , X (N N . Therefore
|Bρ/2 (0)| N ) . = N ! P N (N z | N | ) (N ) Two events (N z and z are either equal or disjoint. Equality holds if and only if (N ) {z c,i : i ≤ kc } = {z c,i : i ≤ kc(N ) } for all c ∈ C with #E c > 1 and additionally
Dilute Particle System with Lennard-Jones Potential
617
: i ≤ k (N ) , #E = 1}. Hence we can pick a subset of {z c,i : i ≤ kc(N ) , #E c = 1} = {z c,i c c (N ) Z consisting of #Z (N ) (N ) ⎛ ⎞ ≥ |n |k exp −N log N ⎞⎛ qc eo(N log(1/ N )) c∈C ⎜ (N ) ⎟ ⎜ (N ) ⎟ k c !⎠ ⎝ kc ⎠! ⎝ c∈C #E c >1
c∈C #E c =1
) elements such that the corresponding events (N z are pairwise disjoint. For the lower bound we used (2.7). Together with (2.9) we obtain, for N ≥ N0 , (0) = 1 P N Y N(X ) ∈ O, Y N(X ) N R,n(Y ),ν |Bρ/2 (0)| N (N ) ≥ |n |k exp −N log N qc eo(N log(1/ N )) N ! | N | N c∈C (N ) = exp k log | N |− N log N qc + N log N − N log | N |+o(N log(1/ N )) .
c∈C
Recall that c∈C
qc =
m pi = L(Y ), # Ai i=1
and hence k (N ) = N L(Y ) + O(1). This gives k (N ) log | N | − N log N qc + N log N − N log | N | c∈C
= (L(Y ) − 1) (N log | N | − N log N ) + O(log | N |) 1 N = −J (Y )N log + O(log ). N N Obviously O(log NN ) is also o(N log(1/ N )). Hence lim inf N →∞
1 (0) log P N Y N ∈ O, Y N N R,n(Y = 1 ≥ −J (Y ), ),ν N log(1/ N )
which finishes the proof of Proposition 2.2 (ii). 2.4. Proof of Proposition 2.2 (i), upper bound. Now we show the upper bound for closed sets. As a preliminary step, we estimate the probability P N (q(Y ˆ N(X ) ) = q) from above for any q ∈ [0, 1]N , where qˆ is as defined in (2.5). By the discrete nature of Y N(X ) , this probability is nonzero only when q lies in the set N N qi (N ) N ∈ N ∪ {0} for all i ≤ N , qi = 0 for all i > N . Q = q ∈ [0, 1] : qi = 1, i i=1
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A. Collevecchio, W. König, P. Mörters, N. Sidorova
Substituting ki = N qi /i and noting that ki ≤ N /i, the cardinality of this set can be estimated by N N N (2N ) N (N ) N #Q = # (k1 , . . . , k N ) ∈ N0 : iki = N ≤ +1 ≤ ≤ e2N . i N! i=1
i=1
(2.10) (N )
(N )
Fix q ∈ Q(N ) . On the event {q(Y ˆ N ) = q}, the points X 1 , . . . , X N form ki = N qi /i connected components of size i, for each i ≤ N , and no connected components of size
N larger than N . The number of ways to decompose {1, . . . , N } into i=1 ki sets such that there are ki sets of cardinality i, for any i, is equal to (
N! ) k . i k !)( i i i i! )
)
We bound the probability that such a decomposition represents the connected compo) nents of {X 1(N ) , . . . , X (N N } by the probability that each partition set is just connected. Therefore, using independence, we obtain P N
N k N! (X ) ) ) q(Y ˆ N )=q ≤ P N {X 1(N ) , . . . , X i(N ) } is connected i . ( i ki !)( i i!ki ) i=1
(N )
(N )
If {X 1 , . . . , X i } is connected, there exists a labelled tree on {1, . . . , i} with root in 1, such that for every edge ( j, k) in the tree, we have ) X k(N ) ∈ B R (X (N j ).
By Cayley’s theorem, see [AZ98, pp. 141–146], the number of labelled trees with i vertices is i i−2 , and therefore we can estimate * (i−1)ki + N 1 i|B (0)| R P N q(Y ˆ N(X ) ) = q ≤ N ! ki !(i!)ki | N | i=1 ki N + O(N ) , ≤ N exp − ki log | N | i
N uniformly in q ∈ Q(N ) , using the convention 0 log 0 = 0 and that i=1 iki = N . ∞ qi Let t ∈ (0, 1). Observe from (2.5) and (2.2) that L(Y N ) = i=1 i on the event {q(Y ˆ N ) = q}. Hence log P N (L(Y N ) ≤ t) = log
P N (q(Y ˆ N ) = q)
q∈Q(N )
i qi /i≤t
⎫⎤ ⎧ ⎡ ∞ ⎬ ⎨ (N ) q i ≤t ⎦ max P N q(Y ˆ N ) = q : q ∈ Q(N ) , ≤ log ⎣ #Q ⎭ ⎩ i i=1
≤ N log N − inf
k∈D N
N i=1
ki log
ki + O(N ), | N |
(2.11)
Dilute Particle System with Lennard-Jones Potential
where we introduced
D N = k ∈ [0, ∞) : N
N
619
iki = N ,
i=1
N
ki ≤ t N .
i=1
We show that inf
k∈D N
N
ki log
i=1
ki 1 , ≥ t N log N + o N log | N | N
for N → ∞.
(2.12)
Indeed, substituting ri = ki /N , it suffices to show that N i=1
1 ri log( N ri ) ≥ t log N + o log , N
N
N iri = 1 and i=1 ri ≤ t. Distinguishing the whenever r1 , . . . , r N ≥ 0 satisfy i=1 alternatives log(1/ri ) ≤ i and ri < e−i and using that x → x log(1/x) is increasing near the origin, we get N i=1
ri log(1/ri ) ≤
N i=1
iri +
N
ie−i ≤ 1 +
i=1
∞
ie−i .
i=1
N
Therefore i=1 ri log ri is bounded from −∞, which proves (2.12). Inserting (2.12) into (2.11) implies that 1 1 , for N → ∞. (2.13) + o N log log P N (L(Y N ) ≤ t) ≤ −(1 − t)N log N N Now let C ⊂ M(0) 1 be a closed set. Let s = inf C J ∈ [0, 1]. If s = 1 then C contains only infinite configurations and so P N (Y N ∈ C) = 0. If s = 0 then the upper bound is obviously satisfied. Assume that s ∈ (0, 1). We have log P N (Y N ∈ C) ≤ log P N (J (Y N ) ≥ s) = log P N (L(Y N ) ≤ 1 − s) 1 1 ≤ −s N log + o N log N N 1 inf J + o(1) , = −N log for N → ∞. C N This finishes the proof of the upper bound for closed sets. 3. Analysis of ϕ(n) In this section we prove Lemma 1.1 and provide some properties of ϕ(n) introduced in Sect. 1.4. Recall that ν0 = inf{x > 0 : v(x) < ∞} and v(ν0 ) = ∞. For n ≥ 1, denote Dn = x = (x1 , . . . , xn ) ∈ (Rd )n : |xi − x j | > ν0 for all i = j , (0) from (2.1). so that ϕ(n) = inf x∈Dn Vn (x). Further recall the definition of N R,n,ν 0
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A. Collevecchio, W. König, P. Mörters, N. Sidorova
Lemma 3.1. (i) For each n, there is a minimiser x (n) ∈ Dn such that Vn (x (n) ) = ϕ(n). ϕ(n) (ii) limn→∞ ϕ(n) n = ϕ˜ ∈ (−∞, 0) and n > ϕ˜ for all n ∈ N. (n,m) (iii) For each n, there is a sequence (x : m ∈ N) in Dn such that Vn (x (n,m) ) → ϕ(n) as m → ∞, and n i=1
(0) δx (n,m) ∈ N R,n,ν , 0 i
for all m ∈ N.
Proof. (i) Denote Dn = {x ∈ Dn : x1 = 0 and |xi | ≤ n R for all i}. Observe that ϕ(n) = inf x∈Dn Vn (x). Indeed, Vn is invariant under parallel translations, which allows to fix the first variable to be zero. Further, since v is strictly negative on (R − ν1 , R) and the system is invariant under rotations and translations, only the points x with connected
n configurations i=1 δxi contribute to the infimum. Indeed, if the configuration has more than one connected component then two of them can be rotated and translated in such a way that exactly one new negative interaction (at distance close to R) occurs. The set Dn is bounded and, by Assumption (V) (2), the function Vn is continuous on Dn . As ∂Dn \Dn has components of distance precisely ν0 we infer from Assumption (V) (1) that Vn (x) → ∞ as x → y ∈ ∂Dn \Dn . Hence Vn has a minimum x (n) on Dn , which is also a minimum on Dn . (ii) We first show subadditivity of the sequence (ϕ(n) : n ∈ N). Fix n, m ∈ N. If y (n) and y (m) denote any two configurations of n resp. m points in Rd , then we obtain a configuration y (n+m) of n + m points by putting the two configurations at distance R to each other, such that they do not interact. Then Vn+m (y (n+m) ) = Vn (y (n) )+ Vm (y (m) ). Passing to the infimum of Vn resp. Vm over y (n) and y (m) yields that ϕ(n + m) ≤ ϕ(n) + ϕ(m). Hence ϕ(n) (ϕ(n) : n ∈ N) is subadditive and so ϕ˜ = limn→∞ ϕ(n) n exists and satisfies ϕ˜ ≤ n , for all n ∈ N. Now suppose that for some n we have ϕ(n) ˜ Let y (n) be a translated and rotated n = ϕ.. copy of x (n) such that there is exactly one pair (i, j) with R > |xi(n) − y (n) j | > R − ν1 . Then, by Assumption (V) (4), ϕ(2n) ≤ V2n (x (n) , y (n) ) < Vn (x (n) ) + Vn (x (n) ) = 2ϕ(n) = 2n ϕ˜ and so ϕ(2n) ˜ which contradicts the fact that ϕ˜ = inf n ϕ(n) 2n < ϕ, n . Further, ϕ˜ ≤ ϕ(2)/2 = 1 min v < 0 and so it remains to prove that ϕ ˜ > −∞. This is easy in the case [0,∞) 2 (n) ν0 > 0, since all the points xi have distance ≥ ν0 to each other, and, by a comparison of volume, each of these points interacts only with a number of other particles that is bounded in n. Hence, the total number of interactions is of order n, and ϕ(n) is of order n as well, resulting in ϕ˜ > −∞. Now suppose ν0 = 0. Let In,i : Dn → R be defined by In,i (x) = for 1 ≤ i ≤ n. Then Vn (x) =
n
n
v(|xi − x j |)
j=1 j =i
i=1 In,i (x).
For each n, let
In = max In,i (x (n) ). 1≤i≤n
(3.1)
The main step of the proof is to show that the sequence (In : n ∈ N) is bounded from below.
Dilute Particle System with Lennard-Jones Potential
621
(n) For each n, choose i(n) ∈ {1, . . . , n} in such a way that the ball Bν2 (xi(n) ) contains the (n) (n) maximal number, k(n), of points x1 , . . . , xn . In particular, no ball of radius ν2 /2 contains more than k(n) of the points x1(n) , . . . , xn(n) . Introduce ν3 = inf{x > 0 : v(x) = 0} (n) , but have and denote by m(n) the number of points x1(n) , . . . , xn(n) that interact with xi(n) distance ≥ ν3 to this point. Since v is nonnegative on [ν2 , ν3 ], we obtain
In ≥ In,i(n) (x (n) ) ≥ (k(n) − 1) min v + m(n) min v. [0,ν2 ]
[0,∞)
Hence m(n) ≥
In − (k(n) − 1) min[0,ν2 ] v . min[0,∞) v
Recall the function s defined in Sect. 1.2. By scaling, for any r > 0, s(r/R) is the minimal number of balls of radius r required to cover a ball of radius R. By definition (n) of s , we have (r/R)d s(r/R) ≤ s . Cover the ball B R (xi(n) ) with a least number of balls of radius ν2 /2. By choice of k(n), each of these balls contains at most k(n) of the (n) points x1(n) , . . . , xn(n) . Hence the total number of points x (n) j in B R (x i(n) ) is bounded by k(n)s(ν2 /2R), which is not larger than k(n)(2R/ν2 )d s . In particular, In − (k(n) − 1) min[0,ν2 ] v ≤ m(n) ≤ k(n)(2R/ν2 )d s min[0,∞) v and so ! In ≥ k(n) min v [0,ν2 ]
+ ν2−d (2R)d s
" min v − min v.
[0,∞)
[0,ν2 ]
By Assumption (V) (5), the expression in brackets is nonnegative. Using that k(n) ≥ 1 we obtain, for all n, In ≥ ν2−d (2R)d s min v =: c > −∞. [0,∞)
For each n, let j (n) be the index where the maximum in (3.1) is attained, then we have (n) (n) ϕ(n) = Vn−1 (x1(n) , . . . , x j (n) , . . . , x n ) + 2In ≥ ϕ(n − 1) + 2c ≥ · · · ≥ 2c(n − 1),
where the hat indicates the dropped component. Hence ϕ˜ ≥ 2c > −∞.
n (iii) It has already been argued that there is a minimiser x (n) ∈ Dn for which i=1 δx (n) i
is connected and contains zero. Obviously, it can be approximated by x (n,m) ∈ Dn having both these properties and additionally having no points at distance R. Since Vn is continuous the statement follows.
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A. Collevecchio, W. König, P. Mörters, N. Sidorova
4. Asymptotics of Z N In this section we complete the proof of Theorem 1.3. Recall from (1.13) that the partition function N !| N |−N Z N (βn , N ) is the expectation of an exponential of (Y N ), for : M(0) 1, → (0, ∞] defined by (Y ) :=
Y (d A) W (A),
where W (A) :=
1 v(|x − y|). # A x,y∈A x = y
We would like to apply Varadhan’s lemma for the large deviation principle for Y N established in Proposition 2.2 and the function . However, for both bounds there are technical obstacles. For the upper bound, compactness of the level sets of the rate function J would be required, which is missing. For the lower bound, upper semicontinuity of would be required. But W is only defined on finite configurations and has no upper-semicontinuous extension to the whole space M(0) 1 . In case of hard core interactions, a further problem is that (Y ) is infinity if Y gives positive weight to configurations A with points close to each other. In Sect. 4.1 we prove the upper bound by estimating (Y N ) from below by (q(Y ˆ N )), where is a bounded and continuous function on a compact space and q(Y ˆ ) = (Y ({A : # A = n}) : n ∈ N). By projection, we derive a large deviation principle for q(Y ˆ N ) from the large deviation principle for Y N . We then apply Varadhan’s lemma to the function . In Sect. 4.2 we prove the lower bound by first restricting the expectation to the event that the measure Y N is concentrated on configurations with a bounded number of well-separated points. On this event coincides with a cut-off approximation, which is continuous and bounded on M(0) 1 . Proposition 2.2 (ii) shows that the restriction of Y N to this event satisfies the same large deviation lower bound. Hence we can apply the lower bound of Varadhan’s lemma on this event, and we finish by showing that the restricted variational formula has the same value as the original one. 4.1. Proof of the upper bound. We endow [0, 1]N with the topology of pointwise convergence and recall that it is compact. Further denote ∞ N qi ≤ 1 , Q = q ∈ [0, 1] : i=1
and note that Q is closed, by Fatou’s lemma, and therefore compact. Let : Q → R be defined by 1 0 ∞ ∞ ∞ ϕ(n) ϕ(n) qn qn = qn (q) = + ϕ˜ 1 − − ϕ˜ + ϕ. ˜ n n n=1
n=1
n=1
We recall from (2.5) the definition of the mapping qˆ : M1 → Q given by qˆn (Y ) = Y ({A : # A = n}). Lemma 4.1. (i) For any real sequence (bn : n ∈ N) converging to zero the mapping q → ∞ n=1 qn bn is continuous on Q. In particular, is continuous.
Dilute Particle System with Lennard-Jones Potential
623
(ii) The sequence (q(Y ˆ N ) : N ∈ N) satisfies a large deviation principle on the space Q with speed N log(1/ N ) and rate function H : Q → [0, ∞) given by H (q) := 1 −
∞ qn n=1
n
.
The rate function H is continuous, and the level sets of H are compact. Proof. (i) Let q (i) → q in Q as i → ∞. For any ε > 0 let n 0 be such that |bn | < ε/4 for all n ≥ n 0 . There is i 0 such that |qn(i) − qn | < ε(2n 0 (maxk |bk | + 1))−1 for all i ≥ i 0 for all n ≤ n 0 . For all i ≥ i 0 we obtain ∞ n=1
qn(i) bn −
∞
qn bn ≤ max |bk | k
n=1
n0 n=1
|qn(i) − qn | +
ε (i) (q + qn ) < ε. 4 n>n n 0
Therefore, is continuous, as ϕ(n)/n → ϕ˜ by Lemma 3.1. (ii) For any n ∈ N, let Cn = {A ∈ N R(0) : # A = n}, then the indicator function since the mapping A → # A is continuous by 1Cn : N R(0) → {0, 1} is continuous 2 Lemma 2.1. Since qˆn (Y ) = 1Cn dY for any Y , the map qˆn is continuous. Hence qˆ is continuous. The statement of the lemma follows now from the contraction principle (see [DZ98, Theorem 4.2.1]) since, for any q ∈ Q, J is constant on the set {Y : q(Y ˆ ) = q} and equal to H (q). The continuity of H follows from (i). The level sets of H are closed, and hence compact as Q is compact. Observe that if # A = n and A({a}) = 1 for all a ∈ A then W (A) ≥ ϕ(n)/n. Because
∞ Y N ∈ M(0) n=1 qˆn (Y N ) = 1, for all N ∈ N, we obtain 1, with probability 1 and (Y N ) ≥
∞ n=1
qˆn (Y N )
ϕ(n) = (q(Y ˆ N )). n
Therefore Z N (β N , N ) ≤
| N | N E N exp −Nβ N (q(Y ˆ N )) . N!
Recall that Nβ N = 1c N log(1/ N ). Observe that inf Q ≥ ϕ˜ > −∞ by Lemma 3.1. Since is continuous and nonnegative and q(Y ˆ N ) satisfies a large deviation principle with speed N log(1/ N ) and rate function H (with compact level sets) by Lemma 4.1, we can use the upper bound in Varadhan’s lemma, see [DZ98, Lemma 4.3.6], which implies 1 1 lim sup log E N exp −Nβ N (q(Y + H . ˆ N )) ≤ − inf Q c N →∞ N log(1/ N ) Using (1.14) we get lim sup N →∞
1 log Z N (β N , N ) ≤ − inf { + c(H − 1)} . Nβ N Q
624
Let q∞ = 1 −
A. Collevecchio, W. König, P. Mörters, N. Sidorova
∞
n=1 qn .
Then we see that
inf { + c(H − 1)} = inf Q
⎧ ⎨
q∈Q ⎩ n∈N
= inf
⎧ ⎨ ⎩
⎫ ⎞ ⎛ qn ⎬ ϕ(n) qn qn ⎠ − c + ϕ˜ ⎝1 − n n ⎭
n∈N
n∈N
qn : q ∈ [0, 1]N∪{∞} , qn θn − c n
n∈N∪{∞}
n∈N
n∈N∪{∞}
⎫ ⎬ qn = 1 , ⎭
(4.1) which is equal to the right hand side of (1.8). 4.2. Proof of the lower bound. Recall that v is continuous with a negative minimum and let ν3 = inf{x > 0 : v(x) = 0} ∈ (ν0 , ν1 ). For any ν ∈ (ν0 , ν3 ), denote vν (x) = min{v(x), v(ν)} for x ≥ 0. For each such ν and n ∈ N consider Wn,ν : N R(0) → R defined by ⎧ k k ⎪ ⎪ ⎨ 1 vν (|xi − x j |) if A = δxi and k ≤ n, Wn,ν (A) = # A i, j=1 i=1 ⎪ i = j ⎪ ⎩ 0 otherwise, and denote n,ν : M(0) 1 → R by n,ν (Y ) =
Y (d A) Wn,ν (A).
Lemma 4.2. For each ν ∈ (ν0 , ν3 ) and n, n,ν is well-defined and continuous. Proof. Observe that −∞ < min[0,∞) v ≤ vν (x) ≤ max[ν,ν3 ] v < ∞ for all x ≥ 0 and so, for each A, n min v ≤ Wn,ν (A) ≤ n max v. [0,∞)
[ν,ν3 ]
Hence n,ν is well-defined. To prove the continuity of n,ν it suffices to show the continuity of Wn,ν . Let A(i) → A in N R(0) as i → ∞. If # A > n then # A(i) > n eventually by Lemma 2.1 and so Wn,ν (A(i) ) = Wn,ν (A) = 0. If # A = k ≤ n then eventually # A(i) = k. It follows
from [Res87, Prop. 3.13] that if A = kj=1 δa j then eventually A(i) = kj=1 δai, j , where ai, j → a j as i → ∞ for all j ≤ k. Now the continuity of Wn,ν obviously follows from the continuity of vν . (0) Note that W and Wn,ν agree on N R,n,ν , for each n ∈ N and ν ∈ (ν0 , ν3 ). Using (1.13) we obtain, for each n and ν, that 3 4 || N (0) E N exp −Nβ N n,ν (Y N ) 1 Y N (N R,n,ν Z N (β N , N ) ≥ ) = 1 . (4.2) N!
Denote
) > ν0 , ∈ M(0) : ν(Y = M(0) Y 1,,v 1,
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where ν(Y ) and n(Y ) have been defined in Proposition 2.2(ii). Let ε > 0 and Y ∈ M(0) 1,,v . Pick ν ∈ (ν0 , min{ν(Y ), ν3 }). Since n(Y ),ν is continuous at Y by Lemma 4.2, there is an open set O ⊂ M(0) 1 such that Y ∈ O and n(Y ),ν (Y ) < n(Y ),ν (Y ) + ε for all Y ∈ O. Recall that log(1/ N ) = cβ N . We obtain, using (4.2), (1.14) and Proposition 2.2(ii), 1 log Z N (β N , N ) N →∞ N log(1/ N ) 1 ≥ 1 + lim inf N →∞ N log(1/ N ) 3 4 (0) = 1 × log E N exp −Nβ N n(Y ),ν (Y N ) 1 Y N ∈ O, Y N N R,n(Y ),ν
lim inf
1 1 n(Y ),ν (Y ) + ε + lim inf N →∞ N log(1/ N ) c (0) = 1 × log P N Y N ∈ O, Y N N R,n(Y ),ν
≥1−
1 ε ≥ − n(Y ),ν (Y ) − + L(Y ). c c (0) Observe that n(Y ),ν (Y ) = (Y ) for each Y , as it is concentrated on N R,n(Y ),ν . Since ε and Y are arbitrary, we get
lim inf N →∞
1 {(Y ) − cL(Y )} . log Z N (β N , N ) ≥ − inf (0) (0) Nβ N Y ∈M (N ) 1,,v
(4.3)
R
(0) Now we explicitly construct a sequence in M(0) 1,,v (N R ) such that the values of − cL along the sequence approach (c). Let ε > 0. For each n ∈ N, choose m(n) in (n,m) such a way that Vn (x (n,m(n)) ) − ϕ(n) has been defined in Lemma 3.1.
n< ε, where x (n) (n,m(n)) (n) Denote y = x and A = j=1 δ y (n) , which is in N R , by Lemma 3.1. For each j
k ∈ N and q ∈ Q, denote
pn(k,q)
⎧ qn if n ≤ k, ⎪ ⎪ ⎪ k ⎨ q j if n = 2k , = 1− ⎪ ⎪ j=1 ⎪ ⎩ 0 otherwise.
Finally, denote k
Yk,q =
2
pn(k,q)
n=1
n 1 δ A(n) −y (n) j n j=1
(0) and observe that Yk,q ∈ M(0) 1,,v (N R ) by Lemma 3.1. We have
L(Yk,q ) =
∞ qˆn (Yk,q ) n=1
n
0 1 ∞ k k pn(k,q) qn −k = = +2 qn . 1− n n n=1
n=1
n=1
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Further, (Yk,q ) =
∞
k
qˆn (Yk,q ) W (A(n) ) =
n=1
≤
k n=1
qn
ϕ(n) + n
ϕ(2k ) 2k
0
2
pn(k,q)
n=1
1−
k
Vn (y (n) ) n 1
qn + ε.
n=1
We obtain inf
(0)
(0)
Y ∈M1,,v (N R )
=
k n=1
{(Y ) − cL(Y )} ≤ (Yk,q ) − cL(Yk,q )
0 1 0 1 k k k ϕ(n) ϕ(2k ) qn −k + −c2 qn qn + ε − c qn . 1− 1− n 2k n n=1
n=1
n=1
This inequality is satisfied for all k and ε and hence we can take the limit as k → ∞ and ε → 0, which gives 0 1 ∞ ∞ ∞ ϕ(n) qn {(Y + ϕ˜ 1 − . ) − cL(Y )} ≤ inf qn qn − c (0) (0) n n Y ∈M (N ) 1,,v
n=1
R
n=1
n=1
Since this is true for all q ∈ Q we can take the infimum. Using (4.1) and recalling (4.3), we get the lower bound. 5. Analysis of the Variational Formula In this section we prove Theorem 1.5. Recall that θκ = ϕ(κ)/κ for κ ∈ N, θ∞ = ϕ, ˜ and denote by g the largest convex function [0, 1] → [ϕ, ˜ 0] whose graph lies below the points (1/κ, θκ ), see Fig. 2. This graph changes its slope precisely in the points (1/κn , θκn ) with 1 ≤ n < η + 1, but may contain more of these points. In particular, the sequence of slopes cn with 1 ≤ n < η + 1 is strictly decreasing in n. As θκ > ϕ˜ by Lemma 3.1 (ii), all slopes cn of g are strictly positive. We write θ + cn (x − 1/κn ) if x ∈ (1/κn+1 , 1/κn ] for some 1 ≤ n < η + 1, g(x) = κn ϕ˜ if x = 0. Using the convention 1/∞ = 0, we can rewrite (1.8) as ⎧ ⎨ c : q ∈ [0, 1]N∪{∞} , (c) = inf qκ θκ − ⎩ κ κ∈N∪{∞}
κ∈N∪{∞}
⎫ ⎬ qκ = 1 . (5.1) ⎭
For each c, denote by I (c) ⊂ N∪{∞} the set of points where the continuous mapping κ → θκ − c/κ from N ∪ {∞} to R attains its minimum. Lemma 5.1. (c) =
min
κ∈N∪{∞}
θκ −
c , κ
the infimum in (5.1) is a minimum, and the set of minimisers is the convex hull of q(i) with i ∈ I (c).
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Proof. Let q ∈ [0, 1]N∪{∞} be such that κ∈N∪{∞} qκ = 1. If q j > 0 for some j ∈ / I (c) then c c c > min , θκ − θκ − qκ θκ − qκ = min κ κ κ κ∈N∪{∞} κ∈N∪{∞} κ∈N∪{∞}
κ∈N∪{∞}
otherwise equality holds.
Lemma 5.2.
⎧ ⎪ϕ˜ if c ∈ (0, cη ], c ⎨ c = − κn + θκn if c ∈ [cn , cn−1 ) for some 2 ≤ n < η + 1, (5.2) θκ − min ⎪ κ κ∈N∪{∞} ⎩−c if c ∈ (c1 , ∞).
Further, • • • • •
if c ∈ (0, cη ) then I (c) = {∞}, if c = c∞ then I (c) = {∞}; this is only applicable if η = ∞ and c∞ > 0, if c ∈ (cn , cn−1 ), with some 2 ≤ n < η + 1, then I (c) = {κn }, if c ∈ (c1 , ∞) then I (c) = {1}, if c = cn for some 1 ≤ n < η + 1 then I (c) = I (n) .
Proof. It is easy to see that min
κ∈N∪{∞}
θκ −
c = min (g(x) − cx). x∈[0,1] κ
Since g is strictly convex and piecewise linear, the minimum on the right hand side can be found by comparing the derivative of g with c. • If c = cn for some 1 ≤ n < η + 1 then the minimum is attained on [1/κn+1 , 1/κn ] and is equal to θκn − κcn . • If c1 < c then the minimum is attained at x = 1 and is equal to −c. • If cn < c < cn−1 for some 2 ≤ n < η + 1 then the minimum is attained at 1/κn and is equal to θκn − κcn . ˜ • If c < cη then the minimum is attained at 0 and is equal to ϕ. • In the case η = ∞, c∞ > 0 we additionally have to consider c = c∞ . In that case the minimum is also attained at 0 and is equal to ϕ. ˜ This proves (5.2) and the remaining statements follow easily.
Theorem 1.5 follows now directly from (5.1) in combination with Lemmas 5.1 and 5.2. 6. A potential with More than One Phase Transition By Theorem 1.5, we always have at least two phases: a high temperature phase where particles do not interact, and a low temperature phase where the connected components of interacting particles are unbounded. We now give an example of a potential where at least one further, intermediate, phase exists. To explain the idea, consider the potential v such that v = ∞ on [0, ν0 ) ∪ (ν0 , R), v = 0 on [R, ∞) and v(ν0 ) = −M for some M > 0 and R > 2ν0 . Obviously it does
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Fig. 3. A potential which violates Assumption (V) (left) and its phase transitions diagram (right)
Fig. 4. Modification of the potential in Fig. 3, which satisfies Assumption (V)
not satisfy Assumption (V), but it provides the correct intuitive picture on which we build our example below. For this potential, configurations have finite energy only if any two of its points are either precisely at distance ν0 or do not interact. The largest configurations in Rd such that all distances are equal to ν0 are regular simplices of d + 1 points. Hence, optimal configurations of n particles are organised in n/(d + 1) such simplices at distances > R to each other and one subset in of such a simplex with i n = n −(d +1)n/(d +1) points. In particular, ϕ(n) = −2M d+1 2 n/(d +1)−2M 2 for any n ∈ N, and ϕ˜ = −M. Hence, at zero temperature, we see a phase in which only the simplices are present. In our modification below, this phase is shifted to positive temperature. In the simplest case, where d = 1, we have ϕ(n) = −2M n2 , and the diagram of the points (1/κ, ϕ(κ)/κ) with κ ∈ N ∪ {∞} is depicted in Fig. 3. The potential v violates Assumption (V)(4), as it is not negative just to the left of R. This is the reason why the slope of the largest convex minorant is equal to zero close to 0, and this is why the phase of simplices is present only at zero temperature. Introducing a (small) negative part in v to the left of R will shift this phase to positive temperature. Now we consider a potential of similar shape, but modified in such a way that it fulfills Assumption (V). We will see that the diagram in Fig. 3 is a degeneration of the phase transitions diagram of this example. Fix T > 6 and M > ε > 0. Consider a continuous potential v (shown in Fig. 4) with support equal to [0, 2T + 6], satisfying (i) (ii) (iii) (iv) (v)
v(x) = +∞ for x ∈ [0, T ] and finite otherwise; v(x) ≥ 3M for x ∈ [T + 1, 2T + 3]; min v = −M = v(T + 1/2); v(x) < 0 for x ∈ [2T + 4, 2T + 6); min{v(x) : x > 2T + 3} = −ε = v(2T + 5).
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The potential v satisfies Assumption (V) with ν0 = T and R = 2T +6. For simplicity we will consider its action on one-dimensional point configurations, i.e., we put d = 1. Our goal is to prove that η = 2. For each n, denote by x1 < · · · < xn a minimiser of Vn , which exists by Lemma 3.1. Obviously, xi+1 − xi > T for all i. In the following lemma we prove that only neighbouring points in this configuration interact, and that the points are split into pairs with strong interaction −M between the points in each pair and weak interaction −ε between the pairs. Lemma 6.1. (i) Precisely n − 1 − n−1 2 of the nearest-neighbour distances x i+1 − x i are equal to T + 21 , the others are equal to 2T + 5. No two successive distances are equal to T + 21 . In particular, if n is even then for odd i, T + 21 xi+1 − xi = 2T + 5 for even i. (ii) For each n ∈ N,
6 5 6 5 n−1 n−1 − 2ε . ϕ(n) = −2M n − 1 − 2 2
(iii) ϕ˜ = −M − ε. Proof. Since xi+1 − xi > T for all i, we have xi+3 − xi > 3T > 2T + 6, and so each point interacts with at most two points on its left and at most two points at its right. Let us show that there are no points with interacting distance between T + 1 and 2T + 3, where the potential is large. Suppose this is not true and x j − xi ∈ [T + 1, 2T + 3] for some j > i (recall that j can only be i + 1 or i + 2). Consider a new configuration y1 < · · · < yn given by yk = xk for k ≤ i and yk = xk + 2T + 6 for k ≥ i + 1, i.e., we separate the first i points of the configuration x from the others. This removes all the three interactions that involve xi and xi+1 : Vn (y1 , . . . , yn ) = Vn (x1 , . . . , xn ) − 2 (v(xi+1 − xi−1 ) + v(xi+1 − xi ) + v(xi+2 − xi )) . Since either v(xi+1 − xi ) or v(xi+2 − xi ) is greater than or equal to 3M and the other two are greater than or equal to −M we obtain Vn (y1 , . . . , yn ) < Vn (x1 , . . . , xn ), which contradicts x being a minimiser. Now we conclude that each point in the configuration x interacts only with its neighbours. Indeed, for any i, we have xi+2 − xi = (xi+2 − xi+1 ) + (xi+1 − xi ) > 2T > T + 1. The above implies that xi+2 − xi > 2T +3. This implies that either xi+1 − xi or xi+2 − xi+1 is greater than T + 1 and hence is greater than 2T + 3, by the above. This in turn implies that xi+2 − xi > 3T + 3 > 2T + 6, i.e., xi+2 and xi do not interact. Therefore, since v assumes its minimum on [T, T + 1] at T + 21 with value −M and its minimum outside [T, T + 1] at 2T + 5 with value −ε, all nearest-neighbour distances are either equal to T + 21 or 2T + 5. For each i, at most one of the distances xi+2 − xi+1 and xi+1 − xi belongs to the interval [T, T + 1]. Hence, no two successive distances are equal to the optimal distance, T + 21 . Because M > ε, the distance T + 21 is more favourable than the distance 2T + 5, hence the number of distances equal to T + 21 is n−1 maximal, i.e., equal to n − 1 − n−1 2 , and the other 2 distances are equal to 2T + 5. From this, it is easy to conclude the proof.
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Fig. 5. Phase transitions diagram
Now we argue that there are exactly two phase transitions. Indeed, for any n ∈ N, we have n ϕ˜ − ϕ˜ if n is odd, ϕ(n) = n ϕ˜ + 2ε if n is even. Hence all points (1/n, ϕ(n)/n) with n odd lie on the straight line passing through (0, ϕ) ˜ and (1, 0), whereas the others lie on the straight line passing through (0, ϕ) ˜ and ϕ˜ M (− 2ε , 0) = ( 21 + 2ε , 0), which lies below the first line because M > ε. Hence we obtain the phase transitions diagram of Fig. 5. We obtain two phase transitions: one at c1 = −ϕ(2) = 2M from singletons to particle clouds with even cardinality, and then at c2 = 2ε to infinite clouds. Acknowledgements. We gratefully acknowledge the financial support by the DFG-Forschergruppe 718 ‘Analysis and stochastics in complex physical systems’. The first author was also supported by the Italian PRIN 2007 grant 2007TKLTSR ‘Computational markets design and agent-based models of trading behavior’. The third author is supported by an Advanced Research Fellowship from EPSRC. We further thank two referees for their valuable contributions.
References [AZ98] [AFS09] [Bil68] [DZ98] [GR79] [Ge94] [Res87] [Ru99] [Th06]
Aigner, M., Ziegler, G.M.: Proofs from THE BOOK. Berlin: Springer, 1998 Au Yeung, Y., Friesecke, G., Schmidt, B.: Minimizing atomic configurations of short range pair potentials in two dimensions: crystallization in the Wulff shape. http://arxiv.org/abs/0909. 0927v1[math.AP], 2009 Billingsley, P.: Convergence of probability measures. New York: John Wiley and Sons, 1968 Dembo, A., Zeitouni, O.: Large deviations techniques and applications. Berlin: Springer, 1998 Gardner, C.S., Radin, C.: The infinite-volume ground state of the lennard-jones potential. J. Stat. Phys. 20, 719–724 (1979) Georgii, H.-O.: Large deviations and the equivalence of ensembles for gibbsian particle systems with superstable interaction. Probab. Theory Relat. Fields 99, 171–195 (1994) Resnick, S.I.: Extreme values, regular variation, and point processes. New York: Springer, 1987 Ruelle, D.: Statistical mechanics: Rigorous results. Singapore: World Scientific, 1999 Theil, F.: A proof of crystallization in two dimensions. Commun. Math. Phys. 262, 209–236 (2006)
Communicated by F. Toninelli
Commun. Math. Phys. 299, 631–649 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1107-7
Communications in
Mathematical Physics
On a Class of Integrable Systems with a Cubic First Integral Galliano Valent Laboratoire de Physique Théorique et des Hautes Energies, Unité associée au CNRS UMR 7589, 2 Place Jussieu, 75251 Paris Cedex 05, France. E-mail:
[email protected] Received: 8 June 2009 / Accepted: 18 May 2010 Published online: 12 August 2010 – © Springer-Verlag 2010
Abstract: A few years ago Selivanova gave an existence proof for some integrable models, in fact geodesic flows on two dimensional manifolds, with a cubic first integral. However the explicit form of these models hinged on the solution of a nonlinear third order ordinary differential equation which could not be obtained. We show that an appropriate choice of coordinates allows for integration and gives the explicit local form for the full family of integrable systems. The relevant metrics are described by a finite number of parameters and lead to a large class of models mainly on the manifolds S2 and H2 . Many of these systems are globally defined and contain as special cases integrable systems due to Goryachev, Chaplygin, Dullin, Matveev and Tsiganov. 1. Introduction Let M be a n-dimensional smooth manifold with metric g(X, Y ) = gi j X i Y j and let T ∗ M be its cotangent bundle with coordinates (x, P), where P is a covector from Tx∗ M. Let us recall that T ∗ M is a smooth symplectic 2n-manifold with respect to the standard 2-form ω = d Pi ∧ d x i which induces the Poisson bracket { f, g} =
n ∂ f ∂g ∂ f ∂g . − ∂ x i ∂ Pi ∂ Pi ∂ x i i=1
In T ∗ M the geodesic flow is defined by the Hamiltonian H = K + V,
K =
n 1 ij g (x) Pi P j , 2 i, j=1
where g i j is the inverse metric of gi j .
V = V (x),
(1)
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An “observable” f : T ∗ M → R, which can be written locally f =
f i1 ,··· ,in (x) Pi1 · · · Pin ,
#( f ) = m,
i 1 +···+i n ≤m
is a constant of motion iff {H, f } = 0. A hamiltonian system is said to be integrable in the Liouville sense if there exist n constants of motion (including H ) generically independent and in pairwise involution with respect to the Poisson bracket. In what follows we will deal exclusively with integrable systems defined on two dimensional manifolds: in this case an integrable system is just made out of two independent observables H and Q with {H, Q} = 0. The general line of attack of this problem is based on the integer m = #(Q). For m = 1, M is a surface of revolution and for m = 2, M is a Liouville surface [3]. For higher values of m only particular examples have been obtained, some of which are in explicit form. For M = S2 and m = 3 the oldest explicit examples (early twentieth century) were due to Goryachev and Chaplygin on the one hand and to Chaplygin on the other hand (see [2, p. 483] and [10] for the detailed references). On the same manifold with m = 4 there is the famous Kovalevskaya system [7] and some extension due to Goryachev (see [9] for the reference). More recently there was a revival of this subject due to several existence theorems due to Selivanova and Kiyohara. Selivanova studied integrable systems both for m = 3 in [8] and for m = 4 in [9] and Kiyohara for any m ≥ 3 in [6]. As observed by Kiyohara, for m = 3 the two classes of models are markedly different. In the last years several new explicit examples for m = 3 were given by Dullin and Matveev [4] and Tsiganov [10]. In this work we will focus on Selivanova’s integrable systems with a cubic first integral discussed in [8]. A natural question raised by her existence theorems is the possibility of a constructive approach. According to the coordinates chosen one has to solve either a third order nonlinear ODE, as in [8], or a fourth order one as in Tsiganov’s work [10]. We will show that one can avoid solving these ODEs: an appropriate coordinate choice allows to get locally the explicit form of the full family of integrable systems, described by finitely many parameters. The final step is then, according to the values taken by the parameters, to determine the manifold M (mainly S2 and H2 ) on which the systems are defined and whether the observables H and Q are globally defined on M. The plan of the article is the following: in Sect. 2 we consider the class of models analyzed by Selivanova with the following leading terms for the cubic observable: Q = p Pφ3 + 2q K Pφ + · · · ,
p ∈ R, q ≥ 0,
and the general differential system resulting in {H, Q} = 0 is given. In Sect. 3 we first integrate the special case where q = 0: the differential system is reduced to a second order non-linear ODE. The global issues are then discussed. In Sect. 4 we consider the general case q > 0. Here we have linearized, by an appropriate choice of the coordinates, the possibly non-linear ODE encountered in other approaches. In Sect. 5, with the explicit local form of the metric, it is then possible (but lengthy because an enumeration of cases is required) to determine on which manifolds the metric is defined. We have checked that all the explicit integrable models given in the literature are indeed recovered as special cases.
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2. Cubic First Integral The general structure of the integrable system, laid down by Selivanova [8], is the following: one starts from the hamiltonian (1) with 1 2 Pθ + a(θ )Pφ2 , V = f (θ ) cos φ + g(θ ), f (θ ) ≡ 0, (2) K = 2 and the cubic observable Q = Q3 + Q1, with
Q 3 = p Pφ3 + 2q K Pφ ,
(3)
p ∈ R, q ≥ 0,
(4)
Q 1 = χ (θ ) sin φ Pθ + (β(θ ) + γ (θ ) cos φ) Pφ .
Lemma 1. The constraint {H, Q} = 0 is equivalent to the following differential system: χ f˙ = γ f,
(a) (b)
χ˙ = −q f,
β˙ = 2q g, ˙
χ g˙ = β f,
(˙ = Dθ ) , a˙ γ˙ + χ a = 2q f˙, aγ + χ = 3( p + qa) f. 2
(5)
Proof. The relation {H, Q} = 0 splits into three constraints {K , Q 3 } = 0,
{K , Q 1 } + {V, Q 3 } = 0,
{V, Q 1 } = 0.
(6)
The first is identically true, the second one is equivalent to the relations (5 b) while the last one is equivalent to (5 a).
The special case q = 0 is rather difficult to obtain as the limit of the general case q = 0, so we will first work it out completely. 3. The Integrable System for q = 0 3.1. Local analysis. We can take p = 1 and obvious integrations give χ = χ0 > 0,
β = β0 ∈ R,
γ = χ0
f˙ , f
g˙ =
β0 f, χ0
a=−
γ˙ , χ0
(7)
and the last equation γ¨ + 2
f˙ γ˙ + 6 f = 0. f
(8)
An appropriate choice of coordinates does simplify matters: Lemma 2. The differential equation for u = f˙ as a function of the variable x = f is given by uu 6 (9) u + cx = 0, c= > 0. = Dx . x χ0
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Proof. The relations in (7) become g =
β0 x , χ0 u
γ = χ0
u , x
a = −u
u x
,
(10)
and (8) gives (9).
The solution of this ODE follows from Lemma 3. The general solution of (9) is given by u=−
ζ 2 + c0 , 2c
(11)
with ζ 3 + 3c0 ζ − 2ρ = 0,
2(ρ − ρ0 ) = 3c2 x 2 ,
(12)
and integration constants (ρ0 , c0 ). uu = ζ. Proof. Let us define ζ = −c x/u. This allows a first integration of (9), giving x From this we deduce cu = −ζ ζ
⇒
which in turn implies
ζ 2 + c0 ζ = 2c2 x which concludes the proof.
2c u = −ζ 2 − c0 ,
⇒
ζ 3 + 3c0 ζ − 2ρ = 0,
(13)
It is now clear that the initial coordinates (θ, φ) chosen on S 2 will not lead, at least generically, to a simple form of the hamiltonian! To achieve a real simplification for the observables the symplectic coordinates change (θ, φ, Pθ , Pφ ) → (ζ, φ, Pζ , Pφ ) gives: Theorem 1. Locally, the integrable system has for its explicit form: ⎧ √ ⎪ ⎨ H = 1 F P 2 + G P 2 + λ F cos φ + μ ζ, ζ φ 2 = Dζ , √4F √ ⎪ ⎩ Q = P 3 − 2λ F sin φ Pζ + ( F ) cos φ Pφ − 2μ Pφ , φ
(14)
with F = −2ρ0 + 3c0 ζ + ζ 3 ,
G = 9c02 + 24ρ0 ζ − 18c0 ζ 2 − 3ζ 4 ,
(15)
and real parameters (λ, μ). Proofs. One may obtain these formulas by elementary computations, some scalings of the parameters and (χ0 → λ, β0 → −μ). Alternatively, one can check that (15) implies the relations G = −12 F,
G = F 2 − 2F F ,
(16)
which allows for a direct check of {H, Q} = 0. As proved in [8] this system does not exhibit any linear or quadratic constant of motion and (H, Q) are algebraically independent.
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3.2. The manifolds. We are now in position to analyze the global geometric aspects related to the metric1 g=
dζ 2 4F + dφ 2 , F G
φ ∈ [0, 2π ).
(17)
One has first to impose the positivity of both F and G for this metric to be riemannian. This gives for ζ some interval I whose end-points are possible singularities of the metric. To ascertain that the metric is defined on some manifold one has to ensure that these singularities are apparent ones and not true curvature singularities by techniques developed in [5]. Let us define, for the cubic F, its discriminant = c03 + ρ02 . Theorem 2. The metric (17): (i) is defined, for < 0, on S2 iff F = (ζ − ζ0 )(ζ − ζ1 )(ζ − ζ2 ), The change of coordinates ζ − ζ0 sn (u, k 2 ) = , u ∈ (0, K ), ζ1 − ζ0
ζ0 < ζ < ζ1 < ζ2 .
k2 =
ζ1 − ζ0 ∈ (0, 1), ζ2 − ζ0
gives for integrable system2 ⎧ 1 D(u) 2 ⎪ 2 ⎪ H= Pu + 2 2 2 Pφ + λ k 2 scd cos φ + μ k 2 s 2 , ⎪ ⎪ ⎨ 2 s c d (scd) 3 Q = 4Pφ − λ sin φ Pu + cos φ Pφ − 2μ Pφ , ⎪ ⎪ ⎪ scd ⎪ ⎩ D(u) = (1 − k 2 s 4 )2 − 4k 2 s 4 c2 d 2 > 0.
(18)
(19)
(ii) is defined, for = 0, on H2 iff F = (ζ − ζ1 )2 (ζ + 2ζ1 ),
−2ζ1 < ζ < ζ1 , (ζ1 > 0).
The change of coordinates ζ = ζ1 (−2 + 3 tanh2 u),
u ∈ (0, +∞),
gives for integrable system3 ⎧ 1 (1 + 3 T 2 ) 2 ⎪ 2 2 ⎪ Pu2 + P ⎨H = φ + λ T (1 − T ) cos φ + μ T , 2 2 S 1 − 3T 2 ⎪ ⎪ ⎩ Q = 4Pφ3 − λ sin φ Pu + cos φ Pφ − 2μ Pφ , T (iii) is not defined on any manifold for > 0. 1 As usual we identify the points with φ = 0 and with φ = 2π . 2 We use the shorthand notation: s, c, d respectively for sn (u, k 2 ), cn (u, k 2 ), dn (u, k 2 ). 3 We use the shorthand notation S, C, T respectively for sinh u, cosh u, tanh u.
(20)
(21)
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Proof of (i). If < 0 the cubic F has three simple real roots ζ0 < ζ1 < ζ2 . If we take ζ ∈ (ζ2 , +∞) then F is positive. The relation G = −12F shows that in this interval G is decreasing from G(ζ2 ) = F 2 (ζ2 ) > 0 to −∞ and will vanish for some ζ > ζ2 . Hence to ensure positivity for F and G we must restrict ζ to the interval (ζ2 , ζ ). Since at ζ = ζ the function F does not vanish while G does, this point is a curvature singularity and the metric cannot be defined on a manifold. The positivity of F is also ensured if we take ζ ∈ (ζ0 , ζ1 ). In this interval G decreases monotonously from G(ζ0 ) to G(ζ1 ) = F 2 (ζ1 ) > 0. Let us analyze the singularities at the end points. For ζ close to ζ0 we have for the approximate metric: dζ 2 4 F 2 (ζ0 ) 2 (22) g≈ + (ζ − ζ0 ) dφ . F (ζ0 ) 4(ζ − ζ0 ) G(ζ0 ) √ The relation (16) gives G(ζ0 ) = F 2 (ζ0 ), so the change of variable ρ = ζ − ζ0 allows to write 4 2 g≈ dρ + ρ 2 dφ 2 , (23) F (ζ0 ) which shows that ρ = 0 is an apparent singularity, called a “nut” in [5]. It is a shortcoming of the polar coordinates used and is removed by switching back to cartesian coordinates. So the point ζ = ζ0 can be added to the manifold and similarly we can add ζ = ζ1 . Geometrically the points ζ0 and ζ1 are the poles of the manifold which is therefore S2 . Each “nut” contributes by 1 to the Euler characteristic [5] giving χ = 2 as it should. Then, using the change of variable (18), it is a routine exercise in elliptic functions theory to operate the symplectic coordinates change (ζ, φ, Pζ , Pφ ) → (u, φ, Pu , Pφ ) which, after several scalings of the observables and of their parameters, gives (19). The strict positivity of D for u ∈ [0, K ] follows from the strict positivity of G for ζ ∈ [ζ0 , ζ1 ]. Let us notice that one can also, by direct computation, check that {H, Q} = 0 from the formulas given in (19).
Proof of (ii). In this case we have F = (ζ + 2ζ1 )(ζ − ζ1 )2 ,
G = −3(ζ + 3ζ1 )(ζ − ζ1 )3 ,
1/3
ζ1 = −ρ0 .
For ζ1 < 0 the positivity of F implies ζ ∈ (2|ζ1 |, +∞) and G decreases and vanishes for ζ = 3|ζ1 | leading to a curvature singularity. The case ζ1 = 0 is also excluded since then G ≤ 0 and the remaining case is ζ1 > 0. The positivity of F and G requires ζ ∈ (−2ζ1 , ζ1 ). The singularity structure is most conveniently discussed thanks to the coordinates change (20) which brings the metric to the form 4 sinh2 u 2 g= du 2 + , u ∈ (0, +∞), (24) dφ 3ζ1 1 + 3 tanh2 u from which we conclude that the manifold is H2 . Then starting from (14), the symplectic change of coordinates (ζ, φ, Pζ , Pφ ) → (u, φ, Pu , Pφ ), and some scalings, gives (21).
Proof of (iii). For > 0 the cubic F has a single real zero ζ0 . The positivity of F requires that ζ ∈ (ζ0 , +∞). Since G = −12F the function G decreases from G(ζ0 ) to −∞. Since G(ζ0 ) > 0 there exists ζ > ζ0 for which G( ζ ) = 0. So positivity restricts ζ ∈ (ζ0 , ζ ) and ζ is a curvature singularity showing that the metric cannot be defined on a manifold.
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Remarks. 1. The integrable system (21) corresponds to the limit of (19) when ζ2 → ζ1 or k 2 → 1. Then the elliptic functions degenerate into hyperbolic functions. Let us emphasis that in this limit the observables behave smoothly while the manifold changes drastically. 2. In [8] Selivanova proved an existence theorem for an integrable system on S 2 with a cubic observable (case (i) of Theorem 1.1). Her observables are ψ 2 (y) ψ 2 (y) 2 Py + Pφ2 + (ψ(y) − ψ (y)) cos φ, 2 2 3 3 Q = Pφ3 − ψ (y) sin φ Py + ψ(y) cos φ Pφ , 2 2 H=
= Dy ,
(25)
where ψ(y) is a solution of the ODE ψ ψ = ψ ψ − 2ψ 2 + ψ 2 + ψ 2 ,
ψ(0) = 0, ψ (0) = 1, ψ (0) = τ. (26)
Comparing (25) and (14) for β = 0 makes it obvious that we are dealing with the same integrable system, up to a local diffeomorphism, which is √ √ G (ζ 2 + c0 ) dζ = ± 3 dy. , (27) ψ(y) = − √ F 2 F We have checked that the ODE (26) is a consequence of the relations (27) and (16). We see clearly that Selivanova’s choice of the coordinate y led to a complicated ODE, very difficult to solve; in fact one should rather find coordinates such that the ODE becomes tractable. Now that we have obtained two integrable systems: the first on S2 and the second one on H2 , let us examine the global properties of the observables. 3.3. Global structure. We will give some details in this section, that will be useful to shorten similar proofs needed in the next sections. Theorem 3. The integrable system given by (19) is globally defined on M = S2 . Proof. Its potential is V = λ k 2 scd cos φ + μ k 2 s 2 . The variable u ∈ (0, K ) and {u = 0, u = K } are the poles of the sphere. The potential is continuously differentiable for all u ∈ [0, K ]. For the observable Q there are apparent singularities at the poles since we have (scd) 1 ∼ u → 0, scd u
(scd) 1 ∼ u → K. scd (K − u)
In fact we need to express Q in terms of globally defined quantities, namely the isometries generators in T ∗ M. The sphere x12 + x22 + x32 = 1 in the charts x1 = sin θ cos φ,
x2 = sin θ sin φ,
x3 = ± cos θ, θ ∈ (0, π ), φ ∈ [0, 2π ),
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G. Valent
has for canonical metric g0 (S2 , can) = d x12 + d x22 + d x32 = dθ 2 + sin2 θ dφ 2 , to be compared to the metric in (19): g = du 2 +
s 2 c2 d 2 dφ 2 . D
As is well known these two metrics must be conformally related. Indeed defining the correspondence θ → u by √ D θ = , sin θ scd
θ ∈ (0, π )
→
u ∈ (0, K ),
which integrates up to θ s tan = H (u) 2 c
H (u) = exp −
K u
√
D − d2 dτ , scd
(28)
we get the relation g = g0 (S2 , can),
=
1 d 2 (c2 + s 2 H 2 )2 . = θ 2 4H 2 D
(29)
The functions H and are C ∞ for u ∈ [0, K ] and strictly positive, showing again that the manifold is indeed S2 . Let us define the so(3) generators in T ∗ S2 : L1 =
sin φ cos φ Pφ , Pu + θ tan θ
L2 = −
cos φ sin φ Pφ , Pu + θ tan θ
L 3 = Pφ . (30)
The integrable system becomes ⎧ θ 2 2 ⎪ ⎨H = L 1 + L 22 + L 23 + λ k 2 scd cos φ + μ k 2 s 2 , 2 √ (scd) − cos θ D ⎪ ⎩ Q = 4 L 3 − λ θ L + ρ cos φ L − 2μ L , . ρ= 1 3 3 3 scd
(31)
From the previous discussion θ is C ∞ on [0, K ]. The function ρ is C ∞ on (0, K ) and the series expansions close to the poles ρ(u) = 2(H (0)2 − 1 − k 2 )u + O(u 3 )
ρ(u) = −2k 2 (K − u) + O((K − u)3 )
show that it remains also C ∞ through them. Hence we conclude that this integrable system is globally defined on S2 .
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Remark. It is important to observe that the Hamiltonian vector field vH =
∂H ∂ ∂H ∂ − i ∂ pi ∂ x i ∂ x ∂ pi
which has for explicit expression 2 2 P 2 P cos θ θ ∂V ∂ ∂V ∂ ∂ φ φ − − − v H = −θ θ Pθ2 + 2 3 ∂ P ∂u ∂ P ∂φ ∂ Pφ sin θ sin θ u u Pφ ∂ ∂ + + θ 2 Pu ∂u sin2 θ ∂φ must be defined everywhere, to avoid a singular Hamiltonian flow. This requires not only the continuity of θ and V but also their differentiabilty. This observation will have important consequences for some models discussed in Sect. 5. Let us now prove Theorem 4. The integrable system (21) is globally defined on M = H2 . Proof. Its potential is V = λ T (1 − T 2 ) cos φ + μ T 2 ,
T = tanh u, u ∈ [0, +∞),
and is differentiable for all values of u and φ, particularly at the “nut” u = 0. The hyperbolic plane H2 :
{x12 + x22 − x32 = −1, (x1 , x2 ) ∈ R2 x3 ≥ 1},
in the chart x1 = sinh χ cos φ
x2 = sinh χ sin φ
x3 = cosh χ ,
χ ∈ [0, +∞), φ ∈ [0, 2π )
has for canonical metric g0 (H2 , can) = d x12 + d x22 − d x32 = dχ 2 + sinh2 χ dφ 2 .
(32)
The metric of (21) is g = du 2 +
sinh2 u dφ 2 , 1 + 3 tanh2 u
which must be conformal to g0 . Defining the correspondence χ → u by √ χ 1 + 4 sinh2 u = , χ ∈ [0, +∞) → u ∈ [0, +∞) sinh χ sinh u cosh u
(33)
(34)
which integrates up to tanh(χ /2) = tanh u H (u)
H (u) = exp −
+∞ u
√ ( 1 + 4 sinh2 τ − 1) dτ , (35) sinh τ cosh τ
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G. Valent
we get g = g0 (H2 , can),
=
cosh2 u(1 − tanh2 u H 2 (u))2 . 4(1 + 3 tanh2 u)H 2 (u)
(36)
The function H , which can be expressed in terms of elementary functions, is C ∞ for u ≥ 0, ensuring that the conformal factor (u) is also C ∞ . Its behaviour at infinity is H (u) = 1 − e−u + O(e−2u )
⇒
(u) =
1 1 + O(e−u ) , 16
hence never vanishes. This gives another proof that the metric (33) does live on H2 . Following the same line as previously, let us define the generators of the so(2, 1) Lie algebra in T ∗ H2 , M1 = sin φ Pu +
cos φ Pφ , tanh u
M2 = cos φ Pu −
sin φ Pφ , tanh u
M3 = Pφ . (37)
The observables (21) become ⎧ 3 ⎨ H = M12 + M22 − 1 − 2 M32 + λ T (1 − T 2 ) cos φ + μ T 2 , C ⎩ Q = 4M33 − λ (M1 − 3T cos φ M3 ) − 2μ M3 , showing that this sytem is globally defined on H2 .
(38)
4. The Integrable System for q > 0 As already observed, if one insists in working with the variable θ , the differential system (5) can be reduced either to a third order [8] or to a fourth order [10] non-linear ODE. The key to a full integration of this system is again an appropriate choice of coordinates on the manifold. Theorem 5. Locally, the integrable system (H, Q) has for explicit form ⎧ √ β0 F 1 G 2 ⎨ 2 Pφ + cos φ + , H= F Pζ + = Dζ , 2ζ 4F √ 2qζ 2qζ √ ⎩ Q = p Pφ3 + 2q H Pφ − F sin φ Pζ − ( F ) cos φ Pφ ,
(39)
with the polynomials F = c0 + c1 ζ + c2 ζ 2 +
p 3 ζ , q
G = F 2 − 2F F .
(40)
Proofs. Starting from (5) the functions β and g are easily determined to be β=
β0 , χ2
g=
β0 . 2qχ 2
(41)
The functions γ and a can be expressed in terms of f and its derivatives with respect to χ as 3 (42) ff . γ = −qχ f , a = −q 2 f f + χ
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Then the last relation in (5) reduces to a third order linear ODE, χ ( f f ) + 9 ( f f ) +
15 6p ff = 3, χ q
which is readily integrated to c1 c0 f = ± c2 + f 1 χ 2 + 2 + 4 , χ χ
f1 =
(43)
p . 4q 3
The remaining functions become q 2 c12 − 4c0 c2 12c0 f 1 6c1 f 1 a= 2 − − − 4c2 f 1 − 3 f 12 χ 2 , f χ6 χ4 χ2 c1 2c0 q − f1 χ 2 + 2 + 4 . γ = f χ χ
(44)
(45)
The observables can be written, up to a scaling of the parameters, in terms of F and G defined by F = 4q 2 χ 4 f 2 = c0 + c1 ζ + c2 ζ 2 + g1 ζ 3 ,
g1 =
p , q
ζ = χ 2,
(46)
G = 16q 2 χ 6 f 2 a = c12 − 4c0 c2 − 12c0 g1 ζ − 6c1 g1 ζ 2 − 4c2 g1 ζ 3 − 3g12 ζ 4 . To simplify matters the symplectic change of coordinates (θ, φ, Pθ , Pφ ) → (ζ, φ, Pζ , Pφ ). gives the required result, up to scalings. Alternatively (46) implies the relations G = −12
p F, q
G = F 2 − 2F F ,
(47)
which allow a direct check of {H, Q} = 0. As proved in [8] this system does not exhibit any other conserved observable linear or quadratic in the momenta, and (H, Q) are algebraically independent.
Remarks. 1. The limit q = 0 is quite tricky: it is why we analyzed it separately in the previous section. 2. It is still possible to come √ back to the coordinate θ but the price to pay is the integration of the relation ζ /F dζ = −dθ, which can be done using elementary functions for c0 = 0. 5. The Manifolds Let us now examine the global geometric aspects of the metric g=
4ζ F ζ dζ 2 + dφ 2 , F G
φ ∈ [0, 2π ),
(48)
taking into account the following observations: 1. The positivity constraints are ζ F(ζ ) > 0 and G(ζ ) > 0. They define the end-points of some interval I for ζ . In some cases, discussed in detail later on, one can obtain extensions beyond some of the end-points.
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2. For the observables to be defined it is required that F ≥ 0 ∀ζ ∈ I . 3. As already observed, any point ζ0 with F(ζ0 ) = 0 and G(ζ0 ) = 0 is a curvature singularity. 4. The point ζ = 0 is a curvature singularity for F(0) = 0 and G(0) = 0. In order to have a complete description of all the possible integrable models, we will present them in three sets: the first one for p = 0, the second one for p > 0 and the third one for p < 0. 5.1. First set of integrable models. For p = 0 we have F = c0 +c1 ζ +c2 ζ 2 = c2 (ζ −ζ1 )(ζ −ζ2 ),
G = c12 −4c0 c2 ,
(c0 , c1 , c2 ) ∈ R3 . (49)
Theorem 6. In this set we have the following integrable models: (i) Iff c2 > 0 and 0 < ζ2 < ζ the metric (48) is defined in H2 and ⎧ 2 2 2 ⎪ ⎪ H = 1 M1 + M2 − M3 + α sinh u cos φ + β , ⎨ u ∈ (0, +∞), 2 ρ + cosh u ρ + cosh u ⎪ ζ2 + ζ1 ⎪Q = H M −αM , ⎩ ρ= ∈ (−1, +∞). 3 1 ζ2 − ζ1 (ii) Iff c2 < 0 and 0 < ζ1 < ζ < ζ2 the metric (48) is defined in S2 and ⎧ 1 L 21 + L 22 + L 23 α ρ sin θ cos φ + β ⎪ ⎪ ⎨H = + , θ ∈ (0, π ), 2 1 + ρ cos θ 1 + ρ cos θ ζ2 − ζ1 ⎪ ⎪Q = H L +αL , ⎩ ρ= ∈ (0, +1). 3 1 ζ2 + ζ 1 (iii) Iff c2 = 0 the metric (48) is defined in R2 and ⎧ Px2 + Py2 2α ρ 2 x + β 1 ⎨ + , H= 2 2 2 2 1 + ρ (x + y ) 1 + ρ 2 (x 2 + y 2 ) ⎩ Q = H L z − α Py , ρ > 0.
(x, y) ∈ R2 ,
(50)
(51)
(52)
In all three cases α and β are free parameters. (iv) All of these models are globally defined on their manifold. Proof of (i). The positivity condition G > 0 shows that F has two real and distinct roots ζ1 < ζ2 , so we will write F = c2 (ζ − ζ1 )(ζ − ζ2 ),
G = c22 (ζ1 − ζ2 )2 .
(53)
Then imposing the positivity of ζ F one has to deal with the iff part of the proof by an enumeration of all possible cases for the triplet (0, ζ1 , ζ2 ), including the possibility of one ζi being zero. Taking into account the remarks given at the beginning of this section, one concludes that for c2 > 0, we must take ζ > ζ2 > 0. The change of coordinates ζ =
ζ2 − ζ1 (ρ + cosh u) , 2
(ζ2 , +∞) → (0, +∞),
ρ=
ζ2 + ζ 1 ζ2 − ζ1
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643
brings the metric (48) to the form g=
ζ2 − ζ1 (ρ + cosh u) du 2 + sinh2 u dφ 2 , 2c2
u ∈ (0, +∞),
(54)
which is conformal to the canonical metric on H2 . Using the definitions (37) we obtain (50), up to scalings.
Proof of (ii). For c2 < 0 positivity requires either 0 < ζ1 < ζ < ζ2 or ζ1 < ζ < ζ2 < 0. In both cases the change of coordinates ζ =
ζ1 + ζ2 (1 + ρ cos θ ) , 2
(ζ1 , ζ2 ) → (π, 0),
ρ=
ζ2 − ζ1 , ζ2 + ζ 1
brings the metric (48) to one and the same form g=
ζ1 + ζ2 (1 + ρ cos θ ) dθ 2 + sin2 θ dφ 2 , 2c2
θ ∈ (0, π ),
(55)
which is conformal to the canonical metric on S2 . Using the so(3) generators in T ∗ S2 one obtains (51), up to scalings.
Proof of (iii). For c2 = 0 we have G = c12 > 0. If c1 < 0 we can write F = |c1 |(ζ1 − ζ ) and positivity requires ζ ∈ (0, ζ1 ). If ζ1 = 0 then ζ = 0 is a curvature singularity because F(0) and G(0) are not vanishing. If c1 > 0 we have F = c1 (ζ − ζ1 ). If ζ1 < 0 positivity requires either ζ > 0, but ζ = 0 is a curvature singularity, or ζ < ζ1 and then F is negative. If ζ1 = 0 the metric becomes 1 2 g= dζ + 4ζ 2 dφ 2 , c1 = 2φ ∈ [0, 2π ) and in H appears a term so to recover flat space we have to take φ /2) which does not define a function in R2 . Eventually, if ζ1 > 0 if of the form cos(φ we take ζ < 0 the point ζ = 0 is singular, so we are left with ζ > ζ1 . The change of coordinates ζ = ζ1 (1 + ρ 2 r 2 ), ρ > 0,
x = r cos φ,
y = r sin φ,
brings the metric (48) to the form g=
4ζ12 ρ 2 (1 + ρ 2 r 2 )(d x 2 + dy 2 ), c1
(x, y) ∈ R2 .
(56)
Using the e(3) Lie algebra generators (Px , Py , L z = x Py − y Px ) we obtain (52), up to scalings.
Proof of (iv). In all of the three cases H and Q have been expressed in terms of globally defined quantities, up to conformal factors which are C ∞ over their coordinates ranges.
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The remaining cases are given by p = 0. It is convenient to rescale F by | p|/q and G by p 2 /q 2 in order to have F = (ζ 3 + f 0 ζ 2 +c1 ζ +c0 ), = sign( p),
G = F 2 − 2F F , G = −12 F, (57)
and for the observables, up to scalings √ ⎧ ⎪ ⎨ H = 1 F P 2 + G P 2 + α F cos φ + β , ζ 2ζ 4 F φ ζ ζ . ⎪ ⎩ Q = P 3 + 2 H P − 2α √ F sin φ P + (√ F ) cos φ P , φ ζ φ φ
(58)
So the metric is still given by (48). We will denote by the discriminant of F according to the sign of . 5.2. Second set of integrable models. For p > 0 or = +1 we have: Theorem 7. The metric (48): (i) is defined, for + < 0, on S2 iff F = (ζ − ζ0 )(ζ − ζ1 )(ζ − ζ2 ),
0 < ζ0 < ζ < ζ1 < ζ2 .
The integrable system, using the notations of Theorem 2 case (i), is ⎧ 1 β scd D(u) 2 2 ⎪ ⎪ H= Pu + 2 2 2 Pφ + αk 2 cos φ + , ⎪ ⎪ 2ζ (u) s c d ζ (u) ζ + + ⎪ + (u) ⎨ (scd) Q = 4Pφ3 + 2H Pφ − α sin φ Pu + cos φ Pφ , ⎪ scd ⎪ ⎪ ⎪ ζ0 ⎪ ⎩ ζ+ (u) = ρ + k 2 sn2 u, u ∈ (0, K ), ρ = > 0. ζ2 − ζ0
(59)
(ii) is defined, for + = 0, on H2 iff F = (ζ − ζ0 )(ζ − ζ1 )2 ,
0 < ζ0 < ζ < ζ1 .
The integrable system, using the notations of Theorem 2 case (ii), is ⎧ 1 T (1 − T 2 ) β 3 ⎪ 2 2 2 ⎪ H = M + α M cos φ + , + M − 1 − ⎪ 1 2 3 ⎪ 2 ⎪ 2ζ (u) C ζ (u) ζ + + + (u) ⎨ (60) Q = 4M33 + 2H M3 − α (M1 − 3T cos φ M3 ) , ⎪ ⎪ ⎪ ⎪ ζ0 ⎪ ⎩ ζ+ (u) = ρ + tanh2 u, u ∈ (0, +∞), ρ= > 0. ζ1 − ζ0 (iii) is not defined on any manifold for + > 0. (iv) The systems (59) and (60) are globally defined on their manifold.
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Proof of (i). The iff part results from a case by case examination of all possible orderings of the 4-plet (0, ζ0 , ζ1 , ζ2 ), including the possibility of one of the ζi being zero. We will not give the full details which can be easily worked out, taking into account the remarks presented at the beginning of Sect. 5. The reader can check that with F = (ζ − ζ0 )(ζ − ζ1 )(ζ − ζ2 ) and 0 < ζ0 < ζ < ζ1 < ζ2 , the polynomial F is positive and vanishes at the end-points (ζ0 , ζ1 ) while G is strictly positive. It follows that ζ = ζ0 and ζ = ζ1 are the poles of the manifold S2 . Operating the same coordinates change as in Theorem 2, case (i), one obtains (59).
Proof of (ii). The polynomial G becomes G = (ζ1 − ζ )3 (3ζ + ζ1 − 4ζ0 ). The change of variable ζ0 ζ = (ζ1 − ζ0 )(ρ + th2 u), ζ ∈ (ζ0 , ζ1 ) → u ∈ (0, +∞), ρ= , ζ1 − ζ0 transforms the observables, up to scalings, into ⎧ 1 α 1 + 3T 2 2 β ⎪ 2 ⎪ ⎪H = Pu + T (1 − T 2 ) cos φ + , Pφ + ⎪ 2 ⎨ 2ζ+ (u) S ζ+ (u) ζ+ (u) 2 (1 − 3T ) ⎪ Q = 4Pφ3 + 2H Pφ − α sin φ Pu − α cos φ Pφ , ⎪ ⎪ T ⎪ ⎩ 2 ζ+ (u) = ρ + tanh u. Using the relations (37) one gets (60).
(61)
Proof of (iii). Examining all the possible cases gives no manifold for the metric.
Proof of (iv). For the first model (resp. the second model) the result follows from Theorem 3 (resp. Theorem 4) and the observation that the conformal factor ζ+ (u) is C ∞ on the interval [0, K ] (resp. [0, +∞)) and never vanishes.
5.3. Third set of integrable models. It is given by p < 0 or = −1. It displays a richer structure and for clarity we will split up the description of the integrable systems into several theorems. Theorem 8. The metric (48) for − < 0 is defined on S2 iff: (i) either F = (ζ − ζ0 )(ζ − ζ1 )(ζ2 − ζ ), ζ0 < ζ1 < ζ < ζ2 (ζ1 > 0). The change of coordinates ζ2 − ζ ζ2 − ζ1 2 sn (u, k ) = , u ∈ (0, K ), k2 = ∈ (0, 1), ζ2 − ζ1 ζ2 − ζ0 gives for the integrable system ⎧ 1 k 2 scd β D(u) 2 ⎪ 2 ⎪ ⎪ H = + α P cos φ + , + P ⎪ u φ 2 2 2 ⎪ 2ζ− (u) s c d ζ− (u) ζ ⎪ − (u) ⎨ (scd) Q = −4 Pφ3 + 2H Pφ + α sin φ Pu + cos φ Pφ , ⎪ scd ⎪ ⎪ ⎪ ζ2 ⎪ ⎪ ⎩ ζ− (u) = k 2 ρ − sn2 u , ρ= > 1. ζ2 − ζ1 This system is globally defined on S2 .
(62)
(63)
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(ii) or F = (ζ0 − ζ )(ζ − ζ1 )(ζ − ζ2 ) and G(0) = 0, 0 < ζ < ζ0 < ζ1 < ζ2 . The integrable system is ⎧ L 23 1 1 h ⎪ 2 2 2 ⎪ H = f (L − cos + L ) + θ f ⎪ 1 2 ⎪ 2 2 3f ⎪ sin2 θ ⎨ √ β sin θ f (64) cos φ + , +α ⎪ ⎪ (cos2 θ )1/3 (cos2 θ )1/3 ⎪ ⎪ ⎪ ⎩ Q = − 4 L 3 + 2H L + 3α (cos θ )1/3 f L + ( f ) cos φ L , 3 1 3 3 9 where f (θ ) = fˆ(cos θ ) with ζ1 ζ2 2/3 2/3 − μ − μ ζ0 ζ0 fˆ(μ) = , 4/3 2/3 μ +μ +1
μ ∈ (−1, +1),
(65)
ˆ and h(θ ) = h(cos θ ) with
ζ ζ + ζ + ζ ζ 4 ζ ζ 1 ζ2 1 2 1 2 1 2 4/3 ˆ μ −2 1+ h(μ) = −μ + + 2 μ2/3 +4 2 . 3 ζ0 ζ0 ζ0 ζ0 2
(66)
The parameter ζ0 is: ζ 1 ζ2 < ζ1 . ζ0 = √ √ ( ζ1 + ζ2 ) 2
(67)
This system exhibits a singular Hamiltonian flow. Proof of (i). The change of variable (62) gives (63) by lengthy but straightforward computations. It is globally defined from Theorem 3 and because the conformal factor ζ− (u) is C ∞ and never vanishes on [0, K ].
Proof of (ii). One has G(0) = (ζ1 − ζ2 )2 ζ02 − 2ζ1 ζ2 (ζ1 + ζ2 ) ζ0 + ζ12 ζ22 . Its vanishing determines uniquely ζ0 in terms of (ζ1 , ζ2 ) as given by (67). At this stage positivity requires ζ ∈ (0, ζ0 ). Let us make the change of variable ζ = ζ0 μ2/3 . The metric becomes ˆ dμ2 4 2 f (μ) 2 g= μ ∈ (0, 1). + 3(1 − μ ) dφ , ˆ 9 (1 − μ2 ) fˆ(μ) h(μ) All the functions in the metric are even functions of μ: we can therefore take μ ∈ (−1, +1) extending the metric beyond μ = 0. One can check that the points μ = ±1 are “nuts” and therefore we get again for manifold S2 . The change of variable μ = cos θ with θ ∈ (0, π ) gives then for result (64). As previously observed in the Remark of Subsect. 3.3, for the Hamiltonian flow to be defined, the function f must be differentiable for all μ ∈ [−1, +1] and this is not true for μ = 0 since f is a function of μ2/3 . Let us notice that no choice of the parameters ζ0 , ζ1 and ζ2 allows to cure f from this pathology.
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Remark. Theorem 8, case (i) does not describe appropriately the special case ζ0 = 0 for which elliptic functions are no longer required. Indeed the coordinates change ζ =
ζ1 + ζ2 ζ1 − ζ2 − cos θ, 2 2
(ζ1 , ζ2 ) → (π, 0),
gives for the metric g = dθ 2 +
sin2 θ dφ 2 , 1 + sin2 θ G(cos θ )
(68)
with G(μ) =
3μ2 + 4ρμ + 1 , 4(ρ + μ)2
ρ=
ζ2 + ζ 1 > 1. ζ2 − ζ1
The integrable system is ⎧ 1 2 sin θ β ⎪ L 1 + L 22 + (1 + G(cos θ ))L 23 + α √ cos φ + , ⎨H = 2 U U √ sin θ ⎪ 3 ⎩ Q = −L 3 + 2H L 3 + 2α U L 1 − α √ cos φ L 3 , U
(69)
U = ρ +cos θ, (70)
on which we recognize the Dullin-Matveev system [4]. Let us proceed to: Theorem 9. (a) The metric (48) for − = 0 is defined on S2 iff: (i) either F = ζ 2 (ζ0 − ζ ), 0 < ζ < ζ0 , and we have ⎧ β 1 ⎨ L 21 + L 22 + 4L 23 + α sin θ cos φ + , θ ∈ (0, π ), H= 2 cos2 θ ⎩ Q = −4 3 L + 2H L + α (cos θ L − 2 sin θ cos φ L ) , 3
3
1
(71)
3
which is the Goryachev-Chaplygin top, globally defined for β = 0. (ii) or F = (ζ − ζ1 )2 (ζ0 − ζ ) and G(0) = 0, 0 < ζ < ζ0 . The integrable system (with ζ0 = ζ1 /4) has the form (64) with the functions 2 4 − μ2/3 ˆ ˆ , h(μ) = (4 − μ2/3 )3 , μ ∈ (−1, +1). (72) f (μ) = 4/3 μ + μ2/3 + 1 This system exhibits a singular Hamiltonian flow. (b) The metric (48) for − = 0 is defined on H2 iff: F = (ζ − ζ1 )2 (ζ0 − ζ ),
0 < ζ1 < ζ < ζ0 .
The integrable system, in the notations of Theorem 2, case (ii), is ⎧ 1 T (1 − T 2 ) β 3 ⎪ 2 2 2 ⎪ H = M + α M cos φ + , + M − 1 − ⎪ 1 2 3 ⎪ 2 ⎪ 2ζ− (u) C ζ− (u) ζ− (u) ⎨ (73) Q = −4M33 + 2H M3 + α (M1 − 3T cos φ M3 ) , ⎪ ⎪ ⎪ ⎪ ζ0 ⎪ ⎩ ζ− (u) = ρ − tanh2 u, u ∈ (0, +∞), ρ= > 1. ζ0 − ζ1 It is globally defined on H2 .
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Proof of (a)(i). We have F = ζ 2 (ζ0 − ζ ) and G = ζ 3 (4ζ0 − 3ζ ) and ζ ∈ (0, ζ0 ) from positivity. Taking for the new variable θ such that ζ = ζ0 cos2 θ we get for the metric sin2 θ 2 2 θ ∈ (0, π/2). (74) g = 4 dθ + dφ , 1 + 3 sin2 θ As it stands the manifold is P 2 (R) (see [1, p. 268]), but we can take θ ∈ (0, π ) extending the manifold to S2 with poles for θ = 0 and θ = π . The observables can be transformed into (71) and we recover the Goryachev-Chaplygin top.
Proof of (a)(ii). In this case we have G(0) = ζ13 (ζ1 − 4ζ0 ) which fixes ζ0 = ζ1 /4. The argument then proceeds as in the proof of Theorem 8, case (ii).
Proof of (b). The proof is identical to the one for Theorem 7, case (ii), except for the change of coordinates, which is now ζ = ζ0 − (ζ0 − ζ1 ) tanh2 u
(ζ0 , ζ1 )
→
(0, +∞),
and leads to (73).
Theorem 10. The metric (48) for − > 0 is defined on S2 iff: F = (ζ0 − ζ )(ζ − ζ1 )(ζ − ζ1 ) and G(0) = 0, 0 < ζ < ζ0 . The integrable system is of the form (64) with the functions μ2/3 − ζζ01 μ2/3 − ζζ01 , μ ∈ (−1, +1), fˆ(μ) = μ4/3 + μ2/3 + 1
(75)
and
2 ζ + ζ + ζ | 4 |ζ |ζ1 |2 ζ 1 1 1 1 1 4/3 ˆ h(μ) = −μ + + 2 μ2/3 + 4 2 . μ −2 1+ 3 ζ0 ζ0 ζ0 ζ0 2
(76)
The value of ζ0 is ζ0 =
|ζ1 |2 ζ1 + ζ 1 + 2|ζ1 |
> 0.
(77)
This system, for generic parameters, exhibits a singular Hamiltonian flow. Proof. We have G(0) = (ζ1 − ζ 1 )2 ζ02 − 2(ζ1 + ζ 1 )|ζ1 |2 ζ0 + |ζ1 |4 . Its vanishing gives two roots for ζ0 , but only (77) is positive. The subsequent analysis is identical to that already given in the proof of Theorem 8, case (ii). For some special values of the parameters f may reduce to a constant, as for the Goryachev top examined just below.
To conclude, let us examine the explicitly known integrable systems, with a metric defined in S2 and with a cubic observable already given in the literature:
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1. The Goryachev-Chaplygin top given by Theorem 9, case (a)(i). 2. The Dullin-Matveev top [4] given in the remark after the proof of Theorem 8. 3. If we restrict, in Theorem 10, the parameters according to ζ0 = −(ζ1 + ζ 1 ) > 0 and ζ0 = |ζ1 |,
⇒
f = 1, h = 4 − μ2 ,
we recover the Goryachev top ⎧ 1 sin θ 4 β ⎪ ⎨H = L 21 + L 22 + L 23 + α cos φ + , 2 1/3 2 2 3 (cos θ ) (cos θ )1/3 (78) ⎪ ⎩ Q = − 4 L 3 + 2H L 3 + 3α (cos θ )1/3 L 1 . 3 9 However, due to the “equatorial” singularities (θ = π/2) of the potential, this system is globally defined only in the trivial case α = β = 0. 4. The two new examples given by Tsiganov in [10] are not defined on a manifold. All of the previously known examples belong to the third set with p < 0. 6. Conclusion We have exhaustively and explicitly constructed all of the integrable models, on two dimensional manifolds, for Selivanova’s models characterized by the following form of the observables ⎧ ⎨ H = 1 P 2 + a(θ )P 2 + f (θ ) cos φ + g(θ ), φ 2 θ (79) ⎩ Q = p P 3 + q P 2 + a(θ )P 2 P + χ (θ ) sin φ P + (β(θ ) + γ (θ ) cos φ) P . φ θ φ φ θ φ The main point which stems from our work is that the coordinates choice is of the utmost delicacy since it determines the structure of the differential equations to be solved eventually. The same difficulty must be overcome when looking for Einstein metrics with symmetry: the Einstein equations reduce to coupled ODE and finding exact solutions relies on an adapted choice of coordinates which may simplify or even linearize the differential system to be integrated. Acknowledgements. We are greatly indebted to K. P. Tod for his kind and efficient help in the analysis of the metric singularities of Sect. 5, and to the Referee for several improvements and corrections induced by his observations.
References 1. Besse, A.L.: Einstein manifolds. In: Classics in Mathematics, Berlin Heidelberg-New York: SpringerVerlag, 2002 2. Bolsinov, A.V., Kozlov, V.V., Fomenko, A.T.: Russ. Math. Surv. 50, 473–501 (1995) 3. Darboux, G.: Leçons sur la théorie générale des surfaces. 3e partie, New York: Chelsea Publishing Company, 1972 4. Dullin, H.R., Matveev, V.S.: Math. Research Lett. 11, 715–722 (2004) 5. Gibbons, G., Hawking, S.: Commun. Math. Phys. 66, 291–310 (1979) 6. Kiyohara, K.: Math. Ann. 320, 487–505 (2001) 7. Kovalevskaya, S.V.: Acta Math. 12, 177–232 (1889) 8. Selivanova, E.N.: Commun. Math. Phys. 207, 641–663 (1999) 9. Selivanova, E.N.: Ann. Global Anal. Geom. 17, 201–219 (1999) 10. Tsiganov, A.V.: J. Phys. A: Math. Gen. 38, 921–927 (2005) Communicated by M. Aizenman
Commun. Math. Phys. 299, 651–676 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1104-x
Communications in
Mathematical Physics
Stability and Duality in N = 2 Supergravity Jan Manschot NHETC, Rutgers University, Piscataway, NJ 08854-8019, USA. E-mail:
[email protected] Received: 17 June 2009 / Accepted: 15 April 2010 Published online: 13 August 2010 – © Springer-Verlag 2010
Abstract: The BPS-spectrum is known to change when moduli cross a wall of marginal stability. This paper tests the compatibility of wall-crossing with S-duality and electric-magnetic duality for N = 2 supergravity. To this end, the BPS-spectrum of D4-D2-D0 branes is analyzed in the large volume limit of Calabi-Yau moduli space. Partition functions are presented, which capture the stability of BPS-states corresponding to two constituents with primitive charges and supported on very ample divisors in a compact Calabi-Yau. These functions are “mock modular invariant” and therefore confirm S-duality. Furthermore, wall-crossing preserves electric-magnetic duality, but is shown to break the “spectral flow” symmetry of the N = (4, 0) CFT, which captures the degrees of freedom of a single constituent. Contents 1. 2. 3. 4. 5. A.
Introduction . . . . . . . . . . . . . . . . BPS-States in N = 2 Supergravity . . . . . D4-D2-D0 BPS-States . . . . . . . . . . . Wall-Crossing in the Large Volume Limit . Conclusion and Discussion . . . . . . . . Two Mock Siegel-Narain Theta Functions
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1. Introduction The study of BPS-states in physics has been very fruitful. Their invariance under (part of the) supersymmetry transformations of a theory makes them insensitive to variations of certain parameters. This allows the calculation of some quantities in a different regime than the regime of interest. BPS-states have been specifically useful in testing various dualities, for example S-duality in N = 4 Yang-Mills theory [43] or in string theory [39]. Another major application is the understanding of the spectrum of supersymmetric
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theories of gravity, leading to the microscopic account of black hole entropy for various supersymmetric black holes in string theory [32,41]. This article considers the BPS-spectrum of N = 2 supergravity theories in 4 dimensions. N = 2 supersymmetry is the least amount of supersymmetry, which allows massive states to be BPS. It appears in string theory by compactifying the 10-dimensional space-time on a compact 6-dimensional Calabi-Yau manifold X . A large class of BPSstates are formed by wrapping D-branes around cycles of X , which might correspond to black hole states if the number of D-branes is sufficiently large. The Witten index (degeneracy counted with (−1) F ) is insensitive to perturbations of the string coupling constant gs , and plays therefore a central role in this paper. It allows to show for certain cases that the magnitude of the index agrees with black hole entropy: log ∼ SBH . The study of D-branes on X revealed many connections to objects in mathematics, like vector bundles, coherent sheaves and derived categories, which helps to understand their nature, see for a review Ref. [2]. The index corresponds from this perspective to the Euler number χ (M) of their moduli space M [43], or an analogous but better defined invariant like Donaldson-Thomas invariants [42]. An intriguing aspect of BPS-states is their behavior as a function of the moduli of the theory. The moduli parametrize the Calabi-Yau X and appear in supergravity as scalar fields. Under variations of the moduli, conservation laws allow BPS-states to become stable or unstable at codimension 1 subspaces (walls) of the moduli space. Such changes in the spectrum indeed occur, and were first observed in 4 dimensions by Seiberg and Witten [38]. Denef [11] has given an illuminating picture of stability in supergravity as multi black hole solutions whose relative distances depend on the value of the moduli at infinity. At a wall, these distances might diverge or become positive and finite. The changes in the degeneracies at a wall show the impact on the spectrum of these processes. Ref. [12] derives formulas for for n-body semi-primitive decay using arguments from supergravity. The notion of stability for D-branes is closely related to the notion of stability in mathematics [16,17]. In this context, Kontsevich and Soibelman [30] derive a very general wall-crossing formula for (generalized) Donaldson-Thomas invariants. Gaiotto et al. [23] shows that this generic formula applied to the indices of 4-dimensional N = 2 quantum field theory, is implied by properties of the field theory. Much evidence exists for the presence of an S-duality and electric-magnetic duality group in N = 2 supergravity [7,44]. S-duality is an S L(2, Z) group which exchanges weak and strong coupling; electric-magnetic duality is the action of a symplectic group on the vector multiplets. These dualities impose strong constraints on the spectrum of the theory. The wall-crossing formulas are very generic on the other hand, and the walls form a very intricate web in the moduli space. It is therefore appropriate to ask: are wallcrossing and duality compatible with each other? This paper analyses this question, concentrating on D4-D2-D0 BPS-states or M-theory black holes, in the large volume limit of Calabi-Yau moduli space. The BPS-objects correspond in this limit to coherent sheaves on a Calabi-Yau 3-fold supported on an ample divisor. The analysis considers the walls, the primitive wall-crossing formula and (part of) the supergravity partition function Zsugra (τ, C, t), which enumerates the indices as a function of D2- and D0-brane charges for fixed D4-brane charge. Zsugra (τ, C, t) captures the changes in the spectrum by wall-crossing. S-duality predicts modularity for this function, which is tested in this paper. The degrees of freedom of a single D4-D2-D0 black hole are related via M-theory to a 2-dimensional N = (4, 0) superconformal field theory (SCFT) [32]. One of the symmetries of the SCFT spectrum is the “spectral flow symmetry” [4,21,31], which are
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certain transformations of the charges, which do not change the value of the moduli at infinity. This imposes additional constraints on the spectrum to the ones imposed by the supergravity duality groups. A single constituent cannot decay any further, and conjectures by [1,5] indicate that the SCFT description of the spectrum (for given charge) might only be valid for a specific value of the moduli. Therefore, interesting dependence of the SCFT spectrum as a function of the moduli at infinity is not expected. This suggests that a natural decomposition for the supergravity partition function with fixed magnetic charge P might be Zsugra (τ, C, t) = ZCFT (τ, C, t) + Zwc (τ, C, t),
(1.1)
where ZCFT (τ, C, t) is the well-studied SCFT elliptic genus [4,21,31,33], and all wallcrossing in the moduli space is captured by Zwc (τ, C, t). ZCFT (τ, C, t) is known to transform as a modular form from arguments of CFT; the modular properties of Zwc (τ, C, t) are however unknown. This paper considers a small part of Zwc (τ, C, t), namely Z P1 ↔P2 P1 +P2 =P ample, primitive
(τ, C, t), which enumerates the indices of composite BPS-configurations with two constituents, with ample and primitive magnetic charges P1 and P2 . An important building block of these functions is the newly introduced “mock Siegel-Narain theta function”. Mock modular forms do not transform exactly as modular forms, but can be made so by the addition of a relatively simple correction term [46], which is applied to mock Siegel-Narain theta functions in the appendix. Using its transformation properties, one can show that the corrected partition function transforms precisely as the SCFT elliptic genus, thereby confirming S-duality. From the analysis follows also that electric-magnetic duality remains present in the theory, but the “spectral flow” symmetry of the SCFT is generically not present. This is not quite unexpected since this is not a symmetry of supergravity. Another indication that the spectral flow symmetry is not present appears in Ref. [1], which explains that the jump in the D4-D2-D0 index by wall-crossing can be larger than the index of a single BPS-object (this effect is known as the entropy enigma [12]). A special property of Z P1 ↔P2 (τ, C, t) is that it does not contribute to the index if the moduli are chosen at the corresponding attractor point. However, Z P1 ↔P2 (τ, C, t) is generically not zero, and therefore Zsugra (τ, C, t) is nowhere equal to ZCFT (τ, C, t) generically. Section 4 explains how these observations are in agreement with conjectures of Refs. [1,5] about the uplift of these BPS-configurations to five dimensions. Although the compatibility with the dualities is expected, it is very interesting to see how it is realized. The stability condition and primitive wall-crossing formula combine in an almost miraculous way to the mock Siegel-Narain theta function, which gives insights in the way wall-crossing is captured by N = 2 BPS partition functions for compact Calabi-Yau 3-folds. An intriguing property of the corrected partition function is that it is continuous as a function of the Kähler moduli t, which is reminiscent of earlier discussions [23,29]. The outline of this paper is as follows. Section 2 reviews briefly the relevant aspects of N = 2 supergravity. Section 3 describes the BPS-states of interest and the expected properties of their partition function. Section 4 is the heart of the paper; it describes the walls and the partition functions capturing wall-crossing. Section 5 finishes with discussions and suggestions for further research. The Appendix defines two mock Siegel-Narain theta functions and gives some of their properties.
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2. BPS-States in N = 2 Supergravity If IIA string theory is compactified on a compact Calabi-Yau 3-fold X , one obtains N = 2 supergravity as the low energy theory in the non-compact dimensions. The most essential part of the field content for this article is the b2 + 1 vector multiplets, which A and complex scalar X A , A = 1, . . . , b + 1 (with b each contain a U (1) gauge field Fμν 2 2 the second Betti number of X ). The gauge fields lead to a vector of conserved charges = (P 0 , P a , Q a , Q 0 )T , a = 1 . . . b2 , which take value in the (2b2 + 2)-dimensional lattice L. The magnetic charges are denoted by P A and electric charges by Q A . The charges arise in IIA string theory as wrapped D-branes on the even homology of X ; the components of represent 6-, 4-, 2- and 0-dimensional cycles. A symplectic pairing is defined on the charge lattice 1 , 2 = −P10 Q 0,2 + P1 · Q 2 − P2 · Q 1 + P20 Q 0,1 . The symplectic inner product is thus ⎛
−1
⎜ I=⎝
−1
1
⎞ ⎟ ⎠,
1 where 1 denotes a b2 × b2 unit matrix. The scalars X A parametrize the Kähler moduli space of the Calabi-Yau X : the complexified Kähler moduli are given by t a = B a + i J a = X a / X 0 . Here, B a and J a are periods of the B-field and the Kähler form respectively.1 The B-field takes values in H 2 (X, R). The Kähler forms are restricted to the Kähler cone C X , which is defined to be the space of 2-forms such that γ J > 0, P J 2 > 0 and X J 3 > 0 for any holomorphic curve γ and surface P ∈ X . An accurate Lagrangian description of supergravity requires that the volume of X is parametrically larger than the Planck length, thus J a → ∞. This article is mainly concerned with this parameter regime. Loop and instanton corrections can here be neglected, such that the prepotential simplifies to the cubic expression F(t) =
1 dabc t a t b t c , 6
where dabc is the triple intersection number of 4-cycles in X . The supergravity Lagrangian is invariant under the electric-magnetic duality group, which acts on the vector multiplets and more specifically on the electric-magnetic fields and moduli. This duality group is essentially a gauge redundancy, which appears by working on the universal covering space of the moduli space instead of the moduli space itself. The group is Sp(2b2 + 2, Z): the group of (2b2 + 2) × (2b2 + 2) matrices K which leave invariant I [44]:2 KT IK = I. 1 The moduli t a will sometimes be viewed as 2-forms instead of scalars. Similarly, the charges can also be viewed as homology cycles or their Poincaré dual forms. 2 Note that we use here a different notation as in e.g. [44], which is more natural from the point of view of geometry.
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The arguments that the group is Sp(2b2 + 2, Z) are valid in the large volume limit. The correct electric-magnetic duality group, which is valid for any value of J , is a subgroup of this and generated by the monodromies around singularities in the moduli space. These generators are generically hard to determine, except for the monodromies in the limit J → ∞. They are simply the translations3 ⎛ ⎞ 1 ⎜ ⎟ ka 1 ⎜ ⎟ ⎜ ⎟ , k ∈ Zb2 . K(k) = ⎜ 1 (2.1) ⎟ b c c dabc k 1 ⎝ 2 dabc k k ⎠ 1 1 c b c b c ka 1 6 dabc k k k 2 dabc k k In addition, an S L(2, Z) duality group is present, which exchanges the weak and strong coupling regime. This group acts on the hypermultiplets, and is most manifest in the IIB description for large Kähler parameters [7]. If time is considered as Euclidean and compactified, another S L(2, Z) duality group appears. This can be seen from the M-theory viewpoint, where the total geometry 1. is R3 × T 2 × X , and T 2 is the product of the time and M-theory circle St1 ×SM 3 A Kaluza-Klein reduction to R , leads to a 3-dimensional N = 4 supergravity theory. Since the physics in R3 is independent of large coordinate reparametrizations of T 2 , it should exhibit an S L(2, Z) duality group. The duality (complex structure) parameter is given by τ = C1 + ie− = C1 + iβ/gs ,
(2.2)
where C1 ∈ R is the component of the RR-potential 1-form along the time direction. 1 , which changes the physical interpretaNote that this S L(2, Z) exchanges St1 and SM tion of the states on both sides of the duality. D2-branes become for example worldsheet instantons and vice versa. The full BPS-spectrum should however be invariant under these S L(2, Z) transformations. These transformations also transform the B- and Cfields into each other, which is easily seen from the M-theory perspective: the B- and C-field are reductions of the M-theory 3-form over different 2-cycles of the torus. The duality transformations are summarized by τ→
aτ + b , C → aC + bB, cτ + d
B → cC + d B,
J → |cτ + d|J
(2.3)
a b ∈ S L(2, Z). Note that this S L(2, Z) is not the weak-strong duality of the c d 4-dimensional supergravity. But it is possible to relate this “M-theory” S L(2, Z) to the S-duality S L(2, Z) of IIB, by a T-duality along the time circle [12]. This transforms C1 into C0 and (2.2) becomes the familiar IIB duality parameter. The physical D4-D2-D0 branes of IIA become D3-D1-D-1 instantons of IIB. Therefore, a test of the M-theory S L(2, Z) is equivalent to testing S-duality, and in the rest of the paper the M-theory S L(2, Z) is referred to as S-duality. The N = 2 supersymmetry algebra contains a central element, the central charge Z () ∈ C. The central charge of a BPS-state is a linear function of its charge and a non-linear function of the Kähler or complex structure moduli of X . Only the complexified Kähler moduli t a appear in Z () for the relevant BPS-states in this article, thus Z (, t).
with
3 Note that the upper or lower indices might label either rows or columns in the matrix.
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The mass M of supersymmetric states is determined by the supersymmetry algebra to be M = |Z (, t)|. In a theory of gravity, a sufficiently massive BPS-state corresponds to a black hole state in the non-compact dimensions. The moduli depend generically on the spatial position t ( x ) in a black hole solution. Their value at the horizon is determined in terms of the charge by the attractor mechanism [19], whereas the value at infinity is imposed as boundary condition. The mass M is determined by the moduli at infinity. The following sections deal with the stability of BPS-states, which is determined by these values at infinity. Also the S L(2, Z) duality group is acting on the complex structure parameter τ of T 2 at infinity. The expression for the central charge as a function of the moduli is generically highly non-trivial. However in the limit J → ∞ it simplifies to [2] Z (, t) = − e−t ∧ , X
where the moduli t and the charge are viewed as forms on X . Alternatively, one can write
1 1 Z (, t) = 1, t a , dabc t b t c , dabc t a t b t c I = T I , 2 6 where we defined the vector of the periods . A very intriguing aspect of BPS-states is their stability. The simplest example is the case with two BPS-objects with primitive charges 1 and 2 . Their total mass is larger than or equal to the mass of a single BPS-object with the same total charge: |Z (1 , t)| + |Z (2 , t)| ≥ |Z (1 + 2 , t)|. The equality is generically not saturated, but for special values of the moduli t = tms , the central charges can align Z (1 , tms )/Z (2 , tms ) ∈ R+ , and the equality holds. These values form a real codimension 1 subspace of the moduli space, appropriately called the “walls of marginal stability”. They decompose the moduli space into chambers. BPS-states might decay or become stable, whenever the moduli cross a wall. Denef [11] has shown how wall-crossing phenomena are manifested in supergravity. The equations of motions allow for BPS-solutions with multiple black holes. The ones of interest for the present discussion are solutions with only two black holes. The relative distance between the two centers is given by
1 , 2 |Z (1 , t) + Z (2 , t)| |x1 − x2 | = G 4 , 2 Im(Z (1 , t) Z¯ (2 , t)) ∞ where |∞ means that the central charges are evaluated at asymptotic infinity in the black hole solution; G 4 is the 4-dimensional Newton constant.4 In the limit G 4 → 0, or equivalently gs → 0, the distance between the centers also approaches 0. This is the regime, where a microscopic analysis is typically carried out, it is the D-brane regime as opposed to the black hole regime. Since distances must be positive, the solution can only exist for 1 , 2 Im(Z (1 , t) Z¯ (2 , t)) > 0.
(2.4)
4 G is the 4-dimensional Newton constant, and is given in terms of IIA and M-theory parameters by 4 6 )3 2 P P G 4 = gs2 α (α VCY and G 4 = P 2π R VCY , respectively.
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Importantly, |x1 − x2 | depends on the moduli: if t approaches a wall of marginal stability Im(Z (1 , t) Z¯ (2 , t)) = 0,
(2.5)
|x1 − x2 | → ∞ and the 2-center solution decays. An implication of the mechanism for stability in supergravity is that single center black holes cannot decay into BPS-configurations with multiple constituents. If the moduli are chosen at the attractor point at infinity, and are thus constant throughout the black hole solution, 2-center solutions cannot exist. Moreover, the moduli flow in a 2-center solution from a stable chamber at infinity, to unstable chambers at the attractor points. In the following, we will analyze wall-crossing between two chambers CA and CB . To avoid ambiguities, one can choose 1 and 2 such that Im(Z (1 , t) Z¯ (2 , t)) < 0 in CB , which is equivalent to the convention in the mathematical literature, see for example [45]. This means that a stable object with charge satisfies Im(Z (2 , t)) Im(Z (, t)) < , Re(Z (2 , t)) Re(Z (, t)) with , 2 > 0. Coherent sheaves are expected to be the proper mathematical description of D-branes in the limit J → ∞ [2]. The charge of the BPS-state is determined by the Chern character of the corresponding sheaf E and of the Aˆ genus of the Calabi-Yau [35] ˆ X ), = ch(i ! E) A(T (2.6) where i : P → X is the inclusion map of the divisor into the Calabi-Yau. Of central interest are the degeneracies of BPS-states with charge . Most useful is actually the index (; t) =
1 Tr H(;t) (2J3 )2 (−1)2J3 , 2
(2.7)
where J3 is a generator of the rotation group Spin(3). (; t) is a protected quantity against variations of gs . The degeneracies are only constant in chambers of the moduli space, but jump if a wall is crossed. This is easily understood from the mechanism for decay in supergravity: the constituents separate, leading to a factorization of the Hilbert spaces, and consequently a loss of the number of states. The change in the index is [12]: (; ts → tu ) = (; tu ) − (; ts ) = −(−1)1 ,2 −1 |1 , 2 | (1 ; tms ) (2 ; tms ). Of course, in crossing a wall towards stability one gains states. Therefore the change of the index is in this case (; tu → ts ) = (−1)1 ,2 −1 |1 , 2 | (1 ; tms ) (2 ; tms ). Wall-crossing occurs more generally between two chambers CA and CB . If 1 and 2 are chosen such that Im(Z (1 , tB ) Z¯ (2 , tB )) < 0 in CB , the change of the index between the two chambers is (; tA → tB ) = (−1)1 ,2 1 , 2 (1 ; tms ) (2 ; tms ).
(2.8)
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This is consistent with jumps of the invariants in mathematics at walls of marginal stability. We can of course choose the points tA and tB more generally and allow them to lie in the same chamber. Then the change in the index is (; tA → tB ) = (−1)1 ,2 1 , 2 (1 ; t) (2 ; t) 1 × sgn(Im(Z (1 ; tA ) Z¯ (2 ; tA ))) 2 − sgn(Im(Z (1 ; tB ) Z¯ (2 ; tB ))) ,
(2.9)
where sgn(z) is defined as sgn(z) = 1 for z > 0, 0 for z = 0, and −1 for z < 0. Note that (; tA → tB ) satisfies a cocycle relation [AC] = [AB] + [BC]. Compatibility of the earlier described dualities with wall-crossing is non-trivial. Consider here the compatibility of electric-magnetic duality. As a gauge redundancy, Sp(2b2 + 2, Z) (or the relevant subgroup) leaves invariant the central charge: Z (; t) = Z (K; Kt) (Kt denotes the transformed vector of moduli), and the indices: (; t) = (K; Kt),
(2.10)
for every ∈ L. Since the walls are determined by the central charges, and 1 , 2 = K1 , K2 , it is clear that wall-crossing does not obstruct the electric-magnetic duality group. Note that generically (; t) = (K; t), and that no symmetry exists in supergravity which relates these two indices. Section 3 comes back to this point. The S L(2, Z)-duality group also implies non-trivial constraints for the degeneracies and their wall-crossing. The test of this duality is however much more involved and the subject of Sect. 4, after general aspects of D4-D2-D0 BPS-states and their partition functions are explained in the next section.
3. D4-D2-D0 BPS-States This section specializes the general considerations of the previous section to the set of states with charge = (0, P, Q, Q 0 ), and discusses the supergravity partition functions for this class of charges. These BPS-states correspond to D4-branes wrapping a divisor in X , with homology class P ∈ H4 (X, Z). This class of BPS-states is well-described in the literature, see for example [4,21,32,36], therefore the review here will only include the most essential parts for the discussion. The divisor is also denoted by P and taken to be very ample, which means among others that it has non-zero positive components in all 4-dimensional homology classes. The intersection form on P leads to a quadratic form Dab = dabc P c for magnetic charges k ∈ H4 (X, Z); the signature of Dab is (1, b2 − 1). The lattice is denoted by . The electric charge Q takes its value in ∗ + P/2 [20,35]. The conjugacy class of Q − P/2 in ∗ / is denoted by μ. If necessary, the dependence of Dab on P will be made explicit, like P · J 2 , otherwise simply J 2 is used. The real and imaginary part of the central charge Z ((P, Q, Q 0 ), t) of these states are 1 P · (J 2 − B 2 ) + Q · B − Q 0 , 2 Im(Z (, t)) = (Q − B P) · J. Re(Z (, t)) =
Stability and Duality in N = 2 Supergravity
The mass |Z (; t)| of BPS-states in the regime P · J 2 |(Q − |(Q − B P) · J | is: |Z (, t)| =
659 1 2 B)
· B − Q 0 |,
1 1 ((Q − B P) · J )2 P · J 2 + (Q − B P) · B − Q 0 + + O(J −2 ). 2 2 P · J2
(3.1)
All but the first term are homogeneous of degree 0 in J , and thus invariant under rescaP)·J )2 is positive definite: (Q − B)2+ . J has thus a natural lings. The combination ((Q−B P·J 2 interpretation as a point of the Grassmannian which parametrizes 1-dimensional subspaces on which Dab is positive definite. It therefore determines a decomposition of ⊗ R into a 1-dimensional positive definite subspace and a (b2 − 1)-dimensional negative definite subspace. For P 0 = 0, P = 0, the transformations (2.1) act on the charges and moduli as 1 Q 0 → Q 0 + k · Q + dabc k a k b P c , 2 Q a → Q a + dabc k b P c , t a → t a + ka , with k a ∈ . As mentioned in the Introduction, the microscopic explanation for the macroscopic entropy SBH = π |Z |2 of a single center D4-D2-D0 black hole was given by Ref. [32] using M-theory. The black hole degrees of freedom are in this case those of an M5-brane which wraps the divisor in X times the torus T 2 . The microscopic counting relied on a 2-dimensional N = (4, 0) CFT, which can be obtained as the reduction of the M5-brane worldvolume theory to T 2 . The magnetic charge P determines mainly the field content of the CFT, whereas the electric charges Q and Q 0 are charges of states within the CFT. The BPS-indices of the single center black hole are the Fourier coefficients of the SCFT elliptic genus ZCFT (τ, C, B) [4,21,31]. To test the compatibility of S-duality in supergravity with wall-crossing, one needs to consider the full supergravity partition function Z(τ, C, t),5 which captures the stability of BPS-states as a function of t. Properties of Z(τ, C, t) are now briefly reviewed, tailored for the present discussion. It is defined by Z(τ, C, t) =
1 Tr H(P,Q,Q 0 ;t) (2J3 )2 (−1)2J3 +P·Q 2 Q0 , Q × exp −2π τ2 |Z (, t)| + 2πiτ1 (Q 0 − Q · B + B 2 /2)+2πiC · (Q − B/2) ,
with τ2 = gβs ∈ R+ , τ1 = C1 ∈ R, t = B +i J ∈ ⊗C and B, C ∈ ⊗R. This function sums over Hilbert spaces with fixed magnetic charge and varying electric charges. This is in agreement with a microcanonical ensemble for magnetic charge and a canonical ensemble for electric charges, which is natural in the statistical physics of BPS black holes [37]. After insertion of (3.1) one finds 1 Z(τ, C, t) = exp(−π τ2 J 2 ) Tr H(P,Q,Q 0 ;t) (2J3 )2 (−1)2J3 +P·Q 2 Q0 , Q × e −τ¯ Qˆ 0¯ + τ (Q − B)2+ /2 + τ¯ (Q − B)2− /2 + C · (Q − B/2) , 5 The subscript “sugra” used in the Introduction will be omitted.
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with Qˆ 0¯ = Q 0¯ + 21 Q 2 , Q 0¯ = −Q 0 and e(x) = exp(2πi x). The modular invariant prefactor exp(−π τ2 J 2 ) is omitted in the following. The partition function has an expansion Z(τ, C, t) = (P, Q, Q 0 ; t) (−1) P·Q Q0, Q
× e −τ¯ Qˆ 0¯ + τ (Q − B)2+ /2 + τ¯ (Q − B)2− /2 + C · (Q − B/2) . Note that the partition function depends in various ways on the Kähler moduli t: they appear in (P, Q, Q 0 ; t), moreover B shifts the electric charges and J determines the decomposition of the lattice into a positive and negative definite subspace of ⊗ R. The sum over Q 0 and Q is unrestricted and might at some point invalidate the estimate used for (3.1), even in the limit J → ∞. To verify that this does not invalidate the analysis, we compute the term O(J −2 ). It is given by −2(Q − B)2+ ( Qˆ 0¯ − 21 (Q − B)2− )/P · J 2 . It follows from the CFT analysis that Qˆ 0¯ is bounded below for a single constituent, therefore Qˆ 0¯ − 21 (Q− B)2− is as well. Moreover, the next section shows that Qˆ 0¯ − 21 (Q− B)2− is also bounded below for stable bound states of 2 constituents. If (Q − B)2+ ( Qˆ 0¯ − 21 (Q − B)2− ) is O(J 2 ), then |Z (, t)|− 21 P · J 2 is at least O(J ). Contributions to the partition function of states for which the approximations for Eq. (3.1) are not satisfied, are thus highly suppressed compared to the states for which they are satisfied, which shows that the analysis is not invalidated. Also for any given value of the charges, one can always increase J to sufficiently large values, such that the approximations are valid. It is very well possible however, that not the whole partition function has a nice Fourier expansion. It is well known that Z(τ, C, t) contains a pole for τ → i∞ and its S L(2, Z) images. It is less clear at this point whether poles in B or C can appear in Z(τ, C, t). Examples of CFT’s where such poles appear, are the characters of massless representations of the N = 4 SCFT algebra [18], and the sigma model with the non-compact target space H3+ [24]. The Fourier expansion of a partition function with poles depends on the integration contour. This is how the partition function of dyons in N = 4 supergravity [40] captures wall-crossing phenomena. However, the stability condition (2.5) for D4-branes on ample divisors show that no wall-crossing as a function of C is present. Moreover, the partition functions for bound states of two constituents, derived in the next section, are not directly suggestive for “wall-crossing by poles”. Therefore, in the following it is assumed that no poles in B or C are present in Z(τ, C, t). The translations K(k) of the electric-magnetic duality group imply a symmetry for the partition function. Using (2.10) and assuming the Fourier expansion, one verifies easily that Z(τ, C, t) −→ (−1) P·k e(C · k/2) Z(τ, C, t), under transformations by K(k). Also using (2.10) one can show a quasi-periodicity in B: Z(τ, C, t + k) = (−1) P·k e(C · k/2) Z(τ, C, t). Additionally, Z(τ, C, t) satisfies a quasi-periodicity in C: Z(τ, C + k, t) = (−1) P·k e(−B · k/2) Z(τ, C, t).
(3.2)
These translations are large gauge transformations of C. A theta function decomposition is not implied by the two periodicities since the Fourier coefficients (; t) explicitly depend on B, and generically (K(k); t) = (; t).
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A distinguishing property of the partition function for this class of BPS-states is that charges multiply either τ or τ¯ , in contrast to, for example, D2- or D6-brane partition functions. Additionally, space-time S-duality suggests that the function transforms as a modular form, such that techniques of the theory modular forms can be usefully applied. Refs. [21,22] present some coefficients ((0, 1, Q, Q 0 ); t) for several Calabi-Yau 3-folds with b2 = 1. These coefficients determine the whole partition function, and confirm modularity in a non-trivial way. However, stability phenomena do not occur in the limit J → ∞ for these Calabi-Yau’s, since b2 = 1. The next section tests modularity, if wall-crossing is present. The arguments from CFT for modularity are very robust. Refs. [4,21,33] derive that the action of the generators of S L(2, Z) on ZCFT (τ, C, t) is given by: 1
3
S :
Z(−1/τ, −B, C + i|τ |J ) = τ 2 τ¯ − 2 ε(S) Z(τ, C, t),
T :
Z(τ + 1, C + B, t) = ε(T ) Z(τ, C, t),
(3.3)
where ε(T ) = e (−c2 (X ) · P/24) and ε(S) = ε(T )−3 [12,33]. Here the analysis of [4,21] is adapted to the supergravity point of view following [12]. The next section gives evidence that the same transformation properties continue to hold for the full supergravity partition function.6 Note that S-duality is consistent with the two periodicities mentioned above. The periodicities and the S L(2, Z) form together a Jacobi group S L(2, Z) (Zb2 )2 . The partition function for single constituents can be decomposed in a vectorvalued modular form and a theta function by arguments from CFT. The indices of the CFT are independent of the moduli at infinity: CFT (; t) = CFT () = (), and obey the “spectral flow symmetry” () = (K(k)). To see this, recall that the D2-brane charges appear in the CFT in a U (1)b2 current algebra, which can be factored out of the total CFT by the Sugawara construction, which implies that the indices satisfy CFT () = CFT (K(k)) [4,21,31]. The name “spectral flow” comes originally from the SCFT of superstrings. In the current context, one could see the flow as a flow of the B-field. As mentioned already after Eq. (2.10), no evidence exists that this is a symmetry of the full spectrum of 4-dimensional supergravity. In fact, Sect. 4 shows that wall-crossing is incompatible with this symmetry at generic points of the moduli space. Since the spectral flow symmetry is present in the spectrum of a single D4-D2-D0 black hole, the theta function decomposition is reviewed here. We define the functions 1 2 h P,Q− 1 P (τ ) = (P, Q, Q 0 ) q Q 0¯ + 2 Q . (3.4) 2
Q 0¯
Using that () = (K(k)), one can show that the invariants (P, Q, Q 0 ) depend only on Qˆ 0¯ and the conjugacy class μ of Q ∈ ∗ , thus (P, Q, Q 0 ) = μ ( Qˆ 0¯ ). Therefore, h P,Q− 1 P (τ ) = h P,Q− 1 P+k (τ ) with k ∈ . This allows a decomposition 2 2 of ZCFT (τ, C, t) into a vector-valued modular form h P,μ (τ ) and a Siegel-Narain theta function μ (τ, C, B): h P,μ (τ ) μ (τ, C, B), (3.5) ZCFT (τ, C, t) = μ∈∗ / 6 Evidence exists that Z(τ, C, t) does only transform as (3.3) under the full group S L(2, Z) if P is prime. Otherwise it transforms as a modular form of a congruence subgroup, whose level is determined by the divisors of P. Consequently, the rest of the article assumes implicitly that P is prime, although it nowhere explicitly enters the calculations.
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with μ (τ, C, B) =
(−1) P·Q e τ (Q − B)2+ /2+ τ¯ (Q − B)2− /2 + C · (Q − B/2) .
Q∈+P/2+μ
(3.6) The dependence of μ (τ, C, B) on the Kähler moduli J is not made explicit. The transformation properties of μ (τ, C, B) are − 1 + (−iτ )b2 /2 (i τ¯ )b2 /2 e(−P 2 /4) S : μ (−1/τ, −B, C) = √ ∗ | /| × e(−μ · ν)ν (τ, C, B),
ν
T : μ (τ + 1, C + B, B) = e (μ + P/2)2 /2 μ (τ, B, C). They satisfy in addition two periodicity relations for B and C with k ∈ : μ (τ, C, B + k) = (−1)k·P e(C · k/2) μ (τ, C, B), μ (τ, C + k, B) = (−1)k·P e(−B · k/2) μ (τ, C, B). All the dependence on τ and the “explicit” dependence on B, C and J of ZCFT (τ, C, t) is captured by the μ (τ, C, B). Note that the μ (τ, C, B) are annihilated by D = i 2 ∂C+ + 21 B+ · ∂C+ − 41 πi B+2 . Z(τ, C, t) is also annihilated by D, if holomorphic ∂τ + 4π anomalies in h P,μ (τ ) are ignored; these are known to arise in similar partition functions for 4-dimensional gauge theory [43]. The transformation properties of μ (τ, C, B) imply that h P,μ (τ ) transforms as a vector-valued modular form: 1 (−iτ )−b2 /2−1 ε(S)∗ e −P 2 /4 S : h P,μ (−1/τ ) = − √ ∗ | /| × e(−δ · μ)h P,δ (τ ), δ∈∗ /
T : h P,μ (τ + 1) = ε(T )∗ e (μ + P/2)2 /2 h P,μ (τ ). From the asymptotic growth of these Fourier coefficients follows the black hole entropy SBH = π 2 (P 3 + c2 (X ) · P) Qˆ ¯ for Qˆ ¯ P 3 + c2 (X ) · P. 3
0
0
4. Wall-Crossing in the Large Volume Limit As explained in Sect. 3, the partition function is expected to exhibit the modular symmetry and electric-magnetic duality in the large volume limit J → ∞. This section constructs the contribution Z P1 ↔P2 (τ, C, t) of bound states of two primitive constituents with primitive D4-brane charges P1 and P2 = 0 to Z(τ, C, t), and tests its modular properties. I take the following Ansatz for the contribution to the index of a bound state of two primitive constituents at a point t in the moduli space: 1 sgn(Im(Z (1 , t) Z¯ (2 , t))) + sgn(1 , 2 ) 1 ↔2 (; t) = 2 ×(−1)1 ,2 −1 1 , 2 (1 )(2 ). (4.1)
Stability and Duality in N = 2 Supergravity
663
The first term of the first line ensures that this Ansatz reproduces the wall-crossing formula (2.9). The non-trivial part of the Ansatz is thus the term sgn(1 , 2 ). This section explains that this is also in agreement with other important physical requirements. Based on the Ansatz, the generating function of the contribution to the index of the bound states is determined in Eq. (4.7). A study of the generating function leads to the following results: – the generating function (4.7) is convergent, – the generating function does not exhibit the modular properties of ZCFT (τ, C, t) (the partition function of a single center black hole with magnetic charge P1 + P2 ), but it can be made so by the addition of a “modular completion” using techniques of mock modular forms. The “completed” generating function (4.10) is proposed as the contribution Z P1 ↔P2 (τ, C, t) of 2-center bound states, which is thus compatible with S-duality. – Z P1 ↔P2 (τ, C, t) has the unexpected property that it is continuous as a function of the moduli, which is reminiscent of earlier work on wall-crossing [23,29]. The generating function is by construction a discontinuous function of the moduli. The combination of the first and second property is essentially a unique consequence of the Ansatz. The agreement of the Ansatz with the supergravity picture is discussed later. We continue now by taking a closer look at the walls of marginal stability. Specializing Eq. (2.5) gives for the walls at J → ∞ (without 1/J corrections) P1 · J 2 (Q 2 − B P2 ) · J − P2 · J 2 (Q 1 − B P1 ) · J = 0.
(4.2)
Note that this wall is independent of the D0-brane charges Q 0,i . And so states decay at this wall, independent of their D0-charge and of their distribution between the constituents. The condition for stability for this class of states is P1 · J 2 (Q 2 − B P2 ) · J − P2 · J 2 (Q 1 − B P1 ) · J < 0, if 1 , 2 > 0. This stability condition is a natural generalization of slope stability for sheaves or bundles on surfaces [15], since P · J 2 replaces the notion of rank. It can be derived from the stability for sheaves [28]. When 1/J corrections are included, one finds that actually many physical walls merge with each other in the limit J → ∞ [13]. We define I(Q 1 , Q 2 ; t) =
P1 · J 2 (Q 2 − B P2 ) · J − P2 · J 2 (Q 1 − B P1 ) · J
, P1 · J 2 P2 · J 2 P · J 2
(4.3)
which is invariant under rescalings of J . It is instructive to look at the symmetries of the wall (4.2). Clearly, it is invariant under the translations K(k) (2.1), if it acts both on the charges and the moduli. However, the wall is not invariant in general if only the charges are transformed. This is only the case for very special situations like P1 ||P2 . The change in the index is therefore not consistent with the spectral flow symmetry. Indeed, already in Sect. 2 we argued that this symmetry is not natural from the supergravity perspective. The fact that the symmetry is broken has major implications for supergravity partition functions, since the decomposition into a vector-valued modular form and theta functions is not valid. We can now see that Eq. (4.1) is in agreement with the supergravity picture. As mentioned before, the picture of stability in supergravity shows that only the single center solution exists if the moduli are chosen at the corresponding attractor point t ().
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Therefore, the index should equal the CFT-index at this point: (; t ()) = CFT (), which is consistent with the account of black hole entropy [32]. More evidence for this idea comes from the conjectures in Refs. [1,5], which suggest a one to one correspondence between connected components of the solution space of multi-centered asymptotic AdS3 ×S 2 solutions and IIA attractor flow trees starting at t () = limλ→∞ D −1 Q+iλP. Note that Z(τ, C, t) does not depend on λ in the limit J → ∞. By the AdS3 /CFT2 correspondence, this also suggests that (, t ()) = CFT (). If this is correct, (4.1) should not contribute to (; t ()). Indeed, computation of I(Q 1 , Q 2 ; t ()) gives P3 (P1 · Q 2 − P1 P 2 P2 P 2
P2 · Q 1 ), and therefore sgn(I(Q 1 , Q 2 ; t ())) − sgn(P · Q) = 0, such that there is never a contribution from bound states at the attractor point using this Ansatz. On the other hand, bound states with two constituents for charges ˜ = ˜ t ()) = CFT (). ˜ Therefore, these considmight exist at t (), and consequently (, erations of BPS-configurations with two constituents show that generically Z(τ, C, t) equals nowhere in the moduli space ZCFT (τ, C, t). i.e. 1 = (0, 0, Q 1 , Q 0,1 ). If one does not A special choice of charges is P1 = 0, move the moduli outside the Kähler cone, then walls for this choice can not be crossed. To see this, recall that Q 1 represents now the support of a coherent sheaf and must therefore represent a holomorphically embedded D2-brane. Therefore, Q 1 · J > 0 for J ∈ C X . The stability condition for (P, Q, Q 0,1 ) → (0, Q 1 , Q 0,1 ) + (P, Q 2 , Q 0,2 ) is given by P · Q 1 Q 1 · J < 0,
(4.4)
which is independent of the B-field. Equation (4.4) may or may not be satisfied for given charges. However, because Q 1 · J cannot change its sign for J ∈ C X , no walls of marginal stability are present in the large volume limit. It is thus consistent to consider only bound states of constituents with non-zero D4-brane charge. To construct the generating function, it is covenient to introduce some notation. For constituent i = 1, 2 with charge i , the corresponding quadratic form is denoted by (Q i )i2 and the conjugacy class of Q i in i∗ /i is μi . 1 , 2 can be written as an inner√ 2 ,P1 ) ∈ product of 2 vectors in 1 ⊕ 2 ⊗ R. Define to this end the unit vector P = (−P P P1 P2 √ 1 ⊕ 2 ⊗ R, then (Q 1 , Q 2 ) · P = Q · P = 1 , 2 / P P1 P2 . In the Appendix, I(Q 1 , Q 2 ; t) is also written as an innerproduct. Since the wall is independent of the D0-brane charge, the index (P, Q, Q 0 ; t) jumps irrespective of the D0-brane charge. For the partition function, we only want to keep track of the magnetic charge of the two constituents and sum over all the electric charge. Therefore, the contribution to the index (P, Q, Q 0 ; t) from bound states of constituents whose D4-brane charges are P1 and P2 includes a sum over the D0- and D2-brane charge: P1 ↔P2 (P, Q, Q 0 ; t) =
(Q 1 ,Q 0,1 )+(Q 2 ,Q 0,2 )=(Q,Q 0 )
1 (sgn(I (Q 1 , Q 2 ; t)) − sgn(1 , 2 )) 2
×(−1)1 ,2 (P1 · Q 2 − P2 · Q 1 ) (P1 , Q 1 , Q 0,1 )(P2 , Q 2 , Q 0,2 ).
The generating function of P1 ↔P2 (P, Q, Q 0 ; t) analogous to (3.4) is h P1 ↔P2 ,Q− 1 P 2 −Q 0 + 12 Q 2 (τ ) = (P, Q, Q ; t) q . This can be expressed in terms of the 0 Q 0 P1 ↔P2 vector-valued modular forms of the last section:
Stability and Duality in N = 2 Supergravity 1
h P1 ↔P2 ,Q− 1 P (τ ) q − 2 Q 2 =
665
2
(−1) P1 ·Q 2 −P2 ·Q 1 (P1 · Q 2 − P2 · Q 1 ) (1 ) (2 )
(Q 1 ,Q 0,1 )+(Q 2 ,Q 0,2 )=(Q,Q 0 ) Q0
1 ¯ +Q 0,2 ¯ × ( sgn(I(Q 1 , Q 2 ; t)) − sgn(1 , 2 ) ) q Q 0,1 2 1 ( sgn(I(Q 1 , Q 2 ; t)) − sgn(1 , 2 ) ) (−1) P1 ·Q 2 −P2 ·Q 1 = 2 Q 1 +Q 2 =Q
1
1
× (P1 · Q 2 − P2 · Q 1 ) h P1 ,μ1 (τ ) h P2 ,μ2 (τ ) q − 2 (Q 1 )1 − 2 (Q 2 )2 . 2
2
(4.5)
Note that the spectral flow symmetry is used here to write h Pi ,μi (τ ) instead of h Pi ,Q i −Pi /2 (τ ). Equation (4.5) can be seen as a major generalization of a similar formula for rank 2 sheaves on a rational surface [27]. To obtain the full generating function, we have to multiply h P1 ↔P2 ,Q− 1 P (τ ) by 2
(−1) P·Q e τ (Q − B)2+ /2 + τ¯ (Q − B)2− /2 + C · (Q − B/2) ,
(4.6)
and sum over Q ∈ ∗ . The various quadratic forms in the exponent combine to e τ (Q − B)2+ /2 + τ¯ (Q − B)21⊕2 − (Q − B)2+ /2 + C · (Q − B/2) , where Q 21⊕2 = (Q 1 )21 + (Q 2 )22 . See the Appendix for more explanation of the notation. The term (Q − B)21⊕2 −2(Q − B)2+ , which multiplies π τ2 in the exponent is not negative definite, but has signature (1, 2b2 −1). An unrestricted sum over all (Q 1 , Q 2 ) ∈ 1 ⊕2 is therefore clearly divergent. However, the presence of sgn(I(Q 1 , Q 2 ; t))− sgn(P · Q) ensures that the function is convergent, which follows from Proposition 1 in the Appendix. Thus the stability condition implies that the quadratic form is negative definite, if evaluated for stable bound states. Performing the sum over Q, one obtains the generating series: h P1 ,μ1 (τ ) h P2 ,μ2 (τ ) μ1⊕2 (τ, C, B), (4.7) μ1⊕2 ∈∗1⊕2 /1⊕2
where 1⊕2 = 1 ⊕ 2 , μ1⊕2 = (μ1 , μ2 ) ∈ ∗1⊕2 /1⊕2 and
μ1⊕2 (τ, C, B) =
(P1 · Q 2 − P2 · Q 1 ) (−1) P1 ·Q 1 +P2 ·Q 2
Q 1 ∈1 +μ1 +P1 /2 Q 2 ∈2 +μ2 +P2 /2
×
1 ( sgn(I(Q 1 , Q 2 ; t)) − sgn(P · Q) ) 2
× e τ (Q − B)2+ /2 + τ¯ (Q − B)21⊕2 − (Q − B)2+ /2 + C · (Q − B/2) , (4.8) with P =
(−P √ 2 ,P1 ) P P1 P2
∈ 1⊕2 ⊗ R.
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The test of S-duality is now reduced to testing modularity for (4.8). Since μ1⊕2 (τ, C, B) is not a sum over the total lattice 1⊕2 , it does not have the nice modular properties of the familiar theta functions. However, Ref. [46] explains that a real-analytic term can be added to a sum over a positive definite cone in an indefinite lattice with signature (n − 1, 1), such that the resulting function transforms as a familiar theta function. Appendix A applies this technique to μ1⊕2 (τ, C, B), and explains in detail how it can be completed to a function μ∗ 1⊕2 (τ, C, B), which transforms as a Siegel-Narain theta function.7 The essential idea of this procedure is to make the replacement √
sgn(z)
−→
2τ2 z
2
e−π u du, 2
(4.9)
0
which interpolates monotonically and continuously between −1 at z = −∞ and 1 at z = +∞. It approaches sgn(z) in the limit τ2 → ∞. To complete μ1⊕2 (τ, C, B) to a modular function, one also needs to replace z sgn(z) by an appriopriate continuous function as explained in the Appendix. Indefinite theta functions are prominent in the work on mock modular forms [46]; μ1⊕2 (τ, C, B) is therefore appropriately called a “mock Siegel-Narain theta function”. By replacing μ1⊕2 (τ, C, B) with μ∗ 1⊕2 (τ, C, B) in Eq. (4.7), we obtain our final proposal of the contribution of 2-center bound states Z P1 ↔P2 (τ, C, t) to Z(τ, C, t): Z P1 ↔P2 (τ, C, t) = h P1 ,μ1 (τ ) h P2 ,μ2 (τ ) μ∗ 1⊕2 (τ, C, B). (4.10) μ1⊕2 ∈∗1⊕2 /1⊕2
From the transformation properties of the three functions it follows that Z P1 ↔P2 (τ, C, t) transforms precisely as the CFT partition function ZCFT (τ, C, t) of the single constituent with D4-brane charge P1 + P2 (3.3)! To see that the weight agrees, note that the weight of μ∗ 1⊕2 (τ, C, B) is 21 (1, 2b2 + 1) = 21 (1, 2b2 − 1) + (0, 1), where 21 (1, 2b2 − 1) is due to the lattice sum and (0, 1) is due to the insertion of P1 · Q 2 − P2 · Q 1 . Combining this with 2 · (0, − 21 b2 − 1) of the vector-valued modular forms h Pi ,μi (τ ), one precisely finds the weight ( 21 , − 23 ) for Z P1 ↔P2 (τ, C, t). A crucial detail is the grading by (−1) P·Q : (−1)(P1 +P2 )·(Q 1 +Q 2 )+(P1 ·Q 2 −P2 ·Q 1 ) = (−1) P1 ·Q 1 +P2 ·Q 2 , such that μ∗ 1⊕2 (τ, C, B) does transform conjugately to h P1 ,μ1 (τ ) h P2 ,μ2 (τ ). Moreover, as was already mentioned above, coexistence of convergence and modularity is essentially a unique consequence of the Ansatz. In particular, the fact that P is independent of the moduli and satisfies P · (J, J ) = P · (B, B) = 0 is essential. We thus observe that all factors in (4.1) combine in a neat way such that Z P1 ↔P2 (τ, C, t) has the same modular properties as Z P1 +P2 (τ, C, t). One could of course object to correcting the partition function by hand and argue that an anomaly appeared for S-duality. However, the correcting factor could also arise automatically in a more physical derivation, for example by perturbative contributions. It is also not so surprising that corrections to the Fourier expansion (3.2) are necessary, since it was derived by assuming that the charges are finite and J → ∞, which is clearly not the case everywhere in the Hilbert space. Note that a physical derivation might lead to a slightly different modular completion of the generating function, since one could always add a real-analytic function with the same transformation properties. This would however not change the crucial properties we have established. 7 Note that the Fourier expansion (3.2) is thus not modular.
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Besides S-duality, there is another very appealing aspect in favor of the correction term. Equation (4.8) is not continuous as a function of the moduli B and J because of the terms sgn(I(Q 1 , Q 2 ; t)). As discussed above, the correction term is essentially a replacement of the discontinuous functions sgn(z) and z sgn(z) by real analytic functions (which approach the original expression in the limit |z| → ∞). The modular invariant partition function is therefore continuous in B and J . This might not be such a coincidence as it seems at first sight. Ref. [29] proposed a continuous and holomorphic generating function for Donaldson-Thomas invariants (or an extension thereof), which captures wall-crossing. Moreover, Ref. [23] describes that continuity of the metric g of the target manifold of a 3-dimensional sigma model, essentially implies the Kontsevich-Soibelman wall-crossing formula. Continuity of Z(τ, C, t) is very intriguing from this perspective, and it would be interesting to investigate whether it plays here an as fundamental role as in these references. The contribution of all 2-constituent BPS-states with primitive, ample charges is easily included in Z(τ, C, t) by the sum Z P1 ↔P2 (τ, C, t). The above analyses P1 +P2 =P ample, primitive
give some evidence that modularity is also preserved if one of the charges is not ample. 5. Conclusion and Discussion The consistency of wall-crossing with S-duality and electric-magnetic duality is tested by analyzing the BPS-spectrum of D4-D2-D0 branes on a compact Calabi-Yau 3-fold X . The stability of composite BPS-states with two primitive constituents is considered, in the large volume limit of the Kähler moduli space. The consistency of electricmagnetic duality with wall-crossing follows rather straightforwardly from the structure of the walls and the primitive wall-crossing formula. From the equations for the walls in the moduli space it can also be seen that wall-crossing is not compatible with the spectral flow symmetry, which appears in the microscopic description of a single D4-D2-D0 object by a CFT [32]. S-duality is tested by the construction of a partition function (4.10) for two constituents, which captures the changes of the spectrum if walls of marginal stability are crossed. The essential building block is a “mock Siegel-Narain theta function”, which might be of independent mathematical interest. The stability condition and the BPS-degeneracies combine in a very intricate way in order to preserve modularity, which is a confirmation of S-duality. The results of this paper are applicable to various problems, for example those related to entropy enigmas [12]. With these are meant BPS-configurations with multiple constituents, whose number of degeneracies is larger than the number of degeneracies of a single constituent with the same charge. Originally, the common thought was that wall-crossing would only have a subleading effect on the degeneracies. Ref. [1] has shown that enigmatic changes in the spectrum can also happen from D4-D2-D0 configurations with 2 constituents, which are considered in this paper. The present work shows that these enigmatic phenomena can be captured by modular invariant partition functions. This might prove useful in future studies on the entropy enigma. For example Eq. (4.10) entropy of two constituents (if their bound state shows that the leading 2 exists) is π 3 (P13 + P23 + c2 · P) Q 0¯ + 21 (Q 1 )21 + 21 (Q 2 )22 extremized with respect to Q 1 and Q 2 , under the constraint Q 1 + Q 2 = Q. This should be compared with the sin gle constituent entropy π
2 3 3 (P
+ c2 · P)(Q 0¯ + 21 Q 2 ). Based on these equations, one
can show the existence of enigmatic configurations, even in the regime
Qˆ 0¯ P3
1, or
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large topological string coupling. This shows that Zwc (τ, C, t) is not necessarily a small correction to ZCFT (τ, C, t) in (1.1). A detailed analysis of the conditions for the first entropy to be larger than the second would be very instructive. This raises the question of the relation of the discussed partition functions in this paper and the OSV-conjecture, which relates the black hole partition function and the one of topological strings [37]. The D4-D2-D0 BPS-degeneracies are also related to mathematically defined invariants. In the large volume limit, the D4-D2-D0 index correspond to the Euler number (or a variant thereof) of the moduli space of coherent sheaves with support on the divisor of the Calabi-Yau. An explicit calculation of these Euler numbers is currently not feasible, but would be magnificent. It would for example provide a more rigorous test of modularity of the partition functions. A more tractable possibility for future work is to replace the index (; t) by a more refined quantity [14] by including the spin dependence (; t, y) = Tr H(;t) (−y)2J3 . This is not a protected quantity, but is nevertheless of interest. The corresponding partition function might still exhibit modular properties, and wall-crossing formulas do exist in the literature for (; t, y) in the context of surfaces [27,45] and also physics [13]. A generalization of Sect. 4 to include these refined invariants should therefore be possible. Another suggestion is to move away from the limit J → ∞ by including finite size corrections. This would also leave the description of the BPS-states as coherent sheaves, and the relations with dualities probably become more intricate. A limitation of this work is that it considers only primitive wall-crossing. One might continue in a similar fashion as Sect. 4 to construct partition functions for BPSconfigurations with more constituents, and test the compatibility of the semi-primitive wall-crossing formula [12] and S-duality in this way. Much more appealing would be a closed expression for the partition function, which does not sum over all possible decays. Such an expression might ultimately allow for a test of the generic Kontsevich-Soibelman wall-crossing formula with respect to S-duality, or even explain the KS-formula in N = 2 supergravity from physical considerations, as was done for N = 2 field theory [23]. Although this paper took in some sense an opposite approach, some lessons might still be learned. The requirement of the dualities implies non-trivial constraints for the indices and wall-crossing formulas. These do not seem constraining enough to deduce the KS-formula. For example, the appearance of mock modular forms instead of normal modular forms was a priori unknown. This can of course be seen as an anomaly for S-duality. On the other hand, it is really pretty close to modularity, and the functions can be made modular by a simple modification as explained in the Appendix. These modifications might appear in a more physical derivation of the partition function in order to preserve S-duality. The correction terms might be determined by a differential equation, similar to the holomorphic anomaly equation of topological strings [3]. Proposition 5 gives the action of D, defined in Sect. 3, on μ∗ 1⊕2 (τ, C, B). This shows that DZ P1 ↔P2 (τ, C, t) includes a term ZCFT,P1 (τ, C, B)ZCFT,P2 (τ, C, B), which is suggestive and reminiscent of earlier work on holomorphic anomaly equations, see for example Ref. [34]. Another consequence of the correction terms is that they make the function continuous as a function of the moduli, although it captures the changes of the spectrum under variations of the moduli. This is quite intriguing, since “continuity” was essential in the field theory derivation of the KS-formula in Ref. [23], more precisely the continuity of the metric of the target space of a 3-dimensional sigma model. The appearance of a continuous partition function in this paper suggests that continuity might be fundamental here too. More investigation is clearly necessary to find out to what extent continuity and the dualities can imply the generic wall-crossing formula [30] for BPS-invariants.
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669
Ref. [29] suggested earlier a continuous, holomorphic generating function for Donaldson-Thomas invariants, and its discussion resembles in some respects Ref. [23]. However, Z P1 ↔P2 (τ, C, t) does not seem to be holomorphic in t. Note that the way Z P1 ↔P2 (τ, C, t) captures stability is quite different from how the partition function of 41 -BPS states (or dyons) of N = 4 supergravity captures stability. That function captures wall-crossing in a very appealing way by poles [40] and a proper choice of the integration contour [8] to obtain Fourier coefficients. In this way, mock modular forms arise via meromorphic Jacobi forms [9]. Section 4 shows that the supergravity partition function is nowhere in moduli space equal to the CFT partition function (except for special cases like a Calabi-Yau with b2 = 1). A natural question is: is the supergravity partition function related to the partition function of a lower dimensional theory, just as the spectrum of a single constituent is captured by the N = (4, 0) SCFT? Ref. [5] (see also [6]) proposes that such a theory might be classically a 2-dimensional sigma model into the moduli space of supersymmetric divisors in the Calabi-Yau, whose “beta function does not vanish for Y 8 different from the attractor point and the Y undergo renormalization group flow till they reach the attractor point, an IR fixed point. Along the flow, the constituents of M5-M5 bound states decouple from each other; each of them has its own IR fixed point corresponding to an AdS3 × S 2 .” The structure of the partition function (4.10) shows the decoupled constituents. It is also in agreement with the suggestion that the theory is not a CFT, since the spectral flow symmetry is not present. On the other hand, Zsugra (τ, C, t) does not equal ZCFT (τ, C, t) at attractor points, which indicates that the microscopic theory (if it exists) is not a CFT, not even at these points. A better understanding of these issues is clearly desired. Another alternative for a microscopic theory is quiver quantum mechanics [10], which arises in the limit gs → 0, and is known to capture bound states in 4 dimensions. A connection between this theory, the D4-D2-D0 bound states and their partition functions might lead to interesting insights. An intriguing implication of the proposed function is wall-crossing as a function of the C-field for the BPS-states one obtains after S-duality. A D4-D2-D0 BPS-state becomes a D3-D1-D-1 instantonic BPS-state after performing a T-duality along the time circle. This does not yet change anything fundamental, stability of this configuration is still captured by B and J . However, S-duality transforms such a configuration to one with instanton D3-branes and fundamental string instantons. Moreover, B and C are interchanged, which implies that the degeneracies of these BPS-states jump as a function of C and J . This is quite interesting since the C-field is generically not considered as a stability parameter, and gives also evidence that B and C should be considered on a more equal footing. The K-theoretic description of the C-fields is however very different in nature than the description of the B-field. Acknowledgements. I would like to thank Dieter van den Bleeken, Wu-yen Chuang, Atish Dabholkar, Emanuel Diaconescu, Davide Gaiotto, Lothar Göttsche and Gregory Moore for fruitful discussions. I owe special thanks to Gregory Moore for his comments on the manuscript. This work is supported by the DOE under grant DE-FG02-96ER40949.
A. Two Mock Siegel-Narain Theta Functions This Appendix computes the transformation properties of the Siegel-Narain mock theta function which appear in Sect. 4. The proofs are similar to those given in [46]. The 8 Y is the vector of normalized 5-dimensional Kähler moduli, which is proportional to J .
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dependence on the Grassmannian, which parametrizes 1-dimensional positive definite subspaces in the lattice , however complicates the discussion. First, properties of a simpler mock Siegel-Narain theta function are analyzed before those of μ∗ 1⊕2 (τ, C, B). Let , 1 and 2 be three lattices with signature (1, b2 − 1). The quadratic forms of the lattices are determined by a cubic form dabc : respectively dabc P c , dabc P1c and 3 > 0. dabc P2c . The vectors P(i) are characteristic vectors of the lattices and positive: P(i) They are related by P = P1 + P2 . The projection of a vector x ∈ ⊗ R on the positive definite subspace is determined by the vector J ∈ ⊗ R: x+ = (x · J/P · J 2 )J , 2 . The positive definite combination x 2 − x 2 is called x− = x − x+ , and x 2 = x+2 + x− + − the majorant associated to J . It is sufficient for this Appendix that J lies in the space 2 C := J ∈ ⊗ R : P(i) · J 2 , P(i) · J > 0, i = 1, 2 . J is thus positive in all three lattices. The direct sum 1 ⊕ 2 is denoted by 1⊕2 with quadratic form Q 21⊕2 = (Q 1 )21 + (Q 2 )22 for Q = (Q 1 , Q 2 ) ∈ ∗1⊕2 . Vectors in 1⊕2 are sometimes given the subscript 1 ⊕ 2, and in i the subscript i. For example, P1⊕2 = P1 + P2 ∈ 1⊕2 . Similarly, μ1⊕2 = μ1 + μ2 ∈ ∗1⊕2 /1⊕2 , and μ = μ1 + μ2 ∈ ∗ / with μi ∈ i∗ /i . With a slight abuse of notation Q 2+ denotes ((Q 1 + Q 2 ) · J )2 /P · J 2 . Define I(Q 1 , Q 2 ; t) as in the main text by I(Q 1 , Q 2 ; t) =
P1 · J 2 (Q 2 − P2 B) · J − P2 · J 2 (Q 1 − P1 B) · J
. P1 · J 2 P2 · J 2 P · J 2
(A.1)
Define additionally the vector (−P2 , P1 ) ∈ 1⊕2 ⊗ R, P= √ P P1 P2
(A.2)
which satisfies P 2 = 1. Definition 1. Let t = B + i J , with B ∈ ⊗ R, and J ∈ C . Then ∗μ1⊕2 (τ, C, B) is defined by: 1 (−1) P1 ·Q 1 +P2 ·Q 2 ∗μ1⊕2 (τ, C, B) = 2 Q∈1⊕2 +μ1⊕2 +P1⊕2 /2
× E I(Q 1 , Q 2 ; t) 2τ2 − E P · Q 2τ2 × e τ (Q − B)2+ /2 + τ¯ ((Q − B)21⊕2 − (Q − B)2+ )/2 + (Q − B/2) · C , (A.3) with
E(z) = 2
z
2 e−π u du = sgn(z) 1 − β(z 2 ) ,
0
where
∞
β(x) = x
1
u − 2 e−π u du,
x ∈ R≥0 .
The moduli in the exponent of (A.3) are determined by t. The “ * ” of ∗μ1⊕2 (τ, C, B) distinguishes this function from μ1⊕2 (τ, C, B), which would be defined by replacing E(z) by sgn(z) in the definition.
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Proposition 1. ∗μ1⊕2 (τ, C, B) is convergent for J ∈ C and B, C ∈ ⊗ R. Proof. First consider the case B = C = 0. The term which multiplies τ2 in the exponent, and thus determines the absolute value of the exponential is ((Q 1 + Q 2 ) · J )2 = Q 21⊕2 − 2Q 2+ . (A.4) P · J2 The signature of this quadratic form is (1, 2b2 − 1) which is problematic for convergence. To show convergence, note that 0 ≤ β(x) ≤ e−π x for all R≥0 and that therefore the terms involving β(x) in (A.3) are convergent. Consider next the terms with sgn(P · Q) − sgn(I(Q 1 , Q 2 ; i J )). There are essentially two possibilities: sgn(P·Q) sgn(I(Q 1 , Q 2 ; i J )) < 0 or > 0. Define the vector Q 2J := Q 21⊕2 − 2
(−P2 · J 2 J, P1 · J 2 J ) s(J ) = ∈ 1⊕2 ⊗ R, P1 · J 2 P2 · J 2 P · J 2 ) and s(J )2 = 1. such that Q · s(J ) = I(Q 1 , Q 2 ; i J One can show that P · s(J ) =
P·J 2 (P1 P2 J )2 P P1 P2 P1 ·J 2 P2 ·J 2
> 0 and P+ = s(J )+ = 0. The
space span(P, s(J )) has signature (1, 1) in 1⊕2 with inner product Q 2J . Therefore 1 P · s(J ) 2 = 1 − (P · s(J )) < 0. P · s(J ) 1 Take now a vector Q ∈ 1⊕2 , which is linearly independent of P and s(J ), then span(Q, P, s(J )) is a space with signature (1, 2). Therefore, Q2 Q·P Q · s(J ) J Q·P 1 P · s(J ) > 0. Q · s(J ) P · s(J ) 1 From this follows directly Q 2J +
2 P · s(J ) (Q · P)2 + (Q · s(J ))2 Q · P Q · s(J ) < < 0. 1 − (P · s(J ))2 1 − (P · s(J ))2
(A.5)
Therefore, if sgn(P · Q) sgn(I(Q 1 , Q 2 ; i J )) < 0 then Q 2J < 0. If Q is a linear combination of P and s(J ), the determinant is zero. From this follows that Q 2J = 0 only for Q = 0, and otherwise Q 2J < 0. The sum for sgn(Q · P) sgn(Q · J ) < 0 is therefore convergent. What is left is the case > 0. Then all the terms vanish identically, and therefore the whole sum is convergent. Inclusion of B and C does not alter the final conclusion. Proposition 2. ∗μ1⊕2 (τ, C, B) transforms under the generators S and T of S L(2, Z) as: S:
i(−iτ )1/2 (i τ¯ )b2 −1/2 2 ∗μ1⊕2 (−1/τ, −B, C) = − ∗ e(−P1⊕2 /4) |1 /1 ||∗2 /2 | e(−μ1⊕2 · ν1⊕2 ) ∗ν1⊕2 (τ, C, B), × ν1⊕2 ∈∗1⊕2 /1⊕2
T : ∗μ1⊕2 (τ + 1, B + C, B) = e((μ1⊕2 + P1⊕2 /2)21⊕2 /2) ∗μ1⊕2 (τ, C, B),
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J. Manschot
Proof. The S-transformation is proven using k∈ f (k) = k∈∗ fˆ(k), with fˆ(k) the Fourier transform of f (k). Therefore, one needs to determine the following Fourier transform:
d 2b2x E I(x1 , x2 ; i J ) 2Im(−1/τ ) 1⊕2 ⊗R
2 2 + π Im(−1/τ¯ )( x1⊕2 − 2x+2 ) + 2πi x · y × exp πiRe(−1/τ¯ )x1⊕2
= d 2b2x E I(x1 , x2 ; i J ) 2Im(−1/τ ) 1⊕2 ⊗R
2 − x+2 )/2τ¯ + x · y , ×e −x+2 /2τ − (x1⊕2
(A.6)
and the one with I(x1 , x2 ; i J ) replaced by Q · P. The following concentrates on the case with I(x1 , x2 ; i J ), the derivation for Q · P is completely analogous. Let Q · s(J ) = I(Q 1 , Q 2 ; i J ) as in Proposition 1, then the following definite quadratic forms can be defined: Q 21⊕2+ = Q 2+ + (Q · s(J ))2 ,
Q 21⊕2− = Q 21⊕2 − Q 21⊕2+ ,
since (J, J ) · s(J ) = 0. Using these quadratic forms, we write 2 2 e −x+2 /2τ − (x1⊕2 − x+2 )/2τ¯ = e −x+2 /2τ − (x · s(J ))2 /2τ¯ − x1⊕2− /2τ¯ . The Fourier transform can be written in the form 2 = e τ y+2 /2 + τ¯ I(y1 , y2 ; i J )2 /2 + τ¯ y1⊕2−
× d 2b2x E I(x1 , x2 ; i J ) 2Im(−1/τ ) 1⊕2 ⊗R
× e −(x − yτ )2+ /2τ − I(x1 − y1 τ¯ , x2 − y2 τ¯ ; i J )2 /2τ¯ − (x − y τ¯ )21⊕2− /2τ¯ . To proceed, one calculates the derivative of the integral
∂ d 2b2x E I(x1 , x2 ; i J ) 2Im(−1/τ ) ∂I(y1 , y2 ; i J ) 1⊕2 ⊗R ×e −(x − yτ )2+ /2τ − I(x1 − y1 τ¯ , x2 − y2 τ¯ ; i J )2 /2τ¯ − (x − y τ¯ )21⊕2− /2τ¯ √ i(−iτ )1/2 (i τ¯ )b2 −1/2 ∂ E I(y1 , y2 ; i J ) 2τ2 . =− ∗ ∂I(y1 , y2 ; i J ) |1 /1 ||∗2 /2 | This is shown by replacing the derivative by −τ¯ ∂I (x1 ,x2 ;i J ) , acting only on the exponent, and performing a partial integration. The equality is then easily established. Since (A.6) is an odd function of y, the integration constant is 0. Therefore (A.6) is equal to
i(−iτ )1/2 (i τ¯ )b2 −1/2 − ∗ E I(y , y ; i J ) 2τ2 1 2 |1 /1 ||∗2 /2 | 2 . (A.7) ×e τ y+2 /2 + τ¯ I(Q 1 , Q 2 ; i J )2 /2 + τ¯ y1⊕2− Using the standard techniques to include B- and C-field dependence etc., one finds the posed transformation law. Note that P ·(Q 1 − B P1 , Q 2 − B P2 ) = P ·(Q 1 , Q 2 ) = P · Q. The proof of the T -transformation is standard.
Stability and Duality in N = 2 Supergravity
Proposition 3. Define D = ∂τ +
673
i 2 4π ∂C+ 1/2
τ2
+ 21 B+ · ∂C+ − 41 πi B+2 , then
D μ1⊕2 (τ, C, B)
is a modular form of weight (2, b2 − 1). Proof. The action of D√on the exponents vanishes, and therefore only the derivative to τ on the functions E(z 2τ2 ) remains. The proposition follows easily from here. Definition 2. With the same input as for Definition 1: μ∗ 1⊕2 (τ, C, B) ⎛ 2 2 1 ⎝ P · J (P1 P2 J ) μ1 (τ, C, B) μ2 (τ, C, B) = √ 2 2 P1 · J P2 · J 2π 2τ2
− P P1 P2 μ1⊕2 (τ, C, B, P) 1 (−1) P1 ·Q 1 +P2 ·Q 2 (P1 · Q 2 − P2 · Q 1 ) 2 Q∈1⊕2 +μ1⊕2 +P1⊕2 /2
× E I(Q 1 , Q 2 ; t) 2τ2 − E P · Q 2τ2 × e τ (Q − B)2+ /2 + τ¯ ((Q − B)21⊕2 − (Q − B)2+ )/2 + (Q − B/2) · C
+
(A.8)
with μi (τ, C, B) as defined by Eq. (3.6), summing over i . μ1⊕2 (τ, C, B, P) is defined by μ1⊕2 (τ, C, B, P) =
(−1) P1⊕2 ·Q
Q∈1⊕2 +P1⊕2 /2+μ1⊕2
×e τ (Q − B)2+ /2 + τ (P · Q)2 /2 + τ¯ (Q − B)21⊕2− /2 + C · (Q − B/2) . In the limit τ2 → ∞, μ∗ 1⊕2 (τ, C, B) approaches μ1⊕2 (τ, C, B), which is defined in Eq. (4.8). This series is convergent because ∗μ1⊕2 (τ, C, B) is convergent. Proposition 4. μ∗ 1⊕2 (τ, C, B) transforms under the generators S and T of S L(2, Z) as: (−iτ )1/2 (i τ¯ )b2 +1/2 2 S : μ∗ 1⊕2 (−1/τ, −B, C) = − ∗ e(−P1⊕2 /4) |1 /1 ||∗2 /2 | e(−μ1⊕2 · ν1⊕2 ) ν∗1⊕2 (τ, C, B), × ν1⊕2 ∈∗1⊕2 /1⊕2
T : μ∗ 1⊕2 (τ + 1, B + C, B) = e((μ1⊕2 + P1⊕2 /2)21⊕2 /2) μ∗ 1⊕2 (τ, C, B).
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Proof. This is a continuation of the proof of Proposition 2. The following Fourier transform needs to be calculated:
d 2b2x (P1 · x2 − P2 · x1 ) E I(x1 , x2 ; i J ) 2Im(−1/τ ) 1⊕2 ⊗R
2 − x+2 )/2τ¯ − x+2 /2τ + x · y , ×e −(x1⊕2
(A.9)
and the one with I(x1 , x2 ; i J ) replaced by P · Q. We again concentrate on the case with I(x1 , x2 ; i J ). It is instructive to write P1 ·x2 − P2 ·x1 as (−P2 , P1 )·x T with x = (x1 , x2 ). The inner product (−P2 , P1 ) · x+ with x+ = x · J J/P · J 2 vanishes. Therefore, T (−P2 , P1 ) · x T = (−P2 , P1 ) · x− + (−P2 , P1 ) · s(J )T x · s(J ) 2 (P1 P2 J )2 T + P·J I(x1 , x2 ; i J ), = (−P2 , P1 ) · x− P ·J 2 P ·J 2 1
2
with s(J ) ∈ 1⊕2 as in the proofof Proposition 1. This shows that the factor P1 ·x 2 − P2 ·
x1 can be replaced by (2πi)−1 (−P2 , P1 ) · ∂ y− + Proposition 2, one finds that (A.9) equals
P·J 2 (P1 P2 J )2 ∂ P1 ·J 2 P2 ·J 2 I (y1 ,y2 ;i J )
. Using
⎡
(−iτ )1/2 (i τ¯ )b2 +1/2 ⎣ 2 − ∗ · y − P · y ) E I (y , y ; i J ) 2τ2 e τ y+2 /2+ τ¯ (y1⊕2 − y+2 )/2 (P 1 2 2 1 1 2 ∗ |1 /1 ||2 /2 | ⎤ √ 2τ2 P · J 2 (P1 P2 J )2 2 2 + e τ y+ /2 + τ I (y1 , y2 ; i J )2 /2 + τ¯ y1⊕2− /2 ⎦. πi τ¯ P1 · J 2 P2 · J 2
Clearly, this Fourier transform leads to a shift in the modular transformation properties. This can be cured if one recalls the transformation properties of the second Eisenstein series: E 2 (−1/τ ) = τ 2 (E 2 (τ ) − π6iτ ). A correction term can be added to E 2 (τ ): E 2∗ (τ ) = E 2 (τ ) − π3τ2 which transforms as a modular form of weight 2. This leads precisely to the term with theta functions in the definition. This means that the discontinuous function z sgn(z), which appears in (4.8), is replaced in μ∗ 1⊕2 (τ, C, B) by the real analytic function F(z) = z E(z) + |z| → ∞.
1 −π z 2 . πe
F(z) approaches z sgn(z) for
Proposition 5. With D as in Proposition 3, P · J 2 (P1 P2 J )2 i ∗ Dμ1⊕2 (τ, C, B) = − √ ϒμ1⊕2 (τ, C, B) P1 · J 2 P2 · J 2 2 2τ2 i + μ1 (τ, C, B)μ2 (τ, C, B) − μ1⊕2 (τ, C, B, P) , 3/2 4π(2τ2 ) with ϒμ1⊕2 (τ, C, B) = (−1) P1⊕2 ·Q (−P2 , P1 ) · Q − I(Q 1 , Q 2 ; t) Q∈1⊕2 +P1⊕2 /2+μ1⊕2
×e τ (Q − B)2+ /2 + τ I(Q 1 , Q 2 ; t)2 /2 + τ¯ (Q − B)21⊕2− /2 + C · (Q − B/2) . Proof. The proof is straightforward. Note that μi (τ, C, B) and ϒμ1⊕2 (τ, C, B) are not mock modular forms. The weights are respectively (1, b2 − 1) and (2, b2 ), such that the weight of Dμ∗ 1⊕2 (τ, C, B) is (5/2, (2b2 + 1)/2) as expected.
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References 1. Andriyash, E., Moore, G. W.: Ample D4-D2-D0 Decay. http://arxiv.org/abs/0806.4960v1 [hep-th], 2008 2. Aspinwall, P. S.: D-branes on Calabi-Yau manifolds. http://arxiv.org/abs/hep-th/0403166v1, 2004 3. Bershadsky, M., Cecotti, S., Ooguri, H., Vafa, C.: Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes. Commun. Math. Phys. 165, 311 (1994) 4. de Boer, J., Cheng, M.C.N., Dijkgraaf, R., Manschot, J., Verlinde, E.: A farey tail for attractor black holes. JHEP 0611, 024 (2006) 5. de Boer, J., Denef, F., El-Showk, S., Messamah, I., Vanden Bleeken, D.: Black hole bound states in AdS3 × S 2 . JHEP 0811, 050 (2008) 6. de Boer, J., Manschot, J., Papadodimas, K., Verlinde, E.: The chiral ring of AdS3/CFT2 and the attractor mechanism. JHEP 0903, 030 (2009) 7. Böhm, R., Günther, H., Herrmann, C., Louis, J.: Compactification of type IIB string theory on Calabi-Yau threefolds. Nucl. Phys. B 569, 229 (2000) 8. Cheng, M.C.N., Verlinde, E.: Dying Dyons Don’t Count. JHEP 0709, 070 (2007) 9. Dabholkar, A., Murthy, S., Zagier, D.: Quantum black holes and mock modular forms. to appear 10. Denef, F.: Quantum quivers and Hall/hole halos. JHEP 0210, 023 (2002) 11. Denef, F.: Supergravity flows and D-brane stability. JHEP 0008, 050 (2000) 12. Denef, F., Moore, G.W.: Split states, entropy enigmas, holes and halos. http://arxiv.org/abs/hep-th/ 0702146v2, 2007 13. Diaconescu, E., Moore, G.W.: Crossing the Wall: Branes vs. Bundles. http://arxiv.org/abs/0706.3193v4 [hep-th], 2007 14. Dimofte, T., Gukov, S.: Refined, Motivic, and Quantum. Lett. Math. Phys. 91, 1 (2010) 15. Donaldson, S.K., Kronheimer, P.B.: The geometry of four-manifolds. Oxford: Oxford University Press, 1990 16. Douglas, M.R., Fiol, B., Romelsberger, C.: Stability and BPS branes. JHEP 0509, 006 (2005) 17. Douglas, M.R.: D-branes, categories and N = 1 supersymmetry. J. Math. Phys. 42, 2818 (2001) 18. Eguchi, T., Taormina, A.: Character formulas for N = 4 superconformal algebra. Phys. Lett. B 200, 315 (1988) 19. Ferrara, S., Kallosh, R., Strominger, A.: N = 2 extremal black holes. Phys. Rev. D 52, 5412 (1995) 20. Freed, D.S., Witten, E.: Anomalies in string theory with D-branes. http://arxiv.org/abs/hep-th/9907189v2, 2000 21. Gaiotto, D., Strominger, A., Yin, X.: The M5-brane elliptic genus: Modularity and BPS states. JHEP 0708, 070 (2007) 22. Gaiotto, D., Yin, X.: Examples of M5-brane elliptic genera. JHEP 0711, 004 (2007) 23. Gaiotto, D., Moore, G.W., Neitzke, A.: Four-dimensional wall-crossing via three-dimensional field theory, http://arxiv.org/abs/0807.4723v3 [hep-th], 2009 24. Gawedzki, K.: Noncompact WZW conformal field theories. http://arxiv.org/abs/hep-th/9110076v1, 1991 25. Gopakumar, R., Vafa, C.: M-theory and topological strings. I,II. http://arxiv.org/abs/hep-th/9809187v2, 1998, http://arxiv.org/abs/hep-th/9812127v1, 1998 26. Göttsche, L., Zagier, D.: Jacobi forms and the structure of Donaldson invariants for 4-manifolds with b+ = 1. Selecta Math., New Ser. 4, 69 (1998) 27. Göttsche, L.: Theta functions and Hodge numbers of moduli spaces of sheaves on rational surfaces. Commun. Math. Physics 206, 105 (1999) 28. Huybrechts, D., Lehn, M.: The geometry of moduli spaces of sheaves. Cambridge: Cambridge Univ. Press, 1996 29. Joyce, D.: Holomorphic generating functions for invariants counting coherent sheaves on Calabi-Yau 3-folds. http://arxiv.org/abs/hep-th/0607039v1, 2006 30. Kontsevich, M., Soibelman, Y.: Stability structures, motivic Donaldson-Thomas invariants and cluster transformations. http://arxiv.org/abs/0811.2435v1[math.AG], 2008 31. Kraus, P., Larsen, F.: Partition functions and elliptic genera from supergravity. JHEP 0701, 002 (2007) 32. Maldacena, J.M., Strominger, A., Witten, E.: Black hole entropy in M-theory. JHEP 9712, 002 (1997) 33. Manschot, J.: On the space of elliptic genera. Comm. Num. Theor. Phys. 2, 803 (2008) 34. Minahan, J.A., Nemeschansky, D., Vafa, C., Warner, N.P.: E-strings and N = 4 topological Yang-Mills theories. Nucl. Phys. B 527, 581 (1998) 35. Minasian, R., Moore, G.W.: K-theory and Ramond-Ramond charge. JHEP 9711, 002 (1997) 36. Minasian, R., Moore, G.W., Tsimpis, D.: Calabi-Yau black holes and (0,4) sigma models. Commun. Math. Phys. 209, 325 (2000) 37. Ooguri, H., Strominger, A., Vafa, C.: Black hole attractors and the topological string. Phys. Rev. D 70, 106007 (2004) 38. Seiberg, N., Witten, E.: Monopole Condensation, And Confinement In N = 2 Supersymmetric Yang-Mills Theory. Nucl. Phys. B 426, 19 (1994) [Erratum-ibid. B 430, 485 (1994)]
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Commun. Math. Phys. 299, 677–708 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1099-3
Communications in
Mathematical Physics
Limiting Absorption Principle for Some Long Range Perturbations of Dirac Systems at Threshold Energies Nabile Boussaid1 , Sylvain Golénia2 1 Département de Mathématiques, Université de Franche-Comté, 16 Route de Gray, 25030 Besançon Cedex,
France. E-mail:
[email protected] 2 Mathematisches Institut, Universität Erlangen-Nürnberg, Bismarckstr. 1 1/2, 91054 Erlangen,
Germany. E-mail:
[email protected] Received: 29 June 2009 / Accepted: 16 February 2010 Published online: 12 August 2010 – © Springer-Verlag 2010
Abstract: We establish a limiting absorption principle for some long range perturbations of the Dirac systems at threshold energies. We cover multi-center interactions with small coupling constants. The analysis is reduced to studying a family of non-self-adjoint operators. The technique is based on a positive commutator theory for non-self-adjoint operators, which we develop in the Appendix. We also discuss some applications to the dispersive Helmholtz model in the quantum regime.
Contents 1. 2.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reduction of the Problem . . . . . . . . . . . . . . . . . . . . . 2.1 The non-self-adjoint operator . . . . . . . . . . . . . . . . . 2.2 From one limiting absorption principle to another . . . . . . 3. Positive Commutator Estimates . . . . . . . . . . . . . . . . . . 4. Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements. . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. Commutator Expansions . . . . . . . . . . . . . . . . . Appendix B. A Non-selfadjoint Weak Mourre Theory . . . . . . . . . Appendix C. Application to Non-relativistic Dispersive Hamiltonians References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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677 682 683 686 689 693 696 696 698 704 706
1. Introduction We study properties of relativistic massive charged particles with spin-1/2 (e.g., electron, positron, (anti-)muon, (anti-)tauon,. . .). We follow the Dirac formalism, see [17]. Because of the spin, the configuration space of the particle is vector valued. To simplify,
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we consider finite dimensional and trivial fiber. Let ν ≥ 2 be an integer. The movement of the free particle is given by the Dirac equation, i
∂ϕ = Dm ϕ, ∂t
in L 2 (R3 ; C2ν ),
where m > 0 is the mass, c the speed of light, the reduced Planck constant, and Dm := c α · P + mc2 β = −ic
3
αk ∂k + mc2 β.
(1.1)
k=1
Here we set α := (α1 , α2 , α3 ) and β := α4 . The αi , for i ∈ {1, 2, 3, 4}, are linearly independent self-adjoint linear maps, acting in C2ν , satisfying the anti-commutation relations: αi α j + α j αi = 2δi, j 1C2ν ,
where i, j ∈ {1, 2, 3, 4}.
For instance, when ν = 2, one may choose the Pauli-Dirac representation: IdCν 0 σi 0 and β = αi = , 0 −IdCν σi 0 0 1 0 −i 1 0 where σ1 = , σ2 = and σ3 = , 1 0 i 0 0 −1
(1.2)
(1.3)
for i = 1, 2, 3. We refer to [66, App. 1.A] for various equivalent representations. In this paper we do not choose any specific basis and work intrinsically with (1.2). We refer to [53] for a discussion of the representations of the Clifford algebra generated by (1.2). We also renormalize and consider = c = 1. The operator Dm is essentially self-adjoint on Cc∞ (R3 ; C2ν ) and the domain of its closure is H 1 (R3 ; C2ν ), the Sobolev space of order 1 with values in C2ν . We denote the closure with the same symbol. Easily, using Fourier transformation and some symmetries, one deduces the spectrum of Dm is purely absolutely continuous and given by (−∞, −m] ∪ [m, ∞). In this Introduction, we focus on the dynamical and spectral properties of the Hamiltonian describing the movement of the particle interacting with n fixed, charged particles. We model them by fixed points {ai }i=1,...,n ∈ R3n with respective charges {z i }i=1,...,n ∈ Rn . Doing so, we tacitly suppose that the particles {ai } are far enough from one another, so as to neglect their interactions. Note we make no hypothesis on the sign of the charges. The new Hamiltonian is given by zi Hγ := Dm + γ Vc (Q), where Vc := vc ⊗ IdC2ν and vc (x) := , |x − ai | k=1,...,n
(1.4) acting on Cc∞ (R3 \{ai }i=1,...,n ; C2ν ), with ai = a j for i = j. The γ ∈ R is the coupling constant. The index c stands for coulombic multi-center. The notation V (Q) indicates the operator of multiplication by V . Here, we identify L 2 (R3 ; C2ν ) L 2 (R3 ) ⊗ C2ν , canonically. Remark the perturbation Vc is not relatively compact with respect to Dm , then one needs to be careful to define a self-adjoint extension for Dm . Assuming √ (1.5) Z := |γ | max (|z i |) < 3/2, i=1,...,n
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the theorem of Levitan-Otelbaev ensures that Hγ is essentially self-adjoint and its domain is the Sobolev space H 1 (R3 ; C2ν ), see [2,45,49,51,52,50] for various gen−1 Z ≤ 118, where eralizations. This condition corresponds to the nuclear charge αat −1 αat = 137.035999710(96). Note that using the Hardy-inequality, the Kato-Rellich theorem will apply till Z < 1/2 and is optimal in the matrix-valued case, see [66, Sect. 4.3] for instance. For Z < 1, one shows there exists only one self-adjoint extension so that its domain is included in H 1/2 (R3 ; C2ν ), see [58]. This covers the nuclear charges up to Z = 137. When n = 1 and Z = 1, this property still holds true, see [23]. Surprisingly enough, when n = 1 and Z > 1, there is no self-adjoint extension with domain included in H 1/2 (R3 ; C2ν ), see [74, Theorem 6.3]. We mention also the work of [68] for Z > 1. In [58], one shows for Z < 1 that the essential spectrum is given by (−∞, −m] ∩ [m, ∞) for all self-adjoint extensions. For all Z , one refers to [30, Prop. 4.8.], which relies on [74]. In [29] one gives some criteria of stability of the essential spectrum for some very singular cases. In [4], one proves there is no embedded eigenvalues for a more general model until the coupling constant Z < 1. For all energies in a compact set included in (−∞, −m) ∩ (m, ∞), [30] obtains some estimates of the resolvent. This implies some propagation estimates and that the spectrum of Hγ is purely absolutely continuous. Similar results have been obtained for magnetic potential of constant direction, see [70] and more recently [64]. In this paper we are interested in uniform estimates of the resolvent at threshold energies. The energy m is called the electronic threshold and −m the positronic threshold. In Theorem (1.2), we obtain a uniform estimation of the resolvent over [−m − δ, −m] ∪ [m, m + δ], see (1.8) and deduce some propagation properties, see (1.9). One difficulty is that in the case n = 1 and z i < 0, it is well known there are infinitely many eigenvalues in the gap (−m, m) converging to the m as soon as γ = 0 (see for instance [66, Sect. 7.4] and references therein). This is a difficult problem and, to our knowledge, this result is new for the multi-center case. There is a larger literature for non-relativist models, e.g., − + V in L 2 (Rn ; C). The question is intimately linked with the presence of resonances at threshold energy, [43,25,57,63,69]. We mention also [14] for applications to Strichartz estimates and [19,20] for applications to scattering theory. We refer to [8,9] for perturbations in divergence form and to [36,37,67] for some more geometrical settings. We also point out some low energy results in the context of non-relativistic quantum electrodynamics, [26,27]. Before giving the main result, we shall discuss some commutator methods. The first stone was set by C.R. Putnam, see [61] and for instance [62, Theorem XIII.28]. Let H be a self-adjoint operator acting in a Hilbert space H . One supposes there is a bounded operator A so that C := [H, i A]◦ > 0,
(1.6)
where “>” means non-negative and injective. The commutator has to be understood in the form sense. When it extends into a bounded operator between some spaces, we denote this extension with the symbol ◦ in the subscript, see Appendix A. The operator A is said to be conjugate to H . One deduces some estimation on the imaginary part of the resolvent, i.e., one finds some weight B, a closed injective operator with dense domain, so that sup
(z)∈R,(z)>0
f, (H − z)−1 f ≤ B f 2 .
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This estimation is equivalent to the global propagation estimate, c.f. [46] and [62, Theorem XIII.25]: R
B −1 eit H f 2 dt ≤ 2 f 2 .
One infers that the spectrum of H is purely absolutely continuous with respect to the Lebesgue measure. In particular, H has no eigenvalue. To deal with the presence of eigenvalues, the fact that A is unbounded and with the 3-body-problem, E. Mourre had the idea to localize the estimates in energy had the idea to localize the estimates in energy and to allow a compact perturbation, see [56]. With further hypotheses, one shows an estimate of the resolvent (and not only on the imaginary part). The applications of this theory are numerous. The theory was immediately adapted to treat the N -body problem, see [60]. The theory was finally improved in many directions and optimized in many ways, see [1] for a more thorough discussion of these matters. We mention also [31,34,32] for recent developments. As we are concerned about thresholds, Mourre’s method does not seem enough, as the estimate of the resolvent is given on an interval which is strictly smaller than the one used in the commutator estimate. In [13] one generalizes the result of Putnam’s approach. Under some conditions, one allows A to be unbounded. They obtain a global estimate of the resolvent. Note this implies the absence of eigenvalue. In [25], in the non-relativistic context, by asking about some positivity on the Virial of the potential, see below, one is able to conciliate the estimation of the resolvent above the threshold energy and the accumulation of eigenvalues under it. In [63], one presents an abstract version of the method of [25]. To give an idea, we shall compare the theories on a non-optimal example. Take H := − + V in L 2 (R3 ), with V being in the Schwartz space. Consider the generator of dilation A := (P · Q + Q · P)/2, where P := −i∇. One looks at the quantity [H, i A]◦ − cH = −(2 − c) − WV (Q), where WV (Q) := Q · ∇V (Q) + cV (Q), with c ∈ (0, 2) and seeks some positivity. The expression WV is called the Virial of V . In [25], one uses extensively that WV (x) ≤ −cx−α for some α, c > 0 and |x| big enough. In [63], one notices that it suffices to suppose that WV (x) ≤ 0 and to take advantage of the positivity of the Laplacian. We take the opportunity to mention that it is enough to suppose that WV (x) ≤ c |x|−2 , for some small positive constant c , see Theorem C.1. Observe also that these methods give different weights. For instance, [25] obtains better weights in the scale of Qα and [63] can obtain singular weights like |Q|, see Appendix B. Finally, [25] deals only with low energy estimates and [63] works globally on [0, ∞). We also point out [39] which relies on commutator techniques and deals with smooth homogeneous potentials. In this article, we revisit the approach of [63] and make several improvements, see Appendix B. Our aim is twofold: to treat the dispersive non-self-adjoint operator and to obtain estimates of the resolvent uniformly in a parameter. At first sight, these improvements are pointless from the standpoint of the Coulomb-Dirac problem we treat. In reality, they are the key-stone of our approach. As a direct by-product of the method, we obtain some new results for dispersive Schrödinger operators. The following V2 term corresponds to the absorption coefficient of the laser energy by the material medium absorption term in the Helmholtz model, see [42] for instance.
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Theorem 1.1. Let n ≥ 3. Suppose that V1 , V2 ∈ L ∞ (Rn ; R) satisfy: (H1) ∇Vi , Q · ∇Vi (Q), Q(Q · ∇Vi )2 (Q) are bounded, for i ∈ {1, 2}. (H2) There are c1 ∈ [0, 2) and c1 ∈ 0, 4(2 − c1 )/(n − 2)2 such that W1 (x) := x · (∇V1 )(x) + c1 V1 (x) ≤
c1 , for all x ∈ Rn , |x|2
and V2 (x) ≥ 0 and − x · (∇V2 )(x) ≥ 0, for all x ∈ Rn . On Cc∞ (Rn ), we define H := − + V (Q), where V := V1 + iV2 . The closure of H defines a dispersive closed operator with domain H 2 (Rn ). We keep denoting it with H . Its spectrum is included in the upper half-plane. The operator H has no eigenvalue in [0, ∞). Moreover, |Q|−1 (H − λ + iμ)−1 |Q|−1 < ∞. (1.7) sup λ∈[0,∞), μ>0
Note we require neither smoothness on the potentials nor that they are relatively compact with respect to the Laplacian. We refer to Appendix C for further comments, the case c1 = 0 and a stronger result. We come back to the main application, namely the operator Hγ defined by (1.4). As the Dirac operator is vector-valued, coulombic interactions are singular and as we are interested in both thresholds, we were not able to use directly the ideas of [25,63]. Indeed, it is unclear for us if one can actually deal with the threshold energy and keep the “positivity” of something close to the quantity [Hγ , i A] − cHγ , for some self-adjoint operator A. We avoid this fundamental problem. First of all we cut-off the singularities of the potential Vc and consider the operator Hγbd = Dm + γ V in Sect. 2. We recover the singularities of the operator by perturbation in Proposition 4.1. In Sect. 2.1, similarly to [21], we make explicit the resolvent of Hγbd − z relative to a spin-down/up decomposition. This transfers the analysis to the one of an elliptic operator of second order, m,v,z , see Sect. 2.1. The drawback is that this operator is dispersive and also depends on the spectral parameter z. We bypass the latter difficulty by studying the family { m,v,ξ }ξ ∈E uniformly in E. In Sect. 2.2, we explain how to deduce the estimation of the resolvent of Hγbd having the one of m,v,z . In Sect. 3, we establish some positive commutator estimates for m,v,z and derive the sought estimates of the resolvent, see Theorem (3.1). For the last step, we rely on the theory developed in Appendix B. The main result of this Introduction is the following one. Theorem 1.2. There are κ, δ, C > 0 such that Q−1 (Hγ − λ − iε)−1 Q−1 ≤ C.
(1.8)
In particular, Hγ has no eigenvalue in ±m. Moreover, there is C so that sup Q−1 e−it Hγ E I (Hγ ) f 2 dt ≤ C f 2 ,
(1.9)
sup
|λ|∈[m,m+δ], ε>0,|γ |≤κ
|γ |≤κ R
where I = [−m − δ, −m] ∪ [m, m + δ] and where E I (Hγ ) denotes the spectral measure of Hγ .
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A more general result is given in Theorem 4.1. In Theorem 4.2 we discuss the weights P1/2 |Q| and in Remark 4.2 the weights |Q|. If one is not interested in the uniformity in the coupling constant, using [30], one can consider all δ > 0 and deduce (1.8). The propagation estimate (1.9) refers to Kato smoothness and it is a well-known consequence of (1.8), see [46]. Using some kernel estimates, one can obtain (1.8) directly for the free Dirac operator, i.e., γ = 0, see for instance [66, Sect. 1.E] and [48]. One may find an alternative proof of this fact in [41] which relies on some positive commutator techniques. In this study, we are mainly interested in long range perturbations of Dirac operators. Concerning limiting absorption principle for short range perturbations of Dirac operators there are some interesting works such as [15] for small perturbations without discrete spectrum or [10] for potentials producing discrete spectrum. These authors were mainly interested in time decay estimates similar to (1.9). In the short range case, the limiting absorption principle is a key ingredient to establish Strichartz estimates for perturbed Dirac type equations see [11,16]. For free Dirac equations there are some direct proofs, see [22,55,54]. Time decay estimates such as (1.9) or Strichartz are crucial tools to establish well posedness results [22,55,54] and stability results [10,11] for nonlinear Dirac equations. The paper is organized as follows. In the second section we reduced the analysis of the resolvent of the Dirac operator perturbed with a bounded potential to the one of family of non-self-adjoint operators. In the third part, we analyze these operators and obtain some estimates of the resolvent. In the fourth part, we state the main results of the paper. For the convenience of the reader, we expose some commutator expansions in Appendix A. In Appendix B, we develop the abstract positive commutator theory. At last in Appendix C, we give a direct application to the theory in the context of the Helmholtz equation. Notation. In the following and denote the real and imaginary part, respectively. The smooth function with compact support are denoted by Cc∞ . Given a complex-valued function F, we denote by F(Q) the operator of multiplication by F. We mention also the notation P = −i∇. We use the standard · := (1 + | · |2 )1/2 . 2. Reduction of the Problem In this section, we study the resolvent of the perturbed Dirac operator Hγbd := Dm + γ V, where V := v ⊗ IdC2ν and v bounded.
(2.1)
In Sect. 4, we explain how to cover some singularities. Due to the method, we will consider only small coupling constants. We will show the limiting absorption principle sup
|λ|∈[m,m+δ], ε>0, |γ |≤κ
Q−1 (Hγbd − λ − iε)−1 Q−1 ≤ C,
(2.2)
for some κ > 0. We notice this is equivalent to sup
λ∈[m,m+δ], ε>0, |γ |≤κ
Q−1 (Hγbd − λ − iε)−1 Q−1 ≤ C.
(2.3)
Indeed, by setting α5 := α1 α2 α3 α4 and using the anti-commutation relation (1.2), we infer α5 (Dm + γ V ) α5−1 = −Dm + γ V.
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Note that α5 is unitary in the Sobolev spaces H s (R3 ; C2n ), for s ∈ R. This gives α5 ϕ(Dm + γ V )α5−1 = ϕ − (Dm − γ V ) , for all ϕ ∈ C(R; C). (2.4) Finally notice that Q commutes to α5 . 2.1. The non-self-adjoint operator. Here, we relate the resolvent of (2.1) in a point z ∈ C\R with the one of some non-self-adjoint Laplacian type operator m,v,z , chosen in (2.8). We fix a compact set I being the area of energy we are concentrating on. In the next section, we explain how to recover a limiting absorption principle for Hγbd over I given the one of m,γ v,z . We consider a potential v ∈ L ∞ (R3 ; R), not necessarily smooth, satisfying v∞ ≤ m/2 and ∇v ∈ L ∞ (R3 ; R3 ).
(2.5)
In particular, (v(Q) − m − z)−1 stabilizes H 1 (R3 ; C2ν ) for all z in C\R. Since β = α4 satisfies (1.2), we deduct that β has the eigenvalues ±1 and the eigenspaces have the same dimension. Let P + be the orthogonal projection on the spin-up part of the space, i.e., on ker(β − 1). Let P − := 1 − P + . Since α j satisfies (1.2), for j ∈ {1, 2, 3}, we get P ± α j P ± = 0. We set: − + α +j := P + α j P − and α − j := P α j P , for j ∈ {1, 2, 3}.
They are partial isometries: + ∗ + − + + − and α − α j = α− j , αj αj = P j α j = P , for j ∈ {1, 2, 3}. The anti-commutation relation (1.2) gives: + − + − + αi− α +j + α − and αi+ α − j αi = 2δi, j P j + α j αi = 2δi, j P , for i, j ∈ {1, 2, 3}.
(2.6) We set Cν± := P ± C2ν . In the direct sum Cν− ⊕ Cν+ , with a slight abuse of notation, one can write 0 α +j 0 IdCν and α j = β= , for j ∈ {1, 2, 3}. 0 −IdCν α− 0 j We now split the Hilbert space H = L 2 (R3 ; C2ν ) into the spin-up and down part: H = H + ⊕ H − , where H ± := L 2 (R3 ; Cν± ) L 2 (R3 ; Cν ).
(2.7)
We define the operator:
m,v,z := α + · P
1 α − · P + v(Q) m − v(Q) + z
(2.8)
on Cc∞ (R3 ; Cν+ ). It is well defined by (2.5). It is closable as its adjoint has a dense domain. We consider the minimal extension its closure. We denote its domain by Dmin ( m,v,z ) and keep the same symbol for the operator. It is well known that even for symmetric operators one needs to be careful with domains as the domain of the adjoint could be much bigger than the one of the closure. In the next proposition, present this problem in our non-symmetric setting.
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Proposition 2.1. Let z ∈ C\R such that (z) ≥ 0. Under the hypotheses (2.5), we have that Dmin ( m,v,z ) = D( ∗m,v,z ) = H 2 (R3 ; Cν+ ) and m,v,z = ∗m,v,z . Proof. We mimic the Kato-Rellich with the more conve approach and compare2 m,v,z 3 ν ˜ nient operator z := 1/(m + z) 1,0,0 . Its domain is H (R ; C+ ) and its spectrum is {(m + z)t | t ∈ [0, ∞)}. We now show there is a ∈ [0, 1) and b ≥ 0 such that ˜ 2 2 B f 2 ≤ a
(2.9) z f + b f , holds true for all f ∈ Cc∞ (R3 , Cν+ ), where B :=
v (α + · ∇v)(Q) ˜z −i
α − · P + v(Q). (m − v + z) (m − v(Q) + z)2
Since v∞ ≤ m/2, (z) ≥ 0 and (z) > 0, we infer a0 := v/(m − v + z)∞ < 1. Set M := (α + · ∇v)(·)/(m − v(·) + z)2 ∞ . Take ε, ε ∈ (0, 1), 2 2 + 1 (α · ∇v)(Q) α − · P f + v(Q) f , f + 1+ 2 ε (m − v(Q) + z) 2 ˜ 2 4M α − · P f 2 + 4v∞ f 2 , ≤ (1 + ε)a02
z f + ε ε 2 4v
2|m + z|2 M 2 ∞ ˜ f 2 . + f + ≤ (1 + ε)a02 + ε
z ε εε
˜ B f 2 ≤ (1 + ε)a02
z
By choosing ε and ε so that the first constant is smaller than 1, (2.9) is fulfilled. ˜ z +μ)−1 2 ≤ a +bμ−2 for μ > 0. Fix μ0 > 0 Now, observe that since z > 0, B(
−1 ˜ ˜ z + μ0 )−1 ) is bijective. Noticing that such that B( z + μ0 ) < 1. Then (1 + B(
˜ z + μ0 ) = m,v,z + μ0 , ˜ z + μ0 )−1 (
Id + B(
we infer that m,v,z + μ0 is bijective from H 2 (R3 ; Cν+ ) onto L 2 (R3 ; Cν+ ). In particular Dmin ( m,v,z ) = H 2 (R3 ; Cν+ ). Directly, one has Dmin ( m,v,z ) ⊂ D( ∗m,v,z ) and
m,v,z ⊂ ∗m,v,z (inclusion of graphs). Take now f ∈ D( ∗m,v,z ). Since m,v,z + μ0 is surjective, there is g ∈ Dmin ( m,v,z ) so that ( m,v,z + μ0 )g = ( ∗m,v,z + μ0 ) f. In particular, ( ∗m,v,z + μ0 )( f − g) = 0. As m,v,z + μ0 is surjective, we derive that ker( ∗m,v,z + μ0 ) = {0}. In particular f = g, Dmin ( m,v,z ) = D( ∗m,v,z ) and m,v,z =
∗m,v,z . As a corollary, we derive: Lemma 2.1. The spectrum of m,v,z is contained in the lower/upper half-plane which does not contain z. In particular, c + z is always in the resolvent set of m,v,z for any c ∈ R.
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Proof. Take now f ∈ H 2 (R3 ; Cν+ ). Since −(z) α − · P f , f, m,v,z f = α − · P f, 2 m − v(Q) + (z) + (z)2
(2.10)
is of the sign of −(z). Since m,v,z is a closed operator having the same domain of its adjoint, the spectrum of m,v,z is contained in the closure of its numerical range, see Lemma B.1. We give a kind of Schur’s Lemma, so as to compute the inverse of the Dirac operator, see also [21,43]. Lemma 2.2. Suppose (2.5). Take z ∈ C\R such that (z) ≥ 0. In the spin-up/down decomposition H = H + ⊕ H − , we have (H1bd − z)−1 = ⎛ ( m,v,z + m − z)−1 ⎜ ⎝ 1 α − · P( m,v,z + m − z)−1 m − v(Q) + z ⎞ 1 −1 + ( m,v,z + m − z) α · P ⎟ m − v(Q) + z ⎟ ⎟. ⎠ 1 1 1 α − · P( m,v,z + m − z)−1 α + · P − m − v(Q) + z m − v(Q) + z m − v(Q) + z Remark 2.1. The operator (H1bd −z)−1 is bounded from L 2 (R3 ; C2ν ) into H 1 (R3 ; C2ν ). However, this improvement in the Sobolev scale does not hold if one looks at the matricial terms separately. There is a real compensation coming from the off-diagonal terms. First note that α − · P( m,v,z + m − z)−1 α + · P is a bounded operator in L 2 (R3 ; Cν− ) and a priori not into H s (R3 ; Cν− ), with s > 0. Indeed, α + · P sends L 2 (R3 ; Cν− ) into H −1 (R3 ; Cν+ ), then ( m,v,z + m − z)−1 to H 1 (R3 ; Cν+ ) and the left α − · P sends it again into L 2 (R3 ; Cν− ). On the other hand, the term ( m,v,z + m − z)−1 is bounded from L 2 (R3 ; Cν+ ) into H 2 (R3 ; Cν+ ), which is much better than expected. Proof. Let f ∈ L 2 (R3 ; C2ν ). By self-adjointness of H1bd , there is a unique ψ ∈ H 1 (R3 ; C2ν ) such that (H1bd − z)ψ = f . We separate the upper and lower spin components and denote f = ( f + , f − ) and ψ = (ψ+ , ψ− ) in H = H + ⊕ H − . We rewrite the equation (Dm + V (Q) − z)ψ = f to get: α + · Pψ− + mψ+ + v(Q)ψ+ − zψ+ = f + , (2.11) α − · Pψ+ − mψ− + v(Q)ψ− − zψ− = f − . From the second line, we get (v(Q) − m − z)ψ− = f − − α − · Pψ+ . Since z is not real, we can take the inverse and infer ψ− = (v(Q) − m − z)−1 ( f − − α − · Pψ+ ). Since ψ− ∈ H 1 , we can apply it to α + ·P and obtain a vector of L 2 (R3 ; Cν+ ). Now, since f − is in L 2 (R3 ; Cν− ) and since (v(Q)−m−z)−1 is bounded, we have α + · P(v(Q)−m−z)−1 f − ∈ H −1 (R3 ; Cν+ ) and since (v(Q) − m − z)−1 α − · Pψ+ is in L 2 (R3 ; Cν− ), we rewrite the system: ⎧ ⎪ ⎨ α+ · P ⎪ ⎩
1 1 α − · P + v(Q) + m − z ψ+ = f + + α + · P f− , m − v(Q) + z m − v(Q) + z − 1 ψ− = α · Pψ+ − f − . m − v(Q) + z
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To conclude it remains to show that m,v,z + m − z is invertible in B(H 1 , H −1 ), so as to invert it in the system. Using (2.10), we have |u, ( m,v,z − z)u| ≥ cu2H 1 . Then ( m,v,z + m − z)uH −1 ≥ cuH 1 and ( m,v,z + m − z)∗ uH −1 ≥ cuH 1 hold. Thus, m,v,z − z is bijective from H 1 onto H −1 . 2.2. From one limiting absorption principle to another. The main motivation for the operator m,v,z is to deduce a limiting absorption principle for Hγbd starting with one for m,γ v,z . Consider the upper right term in Lemma 2.2; the basic idea would be to put by force the weight Q−1 and to say that all terms are bounded. However, we have that 1 Q−1 ( m,v,z + m − z)−1 Q−1 Q α + · PQ−1 . m − v(Q) + z bounded from LAP for m,v,z
unbounded
One needs to take advantage to seek an estimate on a bounded interval of the spectrum. Therefore, we start with a lemma of localization in the momentum space and elicit a solution in Lemma 2.4. Note also that one may consider z < 0 by taking the adjoints in the next two lemmata. We shall also use estimates which are uniform in the coupling constant, due to Proposition 4.1. Lemma 2.3. Set I ⊂ R a compact interval. Let V be a bounded potential and κ > 0. There is an even function ϕ ∈ Cc∞ (R; R) such that the following estimations of the resolvent are equivalent: sup Q−1 ϕ(α · P)(Dm + γ V (Q) − z)−1 ϕ(α · P)Q−1 < ∞,
z∈I ,z>0,|γ |≤κ
sup
z∈I ,z>0,|γ |≤κ
sup
Q−1 (Dm + γ V (Q) − z)−1 ϕ(α · P)Q−1 < ∞,
z∈I ,z>0,|γ |≤κ
Q−1 (Dm + γ V (Q) − z)−1 Q−1 < ∞.
(2.12) (2.13) (2.14)
Proof. It is enough to consider z ∈ (0, 1]. Set J := I × (0, 1] × [−κ, κ], H◦ := α · P and Hγ := Dm + γ V . We choose ϕ1 ∈ Cc∞ (R) with value in [0, 1], being even and equal to 1 in a neighborhood of 0. We define ϕ R (·) := ϕ1 (·/R) and ϕ R := 1 − ϕ R . We first notice that Q ∈ C 1 (H◦ ), see Appendix A. There is a constant C > 0 so that |Q f, α · P f − α · P f, Q f | = | f, (α · ∇·)(Q) f | ≤ C f 2
(2.15)
holds true for all f ∈ Cc∞ (R3 ; C2ν ). This is usually not enough to deduce the C 1 property, see [28]. We use [35, Lemma A.2] with the notations A := H◦ , H := Q, χ n (x) := ϕ(x/n) and with D := Cc∞ (R3 ; C2ν ). The hypotheses are fulfilled and we deduce that Q ∈ C 1 (H◦ ). By the resolvent equality, we have: (Hγ − z)−1 ϕ˜ R (H◦ ) Id + W (H◦ − z)−1 ϕ˜ R (H◦ ) = (H◦ − z)−1 ϕ˜ R (H◦ ) − (Hγ − z)−1 ϕ R (H◦ )W (H◦ − z)−1 ϕ˜ R (H◦ ), (2.16)
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where W := γ V + mβ. Note that the support of ϕ˜ R vanishes as R goes to infinity. We have Q(H◦ − z)−1 ϕ˜ R (H◦ )Q−1 ≤ O(1/R), uniformly in (z, γ ) ∈ J . Indeed, if we commute with Q, the part in (H◦ − z)−1 ϕ˜ R (H◦ ) is a O(1/R) by functional calculus. For the other part, Lemma A.2 gives [Q, (H◦ − z)−1 ϕ˜ R (H◦ )]Q−1 ≤ O(1/R 2 ), uniformly in (z, γ ) ∈ J . Remembering V is bounded and choosing R big enough, we infer there is a constant c ∈ (0, 1), so that W Q(H◦ − z)−1 ϕ˜ R (H◦ )Q−1 ≤ c, uniformly in (z, γ ) ∈ J .
(2.17)
We fix R and choose ϕ := ϕ R . We now prove the equivalence. Observe that Q−1 ϕ(H◦ ) Q is bounded, since Q ∈ C 1 (H◦ ). One infers directly that (2.14) ⇒ (2.13) ⇒ (2.12). It remains to prove (2.12) ⇒ (2.14). Thanks to (2.17), we deduce from (2.16) that: ˜ ◦ )Q−1 Q−1 (Hγ − z)−1 ϕ(H = Q−1 (H◦ − z)−1 ϕ(H ˜ ◦ )Q−1 −Q−1 (Hγ − z)−1 ϕ(H◦ )Q−1 W Q(H◦ − z)−1 ϕ(H ˜ ◦ )Q−1 −1 ˜ ◦ )Q−1 . × Id + W Q(H◦ − z)−1 ϕ(H
(2.18)
Note that the last line and the right part of the second line of the r.h.s. are uniformly bounded in (z, γ ) ∈ J by (2.17). We multiply on the left by the bounded operator Q−1 ϕ(H◦ )Q. The first term of the r.h.s. is bounded uniformly by functional calculus. For the second one, we use (2.12). We infer: sup Q−1 ϕ(H◦ )(Hγ − z)−1 ϕ(H ˜ ◦ )Q−1 < ∞. (z,γ )∈J
Doing like in (2.18), on the left-hand side, we get sup Q−1 ϕ(H ˜ ◦ )(Hγ − z)−1 ϕ(H◦ )Q−1 < ∞. (z,γ )∈J
(2.19)
Finally, to control Q−1 ϕ(H ˜ ◦ )(Hγ − z)−1 ϕ(H ˜ ◦ )Q−1 , we multiply (2.18) on the left −1 ˜ ◦ )Q and deduce the boundedness using (2.19). by the bounded operator Q ϕ(H Lemma 2.4. Take κ ∈ (0, 1] and a compact interval I ⊂ [0, ∞). Suppose (2.5) and that (2.20) sup Q−1 ( m,γ v,z + m − z)−1 Q−1 < ∞
z∈I ,z∈(0,1],|γ |≤κ
hold true. Then, we have sup Q−1 (Dm + γ v(Q) ⊗ IdC2ν − z)−1 Q−1 < ∞.
z∈I ,z>0,|γ |≤κ
(2.21)
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Proof. Set H◦ := α · P and J := I × (0, 1] × [−κ, κ]. By Lemma 2.3, it is enough to show (2.12) for a chosen ϕ. Since ϕ is even and constant in a neighborhood of 0, √ by setting ψ(·) := ϕ( | · |), we have ψ ∈ Cc∞ (R) and that ϕ(H◦ ) = ψ (α · P)2 . In particular, we obtain ϕ(H◦ ) stabilizes H ± and have the right to let it appear in spin decomposition of the resolvent of H of Lemma 2.2. We treat only the upper right corner of the expression as the others are managed in the same way. We need to bound the term: Q−1 ϕ(H◦ )( m,γ v,z + m − z)−1 α + · P
1 ϕ(H◦ )Q−1 m − γ v(Q) + z
= Q−1 ϕ(H◦ )Q Q−1 ( m,γ v,z + m − z)−1 Q−1 1 ϕ(H◦ )Q−1 . Qα + · P m − γ v(Q) + z The middle term is controlled by the hypothesis. Thanks to (2.15), one has that [ϕ(H◦ ), Q] is bounded; hence the first term is bounded. For the last one, we commute: 1 ϕ(H◦ )Q−1 Qα + · P m − γ v(Q) + z 1 Q−1 Qϕ(H◦ )Q−1 = Q α + · P, m − γ v(Q) + z 1 Q−1 Qα + · Pϕ(H◦ )Q−1 . +Q m − γ v(Q) + z We estimate uniformly in (z, γ ) ∈ J . By (2.5) , we get Q(m − γ v(Q) + z)−1 Q−1 is bounded as Q commute with v. By (2.5), we also obtain that Q[α + · P, (m − γ v + z)−1 ]Q−1 is also controlled. At last, it is enough to consider Q∂ j ϕ(H◦ )Q−1 , which is easily bounded by Lemma A.2 for instance. We come to other types of weights. Motivated by the non-relativistic case, see Theorem C.1, we are interested in singular weights like |Q|. But, as noticed in Remark 4.2, the operator |Q|−1 (H bd − z)−1 |Q|−1 is even not bounded. Therefore, we enlarge the space in momentum and try the first reasonable weight, namely P1/2 |Q|. Given z ∈ C\R and using the Hardy inequality, one reaches P−1 |Q|−1 (Hγbd − v − z)−1 |Q|−1 ≤ P−1 |P| · |P|−1 |Q|−1 2 · (Hγbd − v − z)−1 |P| ≤ C(κ)z/|(z)|. By interpolation, one infers P−1/2 |Q|−1 (Hγbd − v − z)−1 |Q|−1 P−1/2 ≤ C(κ)z/|(z)| < ∞. The upper bound seems relatively sharp in z. However, under the same hypotheses as before, we obtain: Lemma 2.5. Take κ ∈ (0, 1] and a compact interval I ⊂ [0, ∞). Suppose (2.5) and that (2.20) hold true. Then, there is C > 0 so that sup P−1/2 |Q|−1 (Dm + γ v(Q) ⊗ IdC2ν − z)−1 |Q|−1 P−1/2 ≤ C.
z∈I ,z>0,|γ |≤κ
(2.22)
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Proof. It is enough to consider z ∈ (0, 1]. Set Hγ := Dm + γ v(Q) ⊗ IdC2ν and J := I × (0, 1] × [−κ, κ]. Let f = ( f + , f − ), with f ± ∈ Cc∞ (R3 \{0}; Cν± ). Lemma 2.2 and (2.5) give a constant C > 0, uniform in (z, γ ) ∈ J , so that: f, (Hγ − z)−1 f ≤ 4/m 2 f − 2 + 2 |Q|−1 ( m,γ v,z + m − z)−1 |Q|−1
2 2 × |Q| f + + |Q|α + · P(m − v(Q) + z)−1 f − 2 +|Q|α + · P(m − v(Q) + z)−1 f −
2 2 ≤ C |Q| f + + |Q|α + · P f − + f − 2 . Note that the Hardy inequality gives that f − ≤ 2|Q|α + · P f − . Then, by commuting |Q| with α + · P over Cc∞ (R3 \{0}; Cν ), we find C > 0, so that:
2 2 sup f, |Q|−1 (Hγ − z)−1 |Q|−1 f ≤ C Id ⊗ P+ f + P ⊗ P− f , (z,γ )∈J
for all f ∈ H 1 (R3 ; C2ν ), since Cc∞ (R3 \{0}; C2ν ) is dense in H 1 (R3 ; C2ν ). Here we identify, L 2 (R3 ; C2ν ) L 2 (R3 ) ⊗ C2ν . We now exchange the role of P+ and P− . Considering the operator α− · P
1 α + · P − v(Q) in L 2 (R3 ; Cν− ), m + v(Q) + z
which leads to the same arguments as for m,−v,z if one identifies Cν− Cν+ , one obtains also that
2 2 f, |Q|−1 (Hγ − z)−1 |Q|−1 f ≤ C Id ⊗ P− f + P ⊗ P+ f , for all f ∈ H 1 (R3 ; C2ν ). By interpolation, 6.4.5.(7)], we infer: f, |Q|−1 (Hγ − z)−1 |Q|−1 f for all f ∈ H 1/2 (R3 ; C2ν ).
e.g., [7, Theorem 4.4.1 and Theorem 2 ≤ C P1/2 f ,
3. Positive Commutator Estimates In the previous section, we saw how to deduce some estimate of the resolvent for Dm + V (Q) starting with some of m,v,z , namely (2.20). First, one technical problem is that these operators depend on the spectral parameter; hence we will study a family of operators uniformly in the spectral parameter. Secondly, we are concerned about the interval [m, m + δ] and we know that there is no such estimate above (m − ε, m) as eigenvalues usually accumulate to m from below. Since the theory developed in Appendix B gives some estimates for z ∈ [0, ∞), we will perform a shift. Therefore, we study the operator
2m,γ v,ξ , uniformly in (γ , ξ ) ∈ E = E(κ, δ) := [−κ, κ] × [0, δ] × (0, 1].
(3.1)
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Here we use a slight abuse of notation identifying C R2 . One should read ξ ∈ [0, δ], ξ ∈ (0, 1] and |γ | ≤ κ. Note the uniformity in the coupling constant is used in Proposition 4.1. To show (2.20), and therefore (2.14) with the help of Lemma 2.3, it is enough to prove the following fact. Note we strengthen the hypothesis (2.5). Theorem 3.1. Suppose that v ∈ L ∞ (R3 ; R) satisfies the hypotheses (H1) and (H2) from Theorem 4.1. Then there are δ, κ, CLAP > 0 such that −1 sup (3.2) |Q| ( 2m,γ v,ξ − z)−1 |Q|−1 ≤ CLAP .
z≥0,z>0,(γ ,ξ )∈E
We will show the theorem in the end of the section. We proceed by checking the hypothesis of Appendix B. We recall (2.6) and fix some notation: S := 1,0,0 = α + · P α − · P = − R3 ⊗ IdCν+ in H 2 (R3 ; Cν+ ) H 2 (R3 ) ⊗ Cν+ , and set S := H˙ 1 (R3 ; Cν+ ), the homogeneous Sobolev space of order 1, i.e., the completion of H 1 (R3 ; Cν+ ) under the norm f S := S 1/2 f 2 . Consider the strongly continuous one-parameter unitary group {Wt }t∈R acting by: (Wt f )(x) = e3t/2 f (et x), for all f ∈ L 2 (R3 ; Cν+ ). This is the C0 -group of dilatation. Easily, by interpolation and duality, one gets Wt S ⊂ S and Wt H s (R3 ; Cν+ ) ⊂ H s (R3 ; Cν+ ), for all s ∈ R.
(3.3)
Consider now its generator A in L 2 (R3 ; Cν+ ). It acts as follows: A=
1 (P · Q + Q · P) ⊗ IdCν+ on Cc∞ (R3 ; Cν+ ) Cc∞ (R3 ) ⊗ Cν+ . 2
By the Nelson Lemma, it is essentially self-adjoint on Cc∞ (R3 ; Cν+ ). In the next proposition, we will choose the upper bound κ of the coupling constant and state the commutator estimates. Proposition 3.1. Let δ ∈ (0, 2m). Suppose that the hypotheses (H1) and (H2) are fulfilled. Then there are c1 , κ > 0 such that D( 2m,γ v,ξ ) = H 2 (R3 ; Cν+ ), ( 2m,γ v,ξ )∗ = 2m,γ v,ξ ,
(3.4)
[ ( 2m,γ v,ξ ), i A]◦ − cv ( 2m,γ v,ξ ) ≥ c1 S > 0, ∓( 2m,γ v, (ξ )±i(ξ ) ) ≥ 0, ∓[( 2m,γ v, (ξ )±i(ξ ) ), i A]◦ ≥ 0,
(3.5) (3.6)
hold true in the sense of forms on H 1 (R3 ; Cν+ ), for all (γ , ξ ) ∈ E. Proof. The first part of (3.6) follows from (2.10). We start with a first restriction on κ. We impose κ ≤ (2m − δ)/v∞ . Hence, δ ≤ 2m − γ v(·) + (ξ ) ≤ 4m, for all (γ , ξ ) ∈ E.
(3.7)
In particular, 0 is not in the essential image of 2m − γ v + (ξ ); Proposition 2.1 gives (3.4).
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We turn to the commutator estimates. It is enough to compute the commutators in the sense of the form on Cc∞ (R3 ; Cν ), since it is a core for 2m,v,ξ and A, !
2m,γ v,ξ , i A = α + · P
1 − α · P, i A + γ [v, i A] 2m − γ v + ξ 1 = 2 α+ · P α− · P 2m − γ v + ξ Q · ∇v(Q) −γ α + · P α − · P − γ Q · ∇v(Q). (2m − γ v + ξ )2
(3.8)
Then, we have [ ( 2m,γ v,ξ ), i A] − cv ( 2m,γ v,ξ ) = 2m − γ v + (ξ ) = (2 − cv ) α + · P α− · P 2 2 2m − γ v + (ξ ) + (ξ ) " 2 # Q · ∇v(Q) 2m − γ v + (ξ ) − (ξ )2 + −γ α · P α− · P 2 2 2m − γ v + (ξ ) + (ξ )2 −γ Q · ∇v(Q) − cv γ v(Q). 16m 2 + 1 δ c S − κ Q · ∇v(Q) ≥ (2 − cv ) S − κ v 2 ≥ c1 S, 2 4 16m + 1 δ |Q| c1 v) where c1 := δ(2−c and by assuming that κ ≤ (4c +Q·∇v(Q)(16m 2 +1)/δ 4 ) . Note the “4” 32m 2 +2 v comes from the Hardy inequality. This gives (3.5). At last, we have:
[ 2m,γ v,ξ , i A] = −2(ξ ) α + 2 2m − γ v + (ξ ) + (ξ )2 − γ Q · ∇v(Q) 2m − γ v + (ξ ) − ·P α · P. 2 2 2m − γ v + (ξ ) + (ξ )2 This is of the sign of −(ξ ), when we further impose κ ≤ δ 2 /(8mQ · ∇v(Q)).
We now bound some commutators. Proposition 3.2. Let δ ∈ (0, 2m). Suppose that the hypotheses (H1) and (H2) are fulfilled. Consider the c1 , κ > 0 from Proposition 3.1. There is c and C depending on cv , δ, κ and v, such that | 2m,γ v,ξ f, Ag − A f, 2m,γ v,ξ g| ≤ c f · ( 2m,γ v,ξ ± i)g,
(3.9)
holds true, for all f, g ∈ H 2 (R3 ; C+ν ) ∩ D(A) and | f, [[ 2m,γ v,ξ , i A]◦ , i A]◦ f | ≤ C f, S f holds true for all f ∈ H 1 (R3 ; Cν+ ).
(3.10)
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Proof. As the domain of A and S are explicit, one easily sees that Cc∞ (R3 ; Cν+ ) is dense in D(A) ∩ D(S) endowed with the norm · + A · + S · . More generally, this also follows from the fact that S ∈ C 1 (A), see Theorem 6.2.10 of [1]. Therefore, it is enough to prove (3.9) on this core. We take κ as in the proof of Proposition 3.1. We first find c > 0, uniform in (γ , ξ ) ∈ E, so that | f, [ 2m,γ v,ξ , i A]◦ g| ≤ c f · g + f · Sg , for all f, g ∈ Cc∞ (R3 ; Cν+ ). (3.11) Taking into account (3.8), observe that 2 Q · ∇v(Q) 2 Q · ∇v(Q) ≤ +κ − γ . 2m − γ v + ξ 2 (2m − γ v + ξ ) δ δ2 It remains to find a, b > 0, which are uniform in (γ , ξ ) ∈ E, such that the following estimation holds: 2m,γ v,ξ f ≥ aS f − b f , for all f ∈ Cc∞ (R3 , Cν+ ). This follows from 2m,γ v,ξ f 2 ≥ a 2 S f 2 − b2 f 2 . Take ε, ε ∈ (0, 1), 2 2 γ (α + · ∇v)(Q) 1 1 − Sf + 1 − α · Pf . 2m − γ v(Q) + ξ ε (2m − γ v(Q) + ξ )2 1 κα + · ∇v(Q) 1 α − · P f 2 , S f 2 + 1 − ≥ (1 − ε) 2 4 ε 1 + 16m δ + · ∇v(Q) + 1 κα (ε − 1) 2 + (ε − 1) κα · ∇v(Q) f 2 . S f ≥ (1 − ε) + ε 1 + 16m 2 2εδ 4 2εε δ 4
2m,v,z f 2 ≥ (1 − ε)
Choosing ε small enough, we infer (3.9). We turn to (3.10). Again, it is enough to compute in the form sense on Cc∞ (R3 ; Cν ), 1 Q · ∇v(Q) − α− · P + 4 α+ · P α ·P 2m − v + z (2m − v + z)2 Q · ∇ Q · ∇v(Q) − (Q · ∇v(Q))2 + + −α · P α · P + 2 α · P (2m − v + z)2 (2m − v + z)3 − 2 α · P + (Q · ∇) v(Q).
[[ 2m,v,z , i A], i A] = 4 α + · P
Note that (H1) ensures that (Q · ∇)2 v(Q) f 2 ≤ 4 |Q|(Q · ∇)2 v(Q)2 S f 2 is controlled by S. Relying again on (3.7), the bound (3.10) follows. We finally turn to the proof of the main result of this section. Proof of Theorem 3.1. We check the hypotheses of Theorem B.1 for H − ( p) := 2m,γ v,ξ and p := (γ , ξ ), and where E is defined in (3.1). Clearly, S ∈ C 1 (A) and (B.4) is given by (3.3). Now, observe that t d ∗ f, (Wt H − ( p) − H − ( p)Wt )g = f, H − ( p)Ws gds Wt−s ds 0 t ∗ = Wt−s f, [H − ( p), i A]Ws gds, 0
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for all f, g ∈ Cc∞ (R3 ; Cν+ ). Using (3.11) and by density, we derive t Wt−s [H − ( p), i A]◦ Ws g ds, for all g ∈ H 2 , Wt H − ( p) − H − ( p)Wt g = 0
(3.12) and where the integral exists in the strong sense. By dividing by t, letting t go to 0 and using the fact that {Wt } is a C0 -group in H and in H 2 , we derive that 2m,γ v,ξ ∈ C 1 (A, H 2 , H ), for all (γ , ξ ) ∈ E. Note that (3.12) ensures that the strong limit of (it)−1 [H − ( p), Wt ] is [H − ( p), A]◦ . By interpolation, we deduce that 2m,γ v,ξ ∈ C 1 (A, H 1 , H −1 ). Now taking in account (3.10), we infer in the same way that
2m,γ v,ξ ∈ C 2 (A, H 1 , H −1 ), for all (γ , ξ ) ∈ E. Using Propositions 3.1 and 3.2, we can apply Theorem B.1. We derive there a finite C so that
sup | f, ( 2m,γ v,ξ − z)−1 f | ≤ C S −1/2 f 2 + S −1/2 A f 2 .
z≥0,z>0,(γ ,ξ )∈E
The Hardy inequality concludes.
4. Main Result In this section, we will prove the main result of this paper and deduce Theorem 1.2. Theorem 4.1. Let γ ∈ R. Suppose that v ∈ L ∞ (R3 ; R) satisfies the hypothesis: (H1) v∞ ≤ m/2 and ∇v, Q · ∇v(Q), Q(Q · ∇v)2 (Q) are bounded. (H2) There are cv ∈ [0, 2) and cv ≥ 0 such that x · (∇v)(x) + cv v(x) ≤
cv , for all x ∈ R3 \{0}. |x|2
Set V1 (Q) := v(Q) ⊗ IdC2n , where L 2 (R3 ; C2ν ) L 2 (R3 ) ⊗ C2ν . (H3) Consider V2 ∈ L 1loc (R3 ; R2ν ) satisfying: Q2 V2 (Q) ∈ B H 1 (R3 ; C2ν ), L 2 (R3 ; C2ν ) . Then, there are κ, δ, C > 0, such that Hγ := Dm + γ V (Q), where V := V1 + V2 , is self-adjoint with domain H 1 (R3 ; C2ν ). Moreover, sup
|λ|∈[m,m+δ], ε>0,|γ |≤κ
Q−1 (Hγ − λ − iε)−1 Q−1 ≤ C.
(4.1)
In particular, Hγ has no eigenvalue in ±m. Moreover, there is C so that Q−1 e−it Hγ E I (Hγ ) f 2 dt ≤ C f 2 , for all f ∈ L 2 (R3 , C4 ), (4.2) sup |γ |≤κ R
where I = [−m −δ, −m]∪[m, m +δ], and where E I (Hγ ) denotes the spectral measure of Hγ .
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Remark 4.1. In [25] and in [63], one takes advantage that the Virial of the potential is negative, in order to prove the limiting absorption principle for some self-adjoint Schrödinger operators, see Remark C.1. Here, we cannot allow this hypothesis as we are also interested in the positronic threshold, i.e., we seek a result for v and −v, see (2.4). We recover the positivity using some Hardy inequality and small coupling constants. Proof of Theorem 4.1. First note that (4.2) is a consequence of (4.1), see [46]. Consider the case V2 = 0. Note that, in Sect. 2, the operator Hγ is denoted by Hγbd . The self-adjointness is clear. We first apply Theorem 3.1 and obtain (3.2). We now choose ξ = z. As |Q| f ≤ Q f , we infer (2.20). In turn, it implies (2.14). Finally, using the unitary transformation α5 , (4.1) follows from (2.4). For a general V2 , we use Proposition 4.1. It remains to explain how to add the singular part V2 of the potential by perturbing the limiting absorption principal. This is somehow standard. Note that unlike [43], for instance, we do not distinguish the nature of the singularity at the threshold energy, as we work with small coupling constants. Proposition 4.1. Assume that Theorem 4.1 holds true for V2 = 0. Take now V2 satisfying (H3). Then there is κ ∈ (0, κ], so that Hγ := Dm + γ (V + V2 )(Q) is self-adjoint with domain H 1 (R3 ; C2ν ), for all |γ | ≤ κ. Moreover, sup Q−1 (Hγ − z)−1 Q−1 < ∞.
z∈[m,m+δ],z>0,|γ |≤κ
Proof. Up to a smaller κ, Kato-Rellich ensures the self-adjointness. We turn to the estimate of the resolvent. Easily, one reduces to the case |(z)| ≤ 1. From the resolvent identity, we have: % $ Q−1 (Hγ − z)−1 Q−1 Q Id + γ V2 (Hγbd − z)−1 Q−1 = Q−1 (Hγbd − z)−1 Q−1 . Considering Lemma 2.4 and Theorem 3.1, the result follows if we can invert the second term of the l.h.s. uniformly in the parameters. Therefore, we show there is κ ∈ (0, κ] so that sup
(z)∈[m,m+δ],(z)∈(0,1],|γ |≤κ
Qγ V2 (Hγbd − z)−1 Q−1 < 1.
Using the identity of the resolvent, we get QV2 (Hγbd − z)−1 Q−1 = QV2 (H0bd − i)−1 Q−1 −QV2 (H0bd − i)−1 Q (γ V − z + i) Q−1 (Hγbd − z)−1 Q−1 . The first term of the r.h.s. is bounded using (H3). To control the last term, remember that z is bounded and use again Lemma 2.4 and Theorem 3.1. It remains to notice that QV2 (H0bd − i)−1 Q = Q2 V2 (H0bd − i)−1 − QV2 (H0bd − i)−1 [H0bd , Q]◦ (H0bd − i)−1
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is bounded. Indeed, the assumption (H3) controls the terms in V2 and Q ∈ C 1 (H0 ) and [H0bd , Q]◦ is bounded, see the proof of Lemma 2.3. At last, Theorem 1.2 is an immediate corollary of Theorem 4.1. Indeed, one has: Example 4.1 (Multi-center). For i = 1, . . . , n, we choose ai ∈ R3 the site of the poles and Z i ∈ R its charge. We set: vc :=
n i=1
zi . | · −ai |
Note that Q · ∇vc (Q) + vc :=
n
ai · ∇v(Q).
i=1
Choose now ϕ ∈ Cc∞ (R3 ) radial with values in [0, 1]. Moreover, we ask that ϕ restricted to the ball B(0, 2 max(|ai |)) is 1. Consider the support large enough, so that ϕv ˜ ∞≤ m/2, where ϕ˜ : 1 − ϕ. Set v := ϕv ˜ c . Straightforwardly, the Hypothesis (H1) and (H2) are satisfied. Note that (H3) follows from the Hardy inequality. Example 4.2 (Smooth homogeneous potentials). In [39], one considers a smooth potential independent of |x| of the form v(x) := v(|x|/x), ˜ with v ∈ C ∞ (S 2 ), see also Remark C.2. Here, by taking cv = 0 in Theorems 4.1 and 4.2, one obtains a relativistic equivalent of this result. We point out that this perturbation is not relatively compact with respect to the Dirac operator. We now discuss singular weights in |Q|. Remark 4.2. It is important to note that unlike in the non-relativistic case, see Theorem C.1, one cannot replace the weights Q in (2.2) by |Q|. Indeed, with the notation of Theorem 4.1, V2 =0 and z ∈ C, consider a function f in Cc∞ (R3 \{0}; C2ν ) and notice −3/2 |Q|(Hγ − z)|Q| f (·/R) 2 tends to 0, as R goes to 0. Therefore, the expression R there is no z ∈ C such that the operator |Q|(Hγ − z)|Q| has a bounded inverse. We finally give a second result with a weight allowing some singularity in |Q|. Using Lemma 2.5 instead of Lemma 2.4 in the proof of Theorem 4.1, we infer straightforwardly: Theorem 4.2. Let γ ∈ R and take v ∈ L ∞ (R3 ; R) satisfying (H1) and (H2). Then, there are κ, δ, C > 0, such that Hγ := Dm + γ v(Q) ⊗ IdC2ν satisfies sup
|λ|∈[m,m+δ], ε>0,|γ |≤κ
P−1/2 |Q|−1 (Hγ − λ − iε)−1 |Q|−1 P−1/2 ≤ C. (4.3)
Moreover, there is C so that sup P−1/2 |Q|−1 e−it Hγ E I (Hγ ) f 2 dt ≤ C f 2 , |γ |≤κ R
(4.4)
where I = [−m − δ, −m] ∪ [m, m + δ] and where E I (Hγ ) denotes the spectral measure of Hγ .
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Keeping in mind Proposition 4.1, one sees that one can only add trivial potentials in the perturbation theory of the limiting absorption principle. Hence, it is an open question whether one can cover Example 4.1 with the weights P1/2 |Q|. Acknowledgements. We would like to thank Lyonel Boulton, Bertfried Fauser, Vladimir Georgescu, Nicolas Jacon, Thierry Jecko, Hubert Kalf, Andreas Knauf, Michael Levitin, François Nicoleau, Heinz Siedentop and Xue Ping Wang for useful discussions. The first author was partially supported by ESPRC grant EP/D054621.
Appendix A. Commutator Expansions This section is a small improvement of [34, App. B], see also [18,40]. We start with some generalities. Given a bounded operator B and a self-adjoint operator A acting in a Hilbert space H , one says that B ∈ C k (A) if t → e−it A Beit A is strongly C k . Given a closed and densely defined operator B, one says that B ∈ C k (A) if for some (hence any) z∈ / σ (B), t → e−it A (B − z)−1 eit A is strongly C k . The two definitions coincide in the case of a bounded self-adjoint operator. We recall a result following from Lemma 6.2.9 and Theorem 6.2.10 of [1]. Theorem A.1. Let A and B be two self-adjoint operators in the Hilbert space H . For z∈ / σ (A), set R(z) := (B − z)−1 . The following points are equivalent to B ∈ C 1 (A): (1) For one (then for all) z ∈ / σ (B), there is a finite c such that |A f, R(z) f − R(z) f, A f | ≤ c f 2 , for all f ∈ D(A). (2) a. There is a finite c such that for all f ∈ D(A) ∩ D(B): |A f, B f − B f, A f | ≤ c B f 2 + f 2 .
(A.1)
(A.2)
b. For some (then for all) z ∈ / σ (B), the set { f ∈ D(A) | R(z) f ∈ D(A) and R(z) f ∈ D(A)} is a core for A. Note that the condition (2.b) could be uneasy to check, see [28]. We mention [35, Lemma A.2] and [33, Lemma 3.2.2] to overcome this subtlety. As (B + i)−1 is a homeomorphism between H onto D(B), (B + i)−1 D(A) is dense in D(B), endowed with the graph norm. Moreover, (A.1) gives (B +i)−1 D(A) ⊂ D(A). Therefore (B + i)−1 D(A) ⊂ D(B) ∩ D(A) are dense in D(B) for the graph norm. Remark that D(B) ∩ D(A) is usually not dense in D(A), see [31]. Note that (A.1) yields the commutator [A, R(z)] extends to a bounded operator, in the form sense. We shall denote the extension by [A, R(z)]◦ . In the same way, since D(B) ∩ D(A) is dense in D(B), (A.2) ensures that the commutator [B, A] extends to a unique element of B D(B), D(B)∗ denoted by [B, A]◦ . Moreover, when B ∈ C 1 (A), one has: ! [B, A]◦ (B − z)−1 . A, (B − z)−1 ◦ = (B − z)−1 H ←D(B)∗
D(B)∗ ←D(B)
D(B)←H
Here we use the Riesz lemma to identify H with its anti-dual H ∗ . We now recall some well known facts on symbolic calculus and almost analytic extensions. For ρ ∈ R, let S ρ be the class of function ϕ ∈ C ∞ (R; C) such that ∀k ∈ N, Ck (ϕ) := sup t−ρ+k |ϕ (k) (t)| < ∞. t∈R
(A.3)
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Equipped with the semi-norms defined by (A.3), S ρ is a Fréchet space. Leibniz’ formula implies the continuous embedding: S ρ · S ρ ⊂ S ρ+ρ . We shall use the following result, e.g., [18]. Lemma A.1. Let ϕ ∈ S ρ with ρ ∈ R. For all l ∈ N, there is a smooth function ϕ C : C → C, such that: ∂ϕ C C (z) ≤ c1 (z)ρ−1−l |(z)|l , ϕ |R = ϕ, (A.4) ∂z suppϕ C ⊂ {x + iy | |y| ≤ c2 x},
(A.5)
C
ϕ (x + iy) = 0, if x ∈ suppϕ,
(A.6)
for some constants c1 , c2 depending on the semi-norms (A.3) of ϕ in S ρ and not on ϕ. One calls ϕ C an almost analytic extension of ϕ. Let A be a self-adjoint operator, ρ < 0 and ϕ ∈ S ρ . By functional calculus, one has ϕ(A) bounded. The Helffer-Sjöstrand Formula, see [38] and [18] for instance, gives that for all almost analytic extensions of ϕ, one has: i ∂ϕ C ϕ(A) = (A.7) (z − A)−1 dz ∧ dz. 2π C ∂z Note the integral exists in the norm topology, by (A.4) with l = 1. Next we come to a commutator expansion. Here B is not necessarily bounded while in [34], one considers j the case B bounded. We denote by ad A (B) the extension of the j th commutator of A p p−1 with B defined inductively by ad A (B) := [ad A (B), A]◦ , when it exists. Proposition A.1. Let k ∈ N∗ and B ∈ C k (A) be self-adjoint. Suppose ad A (B) are bounded operators, for j = 1, . . . , k. Let ρ < k and ϕ ∈ S ρ . Suppose that D(B) ∩ D(Aρ ) is dense in D(Aρ )for the graph norm. Then, the commutator [ϕ(A), B]◦ belongs to B D(Aρ−1 )), H and satisfies j
[ϕ(A), B]◦ =
k−1 1 ( j) j ϕ (A)ad A (B) j! j=1 i ∂ϕ C + (z − A)−k adkA (B)(z − A)−1 dz ∧ dz, 2π C ∂z
(A.8)
where the integral exists for the topology of B(H ). Proof. We cannot use (A.7) directly with ϕ as the integral does not seem to exist. We proceed as in [34]. Take χ 1 ∈ Cc∞ (R; R) with values in [0, 1] and being 1 on [−1, 1]. Set χ R := χ (·/R). As R goes to infinity, χ R converges pointwise to 1. Moreover, {χ R } R∈[1,∞] is bounded in S 0 . We infer ϕ R := ϕ χ R tends pointwise to ϕ and that {ϕ R } R∈[1,∞] is bounded in S ρ . Now, note that [ϕ R (A), B] =
k−1 ∂ϕ C i j R (z − A)− j−1 ad A (B)dz ∧ dz 2π C ∂z j=1 ∂ϕ C i R (z − A)−k adkA (B)(z − A)−1 dz ∧ dz + 2π C ∂z
(A.9)
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in the form sense on D(B). Using (A.4), the integral converges in norm. We write & ( j) j [ϕ R (A), B]◦ on the l.h.s. The first term of the r.h.s. is k−1 j=1 ϕ R (A)ad A (B)/j! Now we let R go to infinity. For the l.h.s. and the first term of the r.h.s., we expand the commutator in (A.9) in the form sense on D(Aρ ) ∩ D(B), take the limit by functional calculus and finish by density in D(Aρ ). For the remainder term of the r.h.s., we use the Lebesgue convergence theorem. It remains to note that the operator of the r.h.s. is in B(Aρ−1 , H ) since ϕ ∈ S ρ . The hypothesis on the density of D(B) ∩ D(Aρ ) in D(Aρ ) could be delicate k to check. It follows by the Nelson Lemma from the fact that the C0 -group {eit A }t∈R stabilizes D(B). We mention that for k = 1, since [B, i A]◦ is bounded, [28, Lemma 2] ensures this invariance of the domain. The rest of the previous expansion is estimated as in [34]. We rely on the following important bound. Let c > 0 and s ∈ [0, 1], there exists some C > 0 so that, for all z = x + iy ∈ {a + ib | 0 < |b| ≤ ca}: s A (A − z)−1 ≤ Cxs · |y|−1 . (A.10) j
Lemma A.2. Let B ∈ C k (A) be self-adjoint. Suppose ad A (B) are bounded operators, for j = 1, . . . , k. Let ϕ ∈ S ρ , with ρ < k. Let Ik (ϕ) be the rest of the development of order k of [ϕ(A), B] in (A.8). Let s, s ∈ [0, 1] such that ρ + s + s < k. Then As Ik (ϕ)As is bounded and it is uniformly bounded when ϕ stays in a bounded subset ρ of S . Let R > 0. If ϕ stays in a bounded subset of {ψ ∈ S ρ | [−R; R] ∩ supp(ϕ) = ∅} then Rk−ρ−s−s As Ik (ϕ)As is uniformly bounded. Proof. We will follow ideas from [18, Lemma C.3.1]. In this proof, all the constants are denoted by C, independently of their value. Given a complex number z, x and y will denote its real and imaginary part, respectively. Since B ∈ C k (A), adkA (B) is bounded. We start with the second assertion. Let ϕ ∈ S ρ , R > 0 such that [−R; R]∩supp(ϕ) = ∅. Notice that, by (A.6), ϕ C (x + iy) = 0 for |x| ≤ R. By (A.10), ∂ϕ C xs 1 xs s s k A Ik (ϕ)A ≤ · d x ∧ dy · ad A (B) · π ∂z |y|k |y| ≤ C(ϕ) xρ+s+s −1−l |y|l |y|−k−1 d x ∧ dy, |x|≥R
|y|≤c2 x
for any l, by (A.4). Recall that dz ∧ dz = −2idx ∧ dy. We choose l = k + 1. We have, As Ik (ϕ)As ≤ C(ϕ) xρ+s+s −k−1 d x ≤ C(ϕ)Rρ+s+s −k . |x|≥R
Since C(ϕ) is bounded when ϕ stays in a bounded subset of S ρ , this yields the second assertion. For the first one, we can follow the same lines, replacing R by 0 in the integrals, and arrive at the result. Appendix B. A Non-selfadjoint Weak Mourre Theory In this section, we adapt ideas coming from [25] and [63] in order to obtain a limiting absorption principle for a family of closed operators {H ± ( p)} p∈E . We ask that they have a common domain (B.1) D := D H + ( p) = D H − ( p) , for all p ∈ E.
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We choose p0 ∈ E and endow D with the graph norm of H + ( p0 ). We also ask that
H + ( p)
∗
= H − ( p), for all p ∈ E.
(B.2)
In particular, we have that D (H ± ( p))∗ = D. In the sequel, we forgo p, when no confusion can arise. Since H ± are densely defined, share the same domain and are adjoint of the other, we have that (H ± ) and (H ± ) are closable operators on D, indeed their adjoints are densely defined. We denote by (H ± ) and by (H ± ) the closure of these operators. It is possible that they are not self-adjoint, albeit they are symmetric. However, D is a core for them. Their domain is possibly bigger than D. We suppose that H + is dissipative, i.e., f, (H + ) f ≥ 0,
for all f ∈ D.
This gives also that (H − ) ≤ 0. By the numerical range theorem (see Lemma B.1), we infer that σ (H ± ) is included in the half-plan containing ±i. Take now a non-negative selfadjoint operator S, independent of p ∈ E, with form domain G := D(S 1/2 ) ⊃ D. We assume that S is injective. We have f, S f > 0 for all f ∈ G \{0} and simply write S > 0. One defines S as the completion of G under the norm f 2S := f, S f . We obtain G ⊂ S with dense and continuous embedding. Moreover, since G = S 1/2 −1 H , S is also the completion of H under the norm given by S 1/2 S 1/2 −1 · . We use the Riesz Lemma to identify H with H ∗ , its anti-dual. The adjoint space S ∗ of S is exactly the domain of S 1/2 S −1/2 in H H ∗ . Note that S −1 is an isomorphism between S and S ∗ . We get the following scale with continuous and dense embeddings: S∗ ↓ & D −→ G −→ H H ∗ −→ G ∗ −→ D ∗ . & ↓ S
(B.3)
To perform this analysis, we consider an external operator, the conjugate operator. Let A be a self-adjoint operator in H . We assume S ∈ C 1 (A). Let Wt := eit A be the C0 -group associated to A in H . We ask: Wt G ⊂ G and Wt S ⊂ S , for all t ∈ R.
(B.4)
By duality, we have Wt stabilizes G ∗ and also S ∗ (but may be not D or D ∗ ). The restricted group to these spaces is also a C0 -group. We denote the generator by A with the subspace in subscript. Given Hi ⊂ H j are two of those spaces, one easily shows that A|Hi ⊂ A|H j and that A|H j is the closure of A|Hi in H j . Moreover, one has D(A|Hi ) =
$
% f ∈ D A|H j ∩ Hi such that A|H j f ∈ Hi .
(B.5)
We now explain how to check the second hypothesis of (B.4), see also [63]. We mention this result is due to [24] when D(S) ⊂ D(A).
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Remark B.1. The second invariance of the domains of (B.4) follows from the first one and from |S f, A f − A f, S f | ≤ cS 1/2 f 2 , for all f ∈ D(S) ∩ D(A).
(B.6)
As (S + i)−1 is a homeomorphism between H onto D(S), (S + i)−1 D(A) is dense in D(S), endowed with the graph norm. Moreover, since S ∈ C 1 (A), one has (S + i)−1 D(A) ⊂ D(A). Therefore (S + i)−1 D(A) ⊂ D(S) ∩ D(A) are dense in D(S), hence in G and in S . The commutator [S, A] has a unique extension to an element of B(S , S ∗ ), in the form sense. We denote it by [S, A]◦ . Take now f ∈ G ∩ D(A), which is a dense set in G . On one hand we have τ → Wτ f 2S is bounded when τ is in a compact set (since G → S . On the other hand, the Gronwall Lemma concludes by noticing: t |t| Wt f 2S = f, S f + Wτ f, [S, i A]◦ Wτ f dτ ≤ S 1/2 f 2 + c Wτ f 2S dτ. 0
0
Let K ⊂ H be a space which is stabilized by Wt . Consider L ∈ B(K , K ∗ ). We say that L ∈ C k (A; K , K ∗ ), when t → W−t L Wt is strongly C k from K into K ∗ . When K = H , using the resolvent equality, one observes that this class is the same as C k (A), see for instance [1][Theorem 6.3.4 a.]. Theorem B.1. Let H ± = H ± ( p), with p ∈ E as above. Let A be self-adjoint such that (B.4) holds true. Suppose that H ± ∈ C 2 (A; G , G ∗ ) and that there is a constant c, independent of p, such that |H ∓ f, Ag − A f, H ± g| ≤ c f · (H ± ± i)g, for all f, g ∈ D ∩ D(A). (B.7) Take c1 ≥ 0 independent of p and assume that [ (H ± ), i A]◦ − c1 (H ± ) ≥ S > 0, ±c1 [(H ± ), i A]◦ ≥ 0, ±(H ± ) ≥ 0,
(B.8) (B.9)
in the sense of forms on G . Suppose also there exists C > 0 independent of p ∈ E such that ± ! ! f, H , A , A f ≤ CS 1/2 f 2 , for all f ∈ G . (B.10) ◦ ◦ Then, there are c and μ0 > 0, both independent of p, such that
| f, (H ± − λ ± iμ)−1 f | ≤ c S −1/2 f 2 + S −1/2 A f 2 ≤ c f D(A|S ∗ ) , (B.11) for all p ∈ E, μ ∈ (0, μ0 ) and λ ≥ 0, in the case c1 > 0 and λ ∈ R if c1 = 0. In the self-adjoint setting, the case c1 = 0 is treated in [12,13]. Comparing with [63], which deals with the case of one self-adjoint operator and for c1 > 0, we give some few improvements. First, we do not ask D to be the domain of S. Moreover, we drop the hypothesis that the first commutator [H, i A]◦ is bounded from below. For the latter, we use more carefully the numerical range theorem in our proof. Finally, unlike [63], we shall not go into interpolation theory so as to improve the norm in the limiting absorption principle. Indeed, in the context of the model we are considering here, we
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reach the weights we are interested in without it. We stick to an intermediate and explicit result, which is closer to [41]. Therefore, for the sake of clarity, we present then the easiest proof possible and pay important care about domains. We also mention that there exists another Mourre-like theory for non-self-adjoint operators, [3,65]. Proof. We focus on the case c1 > 0; as for the case c1 = 0, one replaces “λ ≥ 0” by “λ ∈ R”. Since H ± ∈ C 1 (A, G , G ∗ ), by the resolvent equality, we obtain ! ! ± (H ± i)−1 , Wt = − (H ± ± i)−1 H ± , Wt (H ± ± i)−1 . H ←−G ∗
G ∗ ←−G
G ←−H
We take the derivative with respect to t. It exists strongly in H , then H ± ∈ C 1 (A). In particular, as in Remark B.1, one has D ∩ D(A) dense in D for the graph norm. Thus, (B.7) gives [H ± , i A]◦ ∈ B(D, H ). We define Hε± := H ± ± iε[H ± , i A]◦ with the domain D for ε ≥ 0. Since H ± ± i is bijective, by writing Hε± ± i = common ± 1 ± iε[H , i A]◦ (H ± ± i)−1 (H ± ± i) and using (B.7), we get there is ε0 such that for all |ε| ≤ ε0 and all p ∈ E. Therefore (Hε± ± i)∗ is Hε± ( p) ± i is bijective and closed ± ∗ also bijective from D (Hε ) onto H . Now since (Hε± ± i)∗ is an extension of Hε∓ ∓ i which is also bijective, we infer the equality of the domains and that (Hε± )∗ = Hε∓ for ε ≤ ε0 . Since H ± ∈ C 1 (A; G , G ∗ ), we obtain that (H ± ) and (H ± ) are in C 1 (A; G , G ∗ ). In this space we have [H ± , A]◦ = [ (H ± ), A]◦ + i[(H ± ), A]◦ . Now, take f ∈ G . Take ε, λ, μ ≥ 0. We get: ' ( ' ( −c1 ε f, (Hε± − λ ± iμ) f ± f, (Hε± − λ ± iμ) f ( ' ( ' = −c1 ε f, (H ± )∓ε[(H ± ), i A]◦ −λ f ± f, (H ± )±μ ± ε[ (H ± ), i A]◦ f ' ( = ε f, [ (H ± ), i A]◦ − c1 (H ± ) f + (c1 λε + μ) f 2 ' ( ± f, c1 ε2 [(H ± ), i A]◦ + (H ± ) f ≥ (c1 λε + μ) f 2 + εS 1/2 f 2 .
(B.12)
We start with a crude bound. For ε, μ > 0, we get: (c1 ε + 1) (Hε± − λ ± iμ) f G ∗ ≥ min(c1 λε + μ, ε) f G . Since Hε± − λ ± iμ ∈ B(G , G ∗ ) and since they are adjoint of the other, we infer the injectivity and that the ranges are closed. They are bijective and the inverse is bounded by the open mapping theorem. ± ± −1 G± exists in B(G ∗ , G ), for λ ≥ 0 and ε, μ > 0. ε := G ε (λ, μ) = (Hε − λ ± iμ)
Here we lighten the notation but keep in mind the dependency in λ and μ. Moreover, G ± ε B(G ∗ ,G ) ≤ (c1 ε + 1)/ min(c1 λε + μ, ε), for λ ≥ 0 and ε, μ > 0. (B.13) This bound seems not enough to lead the whole analysis. Then, we first restrict the domain of G ± ε to H and improve it. Since this inequality (B.12) holds also true on the common domain of Hε± (and of its adjoint), we can apply the numerical range theorem, Lemma B.1. Since S ≥ 0, we get the spectrum of Hε+ − λ + iμ is contained in the lower half-plane delimited by the equation y ≤ −c1 εx − μ. Hence, for ε ∈ (0, ε0 ] and μ > 0,
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Hε± − λ ± iμ is bijective and by taking ε0 smaller, one has the distance from 0 to the boundary of the cone bigger than μ/2. Then, G ± ε B(H ) ≤ 2/μ, for μ > 0 and ε ∈ [0, ε0 ].
(B.14)
∗ ∓ Note also that (G ± ε ) = G ε . Take ε, μ > 0. We fix f ∈ H and set: ' ( Fε± := f, G ± ε f .
Since G ± ε H ⊂ D ⊂ S and using (B.12), we infer
' 1/2 ± 2 ± ± ( 1 'G ± f, (H ± − λ ± iμ)G ± f ( S G ε f ≤ c1 G ± ε f, (Hε − λ ± iμ)G ε f + ε ε ε ε 1 ± ≤ max c1 , Fε . (B.15) ε
2 ≤ F ± /ε for all ε ∈ (0, ε0 ]. Hence up to a smaller ε0 > 0, we obtain S 1/2 G ± ε f ε Moreover, if f ∈ D(S −1/2 ), we obtain ) Fε± ± −1/2 F ≤ f √ S −1/2 f S 1/2 G ± ε ε f ≤ S ε and deduce
2 ± 1 F ≤ S −1/2 f , for all ε ∈ (0, ε0 ]. ε ε
(B.16)
1 ± We now show that G ± ε ∈ C (A). First note that G ε is a bijection from H onto D. Then by taking the adjoint, it is also a bijection from D ∗ onto H. Remember now that Wt stabilizes G and G ∗ . By the resolvent equality in B(H ), we have: ± ! ± H ± iε[H ± , i A], Wt G± . [G ± ε , Wt ] = − G ε ε
H ←−G ∗
G ∗ ←−G
G ←−H
Let us now take the derivative in 0. Since H ± and [H ± , i A] are in C 1 (A; G , G ∗ ) (the former being in C 2 (A; G , G ∗ )), the right hand side has a strong limit for all elements in H . Hence, Hε± ∈ C 1 (A). As in Remark B.1, it follows that G ± ε D(A) ⊂ D(A) ∩ D and one can safely expand the commutator in the next computation. Take f ∈ D(A). * + ' ( d d ± ± ± Fε = f, G ± f = ±i G ∓ ε ε f, [H , i A]◦ G ε f dε dε ' ∓ ( ' ( ' ∓ ! ± ( = ± G ε f, A f ∓ A f, G ∓ ε f − ε G ε f, [H, i A]◦ , i A G ε f . Here the last commutator in taken in the form sense. Now use three times (B.15) and the bound (B.10), which is uniform in p ∈ E, then integrate to obtain ε , ± ± |F | s F − F ± ≤ 2 √ S −1/2 A f + C Fs± ds, for 0 < ε ≤ ε ≤ ε0 , ε ε s ε (B.17) and for all f ∈ D(S −1/2 A) ∩ D(A).
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We give a first estimation. Using (B.16) and the Gronwall Lemma, see [1, Lemma 7.A.1] with θ = 1/2 or [59, Lemma 2.6] with p = 1/2, we infer there are some constants C, C , C , C , independent of ε ∈ (0, ε0 ], λ ≥ 0, μ > 0 and of p ∈ E, so that / ε0 . 2 ± 1 − 1 C(η−ε0 ) −1/2 F ≤ eC(ε−ε0 ) F ± 1/2 + 2 S dη e A f √ ε ε0 η ε 2 √ √ 2 ≤ C Fε±0 + ε − ε0 S −1/2 A f √ 2 1 −1/2 −1/2 2 2 √ ≤C f + ε − ε0 S A f ≤ C f 2 ˜ ∗ S S ε0
(B.18)
for f ∈ D(S −1/2 ) ∩ D(S −1/2 A) ∩ D(A), and where S˜ ∗ is the completion of D(A|S ∗ ) 2 2 under the norm f 2 ˜ ∗ := S −1/2 f + S −1/2 A f . Here one notices that the norm S is well defined for elements of D(A|S ∗ ) by taking into account (B.5). We now plug this back in (B.17). Since the inverse of the square root is integrable around 0, we find C with the same independence so that ± F − F ± ≤ ε
ε
ε
ε
√ √ √ C 2 √ + CC ds f 2 ˜ ∗ = C ε − ε f 2 ˜ ∗ . S S s
Then, {Fε± }ε∈(0,ε0 ] is a Cauchy sequence. We denote by F0±+ the limit, as ε goes to 0. It remains to notice that F0±+ = F0± . Indeed, using (B.14) and (B.7), one has the stronger fact that ± ± ± ± −1 G ± 0 − G ε B(H ) ≤ εG ε B(H ) · [H ( p), i A](H ( p) − λ ± iμ) B(H ) ≤
This gives us (B.11).
cε . μ2
For the convenience of the reader, we give a proof of the following well known fact: Lemma B.1 (Numerical Range Theorem). Let H be a closed operator. Suppose that D := D(H ) = D(H ∗ ). The numerical range of H is defined by N := { f, H f with f ∈ D and f = 1}. We have that σ (H ) ⊂ N , the closure of N . Moreover, if λ∈ / σ (H ), then (H − λ)−1 ≤ 1/d(λ, N ). Proof. Let λ ∈ / N . There is c := d(λ, N ) > 0, such that | f, H f − λ| ≥ c. Then, (H − λ) f ≥ c f , (H ∗ − λ) f ≥ c f , for all f ∈ D and f = 1. From the second part, we get the range of (H − λ) is dense. Then, since H is closed, the first part gives that the range of (H − λ) is closed. Hence, using again the first inequality, H − λ is bijective. The open mapping theorem concludes.
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Appendix C. Application to Non-relativistic Dispersive Hamiltonians In this section, we give an immediate application to the theory exposed in Appendix B. We do not discuss the uniformity with respect to the external parameter. The latter would be used in the heart of our approach, see Sect. 3. We discuss shortly the Helmholtz equation, see [5,6,72,73]. In [65], one studies the size of the resolvent of Hh := −h 2 + V1 (Q) − ihV2 (Q), as h → 0. This operator models accurately the propagation of the electromagnetic field of a laser in material medium. The important improvement between [65] and the previous ones, is that he allows V2 to be a smooth function tending to 0 without any assumption on the size of V2 ∞ . Note he supposes the coefficients are smooth as some pseudo-differential calculus is used to apply the non self-adjoint Mourre theory he develops. Then, he discusses trapping conditions in the spirit of [72]. Here, we will stick to the quantum case and choose h = −1. To simplify the presentation and expose some key ideas of Sect. 3, we focus on L 2 (Rn ; C), with n ≥ 3. For dimensions 1 and 2, one needs to adapt the first part of (H2) and the weights in (C.1). Theorem C.1. Suppose that V1 , V2 ∈ L 1loc (Rn ; R) satisfy: (H0) Vi are -operator bounded with a relative bound a < 1, for i ∈ {1, 2}. (H1) ∇Vi , Q · ∇Vi (Q) are in B(H 2 (Rn ); L 2 (Rn )) and Q(Q · ∇Vi )2 (Q) is bounded, for i ∈ {1, 2}. (H2) There are c1 ∈ [0, 2) and c1 ∈ 0, 4(2 − c1 )/(n − 2)2 such that WV1 (x) := x · (∇V1 )(x) + c1 V1 (x) ≤
c1 , for all x ∈ Rn , |x|2
and V2 (x) ≥ 0 and − c1 x · (∇V2 )(x) ≥ 0, for all x ∈ Rn . On Cc∞ (Rn ), we define H := − + V (Q), where V := V1 + iV2 . The closure of H defines a dispersive closed operator with domain H 2 (Rn ). We keep denoting it with H . Its spectrum is included in the upper half-plane. Moreover, H has no eigenvalue in [0, ∞) and |Q|−1 (H − λ + iμ)−1 |Q|−1 < ∞. (C.1) sup λ∈[0,∞), μ>0
If c1 = 0, H has no eigenvalue in R and (C.1) holds true for λ ∈ R. The quantity WV1 is called the virial of V1 . For h fixed and for a compact I included in (0, ∞), [65] shows some estimates of the resolvent above I. Here we deal with the threshold 0 and with high energy estimates. On the other hand, as he avoids the threshold, he reaches some very sharp weights. As mentioned above, one can improve the weights |Q| to some extent by the use of Besov spaces, see [63]. In [65] one makes an hypothesis on the sign of V2 but not on the one of x · (∇V2 )(x). Note that if one supposes c1 = 0, we are also in this situation. We take the opportunity to point out [71], where one discusses the presence of possible eigenvalues in 0 for non self-adjoint problems.
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Remark C.1. Taking V2 = 0, we can compare the results with [25,63]. In [25], one uses in a crucial way that WV1 (x) ≤ −cxα in a neighborhood of infinity, for some α, c > 0. In [63], one remarks that the condition WV1 (x) ≤ 0 is enough to obtain the estimate. Here we mention that the condition (H2) is sufficient. Note this example is not explicitly discussed in [63] but is covered by his abstract approach. In [12], for the special case c1 = 0, one uses extensively the condition (H2). This implies (C.1) for λ ∈ R. Remark C.2. Unlike in [65], we stress that V is not supposed to be a relatively compact perturbation of H and that the essential spectrum of H can be different from [0, ∞). In [39], see also [12], one studies V2 = 0 and V1 (x) := v(x/|x|), with v ∈ C ∞ (S n−1 ). We improve the weights of [39, Theorem 3.2] from Q to |Q|. We can also give a non-self-adjoint version. Consider V1 satisfying (H1) and being relatively compact with respect to and V2 (x) := v(x/|x|), where v ∈ C 0 (S n−1 ), non-negative. If v −1 (0) is non-empty, one shows [0, ∞) is included in the essential spectrum of H by using some Weyl sequences. Proof of Theorem C.1. Using (H0) and adapting the proof of Kato-Rellich, e.g., [62, Theorem X.12], one obtains easily D(H ) = D(H ∗ ) = H 2 (Rn ). Let S := cs (− )1/2 , with cs := 2 − c1 − (n − 2)2 c1 /4 > 0. Set S := H˙ 1 (Rn ), the homogeneous Sobolev space of order 1, i.e., the completion of H 1 (Rn ) under the norm f S := S 1/2 f 2 . Consider the strongly continuous one-parameter unitary group {Wt }t∈R acting by: (Wt f )(x) = ent/2 f (et x), for all f ∈ L 2 (R3 ). This is the C0 -group of dilatation. By interpolation and duality, one derives Wt S ⊂ S and Wt H s (R3 ) ⊂ H s (Rn ), for all s ∈ R. Consider now its generator A in L 2 (Rn ). By the Nelson lemma, it is essentially self-adjoint on Cc∞ (Rn ) and acts as follows: A = (P · Q + Q · P)/2 on Cc∞ (Rn ). By computing on Cc∞ (Rn ) in the form sense, we obtain that [ (H ), i A] − c1 (H ) = −(2 − c1 ) − WV1 ≥ S,
(C.2)
here we used the Hardy inequality for the last step. Furthermore, (H ) = V2 (Q) ≥ 0, [(H ), i A] = −Q · ∇(V2 )(Q),
(C.3)
[[H, i A], i A] = −4 + (Q · ∇V )2 (Q).
(C.4)
and also
Since Wt stabilizes G := H 1 and as (C.2), (C.3) and (C.4) extend to bounded operators from H 1 into H −1 , we infer that H and H ∗ are in C 2 (A; H 1 , H −1 ) and also (B.8) and (B.9). Now since Cc∞ (R3 ) is a core for H , H ∗ and A, (C.2) and (C.3) give (B.7), with notation H + = H and H − = H ∗ . In addition (B.10) follows from the Hardy inequality and (H1), as (Q · ∇)2 v(Q) f 2 ≤ c |Q|(Q · ∇)2 v(Q)2 S f 2 . Therefore, we can apply Theorem B.1 and derive the weight |Q| by the Hardy inequality. Finally, we recall the Hardy inequality. Take E a finite dimensional vector space. One has: 2 1 n−2 2 ∞ n |x| f (x) d x ≤ f, − f , where n ≥ 3 and f ∈ Cc (R ; E). n 2 R (C.5)
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63. Richard, S.: Some improvements in the method of the weakly conjugate operator. Lett. Math. Phys. 76(1), 27–36 (2006) 64. Richard, S., Tiedrade Aldecoa, R.: On the spectrum of magnetic Dirac operators with Coulomb-type perturbations. J. Funct. Anal. 250(2), 625–641 (2007) 65. Royer, J.: Limiting absorption principle for the dissipative Helmholtz equation, http://arxiv.org/abs/0905. 0355v2 [math,AP], 2009 66. Thaller, B.: The Dirac equation, Texts and Monographs in Physics, Berlin: Springer-Verlag, 1992 67. Vasy, A., Wunsch, J.: Positive commutators at the bottom of the spectrum. http://arxiv.org/abs/0909. 4583v2 [math.AP], 2009 68. Voronov, B.L., Gitman, D.M., Tyutin, I.V.: The Dirac Hamiltonian with a superstrong Coulomb field. Teoret. Mat. Fiz. 150(1), 41–84 (2007) 69. Yafaev, D.R.: The low energy scattering for slowly decreasing potentials. Commun. Math. Phys. 85(2), 177–196 (1982) 70. Yokoyama, K.: Limiting absorption principle for Dirac operator with constant magnetic field and longrange potential. Osaka J. Math. 38(3), 649–666 (2001) 71. Wang, X.P.: Number of eigenvalues for a class of non-selfadjoint Schrödinger operators, Available at http://hal.archives-ouvertes.fr/docs/00/37/30/28/DDF/complex-eigenvalues.pdf, 2009 72. Wang, X.P.: Time-decay of scattering solutions and classical trajectories. Annales de l’I.H.P., Sect. A 47(1), 25–37 (1987) 73. Wang, X.P., Zhang, P.: High-frequency limit of the Helmholtz equation with variable refraction index. J. Funct. Ana. 230, 116–168 (2006) 74. Xia, J.: On the contribution of the Coulomb singularity of arbitrary charge to the Dirac Hamiltonian. Trans. Amer. Math. Soc. 351, 1989–2023 (1999) Communicated by I.M. Sigal
Commun. Math. Phys. 299, 709–740 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1109-5
Communications in
Mathematical Physics
Wigner Measures in Noncommutative Quantum Mechanics C. Bastos1 , N. C. Dias2,3 , J. N. Prata2,3 1 Departamento de Física and Instituto de Plasmas e Fusão Nuclear, Instituto Superior Técnico,
Avenida Rovisco Pais 1, 1049-001 Lisboa, Portugal. E-mail:
[email protected] 2 Departamento de Matemática, Universidade Lusófona de Humanidades e Tecnologias,
Av. Campo Grande 376, 1749-024 Lisboa, Portugal
3 Grupo de Física Matemática, Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal.
E-mail:
[email protected];
[email protected] Received: 25 July 2009 / Accepted: 26 April 2010 Published online: 15 August 2010 – © Springer-Verlag 2010
Abstract: We study the properties of quasi-distributions or Wigner measures in the context of noncommutative quantum mechanics. In particular, we obtain necessary and sufficient conditions for a phase-space function to be a noncommutative Wigner measure, for a Gaussian to be a noncommutative Wigner measure, and derive certain properties of the marginal distributions which are not shared by ordinary Wigner measures. Moreover, we derive the Robertson-Schrödinger uncertainty principle. Finally, we show explicitly how the set of noncommutative Wigner measures relates to the sets of Liouville and (commutative) Wigner measures. 1. Introduction In this work we address several features of phase-space quasi-distributions in the context of a canonical noncommutative extension of non-relativistic quantum mechanics. This type of system has been recently considered by various authors [3–5,8,16,21,30,38,47] as a simplified model for the more elaborate noncommutative field theories [23]. The motivation comes mainly from string theory [54] and noncommutative geometry [15,44]. Although one may address noncommutative quantum mechanics with a standard operator formulation, we feel that the Weyl-Wigner formulation is more adequate. Indeed, (i) it places position and momentum on equal footing, (ii) the extra noncommutativity is trivially encapsulated in a modified Moyal product, (iii) the passage from noncommutative to ordinary quantum mechanics is more transparent, (iv) the difference between ordinary and noncommutative quantum mechanical systems is manifest when one considers the sets of quasi-distributions of the two theories. In [4] we initiated a systematic study of the Weyl-Wigner formulation of noncommutative quantum mechanics. There, we mainly focused on the algebraic structure of the theory. The aim of the present work is to further develop this formulation by addressing a number of issues related to the characterization of the set of states of the theory. These will be called the noncommutative Wigner measures (NCWMs).
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In connection with (iii) and (iv), some of us studied the emergence of ordinary quantum mechanics in the realm of noncommutative quantum mechanics [21]. A noncommutative Brownian particle was placed in interaction with an external heat bath of harmonic oscillators. We were looking for an estimate for the time scale of this noncommutative-commutative transition. Using a decoherence approach we were able to produce such an estimate for this simple system. In more general cases, however, we are required to established rigorous criteria for assessing whether such transition takes place. These criteria hinge upon the difference between the states of a system in ordinary and in noncommutative quantum mechanics. This is analogous to the comparison between classical and quantum states. The phase space framework is more suitable to address this issue. Classical states are described by positive Liouville measures in phase-space and they need not satisfy the Heisenberg uncertainty relations. In contrast, quantum mechanical phase-space quasi-distributions (Wigner measures) need not be positive, but must comply with the Heisenberg constraints. The difference between the space of states in ordinary and noncommutative quantum mechanics is also more transparent in the phase-space framework. In the standard operator formulation of noncommutative quantum mechanics the space of states is the usual Hilbert space of square integrable functions. From this point of view, the space of states of ordinary and noncommutative quantum mechanics coincide and one cannot tell one from the other. In contrast with this situation, if one resorts to the Weyl-Wigner formulation, one obtains two disparate phase-space representations: the star-product and the set of phase-space quasi-distributions are different for the two theories. This constitutes a clear advantage if one wants to study the emergence of ordinary quantum mechanics from a more fundamental noncommutative quantum theory [21]. In this respect, our work here will culminate in Lemma 4.14 and Fig. 1. These reveal that, in the wider set of phase-space real and normalizable functions, one can find the subsets of Liouville measures (the classical states), the Wigner measures (the quantum states) and the noncommutative Wigner measures (the noncommutative quantum states), and that these have non-trivial mutual intersections. This constitutes a natural framework to study transitions from one theory to another. A word of caution is however in order. We will prove Lemma 4.14 for 2-dimensional systems only. However, we believe that similar results hold for higher dimensions. The reason is that the d = 2 case can be obtained from the higher dimensional ones by tracing out suitable degrees of freedom. We hope to return to this issue in a future work. Another interesting problem that one faces when dealing with noncommutative quantum mechanics or noncommutative field theories is the fact that the noncommutative corrections are too small to be observed experimentally. This arises from upper bounds on the noncommutative parameters such as those found in refs. [8,14]. However our results stated in Theorems 4.2 and 4.3 and in Lemmata 4.4 and 4.5 may provide a framework to obtain physical predictions which can be tested experimentally. Indeed, in Lemma 4.4, we construct certain NCWMs which maximize a certain functional. These measures are not Wigner measures. Moreover, they look like the tensor product of two lower dimensional states. It is then natural to look for entangled combinations of such states as the one in (4.40). It is well known that entanglement “amplifies” the quantum nature of a system, so that its quantumness can be observed macroscopically. We hope that the entangled states of the form (4.40) which exist only in noncommutative quantum mechanics may play this role here. In a certain sense, the formalism seems ripe to develop a continuous variable quantum computation and quantum information for noncommutative quantum mechanics [32].
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Here is a brief outlook of our work. In Sect. 2, we recapitulate certain aspects of the Weyl-Wigner formulation which will be generalized to noncommutative quantum mechanics in the sequel. In Sect. 3 we address the Weyl-Wigner formulation of a d-dimensional noncommutative quantum mechanical system. We define the notion of NCWM and state some of its properties in d dimensions. In particular, (i) we obtain the uncertainty principle for NCWM in the Robertson-Schrödinger form; (ii) we establish an upper bound on the purity of a NCWM; (iii) we derive the Hudson Theorem which classifies the set of positive, pure state NCWMs; (iv) we obtain necessary and sufficient conditions for a Gaussian to be a NCWM or a pure state NCWM (Littlejohn’s Theorem); (v) we define the linear transformations which map a NCWM to another NCWM and which leave the uncertainty principle unchanged. In Sect. 4, we specialize to the case d = 2: (i) we show that the marginal distributions are not necessarily non-negative and that they must satisfy certain purity-type constraints which are not required in ordinary quantum mechanics; (ii) we show that these bounds can be saturated by certain states which have the structure of a tensor product of two lower dimensional quasi-distributions; (iii) we derive necessary and sufficient conditions for a phase-space function to be a bona fide NCWM; (iv) we establish the relation between the states in classical mechanics, and in ordinary and noncommutative quantum mechanics (see Fig. 1). 2. Weyl-Wigner Formulation of Quantum Mechanics In this section we review some basic facts about the Weyl-Wigner formulation of ordinary quantum mechanics which will be relevant for the sequel. Our analysis is restricted to the case of flat phase-spaces. The reader is referred to Refs. [6,7,17–20,24,26,29,49,53,56, 57] for a more detailed presentation and to Refs. [10,27,28,41,58] for the generalization of the formalism to the non-flat case. Let us then settle down the preliminaries: we consider a d-dimensional dynamical system, such that its classical formulation lives in the flat phase-space T ∗ M (Rd )∗ × Rd R2d . A global Darboux chart can be defined on T ∗ M, ξ = (R, ) = (Ri , i ), i = 1, . . . , d, ξα = Rα , α = 1, . . . , d and ξα = α−d , α = d + 1, . . . , 2d
(2.1)
in terms of which the sympletic structure reads d Ri ∧ di . In the sequel the Latin letters run from 1 to d (e.g. i, j, k, . . . = 1, . . . , d), whereas the Greek letters stand for phasespace indices (e.g. α, β, γ , . . . = 1, . . . , 2d), unless otherwise stated. Moreover, summation over repeated indices is assumed. Upon quantization, the set {ξˆα , α = 1, . . . , 2d} satisfies the commutation relations of the standard Heisenberg algebra: ξˆα , ξˆβ = i h¯ jαβ ,
J=
0d×d Id×d −Id×d 0d×d
,
(2.2)
where jαβ are the components of the matrix J. Moreover { Rˆ i , i = 1, . . . , d} constitute a complete set of commuting observables. Let us denote by |R the general eigenstate of Rˆ associated to the array of eigenvalues Ri , i = 1, . . . , d and spanning the Hilbert space H = L 2 (Rd , d R) of complex valued functions ψ : Rd −→ C (ψ(R) = R|ψ), which are square integrable with respect to the standard Lesbegue measure d R. The scalar product in H is given by:
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(ψ, φ)H =
Rd
ψ(R)φ(R)d R,
(2.3)
where the over-bar denotes complex conjugation. We now introduce the Gel’fand triple of vector spaces [1,2,9,31,50]: S (Rd ) ⊂ H ⊂ S (Rd ),
(2.4)
where S (Rd ) is the space of all complex valued functions t (R) that are infinitely smooth and, as ||R|| → ∞, they and all their partial derivatives decay to zero faster than any power of 1/||R||. S (Rd ) is the space of rapid descent test functions [36,37,59], and S (Rd ) is its dual, i.e. the space of tempered distributions. In analogy with (2.4) let us also introduce the triple: S (R2d ) ⊂ F = L 2 (R2d , d Rd) ⊂ S (R2d ),
(2.5)
where F is the set of square integrable phase-space functions with scalar product: 1 (F, G)F = F(ξ )G(ξ )dξ. (2.6) (2π h¯ )d R2d Finally, let Sˆ be the set of linear operators admitting a representation of the form [24]: ˆ = A K (R, R )ψ(R )d R , (2.7) Aˆ : S (Rd ) −→ S (Rd ); ψ(R) −→ ( Aψ)(R) Rd
ˆ ∈ S (R2d ) is a distributional kernel. The elements of where A K (R, R ) = R| A|R Sˆ are named generalized operators. The Weyl-Wigner transform is the linear one-to-one invertible map [11,24,53]: Wξ : Sˆ −→ S (R2d );
h¯ h¯ y, R − y)dy 2 2 h h ¯ ¯ ˆ − ydy, = h¯ d e−i·y R + y| A|R 2 2 Rd
ˆ = h¯ d Aˆ −→ A(R, ) = Wξ ( A)
Rd
e−i·y A K (R +
(2.8) where the Fourier transform is taken in the usual generalized way1 and the second form of the Weyl-Wigner map in terms of Dirac’s bra and ket notation is more standard. There are two important restrictions of Wξ : (1) The first one is to the vector space Fˆ of Hilbert-Schmidt operators on H , which admit a representation of the form (2.7) with A K (R, R ) ∈ F , regarded as an algebra with respect to the standard operator product, which is an inner operation in Fˆ . ˆ and ˆ B) ˆ ˆ ≡ tr ( Aˆ † B) In this space we may also introduce the inner product ( A, F the Weyl-Wigner map Wξ : Fˆ −→ F becomes a one-to-one invertible unitary transformation. 1 The Fourier transform T of a generalized function B ∈ S (Rn ) (for n ≥ 1) is another generalized F function TF [B] ∈ S (Rn ) which is defined by TF [B], t = B, TF [t] for all t ∈ S (Rn ) [36,37,59], and where A, t denotes the action of a distribution A ∈ S (Rn ) on the test function t ∈ S (Rn ).
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(2) The second one is to the enveloping algebra Aˆ(H ) of the Heisenberg-Weyl Lie ˆ ˆ and Iˆ algebra which contains all polynomials of the fundamental operators R, modulo the ideal generated by the Heisenberg commutation relations. In this case the Weyl-Wigner transform Wξ : Aˆ(H ) −→ A (R2d ) becomes a one-to-one invertible map from Aˆ(H ) to the algebra A (R2d ) of polynomial functions on R2d . In ˆ = R and Wξ () ˆ = . particular Wξ ( Iˆ) = 1, Wξ ( R) The previous restrictions can be promoted to isomorphisms, if F and A (R2d ) are endowed with a suitable product. This is defined by: ˆ h¯ Wξ ( B) ˆ := Wξ ( Aˆ B) ˆ Wξ ( A)
(2.9)
ˆ Bˆ ∈ Sˆ such that Aˆ Bˆ ∈ Sˆ . The -product admits the kernel representation: for A, 1 A(ξ )B(ξ ) A(ξ ) h¯ B(ξ ) = (π h¯ )2d R2d R2d
T 2i (2.10) ξ − ξ J ξ − ξ dξ dξ × exp − h¯ and is an inner operation in F as well as in A (R2d ). The previous formula is also valid if we want to compute A h¯ B with A ∈ F and B ∈ A (R2d ), in which case A h¯ B ∈ S (R2d ). On the other hand, if A ∈ A (R2d ) and B ∈ A (R2d ) ∪ F the
-product can also be written in the well-known form [35,45]: i h¯
A(ξ ) h¯ B(ξ ) = A(ξ )e 2 where ∂ξα A(ξ ) =
←
→
∂ ξα jαβ ∂ ξβ
B(ξ ),
(2.11)
∂ ∂ξα
A(ξ ) and jαβ are the components of the sympletic matrix (2.2). When the Weyl-Wigner map is applied to a density matrix ρˆ ∈ Fˆ , we get the celebrated Wigner measure or quasi-distribution: f C (ξ ) =
1 Wξ (ρ)(ξ ˆ ). (2π h¯ )d
(2.12)
The superscript “C” will become clear in the sequel. If the system is in a pure state ρˆ = |ψ >< ψ|, then the Wigner measure reads: 1 C e−2i y·/h¯ ψ(R − y)ψ(R + y)dy. (2.13) f (R, ) = (π h¯ )d Rd Mixed states are just convex combinations of the latter. It is important to recapitulate some properties of Wigner measures. They are real and normalized phase-space functions which admit marginal ditributions. For instance, for a pure state (2.13) these read: P R (R) = f C (R, )d = |ψ(R)|2 ≥ 0, Rd (2.14) 2 ˆ P () = f C (R, )d R = |ψ()| ≥ 0, Rd
ˆ where ψ() is the Fourier transform of ψ(R). However, even though their marginal distributions are bona fide probability densities for position or momentum, the Wigner
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measures themselves cannot be regarded as joint probability distributions for position and momentum. The reason is that they are not necessarily everywhere non-negative. This is usually regarded as a manifestation of Heisenberg’s uncertainty principle. Nevertheless, the existence of strictly positive Wigner measures is not precluded. It is a difficult (and unfinished) task to classify the entire set of Wigner functions which are everywhere non-negative. An important step in that direction is Hudson’s beautiful theorem [39,55]: Theorem 2.1 (Hudson, Soto, Claverie). Let ψ ∈ L 2 (Rd , d R) be a state vector. The Wigner measure of ψ is non-negative iff ψ is a Gaussian state. There is thus far no analogous result for mixed states (see [12] for some attempts at classifying positive Wigner measures of mixed states). A class of positive Wigner measures of mixed states can be constructed by convoluting Wigner measures with suitable kernels (see below). Another important property of Wigner measures is the following bound: 2 1 f C (ξ ) dξ ≤ . (2.15) (2π h¯ )d R2d The equality holds iff the state is pure. For this reason, one calls the integral on the left-hand side the purity of the system. Equally interesting is the issue of whether the previous properties are necessary and sufficient conditions for a phase-space function to be a Wigner measure. The answer is no. In fact these are just necesssary conditions. A phase-space function f (ξ ) is a Wigner measure iff there exists b(ξ ) ∈ F such that: (i) |b(z)|2 dξ = 1, (2.16) R2d
(ii) f (ξ ) = b(ξ ) h¯ b(ξ ).
(2.17)
If the state is pure, then we may take b(ξ ) = (2π h¯ )d/2 f (ξ ), i.e.: C C f pur e (ξ ) h¯ f pur e (ξ ) =
1 f C (ξ ). (2π h¯ )d pur e
(2.18)
These necessary and sufficient requirements are equivalent to another set of conditions called the KLM conditions. To state the latter, we first need the concept of symplectic Fourier transform: Definition 2.2. Let f (ξ ) ∈ F . We define its symplectic Fourier transform according to f˜J (a) = f (ξ ) exp −ia T Jξ dξ. (2.19) R2d
The superscript “J” will be useful for the sequel. The formula (2.19) can be inverted: 1 ˜J (a) exp ia T Jξ da. (2.20) f f (ξ ) = (2π )2d R2d Definition 2.3. The symplectic Fourier transform f˜J (a) is said to be of the α-positive type if the m × m matrix with entries
Wigner Measures in Noncommutative Quantum Mechanics
iα M jk = f˜J (a j − ak ) exp − akT Ja j 2
715
(2.21)
is hermitian and non-negative for any positive integer m and any set of m points a1 , . . . , am in the dual of the phase-space. By abuse of language we sometimes say that f (ξ ) is of the α-positive type. With these definitions one can state the KLM (Kastler, Loupias, Miracle-Sole [40,43,48]) conditions, equivalent to (2.16, 2.17): Theorem 2.4. The phase-space function f (ξ ) is a Wigner measure, iff its symplectic Fourier transform f˜J (a) satisfies the KLM conditions: (i) f˜J (0) = 1, (ii) f˜J (a) is continuous and of h¯ -positive type.
(2.22) (2.23)
The concept of Narcowich-Wigner (NW) spectrum is useful in this context. Definition 2.5. The Narcowich-Wigner spectrum of a phase-space function f (ξ ) ∈ F is the set:
W ( f ) = α ∈ R f˜J (a) is of the α-positive type . (2.24) Consequently, one may say that if f is a Wigner measure, then h¯ ∈ W ( f ). If we analyze carefully the equivalence between the necessary and suffcient conditions (2.16, 2.17) and the set of KLM conditions (2.22, 2.23), we conclude that there is nothing special about h¯ . In fact, it is trivial to conclude that: Theorem 2.6. Let f ∈ F , f˜J be its symplectic Fourier transform, and α ∈ R\ {0}. Then the following sets of conditions are equivalent: (i) f˜J (a) is continuous, f˜J (0) = 1 and f˜J (a) is of the α-positive type. (2.25) (ii) There exists b(ξ ) ∈ F such that R2d |b(ξ )|2 dξ = 1 and f (ξ ) = b(ξ ) α b(ξ ). Here α denotes the Moyal product (2.10, 2.11) with h¯ replaced by α. The case α = 0 is singular. Functions f˜J (a) which are of 0-positive type correspond, according to Bochner’s theorem, to phase-space functions f (ξ ) which are everywhere non-negative. From the previous theorem and the fact that a(ξ ) α b(ξ ) = b(ξ ) −α a(ξ ), we can check that: Corollary 2.7. Let f ∈ F . If α ∈ W ( f ), then −α ∈ W ( f ). Let f g denote the convolution: ( f g)(ξ ) :=
R2d
f (ξ − ξ )g(ξ )dξ .
(2.26)
We use this somewhat unusual notation for the convolution to avoid confusion with the star product. An important result concerning the convolution of phase-space functions was proved in ref. [48], using the fact that the Schur (or Hadamard) product of hermitian and nonnegative matrices is again a hermitian and non-negative matrix:
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Theorem 2.8. The NW spectrum of the convolution f g contains all elements of the form α1 + α2 with α1 ∈ W ( f ) and α2 ∈ W (g). Nevertheless, it is not true that W ( f g) = {α1 + α2 |α1 ∈ W ( f ), α2 ∈ W (g) }. Counter-examples can be found in [12]. An immediate consequence of Corollary 2.7 and Theorem 2.8 is that we can construct positive Wigner measures by convoluting a Wigner measure f C (ξ ) with another Wigner measure g C (ξ ), which, in addition to being of h¯ -positive type, is also of the 0-positive or 2h¯ -positive type. In ref. [48] functions g C (ξ ) with these characteristics were explicitly constructed. Moreover, by resorting to this concept of h¯ -positivity, Narcowich stated necessary and sufficient conditions for a Gaussian to be a Wigner measure: Lemma 2.9. Let A be a real, symmetric, positive defined 2d × 2d matrix. Then the Gaussian det A T f (ξ ) = exp −(ξ − ξ ) A(ξ − ξ ) (2.27) 0 0 π 2d is a Wigner measure iff the matrix B = A−1 + i h¯ J is a non-negative matrix in C2d . Note that these conditions are equivalent to a Wigner measure satisfying the RobertsonSchrödinger form of the uncertainty principle [33,34,51,52]. One is also able to tell when a Gaussian represents a pure state [42]: Theorem 2.10 (Littlejohn). The Gaussian in (2.27) is the Wigner measure of a pure state iff there exists a symplectic matrix P ∈ Sp(2d; R) such that A = P T P. The NW spectrum of a pure state was completely characterized in [22]: Theorem 2.11. Let ψ ∈ L 2 (Rd , d R) be a state vector and f ψ the associated Wigner measure. If ψ is a Gaussian, then its NW spectrum reads W ( f ψ ) = [−h¯ , h¯ ]. If ψ is non-Gaussian, then W ( f ψ ) = {−h¯ , h¯ }. 3. Weyl-Wigner Formulation of Noncommutative Quantum Mechanics In noncommutative quantum mechanics, one replaces the Heisenberg algebra (2.2) by an extended Heisenberg algebra: h¯ Id×d , (3.1) zˆ α , zˆ β = i h¯ ωαβ , α, β = 1, . . . 2d, = h¯ −1 −h¯ Id×d N where zˆ = (q, ˆ p) ˆ stand for the physical position and momentum variables and , N are d × d constant antisymmetric real matrices whose entries θi j , ηi j , with dimensions (length)2 and (momentum)2 , measure the noncommmutativity in the spatial and momentum sectors, respectively. The matrix has entries ωαβ . We shall tacitly assume that: θi j ηkl < h¯ 2 ,
1 ≤ i < j ≤ d, 1 ≤ k < l ≤ d.
(3.2)
This condition, which is compatible with experimental results [8,14], ensures that the skew-symmetric, bilinear form ω(z, u) = z T u = z α ωαβ u β
(3.3)
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is non-degenerate (see Lemma 6.1 in the Appendix). By a linear version of Darboux’s Theorem, any skew-symmetric bilinear form can be cast in a “normal” form [13,34] under a linear transformation. This is a sort of symplectic Gram-Schmidt orthogonalization process. We shall denote it by Darboux (D) transformation. In practical terms, since is even-dimensional and non-degenerate, this means that under the D transformation: zˆ = Tˆ (ξˆ ) = Sξˆ ,
(3.4)
we obtain the Heisenberg algebra (2.2) for the variables ξˆ . Here S is a 2d × 2d constant real matrix. From (2.2, 3.1, 3.4), we conclude that: SJST = .
√ This implies that det S = ± det . But in fact it can be shown that: √ det S = det = |P f ()| > 0,
(3.5)
(3.6)
where P f () denotes the Pfaffian of . This result is proved in the Appendix. The D transformation is not unique. Indeed, if S is a solution of (3.5), then SL with L ∈ Sp(2d; R) is equally a solution. Nevertheless, in [4] we proved that all physical predictions (in particular all traces of trace-class operators) are invariant under different choices of D maps. Using this transformation, we constructed a Weyl-Wigner formulation for noncommutative systems by resorting to an extended Weyl-Wigner map: Wzξ : Sˆ −→ S (R2d ),
ˆ = T ◦ Wξ ◦ Tˆ −1 . Aˆ −→ Wzξ ( A)
(3.7)
Since the D transformation (3.4) is linear, there are no ordering ambiguities and for all practical purposes, T and Tˆ are the same transformation: T (ξ ) = Sξ , ξ ∈ T ∗ M. The extended Weyl-Wigner map is independent of the particular D transformation. This means that: Wzξ = T ◦ Wξ ◦ Tˆ −1 = T ◦ Wξ ◦ Tˆ −1 = Wzξ , z = T (ξ ) = Sξ = T (ξ ) = S ξ , (3.8)
where the matrices S, S are both solutions of (3.5) and ξˆ and ξˆ both obey the Heisenberg algebra. Let us denote by D (2d; R) the set of all real 2d × 2d matrices S which satisfy (3.5). Notice that D (2d; R) is not a subgroup of Gl(2d; R), as it is not even closed under matrix multiplication. One of consequences of this definition is that the -product in noncommutative quantum mechanics becomes [4]: → i h¯ ← ξ ˆ ˆ ξ ˆ ξ ˆ Wz ( A · B) := Wz ( A) Wz ( B) = A(z) B(z) = A(z) exp ∂ z α ωαβ ∂ z β B(z), 2 (3.9) ξ ˆ ξ ˆ for Wz ( A) := A ∈ A (R2d ) and Wz ( B) :=∈ A (R2d ) ∪ F . This -product admits the kernel representation:
A(z) B(z) =
1 (π h¯ )2d det
2i (z − z )T −1 (z − z) dz dz , A(z )B(z ) exp × h¯ R2d R2d (3.10)
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for A, B ∈ A (R2d ) ∪ F . From (3.1, 3.9), we conclude that the -product is of the form: A(z) B(z) = A(z) h¯ θ η B(z),
(3.11)
where → i h¯ ← ∂ z α jαβ ∂ z β B(z), A(z) h¯ B(z) = A(z) exp 2 ⎛ ← ⎞ → i ∂ ∂ ⎠ A(z) θ B(z) = A(z) exp ⎝ θi j B(z), 2 ∂qi ∂q j ⎛ ← ⎞ → i ∂ ∂ ⎠ A(z) η B(z) = A(z) exp ⎝ ηi j B(z). 2 ∂ pi ∂pj
(3.12) (3.13)
(3.14)
If or N are non-degenerate (for instance for d = 2), we may equally write a kernel representation for θ , η : 1 a(q )b(q ) exp 2i(q −q )T −1 (q −q) dq dq , a(q) θ b(q) = d π det Rd Rd (3.15) 1 T −1 c( p )d( p ) exp 2i( p− p ) N ( p − p) dp dp , c( p) η d( p) = d π det N Rd Rd for a, b ∈ L 2 (Rd , dq) ∪ A (Rd ) and c, d ∈ L 2 (Rd , dp) ∪ A (Rd ). The states of the system are adequately represented by what we called the noncommutative Wigner measures (NCWM) [4]. The latter can be regarded as the composition of an ordinary Wigner measure with a D transformation (up to a multiplicative normalization constant): Definition 3.1. The noncommutative Wigner measure associated with a state with density matrix ρˆ is a phase-space function of the form f N C (z) :=
1 1 ˆ = W ξ (ρ) f C (S−1 z), |P f ()| (2π h¯ )d |P f ()| z
(3.16)
where f C is the Wigner measure associated with ρˆ and S ∈ D (2d; R) is a D transformation. If f C is the Wigner measure associated with a pure state ρˆ = |ψ >< ψ|, then f N C is said to be a pure state noncommutative Wigner measure. In particular, one can compute the expectation value of an operator Aˆ in the state ρˆ according to: E Aˆ = tr ( Aˆ ρ) ˆ =
R2d
f N C (z)A(z) dz,
(3.17) ξ
ˆ where f N C is of the form (3.16) with f C = (2π h¯ )−d Wξ (ρ) ˆ and A(z) = Wz ( A).
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Here is an alternative characterization of a NCWM which was poved in [4]: Proposition 3.2. A function f (z) ∈ F is a NCWM iff there exists b(z) ∈ F such that: (i)
|b(z)|2 dz = 1,
(3.18)
(ii) f (z) = b(z) b(z).
(3.19)
R2d
With the previous definitions we may prove the following proposition. Proposition 3.3. Let f N C (z) be a NCWM. Then the inequality
R2d
g(z) g(z) f N C (z) dz ≥ 0,
(3.20)
holds for any symbol g(z) for which the left-hand side exists. Proof. In this proof we shall use the cyclic property
R2d
A(z) B(z) dz =
R2d
A(z)B(z) dz,
(3.21)
which was proved in [4]. From (3.19), we have:
R2d
=
(z) dz =
R 2 = b(z) g(z) dz ≥ 0. 2d
g(z) g(z) b(z) b(z) dz R2d g(z) g(z) b(z) b(z) dz = b(z) g(z) g(z) b(z) dz
g(z) g(z) f
NC
R2d
(3.22)
R2d
Proposition 3.4. Any NCWM f N C (z) satisfies the bound:
R2d
f N C (z)
2
dz ≤
1 . (2π h¯ )d |P f ()|
(3.23)
The equality holds iff f N C is a pure state NCWM. Proof. Let us compute the left-hand side of (3.23) using (3.16):
2 2 1 f N C (z) dz = f C (S−1 z) dz det R2d R2d 2 2 det S 1 f C (ξ ) dξ = f C (ξ ) dξ. = |P f ()| R2d det R2d
(3.24)
In the last step we used (3.6). From (2.15), the result of the proposition follows immediately.
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Let us now derive the uncertainty principle for NCWMs in the Robertson-Schrödinger form [33,34,51,52]. Let: < zˆ α >:= z α f N C (z) dz, α = 1, . . . , 2d. (3.25) R2d
We also define τˆα := zˆ α − < zˆ α >,
α = 1, . . . , 2d.
Moreover, let be the covariance matrix with entries: σαβ = τα τβ f N C (z) dz, α, β = 1, . . . , 2d R2d
(3.26)
(3.27)
We then have: Proposition 3.5. Let f N C be a NCWM and let be its covariance matrix. Then it obeys the following uncertainty principle. The matrix +
i h¯ 2
(3.28)
is non-negative. Proof. Consider an arbitrary set of 2d complex constants aα (α = 1, . . . , 2d). From (3.20), we have: NC i h¯ (3.29) 0≤ (aα τα ) aβ τβ f (z) dz = aα σαβ + αβ aβ , 2 R2d which yields the result. Such uncertainty relations can be derived in a similar fashion for pairs of noncommuting essentially self-adjoint operators, with some common domain. Theorem 3.6 (Hudson’s Theorem for NCWM). Let ψ ∈ L 2 (Rd , d R) be a state vector. The noncommutative Wigner measure of ψ is non-negative iff ψ is a Gaussian state. Proof. The result of the theorem follows from (3.16). Indeed f N C (z) is everywhere nonnegative iff f C (ξ ) is everywhere non-negative. The rest is an immediate consequence of Theorem 2.1. Next, we consider the analogs of Lemma 2.9 and Theorem 2.10. Lemma 3.7. Let C be a real, symmetric, positive defined 2d × 2d matrix. Then the Gaussian det C T f (z) = exp −(z − z ) C(z − z ) (3.30) 0 0 π 2d is a NCWM iff the matrix D = C−1 + i h¯ is non-negative in C2d .
Wigner Measures in Noncommutative Quantum Mechanics
Proof. Let us define g(ξ ) = |P f ()| f (Sξ ). Then from (3.30), we have: det(C) T g(ξ ) = exp −(Sξ − z ) C(Sξ − z ) 0 0 π 2d det A T = exp −(ξ − ξ ) A(ξ − ξ ) , 0 0 π 2d
721
(3.31)
where ξ0 = S−1 z 0 and A = ST CS. Also, from the definition of the matrix A, it follows that det A = (det S)2 det C = det(C). Clearly, A is equally a real, symmetric, positive defined matrix. From Lemma 2.9 and the definition (3.16) of NCWM, we conclude that f is a NCWM iff the matrix B = A−1 + i h¯ J = S−1 C−1 + i h¯ SJST (ST )−1 = S−1 D(ST )−1 (3.32) is non-negative in C2d . This is of course equivalent to D being non-negative, as the matrices S are real. Obviously, the Gaussian is simultaneously a Wigner measure and a NCWM iff both D and C−1 + i h¯ J are non-negative. Moreover, notice that, since the covariance matrix of the Gaussian is 21 C −1 , we conclude that the uncertainty principle (Proposition 3.5) is a necessary and sufficient condition for a Gaussian to be a NCWM. For more general states however, it is necessary but not sufficient. Theorem 3.8 (Littlejohn’s Theorem for NCWM). The Gaussian in (3.30) is the NCWM of a pure state iff there exists B ∈ D (2d; R) such that C−1 = BT B. Proof. From (3.16, 3.30) and Theorem 2.10, we conclude that (3.30) is the NCWM of a pure state iff there exists a matrix P ∈ Sp(2d; R) such that A = P T P, i.e. C = −1 T −1 T B B , with B−1 = PS−1 . From (3.5), we conclude that B−1 B−1 = J. In other words: B ∈ D (2d; R). Before we conclude this section let us briefly discuss the set of diffeomorphisms that transform a NCWM into another NCWM. Definition 3.9. We denote by Sp (2d; R) the set of noncommutative symplectic transformations. These are the 2d × 2d real, constant matrices M that satisfy: MMT = .
(3.33)
These correspond to the linear transformations which leave the skew-symmetric bilinear form (3.3) invariant. It is not very difficult to check that Sp (2d; R) is a group with respect to matrix multiplication. We also define the group of noncommutative symplectomorphisms to be the set of automorphisms φ of R2d such that dφ(z) ∈ Sp (2d; R) for all z ∈ R2d . Lemma 3.10. The groups Sp (2d; R) and Sp(2d; R) are isomorphic. Proof. Fix some element S ∈ D (2d; R). Then the map φS : Sp(2d; R) −→ Sp (2d; R) P −→ φS (P) = SPS−1 is a Lie group isomorphism.
(3.34)
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Lemma 3.11. Noncommutative symplectic transformations map NCWM’s to NCWM’s. Proof. Let M ∈ Sp (2d; R) and let f N C be some NCWM. Then there exist a Wigner measure f C and S ∈ D (2d; R) such that (3.16) holds. Under the noncommutative symplectic transformation z −→ Mz, we obtain −1 1 1 NC NC C −1 C −1 M S S Mz = f (z) −→ f (Mz) = z . f f |P f ()| |P f ()| T On the other hand M−1 S J M−1 S = , which means that M−1 S ∈ D (2d; R). It follows that f N C (Mz) is again a NCWM. Lemma 3.12. The uncertainty principle is invariant under noncommutative symplectic transformations. Proof. Let M ∈ Sp (2d; R) and zˆ = Mˆz . We then have τˆ = Mτˆ and = MMT . For arbitrary a ∈ C2d : † i h¯ i h¯ i h¯ + M T a ≥ 0. a † + a = a † MMT + a = M T a 2 2 2 (3.35) 4. Properties of Noncommutative Wigner Measures in Two Dimensions In this section we shall focus on the d = 2 case. The extended Heisenberg algebra in two dimensions reads: θ E I2×2 h ¯ zˆ α , zˆ β = i h¯ ωαβ , α, β = 1, · · · 4, = . (4.1) −I2×2 hη¯ E The 2 × 2 matrix E has entries 11 = 22 = 0, 12 = −21 = 1. The real, constant, positive parameters θ, η measure the noncommutativity in phase-space. In this case, we have from (3.2,3.6): det S = |P f ()| = 1 − ζ,
ζ =
θη < 1, h¯ 2
where S ∈ D (4; R). In particular, the bound on the purity (3.23) reads: 2 1 f N C (z) dz ≤ . √ 4 (2π 1 − ζ )2 R In this paper we shall frequently use the following D map: λI2×2 − 2λθ h¯ E S= , η 2μh¯ E μI2×2 where λ, μ are adimensional real constants such that √ 1+ 1−ζ . μλ = 2
(4.2)
(4.3)
(4.4)
(4.5)
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From (4.2,4.4), we conclude that the D map admits the inverse: μI2×2 2λθ h¯ E 1 −1 S =√ . η 1 − ζ − 2μh¯ E λI2×2
(4.6)
4.1. Marginal distributions. Our aim is now to derive some properties of NCWMs for d = 2. Let us start by defining the position and momentum marginal distributions: Definition 4.1. The position and momentum marginal distributions of a NCWM f N C are defined by: f N C (q, p) dp P p ( p) = f N C (q, p) dq. (4.7) Pq (q) = R2
R2
It is important to remark that, albeit real and normalized, these distributions cannot be regarded as true probability densities as was the case with the commutative counterparts. This is not surprising given that the position variables do not commute amongst themselves and neither do the momenta. Therefore we cannot expect to obtain a joint probability density for, say, q1 and q2 . This manifests itself in the fact that Pq and P p may (and usually do) take on negative values. This is an immediate consequence of the following theorem: Theorem 4.2. The position and momentum marginal distributions of the NCWM associated with the pure state ρˆ = |ψ >< ψ| can be written as: Pq (q) = φ(q) θ φ(q),
(4.8)
P p ( p) = χ ( p) η χ ( p),
(4.9)
where φ(q) =
1 q ψ λ λ
χ ( p) =
1 ψˆ μ
p . μ
(4.10)
Here λ, μ are the adimensional parameters that appear in the D transformation (4.4) ˆ and ψ() is the Fourier transform of ψ(R) =< R|ψ >. Proof. The NCWM of a pure state is given by (3.16) with (cf. (2.13)): 1 e−2i y·/h¯ ψ(R − y)ψ(R + y) dy. f C (R, ) = (π h¯ )2 R2
(4.11)
We then have: Pq (q) =
1 (π h¯ 1 − ζ )2 × e−2i y·(q, p)/h¯ ψ [R(q, p) − y]ψ [R(q, p) + y] dydp. √
R2 R2
(4.12)
We next perform the change of variables: u = λ [R(q, p) + y] ,
v = λ [R(q, p) − y] ,
(4.13)
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and obtain:
1 φ(u)φ(v) Pq (q) = (π θ )2 R2 R2
h¯ η 2h¯ μλ i − Eq − 1 − ζ E(u + v) du dv. (u −v) · × exp − √ θ 2μλh¯ θ h¯ 1 − ζ (4.14)
Here φ(q) is given by (4.10). Notice also that from (4.5), we have: η 2h¯ 2h¯ μλ − = 1 − ζ. θ 2μλh¯ θ
(4.15)
Consequently:
1 2i T Pq (q) = φ(u)φ(v) exp − (u − q) E(q − v) dudv, (π θ )2 R2 R2 θ
(4.16)
which is the kernel representation of the θ -product (3.15). Likewise, let us consider the Wigner measure (4.11) in the momentum representation: 1 ˆ − )ψ(k ˆ + ) dk. (4.17) f C (R, ) = e−2ik·R/h¯ ψ(k (π h¯ )2 R2 Following exactly the same procedure, we arrive at the kernel representation of χ ( p) η χ ( p), where χ ( p) is given by (4.10). The nonlocal nature of the -products in (4.8, 4.9) entails that the marginal distributions may take on negative values. Moreover, Theorem 4.2 has another important consequence. If we look at Eq. (4.8), we notice that Pq (q) is a positive, normalized element of the -algebra with θ -product. Furthermore, the matrix E coincides with the sympletic matrix J of a two-dimensional phase space. We thus conclude that, if we make the correspondence E ↔ J, q1 ↔ R, q2 ↔ , θ ↔ h¯ for d = 1, then the expression (4.8) states that Pq (q) can be written as (cf. (2.13, 2.16, 2.17)): Pq (q) = s j f jθ (q1 , q2 ), (4.18) j
with 0 ≤ s j ≤ 1, ∀ j,
s j = 1,
(4.19)
j
and f jθ (q1 , q2 ) =
1 e−2i yq2 /θ φ j (q1 − y)φ j (q1 + y) dy, πθ R
(4.20)
for some set of wave functions φ j ∈ L 2 (R, dq1 ) which can be chosen to be orthonormal: φ j (q1 )φk (q1 ) dq1 = δ jk . (4.21) R
We shall call functions of the form (4.18–4.20) θ -Wigner measures.
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If instead of a pure state we consider a mixed state: f N C (z) = pi f iN C (z),
(4.22)
i
then Eq. (4.8) would be replaced by: Pq (q) =
pi φi (q) θ φi (q).
(4.23)
i
This is then a convex combination of θ -Wigner measures, which means that it must again be of the form (4.18). Analogous conclusions can be drawn for the momentum marginal distribution, η P p ( p) = rk f k ( p1 , p2 ), (4.24) k
with 0 ≤ rk ≤ 1, ∀k,
rk = 1,
(4.25)
k
and η
f k ( p1 , p2 ) =
1 e−2i yp2 /η χk (y − p1 )χk (y + p1 ) dy. πη R
The functions χk ∈ L 2 (R, dp1 ) can also be chosen to be orthonormal: χk ( p1 )χl ( p1 ) dp1 = δkl . R
(4.26)
(4.27)
The functions of the form (4.24–4.26) are called η-Wigner measures. From (2.15, 4.18, 4.24) we conclude that: Theorem 4.3. The marginal distributions of a NCWM satisfy the bounds: 2 1 , Pq (q) dq ≤ 2 2π θ R 2 1 . P p ( p) dp ≤ 2 2π η R
(4.28) (4.29)
These bounds have no counterpart in ordinary quantum mechanics. We shall denote the integrals in (4.28, 4.29) by θ -purity and η-purity, respectively. The integral (4.3) will be simply called purity. An interesting question regards the saturation of the inequalities (4.28, 4.29). The following lemma shows that it is possible to maximize them. Lemma 4.4. The states of the form: f N C (q, p) =
1 1−ζ
2 q + (θ/h¯ )E p θ f φθ1 f φθ2 (q), √ h¯ 1−ζ
(4.30)
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C. Bastos, N. C. Dias, J. N. Prata
where f φθi (q) is the θ -Wigner measure associated with the function φi ∈ L 2 (R, dq1 ) (i = 1, 2), maximize the θ -purity (4.28). Likewise, the states of the form: f
NC
1 (q, p) = 1−ζ
2 p + (η/h¯ )Eq η η f χη2 ( p), f χ1 √ h¯ 1−ζ
(4.31)
η
where f χi (q) is the η-Wigner measure associated with the function χi ∈ L 2 (R, dp1 ) (i = 1, 2), maximize the η-purity (4.29). Proof. First of all, let us check that (4.30) is indeed a NCWM. Consider the normalized wavefunction: 2 e−2iλxq2 /θ φ1 (λq1 − x)φ2 (λq2 + x) d x, (4.32) ψ(q) = λ π θ R2 where λ is as in Eq. (4.4). The corresponding NCWM reads:
2 2i 2i exp − y1 1 − y2 2 h¯ h¯ πθ R R R R 2iλ 2iλ x1 (R2 − y2 ) − x2 (R2 + y2 ) × φ1 (λ(R1 + y1 )−x2)φ1 (λ(R1 − y1 ) − x1 ) + θ θ
f N C (q, p) =
λ √ π h¯ 1 − ζ
2
× φ2 (λ(R1 − y1 ) + x1 )φ2 (λ(R1 + y1 ) + x2 ) dy1 dy2 d x1 d x2 .
(4.33)
Upon integration over y2 , x2 , we obtain:
2λ 2i 2iλ θ NC R2 2x1 + exp − y1 1 + 2 f (q, p) = √ 2 θ h¯ h¯ λ R R π h¯ 1 − ζ θ 2 φ1 (λ(R1 − y1 ) − x1 ) × φ1 λ(R1 + y1 ) + x1 + h¯ λ θ × φ2 (λ(R1 − y1 ) + x1 )φ2 λ(R1 + y1 ) − x1 − 2 dy1 d x1 . (4.34) h¯ λ We then perform the substitution y1 =
v−u , 2λ
x1 = −
u + v 2
−
θ 2 , 2h¯ λ
(4.35)
with the Jacobian (2λ)−1 . The result is: 2 θ θ 1 θ f φθ1 λR + E f φθ2 λR − E . (4.36) f N C (q, p) = 1−ζ 2h¯ λ 2h¯ λ h¯ Taking into account the D transformation (4.4), we obtain (4.30). By construction this is indeed a NCWM. It remains to prove that (4.30) maximizes the θ -purity (4.28): Pq (q) =
2 q + (θ/h¯ )E p 1 θ f φθ2 (q) f φθ1 dp = f φθ2 (q). √ 1−ζ h¯ 1−ζ R2
(4.37)
Wigner Measures in Noncommutative Quantum Mechanics
We then have:
R2
2 Pq (q) dq =
R2
f φθ2 (q)
727
2
dq =
1 . 2π θ
(4.38)
In the last step, we used the fact that f φθ2 is a θ -Wigner measure (4.20). The proof that the states (4.31) saturate the η-purity is analogous. Lemma 4.5. States of the form (4.30, 4.31) cannot be Wigner measures. Proof. Let us compute the purity of the states of the form (4.30): 2 f N C (q, p) dqdp R2 R2
4
2 θ q + (θ/h¯ )E p 2 θ f φθ1 f φ2 (q) dqdp √ 1−ζ h¯ 1 − ζ R2 R2 2 2 2 1 1 θ f φθ1 (q ) f φθ2 (q) dqdq = > . = √ √ 2 (2π h¯ )2 (2π h¯ 1 − ζ ) h¯ 1 − ζ R2 R2 (4.39)
=
√
We obtain the same result if we use instead the states (4.31). From (2.15) this proves the lemma. Notice that we may apply Hudson’s theorem to the marginal distributions whenever we have states of the form (4.30) or (4.31): the marginal distributions Pq (q) or P p ( p) are positive iff φ2 or χ2 are Gaussian, respectively. Before we conclude this section, a brief remark is in order. States of the form (4.30, 4.31) look like the tensor product of one-dimensional states. It then seems natural to look for entangled states of the form 2 q + (θ/h¯ )E p θ 1 f N C (q, p) = r f φθ1 f φθ2 (q) √ 1 − ζ h¯ 1−ζ
q + (θ/h¯ )E p f ψθ 2 (q) , + (1 − r ) f ψθ 1 (4.40) √ 1−ζ with 0 < r < 1. As we mentioned in the Introduction, this may be the starting point for a theory of noncommutative quantum information and quantum computation for continuous variables [32]. Hopefully, this may lead to qualitatively new predictions that could signal the existence of noncommutativity in the physical world. This will be the subject of a future work. 4.2. Noncommutative Narcowich-Wigner spectrum. An interesting question is that of constructing the analog of the KLM conditions for noncommutative Wigner measures. Let us start by defining the noncommutative version of the symplectic Fourier transform: Definition 4.6. Let f ∈ F . Its noncommutative symplectic Fourier transform is defined by: f˜ (a) = f (z) exp ia T −1 z dz. (4.41) R4
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C. Bastos, N. C. Dias, J. N. Prata
The inverse formula is:
1 ˜ (a) exp −ia T −1 z da. f (z) = f √ (2π 1 − ζ )4 R4
(4.42)
As usual we define the convolution of f ∈ L p and g ∈ L q with p −1 + q −1 = 1: f (z − z )g(z )dz = f (z )g(z − z ) dz . (4.43) ( f g)(z) = R4
R4
From (4.41–4.43) it is easy to prove that the noncommutative symplectic Fourier transform of the convolution amounts to pointwise multiplication: ( f g) = f˜ (a) · g˜ (a).
(4.44)
We then propose the following noncommutative generalization of Definition 2.3: Definition 4.7. We say that f˜ (a) is of the (α, β, γ )-positive type, if the m × m matrix with entries i T ˜ N jk ≡ f (a j − ak ) exp (4.45) a (α, β, γ )a j 2 k is hermitian and non-negative for any positive integer m and any set of m points a1 , . . . , am in the dual of the phase space. The matrix is defined by: γ E −αI2×2 (α, β, γ ) = . (4.46) αI2×2 βE Then the following theorem holds: Theorem 4.8. The function f ∈ F is a NCWM iff its noncommutative symplectic Fourier transform f˜ (a) satisfies the set of noncommutative KLM conditions: (i) f˜ (0) = 1, (ii) f˜ (a) is continuous and of the (h¯˜ , θ˜ , η)-positive ˜ type,
(4.47)
h¯˜ = (1 − ζ )−1 h¯ ,
(4.49)
(4.48)
where θ˜ = (1 − ζ )−1 θ,
η˜ = (1 − ζ )−1 η.
Proof. The function f ∈ F is a NCWM iff there exists a Wigner measure g and a matrix S ∈ D (2d; R) such that f (z) =
1 g(S−1 z). |P f ()|
(4.50)
The function g(ξ ) thus satisfies the KLM conditions (2.22, 2.23). Let us compute the noncommutative symplectic Fourier transform of (4.50). From (2.19, 3.5, 3.6) and the fact that J−1 = JT = −J, we get: 1 ˜ f (a) = g(S−1 z) exp ia T −1 z dz |P f ()| R4 det S = g(ξ ) exp ia T −1 Sξ dξ |P f ()| R4 g(ξ ) exp −i(S−1 a)T Jξ dξ = g˜ J (S−1 a). (4.51) = R4
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729
From this equation it follows that conditions (2.22) and (4.47) are equivalent. For any positive integer m and any set of points a1 , . . . , am in the dual of the phase-space, let us consider the matrices: i h¯ T J (4.52) M jk = g˜ (a j − ak ) exp − ak Ja j . 2 If we define bi = Sai (i = 1, . . . , m), we get from (4.51): i h¯ T −1 ˜ b bj , M jk = f (b j − bk ) exp 2 k
(4.53)
where we used (cf. (3.5)): − (ST )−1 JS−1 = (SJST )−1 = −1 .
(4.54)
Now notice that: i h¯ −1 i = 2 2
ηE ˜ −h¯˜ I2×2 ˜h¯ I2×2 θ˜ E
That is: M jk = f˜ (b j − bk ) exp
=
i (h¯˜ , θ˜ , η). ˜ 2
i T ˜ ˜ j . bk (h¯ , θ˜ , η)b 2
(4.55)
(4.56)
˜ The function g˜ J (a) is then of the h¯ -positive type, iff f˜ (a) is of the (h¯˜ , θ˜ , η)-positive type. Definition 4.9. The noncommutative Narcowich-Wigner (NCNW) spectrum of f ∈ F is the set:
W N C ( f ) = (α, β, γ ) ∈ R3 f˜ (a) is of (α, β, γ )-positive type . (4.57) Obviously, if f is a NCWM, then (h¯˜ , θ˜ , η) ˜ ∈W
N C ( f ).
From this definition, we may prove the analog of Theorem 2.8: Theorem 4.10. The NCNW spectrum of the convolution f g of f, g ∈ F contains all elements of the form (α1 + α2 , β1 + β2 , γ1 + γ2 ), with (α1 , β1 , γ1 ) ∈ W
NC ( f )
and (α2 , β2 , γ2 ) ∈ W
(4.58)
N C (g).
Proof. The theorem follows immediately from the facts that (i) the Schur (or Hadamard) product of two hermitian and non-negative matrices is again a hermitian, non-negative matrix, and (ii) the matrix (α, β, γ ) is additive with respect to its arguments: (α1 + α2 , β1 + β2 , γ1 + γ2 ) = (α1 , β1 , γ1 ) + (α2 , β2 , γ2 ).
(4.59)
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The interpretation of (α, β, γ )-positive functions is slightly less straightforward than that of α-positive functions in the commutative case. Indeed, we know from Theorem 2.6, that if g(ξ ) is properly normalized and if its symplectic Fourier transform is continuous and of the α-positive type (with α = 0), then we can just think of it as a Wigner function, where h¯ has been replaced by α. In other words, there exists b(ξ ) ∈ F such that (2.25) holds. In the noncommutative case, the relation between (α, β, γ )-positive functions and elements of the form b(z) α β γ b(z) is less clear. To begin with, if we consider a function f (z) = b(z) h¯ θ η b(z), then Eq. (4.49) reveals that f is not of the (h¯ , θ, η)positive type but rather of the (h¯˜ , θ˜ , η)-positive ˜ type. Moreover, there is an additional complication. If we look carefully at the noncommutative symplectic Fourier transform (4.41) (and contrary to what happens with the commutative symplectic Fourier transform (2.19)), it depends on the parameters h¯ , θ, η, even if we are considering functions of (α, β, γ )-positive type, with (α, β, γ ) = (h¯˜ , θ˜ , η). ˜ To circumvent this difficulty, let us consider a function f ∈ F of the form: f (z) = b(z) α β γ b(z),
(4.60)
with R4
|b(z)|2 dz = 1,
(4.61)
where α, β, γ have the dimensions of h¯ , θ, η, respectively and where α 2 = βγ .
(4.62)
We may thus construct a new complete Weyl-Wigner formulation by defining: βE αI2×2 . α = −αI2×2 γ E
(4.63)
We also define appropriate D maps S : S JS T = ,
det S = |1 − ζ |,
ζ =
βγ . α
(4.64)
Our whole construction of noncommutative quantum mechanics in phase-space [4] goes through with (h¯ , θ, η) replaced by (α, β, γ ). Consequently if f is of the form (4.60, 4.61), there must exist a normalized function g(ξ ) of α-positive type such that: 1 g(S −1 z). f (z) = P f ( )
(4.65)
The corresponding noncommutative symplectic Fourier transform is: 1 −1 T −1 f (z) exp ia T −1 z dz = g(S z) exp ia z dz f˜ (a) = P f ( ) R4 R4 = g(ξ ) exp ia T −1 S JT Jξ dξ = g˜ J (JS T −1 a). (4.66) R4
Wigner Measures in Noncommutative Quantum Mechanics
731
Since g˜ J is of α-positive type, for any positive integer m and any set of points b1 , . . . , bm , the matrices iα T J (4.67) M jk = g˜ (b j − bk ) exp − bk Jb j , 2 are hermitian and non-negative. If we define bi = JS T −1 ai , (i = 1, . . . , m), we get from (4.66): iα T −1 T −1 (4.68) M jk = f˜ (a j − ak ) exp ak S JS a j . 2 From (4.64) it then follows that: iα T −1 −1 M jk = f˜ (a j − ak ) exp ak a j . 2
(4.69)
α−1 −1 = (α , β , γ ),
(4.70)
Now notice that
where α =
2 α h¯˜ (1 + ζ ) − ηβ ˜ − θ˜ γ 2α h¯˜ θ˜ − h¯˜ β − θ˜ 2 γ , , β = h¯ (1 − ζ ) h¯ 2 2
γ =
2α h¯˜ η˜ − h¯˜ γ − η˜ 2 β . h¯ 2
(4.71)
This system is easily inverted: 1 2h¯ ζ α − ηβ − θ ζ γ , (ηβ + θ γ ), β = η h¯ 2h¯ ζ α − θ γ − ζ ηβ γ = . θ
α = (1 + ζ )α −
(4.72)
Consequently, for α, β, γ such that (4.62) holds, we have: Theorem 4.11. If f (z) ∈ F is such that (4.60, 4.61) holds for some b(z) ∈ F , then f is such that f˜ (0) = 1 and f˜ is continuous and of the (α , β , γ )-positive type, with α , β , γ given by (4.71). Conversely, if f˜ (0) = 1 and f˜ is continuous and of the (α , β , γ )-positive type, then there exists b(z) ∈ F such that (4.60, 4.61) hold, for α, β, γ , given by (4.72) as long as α 2 = βγ . Corollary 4.12. If (α , β , γ ) ∈ W
N C ( f ),
then (−α , −β , −γ ) ∈ W
N C ( f ).
Proof. Again this is a consequence of (i) the fact that a(z) α β γ b(z) = b(z) −α
−β −γ a(z), (ii) Theorem 4.11, and (iii) the fact that under the replacement (α, β, γ ) → (−α, −β, −γ ) in (4.71), we get (α , β , γ ) → (−α , −β , −γ ).
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Remark 4.13. If we set β = γ = 0 and α = h¯ in (4.71), we obtain (α , β , γ ) = (1 − ζ )−2 (h¯ (1 + ζ ), 2θ, 2η). Consequently, if f is a Wigner measure, then (h¯˜ (1 + ζ ), 2θ˜ , 2η) ˜ ∈W 1−ζ
NC
( f ).
(4.73)
Obviously, from Bochner’s theorem, if (0, 0, 0) ∈ W N C ( f ), then f is everywhere non-negative. Theorems 4.10 and 4.11, then suggest a way of constructing functions which are simultaneously commutative and noncommutative Wigner measures. Indeed, let f, g ∈ F be such that (i) f (z) is a NCWM, i.e. (cf. Theorem 4.8 and Corollary 4.12):
(h¯˜ , θ˜ , η), ˜ (−h¯˜ , −θ˜ , −η) ˜ ⊆W
NC
( f ).
(4.74)
(ii) g(z) is such that
(0, 0, 0), (α , β , γ ), (−α , −β , −γ ) ⊆ W
NC
(g),
(4.75)
where α , β , γ are given by (α , β , γ ) =
(2h¯˜ ζ, (1 + ζ )θ˜ , (1 + ζ )η) ˜ . 1−ζ
(4.76)
From Theorem 4.10, we conclude that the NCNW spectrum of the convolution f g contains the elements: (h¯˜ (1 + ζ ), 2θ˜ , 2η) ˜ (h¯˜ , θ˜ , η) ˜ + (α , β , γ ) = , 1−ζ
(h¯˜ , θ˜ , η) ˜ + (0, 0, 0) = (h¯˜ , θ˜ , η). ˜ (4.77)
The first element means that the convolution is a Wigner measure (4.73), whereas the second one entails that it is equally a NCWM (4.74). Functions g satisfying (4.75) are easy to construct. Indeed from (4.60,4.71), we conclude that any function of the form g(z) = b(z) θ η b(z),
b(z) ∈ F
(4.78)
contains ±(α , β , γ ) (4.76) in its NCNW spectrum. Moreover, it is easy to check that if b(z) is a Gaussian, then g(z) in (4.78) is positive, as it is another Gaussian. Notice that we can safely set α = 0. Indeed it always appears in all the formulae in the combination α (4.63) which is regular as α ↓ 0. 4.3. Constructing functions in F C , F N C and L . Our purpose now is to investigate how the sets of Wigner measures (F C ), noncommutative Wigner measures (F N C ) and Liouville measures (L ) relate to each other. The latter is the set of real, normalized
Wigner Measures in Noncommutative Quantum Mechanics
733
Fig. 1. Different sets of functions and their intersection
phase-space functions, which are everywhere non-negative, i.e. functions whose symplectic Fourier transform is of 0-positive type or whose noncommutative symplectic Fourier transform is of (0, 0, 0)-positive type. Let us then define the sets: 1 = F C \(F N C ∪ L ), 2 = F N C \(F C ∪ L ), 3 = L \(F C ∪ F N C ), 4 = (F C ∩ F N C )\L , 5 = (F C ∩ L )\F N C , 6 = (F N C ∩ L )\F C , 7 = F C ∩ F N C ∩ L . (4.79) The remainder of this section is devoted to proving Lemma 4.14. We depicted the content of the lemma in Fig. 1. Lemma 4.14. The sets i (i = 1, . . . , 7) are all non-empty. Proof. To prove the lemma we shall construct explicitly families of functions in each of the sets, by resorting to the properties of functions in F C , F N C and L . Let us start with the simplest case: A function in 3 . The function 2 q p2 1 f 3 (q, p) = 2 exp − − , π ab a b
a, b > 0, ab < h¯ 2 (1 − ζ )
(4.80)
belongs to 3 . It is obvious that f 3 ∈ L , since it is real, normalized and everywhere positive. To prove that it does not belong to F C ∪ F N C , let us compute its purity: 1 1 1 > > . (4.81) [ f 3 (q, p)]2 dqdp = √ 2 2 (2π ) ab (2π h¯ )2 (2π h¯ 1 − ζ ) R2 R2 Consequently, from (2.15,4.3), we conclude that f 3 ∈ 3 .
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C. Bastos, N. C. Dias, J. N. Prata
A function in 5 . The function f 5 (q, p) =
2 ap 2 1 q , 00 1 2 2 2aθ 2 2aθ 2 h¯ (1−ζ ) h¯ 1−ζ (4.89) belongs to 2 . If we choose φ1 (q1 ) =
2a 41 π
exp(−aq12 ) and φ2 (q1 ) =
32a 3 π
1 4
q1 exp(−aq12 ) and
substitute in (4.30), we obtain f 2 . From Lemma 4.5 we conclude that f 2 ∈ F N C \F C . q2 2 1 < 4a , we conclude that f 2 ∈ / L. Since f 2 is negative inside the ellipse q12 + 2aθ A function in 7 . Any function of the form f 7 (z) = ( f 6 g)(z) with f 6 an element of 6 and 2 1 q p2 g(q, p) = 2 exp − − , c ≥ θ, d ≥ η (4.90) π cd c d belongs to 7 . Let us choose α, β > 0 such that: c=
1 + α2 θ 2 , 2α
d=
1 + β 2 η2 . 2β
(4.91)
With this choice, c, d automatically satisfy c ≥ θ and d ≥ η. Moreover, we define: 2 b(q, p) = (4.92) αβ exp −αq 2 − βp 2 . π Using the kernel representations (3.15) of the star-products it is straightforward to show that: g(z) = b(z) θ η b(z).
(4.93)
From our discussion in Remark 4.13, we know that under these circumstances, the convolution of f 6 and g is simultaneously a Wigner measure and a NCWM. Finally, since g(z) is positive, its convolution with another positive function is again positive. A function in 4 . The function % $ 2 2 θ 2θ p 2 1 2q 2 2 q − E p − 1 exp − − q · Ep , f 4 (q, p) = − 3(π h¯ )2 3θ 3θ 3h¯ h¯ 3h¯ 2 (4.94) is a function of 4 .
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C. Bastos, N. C. Dias, J. N. Prata
Let us choose g(z) of the form (4.90) with c = θ and d = η. Moreover, let us consider 1 the function f 2 (z) in (4.89) with a = 2θ . Then it is easy to show that f 4 (z) = (g f 2 )(z). From Remark 4.13, we conclude that f 4 ∈ F C ∪ F N C . However, f 4 is negative for 2 q − hθ¯ E p < 3θ / L . This completes the proof of the lemma. 2 , which means that f 4 ∈ Remark 4.15. The function f 6 in (4.88) reveals that functions of the form (4.30) saturate the θ -purity but not the η-purity. Indeed by a simple calculation, we obtain: 1 θ 2 2 2 2 exp − ( p P p ( p) = f 6 (q, p) dq = + 4a θ p ) . 1 2 π h¯ 2 (2 − ζ ) 2a h¯ 2 (2 − ζ ) R2 (4.95) Consequently
R2
2 P p ( p) dp =
ζ . 2π η(2 − ζ )
(4.96)
Since ζ < 1, we conclude that this is strictly smaller than 2π1 η . By the same token, we can show that states of the form (4.31), albeit maximizing the η-purity, need not saturate the θ -purity. Acknowkedgements. The authors would like to thank O. Bertolami for useful discussions and for reading the manuscript. The work of CB is supported by Fundação para a Ciência e a Tecnologia (FCT) under the fellowship SFRH/BD/24058/2005. The work of NCD and JNP was partially supported by the grant PTDC/MAT/69635/2006 of the FCT.
5. Appendix In this Appendix we prove that for a D transformation, the associated matrix S ∈ D (2d; R) satisfies: √ det S = det = |P f ()| > 0. (5.1) To prove this, we first derive the following lemma: Lemma 5.1. Under the assumption (3.2), the sign of the Pfaffian of the matrix reads: sign (P f ()) = (−1)d(d−1)/2 .
(5.2)
Proof. Let (ωαβ ) (α, β = 1, · · · , 2d) denote the elements of . The Pfaffian of can be obtained from the following recursive formula [46]: P f () =
2d α=2
(−1)α ω1,α P f (1, ˆ αˆ ),
(5.3)
st th where 1, ˆ αˆ denotes the matrix with both the 1 and α rows and columns removed. From (3.1), we get:
P f () =
d θ1i d+1 (−1)i P f (1, P f (1, ˆ ). ˆ iˆ ) + (−1) ˆ d+1 h¯ i=2
(5.4)
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A term which is independent of the elements of and N can only be found in (−1)d+1 P f (1, ˆ ). Suppose that d ≥ 3. If we apply the recursive formula (5.3) again ˆ d+1 we obtain a term of the form (−1)d+1 (−1)d P f (A2 ) where A2 is obtained from by removing the 1st , 2nd , (d + 1)th and (d + 2)th rows and columns. After i steps we obtain a term (−1)d+1 (−1)d · · · (−1)d+2−i P f (Ai ) where Ai is obtained from by removing the 1st , 2nd , . . . , ith , and (d+1)th , (d+2)th , . . . , (d+i)th rows and columns. We terminate this process when i = d − 2. We thus obtain: ⎛ ⎞ θd−1,d 0 1 0 h¯ ⎜ θd,d−1 ⎟ ⎜ 0 0 1 ⎟ (−1)d+1 (−1)d · · · (−1)4 P f ⎜ h¯ ηd−1,d ⎟ ⎝ −1 ⎠ 0 0 h¯ ηd,d−1 0 −1 0 h¯ (d+1 θd−1,d ηd−1,d = − 1 (−1) i=4 i . (5.5) h¯ 2 Thus the term independent of the elements of and N is (−1)d(d−1)/2 . We leave to the reader the simple task of verifying that this result also holds when d = 2. Let us now turn to the θ and η dependent terms. We resort to the definition of the Pfaffian [46]: 1 d P f () = d sgn(σ )i=1 ωσ (2i−1),σ (2i) , (5.6) 2 d! σ ∈S2d
where S2d is the symmetric group and sgn(σ ) is the signature of the permutation σ . Moreover, we use the following notation. If d = 2, for instance, then we consider the permutations of the set {1, 2, 3, 4}. As an example, consider σ = {3, 1, 4, 2}. Then we write: σ (1) = 3, σ (2) = 1, σ (3) = 4, and σ (4) = 2. d ω Suppose that in the string i=1 ¯ −1 , σ (2i−1),σ (2i) we pick k elements of the matrix h −1 p elements of the matrix h¯ N and l elements of the matrix I or −I. Then, of course k + l + p = d.
(5.7)
If we pick l elements from I or −I, then the remaining k + p terms can only be taken from when 2l lines and rows have been eliminated. In particular, we remove l lines and rows from h¯ −1 . That leaves us with (d − l − 1)(d − l)/2 non-vanishing independent paramd ω eters in h¯ −1 . Each time we choose one of the latter for our string i=1 σ (2i−1),σ (2i) , we have to eliminate another 2 lines and 2 columns. So if we pick k elements out of the (d − l − 1)(d − l)/2 non-vanishing independent elements of h¯ −1 , we remove 2k lines and columns. We are left with (d −l − 2k − 1)(d −l − 2k)/2 non-vanishing independent elements. But this is only possible if 2k ≤ d − l.
(5.8)
2 p ≤ d − l.
(5.9)
A similar argument leads to
Now, (5.8) and (5.9) are only compatible with (5.7) if k=p=
d −l . 2
(5.10)
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C. Bastos, N. C. Dias, J. N. Prata
This means that in each string we have exactly the same number of elements of h¯ −1 and h¯ −1 N. This proves that P f () = (−1)d(d−1)/2 + P[d/2] ,
(5.11)
where P[d/2] is a homogeneous polynomial of degree [d/2] (the integral part of d/2) in the dimensionless variables θi j ηkl /h¯ 2 with 1 ≤ i < j ≤ d and 1 ≤ k < l ≤ d. Let us define:
ζ = max θi j ηkl /h¯ 2 , 1 ≤ i < j ≤ d, 1 ≤ k < l ≤ d . (5.12) which yields the contribution (−1)d(d−1)/2 to the Pfaffian and Let σ be the permutation := S \ σ . We thus have: let S2d 2d 1 d d ≤ 1 P[d/2] = sgn(σ ) ω i=1 ωσ (2i−1),σ (2i) . σ (2i−1),σ (2i) i=1 2d d! 2d d! σ ∈S σ ∈S 2d
2d
(5.13) If a string then
d ω i=1 σ (2i−1),σ (2i)
h −1
contains k elements of ¯
and k elements of h¯ −1 N,
d ωσ (2i−1),σ (2i) ≤ ζ k < ζ, i=1
where we used ζ < 1. Since there are d! − 1 < d! elements in P[d/2] < ζ < 1. 2d
(5.14) , S2d
we conclude that (5.15)
This yields the desired result. An immediate consequence of this lemma is that the matrix is invertible and the skew-symmetric form (3.3) is non-degenerate as advertised. Proposition 5.2. A matrix S associated with a D transformation has positive determinant. Proof. We use the well known formula [46]: P f BABT = det(B)P f (A),
(5.16)
which holds for any 2d × 2d skew-symmetric matrix A and any 2d × 2d matrix B. If we apply this formula to (3.5), we obtain: det(S)P f (J) = P f (). For an arbitrary d × d matrix M we have [46]: 0 M = (−1)d(d−1)/2 det M. Pf −MT 0
(5.17)
(5.18)
We conclude that P f (J) = (−1)d(d−1)/2 .
(5.19)
From (5.17,5.19) and Lemma 6.1, we infer that det(S) > 0.
(5.20)
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Commun. Math. Phys. 299, 741–763 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1108-6
Communications in
Mathematical Physics
Fractional Loop Group and Twisted K-Theory Pedram Hekmati1 , Jouko Mickelsson1,2 1 Department of Theoretical Physics, Royal Institute of Technology, SE-106 91 Stockholm, Sweden.
E-mail:
[email protected];
[email protected] 2 Department of Mathematics and Statistics, University of Helsinki, FI-00014 Helsinki, Finland
Received: 3 August 2009 / Accepted: 2 April 2010 Published online: 14 August 2010 – © Springer-Verlag 2010
Abstract: We study the structure of abelian extensions of the group L q G of q-differentiable loops (in the Sobolev sense), generalizing from the case of the central extension of the smooth loop group. This is motivated by the aim of understanding the problems with current algebras in higher dimensions. Highest weight modules are constructed for the Lie algebra. The construction is extended to the current algebra of the supersymmetric Wess-Zumino-Witten model. An application to the twisted K-theory on G is discussed. 1. Introduction The main motivation for the present paper comes from trying to understand the representation theory of groups of gauge transformations in higher dimensions than one. In the case of a circle, the relevant group is the loop group LG of smooth functions on the unit circle S 1 taking values in a compact Lie group G. In quantum field theory one considers representations of a central extension LG of LG; in the case when G is semisimple, this corresponds to an affine Lie algebra. The requirement that the energy is bounded from below leads to the study of highest weight representations of LG. This part of the representation theory is well understood, [6]. In higher dimensions much less is known. Quantum field theory gives us a candidate for an extension of the gauge group Map(M, G), the group of smooth mappings from a compact manifold M to a compact group G. The extension is not central, but by an abelian ideal. The geometric reason for this is that the curvature form of the determinant line bundle over the moduli space of gauge connections (the Chern class of which is determined by a quantum anomaly) is not homogeneous; it is not invariant under left (or right) translations, [11]. There are two main obstructions when trying to extend the representation theory of affine Lie algebras to the case of Map(M, g). The first is that there is no natural polarization giving meaning to the highest weight condition; on S 1 the polarization is given by the decomposition of loops to positive and negative Fourier modes. The second
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obstruction has to do with renormalization problems in higher dimensions. On the circle, with respect to the Fourier polarization, one can use methods of canonical quantization for producing representations of the loop group; the only renormalization needed is the normal ordering of quantities quadratic in the fermion field, [9,17]. In higher dimensions further renormalization is needed, leading to an action of the gauge group, not in a single Hilbert space, but in a Hilbert bundle over the space of gauge connections, [12]. In this paper we make partial progress in trying to resolve the two obstructions above. We consider instead of LG the group L q G of loops which are not smooth but only differentiable of order 0 < q < ∞ in the Sobolev sense, the fractional loop group. In the range 21 ≤ q the usual theory of highest weight representations is valid, the cocycle determining the central extension is well defined down to the critical order q = 21 . However, for q < 21 we have to use again a “renormalized” cocycle defining an abelian extension of the group L q G, similar as in the case of Map(M, G). The renormalization means that the restriction of the 2-cocycle to the smooth subgroup LG ⊂ L q G is equal to the 2-cocycle c for the central extension (affine Kac-Moody algebra) plus a coboundary δη of a 1-cochain η (the renormalization 1-cochain). The 1-cochain is defined only on LG and does not extend to L q G when q < 1/2, only the sum c + δη is well-defined on L q G. The important difference between L q G and Map(M, G) is that in the former case we still have a natural polarization of the Lie algebra into positive and negative Fourier modes and we can still talk about highest weight modules for the Lie algebra L q g. Because of the existence of the highest weight modules for L q g we can even define the supercharge operator Q for the supersymmetric Wess-Zumino-Witten model. In the case of the central extension the supercharge is defined as a product of a fermion field on the circle and the gauge current; this is well defined because the vacuum is annihilated both by the negative frequencies of the fermion field and the current, thus when acting on the vacuum only the (finite number of) zero Fourier modes remain. This property is still intact in the case of the abelian extension of the current algebra. We can also introduce a family of supercharges, by a “minimal coupling” to a gauge connection on the loop group. In the case of central extension the connections on LG can be taken to be left invariant and they are written as a fixed connection plus a left invariant 1-form A on LG. The form A at the identity element is identified as a vector in the dual Lg∗ which again is identified, through an invariant inner product, as a vector in Lg. This vector in turn defines a g-valued 1-form on the circle. The left translations on LG induce the gauge action on the potentials A. Modulo the action of the group G of based loops, the set of vector potentials on the circle is equal to the group G of holonomies. In this way the family of supercharges parametrized by A defines an element in the twisted K-theory of G. Here the twist is equal to an integral 3-cohomology class on G fixed by the level k of the loop group representation, [13]. In the case of L q G and the abelian extension, the connections are not invariant under the action of L q G and thus we have to consider the larger family of supercharges parametrized by the space A of all connections of a circle bundle over L q G. This is still an affine space, the extension of L q G acts on it. The family of supercharges transforms equivariantly under the extension and it follows that it can be viewed as an element in twisted K-theory of the moduli stack A// L q G. This replaces the G-equivariant twisted K-theory on A// LG in case of the central extension LG, the latter being equivalent to twisted G-equivariant K-theory on the group G of gauge holonomies. The paper is organized as follows. In Sect. 2 we introduce the fractional loop group and consider its role as the gauge group of a fractional Dirac-Yang-Mills system. It turns
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out that the natural setting is a spectral triple in the sense of non-commutative geometry. Interestingly enough, similar attempts have been made recently, [4]. We move on to discuss the embedding L q G ⊂ G L p in Sect. 3 and the construction of Lie algebra cocycles in Sect. 4. Finally the last two sections are devoted to extending the current algebra of the supersymmetric WZW model to the fractional case and discussing its application to the twisted K-theory of G.
2. Fractional Loop Group Let G denote a compact semisimple Lie group and g its Lie algebra. Fix a faithful representation ρ : G → G L(V ) in a finite dimensional complex vector space V . Definition 2.1. The fractional loop group L q G for real index Sobolev space, L q G := H q (S 1 , G) = {g ∈ Map(S 1 , G) | g22,q =
1 2
< q is defined to be the
(1 + k 2 )q |ρ(gk )|2 < ∞},
k∈Z
where |ρ(gk )| is the standard matrix norm of the k th Fourier component of g : S 1 → G. The group operation is given by pointwise multiplication (g1 g2 )(x) = g1 (x)g2 (x). There is a natural Hilbert Lie group structure on L q G for 21 < q. It is defined by the Hilbert space completion of the Lie algebra of smooth maps C ∞ (S 1 , g) with respect to the Sobolev inner product. The exponential map exp : H q (S 1 , g) → H q (S 1 , G) provides a local chart near the identity and is extended to an atlas by left translations. For our purposes however, we will use a Banach topology on the Lie algebra, L q g = {X ∈ L ∞ (S 1 , g) | ||X || = ||X ||∞ + ||X ||2,q < ∞ }, where L ∞ is the set of measurable essentially bounded functions and || ||∞ denotes the supremum norm. This induces a Banach Lie group structure on L q G, [17] and is a more appropriate topology for the applications in this paper. In fact, we have natural inclusions of Lie groups LG ⊂ L q G ⊂ L c G, where LG is the smooth loop group and L c G is the Banach-Lie group of continuous loops in G, see Sect. 6. In the next section, we will modify the definition of L q G with respect to a different norm, to allow for values 0 < q ≤ 21 . In Yang-Mills theory on the cylinder S 1 × R, the loop group LG appears as the group of (time-independent) local gauge transformations. Quantization of massless chiral fermions in external Yang-Mills fields breaks the local gauge symmetry, leading to a central extension of LG. In order to make sense of this in the fractional setting, we need a notion of fractional differentiation. The study of fractional calculus dates back to early 18th century and a comprehensive review can be found in [18]. The transition to fractional calculus is by no means unique. There are several competing definitions, but many are known to coincide on overlapping domains. For functions on the real line the Riemann-Liouville fractional derivative is defined by q
Da ψ(x) =
dn dxn
1 (n − q)
a
x
ψ(y) dy (x − y)q−n+1
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for n > q and x > a. On the circle however this definition proves inconvenient since periodic functions are not mapped onto periodic ones. An operator that does preserve periodicity is the Weyl fractional derivative, iqπ q ψ(x) = D (ik)q ψk eikx = e 2 sgn(k) |k|q ψk eikx k∈Z
k∈Z
q and Daq coincide for all q ∈ R. In fact, by extending to the real line one shows that D for −1 < q and a = −∞ on an appropriate domain. For our purposes, we need a selfadjoint operator on a dense domain in H = L 2 (S 1 , V ). The fractional Dirac operator on the circle is therefore defined by D q ψ(x) = sgn(k)|k|q ψk eikx , k∈Z
where the complex phase has been replaced by the sign function. This has the consequence that D q ◦ Dr = D q+r , but we have instead D q ◦ Dr = |D q+r |. For odd integers q > 1, the fractional Dirac operator is simply the q th power of the rotation operator −i ddx on the circle. The domain of D q is the Sobolev space H q (S 1 , V ). It is by construction an unbounded, self-adjoint operator with discrete spectrum {sgn(k)|k|q }k∈Z and a complete set of eigenstates in H. However, the Leibniz rule is no longer satisfied D q (ψφ) = sgn(k)|k|q ψm φk−m eikx k,m∈Z q
= (D ψ)φ + ψ(D q φ) sgn(m)|m|q + sgn(k − m)|k − m|q ψm φk−m eikx , = k,m∈Z
unless q = 1. We introduce interactions by imposing local gauge invariance. The covariant derivq ative D A = D q + A should transform equivariantly under gauge transformations g −1 D A g = D q + g −1 [D q , g] + g −1 Ag = D Ag . q
q
This motivates the following definition of fractional Yang-Mills connections on the circle; A = α[D q , β], α, β ∈ H q (S 1 , g). The fractional loop group L q G acts on A by (g, A) → A g = g −1 Ag + g −1 [D q , g], and the infinitesimal gauge action is given by (X, A) → L X A = [A, X ] + [D q , X ]. For values 21 < q ≤ 1, there is a geometric interpretation of this data in the non-commutative geometry sense. What we have is precisely a spectral triple, namely a Dirac operator D q , a Hilbert space H and an associative ∗-algebra L q C, [2]. Here L q C = H q (S 1 , C) is an algebra for 21 < q by the Sobolev multiplication theorem.
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Proposition 2.2. [D q , X ] is a bounded operator for all X ∈ L q C and 0 < q ≤ 1,
q
[D , X ] = sup [D q , X ]ψ < ∞. ψ∈H ψ=1
Proof. By expanding in Fourier series, [D q , X ]ψ = X m ψk−m sgn(k)|k|q − sgn(k − m)|k − m|q ei(k+m)x , k,m∈Z
it follows that
q 2
[D , X ]ψ 2 = |X m ψk−m |2 sgn(k)|k|q − sgn(k − m)|k − m|q m,k∈Z
=
2 |X m ψn |2 sgn(n + m)|n + m|q − sgn(n)|n|q .
m,n∈Z
Since the sequence |ψn |2 belongs to l 1 , the sum converges if 2 |X m |2 sgn(n + m)|n + m|q − sgn(n)|n|q < C m∈Z
is uniformly bounded by some constant C. To establish this, we rewrite 2 |X m |2 sgn(n + m)|n + m|q − sgn(n)|n|q m∈Z
=
m∈Z
n
q
n
q 2 m
|X m |2 m 2q sgn 1 + .
1 + − n m m
For 0 < q ≤ 1, this sum is bounded for all n ∈ Z since 1 |1 + x|q − |x|q f (x) = sgn 1 + x is a bounded function on the real line.
We have an immediate corollary: Corollary 2.3. For 21 ≤ q ≤ 1 the space L q C is the algebra of essentially bounded measurable loops X such that [D q , X ] is a bounded operator. The inverse statement follows from the observation that taking ψ(x) = 1, the constant loop in the proof above, the norm ||[D q , X ]ψ|| is equal to ||X ||2,q . Similarly one verifies that [|D q |, X ] is bounded for all X ∈ L q C. Recall that a spectral triple is p + -summable if |D q |− p belongs to the Dixmier ideal L1+ . This means that for some real number p ≥ 1, 1 λk < ∞, N →∞ log(N ) N
lim
k=1
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where λk ≥ λk+1 ≥ λk+2 . . . are eigenvalues of |D q |− p listed in descending order. In our case λk =
1 kq p
for k = 1, 2, . . .. This gives 1 1 N →∞ log(N ) kq p N
lim
k=1
which is finite if and only if qp ≥ 1. Moreover, since
2
2n
n |X m ψk−m |2 |k|q − |k − m|q
ad|D|q (X )ψ = m,k∈Z
diverges for n > 1, we conclude that the spectral triple is not tame. It is also interesting to note that even though the algebra L q C is commutative, the spectral dimension p of the circle is strictly larger than one when q < 1. Fractional differentiability has been studied systematically in the more general context of θ -summable spectral triples in [5]. However, their definition of fractional differentiability, although similar, differs from ours which is geared to the special case of loop groups and L p -summable spectral triples. 3. Embedding of L q G in G L p When dealing with representations of the loop group LG, one is lead to consider central extensions by the circle 1 → S1 → LG → LG → 1. satisfy In Fourier basis, the generators Sna of the Lie algebra Lg c [Sna , Smb ] = λabc Sn+m + knδ ab δn,−m ,
where k is the central element (represented as multiplication by a scalar in an irreducible representation). Here the upper index refers to a normalized basis (with respect to an invariant nondegenerate bilinear form) of the Lie algebra g of the group G. The λabc ’s are the structure constants in this basis. We have adopted the Einstein summation convention meaning that an index appearing twice in a term is summed over all its possible values. When trying to extend the central extension to the fractional setting, one runs immediately into a problem. For infinite linear combinations of the Fourier modes Sna the central term blows up. A precise condition for the divergence can be formulated by regarding L q G as a group of operators in a Hilbert space. Let ρ : G → G L(V ) denote a representation of G as in the previous section. Elements in the fractional loop group act as multiplication operators in H = L 2 (S 1 , V ) by pointwise multiplication, (Mg ψ)(x) = ρ(g(x))ψ(x) for all g ∈ L q G, ψ ∈ H. In fact, M : L q G → G L(H), g → Mg defines a continuous embedding into the general linear group. This statement is somewhat crude however.
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Below we show that L q G is actually contained in a subgroup G L p of G L. Recall that Dq the sign operator = |D q | defines an orthogonal decomposition H = H+ ⊕ H− into positive and negative Fourier modes. We use the convention that the zero mode of D q is on the positive side of the spectrum. Introduce the Schatten class 1 2p L 2 p = {A ∈ B(H) | A2 p = Tr(A† A) p < ∞} which is a two-sided ideal in the algebra of bounded operators B(H). In particular, the first Schatten class L 1 is the space of trace class operators and L 2 is the space of Hilbert-Schmidt operators. We will use an equivalent norm which is better suited for computations A2 p =
Aφk
2p
1 2p
,
k∈Z
where {φk }k∈Z is an orthonormal basis in H. The subgroup G L p ⊂ G L(H) is defined by G L p = {A ∈ G L(H) | [, A] ∈ L 2 p }. Writing elements in G L(H) in block form with respect to the Hilbert space polarization A++ A−+
A+− , A−−
0 −A−+
A+− 0
A= the condition [, A] = 2
∈ L2p
simply means that the off-diagonal blocks are not “too large”. Given the topology defined by the norm |A| p = A++ + A+− 2 p + A−+ 2 p + A−− , where a = sup aψ ψ=1
denotes the operator norm, G L p is a Banach Lie group with the Lie algebra gl p = {X ∈ B(H) | [, X ] ∈ L 2 p }. Proposition 3.1. If p ≥
1 2q ,
then L q G is contained in G L p .
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Proof. In order to avoid cumbersome notation, we write g(x) instead of ρ(g(x)). Expanding in Fourier series g(x) = k∈Z gk eikx , we have
Mg eikx =
gm ei(m+k)x =
m∈Z
gm−k eimx
m∈Z
so that (Mg )mk = gm−k . Moreover eikx =
⎧ k ikx ⎨ |k| e k = 0 ⎩eikx
k=0
.
We check that [, Mg ] 2 p is finite;
2 p
ikx
[, Mg ] 2 p = M ]e
[,
g 2p k∈Z
m k
2 p imx 2 p
=
|m| − |k| |gk−m e | m,k∈Z
(n + k) k
2 p 2p
=
|n + k| − |k| |gn | n,k∈Z = 22 p |n||gn |2 p n∈Z
=2
2p
≤2
2p
1
||n| 2 p gn |2 p
n∈Z
1
(n 2 ) 2 p |gn |2 + C g ,
n∈Z 1
where C g is the finite part of the sum containing all terms where |n| 2 p |gn | ≥ 1. Thus the sum converges if the Sobolev norm g22,
1 2p
=
1
(1 + n 2 ) 2 p |gn |2
n∈Z
is finite, which is the case when p≥
1 . 2q
The same arguments apply without modification to the Lie algebra L q g → gl p . We see here precisely how the degree of differentiability q is related to the Schatten index p. Furthermore, this allows us to extend the definition of L q G to all real values 0 < q < ∞:
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Definition 3.2. The fractional loop group Lq G is defined to be the group of continuous 1 1 loops contained in G L p , where p = max 2 , 2q . We shall use the induced Banach structure on the Lie algebra L q g coming from the embedding, defined by the norm X ∞ + |X | p . This endows L q G with a Banach Lie group structure. Remark 3.3. By abuse of notation, we use the same label L q G as in Definition 2.1 for this slightly larger fractional loop group. Actually, it follows by Proposition 3.1 that the group in Definition 2.1 is continuously embedded in L q G, for q > 21 . For the remainder of this paper, we shall adopt Definition 3.2 for the fractional loop group. Also, note that for q ≥ 1, L q G consists of continuous loops contained in G L 1 , that is operators whose 2 off-diagonal blocks are trace class. This is the critical value for the Schatten index p, since for lower values than 21 the spaces L 2 p are no longer ideals. can be written The cocycle defining the central extension Lg c0 (X, Y ) =
1 Tr ([[, X ], [, Y ]]) = Tr (X +− Y−+ − Y+− X −+ ) 8
for X, Y ∈ Lg. This is finite so long as the off-diagonal blocks are Hilbert-Schmidt, i.e. [, X ] and [, Y ] belong to L 2 . However for p > 1, corresponding to differentiability q less than 21 , the operators X +− Y−+ and Y+− X −+ are no longer trace-class, because a, b ∈ L 2 p implies that ab ∈ L p by Hölder inequality. This is the reason behind the divergence. In order to make sense of the cocycle for higher p, regularization is required. We introduce the Grassmannian Gr p [11], which is a smooth Banach manifold parametrized by idempotent hermitian operators F such that F − ∈ L 2 p . Points on Gr p may also be thought of as closed subspaces W ⊂ H such that the orthogonal projection pr + : W → H+ is Fredholm and pr − : W → H− is in L 2 p . Let η(X ; F) denote a 1-cochain parametrized by points on Gr p . For a suitable choice of η, adding the coboundary to the original cocycle c p (X, Y ; F) = c0 (X, Y ) + (δη)(X, Y ; F), we obtain a well-defined cocycle on gl p for some fixed p. The Lie algebra cohomology coboundary operator δ is defined by Palais’ formula in Sect. 4. Although each term on the right diverges separately, the sum will be finite. In the case p = 2, we could take η(X ; F) = −
1 Tr ([X, ][F, ]) 16
which gives c2 (X, Y ; F) = c0 (X, Y ) + (δη)(X, Y ; F) =
1 Tr ([[, X ], [, Y ]]( − F)) . 8
An important consequence of this so called “infinite charge renormalization” is that the will be replaced by an abelian extension by the infinite-dimensional central extension Lg ideal Map(Gr p , C), 0 → Map(Gr p , C) → L q g → L q g → 0.
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4. Lie Algebra Cocycles In this section we will construct Lie algebra cocycles for gl p for all p ≥ 21 , which by restriction yield cocycles on L q g. In particular, we show that the cocycles respect the decomposition of L q g = L q g+ ⊕ L q g− into positive and negative Fourier modes on the circle. The computation is done in the non-commutative BRST bicomplex, where the classical de Rham complex is replaced by a graded differential algebra (, d). Let Dq ∗ 2 = 1. We = |D q | denote the sign operator on the circle satisfying = and introduce a family of vector spaces 0 = {X ∈ B(H) | X − X ∈ L 2 p }, 2k−1 = {X ∈ B(H) | X + X ∈ L 2 p , X − X ∈ L 2k
2k = {X ∈ B(H) | X + X ∈ L 2 p , X − X ∈ L 2k
2p 2k−1 2p 2k+1
}, },
k and set = ⊕∞ k=0 . The exterior differentiation is defined by [, X ], if X ∈ 2k dX = {, X }, if X ∈ 2k−1
and satisfies d 2 = 0. Here {X, Y } = X Y + Y X is the anticommutator. The space k consists of linear combinations of k-forms X 0 d X 1 d X 2 . . . d X k with X i ∈ 0 . If X ∈ k and Y ∈ l , then X Y ∈ k+l , d X ∈ k+1 , d(X Y ) = (d X )Y + (−1)k X dY which follows by the generalized Hölder inequality for Schatten ideals. This ensures that (, d) is a N-graded differential algebra. Integration of forms is substituted by a graded trace functional Str(X ) = TrC ( X ), for X ∈ k and k ≥ p, where is a grading operator on H and TrC (X ) = 21 Tr(X + X ) is the conditional trace, [2]. In case of an even Fredholm module (k even), anticommutes with and in case of an odd Fredholm module (k odd), = 1. Next we define the Lie algebra chain complex (C, δ). Since 0 = gl p and Gr p ⊂ 1 , we interpret B ∈ 1 as generalized connection 1-forms and define the infinitesimal gauge action by 0 × 1 → 1 , (X, B) → L X B = [B, X ] + d X. Indeed for any F ∈ Gr p we have F = g −1 g for some g ∈ G L p , since G L p acts transitively on the Grassmannian. Thus F − = g −1 dg corresponds to flat connections. The abelian group Map(1 , ) is naturally a 0 -module under the action d −t X t X
f e 0 × Map(1 , ) → Map(1 , ), (X, f ) → L X f (B) = Be +td X . t=0 dt Define the space of k-chains C k as alternating multilinear maps ω : 0 × · · · × 0 → Map(1 , ) k
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751
k and set C = ⊕∞ k=0 C . The coboundary operator is given by Palais’ formula
δω(X 1 , . . . , X k ; B) =
k (−1) j+1 L X j ω(X 1 , . . . , Xˆ j , . . . , X k ; B) j=1
+
(−1)i+ j ω([X i , X j ], X 1 , . . . , Xˆ i , . . . , Xˆ j , . . . , X k ; B)
i< j
and satisfies δ 2 = 0. Here Xˆ j means that the variable X j is omitted. This provides us with a double complex. For our purposes, we need cocycles of degree 2 in the Lie algebra cohomology. There is a straightforward way to compute such cocycles parametrized by flat connections B = F − . Given a bicomplex with commuting differentials, there is an associated singly graded complex with differential D = δ + (−1)k d. The Lie algebra 2-cocycle is given by 2 p+3 22 p c˜ p (X, Y ; B) = Str g −1 Dg (X, Y ) [2 p+1,2] (2 p + 1) ! p 2p −1 2 p+1−k −1 −1 k −1 = 2 Str (g dg) (g δg)(g dg) (g δg) (X, Y ) =2
2p
Str
k=0 p
(−1)
k
B
2 p+1−k
XB Y − B k
2 p+1−k
k
!
YB X
k=0
for all p ≥ 0. By (. . .)[ j,k] we mean the component of degree ( j, k) in (d, δ) cohomology. That this is a cocycle follows from δStr(g −1 Dg)2 p+3 = Str D(g −1 Dg)2 p+3 = Str(g −1 Dg)2 p+4 = 0, where we have used that Str(g −1 Dg)even = 0. Although c˜ p (X, Y ; B) does not vanish when X, Y ∈ L q g− , we have: Theorem 4.1. The cocycle c˜ p is cohomologous to a cocycle c p which has the property that c p (X, Y ; B) = 0 if both X and Y are in L q g− (or both in L q g+ ). Remark 4.2. Note that we can replace the abelian ideal of smooth functions of the variable B by the space of functions on L q G by using the embedding of L q G to the Grassmannian Gr p given by g → B = g −1 [, g]. The proof of Theorem 4.1 is by direct computation and is shifted to the Appendix. Theorem 4.3. The cocycle c p in the Appendix, when restricted to the Lie algebra of smooth loops, is cohomologous to the cocycle defining the standard central extension of the loop algebra Lg. Proof. Define η p (X ; B) = 22 p+1 Tr B 2 p+1 d X . Then by direct computation, similar to the one in the Appendix and which will not be repeated here, c p+1 (X, Y ; B) = c p (X, Y ; B) − (δη p )(X, Y ; B). Thus c p is cohomologous to c p−1 and by induction to the cocycle c0 . But 1 Tr (X dY ) 2 which is precisely the cocycle defining the central extension of the loop algebra. c0 (X, Y ; B) =
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P. Hekmati, J. Mickelsson
We note that the 1-cochain η(X ; F) in Sect. 3 is simply the sum − where B = F − .
p−1 k=0
ηk (X ; B),
Remark 4.4. Since the Lie algebra cocycle vanishes on L q g− one can define generalized Verma modules as the quotient of the universal enveloping algebra U( L q g) by the left ideal generated by L q g− and by the elements h − λ(h), where the h’s are elements in a Cartan subalgebra h of g and λ ∈ h∗ is a weight. In the case of the central extension of the smooth loop algebra, for dominant integral weights one can construct an invariant hermitian form in the Verma module; there is a subquotient (including the highest weight vector) which carries an irreducible unitarizable representation of the Lie algebra. However, in the case of L q g for q < 1/2 this is not possible: due to the large abelian ideal in the extension of L q g we cannot construct any invariant hermitian semidefinite form on the Verma module. Whether this is possible at all is an open question, although we conjecture that the answer is negative. 5. Generalized Supersymmetric WZW Model Let us recall the construction of a family of supercharges Q(A) for the supersymmetric Wess-Zumino-Witten model in the setting of representations of the smooth loop algebra, [3,8,13]. Here we assume that G is a connected, simply connected simple compact Lie group of dimension N . Let Hb denote the “bosonic” Hilbert space carrying an irreducible of level k. The level k is a unitary highest weight representation of the loop algebra Lg non-negative integer and we introduce k = 2θ 2 k, where θ is the length of the longest root of G. The generators of the loop algebra in the Fourier basis Tna satisfy c [Tna , Tmb ] = λabc Tn+m +
k ab nδ δn,−m , 4
where n ∈ Z and a = 1, 2, . . . , N . We fix an orthonormal basis T a in g, with respect to the Killing form, so that the structure constants λabc are completely antisymmetric and the Casimir invariant C2 = λabc λacb equals −N . Moreover, we have a (Tna )∗ = −T−n .
The fermionic Hilbert space H f carries an irreducible representation of the canonical anticommutation relations (CAR) {ψna , ψmb } = 2δ ab δn,−m , N
a . The Fock vacuum is a subspace of H of dimension 2[ 2 ] . It carries where (ψna )∗ = ψ−n f an irreducible representation of the Clifford algebra spanned by the zero modes ψ0a and acts in H f through the lies in the kernel of all ψna with n < 0. The loop algebra Lg minimal representation of level h ∨ , the dual Coxeter number of G. The operators are explicitly realized as bilinears in the Clifford generators
1 b K na = − λabc : ψn−m ψmc : 4 and satisfy c [K na , K mb ] = λabc K n+m +
h ∨ ab nδ δn,−m , 4
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where h ∨ = 2θ 2 h ∨ . The normal ordering : : indicates that operators with positive Fourier index are placed to the left of those with negative index. In case of fermions a ψ b := −ψ b ψ a if n > 0. The full Hilbert space there is a change of sign, : ψ−n n n −n of level k + h ∨ , with H = Hb ⊗ H f carries a tensor product representation of Lg a a a generators Sn = Tn ⊗ 1 + 1 ⊗ K n . The supercharge operator is defined by 1 a a Q = iψna T−n + K −n 3 and squares to the free Hamilton operator h = Q 2 of the supersymmetric WZW model a h = − : Tna T−n :+
k¯ N a ¯ f + N, : nψna ψ−n = h b + 2kh :+ 2 24 24
∨
where k¯ = k +h 4 . Interaction with external g-valued 1-forms A on the circle is introduced by minimal coupling ¯ na Aa−n , Q(A) = Q + i kψ where Aan are the Fourier coefficients of A in the basis Tna , satisfying (Aan )∗ = −Aa−n . This provides us with a family of self-adjoint Fredholm operators Q(A) that is equivariant with respect to the action of LG, S(g)−1 Q(A)S(g) = Q(A g ) with A g = g −1 Ag + g −1 dg. Infinitesimally this translates to a abc c ¯ ψn+m Ab−m ) = −Lan Q(A), [Sna , Q(A)] = i k(nψ n +λ
where Lan denotes the Lie derivative (infinitesimal gauge transformation) in the direction X = Sna . The interacting Hamiltonian h(A) = Q(A)2 is given by h(A) = h − k¯ 2Sna Aa−n + k¯ Aan Aa−n = h + h int . Next we consider extending this construction to the fractional setting. As previously mentioned, this necessarily entails certain regularization. Let us first denote by S0 the representation of the smooth loop algebra with commutation relations [S0 (X ), S0 (Y )] = S0 ([X, Y ]) + c0 (X, Y ). We proceed by adding a 1-cochain S(X ) = S0 (X ) + η(X ; B), Q = Q 0 + η(ψ; B), a ; B) = ψ a ηa and we denote the where in component notation η(ψ; B) = ψna η(T−n n −n original supercharge by Q 0 from now on. Here B = g −1 [, g] parametrizes points on the Grassmannian, as in Sect. 4, with g ∈ L q G. We write also L X = X na La−n for an element X ∈ L q g. Adding cochains of the type η, which are functions of the variable B, means that we are extending the original loop algebra by Fréchet differentiable functions of B. Since gauge transformations are acting on B by the formula B → g −1 Bg + g −1 [, g],
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or infinitesimally as B → [B, X ] + [, X ] for X ∈ L q g, the commutator of S(X ) by any Fréchet differentiable function f of B is given as [S(X ), f (B)] = L X f (B). The other new commutation relations will be [S(X ), S(Y )] [S(X ), ψ(Y )] [S(X ), Q] {Q, ψ(Y )}
= S([X, Y ]) + c(X, Y ; B), = ψ([X, Y ]), = ic(X, ψ; B), = 2i S(Y ),
where c(X, Y ; B) = c0 (X, Y ) + (δη)(X, Y ; B) converges for an appropriate choice of η, according to Theorem 4.3. Moreover, we set h = Q 2 , where Q 2 = Q 20 − 2S(η) − η2 + iLψ η(ψ; B),
" # a and S(η) = ηa S a . In the Fourier basis, the generators ψ a , S b , Q, h η2 = ηna η−n n −n n m satisfy the following commutation relations: {ψna , ψmb } = 2δ ab δn,−m , c a,b + cn,m (B), [Sna , Smb ] = λabc Sn+m c [Sna , ψmb ] = λabc ψn+m , a a {ψn , Q} = 2i Sn , a,b [Sna , Q] = icn,−m (B)ψmb , a,b [ψna , h] = 2cn,−m (B)ψmb , a,b a,b [h, Sna ] = 2Smb cn,−m (B) − ψmb ψ pc Lc− p cn,−m (B), [Q, h] = 0, a,b where cn,m (B) = c(Sna , Smb ; B). For any Fréchet differentiable function f = f (B),
[Sna , f ] = Lan f, [Q, f ] = iψna La−n f, [h, f ] = −2Sna La−n f + ψna ψqd Ld−q La−n f. a,b (B) = knδ ab δn,−m , one recovers the corresponding subalgebra In the smooth case, cn,m of the superconformal current algebra, [7]. Let us consider highest weight representations of the loop algebra generated by S and S0 respectively. Since they differ by a coboundary, one can explicitly relate their vacua by restricting to the subalgebra of smooth loops Lg ⊂ L q g. Indeed, we have
S(X )| >= 0, S0 (X )|0 >= 0 for all X ∈ Lg− , which implies S(X )|0 >= (S0 (X ) + η(X ; B)) |0 >= η(X ; B)|0 > = 0. However if (δη)(X, Y ; B) = 0 for all X, Y ∈ Lg− , then η restricts to a 1-cocycle on Lg− and can in fact be written η(X ; B) = L X (B) for some function of the variable
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B on the smooth Grassmannian consisting of points g −1 [, g] for g ∈ LG. For the cochain η in Theorem 4.3 one can choose (B) ∼ Tr B 2 p+1 and the vacua are linked according to | >= e−(B) |0 > . Indeed for all X ∈ Lg− , we have S(X )| > = e−(B) (S(X ) − L X (B)) |0 > = e−(B) (S0 (X ) + η(X ; B) − L X (B)) |0 >= 0. 6. Twisted K-Theory and the Group L q G We want to make sense of a family of supercharges Q(A) which transforms equivariantly under the action of the abelian extension L q G of the fractional loop group L q G. This should generalize the construction of the similar family in the case of the central extension of the smooth loop group. Let us recall the relevance of the latter for twisted K-theory over G of level k + h ∨ . We fix G to be a simple compact Lie group throughout this section. One can think of elements in K ∗ (G, k + h ∨ ) as maps f : A → Fr ed(H), to Fredholm operators in a Hilbert space H, with the property f (A g ) = gˆ −1 f (A)g. ˆ Here g ∈ LG and A is the space of smooth g-valued vector potentials on the circle. The moduli space A/ G (where G is the group of based loops) can be identified as G. Actually, one can still use the equivariantness under constant loops so that we really deal with the case of G-equivariant twisted K-theory K G∗ (G, k +h ∨ ). For odd/even dimensional groups one gets elements in K 1 /K 0 . The real motivation here is to try to understand the corresponding supercharge operator Q arising from Yang-Mills theory in higher dimensions. If M is a compact spin manifold the gauge group Map(M, G) can be embedded in U p for any 2 p > dimM; this was used in [10] for constructing a geometric realization for the extension of Map(M, G) arising from quantization of chiral Dirac operators in background gauge fields. This is an analogy for our embedding of L q G in U p for p > 1/2q, the index 1/2q playing the role of the dimension of M. The more modest aim here is to show that there is a true family of Fredholm operators which transforms covariantly under L q G. The operators are parametrized by 1-forms on L q G and generalize the family of Fredholm operators Q(A) from the smooth setting to the fractional case. Hopefully, this will help us understand the renormalizations needed for the corresponding problem in gauge theory on a manifold M. Let us denote by L c G the Banach-Lie group of continuous loops in G. The natural topology of L c G is the metric topology defined as d( f, g) = sup dG ( f (x), g(x)), x∈S 1
where dG is the distance function on G determined by the Riemann metric. Local charts on L c G are given by the inverse of the exponential function; at any point f 0 ∈ L c G we can map a sufficiently small open ball around f 0 to an open ball at zero in L c g by f → log( f 0−1 f ). In the smooth version LG the topology is locally given by the topology on Lg; the topology of the vector space Lg is defined by the family of seminorms ||X ||n =
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supx∈S 1 |X (n) (x)| for a fixed norm | · | on g. More precisely, we can define a family of distance functions on LG by d0 ( f, g) = sup dG ( f (x), g(x)) x∈S 1
for n = 0, and dn ( f, g) = sup | f (n) (x) − g (n) (x)| x∈S 1
for n > 0, where f (n) is the nth derivative with respect to the loop parameter: We identify the first derivative as a function with values in g by left translation f → f −1 f and then all the higher derivatives are g-valued functions on the circle. The metric is then defined as dn ( f, g) 2−n . d( f, g) = 1 + dn ( f, g) n≥0
The subgroup LG ⊂ L c G is dense in the topology of the latter. For this reason the cohomology of L c G is completely determined by restriction to LG. Actually, we have a stronger statement: Lemma 6.1 (Carey-Crowley-Murray). The group L c G of continuous loops is homotopy equivalent to the smooth loop group LG. Proof. We may assume that G is connected; otherwise, one repeats the proof for each component of G. When G is connected the full loop group is a product of G and the group G of based loops (whether continuous, smooth, or of type L q ). So we restrict to the group of based loops. As shown in [1], the groups c G and G are weakly homotopic, i.e. the inclusion G ⊂ c G induces an isomorphism of the homotopy groups. According to Theorem 15 by R. Palais, [15] a weak homotopy equivalence of metrizable manifolds implies homotopy equivalence. Actually, in [1] the authors use the CW property of the loop groups for the last step. The CW property in the case of c G is a direct consequence of Theorem 3 in [14].
Lemma 6.2. The group L q G is homotopy equivalent to LG and thus also to L c G. Proof. The proof in [1] can be directly adapted from the smooth setting to the larger group L q G. Let M be a compact manifold with base point m 0 and C((I n , ∂ I n ), (M, m 0 )) the set of continuous maps from the n-dimensional unit cube to M such that the boundary of the cube is mapped to m 0 . The key step in their proof is the observation that in the homotopy class of any map g ∈ C((I n , ∂ I n ), (M, m 0 )) there exists a smooth map; in addition, a homotopy can be given in terms of a differentiable map. Taking M = G and thinking of g as a representative for an element in the homotopy group πn−1 (c G). Since g is homotopic to a smooth map, it also represents an element in the smooth homotopy group of G and thus also an element in the (n − 1)th homotopy group of q . In addition, a continuous homotopy is equivalent to a smooth homotopy. The embedding of LG ⊂ L q G is continuous in their respective topologies [this follows from the Sobolev norm estimates in the proof of Proposition 3.1] and therefore the representatives for the homotopy groups of the former are mapped to representatives of the homotopy groups of the latter.
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Let F : L c G → LG be a smooth homotopy equivalence. We define θ ( f ; g) = F( f )−1 F( f g) for f, g ∈ L c G. This is a 1-cocycle in the sense of θ ( f ; gg ) = θ ( f ; g)θ ( f g; g ). For any g ∈ LG we then have ˆ f ; g)−1 Q 0 θˆ ( f ; g) = Q 0 + i k¯ , θ( where θˆ is the lift of θ to the central extension LG. Here ∂ is the differentiation with respect to the loop parameter. Since the homotopy F is smooth we can define a Lie algebra cocycle dθ ( f ; X ) =
d |t=0 θ ( f ; et X ) dt
with values in Lg for X ∈ L c g. This is a 1-cocycle in the sense that dθ ( f ; [X, Y ]) + L X dθ ( f ; Y ) − LY dθ ( f ; X ) = [dθ ( f ; X ), dθ ( f ; Y )]. Let next Q(A) = Q 0 + i k¯ < ψ, A > be a perturbation of Q 0 by a function A : → L c g. The group L c G acts on A by right translation, (g · A)( f ) = A( f g). ˆ Denote by (g) the operator consisting of the right translation on functions of f and of ˆθ (·; g) acting on values of functions in the Hilbert space H. Then LcG
−1 ˆ ˆ (g) Q(A)(g) = Q(A g ),
where (A g )( f ) = θ ( f ; g)−1 A( f g)θ ( f ; g) + θ ( f ; g)−1 ∂θ ( f ; g).
(1)
Since the group LG acts in H through its central extension LG, the Lie algebra L c g c acts through its abelian extension by Map(L G, iR), the extension being defined by the 2-cocycle ( f ; X ), dθ ( f ; Y )] − dθ ( f ; [X, Y ]) − L X dθ ( f ; Y ) + LY dθ ( f ; X ). ω( f ; X, Y ) = [dθ For an infinitesimal gauge transformation X , the formula (1) leads to δ X A = [A, dθ ( f ; X )] + ∂θ ( f ; X ) + L X A,
(2)
which should be compared with δ X A = [A, X ] + ∂ X in the smooth case, for constant functions A : LG → Lg. The Lie algebra cohomology of any Lie algebra, with coefficients in the module of smooth functions on the Lie group G, is by definition the same as the de Rham cohomology of G. Indeed, take a cocycle c in de Rham cohomology on G. It is an alternating multilinear form on the space of vector fields on G, with values in the space of smooth functions on G. We can restrict it to left invariant vector fields on G; but the left invariant vector fields are just elements of the Lie algebra of G. So we obtain an alternating multilinear form on the Lie algebra of G, with values in the space of smooth functions on G. Looking at the definition of a de Rham cocycle (in terms of smooth vector fields) one
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sees that the cocycle condition is exactly the same as the Lie algebra cocycle condition, with values in C ∞ (G), with the standard action of the Lie algebra (as derivations) on functions. Thus we get a linear map from the space of de Rham cocycles to Lie algebra cocycles. Exact cocycles on the de Rham side map to exact Lie algebra cocycles, so we have a map between the cohomologies. This is an isomorphism since the de Rham forms are uniquely determined by the restriction to left invariant vector fields (at each point on G the left invariant vector fields form a basis). We apply this to the loop group LG together with the fact that the standard central extension of Lg, when viewed as a left invariant 2-form on LG, generates H 2 (LG, R). Thus the Lie algebra cohomology of Lg with coefficients in the module of smooth functions on LG is one dimensional. By Lemma 6.2 we have Lemma 6.3. For a simple compact Lie group G the Lie algebra cohomology in degree 2 of L c g with coefficients in the module of smooth functions on L c G (with respect to the Banach manifold structure) is one dimensional and the class of a cocycle is fixed by the restriction to the smooth version LG. Remark 6.4. We can easily produce an explicit homotopy connecting the standard 2-cocycle on Lg to the restriction of the 2-cocycle on L c g to the smooth subalgebra Lg. Let Fs : LG → LG be any one-parameter family of maps connecting the identity F1 to the map F0 = F ◦ i : LG → LG, where i : LG → L c G is the inclusion and F : L c G → LG is the homotopy equivalence used before, 0 ≤ s ≤ 1. On LG we define 1 η= Tr [, Fs−1 δ Fs ]Fs−1 ∂s Fs . 0
1 2 Tr (X [, Y ]),
Denote c0 (X, Y ) = the standard Lie algebra 2-cocycle on Lg, and 1 c1 (X, Y ) = 2 Tr (dθ ( f ; X )[, dθ ( f ; Y )]) the 2-cocycle coming from the homotopy F0 . Then one checks that, for the restriction of c1 to LG, c0 − c1 = δη. Proposition 6.5. The unitary subgroup U p ⊂ G L p for each p ≥ topic to U 1 .
1 2
is smoothly homo-
2
Proof. We prove the claim inductively by constructing a homotopy equivalence from U p to U p/2 for all p ≥ 1. Let g ∈ U p with α β g= . γ δ Let us denote x = αγ ∗ − βδ ∗ ∈ L 2 p . Define the unitary operator h(g) 0 −x/2 . h(g) = exp ∗ x /2 0 Let F(g) = h(g)−1 g. Using x 2 ∈ L p , by a direct computation one checks that the upper right block in this operator is equal to 1 β + (αγ ∗ − βδ ∗ )δ mod L p . 2
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Using the unitarity relations α ∗ β + γ ∗ δ = 0 and β ∗ β + δ ∗ δ = 1 we see that the above operator is equal to ββ ∗ β. Since β ∈ L 2 p , by the operator Hölder inequalities ββ ∗ β is in L 2 p/3 ⊂ L p . Thus F(g) ∈ U p/2 . This map is homotopic to the identity map in U p : one just needs to replace the operator x by t x, where 0 ≤ t ≤ 1. At t = 0 we then have Ft (g) = g. Since the blocks of F(g) are rational functions in the blocks of g without singularities, the map g → F(g) is smooth in the natural Banach manifold structures of the groups U p and U p/2 . The same argument shows that the identity map on Ur contracts to a map to the subgroup U p/2 for any p ≥ r ≥ p/2.
According to [17] the restriction of the standard 2-cocycle of the Lie algebra of U1 (denoted by Ur es in [17]) to the subalgebra Lg is the cocycle defining an affine Kac-Moody algebra. In combination with Lemma 6.3 and Proposition 6.5 we conclude: Proposition 6.6. The restriction from L c G to L q G of the Lie algebra cocycle ω is equivalent to the Lie algebracocycle of Theorem 4.1 (obtained by restriction of c p from G L p
to L q G with p = max
1 1 2 , 2q
).
Remark 6.7. The discussion in Remark 6.4 applies here as well. By a similar formula one can produce an explicit homotopy between the standard 2-cocycle on the Lie algebra of U 1 and the cocycle obtained from the restriction from U p to the subgroup U 1 , for 2
2
p ≥ 21 .
We close this section by showing that in the fractional setting, the transformation (2) of the field A : L q G → L q g under L q G $ has a more geometric interpretation. Using the inner product <X, Y>= S 1 (X (x), Y (x))g d x in L q g, where (·, ·)g is an invariant inner product in g, and the fact that a Lie group is a parallelizable manifold, we can think of A as a 1-form on the loop group L q G. We denote by A the space of all Fréchet differentiable 1-forms on L q G. There is a circle bundle P over L q G with connection and curvature; the curvature ω is given by the cocycle of the abelian extension, ω(g; X, Y ) where the elements X, Y ∈ Tg (L q G) are identified as left invariant vector L q G the extension of L q G fields (elements of the Lie algebra of L q G). Let us denote by by the abelian normal subgroup Map(L q G, S 1 ) corresponding to the given Lie algebra extension. Conversely, starting from the abelian extension L q G, viewed as a principal bundle over L q G with fiber Map(L q G, S 1 ), we can recover the geometry of the the circle bundle P. The connection in P is obtained as follows. First, the connection form in L q G is given as = Adg−1 prc (d gˆ gˆ −1 ), ˆ where prc is the projection onto the abelian ideal Map(L q G, iR). Remark 6.8. In the case of the central extension LG of LG the adjoint action on prc (•) is trivial but in the case of the abelian extension of L q G it is needed in order to guarantee that the connection form is tautological in the vertical directions in the tangent bundle T P. The connection ∇ on P is then defined using the identification of P as the subbundle of L q G → L q G with fiber S 1 consisting of constant functions in Map(L q G, S 1 ),
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that is, at each base point g ∈ L q G we have the homomorphism γ sending a function f ∈ Map(L q G, S 1 ) to the value of f at the neutral element. This induces a homomorphism dγ : Map(L q G, iR) → iR defining the iR-valued 1-form ψ = dγ ◦, defining the covariant differentiation ∇ in the associated complex line bundle L . An arbitrary connection in the bundle P is then written as a sum ∇ + A with A ∈ A. Theorem 6.9. The transformation of ∇ + A under an infinitesimal right action of (X, α) ∈ L q g induces the action δ A = ωg (X, •) + L X A + L• α. Here the Lie derivative acting on the from A is composed from the directional derivative (by the left invariant vector field X ) on the argument of A and the commutator [A, θ (g; X )]. That is, under the infinitesimal transformation δ(X,0) the transformation is the same as in (2) using the identification of elements in L q g as elements in its dual through the inner product < ·, · > . The function α can be interpreted as an infinitesimal gauge transformation in the line bundle L over L q G. Proof. Let (X, α) ∈ L q g. Infinitesimally, the right action is generated by left invariant vector fields. Thus the infinitesimal shift in the direction (X, α), of the connection evaluated at the tangent vector Y is the commutator [∇Y , ∇ X + α]. The first term in the commutator is equal to the curvature of the circle bundle evaluated in the directions X, Y. This is in turn equal to ωg (X, Y ). Since the covariant derivative ∇Y acts as the Lie derivative LY on functions, we obtain δ(X,α) (∇ + A) = ωg (X, •) + L X A + L• α.
(3)
It follows that we can morally interpret Q(A) as a Dirac operator on the loop group L q G twisted by a complex line bundle. In the case of a central extension the above formulas (on level k + h ∨ ) reproduce the classical gauge action, after identification of the dual Lg∗ as Lg using the Killing form, 1 A → [A, X ] + (k + h ∨ )d X, 4
(4)
for a left invariant 1-form A. Namely, the right Lie derivative in (3), when acting on left invariant forms, produces the commutator term [A, X ] and whereas the shift by d X is coming from the central extension ωg (X, •) = c0 (X, •). The third term on the right-hand-side of Eq. (3) is absent since in the case of central extension we can take α = constant. Going the other way, starting from the action on A in terms of the cocycle ω, we may take the restriction to the Lie algebra of smooth loops and the connection ∇ becomes a sum ∇ = ∇0 + η, where ∇0 is the connection in P defined by the standard central extension of the loop group and η is a 1-form on LG. The form η is actually the 1-cochain relating the central extension to the abelian extension of LG. Writing now a = η + A and S(X ) = S0 (X ) + η(X ) we recover the gauge action formula a → [a, X ] + 41 (k + h ∨ )d X with respect to S0 (X ). Thus the addition of η can be viewed as a renormalization needed for extending from the case of smooth loop algebra to the fractional loops.
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Acknowledgements. The authors would like to thank the Erwin Schrödinger Institute for Mathematical Physics for hospitality were this work was initiated. The work of the second author was partially supported by the grant 516 8/07-08 from Academy of Finland.
Appendix: Proof of Theorem 4.1 Define a 1-cochain by η˜ p (X ; B) = Str B 2 p+1 X for p ≥ 0. Using Palais’ formula the coboundary is given by (δ η˜ p )(X, Y ; B) =
2p
Str B k [B, X ]B 2 p−k Y − B k [B, Y ]B 2 p−k X
k=0 2p
+
Str B k dX B 2 p−k Y − B k dY B 2 p−k X −Str B 2 p+1 [X, Y ] .
k=0
By cyclicity of trace we can rewrite the first sum, 2p
Str B k [B, X ]B 2 p−k Y − B k [B, Y ]B 2 p−k X
k=0 2p = Str B[X, B 2 p Y ] − B[Y, B 2 p X ] Str
k=1
× B [B, X ]B k
2 p−k
Y − B [B, Y ]B 2 p−k X . k
(5)
This can be simplified further. Using [a, bc] = [a, b]c + b[a, c] repeatedly we get B[X, B 2 p Y ] = B 2 p+1 [X, Y ] +
2 p−1
B 2 p−k [X, B]B k Y,
(6)
k=0
and inserting (6) into (5) yields p−1 2 2Str B 2 p+1 [X, Y ] + Str B 2 p−k [X, B]B k Y − B 2 p−k [Y, B]B k X k=0
+
2p
Str B k [B, X ]B 2 p−k Y − B k [B, Y ]B 2 p−k X = 2Str B 2 p+1 [X, Y ] .
k=1
Thus the expression for the coboundary δ η˜ p reduces to ⎛ ⎞ 2p (δ η˜ p )(X, Y ; B) = Str ⎝ B 2 p+1 [X, Y ] + B k dX B 2 p−k Y − B k dY B 2 p−k X ⎠ . k=0
(7)
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We split the remaining sum into even and odd powers. When k = 2m is even, p
Str B 2m dX B 2 p−2m Y − B 2m dY B 2 p−2m X
m=0
=
p
Str B 2m dX B 2 p−2m Y + B 2m Y B 2 p−2m dX ,
(8)
m=0
and when k = 2m − 1 is odd, Str B 2m−1 dX B 2 p−2m+1 Y − B 2m−1 dY B 2 p−2m+1 X
p m=1
p
=
Str B 2m−1 X B 2 p−2m+2 Y − B 2m X B 2 p−2m+1 Y ,
(9)
m=1
where we have used dB 2m = 0, dB 2m+1 = −B 2m+2 and “integration by parts” using d(B 2m−1 X B 2 p−2m+1 Y ) = −B 2m X B 2 p−2m+1 Y − B 2m−1 dX B 2 p−2m+1 Y + B 2m−1 X B 2 p−2m+2 Y + B 2m−1 X B 2 p−2m+1 dY. Finally we show that the sum in c˜ p (X, Y ; B) for values k ≥ 1 equals (9), up to the normalization factor 22 p ; p (−1)k Str B 2 p−k+1 X B k Y − B 2 p−k+1 Y B k X k=1
=
p/2
Str B 2 p−2m+2 Y B 2m−1 X − B 2 p−2m+2 X B 2m−1 Y
m=1 p/2
+
=
Str B 2 p−2m+1 X B 2m Y − B 2 p−2m+1 Y B 2m X
m=1 p/2
Str B 2m−1 X B 2 p−2m+2 Y − B 2m X B 2 p−2m+1 Y
m=1 p/2
+
Str B 2 p−2m+1 X B 2m Y − B 2 p−2m+2 X B 2m−1 Y .
m=1
Shifting the index m = n − p
p 2
in the last sum in (10) we get
Str B 3 p−2n+1 X B 2n− p Y − B 3 p−2n+2 X B 2n− p−1 Y ,
n=( p+2)/2
and reversing the summation order using β α
x ±2n =
β α
x ±2(α+β)∓2n
(10)
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this becomes Str B 2n−1 X B 2 p−2n+2 Y − B 2n X B 2 p−2n+1 Y .
p n=( p+2)/2
Altogether this is precisely the odd part of the sum in (7). Now we define a new cocycle c p (X, Y ; B) = c˜ p (X, Y ; B) − 22 p (δ η˜ p )(X, Y ; B) p Str B 2m dX B 2 p−2m Y − B 2m dY B 2 p−2m X = −22 p = −22 p
m=0 p
Str B 2m dX B 2 p−2m Y + B 2m Y B 2 p−2m dX .
m=0
We note that this cocycle vanishes when both X, Y have only negative (or positive) Fourier components, because the trace of B n dX B 2m dY and B 2m dX B 2n Y are zero when X, Y are upper (or lower) triangular matrices with respect to the grading. References 1. Carey, A.L., Crowley, D., Murray, M.: Principal bundles and the Dixmier-Douady class. Commun. Math. Phys. 193, 171–196 (1998) 2. Connes, A.: Noncommutative Geometry. New York-London: Academic Press, 1994 3. Freed, D., Hopkins, M., Teleman, C.: Twisted K-theory and Loop Group Representations. http://arxiv. org/abs/math/0312155v2[math.AT], 2005 4. Herrmann, R.: Gauge Invariance in Fractional Field Theories. Phys. Lett. A 372, 5515–5522 (2008) 5. Jaffe, A.: Quantum Harmonic Analysis and Geometric Invariants. Adv. Math. 143, 1–110 (1999) 6. Kac, V.G.: Infinite-Dimensional Lie Algebras. Cambridge: Cambridge University Press, 1990 7. Kac, V.G., Todorov, T.: Superconformal Current Algebras and their Unitary Representations. Commun. Math. Phys. 102, 337–347 (1985) 8. Landweber, G.: Multiplets of Representations and Kostant’s Dirac Operator for Equal Rank Loop Groups. Duke Math. J. 110(1), 121–160 (2001) 9. Lundberg, L.-E.: Quasi-free “Second Quantization”. Commun. Math. Phys. 50(2), 103–112 (1976) 10. Mickelsson, J., Rajeev, S.: Current Algebras in d + 1 Dimensions and Determinant Bundles over InfiniteDimensional Grassmannians. Commun. Math. Phys. 112(4), 653–661 (1987) 11. Mickelsson, J.: Current Algebras and Groups. New York-London: Plenum Press, 1989 12. Mickelsson, J.: Wodzicki Residue and Anomalies of Current Algebras. In: Integrable models and strings (Espoo, 1993), Lecture Notes in Phys., 436, Berlin: Springer, 1994, pp. 123–135 13. Mickelsson, J.: Gerbes, (Twisted) K-theory, and Supersymmetric WZW Model. In: Infinite dimensional groups and manifolds, IRMA Lect. Math. Theor. Phys., 5, Berlin: de Gruyter, 2004, pp. 93–107 14. Milnor, J.: On spaces having the homotopy type of a CW complex. Trans. Amer. Math. Soc. 90(2), 272–280 (1959) 15. Palais, R.: Homotopy theory of infinite dimensional manifolds. Topology 5, 1–16 (1966) 16. Pickrell, D.: On the Mickelsson-Faddeev Extension and Unitary Representations. Commun. Math. Phys. 123(4), 617–625 (1989) 17. Pressley, A., Segal, G.: Loop Groups. Oxford: Clarendon Press, 1986 18. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives, Theory and Applications. London: Gordon and Breach Science Publishers, 1993 Communicated by Y. Kawahigashi
Commun. Math. Phys. 299, 765–782 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1113-9
Communications in
Mathematical Physics
Uniqueness of Bounded Solutions for the Homogeneous Landau Equation with a Coulomb Potential Nicolas Fournier LAMA UMR 8050, Faculté de Sciences et Technologies, Université Paris Est, 61, Avenue du Général de Gaulle, 94010 Créteil Cedex, France. E-mail:
[email protected] Received: 3 September 2009 / Accepted: 8 April 2010 Published online: 15 August 2010 – © Springer-Verlag 2010
Abstract: We prove the uniqueness of bounded solutions for the spatially homogeneous Fokker-Planck-Landau equation with a Coulomb potential. Since the local (in time) existence of such solutions has been proved by Arsen’ev–Peskov (Z. Vycisl. Mat. i Mat. Fiz. 17:1063–1068, 1977), we deduce a local well-posedness result. The stability with respect to the initial condition is also checked. 1. Introduction We consider the spatially homogeneous Landau equation for a Coulomb potential. This equation of kinetic physics, also called the Fokker-Planck-Landau equation, has been derived from the Boltzmann equation by Landau. It describes the density f t (v) of particles with velocity v ∈ R3 at time t ≥ 0 in a spatially homogeneous dilute plasma: ∂t f t (v) =
3 1 ∂i ai j (v − v ∗ ) f t (v ∗ )∂ j f t (v) − f t (v)∂ ∗j f t (v ∗ ) dv ∗ . (1) 2 R3 i, j=1
Here ∂t = matrix
∂ ∂t , ∂i
=
∂ ∂vi
, ∂i∗ =
∂ ∂vi∗
and for z ∈ R3 , a(z) is the symmetric nonnegative
ai j (z) = |z|−3 (|z|2 δi j − z i z j ).
(2)
We refer to Villani [14,15], Alexandre-Villani [1] and the references therein for more information on this equation, which has been widely used in plasma physics. Let us mention hold a priori, that is for t ≥ that conservationof mass, momentum and kinetic energy 2 . We classically may assume 0, ϕ(v) f t (v)dv = ϕ(v) f 0 (v)dv, for ϕ(v) = 1, v, |v| without loss of generality that f 0 (v)dv = 1. Another fundamental estimate, that we will not use here, is the decay of entropy: f t (v) log f t (v)dv ≤ f 0 (v) log f 0 (v)dv for all t ≥ 0.
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Assume that in a dilute gas or plasma, particles collide by pairs, due to a repulsive force proportional to 1/r s , where r stands for the distance between the two particles. Then if s ∈ (2, ∞), the velocity distribution solves the corresponding Boltzmann equation [14,15]. But if s = 2, the Boltzmann equation is meaningless [14] and is often replaced by the Landau equation (1). However, there are also many mathematical works on the Landau equation where |z|−3 is replaced by |z|γ in (2), with γ = (s − 5)/(s − 1) ∈ [−3, 1). One usually speaks of hard potentials when γ ∈ (0, 1) (i.e. s > 5), Maxwell molecules when γ = 0 (i.e. s = 5), soft potentials when γ ∈ (−3, 0) (i.e. s ∈ (2, 5)), Coulomb potential when γ = −3 (i.e. s = 2). When γ > −3, the Landau equation can be seen as an approximation of the corresponding Boltzmann equation in the asymptotic of grazing collisions [14]. This can help to understand the effect of grazing collisions in the Boltzmann equation without cutoff and is also of interest numerically. But it seems that only the Landau equation with a Coulomb potential has considerable importance in plasma physics. The existence theory for the homogeneous Landau equation is quite complete. In [14], Villani has proved the global existence of weak solutions to the homogeneous Landau equation for all possible potentials γ ∈ [−3, 1), for any initial condition with finite mass, energy and entropy. He also showed in this paper that the solution to the Landau equation can be seen as the limit of a sequence of solutions to some suitable Boltzmann equations. It is worth noting that for γ ∈ [−2, 1), the tools used in [14] are quite classical. But for γ ∈ [−3, −2), Villani uses some very fine a priori estimates provided by the entropy dissipation. The paper of Alexandre-Villani [1] contains some existence results in the much more difficult inhomogeneous case. Uniqueness for the Landau equation is much less well-understood, even in the spatially homogeneous case. To our knowledge, this problem is still completely open in the realistic Coulomb case. Of course, uniqueness is very important, even from the physical point of view: if uniqueness is not holding, this means that the equation is not well-posed and thus that some additional physical conditions have to be added. Let us summarize the situations in which uniqueness for the homogeneous Landau equation is known to hold. The initial condition f 0 is always supposed to have finite mass and energy, f 0 (v)(1 + |v|2 )dv < ∞. One has global uniqueness when γ = 0, see Villani [13], when γ ∈ (0, 1) and f 02 (v)(1 + |v|q )dv < ∞ for some q > 5γ + 15, see Desvillettes-Villani [5], and when γ ∈ (−2, 0] and [ f 0 (v)|v|q + f 0 (v) log f 0 (v)]dv < ∞ for some q > γ 2 /(2 + γ ), see [8]. One has local (in time) uniqueness when γ ∈ (−3, −2] and f 0 ∈ L p for some p > 3/(3 + γ ), see [8]. The goal of this paper is to extend this final result to the case of a Coulomb potential γ = −3, showing local uniqueness for bounded initial conditions with finite mass and energy. As a matter of fact, we will show that for any T , uniqueness holds in the space L 1 ([0, T ], L ∞ (R3 )). But the only known result that provides existence of such solutions is that of Arsen’ev-Peskov [2]: for f 0 bounded, one can find T∗ ( f 0 ) > 0 and a solution to (1) lying in L ∞ ([0, T∗ ( f 0 )] × R3 ). Thus our uniqueness result is not so satisfying at the moment, since it concerns a functional space in which solutions are known to belong only for a bounded time interval. The Coulomb case is really more difficult than the case γ > −3, essentially because |z|−3 is not integrable near 0. Thus while the global scheme of the proof is the same as in [8], the central computations are much more delicate and borderline. It seems that Villani’s existence results can be extended to the case γ ∈ (−4, −3], see [14, p 284], but we are here really at the boundary of our possibilities.
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767
Let us finally mention that the starting point of our proof is the famous work of Tanaka [12], who proved the first uniqueness result for the Boltzmann equation without cutoff (for Maxwell molecules). We already used Tanaka’s approach to study uniqueness for the Boltzmann equation without cutoff for hard and soft potentials [6,7]. 2. Main Result Let P be the set of probability measures on R3 and P2 = { f ∈ P, m 2 ( f ) < ∞},
where
m2( f ) =
R3
|v|2 f (dv).
For a measurable family ( f t )t∈[0,T ] ⊂ P, we say that ( f t )t∈[0,T ] ∈ L ∞ ([0, T ], P2 )
if
sup m 2 ( f t ) < ∞.
[0,T ]
Observe that any reasonable solution to (1) belongs to L ∞ ([0, T ], P2 ), because m 2 ( f t ) = m 2 ( f 0 ). When f ∈ P has a bounded density, we say that f ∈ L ∞ ; we also denote by f its density and by || f ||∞ its L ∞ -norm. For a measurable family ( f t )t∈[0,T ] ⊂ P, we say that T 1 ∞ || f t ||∞ dt < ∞. ( f t )t∈[0,T ] ∈ L ([0, T ], L ) if 0
We denote by Cb2 0 to 2. For ϕ ∈
R3
functions ϕ : → R with bounded derivatives of order 3 ∈ R , we introduce
the set of C 2
Cb2
and
v, v ∗
Lϕ(v, v ∗ ) =
3 3 1 ai j (v − v ∗ )∂i2j ϕ(v) + bi (v − v ∗ )∂i ϕ(v), 2 i, j=1
where bi (z) =
3
(3)
i=1
∂ j ai j (z) = −2|z|−3 z i .
(4)
j=1
Definition 1. We say that ( f t )t∈[0,T ] is a weak solution to (1) starting from f 0 ∈ P2 if ( f t )t∈[0,T ] ∈ L ∞ ([0, T ], P2 ) ∩ L 1 ([0, T ], L ∞ ) and if for any ϕ ∈ Cb2 , any t ∈ [0, T ], t ϕ(v) f t (v)dv = ϕ(v) f 0 (v)dv + f t (v) f t (v ∗ )Lϕ(v, v ∗ )dvdv ∗ ds. R3
R3
0
R3 R3
(5) For ϕ ∈ Cb2 , one has |Lϕ(v, v ∗ )| ≤ Cϕ (|v −v ∗ |−1 +|v −v ∗ |−2 ) for some constant Cϕ . Thus (8) and our conditions on ( f t )t∈[0,T ] ensure us that all the terms are well-defined in (5). The weak formulation (5) is standard and can be found in [14, Eq. (36)]. We will widely use the Wasserstein distance W2 , defined for f, f˜ ∈ P2 , by
W22 ( f, f˜) = inf E[|V − V˜ |2 ], V ∼ f, V˜ ∼ f˜ |v − v| ˜ 2 R(dv, d v), ˜ R ∈ H( f, f˜) . = inf R 3 ×R 3
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N. Fournier
Here V ∼ f means that V is a R3 -valued random variable with law f and H( f, f˜) is the set of all probability measures on R3 × R3 with marginals f and f˜. The set (P2 , W2 ) is a Polish space and its topology is slightly stronger than the weak topology, see Villani [16, Theorem 7.12]. It is well-known [16, Chap. 1] that the infimum ˜ can find R ∈ H( f, f˜) and V ∼ f, V˜ ∼ f such that is reached: for f, f ∈ P2 , we 2 2 ˜ W2 ( f, f ) = R3 ×R3 |v − v| ˜ R(dv, d v) ˜ = E[|V − V˜ |2 ]. Our main result reads as follows. Theorem 2. Let T > 0. (i) For f 0 ∈ P2 , there is at most one weak solution to (1) starting from f 0 and belonging to L ∞ ([0, T ], P2 ) ∩ L 1 ([0, T ], L ∞ ). (ii) Assume that we have some weak solutions ( f t )t∈[0,T ] and ( f tn )t∈[0,T ] to (1), all T belonging to L ∞ ([0, T ], P2 ) ∩ L 1 ([0, T ], L ∞ ). If supn 0 || f tn ||∞ dt < ∞ and limn W2 ( f 0n , f 0 ) = 0, then limn sup[0,T ] W2 ( f tn , f t ) = 0. Arsen’ev-Peskov [2] have proved the following existence result. Let A > 0 be fixed. There exist some constants T A > 0 and C A depending only on A (with lim A 0 T A = +∞) such that for any f 0 ∈ L ∞ ∩ P2 with || f 0 ||∞ ≤ A, there exists a weak solution ( f t )t∈[0,T A ] to (1) satisfying sup[0,T A ] || f t ||∞ ≤ C A . Theorem 2 ensures us that this solution is unique and continuous with respect to the initial condition. The result in [2] d is based on the (formally easy) estimate dt || f t ||∞ ≤ C(1 + || f t ||2∞ ). In the next section, we establish some fundamental regularity estimates on the coefficients a and b of the Landau equation and we recall a well-known generalization of the Gronwall Lemma. The proof of Theorem 2 is handled in Sects. 4 and 5. In the whole paper, C stands for a universal constant, whose value changes from line to line. 3. Preliminaries We will study the Landau equation through a stochastic differential equation whose coefficients are b (recall (4)) and σ , defined for z ∈ R3 by ⎛ ⎞ z 2 −z 3 0 −3 z3 ⎠ . (6) σ (z) = |z| 2 ⎝ −z 1 0 0 z 1 −z 2 For all z ∈ R3 , one has σ (z).(σ (z))t = a(z), recall (2). Lemma 3. For any z, z˜ ∈ R3 ,
|σ (z) − σ (˜z )|2 ≤ C min |z − z˜ |2 (|z|−3 + |˜z |−3 ); |z|−1 + |˜z |−1 ,
|b(z) − b(˜z )| ≤ C min |z − z˜ |(|z|−3 + |˜z |−3 ); |z|−2 + |˜z |−2 .
Proof. First, we have |σ (z)| ≤ |z|−1/2 , whence |σ (z) − σ (˜z )|2 ≤ 2(|z|−1 + |˜z |−1 ). Next, |σ (z) − σ (˜z )| ≤ |z|−3/2 − |˜z |−3/2 .|z| + |z − z˜ |.|˜z |−3/2 ≤
3 |z|.|z − z˜ | max{|z|−5/2 , |˜z |−5/2 } + |z − z˜ |(|z|−3/2 + |˜z |−3/2 ). 2
Landau Equation with a Coulomb Potential
769
By symmetry, we deduce that
3 min{|z|, |˜z |} max{|z|−5/2 , |˜z |−5/2 } + |z|−3/2 + |˜z |−3/2 2 5 ≤ |z − z˜ | |z|−3/2 + |˜z |−3/2 . 2
|σ (z) − σ (˜z )| ≤ |z − z˜ |
We also have |b(z)| ≤ 2|z|−2 , so that |b(z) − b(˜z )| ≤ 2(|z|−2 + |˜z |−2 ). Finally, |b(z) − b(˜z )| ≤ 2 |z|−3 − |˜z |−3 .|z| + 2|z − z˜ |.|˜z |−3 ≤ 6|z| max{|z|−4 , |˜z |−4 }|z − z˜ | + 2|z − z˜ |(|z|−3 + |˜z |−3 ), whence by symmetry, |b(z) − b(˜z )| ≤ 6 min{|z|, |˜z |} max{|z|−4 , |˜z |−4 }|z − z˜ | + 2|z − z˜ |(|z|−3 + |˜z |−3 ) ≤ 8|z − z˜ |(|z|−3 + |˜z |−3 ), which ends the proof.
Next, we state some easy estimates of constant use in the paper. Lemma 4. Let α ∈ (−3, 0]. There is a constant Cα such that for all g ∈ P ∩ L ∞ , all ∈ (0, 1], |v − v ∗ |α g(v ∗ )dv ∗ ≤ 1 + Cα ||g||∞ , (7) sup
v∈R3 R3
R3 R3
sup v,w∈R3
|v − v ∗ |α g(v)g(v ∗ )dvdv ∗ ≤ 1 + Cα ||g||∞ ,
|v−v ∗ |≤
|w − v ∗ |α g(v ∗ )dv ∗ ≤ Cα ||g||∞ 3+α .
There is a constant C such that for all g ∈ P ∩ L ∞ , all ∈ (0, 1], |v − v ∗ |−3 g(v ∗ )dv ∗ ≤ 1 + C||g||∞ log(1/). sup v∈R3 |v−v ∗ |≥
(8) (9)
(10)
Proof. Below, the variable u belongs to R3 . Recall that there is a constant C such that for ∈ (0, 1], |u|−3 du = C log(1/), (11) ≤|u|≤1
and that for α ∈ (−3, 0], there is a constant Cα such that for all ∈ (0, 1], all u 0 ∈ R3 , |u + u 0 |α du ≤ Cα 3+α . (12) |u|≤
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N. Fournier
Since g has mass 1 and α ∈ (−3, 0], for any v ∈ R3 ,
∗ α
R3
∗
∗
g(v )dv + |v − v ∗ |α g(v ∗ )dv ∗ |v−v ∗ |≤1 ≤ 1 + ||g||∞ |v − v ∗ |α dv ∗ |v−v ∗ |≤1 = 1 + ||g||∞ |u|α du,
|v − v | g(v )dv ≤
∗
∗
|v−v ∗ |≥1
|u|≤1
whence (7) due to (12) with = 1. Inequality (8) follows from (7) because g has mass 1. Next (9) is deduced from (12): for v, w ∈ R3 and ∈ (0, 1], |v−v ∗ |≤
|w − v ∗ |α g(v ∗ )dv ∗ ≤ ||g||∞
|v−v ∗ |≤
= ||g||∞
|u|≤
|w − v ∗ |α dv ∗
|u + (v − w)|α du.
Finally, for any v ∈ R3 , |v−v ∗ |≥
|v − v ∗ |−3 g(v ∗ )dv ∗ ≤
|v−v ∗ |≥1
≤ 1 + ||g||∞
from which (10) follows using (11).
g(v ∗ )dv ∗ + ||g||∞ ≤|u|≤1
≤|v−v ∗ |≤1
|v − v ∗ |−3 dv ∗
|u|−3 du,
We also consider the increasing continuous function : [0, ∞) → R+ defined by (x) = x(1 − 1{x≤1} log x).
(13)
The following remark will allow us to apply the Jensen inequality. Remark 5. We can find a concave increasing continuous function : R+ → R+ such that for all x ≥ 0, (x)/2 ≤ (x) ≤ 2 (x). Proof. Choose (x) = x(1 − log x) for x ∈ [0, 1/2] and (x) = x log 2 + 1/2 for x ≥ 1/2. The two next lemmas contain the fundamental computations of the paper. Lemma 6. For any g ∈ P ∩ L ∞ , for all v, v˜ ∈ R3 , R3
|σ (v − v ∗ ) − σ (v˜ − v ∗ )|2 g(v ∗ )dv ∗ ≤ C(1 + ||g||∞ ) (|v − v| ˜ 2 ),
(14)
|b(v − v ∗ ) − b(v˜ − v ∗ )|g(v ∗ )dv ∗ ≤ C(1 + ||g||∞ ) (|v − v|). ˜
(15)
R3
Landau Equation with a Coulomb Potential
771
Proof. We denote by I the left hand side of (14). Using Lemma 3, |σ (v − v ∗ ) − σ (v˜ − v ∗ )|2
≤ C min |v − v| ˜ 2 (|v − v ∗ |−3 + |v˜ − v ∗ |−3 ); |v − v ∗ |−1 + |v˜ − v ∗ |−1 . Thus
I ≤ C1{|v−v|≥1} ˜
R3
(|v − v ∗ |−1 + |v˜ − v ∗ |−1 )g(v ∗ )dv ∗
+ C1{|v−v|≤1} ˜ + C1{|v−v|≤1} ˜
R3
R3
+ C1{|v−v|≤1} ˜
R3
1{|v−v ∗ |≥|v−v| ˜ 2 (|v − v ∗ |−3 + |v˜ − v ∗ |−3 )g(v ∗ )dv ∗ ˜ 2 ,|v−v ˜ ∗ |≥|v−v| ˜ 2 } |v − v| ∗ −1 1{|v−v ∗ |≤|v−v| + |v˜ − v ∗ |−1 )g(v ∗ )dv ∗ ˜ 2 } (|v − v | ∗ −1 1{|v−v + |v˜ − v ∗ |−1 )g(v ∗ )dv ∗ ˜ ∗ |≤|v−v| ˜ 2 } (|v − v |
=: C(I1 + I2 + I3 + I4 ).
First, (7) with α = −1 implies that ≤ C(1 + ||g||∞ ) (|v − v| ˜ 2 ). I1 ≤ C(1 + ||g||∞ )1{|v−v|≥1} ˜ Next, using (10) with = |v − v| ˜ 2 yields 2 |v − v| ˜ I2 ≤ 1{|v−v|≤1} ˜ +
|v − v ∗ |−3 g(v ∗ )dv ∗ ∗ −3 ∗ ∗ |v˜ − v | g(v )dv |v−v ∗ |≥|v−v| ˜2
|v−v ˜ ∗ |≥|v−v| ˜2
|v − v| ˜ 2 (2 + C||g||∞ log(1/|v − v| ˜ 2 )) ≤ 1{|v−v|≤1} ˜ ≤ C(1 + ||g||∞ )1{|v−v|≤1} |v − v| ˜ 2 (1 − log(|v − v| ˜ 2 )) ˜ ≤ C(1 + ||g||∞ ) (|v − v| ˜ 2 ). Finally, we deduce from (9) with α = −1 and = |v − v| ˜ 2 that (|v − v| ˜ 2 )3−1 ≤ C||g||∞ |v − v| ˜ 2 ≤ C||g||∞ (|v − v| ˜ 2 ). I3 + I4 ≤ C||g||∞ 1{|v−v|≤1} ˜ We now denote by J the left hand side of (15). By Lemma 3, |b(v − v ∗ ) − b(v˜ − v ∗ )|
≤ C min |v − v|(|v ˜ − v ∗ |−3 + |v˜ − v ∗ |−3 ); |v − v ∗ |−2 + |v˜ − v ∗ |−2 . Thus
(|v − v ∗ |−2 + |v˜ − v ∗ |−2 )g(v ∗ )dv ∗ + C1{|v−v|≤1} 1{|v−v ∗ |≥|v−v| ˜ − v ∗ |−3 + |v˜ − v ∗ |−3 )g(v ∗ )dv ∗ ˜ ˜ 2 ,|v−v ˜ ∗ |≥|v−v| ˜ 2 } |v − v|(|v R3 ∗ −2 + C1{|v−v|≤1} 1{|v−v ∗ |≤|v−v| + |v˜ − v ∗ |−2 )g(v ∗ )dv ∗ ˜ ˜ 2 } (|v − v | R3 ∗ −2 + C1{|v−v|≤1} 1{|v−v + |v˜ − v ∗ |−2 )g(v ∗ )dv ∗ ˜ ˜ ∗ |≤|v−v| ˜ 2 } (|v − v |
J ≤ C1{|v−v|≥1} ˜
R3
R3
=: C(J1 + J2 + J3 + J4 ).
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N. Fournier
Using (7) with α = −2, we get J1 ≤ C(1 + ||g||∞ )1{|v−v|≥1} ≤ C(1 + ||g||∞ ) (|v − v|). ˜ ˜ Next, (10) with = |v − v| ˜ 2 yields
+
|v−v ˜ ∗ |≥|v−v| ˜2
|v − v ∗ |−3 g(v ∗ )dv ∗ ∗ −3 ∗ ∗ |v˜ − v | g(v )dv
J2 ≤ 1{|v−v|≤1} |v − v| ˜ ˜
|v−v ∗ |≥|v−v| ˜2
|v − v|[2 ˜ + C||g||∞ log(1/|v − v| ˜ 2 )] ≤ 1{|v−v|≤1} ˜ ≤ C(1 + ||g||∞ )1{|v−v|≤1} |v − v|[1 ˜ − log(|v − v|)] ˜ ˜ ≤ C(1 + ||g||∞ ) (|v − v|). ˜ Finally, J3 + J4 ≤ C||g||∞ 1{|v−v|≤1} (|v − v| ˜ 2 )3−2 ≤ C||g||∞ (|v − v|) ˜ by (9) with ˜ 2 α = −2 and = |v − v| ˜ . ˜ Then Lemma 7. Consider g, g˜ ∈ P2 ∩ L ∞ and Q, R ∈ H(g, g). ∗ |v − v|.|b(v ˜ − v ∗ ) − b(v˜ − v˜ ∗ )|Q(dv, d v)R(dv ˜ , d v˜ ∗ ) 3 3 3 3 R ×R R ×R ≤ C(1 + ||g + g|| ˜ ∞) |v − v| ˜ 2 Q(dv, d v) ˜ R 3 ×R 3 + |v ∗ − v˜ ∗ |2 R(dv ∗ , d v˜ ∗ ) . R 3 ×R 3
(16)
Proof. We denote by K the left hand side of (16) and by δ(v, v, ˜ v ∗ , v˜ ∗ ) = |v − v|. ˜ ∗ ∗ |b(v − v ) − b(v˜ − v˜ )|. Due to Lemma 3, δ is smaller than ˜ + |v ∗ − v˜ ∗ |)(|v − v ∗ |−3 C(|v − v| ˜ + |v ∗ − v˜ ∗ |) min (|v − v|
+ |v˜ − v˜ ∗ |−3 ); |v − v ∗ |−2 + |v˜ − v˜ ∗ |−2 . Hence we can write ∗ −v˜ ∗ |≥1} (|v − v| δ(v, v, ˜ v ∗ , v˜ ∗ ) ≤ C1{|v−v|+|v ˜ + |v ∗ − v˜ ∗ |)(|v − v ∗ |−2 + |v˜ − v˜ ∗ |−2 ) ˜ ∗ −v˜ ∗ |≤1} 1{|v−v ∗ |≥|v−v| ˜ 2 (|v − v ∗ |−3 + |v˜ − v˜ ∗ |−3 ) + C1{|v−v|+|v ˜ ˜ 4 ,|v− ˜ v˜ ∗ |≥|v−v| ˜ 4 } |v − v|
∗ ∗ 2 ∗ −3 ∗ −v˜ ∗ |≤1} 1{|v−v ∗ |≥|v−v| + |v˜ − v˜ ∗ |−3 ) + C1{|v−v|+|v ˜ ˜ 4 ,|v− ˜ v˜ ∗ |≥|v−v| ˜ 4 } |v − v˜ | (|v − v | ∗ −v˜ ∗ |≤1} 1{|v−v ∗ |≤|v−v| + C1{|v−v|+|v ˜ + |v ∗ − v˜ ∗ |)(|v − v ∗ |−2 + |v˜ − v˜ ∗ |−2 ) ˜ ˜ 4 } (|v − v| ∗ −v˜ ∗ |≤1} 1{|v− ˜ + |v ∗ − v˜ ∗ |)(|v − v ∗ |−2 + |v˜ − v˜ ∗ |−2 ) + C1{|v−v|+|v ˜ ˜ v˜ ∗ |≤|v−v| ˜ 4 } (|v − v|
=: C
5 1
Thus K ≤C First,
δi (v, v, ˜ v ∗ , v˜ ∗ ).
5 1
∗ , d v˜ ∗ ). K i , where K i = R3 ×R3 R3 ×R3 δi (v, v, ˜ v ∗ , v˜ ∗ )Q(dv, d v)R(dv ˜
δ1 (v, v, ˜ v ∗ , v˜ ∗ ) ≤ (|v − v| ˜ + |v ∗ − v˜ ∗ |)2 (|v − v ∗ |−2 + |v˜ − v˜ ∗ |−2 ) ≤ 2(|v − v| ˜ 2 + |v ∗ − v˜ ∗ |2 )(|v − v ∗ |−2 + |v˜ − v˜ ∗ |−2 ).
Landau Equation with a Coulomb Potential
773
As a consequence, |v − v| ˜ 2 Q(dv, d v) ˜ (|v − v ∗ |−2 + |v˜ − v˜ ∗ |−2 )R(dv ∗ , d v˜ ∗ ) K1 ≤ 2 R 3 ×R 3 R 3 ×R 3 |v ∗ − v˜ ∗ |2 R(dv ∗ , d v˜ ∗ ) (|v − v ∗ |−2 + |v˜ − v˜ ∗ |−2 )Q(dv, d v). ˜ +2 R 3 ×R 3
R 3 ×R 3
=: 2K 1,1 + 2K 1,2 .
Since now R has marginals g and g, ˜ we deduce from (7) with α = −2 that |v − v| ˜ 2 Q(dv, d v) ˜ K 1,1 = R 3 ×R 3 ∗ −2 ∗ ∗ ∗ −2 ∗ ∗ × |v − v | g(v )dv + |v˜ − v˜ | g( ˜ v˜ )d v˜ R3 R3 |v − v| ˜ 2 Q(dv, d v)[1 ˜ + C||g||∞ + 1 + C||g|| ˜ ∞] ≤ 3 3 R ×R |v − v| ˜ 2 Q(dv, d v) ˜ ≤ C(1 + ||g + g|| ˜ ∞) R 3 ×R 3 ≤ C(1 + ||g + g|| ˜ ∞ ) |v − v| ˜ 2 Q(dv, d v) ˜ . R 3 ×R 3
Similarly,
K 1,2 ≤ C(1 + ||g + g|| ˜ ∞ )
R 3 ×R 3
∗
∗ 2
∗
∗
|v − v˜ | R(dv , d v˜ ) .
Next, δ2 (v, v, ˜ v ∗ , v˜ ∗ ) ≤ 1{|v−v|≤1} 1{|v−v ∗ |≥|v−v| ˜ 2 (|v − v ∗ |−3 + |v˜ − v˜ ∗ |−3 ). ˜ ˜ 4 ,|v− ˜ v˜ ∗ |≥|v−v| ˜ 4 } |v − v|
Thus, (10) with = |v − v| ˜ 4 yields 2 Q(dv, d v)1 ˜ {|v−v|≤1} |v − v| ˜ R(dv ∗ , d v˜ ∗ ) K2 ≤ ˜ R 3 ×R 3 R 3 ×R 3 ∗ −3 ∗ −3 |v − v | + 1 | v ˜ − v ˜ | × 1{|v−v ∗ |≥|v−v| 4 ∗ 4 ˜ } {|v− ˜ v˜ |≥|v−v| ˜ } = Q(dv, d v)1 ˜ {|v−v|≤1} |v − v| ˜2 ˜ R 3 ×R 3 × |v − v ∗ |−3 g(v ∗ )dv ∗ + |v˜ − v˜ ∗ |−3 g( ˜ v˜ ∗ )d v˜ ∗ |v−v ∗ |≥|v−v| ˜4 |v− ˜ v˜ ∗ |≥|v−v| ˜4 ≤ Q(dv, d v)1 ˜ {|v−v|≤1} |v − v| ˜2 ˜ R 3 ×R 3 × 1 + C||g||∞ log(1/|v − v| ˜ 4 ) + 1 + C||g|| ˜ ∞ log(1/|v − v| ˜ 4) ≤ C(1 + ||g + g|| ˜ ∞) Q(dv, d v)1 ˜ {|v−v|≤1} |v − v| ˜ 2 [1 + log(1/|v − v| ˜ 2 )] ˜ 3 3 R ×R (|v − v| ˜ 2 )Q(dv, d v). ˜ ≤ C(1 + ||g + g|| ˜ ∞) R 3 ×R 3
774
N. Fournier
Remark 5 and the Jensen inequality allow us to conclude that 2 ˜ ∞ ) |v − v| ˜ Q(dv, d v) ˜ . K 2 ≤ C(1 + ||g + g|| R 3 ×R 3
The third term K 3 is bounded symmetrically. For the fourth term, we first notice that ∗ −2 ˜ v ∗ , v˜ ∗ ) ≤ 1{|v−v|≤1} 1{|v−v ∗ |≤|v−v| + |v˜ − v˜ ∗ |−2 ], δ4 (v, v, ˜ ˜ 4 } [|v − v |
whence K4 ≤
R 3 ×R 3
Q(dv, d v)1 ˜ {|v−v|≤1} ˜
|v−v ∗ |≤|v−v| ˜4
[|v − v ∗ |−2 + |v˜ − v˜ ∗ |−2 ]R(dv ∗ , d v˜ ∗ ).
But for all v, v, ˜ using (9) with α = −2 and = |v − v| ˜ 4, |v − v ∗ |−2 R(dv ∗ , d v˜ ∗ ) |v−v ∗ |≤|v−v| ˜4 |v − v ∗ |−2 g(v ∗ )dv ∗ ≤ C||g||∞ |v − v| ˜ 4. = |v−v ∗ |≤|v−v| ˜4
Using now the Hölder inequality (with p = 5 and q = 5/4), then (9) with α = 0, = |v − v| ˜ 4 and finally (7) with α = −5/2, |v˜ − v˜ ∗ |−2 R(dv ∗ , d v˜ ∗ ) |v−v ∗ |≤|v−v| ˜4
≤
∗
|v−v ∗ |≤|v−v| ˜4
=
|v−v ∗ |≤|v−v| ˜4
∗
1/5
R(dv , d v˜ ) g(v ∗ )dv ∗
1/5 R3
R 3 ×R 3
∗ −5/2
|v˜ − v˜ |
∗
∗
4/5
R(dv , d v˜ )
|v˜ − v˜ ∗ |−5/2 g( ˜ v˜ ∗ )d v˜ ∗
4/5
1/5 ≤ C||g||∞ [|v − v| ˜ 4 ]3 (1 + C||g|| ˜ ∞ )4/5 ≤ C(1 + ||g + g|| ˜ ∞ )|v − v| ˜ 12/5 , 1/5
because ||g||∞ (1 + ||g|| ˜ ∞ )4/5 ≤ 1 + ||g||∞ + ||g|| ˜ ∞ ≤ 1 + 2||g + g|| ˜ ∞ . We thus have shown that K 4 ≤ C(1 + ||g + g|| ˜ ∞) Q(dv, d v)1 ˜ {|v−v|≤1} (|v − v| ˜ 4 + |v − v| ˜ 12/5 ) ˜ R 3 ×R 3 Q(dv, d v)|v ˜ − v| ˜2 ≤ C(1 + ||g + g|| ˜ ∞) R 3 ×R 3 2 ≤ C(1 + ||g + g|| ˜ ∞ ) |v − v| ˜ Q(dv, d v) ˜ . R 3 ×R 3
The last term K 5 is treated symmetrically, which ends the proof.
We conclude this section by recalling a generalization of the Gronwall Lemma of which the proof can be found in Chemin [4, Lemma 5.2.1, p. 89].
Landau Equation with a Coulomb Potential
775
T Lemma 8. Let T > 0 and γ : [0, T ] → R+ satisfy 0 γ (s)ds < ∞. Recall that 1 was defined by (13) and set M(x) = x (1/ (y))dy for x > 0. Consider a bounded measurablefunction ρ : [0, T ] → R+ such that, for some a ≥ 0, for all t ∈ [0, T ], t ρ(t) ≤ a + 0 γ (s) (ρ(s))ds. (i) If a = 0, then ρ(t) = 0 for all t ∈ [0, t T ]. (ii) If a > 0, then M(a) − M(ρ(t)) ≤ 0 γ (s)ds for all t ∈ [0, T ]. 4. An Integral Inequality Theorem 2 will be easily deduced, in the next section, from Lemma 8 and the following result. Recall that was defined in (13). Theorem 9. There is a constant C such that for any pair of weak solutions ( f t )t∈[0,T ] and ( f˜t )t∈[0,T ] to (1), there is a bounded function ρ : [0, T ] → R+ satisfying, for all t ∈ [0, T ], t 2 2 ˜ ˜ W2 ( f t , f t ) ≤ ρ(t) and ρ(t) ≤ W2 ( f 0 , f 0 ) + C (1 + || f s + f˜s ||∞ ) (ρ(s))ds. 0
From now on, T > 0 and the two weak solutions ( f t )t∈[0,T ] , ( f˜t )t∈[0,T ] to (1), both belonging to L ∞ ([0, T ], P2 ) ∩ L 1 ([0, T ], L ∞ ) are fixed. We follow closely the scheme of proof of [8]: first, we introduce two coupled Landau stochastic processes, the first one associated with f , the second one associated with f˜, in such a way that they remain as close to each other as possible. The probabilistic interpretation of the Landau equation we use here has been introduced by Funaki [9], Guérin [10] and is inspired by the work of Tanaka [12]. For all s ∈ [0, T ], we denote by Rs ∈ H( f s , f˜s ) the (unique) probability2 measure on 3 ˜ Rs (dv, d v). ˜ R × R3 with marginals f s and f˜s such that W22 ( f s , f˜s ) = R3 ×R3 |v − v| On some probability space, we consider a three-dimensional white noise W (dv, d v, ˜ ds) on R3 × R3 × [0, T ] with covariance measure Rs (dv, d v)ds. ˜ This means that W = (W1 , W2 , W3 ), where W1 , W2 and W3 are three independent white noises on R3 ×R3 ×[0, T ] with covariance measure Rs (dv, d v)ds, ˜ see Walsh [17] for definitions. We also need two R3 -valued random variables V0 , V˜0 with laws f 0 , f˜0 , independent of W , such that W22 ( f 0 , f˜0 ) = E[|V0 − V˜0 |2 ]. We consider the two following R3 -valued stochastic differential equations. t t σ (Vs − v)W (dv, d v, ˜ ds) + b(Vs − v) f s (v)dvds, (17) Vt = V0 + 0 R 3 ×R 3 0 R3 t t σ (V˜s − v)W ˜ (dv, d v, ˜ ds) + b(V˜s − v) ˜ f˜s (v)d ˜ vds, ˜ (18) V˜t = V˜0 + 0
R 3 ×R 3
0
R3
b, σ being defined by (4) and (6). We set Ft = σ {V0 , V˜0 , W ([0, s] × A), s ∈ [0, t], A ∈ B(R3 × R3 )}. Given the solution ( f t )t∈[0,T ] , (17) can be seen as a classical stochastic differential equation. Indeed, a simple computation of the quadratic variation of the martingale part shows that (17) is rewritten: t t Vt = V0 + σ fs (Vs )d Bs + b fs (Vs )ds. 0
0
776
N. Fournier
Here b fs (v) = v ∗ ∈R3 b(v − v ∗ ) f s (v ∗ )dv ∗ , σ fs (v) is a square root of v ∗ ∈R3 a(v − v ∗ ) f s (v ∗ )dv ∗ and (Bt )t∈[0,T ] is a standard 3-dimensional Brownian motion. Once this is seen, (17) is nothing but the standard probabilistic interpretation of the Landau equation. The same argument applies to (18). The white noise allows us to couple the two Brownian motions (the one used in (17) and the one used in (18)) in such a way that the two solutions (Vt )t∈[0,T ] and (V˜t )t∈[0,T ] remain close to each other. Proposition 10. (i) There exists a unique pair (Vt )t∈[0,T ] , (V˜t )t∈[0,T ] of continuous (Ft )t∈[0,T ] -adapted processes solving (17) and (18). (ii) For all t ∈ [0, T ], L(Vt ) = f t and L(V˜t ) = f˜t . We admit this proposition for a while. Proof of Theorem 9. By Proposition 10-(ii), we have W22 ( f t , f˜t ) ≤ E[|Vt − V˜t |2 ]. We thus compute this last quantity carefully. The marginals of Rs being f s and f˜s , we may rewrite (17) and (18) as
Vt = V0 + V˜t = V˜0 +
t 0
R 3 ×R 3
0
R 3 ×R 3
t
σ (Vs − v)W (dv, d v, ˜ ds) + σ (V˜s − v)W ˜ (dv, d v, ˜ ds) +
t 0
R 3 ×R 3
0
R 3 ×R 3
t
b(Vs − v)Rs (dv, d v)ds, ˜ b(V˜s − v)R ˜ s (dv, d v)ds. ˜
Using the Itô formula and taking expectations, we obtain E[|Vt − V˜t |2 ] = E[|V0 − V˜0 |2 ] 3 t + i,l=1 0 t
R 3 ×R 3
E
2
σil (Vs − v) − σil (V˜s − v) ˜
Rs (dv, d v)ds ˜
E b(Vs − v) − b(V˜s − v) ˜ .(Vs − V˜s ) Rs (dv, d v)ds ˜ 0 R 3 ×R 3 t t As ds + 2 Bs ds. = W22 ( f 0 , f˜0 ) + +2
0
0
Let Q s (dv, d v) ˜ be the law of the couple (Vs , V˜s ). Using (16) and that Rs , Q s ∈ ˜ H( f s , f s ), |Bs | ≤
R 3 ×R 3 R 3 ×R 3
|v − v|.|b(v ˜ − v ∗ ) − b(v˜ − v˜ ∗ )|Q s (dv, d v)R ˜ s (dv ∗ , d v˜ ∗ )
≤ C(1 + || f s + f˜s ||∞ ) |v − v| ˜ 2 Q s (dv, d v) ˜ + × R 3 ×R 3
R 3 ×R 3
|v ∗ − v˜ ∗ |2 Rs (dv ∗ , d v˜ ∗ ) .
Landau Equation with a Coulomb Potential
777
Next, using (14), As = |σ (v − v ∗ ) − σ (v˜ − v˜ ∗ )|2 Q s (dv, d v)R ˜ s (dv ∗ , d v˜ ∗ ) R 3 ×R 3 R 3 ×R 3 |σ (v − v ∗ ) − σ (v − v˜ ∗ )|2 Q s (dv, d v)R ˜ s (dv ∗ , d v˜ ∗ ) ≤2 3 3 3 3 R ×R R ×R |σ (v − v˜ ∗ ) − σ (v˜ − v˜ ∗ )|2 Q s (dv, d v)R ˜ s (dv ∗ , d v˜ ∗ ) +2 3 3 3 3 R ×R R ×R |σ (v − v ∗ ) − σ (v − v˜ ∗ )|2 f s (v)dv Rs (dv ∗ , d v˜ ∗ ) =2 R 3 ×R 3 R 3 |σ (v − v˜ ∗ ) − σ (v˜ − v˜ ∗ )|2 f˜s (v˜ ∗ )d v˜ ∗ Q s (dv, d v) ˜ +2 R 3 ×R 3 R 3 (|v ∗ − v˜ ∗ |2 )Rs (dv ∗ , d v˜ ∗ ) ≤ C(1 + || f s ||∞ ) R 3 ×R 3 (|v − v| ˜ 2 )Q s (dv, d v). ˜ + C(1 + || f˜s ||∞ ) R 3 ×R 3
Due to Remark 5 and the Jensen inequality, As ≤ C(1 + || f s + f˜s ||∞ ) 2 |v − v| ˜ Q s (dv, d v) ˜ + × R 3 ×R 3
∗
R 3 ×R 3
We now set ρ(t) := E[|Vs − V˜s |2 ]. Since 2 ∗ ∗ 2 ∗ ∗ ˜ W2 ( f s , f s ) = |v − v˜ | Rs (dv , d v˜ ) ≤ R 3 ×R 3
∗ 2
∗
∗
|v − v˜ | Rs (dv , d v˜ )
R 3 ×R 3
.
|v − v| ˜ 2 Q s (dv, d v) ˜ = ρ(t),
and since is increasing, we have shown that t 2 2 ˜ ˜ W2 ( f t , f t ) ≤ ρ(t) ≤ W2 ( f 0 , f 0 ) + C (1 + || f s + f˜s ||∞ ) (ρ(s)) ds. 0
It only remains to check that ρ is bounded on [0, T ]. But ρ(t) ≤ 2E[|Vt |2 ] + 2E[|V˜t |2 ] = 2m 2 ( f t ) + 2m 2 ( f˜t ) by Proposition 10-(ii) and ( f t )t∈[0,T ] , ( f˜t )t∈[0,T ] ∈ L ∞ ([0, T ], P2 ) by assumption. It remains to give the Proof of Proposition 10. We only check the results for (17), the study of (18) being the same. Step 1. For x0 ∈ R3 and for X = (X t )t∈[0,T ] a R3 -valued progressively measurable process, we introduce the R3 -valued progressively measurable process ( (x0 , X )t )t∈[0,T ] defined by t t (x0 , X )t = x0 + σ (X s − v)W (dv, d v, ˜ ds) + b(X s − v) f s (v)dvds. 0
R 3 ×R 3
0
R3
778
N. Fournier
The goal of this step is to prove that ( (x0 , X )t )t∈[0,T ] is automatically continuous and that T 2 T 2 2 E sup | (x0 , X )t | ≤ C |x0 | + || f s ||∞ ds + || f s ||∞ ds , [0,T ]
0
0
which is finite thanks to the conditions imposed on f . We observe, since the first marginal of Rs is f s and using (7) with α = −1, that a.s.,
T
R 3 ×R 3
0
T
|σ (X s − v)|2 Rs (dv, d v)ds ˜ ≤
R3
0 T
≤
0
|X s − v|−1 f s (v)dvds
C(1 + || f s ||∞ )ds < ∞.
From (7) with α = −2, T |b(X s − v)| f s (v)dvds R3
0
≤2
T
R3
0
|X s − v|
−2
f s (v)dvds ≤ 0
T
C(1 + || f s ||∞ )ds < ∞.
The a.s. continuity of (x0 , X ) on [0, T ] follows and the mean square estimate is easily deduced from the Doob inequality. Step 2. We now aim to show that for t ∈ [0, T ], for X, Y two progressively measurable processes,
t := E | (x0 , X )t − (x0 , Y )t |2 t
≤ C (1 + || f s ||∞ ) (E[| (x0 , X )s − (x0 , Y )s |2 ]) + (E[|X s − Ys |2 ]) ds, 0
where was defined in (13). Using the Itô formula and taking expectations, we derive
t =
3 t 3 3 i,l=1 0 R ×R t
+2 0
R 3 ×R 3
E (σil (X s − v) − σil (Ys − v))2 Rs (dv, d v)ds ˜
E [(b(X s − v) − b(Ys −v)) .( (x0 , X )s − (x0 , Y )s )] f s (v)dvds.
Due to (14) and (15) and since the first marginal of Rs is f s , t 2
t ≤ C E |σ (X s − v) − σ (Ys − v)| f s (v)dv ds 0 R3 t +C E | (x 0 , X )s − (x 0 , Y )s | |b(X s − v) − b(Ys − v)| f s (v)dv ds R3 0 t ≤C (1 + || f s ||∞ )E (|X s − Ys |2 ) + | (x0 , X )s − (x0 , Y )s | (|X s − Ys |) ds. 0
Landau Equation with a Coulomb Potential
779
But since x → (x) is non-decreasing and x (x) ≤ (x 2 ), one has, for all u, v ≥ 0, u (v) ≤ 1{u≤v} v (v) + 1{u≥v} u (u) ≤ (u 2 ) + (v 2 ). We thus obtain t (1 + || f s ||∞ )E (|X s − Ys |2 ) + (| (x0 , X )s − (x0 , Y )s |2 ) ds
t ≤ C 0 t
≤C (1 + || f s ||∞ ) (E[|X s − Ys |2 ]) + (E[| (x0 , X )s − (x0 , Y )s |2 ]) ds, 0
the last inequality following from Remark 5 and the Jensen inequality. Step 3. We now check the uniqueness for (17). Consider two solutions V = (V0 , V ) and V˜ = (V0 , V˜ ), and set ρ(t) = E[|Vt − V˜t |2 ], which is bounded on [0, T ] due to Step 1. Using Step 2, we deduce that t ρ(t) ≤ γ (s) (ρ(s))ds, 0
where γ (s) = C(1 + || f s ||∞ ) ∈ L 1 ([0, T ]). Lemma 8 yields that ρ(t) = 0, whence Vt = V˜t a.s., for all t ∈ [0, T ]. The continuity obtained in Step 1 guarantees us that a.s., (Vt )t∈[0,T ] = (V˜t )t∈[0,T ] . Step 4. We now prove the existence of a solution to (17) using a Picard iteration. We define V 0 by Vt0 = V0 and then by induction V n+1 = (V0 , V n ). We then set ρn,k (t) = sup[0,t] E[|Vsn+k − Vsn |2 ], which is uniformly bounded on [0, T ] due to Step 1. Step 2 yields t ρn+1,k (t) ≤ γ (s)[ (ρn+1,k (s)) + (ρn,k (s))]ds, 0
where γ (s) = C(1 + || f s ||∞ ) ∈ L 1 ([0, T ]). Thus for ρn (t) := supk ρn,k (t), we get t ρn+1 (t) ≤ 0 γ (s)[ (ρn (s)) + (ρn+1 (s))]ds. Finally, setting ρ(t) := lim supn ρn (t), t we deduce that ρ(t) ≤ 2 0 γ (s) (ρ(s))ds, whence ρ(T ) = 0 by Lemma 8. We have proved that lim sup sup sup E[|Vtn+k − Vtn |2 ] = 0, n
k [0,T ]
so that the sequence (Vtn )t∈[0,T ] is Cauchy in L ∞ ([0, T ], L 2 ()). Hence there is a process (Vt )t∈[0,T ] such that limn sup[0,T ] E[|Vt − Vtn |2 ] = 0. To conclude this step, it suffices to prove that κn (t) := E[| (V0 , V n )t − (V0 , V )t |2 ] tends to 0 for each t ∈ [0, T ]. This will allow us to pass to the limit in V n+1 = (V0 , V n ) and to get V = (V0 , V ), so that V will solve (17). The a.s. continuity of V will then be deduced from Step 1. We set n := sup[0,T ] E[ (|Vsn − Vs |2 )], which tends to 0 by Remark 5 and the Jensen inequality. Using Step 2 again, we immediately obtain, for t ∈ [0, T ], t κn (t) ≤ γ (s)(n + (κn (s)))ds. 0
780
N. Fournier
For t κ(t) = lim supn κn (t) (which is bounded due to Step 1), we get κ(t) ≤ 0 γ (s) (κ(s))ds. Lemma 8 thus yields κ(t) = 0 for all t ∈ [0, T ], which concludes this step. Step 5. It remains to prove that for all s ∈ [0, T ], L(Vs ) = f s , for V the unique solution of (17). We set gs = L(Vs ) for all s ∈ [0, T ] and we first observe that (gt )t∈[0,T ] solves the linear Landau equation: for any ϕ ∈ Cb2 (R3 ), t ϕ(x)gt (d x) = ϕ(x) f 0 (d x) + Lϕ(x, v)gs (d x) f s (v)dvds, (19) R3
R3
0
R 3 ×R 3
with L defined by (3). It suffices to apply the Itô formula, to take expectations, to use that the first marginal of Rs is f s and that σ.σ t = a. See [8, around Eq. (2.4)] for the detailed computation. Assume for a moment that there is uniqueness for the linear equation (19). Since ( f t )t∈[0,T ] is a weak solution to (1), it is also a weak solution to (19). We deduce that for all t ∈ [0, T ], gt = f t . To prove the uniqueness for (19), we use a result of Horowitz-Karandikar [11, Theorem B.1], see also Bath-Karandikar [3, Theorem 5.2]. Consider, for t ∈ [0, T ], x ∈ R3 and ϕ ∈ Cb2 , the operator At ϕ(x) := R3 Lϕ(x, v) f t (v)dv. A stochastic process (X t )t∈[t0 ,T ] is said to solve the martingale problem for (Cb2 , At ) if for all ϕ ∈ Cb2 , t the process ϕ(X t ) − t0 As ϕ(X s )ds defined for t ∈ [t0 , T ] is a martingale. To apply [11, Theorem B.1], we have to check that: (i) there is a countable family (ϕk )k≥1 ⊂ Cb2 such that for all t ∈ [0, T ], {(ϕk , At ϕk )}k≥1 is dense in {(ϕ, At ϕ), ϕ ∈ Cb2 } for the bounded-pointwise convergence; (ii) for any (t0 , x0 ) in [0, T ] × R3 , there exists a unique (in law) solution (X t )t∈[t0 ,T ] to the martingale problem for (Cb2 , At ) such that X t0 = x0 . We now verify these two points. First consider a countable family of functions (ϕk )k≥1 ⊂ Cb2 , dense in Cb2 for the norm |||ϕ||| := ||ϕ||∞ + ||Dϕ||∞ + ||D 2 ϕ||∞ . Then point (i) easily follows (with the uniform convergence instead of the boundedpointwise convergence) from the estimate |At ϕ(x)| ≤ R3 |Lϕ(x, v)| f t (v)dv ≤ C|||ϕ||| R3 (|x − v|−1 + |x − v|−2 ) f t (v)dv ≤ C|||ϕ|||(1 + || f t ||∞ ) due to (7) with α = −1 and α = −2. To prove (ii), observe that the martingale problem for (Cb2 , At ) with X t0 = x0 corresponds to the stochastic differential equation t t σ (X s − v)W (dv, d v, ˜ ds) + b(X s − v) f s (dv)ds, X t = x0 + t0
R 3 ×R 3
t0
R3
for which we have shown the strong existence and uniqueness (only in the case t0 = 0 and x0 = V0 , but the generalization is straightforward). Point (ii) follows. 5. Conclusion Our main result is easily deduced from Theorem 9 and Lemma 8. Proof of Theorem 2. Let T > 0 be fixed. Point (i). Assume that we have two weak solutions ( f t )t∈[0,T ] , ( f˜t )t∈[0,T ] to (1), with f˜0 = f 0 , both belonging to L ∞ ([0, T ], P2 ) ∩ L 1 ([0, T ], L ∞ ). Theorem 9 implies that
Landau Equation with a Coulomb Potential
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there is a bounded function ρ : [0, T ] → R+ such that W22 ( f t , f˜t ) ≤ ρ(t) for all t t ∈ [0, T ] satisfying ρ(t) ≤ 0 γ (s) (ρ(s))ds, with γ (s) = C(1 + || f s + f˜s ||∞ ) ∈ L 1 ([0, T ]). Lemma 8-(i) implies that ρ(t) = 0, whence W2 ( f t , f˜t ) = 0, for all t ∈ [0, T ]. Thus ( f t )t∈[0,T ] = ( f˜t )t∈[0,T ] . Point (ii). Let now ( f t )t∈[0,T ] , ( f tn )t∈[0,T ] be a family of weak solutions such that RT := T supn 0 (1 + || f sn + f s ||∞ )ds < ∞ and an := W22 ( f 0n , f 0 ) → 0. Applying Theorem 9 we get W22 ( f tn , f t ) ≤ ρn (t), for some bounded function ρn : [0, T ] → R+ satisfying t ρn (t) ≤ an + C(1 + || f sn + f s ||∞ ) (ρn (s))ds. 0
1 Lemma 8-(ii) implies M(an ) − M(ρn (t)) ≤ C RT , where M(x) = x (1/ (y))dy is decreasing on (0, ∞) and satisfies lim x 0 M(x) = +∞. As a consequence, lim inf M( sup ρn (t)) = lim inf inf M(ρn (t)) ≥ lim inf M(an ) − C RT = +∞. n
[0,T ]
n
[0,T ]
n
This implies that limn sup[0,T ] ρn (t) = 0, and finally, limn sup[0,T ] W22 ( f tn , f t ) = 0. Acknowledgements. I am very grateful to the anonymous referee for his support and careful reading. L’auteur de ce travail a bénéficié d’une aide de l’Agence Nationale de la Recherche portant la référence ANR-08BLAN-0220-01.
References 1. Alexandre, R., Villani, C.: On the Landau approximation in plasma physics. Ann. Inst. H. Poincaré Anal. Non Linéaire 21, 61–95 (2004) 2. Arsen’ev, A.A., Peskov, N.V.: The existence of a generalized solution of Landau’s equation. Z. Vycisl. Mat. i Mat. Fiz. 17, 1063–1068 (1977) 3. Bhatt, A.G., Karandikar, R.L.: Invariant measures and evolution equations for Markov processes characterized via martingale problems. Ann. Probab. 21, 2246–2268 (1993) 4. Chemin, J.Y.: Fluides parfaits incompressibles. Astérisque No. 230, 1995, 177 pp 5. Desvillettes, L., Villani, C.: On the spatially homogeneous Landau equation for hard potentials, Part I : existence, uniqueness and smothness. Comm. Part. Diff. Eqs. 25, 179–259 (2000) 6. Fournier, N., Mouhot, C.: On the well-posedness of the spatially homogeneous Boltzmann equation with a moderate angular singularity. Commun. Math. Phys. 289, 803–824 (2009) 7. Fournier, N., Guérin, H.: On the uniqueness for the spatially homogeneous Boltzmann equation with a strong angular singularity. J. Stat. Phys. 131, 749–781 (2008) 8. Fournier, N., Guérin, H.: Well-posedness of the spatially homogeneous Landau equation for soft potentials. J. Funct. Anal. 256, 2542–2560 (2009) 9. Funaki, T.: The diffusion approximation of the spatially homogeneous Boltzmann equation. Duke Math. J. 52, 1–23 (1985) 10. Guérin, H.: Solving Landau equation for some soft potentials through a probabilistic approach. Ann. Appl. Probab. 13, 515–539 (2003) 11. Horowitz, J., Karandikar, R.L.: Martingale problems associated with the Boltzmann equation. In: Seminar on Stochastic Processes, 1989 (San Diego, CA, 1989), Progr. Probab., 18, Boston, MA: Birkhauser Boston, 1990, pp. 75–122 12. Tanaka, H.: Probabilistic treatment of the Boltzmann equation of Maxwellian molecules. Z. Wahrsch. Und Verw. Gebiete 46, 67–105 (1978/79) 13. Villani, C.: On the spatially homogeneous Landau equation for Maxwellian molecules. Math. Models Methods Appl. Sci. 8, 957–983 (1998) 14. Villani, C.: On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations. Arch. Rat. Mech. Anal. 143, 273–307 (1998) 15. Villani, C.: A review of mathematical topics in collisional kinetic theory. Handbook of mathematical fluid dynamics, Vol. I, Amsterdam: North-Holland, 2002, pp. 71–305
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16. Villani, C.: Topics in Optimal Transportation. Graduate Studies in Mathematics Vol. 58, Providence, RI: Amer. Math. Soc., 2003 17. Walsh, J.B.: An introduction to stochastic partial differential equations. École d’été de Probabilités de Saint-Flour XIV, Lect. Notes in Math. 1180, Berlin: Springer, 1986, pp. 265–437 Communicated by P. Constantin
Commun. Math. Phys. 299, 783–792 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1114-8
Communications in
Mathematical Physics
The Structure of Parafermion Vertex Operator Algebras: General Case Chongying Dong1,2, , Qing Wang3, 1 Department of Mathematics, University of California, Santa Cruz, CA 95064, USA 2 School of Mathematics, Sichuan University, Chengdu 610065, China 3 School of Mathematical Sciences, Xiamen University, Xiamen 361005, China.
E-mail:
[email protected] Received: 15 September 2009 / Accepted: 26 March 2010 Published online: 15 August 2010 – © The Author(s) 2010. This article is published with open access at Springerlink.com
Abstract: The structure of the parafermion vertex operator algebra associated to an integrable highest weight module for any affine Kac-Moody algebra is studied. In particular, a set of generators for this algebra has been determined. 1. Introduction This paper is a continuation of our study of the parafermion vertex operator algebra associated to an integrable highest weight module for an arbitrary affine Kac-Moody algebra. We determine a set of generators for these algebras. If the affine Kac-Moody algebra is A(1) 1 , this result was obtained previously in [6]. The parafermion algebra was first studied in [27] in the context of conformal field theory. It was clarified in [7] that the parafermion algebras are essentially the Z -algebras introduced and studied earlier in [19–21] in the process of studying the representation theory for the affine Kac-Moody Lie algebras. As it proved in [7], the parafermion algebras generate certain generalized vertex operator algebras. The partition functions for the parafermion conformal field theory have been given in [13] and [12] in connection with the partition functions associated to the integrable representations for the affine KacMoody Lie algebras. We refer the reader to [1,12,13,15,24,26,27] for various aspects of parafermion conformal field theory. The parafermion vertex operator algebras which have roots in the parafermion conformal field theory are realized as commutants of the Heisenberg vertex operator subalgebras in the vertex operator algebras associated to the integrable highest weight modules for the affine Kac-Moody Lie algebra [11,14,18]. More precisely, let L(k, 0) be the level k integrable highest weight module for affine Kac-Moody algebra g associated Supported by NSF grants, and a Faculty research grant from the University of California at Santa Cruz. Supported by China NSF grants (No.10931006, No.10926040), and Natural Science Foundation of
Fujian Province, China (No.2009J05012).
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to a finite dimensional simple Lie algebra g. Then L(k, 0) has a vertex operator subalgebra Mh(k, 0) generated by the Cartan subalgebra h of g. The commutant K (g, k) of Mh(k, 0) in L(k, 0) is called the parafermion vertex operator algebra. Although the parafermion field theory has been studied for more than two decades, the mathematical investigation of the parafermion vertex operator algebras have been limited by a lack of understanding of the structural theory of these algebras. The goals of this paper and [4–6] are to alleviate this situation. More importantly, it is widely believed that K (g, k) should give a new class of rational, C2 -cofinite vertex operator algebras although this can only been proved in the case g = sl2 and k ≤ 6 [5]. Most well known vertex operator algebras such as lattice vertex operator algebras [2,3,10], the affine vertex operator algebras [7,11,23] and the Virasoro vertex operator algebras [11,25] can be understood well using the underline lattices or Lie algebras. Unfortunately, the structures of parafermion vertex operator algebras with weight one subspaces being zero are much more complicated. It seems that a determination of a set of generators is the first step in understanding parafermion vertex operator algebras and their representation theory. It is well known that L(k, 0) is the irreducible quotient of the generalized Verma module V (k, 0) (see Sect. 2). So the structure of L(k, 0) can be determined by studying the maximal submodule of V (k, 0) or the maximal ideal of V (k, 0) which is also a vertex operator algebra. The same idea can also be applied to the study of parafermion vertex operator algebras. In fact, the Heisenberg vertex operator algebra Mh(k, 0) is also a subalgebra of V (k, 0) and the parafermion vertex operator algebra K (g, k) is the simple quotient of the commutant N (g, k) of Mh(k, 0) in V (k, 0). The main part of this paper is to determine a set of generators for V (k, 0)(0) which is the weight zero subspace of V (k, 0) under the action of the Cartan subalgebra h. The generators for N (g, k) and K (g, k) will be found easily then. Also, the maximal ideal of N (g, k) is generated by one vector. This result is similar to that for the maximal ideal of V (k, 0). It is worth pointing out that the structure theory for the parafermion vertex operator algebra is similar to the structure theory for the finite dimensional Lie algebras or Kac-Moody Lie algebras. The building block of the Kac-Moody Lie algebras is the 3-dimensional simple Lie algebra sl2 associated to any real root. The generator results for the parafermion vertex operator algebras given in this paper and [6] show that the (1) parafermion vertex operator algebras associated to the affine Lie algebra A1 are also the building block of general parafermion vertex operator algebras. We hope this fact will be important in the future study of the representation theory for the parafermion vertex operator algebra. So a complete understanding of representation theory of K (g, k) in the case g = sl2 becomes necessary. The paper is organized as follows. In Sect. 2, we give the construction of the vertex operator algebra V (k, 0) associated to the affine Kac-Moody algebra g from [11]. V (k, 0) has a vertex operator subalgebra V (k, 0)(0) which is the space of h-invariants of V (k, 0). A foundational result in this section is to determine a set of of generators for V (k, 0)(0). In Sect. 3, we give a set of generators for the vertex operator algebra N (g, k) which is the commutant of the Heisenberg vertex operator algebra Mh(k, 0) in α generated by ωα , Wα3 (which V (k, 0). We also discuss the vertex operator subalgebra P α is isomorphic is defined in Sect. 3) associated to any positive root α and prove that P θ,θ to N (sl2 , kα ), where kα = α,α k and θ is the highest root. In fact, N (g, k) is generated α for positive roots α. In Sect. 4, we give a set of generators for the parafermion by P vertex operator algebra K (g, k) which is the simple quotient of N (g, k). We prove that
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the maximal ideal of N (g, k) is generated by the vector x−θ (0)k+1 xθ (−1)k+1 1, where α xθ is the root vector associated to θ (see Sect. 2). We also show that the image Pα of P in K (g, k) is isomorphic to K (sl2 , kα ) and K (g, k) is generated by Pα for positive roots α. That is, K (sl2 , kα ) are the building blocks of K (g, k). We expect the reader to be familiar with the elementary theory of vertex operator algebras as found, for example, in [10] and [18]. We thank the referees for many excellent suggestions which improve and simplify the proof of Theorem 2.1 greatly. 2. Vertex Operator Algebras V (k, 0) and V (k, 0)(0) Let g be a finite dimensional simple Lie algebra with a Cartan subalgebra h. Let be the corresponding root system and Q the root lattice. Let , be an invariant symmetric nondegenerate bilinear form on g such that α, α = 2 if α is a long root, where we have identified h with h∗ via , . As in [15], we denote the image of α ∈ h∗ in h by tα . That is, α(h) = tα , h for any h ∈ h. Fix simple roots {α1 , · · · , αl } and denote the highest root by θ. Let gα denote the root space associated to the root α ∈ . For α ∈ + , we fix 2 x±α ∈ g±α and h α = α,α tα ∈ h such that [xα , x−α ] = h α , [h α , x±α ] = ±2x±α . That 0 1 α , x−α to is, g = Cxα + Ch α + Cx−α is isomorphic to sl2 by sending xα to 0 0 1 0 0 0 θ,θ θ,θ . Then h α , h α = 2 α,α and xα , x−α = α,α for all and h α to 0 −1 1 0 α ∈ . Let g = g ⊗ C[t, t −1 ] ⊕ CK be the corresponding affine Lie algebra. Let k ≥ 1 be an integer and g
V (k, 0) = Vg(k, 0) = I ndg⊗C[t]⊕C K C the induced g-module such that g ⊗ C[t] acts as 0 and K acts as k on 1 = 1. We denote by a(n) the operator on V (k, 0) corresponding to the action of a ⊗ t n . Then [a(m), b(n)] = [a, b](m + n) + ma, bδm+n,0 k for a, b ∈ g and m, n ∈ Z. Let a(z) = n∈Z a(n)z −n−1 . Then V (k, 0) is a vertex operator algebra generated by a(−1)1 for a ∈ g such that Y (a(−1)1, z) = a(z) with the vacuum vector 1 and the Virasoro vector l α, α 1 xα (−1)x−α (−1)1 ωaff = h i (−1)h i (−1)1 + 2(k + h ∨ ) θ, θ i=1
α∈
dim g ∨ is the dual Coxeter number of central charge kk+h ∨ (e.g. [17,18, Sect. 6.2]), where h of g and {h i |i = 1, . . . , l} is an orthonormal basis of h. We will use the standard notation for the component operators of Y (u, z) for u ∈ V (k, 0). That is, Y (u, z) = n∈Z u n z −n−1 . From the definition of vertex operators, we immediately see that (a(−1)1)n = a(n) for a ∈ g. So in the rest of paper, we will
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use both a(n) and (a(−1)1)n for a ∈ g and use u n only for general u without further explanation. For λ ∈ h∗ , set V (k, 0)(λ) = {v ∈ V (k, 0)|h(0)v = λ(h)v, ∀ h ∈ h}. Then we have V (k, 0) = ⊕λ∈Q V (k, 0)(λ).
(2.1)
Since [h(0), Y (u, z)] = Y (h(0)u, z) for h ∈ h and u ∈ V (k, 0), from the definition of affine vertex operator algebra, we see that V (k, 0)(0) is a vertex operator subalgebra of V (k, 0) with the same Virasoro vector ωaff and each V (k, 0)(λ) is a module for V (k, 0)(0). Our first theorem is on a set of generators for V (k, 0)(0). Theorem 2.1. The vertex operator algebra V (k, 0)(0) is generated by vectors αi (−1)1 and x−α (−2)xα (−1)1 for 1 ≤ i ≤ l, α ∈ + . Proof. First note that V (k, 0)(0) is spanned by the vectors a1 (−m 1 ) · · · as (−m s )xβ1 (−n 1 )xβ2 (−n 2 ) · · · xβt (−n t )1, where ai ∈ h, β j ∈ , m i > 0, n j > 0 and β1 + β2 + · · · + βt = 0. Let U be the vertex operator subalgebra generated by αi (−1)1 and x−α (−2)xα (−1)1 for 1 ≤ i ≤ l, α ∈ + . Clearly, αi (−1)1 and x−α (−2)xα (−1)1 ∈ V (k, 0)(0) for 1 ≤ i ≤ l, α ∈ + . It suffices to prove that V (k, 0)(0) ⊂ U. Since (h(−1)1)n = h(n) for h ∈ h, we see that h(n)U ⊂ U for h ∈ h and n ∈ Z. So we only need to prove u = xβ1 (−n 1 )xβ2 (−n 2 ) · · · xβt (−n t )1 ∈ U with β1 + β2 + · · · + βt = 0. We will prove it by induction on t. Clearly, t ≥ 2. If t = 2, it follows from Theorem 2.1 in [6] that x−α (−m)xα (−n)1 ∈ U for m, n > 0. Note that if m ≥ 0, then x−α (m)xα (n)1 = −h α (m + n)1 + mkxα , x−α δm+n,0 1 ∈ U. We claim that x−α (m)xα (n)U ⊂ U for all m, n ∈ Z. Let u ∈ U. From Proposition 4.5.7 of [18], there exist nonnegative integers p, q such that q p m −q q x−α (m)xα (n)u = (x−α (m − q − i + j)xα (−1)1)n+q+i− j u. i j i=0 j=0
Since x−α (m − q − i + j)xα (−1)1 ∈ U , the claim follows. From now on, we assume that t > 2 and that xβ1 (−n 1 )xβ2 (−n 2 ) · · · xβν (−n ν )1 ∈ U with β1 + β2 + · · · + βν = 0 for 2 ≤ ν ≤ t − 1 and n i > 0. We have to show that xβ1 (−n 1 )xβ2 (−n 2 ) · · · xβt (−n t )1 ∈ U with β1 + β2 + · · · + βt = 0. We divide the proof into two cases.
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Case 1. There exist 1 ≤ i, j ≤ t such that βi + β j ∈ . Note that if xβ1 (−n 1 )xβ2 (−n 2 ) · · · xβt (−n t )1 ∈ U, then xβi1 (−n i1 )xβi2 (−n i2 ) · · · xβit (−n it )1 ∈ U by the induction assumption, where (i 1 , . . . , i t ) is any permutation of (1, . . . , t). Without loss of generality, we may assume that β1 + β2 ∈ . Let k, n be positive integers such that −k + n = −n 2 and n > n i for i ≥ 3. Let w = xβ1 +β2 (−k)xβ3 (−n 3 ) · · · xβt (−n t )1 with β1 + β2 + · · · + βt = 0. Then w ∈ U by the induction assumption and xβ1 (−n 1 )x−β1 (n)w ∈ U by the claim. Let [x−β1 , xβ1 +β2 ] = λxβ2 for some nonzero λ. Then xβ1 (−n 1 )x−β1 (n)w = λxβ1 (−n 1 )xβ2 (−n 2 )xβ3 (−n 3 ) · · · xβt (−n t )1 + xβ1 (−n 1 )xβ1 +β2 (−k)[x−β1 , xβ3 ](n − n 3 )xβ4 (−n 4 ) · · · xβt (−n t )1 + · · · + xβ1 (−n 1 )xβ1 +β2 (−k)xβ3 (−n 3 ) · · · [x−β1 , xβt ](n − n t )1. Since n − n i > 0 for i ≥ 3, we see that xβ1 (−n 1 )xβ1 +β2 (−k)[x−β1 , xβ3 ](n − n 3 )xβ4 (−n 4 ) · · · xβt (−n t )1 + · · · + xβ1 (−n 1 )xβ1 +β2 (−k)xβ3 (−n 3 ) · · · [x−β1 , xβt ](n − n t )1 lies in U by the induction assumption. As a result, xβ1 (−n 1 )xβ2 (−n 2 )xβ3 (−n 3 ) · · · xβt (−n t )1 ∈ U.
Case 2. For any 1 ≤ i, j ≤ t, βi + β j ∈ / . We claim that there exist 1 ≤ i , j ≤ t such that βi + β j = 0. Otherwise, βi + β j = 0 for all i, j. This implies that βi , β j ≥ 0 for all i, j, thus β1 , tj=2 β j ≥ 0. On the other hand, since tj=2 β j = −β1 , we have β1 , tj=2 β j < 0, a contradiction. Without loss of generality, we may assume β1 + β2 = 0. Then β3 + · · · + βt = 0. By the induction assumption, xβ3 (−n 3 ) · · · xβt (−n t )1 ∈ U. It is immediate that xβ1 (−n 1 )xβ2 (−n 2 )xβ3 (−n 3 ) · · · xβt (−n t )1 ∈ U from the claim. The proof is complete. Remark 2.2. Theorem 2.1 has been obtained in [6] previously in the case g = sl2 . The proof given here simplifies the proof of Theorem 2.1 in [6]. Next we discuss some automorphisms of vertex operator algebras V (k, 0) and V (k, 0)(0) for later purposes. It is well known that the automorphism group Aut (V (k, 0)) is isomorphic to the automorphism group Aut (g). In fact, if σ ∈ Aut (g), then σ lifts to an automorphism of V (k, 0) in the following way: σ (x1 (−n 1 ) · · · xs (−n s )1) = (σ x1 )(−n 1 ) · · · (σ xs )(−n s )1 for xi ∈ g and n i > 0. Let W (g) be the Weyl group of g. Then W (g) can naturally be regarded as a subgroup of Aut (g) [16]. It is easy to see that if σ (h) = h, then σ (V (k, 0)(0)) = V (k, 0)(0) and the restriction of σ to V (k, 0)(0) gives an automorphism of V (k, 0)(0). In particular, any Weyl group element gives an automorphism of V (k, 0)(0). This fact will be used in later sections.
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3. Vertex Operator Algebra N(g, k) Let Vh(k, 0) be the vertex operator subalgebra of V (k, 0) generated by h(−1)1 for h ∈ h with the Virasoro element ωh =
l 1 h i (−1)h i (−1)1 2k i=1
of central charge l, where {h 1 , · · · h l } is an orthonormal basis of h as before. For λ ∈ h∗ , let Mh(k, λ) denote the irreducible highest weight module for h with a highest weight vector vλ such that h(0)vλ = λ(h)vλ for h ∈ h. Then Vh(k, 0) is identified with Mh(k, 0). Recall V (k, 0)(λ) from Sect. 2. Both V (k, 0) and V (k, 0)(λ), λ ∈ Q are completely reducible Vh(k, 0)-modules. That is, V (k, 0) = ⊕λ∈Q Mh(k, λ) ⊗ Nλ , V (k, 0)(λ) = Mh(k, λ) ⊗ Nλ ,
(3.1) (3.2)
where Nλ = {v ∈ V (k, 0) | h(m)v = λ(h)δm,0 v for h ∈ h, m ≥ 0} is the space of highest weight vectors with highest weight λ for h. Note that N (g, k) = N0 is the commutant [11, Theorem 5.1] of Vh(k, 0) in V (k, 0). The commutant N (g, k) is a vertex operator algebra with the Virasoro vector ω = dim g ωaff − ωh whose central charge is kk+h ∨ − l. Recall from Sect. 2, the 3-dimensional subalgebra gα for α ∈ + . Then the restricθ,θ tion , gα of the bilinear form , to gα is equal to α,α (, ), where (, ) is the standard nondegenerate symmetric invariant bilinear form on gα such that (h α , h α ) = 2. As a θ,θ result, V (k, 0) is a module for gα = gα ⊗ C[t, t −1 ] ⊕ CK of level kα = α,α k as α we regard V (k, 0) as a module for the subalgebra g of g. In other words, V (k, 0) is a gα -module of level 2k or 3k if α is a short root. Following [5], we let ωα =
1 (−kh α (−2)1 − h α (−1)2 1 + 2kxα (−1)x−α (−1)1), 2k(k + 2)
(3.3)
Wα3 = k 2 h α (−3)1 + 3kh α (−2)h α (−1)1 + 2h α (−1)3 1 − 6kh α (−1)xα (−1)x−α (−1)1 + 3k 2 xα (−2)x−α (−1)1 − 3k 2 xα (−1)x−α (−2)1
(3.4)
ωα , Wα3
as in (3.3) and (3.4) if α ∈ + is a long root. If α is a short root, we also define by replacing k by kα . α be the vertex operator subalgebra of N (g, k) generated by ωα and Wα3 . Then Let P α is isomorphic to the W -algebra W (2, 3, 4, 5) [1] by [6, Theorem 3.1] with k replaced P α is isomorphic to N (sl2 , kα ). by kα , i.e., P The first main theorem of this paper is about the generators of N (g, k). Theorem 3.1. The vertex operator algebra N (g, k) is generated by dim g − l vectors α for α ∈ + . ωα and Wα3 for α ∈ + . That is, N (g, k) is generated by subalgebras P
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Proof. We first prove that V (k, 0)(0) is generated by vectors αi (−1)1, ωα and Wα3 for i = 1, . . . , l and α ∈ + . In fact, let U be the vertex operator subalgebra generated by h(−1)1, ωα and Wα3 for h ∈ h and α ∈ + . Then x−α (−1)xα (−1)1 ∈ U and ωaff ∈ U . Moreover, from the expression of Wα3 , we see that x−α (−1)xα (−2)1 − x−α (−2)xα (−1)1 ∈ U . Set L aff (n) = (ωaff )n+1 ; we have [L aff (m), a(n)] = −na(m + n) for m, n ∈ Z, a ∈ g. Thus, L aff (−1)x−α (−1)xα (−1)1 = x−α (−2)xα (−1)1 + x−α (−1)xα (−2)1 ∈ U. Together with x−α (−1)xα (−2)1 − x−α (−2)xα (−1)1 ∈ U , we get x−α (−2)xα (−1)1 ∈ U , and so U is equal to V (k, 0)(0) by Theorem 2.1. Next we show that ωα , Wα3 ∈ N (g, k) for α ∈ . Since h α , h α = 0, we have decomposition h = Ch α ⊕ (Ch α )⊥ , where (Ch α )⊥ is the orthogonal complement of Ch α with respect to , . From [6], we know that h α (n)ωα = h α (n)Wα3 = 0 for n ≥ 0. If u ∈ (Ch α )⊥ , we clearly have u(n)ωα = u(n)Wα3 = 0 for n ≥ 0. This implies that ωα , Wα3 ∈ N (g, k). Notice that Y (u, z 1 )Y (v, z 2 ) = Y (v, z 2 )Y (u, z 1 ) for u ∈ Mh(k, 0) and v ∈ N (g, k). Since V (k, 0)(0) = Mh(k, 0) ⊗ N (g, k), h(−1)1 ∈ Mh(k, 0) for h ∈ h, and ωα , Wα3 ∈ N (g, k) for α ∈ + , we conclude that N (g, k) is generated by ωα , Wα3 for α ∈ + . Remark 3.2. Using the Z -algebra introduced and studied in [19] and [20,21], we can rewrite ωα and Wα3 in terms of Z -operators Z α (m) and Z −α (n). It is not too hard to see that ωα = aα Z α (−1)Z −α (−1)1 and Wα3 = bα Z α (−2)Z −α (−1)1 + cα Z −α (−2)Z α (−1)1 for some constants aα , bα , cα ∈ C. One could determine these constants explicitly using the definition of Z -operators. Remark 3.3. The vertex operator algebra N (g, k) and its quotient K (g, k) are of moonshine type. That is, their weight zero subspaces are 1-dimensional and weight one subspaces are zero. Remark 3.4. We already know that each ωα is a Virasoro element and how to compute the Lie brackets [Y (ωα , z 1 ), Y (Wα3 , z 2 )], [Y (Wα3 , z 1 ), Y (Wα3 , z 2 )] for α ∈ + . It is important to calculate the Lie brackets for vertex operators associated to vectors in different Pα . This will be done in a sequel to this paper where the representation theory will be investigated. Following the discussion given at the end of Sect. 2, we see that any Weyl group element gives an automorphism of N (g, k). 4. Parafermion Vertex Operator Algebras K (g, k) It is well known that the vertex operator algebra V (k, 0) has a unique maximal ideal J generated by a weight k + 1 vector xθ (−1)k+1 1 [17], where θ is the highest root of g. The quotient vertex operator algebra L(k, 0) = V (k, 0)/J is a simple, rational vertex operator algebra associated to the affine Lie algebra g. Again, the Heisenberg vertex operator algebra Vh(k, 0) generated by h(−1)1 for h ∈ h is a simple subalgebra of L(k, 0) and L(k, 0) is a completely reducible Vh(k, 0)-module. We have a decomposition L(k, 0) = ⊕λ∈Q Mh(k, λ) ⊗ K λ
(4.1)
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as modules for Vh(k, 0), where K λ = {v ∈ L(k, 0) | h(m)v = λ(h)δm,0 v for h ∈ h, m ≥ 0}. Set K (g, k) = K 0 . Then K (g, k) is the commutant of Vh(k, 0) in L(k, 0) and is called the parafermion vertex operator algebra associated to the irreducible highest weight module L(k, 0) for g. As we mentioned in the Introduction, K (g, k) are conjectured to be rational, C2 -cofinite vertex operator algebras. As a Vh(k, 0)-module, J is completely reducible. From (3.1), J = ⊕λ∈Q Mh(k, λ) ⊗ (J ∩ Nλ ). In particular,
I = J ∩ N (g, k) is an ideal of N (g, k) and K (g, k) ∼ I. Fol= N (g, k)/
lowing the same proof as [5, Lemma 3.1], we know that I is the unique maximal ideal of N (g, k). Thus K (g, k) is a simple vertex operator algebra. We still use ωaff , ωh, ωα , Wα3 to denote their images in L(k, 0) = V (k, 0)/J . Remark 4.1. In the case k = 1, it follows from the construction of L(1, 0) [9,10] that ω = 0 and K (g, k) = C if g is of ADE type. The following result is a direct consequence of Theorem 3.1. Theorem 4.2. The simple vertex operator algebra K (g, k) is generated by ωα , Wα3 for α ∈ + . / N (g, k). Next, we study the ideal
I of N (g, k) in detail. The vector xθ (−1)k+1 1 ∈ From [5, Theorem 3.2] we know that h θ (n)x−θ (0)k+1 xθ (−1)k+1 1 = 0 for n ≥ 0. It is clear that if h ∈ h satisfies h θ , h = 0, then h(n)x−θ (0)k+1 xθ (−1)k+1 1 = 0 for n ≥ 0. So we have proved the following Lemma 4.3. x−θ (0)k+1 xθ (−1)k+1 1 ∈
I. Furthermore, we have Proposition 4.4. The maximal ideal
I of N (g, k) is generated by x−θ (0)k+1 xθ (−1)k+1 1. Proof. The proof is similar to that of [6, Theorem 4.2 (1)]. Recall gθ = Cxθ + Ch θ + Cx−θ is a subalgebra of g isomorphic to sl2 .V (k, 0) is an gθ -module where a ∈ gθ acts as a(0). Each weight subspace of the vertex operator algebra V (k, 0) is a finite dimensional gθ -module and V (k, 0) is completely reducible as a module for gθ . Consider the gθ -submodule X of V (k, 0) generated by xθ (−1)k+1 1. Since xθ (0)xθ (−1)k+1 1 = 0 and h θ (0)xθ (−1)k+1 1 = 2(k + 1)xθ (−1)k+1 1, xθ (−1)k+1 1 is a highest weight vector with highest weight 2(k + 1) for gθ . Then X is an irreducible gθ -module with basis x−θ (0)i xθ (−1)k+1 1, 0 ≤ i ≤ 2(k + 1) from the representation theory of sl2 . This implies that the ideal J of the vertex operator algebra V (k, 0) can be generated by any nonzero vector in X . In particular, J is generated by x−θ (0)k+1 xθ (−1)k+1 1. Then J is spanned by u n x−θ (0)k+1 xθ (−1)k+1 1 for u ∈ V (k, 0) and n ∈ Z by [8, Cor. 4.2] or [22, Prop. 4.1]. Since vm u ∈ V (k, 0)(λ + μ) for v ∈ V (k, 0)(λ), u ∈ V (k, 0)(μ), λ, μ ∈ Q and m ∈ Z, we see that J ∩V (k, 0)(0) is spanned by vectors of the form u n x−θ (0)k+1 xθ (−1)k+1 1 with u ∈ V (k, 0)(0). Let u = v ⊗ w ∈ V (k, 0)(0) = Mh(k, 0) ⊗ N (g, k) with v ∈ Mh(k, 0) and w ∈ N (g, k). Then Y (u, z) = Y (v, z) ⊗ Y (w, z) acts on Mh(k, 0) ⊗ N (g, k). As a result, we have that
I is spanned by wn x−θ (0)k+1 xθ (−1)k+1 1 for w ∈ N (g, k) and n ∈ Z. That is, the ideal
I of the vertex operator algebra N (g, k) is generated by x−θ (0)k+1 xθ (−1)k+1 1. The proof is complete.
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For α ∈ + , we let Pα be the vertex operator subalgebra of K (g, k) generated by α . A natural question is whether or not Pα is a ωα and Wα3 . Then Pα is a quotient of P simple vertex operator algebra. For this purpose, we recall our discussion earlier on the automorphisms of the vertex operator algebra V (k, 0) and N (g, k). That is, any Weyl group element gives an automorphism of V (k, 0) and N (g, k). Clearly, any automorphism σ of V (k, 0) induces an automorphism of L(k, 0) as σ maps the unique maximal ideal J to J . If σ ∈ W (g), then σ preserves the unique maximal ideal
I and σ gives an automorphism of the parafermion vertex operator algebra K (g, k), Now let α ∈ + be a long root. Then there exists σ ∈ W (g) such that σ θ = α [16]. As a result, σ (x−θ (0)k+1 xθ (−1)k+1 1) = ax−α (0)k+1 xα (−1)k+1 1 I. for some constant a. This implies from Lemma 4.3 that x−α (0)k+1 xα (−1)k+1 1 ∈
Using [6, Theorem 4.2] we obtain: Proposition 4.5. For any long root α ∈ + , the vertex operator subalgebra Pα of K (g, k) is a simple vertex operator algebra isomorphic to the parafermion vertex operator algebra K (sl2 , k) associated to sl2 . We next deal with short roots α ∈ + . As we mentioned already V (k, 0) is a level kα -module for the affine algebra gα . We need a different method to prove the following which is a generalization of Proposition 4.5. Proposition 4.6. Let α ∈ + . Then the vertex operator subalgebra Pα of K (g, k) is a simple vertex operator algebra isomorphic to the parafermion vertex operator algebra K (sl2 , kα ) associated to sl2 . Proof. As in the proof of Proposition 4.5 we only need to prove that I. x−α (0)kα +1 xα (−1)kα +1 1 ∈
Clearly, L(k, 0) is an integrable module for gα as xα (−1) is locally nilpotent on L(k, 0). In particular, the vertex operator subalgebra U of L(k, 0) generated by gα is an integrable highest weight module. That is, U is isomorphic to L(kα , 0) associated to the affine algebra gα . As a result, we have xα (−1)kα +1 1 ∈ J . It follows then immediately that x−α (0)kα +1 xα (−1)kα +1 1 ∈
I, as desired. Remark 4.7. We expect from Proposition 4.6 that the role of K (sl2 , kα ) played in the theory of the parafermion vertex operator algebra is similar to the role of sl2 played in the theory of Kac-Moody Lie algebras. So a study of structural and representation theory for K (sl2 , kα ) becomes extremely important for general parafermion vertex operator algebras. Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
References 1. Blumenhagen, R., Eholzer, W., Honecker, A., Hornfeck, K., Hübel, R.: Coset realization of unifying W -algebras. Int. J. Mod. Phys. A10, 2367–2430 (1995)
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2. Borcherds, R.E.: Vertex algebras, Kac-Moody algebras, and the Monster. Proc. Natl. Acad. Sci. USA 83, 3068–3071 (1986) 3. Dong, C.: Vertex algebras associated with even lattices. J. Algebra 160, 245–265 (1993) 4. Dong, C., Lam, C.H., Yamada, H.: W -algebras in lattice vertex operator algebras. In: Lie Theory and Its Applications in Physics VII, ed. by H.-D. Doebner, V. K. Dobrev, Proc. of the VII International Workshop, Varna, Bulgaria, 2007, Bulgarian J. Phys. 35 supplement, 25–35 (2008) 5. Dong, C., Lam, C.H., Yamada, H.: W-algebras related to parafermion algebras. J. Algebra 322, 2366– 2403 (2009) 6. Dong, C., Lam, C.H., Wang, Q., Yamada, H.: The structure of parafermion vertex operator algebras. J. Algebra 323, 371–381 (2010) 7. Dong, C., Lepowsky, J.: Generalized Vertex Algebras and Relative Vertex Operators. Progress in Math., Vol. 112, Boston, MA: Birkhäuser, 1993 8. Dong, C., Mason, G.: On quantum Galois theory. Duke Math. J. 86, 305–321 (1997) 9. Frenkel, I.B., Kac, V.: Basic representations of affine Lie algebras and dual resonance models. Invent. Math 62, 23–66 (1980) 10. Frenkel, I.B., Lepowsky, J., Meurman, A.: Vertex Operator Algebras and the Monster. Pure and Applied Math., Vol. 134, Boston, MA: Academic Press, 1988 11. Frenkel, I.B., Zhu, Y.: Vertex operator algebras associated to representations of affine and Virasoro algebras. Duke Math. J. 66, 123–168 (1992) 12. Gepner, D.: New conformal field theory associated with Lie algebras and their partition functions. Nucl. Phys. B290, 10–24 (1987) 13. Gepner, D., Qiu, Z.: Modular invariant partition functions for parafermionic field theories. Nucl. Phys. B285, 423–453 (1987) 14. Goddard, P., Kent, A., Olive, D.: Unitary representations of the Virasoro and super-Virasoro algebras. Commun. Math. Phys. 103, 105–119 (1986) 15. Hornfeck, K.: W -algebras with set of primary fields of dimensions (3, 4, 5) and (3, 4, 5, 6). Nucl. Phys. B407, 237–246 (1993) 16. Humphreys, J.: Introduction to Lie Algebras and Representation Theory. New York: Spring-Verlag, 1987 17. Kac, V.G.: Infinite-dimensional Lie Algebras. 3rd ed., Cambridge: Cambridge University Press, 1990 18. Lepowsky, J., Li, H.: Introduction to Vertex Operator Algebras and Their Representations. Progress in Math., Vol. 227, Boston, MA: Birkhäuser, 2004 (1) 19. Lepowsky, J., Primc, M.: Structure of the standard modules for the affine Lie algebra A1 . Contemp. Math. 46, Providence, RI: Amer. Math. Soc., 1985 20. Lepowsky, J., Wilson, R.L.: A new family of algebras underlying the Rogers-Ramanujan identities and generalizations. Proc. Natl. Acad. Sci. USA 78, 7245–7248 (1981) 21. Lepowsky, J., Wilson, R.L.: The structure of standard modules, I: Universal algebras and the RogersRamanujan identities. Invent. Math. 77, 199–290 (1984) 22. Li, H.: An approach to tensor product theory for representations of a vertex operator algebra, Ph.D. thesis, Rutgers University, 1994 23. Meurman, A., Primc, M.: Vertex operator algebras and representations of affine Lie algebras. Representation of Lie groups, Lie algebras and their quantum analogues. Acta Appl. Math. 44, 207–215 (1996) 24. Wang, P., Ding, X.: W -algebra constructed from Zk -parafermion through Nahm’s normal ordering product. Comm. Theor. Phys. 4, 155–189 (1995) 25. Wang, W.: Rationality of Virasoro vertex operator algebras. Int. Math. Res. Notices 7, 197–211 (1993) 26. West, P.: W strings and cohomology in parafermionic theories. Phys. Lett. B 329, 199–209 (1994) 27. Zamolodchikov, A.B., Fateev, V.A.: Nonlocal (parafermion) currents in two-dimensional conformal quantum field theory and self-dual critical points in Z N -symmetric statistical systems. Sov. Phys. JETP 62, 215– 225 (1985) Communicated by Y. Kawahigashi
Commun. Math. Phys. 299, 793–824 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1092-x
Communications in
Mathematical Physics
On the Classification of Automorphic Lie Algebras Sara Lombardo, Jan A. Sanders Department of Mathematics, Faculty of Sciences, Vrije Universiteit, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands. E-mail:
[email protected];
[email protected] Received: 7 December 2009 / Accepted: 16 March 2010 Published online: 24 July 2010 – © The Author(s) 2010. This article is published with open access at Springerlink.com
Abstract: The problem of reduction of integrable equations can be formulated in a uniform way using the theory of invariants. This provides a powerful tool of analysis and it opens the road to new applications of Automorphic Lie Algebras, beyond the context of integrable systems. In this paper it is shown that sl2 (C)–based Automorphic Lie Algebras associated to the icosahedral group I, the octahedral group O, the tetrahedral group T, and the dihedral group Dn are isomorphic. The proof is based on techniques from classical invariant theory and makes use of Clebsch-Gordan decomposition and transvectants, Molien functions and the trace-form. This result provides a complete classification of sl2 (C)–based Automorphic Lie Algebras associated to finite groups when the group representations are chosen to be the same and it is a crucial step towards the complete classification of Automorphic Lie Algebras.
1. Introduction 1.1. Automorphic Lie Algebras. In a recent study [LM05] Lombardo and Mikhailov showed that it is possible to treat algebraic reductions of integrable equations in a systematic way starting from a class of infinite dimensional Lie algebras, called Automorphic Lie Algebras. The authors set up a formalism to treat zero curvature representations (or Lax representations) related to systems of linear equations (Lax pairs) with rational dependence on the spectral parameter λ. Such linear systems are naturally associated to nonlinear partial differential equations (PDEs). Roughly speaking, Automorphic Lie Algebras are infinite dimensional Lie algebras generalising graded Lie algebras [Kac90]. They can be defined (omitting the details, which can be found in [LM05]) starting from any semisimple Lie algebra g as follows: let G ⊂ Aut (C(λ) ⊗ g), where Aut (C(λ) ⊗ g) stands for the group of automorphisms of the algebra C(λ) ⊗ g, g ⊂ gln is a finite dimensional semisimple Lie algebra and C(λ) is the field of rational functions in the complex variable λ with values in C.
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A Lie algebra (C(λ) ⊗ g)G is called automorphic, if its elements a ∈ (C(λ) ⊗ g)G are invariant, g(a) = a, with respect to all automorphisms g ∈ G, i.e. (C(λ) ⊗ g)G = {a ∈ (C(λ) ⊗ g) | g(a) = a , ∀g ∈ G ⊂ Aut (C(λ) ⊗ g)}. Originally motivated by the problem of reduction of Lax pairs, Automorphic Lie Algebras are interesting objects in their own right. So much so that a classification is now both a mathematical question and a tool in the reduction problem, and therefore in applications to the theory of integrable systems and beyond. A first step towards classification of Automorphic Lie Algebras was presented in [LM05], where automorphic algebras associated to finite groups where considered. These groups are the five groups of Klein’s classification, namely, the cyclic group Z/n, the dihedral group Dn , the tetrahedral group T, the octahedral group O and the icosahedral group I. In [LM05] the authors classify automorphic algebras associated to the dihedral group Dn , starting from the finite dimensional algebra sl2 (C). Examples of Automorphic Lie Algebras based on sl3 (C) were also discussed. The aim of this paper is to complete the classification for the case g = sl2 (C) and sketch a classification programme for Automorphic Lie Algebras associated to finite groups more generally. A key feature of this approach is the study of these algebras in the context of classical invariant theory. Indeed, the problem of reduction can be formulated in a uniform way using the theory of invariants. This gives us a powerful tool of analysis on one hand; on the other it opens the road to new applications of these algebras, beyond the context of integrable systems. Moreover, it turns out that in the explicit case we present here, where the underlying Lie algebra is sl2 (C), we can compute Automorphic Lie Algebras only using geometric data. We prove in particular that Automorphic Lie Algebras associated to the Platonic groups T, O, I and Dn are isomorphic in the α case, where α stands here for the orbit of the group G defined by the zeros of the invariant α. Indeed, the following main result holds: Theorem. Let G be either one of the groups T, O, I or Dn . Then, the Automorphic Lie Algebras sl2 (C; α)G are isomorphic as Lie algebras. Here and in what follows sl2 (C; φ)G denotes the G–Automorphic Lie Algebra based on sl2 (C) with homogeneous coefficients having poles at the zeros of φ. This result allows us to provide a complete classification of sl2 (C)–based Automorphic Lie Algebras associated to finite groups when the group representations are chosen to be the same and it is a crucial step towards the complete classification of Automorphic Lie Algebras. This fact, i.e. that the Automorphic Lie Algebras are independent from the group is not quite what one would expect from the topological point of view. Indeed, if one divides out the group action one usually obtains an orbifold, but a manifold in the case of I. This distinction is not visible at the level of the algebra and therefore not on the level of the integrable systems that follow from the reduction procedure. In the next section we sketch the origin of Automorphic Lie Algebra and illustrate their relation with the theory of integrable systems. 1.2. Algebraic reductions and Automorphic Lie Algebras. Many integrable equations are obtained as reductions of larger systems. The fact that this is true for many equations of interest in applications makes of the reduction problem one of the central problems in the theory of integrable systems since its early days. A wide class of (algebraic)
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reductions can be studied in terms of reduction groups [Mik81], that is, reductions can be associated to a discrete symmetry group of the corresponding linear problem (Lax pair), either given by the physical system or simply forced on the solutions. The simplest example of such a symmetry is the conjugation for self-adjoint operators. The requirement that a Lax pair is invariant with respect to a reduction group imposes certain algebraic constraints on the Lax operators and therefore it yields a reduction. As an illustration, consider for instance a fairly general Lax pair L = 0, M = 0, where L = ∂x − X (x, t, λ),
M = ∂t − T (x, t, λ),
and 1 X (x, t, λ) = X 0 (x, t) + X 1 (x, t)λ + X −1 (x, t) , λ 1 1 T (x, t, λ) = T0 (x, t) + T1 (x, t)λ + T−1 (x, t) + T2 (x, t)λ2 + T−2 (x, t) 2 λ λ are n × n matrix functions of x, t and of the spectral parameter λ. The consistency condition t x = xt implies that (for all values of λ) X t − Tx + [X, T ] = 0, i.e. a system of 5 n 2 nonlinear differential equations amongst the entries of the matrices of the Lax pair. A natural question arises: how to reduce it in a systematic way? The system can be reduced imposing symmetry conditions: considering, as an example, reductions associated to the dihedral group Dn , the symmetry constraints read X (λ) = S X (ωλ) S −1 , T (λ) = S T (ωλ) S −1 ,
X (λ) = −X T (1/λ), ωn = 1, T (λ) = −T T (1/λ)
(see [Mik81,LM04] for details). Solutions of the reduced system have been recently investigated in [BM]. This purely algebraic reduction technique, first formulated by Mikhailov (see [Mik81]) and later developed in [MSY87,LM05], has been successfully applied both in classical (e.g. [GKV07a,GKV07b,GGK01,GGIK01,HSAS84,LM04]) and quantum integrable systems theory (e.g. [Bel80,Bel81]) and it essentially consists of finding invariant elements of the Lie algebra over the ring of rational functions used in the Lax pair representation of an integrable equation. These invariant elements form an infinite dimensional Lie algebra known as the Automorphic Lie Algebra [LM05]. Indeed, a reduction group can be seen also as a group representation G of a (sub)group G of the group of automorphisms of the infinite dimensional Lie algebra underlying the Lax pair, e.g. G ⊂ Aut (C(λ)⊗g), where we recall that g ⊂ gln is a finite dimensional semisimple Lie algebra and C(λ) is the field of rational functions in the complex variable λ with values in C (see [LM05] for details). In this latter framework reductions corresponding to G are nothing but a restriction of the Lax pair to the corresponding automorphic (i.e. invariant) subalgebras (C(λ) ⊗ g)G . The paper is organised as follows: in the Introduction we recall basic definitions and facts from the theory of Automorphic Lie Algebras and consider them in the framework of algebraic reductions of integrable systems. In Sect. 2 the set up is defined. Section 3 describes the tool of transvection and recalls basic notions from invariant theory; here we prove a few lemmas in full generality for later use. Section 4 deals with the classification of sl2 (C)–based Automorphic Lie Algebras associated to finite groups. The framework
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is defined in Sects. 4.1, 4.2 and 4.3; in particular covariant algebras gˆ G are derived in Sect. 4.2 while Sect. 4.4 describes the general form of their structure constants. In Sect. 5 we define homogeneous elements, therefore we go from the variables X and Y to λ = YX (or λ = YX ); fixing an orbit of the group G we define a basis of homogeneous elements with –divisors and their structure constants (Sect. 5.3). The normal form of sl2 (C)– based Automorphic Lie Algebras is given in Sect. 6. Section 7 compares the results with the averaging method, while Sect. 8 gives explicit expressions for homogeneous bases associated to finite groups. The last Sect. 9 contains concluding remarks while the Appendices contain the case of Z/n (see Appendix A) and the detailed derivation of homogeneous elements (see Appendices B). The case a of different –divisor is the content of the last Appendix C. Many of the necessary computations were done using the FORM package [Ver00].
2. Set Up We consider automorphic algebras in the context of classical invariant theory, that is the study of invariants of the action of sl2 (C) on binary forms. It turns out that in the explicit case we present here, where the underlying Lie algebra is sl2 (C), we can compute the Automorphic Lie Algebra only using geometric data. Let us define the set up: • Let g be a Lie algebra and let ρk be a representation of g in glk (C); • Let C[X, Y ] be the ring of polynomials in X and Y (later we will define λ = λ = YX ); • Consider gˆ = C[X, Y ] ⊗ g and let ρˆk be the representation induced by ρk :
X Y
or
ρˆk ( p(X, Y ) ⊗ m) = p(X, Y ) ⊗ ρk (m); • Let G be a finite group; let σ be a faithful projective representation of G in C2 , that is σ is a mapping from G to G L 2 (C) obeying the following: σ (g) σ (h) = c(g, h) σ (g, h), ∀g, h ∈ G, where c(g, h) : G × G → C is a 2-cocycle over C , the multiplicative group of C (see for example [Zhm92]). This restricts G to be one of the following groups [Kle56,Kle93]: Z/m, Dm , T , O , I,
(1)
i.e. the cyclic group Z/m, the dihedral group Dm , the tetrahedral group T, the octahedral group O and the icosahedral group I; we refer to Dm , T, O, I as to Platonic groups; • Let τk be a faithful projective representation of G in Ck ; • Define the linear representation πk by πk (g) M = τk (g) M τk (g −1 ),
M ∈ glk (C).
It follows then that πk (g)ρk ([m 1 , m 2 ]) = [πk (g)ρk (m 1 ), πk (g)ρk (m 2 )], ∀ m i ∈ g; • σ ⊗ πk defines a projective representation of G in C[X, Y ] ⊗ glk (C).
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In a diagram the situation can be represented as follows: G
σ G L 2 (C)
πk
? Aut (glk (C)) Let g ∈ G, mˆ = p(X, Y ) ⊗ m ∈ gˆ . Composing the Lie algebra representation and the group representation we write ˆ = p(σ (g −1 )(X, Y )) ⊗ τk (g) ρk (m) τk (g −1 ). (σ ⊗ πk )(g) ρˆk (m) In this paper however we will be concerned with the case g = sl2 (C) and we take k = 2 and τ2 = σ . Furthermore we write ρ2 = ρ, the standard representation of sl2 (C). These simplifying assumptions can no longer be made when considering higher dimensional Lie algebras. See the discussion in Sect. 9. Let χ be a one-dimensional representation of G. Definition 2.1. Let V be a G-module. Then v ∈ V is χ –covariant if g v = χ (g) v, ∀g ∈ G. If χ is trivial then v is called invariant. In the literature, covariants are also called semi-invariants, or relative invariants. We χ denote the space of χ –covariants as VG and the space of covariants by VG . Our first goal is to compute the Lie algebra of covariants gˆ G . Let g be sl2 (C) and let e+ , e− , e0 be a basis, obeying the relations: [e+ , e− ] = e0 , [e0 , e± ] = ±2e± . Theorem 2.1. For all G in the list (1), A = Y 2 e− + X Y e0 − X 2 e+ ∈ gˆ G , i.e. A is invariant. ∂ Proof. Since (ad(e− ) − X ∂Y )A = 0 and (ad(e+ ) − Y ∂∂X )A = 0, the form is invariant under the usual action of S L 2 (C) on binary forms.
Remark 2.1. The theorem holds true in the case that G = S L 2 (C), that is a X + bY X ab . = , σ τ2 = cX + dY Y cd Remark 2.2. This is just the adjoint representation of sl2 (C); it is not clear to us how this object could be conceived in λ-language, since it cannot be associated to a homogeneous invariant element, as is shown in Sect. 5. See however Remark 3.1, Sect. 3. 3. Transvectants In classical invariant theory the basic computational tool is the transvectant: given any two covariants, it is possible to construct a number of (possibly) new covariants by computing transvectants. As a simple example consider two linear forms aY + bX , cY + d X ; their first transvectant is the determinant of the coefficients, i.e. ad − cb. Similarly, the discriminant a0 a2 − a12 of a quadratic form a0 Y 2 + 2a1 X Y + a2 X 2 is the second transvectant of the quadratic form itself.
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In this section we will adapt the idea of transvection to compute invariant algebras. We start from the classical work by Klein about automorphic functions and generalise it to the context of automorphic algebras. To do so, we need first to recall some definitions and facts about transvectants and generalise some of the concepts to the present set up. Definition 3.1. A groundform is a covariant α with its divisor of zeros equal to an exceptional (or degenerate) orbit. ω 0 Example 3.1. Let ω be an elementary 3rd root of unity and let σ (s) = , 0 ω−1 0i σ (r ) = define a projective representation of the group D3 . To obtain an excepi 0 tional orbit we start with an eigenvector of either σ (s) or σ (r ); take for instance the σ (s) eigenvector (1, 0), corresponding to X ; it follows then that σ (s)(1, 0) = ω(1, 0) and σ (r )(1, 0) = i(0, 1), corresponding to X and Y , respectively. The exceptional orbit consists therefore of X and Y and we take as a groundform its multiplicative average X Y . Consider now an eigenvector of σ (r ), e.g. (1, 1), then the exceptional orbit under σ (s) is given by X + Y , ωX + ω−1 Y , ω−1 X + ωY , and the groundform is nothing but its multiplicative average (X + Y )(ωX + ω−1 Y )(ω−1 X + ωY ) = X 3 + Y 3 . Similarly, starting with the eigenvector (1, −1), one obtains the groundform X 3 − Y 3 . The computations of groundforms for all other groups of the list (1) can be found, for instance, in [Dol], [Lam86, II.6] and [Kle93]. χ
Definition 3.2. Let φ ∈ Cn [X, Y ]G and let φk,l = tant of φ with an arbitrary form F ∈ Fkφ = (φ, F)k =
k i=0
ψ gˆ G
(−1)i
∂ k+l φ ; ∂ X k ∂Y l
we define the k th –transvec-
as
k φi,k−i Fk−i,i , i
χψ
Fkφ ∈ gˆ G .
Example 3.2. (Classical Invariant Theory) In the definition above F could as well belong to Pm [X, Y ]. It follows from the classical theory [Kle56,Kle93] that if G is either T, O or I then the groundforms are given by α, β = (α, α)2 , γ = (α, β)1 . If one denotes the degree of a form α by ωα it follows that (see Table 1) ωβ = 2 ωα − 4, ωγ = 3 ωα − 6. If G is Dm then β = (α, α)2 and it has to be computed independently (see Table 2). The degree of β is the number of faces of the Platonic solid and determines its name. We observe that ωα − ωγ + ωβ = 2, the Euler index.
Table 1. Degrees of the groundforms of I, O, T G I O T
ωα 12 6 4
ωβ = 2ωα − 4 20 8 4
ωγ = 3ωα − 6 30 12 6
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Table 2. Degrees of the groundforms of Dm ωα 2
G Dm
ωβ m
ωγ = ωα + ωβ − 2 m
Remark 3.1. In λ-language, β corresponds to the Schwarzian of α. Lemma 3.1. Fkφ = k!X −k
k ωφ − k + l ωF − l φ0,k−l F0,l . (−1)l l k −l l=0
Proof. See [Olv99, p. 90], and references [Gun86] and [Ovs97] therein.
Lemma 3.2. Let F be an invariant form and let φ ∈ C[X, Y ] with ωφ ≥ 2. Then one has F1φ = X −1 ωFφY F − ωφ φFY , (2) ωF ωφ (3) φY Y F − (ωφ − 1)(ωF − 1)φY FY + φFY Y , F2φ = 2X −2 2 2 Fiφ = 0 for i ≥ 3 if F is quadratic. Proof. It follows immediately from Lemma 3.1.
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Example 3.3. Let F be the invariant form A given in Theorem 2.1 and let φ be α = X Y , which happens to be a groundform of the dihedral group Dm (see Sect. 4.3.4). It follows then that 0 −2 X 2 ρ(A1α ) = , −2 Y 2 0 −2 0 , ρ(A2α ) = 0 +2 ρ(Aiα ) = 0
for i ≥ 3,
where ρ is the standard representation in sl2 (C). Let g be the Lie algebra sl2 (C) with the usual basis and commutation relations: [e0 , e± ] = ±2e± , [e+ , e− ] = e0 .
(5) (6)
Lemma 3.3. Let A be the invariant form given in Theorem 2.1, then X −1 [A, AY ] = −2A, X −1 [A, AY Y ] = −2AY and X −1 [AY , AY Y ] = −2AY Y . Proof. We only prove the first relation, the other proofs are even simpler: X −1 [A, AY ] = X −1 [Y 2 e− + X Y e0 − X 2 e+ , 2Y e− + X e0 ] = 2Y [Y e0 − X e+ , e− ] + [Y 2 e− − X 2 e+ , e0 ] = Y 2 [e0 , e− ] − 2X Y [e+ , e− ] − X 2 [e+ , e0 ] = −2Y 2 e− − 2X Y e0 + 2X 2 e+ = −2A.
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Corollary 3.1. If we give A and the Y derivative odd grading, then the A, AY and AY Y form a Z/2-graded Lie algebra, with grading |A| = 1, |AY | = 0, |AY Y | = 1. j
Theorem 3.1. Aφ , j = 0, 1, 2 span a Z/2-graded Lie algebra with the following commutation relations: [A1φ , A] = −2ωφ φ A, [A, A2φ ] = 2(ωφ − 1) A1φ , [A1φ , A2φ ] = 2ωφ φ A2φ −
4 (φ, φ)2 A. ωφ − 1
j
Proof. Express Aφ , j = 1, 2 in A, AY and AY Y : see Lemma 3.2 and use Lemma 3.3 to j
compute the commutators. The induced grading is |Aφ | = 1 + |φ| + j mod 2, where |φ| is to be defined in Definition 4.2. Remark 3.2. The research leading to this paper started with the observation that if one defined (using transvection) A0 = −2Y α X e− − (X α X − Y αY ) e0 − 2X αY e+ , A− = Y 2 e− + X Y e0 − X 2 e+ , A+ = −α 2X e− + α X αY e0 + αY2 e+ , where α is the groundform of Example 3.2, then one has [A+ , A− ] = ωα α A0 , [A0 , A± ] = ±2ωα α A± . Using the methods described in Sect. 5 one can now construct an Automorphic Lie Algebra. This observation was presented at the NEEDS 2009 conference. What it is not clear from this construction, however, is that this is the only possible Automorphic Lie Algebra. The rest of the paper is intended to show that this is indeed the case. For later use (cf. Remark 8.1) we mention that det(ρ(A± )) = 0 and det(ρ(A0 )) = −ωα2 α 2 . 4. Automorphic Lie Algebras Associated to Finite Groups Let G ⊂ Aut (C(λ) ⊗ g). A Lie algebra (C(λ) ⊗ g)G is called automorphic, if its elements a ∈ (C(λ) ⊗ g)G are invariant g(a) = a with respect to all automorphisms g ∈ G, i.e. (C(λ) ⊗ g)G = {a ∈ (C(λ) ⊗ g) | g(a) = a , ∀g ∈ G ⊂ Aut (C(λ) ⊗ g)}. We consider G to be a finite group; in particular, let G be one of the groups in the list (1); we aim for a complete classification for the case g = sl2 (C) using geometric data. This leads us to sketch a classification programme for Automorphic Lie Algebras associated to finite groups more in general, that is beyond sl2 (C). A key feature of this approach is the study of these algebras in the context of classical invariant theory. Indeed, the problem of reduction can be formulated in a uniform way using the theory of invariants. We consider first the problem of invariants starting from C[X, Y ]; through a homogenisation we will then map it to C(λ), where λ = X/Y (or λ = Y/ X ). Our first task is to compute the Lie algebra of covariants gˆ G .
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4.1. The Trace-form. Given a representation ρ of sl2 (C), we can define the trace-form ·, · by X, Y = tr (ρ(X)ρ(Y)). Lemma 4.1. Let ρ be thestandard representation in gl2 (C) and φ, ψ ∈ C[X, Y ] with X Y −X 2 and ωφ,ψ ≥ 2. Then ρ(A) = Y 2 −X Y 1. 2. 3. 4. 5. 6.
A, A = 0. A, A1φ = 0. A1φ , A1ψ = 2 ωφ ωψ φ ψ. A, A2φ = −4 ω2φ φ. A1φ , A2ψ = −4(ωψ − 1)(φ, ψ)1 . A2φ , A2ψ = −4(φ, ψ)2 .
Proof. We prove here items 5 and 6 and leave the other relations to the reader. −2X φY −ψ X Y −ψY Y −X φ X + Y φY 1 2 Aφ , Aψ = 2 tr −2Y φ X −Y φY + X φ X ψX X ψX Y = 4(−φY (Y ψ X Y + X ψ X X ) + φ X (X ψ X Y + Y ψY Y )) = 4(ωψ − 1)(−φY ψ X + φ X ψY ) = −4(ωψ − 1)(φ, ψ)1 , −φ X Y −φY Y −ψ X Y −ψY Y A2φ , A2ψ = 4tr φX X φX Y ψX X ψX Y = 4 (−φY Y ψ X X + 2φ X Y ψ X Y − φ X X ψY Y ) = −4(φ, ψ)2 . Observe that apparently | A, B | = |A| + |B| mod 2.
Example 4.1. Consider the same setting as in Example 3.3, namely A is the invariant j form in Theorem 2.1 and α = X Y ; it follows then that all trace-forms Aiα , Aα vanish with the exception of 0 −2X 2 0 −2X 2 1 1 Aα , Aα = tr −2Y 2 0 −2Y 2 0 = 4X 2 Y 2 + 4X 2 Y 2 = 2 ωα2 α 2 , −2 0 −2 0 A2α , A2α = tr 0 2 0 2 = 4+4 = −4(α, α)2 . Remark 4.1. Instead of the trace-form, one could proceed in a more abstract fashion and use the Killing form of sl2 (C). One can then compute the Killing form on the Y -derivatives of A and, by C[X, Y ]-linearity, on the transvectants of A.
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4.2. Stanley and Clebsch-Gordan decompositions. An essential step in the construction of the algebra of covariants gˆ G is to find a basis for the covariant matrices. This is done in this section by tensoring the Stanley basis of the covariant polynomials with the selfadjoint representation of sl2 (C). Definition 4.1. (Spherical polynomial rings) Let σ be a faithful projective representation of a finite group G on C2 . Let α, β ∈ C[X, Y ]G and γ = (α, β)1 and assume that every covariant can be written as an element in C[α, β, γ ]. As before we consider α and β as the even elements, and γ as the odd one (this supposes that γ ∈ / C[α, β]). We assume (α, γ )1 ∈ C[α, β] and (β, γ )1 ∈ C[α, β] and we notice that ωγ γ 2 = ωβ β(α, γ )1 − ωα α(β, γ )1 ∈ C[α, β]. This implies that the ring of covariants has the following Stanley decomposition: C[X, Y ]G = C[α, β] ⊕ C[α, β]γ .
(7)
We call such a ring a spherical polynomial ring. Remark 4.2. For I, O and T, the Z/2-grading of β is automatically 0. In the case of Dm , where β is not the second transvectant of α, we put |β| = 0. In both cases the Z/2-grading of γ is 1. Theorem 4.1. (Molien, [Mol97]) Let σ : G → G L 2 (C) be a representation of a finite group G over C. Then the Poincaré series of the ring of invariants is given by P(C[X, Y ]G , t) =
1 1 . |G| det(1 − σ (g −1 ) t) g∈G
Proof. See [Smi95, Sect. 4.3].
Remark 4.3. The function P(C[X, Y ]G , t) is called the Molien function of the invariants of the representation σ . The Molien functions of the covariants of the faithful projective representations of the Platonic groups follow from (7): P(C[X, Y ]G , t) =
1 + t ωγ . (1 − t ωα )(1 − t ωβ )
Theorem 4.2. The faithful projective representations of the Platonic groups give rise to spherical polynomial rings of covariants. Remark 4.4. The group Z/n does not fit in since α = X , β = Y , and so γ is constant. The ring of covariants is C[α, β]. The covariant Lie algebra is spanned by A2α 2 , A2αβ and A2β 2 , that is, by the orginal sl2 (C). The details are given in Appendix A. Definition 4.2. We assign a Z/2-grading on (C[α, β] ⊕ C[α, β](α, β)1 ) ⊗ A by setting |α| = 0 (|β| = 0 in the case of Dm ), |A| = 1, |(φ, ψ) j | = |φ| + |ψ| + j mod 2.
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Theorem 4.3. Let A be an invariant quadratic form with coefficients in a G-module and assume α, β ∈ C[X, Y ]G with ωα,β ≥ 2 and γ = (α, β)1 ∈ / C[α, β]. Then it follows from the Clebsch-Gordan decomposition theorem (see [SVM07]) that (C[α, β] ⊕ C[α, β]γ ) ⊗ sl2 (C) = C[α, β] ⊗ (A ⊕ A2α ⊕ A1α ⊕ A2β ⊕ A1β ⊕ A2γ ). (8) Proof. Let φ ∈ C[X, Y ] be a polynomial in X and Y such that ωφ ≥ 2. According to the Clebsch-Gordan decomposition theorem one has φ ⊗ A = φ A ⊕ A1φ ⊕ A2φ . We compute the direct sum decomposition of (C[α, β] ⊕ C[α, β]γ ) ⊗ A; to this end we ω introduce the following hybrid notation: we associate φ with its Poincaré function tφ φ 2 . Then the Clebsch-Gordan decomposition becomes and, similarly, A with tA ω
ω
ω
ω −2
2 2 = tφ φ tA + tAφ1 + tAφ2 tφ φ ⊗ tA φ φ
.
We are now ready to compute the generating function of the left and right hand side of (8): ω
1 + tγ γ
2 ⊗ tA ω (1 − tαωα )(1 − tβ β ) ω ω tβ β tγ γ tαωα 2 = 1+ ⊗ tA ωβ + ωβ + ωβ ωα ωα 1 − tβ (1 − tα )(1 − tβ ) (1 − tα )(1 − tβ ) ⎞ ⎛ ωβ ω ωα ω tA1 tAγ1 tA 1 1 + tγ γ γ β α 2 ⎠ ⎝ + = ω tA + ω + ω ω (1 − tαωα )(1 − tβ β ) 1 − tβ β (1 − tαωα )(1 − tβ β ) (1 − tαωα )(1 − tβ β ) ⎛ ω −2 ⎞ ωγ −2 ωα −2 tAβ2 t tA 2 A2γ ⎜ β ⎟ α +⎝ ωβ + ωβ + ω ⎠ ωα ωα 1 − tβ (1 − tα )(1 − tβ ) (1 − tα )(1 − tβ β ) ω
=
ω −2
2 + t β + t ωα + t β tA A1α A1β A2β
ω −2
2 +t β (1 + t ωα −2 )(tA A2
β
ω −2
− tαωα tAβ2
β
ω −2
ωα −2 + tA + tAγ2 2 α
γ
ω −2
ωα −2 + tA + tAγ2 2 α
γ
ω (1 − tαωα )(1 − tβ β ) ω −2
=
ω
ω
ω (1 − tαωα )(1 − tβ β )
ω
=
ω
2 + t γ t 2 + t β − t ωα t β + t ωα + t γ + t β tA γ A α A1 A1α A1γ A1β A2β β
ωα −2 + tA ) 2
ω (1 − tαωα )(1 − tβ β )
α
.
Remark 4.5. The factorization in the last step seems to be connected to Corollary 4.1. See also [Spr87, Sect. 8].
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The underlying relations which are used to get rid of the minus signs are: 2γ A = ωβ β A1α − ωα α A1β ,
(ωα − 1)(ωβ − 1) 1 ωα − ωβ ωβ ωα 2 2 Aγ = (α, β) A + β Aα − α A2β , ωγ ωγ 2 2 where we know that (α, β)2 A ∈ C[α, β]A ⊕ C[α, β]γ A ⊂ C[α, β](A ⊕ A1α ⊕ A1β ). Now that the counting is in order, it suffices to show that the right-hand side is indeed a direct sum. To this end assume that F1 (α, β)A + F2 (α, β)A2α + F3 (α, β)A1α + F4 (α, β)A2β + F5 (α, β)A1β + F6 (α, β)A2γ = 0. Then, taking the trace-form with A, we obtain 0 = A, F2 (α, β)A2α + A, F4 (α, β)A2β + A, F6 (α, β)A2γ ωα ωβ ωγ α F2 (α, β) − 4 β F4 (α, β) − 4 γ F6 (α, β). = −4 2 2 2 This implies F6 = 0 and ω2α α F2 (α, β) + ω2β β F4 (α, β) = 0. Next we take the traceform with A1α to obtain 0 = A1α , F3 (α, β)A1α + A1α , F4 (α, β)A2β + A1α , F5 (α, β)A1β = 2ωα2 α 2 F3 (α, β) − 4(ωβ − 1)γ F4 (α, β) + 2ωα ωβ αβ F5 (α, β). This implies F4 = 0 (and therefore F2 = 0) and ωα α F3 (α, β) + ωβ β F5 (α, β) = 0. Finally, taking the trace-form with A2α , we obtain 0 = F1 (α, β)A, A2α + F5 (α, β)A1β , A2α ωα α F1 (α, β) + 4(ωβ − 1)γ F5 (α, β). = −4 2 This implies F5 = F1 = 0, and therefore F3 = 0. This shows that indeed the sum is direct, as claimed. This concludes the S L 2 (C) part of the proof. Finally, the Molien function for each group I, O, T and Dm for the given matrix representation has been computed and it coincides with our result in all cases. The Lie algebra generated by A, A2α ,A1α , A2β , A1β , A2γ with coefficient ring C[α, β] is the algebra of covariants gˆ G , where we recall that gˆ = C[X, Y ] ⊗ g. Corollary 4.1. It follows from the grading that ((C[α, β] ⊕ C[α, β](α, β)1 ) ⊗ A)0 = C[α, β] ⊗ (A1α ⊕ A1β ⊕ A2γ )
(9)
((C[α, β] ⊕ C[α, β](α, β)1 ) ⊗ A)1 = C[α, β] ⊗ (A ⊕ A2α ⊕ A2β ).
(10)
and
It also follows from Theorem 3.1 that C[α, β] ⊗ (A1α ⊕ A1β ⊕ A2γ ) is a Lie subalgebra. We find later that these are the only elements that can be mapped to invariant homogeneous elements with divisors α or β in the Automorphic Lie Algebra (see Sect. 5).
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4.3. Description of the group actions. We describe the group action on C2 for later use. In those cases where the action is not given explicitly, it is given implicitly, since it permutes the zeros of the groundforms. Notice that the degrees of the groundforms correspond with the number of vertices, edges and faces in the cases of the groups I, O and T. 4.3.1. Icosahedral group I An icosahedron is a convex regular polyhedron (a Platonic solid) with twenty triangular faces, thirty edges and twelve vertices. A regular icosahedron has sixty rotational (or orientation-preserving) symmetries; the set of orientation-preserving symmetries forms a group referred to as I; I is isomorphic to A5 , the alternating group of even permutations of five objects. As an abstract group it is generated by two elements, s and r , satisfying the identities s 5 = r 2 = id,
(s r )3 = id.
The I–groundforms are given by α = X Y (X 10 + 11X 5 Y 5 − Y 10 ), β = (α, α)2 and γ = (α, β)1 . Since the degree of α, ωα , is equal to 12 it follows that ωβ = 2 ωα −4 = 20 and ωγ = 3 ωα − 6 = 30. 4.3.2. Octahedral group O. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex; it has six vertices and eight edges. A regular octahedron has twenty-four rotational (or orientation-preserving) symmetries. A cube has the same set of symmetries, since it is its dual. The group of orientationpreserving symmetries is denoted by O and it is isomorphic to S4 , or the group of permutations of four objects, since there is exactly one such symmetry for each permutation of the four pairs of opposite sides of the octahedron. As an abstract group it is generated by two elements, s and r , satisfying the identities s 4 = r 2 = id,
(s r )3 = id.
The classical O–groundforms are given by α = X Y (X 4 − Y 4 ), β = (α, α)2 and γ = (α, β)1 . Since ωα = 6 it follows that ωβ = 2 ωα − 4 = 8 and ωγ = 3 ωα − 6 = 12. 4.3.3. Tetrahedral group T. A regular tetrahedron is a regular polyhedron composed of four equilateral triangular faces, three of which meet at each vertex. It has four vertices and six edges. A regular tetrahedron is a Platonic solid; it has twelve rotational (or orientation-preserving) symmetries; the set of orientation-preserving symmetries forms a group referred to as T, isomorphic to the alternating subgroup A4 . As an abstract group it is generated by two elements, s and r , satisfying the identities s 3 = r 2 = id,
(s r )3 = id.
√ The T–groundforms are given by α = X 4 − 2i 3X 2 Y 2 + Y 4 , β = (α, α)2 and γ = (α, β)1 . Since ωα = 4 it follows that ωβ = 2 ωα − 4 = 4 and ωγ = 3 ωα − 6 = 6.
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4.3.4. Dihedral group Dn . Let us now turn our attention to the symmetry group of a dihedron, the dihedral group Dn ; this is fundamentally different from the previous cases, since β = (α, α)2 and it has to be computed independently; however the rest of the procedure is the same. As in the case of Z/n (see Appendix A), in order to get an action of Dn on the spectral parameter λ we need to act with Dm on the X, Y -plane, where m = n if n is odd, and m = 2n when n is even. The dihedral group Dm is the group of rotations and reflections of the plane which preserve a regular polygon with m vertices. It is generated by two elements, s and r , satisfying the identities s m = r 2 = id,
r s r = s −1 .
ω 0 , with 0 ωm−1
We take a projective representation of the group, defined by σ (s) = 0i th . Let α = X Y , β = 21 (X m + Y m ) ω an elementary m root of unity, and σ (r ) = i 0 and γ = (α, β)1 , i.e. γ = (α, β)1 = αY β X − α X βY =
m m (X − Y m ), 2
that is ωα = 2, ωβ = m, ωγ = 2 + m − 2 = m and one has γ 2 − m 2 β 2 = −m 2 α m . Then σ (s)α = α, σ (r )α = i 2 α, σ (s)β = β, σ (r )β = i m β. The action of γ is given by the product of α and β. Example 4.2. (Dn , with n = 2) Let α = X Y and let A be the invariant form given in Theorem 2.1; recall also Example 3.3; one has 0 −2 X 2 ρ(A1α ) = , −2 Y 2 0 4 2Y − 2X 4 −4X Y 3 , ρ(A1β ) = −4X 3 Y 2X 4 − 2Y 4 0 48Y 2 ρ(A2γ ) = . 48X 2 0
4.4. Structure constants. In this section we use the transvectant formula and the traceform to derive the commutation relations of the even part of the covariant algebra gˆ G . We first present a few technical lemmas, to be proven by checking them for each group. Lemma 4.2. For the groups T, O, I and Dm , pG (β) = Proof. See Table 3.
(α,γ )1 β
∈ C[β].
Table 3. pG (β) for the groups I, O, T and Dm G pG (β)
I − 300 121 β
O 48 β − 25
T − 43 β
Dm m2
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Lemma 4.3. For the groups I, O, T and Dm , qG (α) = Proof. See Table 4.
(β,γ )1 α
∈ C[α].
Corollary 4.2. ωγ γ 2 = −ωα α 2 qG (α) + ωβ β 2 pG (β). Lemma 4.4. For the groups T, O, I and Dm , 2(ωγ − 1)2 pG (β)qG (α) + ωα ωβ (γ , γ )2 = 0.
(11)
Proof. See Table 5 for explicit expressions of (γ , γ )2 . Substituting these relations and those of Tables 3 and 4 into (11) proves the lemma. The following lemma will be used later in Theorem 8.1. Lemma 4.5. 2(ωα − 1)2 pG (β) = −ωβ ωγ (α, α)2 . Proof. By inspection of Table 3.
We are now ready to derive the commutation relations of the even part of the covariant algebra gˆ G : Theorem 4.4. The commutation relations on C[α, β] ⊗ (A1α ⊕ A1β ⊕ A2γ ) are [A1α , A1β ] = 2ωβ βA1α − 2ωα αA1β ,
(12)
pG (β) 1 Aβ + 2ωα αA2γ , ωβ q (α) 1 [A1β , A2γ ] = 4(ωγ − 1) G Aα + 2ωβ βA2γ . ωα [A1α , A2γ ] = 4(ωγ − 1)
(13) (14)
Proof. For the first commutation relation (12) one has [A1α , A1β ] = [(α, A)1 , (β, A)1 ] = X −2 [2αY A − ωα αAY , 2βY A − ωβ βAY ] = X −2 (−2ωβ αY β[A, AY ] − 2ωα αβY [AY , A]) = −2X −1 (−2ωβ αY β + 2ωα αβY )A = 2ωβ β A1α − 2ωα α A1β . Table 4. qG (α) for the groups I, O, T and Dm G qG (α)
I −101198592000 α 3
O 34560000 α 2
T √ −3538944 i 3 α
Dm 1 3 m−2 2m α
Table 5. (γ , γ )2 for the groups T, O, I and Dm G (γ , γ )2
I −1758430080000 α 3 β
O 334540800 α 2 β
T √ −14745600 i 3 α β
Dm − 21 m 4 (m − 1)2 α m−2
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It follows from this relation that [A1α , A1β ], A2γ = −8ωβ (ωγ − 1)β(α, γ )1 + 8ωα (ωγ − 1)α(β, γ )1 = 8ωγ (ωγ − 1)γ 2 .
(15)
Consider next the matrix of the trace-forms of the basis A1α , A1β and A2γ :
⎞ 2ωα ωβ αβ −4(ωγ − 1)(α, γ )1 2ωα2 α 2 ⎝ 2ωα ωβ αβ 2ωβ2 β 2 −4(ωγ − 1)(β, γ )1 ⎠ ; −4(ωγ − 1)(α, γ )1 −4(ωγ − 1)(β, γ )1 4(γ , γ )2 ⎛
its determinant is −32(ωγ − 1)2 ωγ2 γ 4 . This implies that the trace-form is nondegenerate and that we can prove the commutation relations by taking the trace-form with the basis elements. When this results in identities, we have a proof. Taking the trace-form of (13) with A1α we see that it trivialises (i.e. it is equal to zero): 0 = 4(ωγ − 1)
(α, γ )1 1 1 Aα , Aβ + 2ωα α A1α , A2γ ωβ β
= 8ωα (ωγ − 1)(α, γ )1 α − 8ωα (ωγ − 1)α(α, γ )1 . Taking the trace-form of (13) with A1β results in the relation for γ 2 : A1β , [A1α , A2γ ] = 4(ωγ − 1)
(α, γ )1 1 1 Aβ , Aβ + 2ωα α A1β , A2γ ωβ β
implies 8ωγ (ωγ − 1) γ 2 = 8ωβ (ωγ − 1)(α, γ )1 β − 8(ωγ − 1)ωα α (β, γ )1 . Taking the trace-form of (13) with A2γ results in 0 = −2(ωγ − 1)2 (α, γ )1 (β, γ )1 − ωα ωβ α β (γ , γ )2 , and this follows from Lemma 4.4. The proof of (14) follows exactly the same pattern. Remark 4.6. The commutation relations show that they only depend on the relation among the groundforms indicating that they are determined by the geometry of the curve given in Corollary 4.2. So far the covariant matrices are functions of X and Y ; in the following we define invariant homogeneous elements in the local coordinate λ = X/Y (or λ = Y/ X ). 5. Towards Lax Pairs: Homogenisation χ
Suppose we are given an element mˆ ∈ gˆ G , with mˆ of degree k in X and Y and χ φ ∈ C[X, Y ]G of the same degree; we can now consider mφˆ as an element in g with coefficients in the field of rational functions C(λ), where λ = X/Y . To map gˆ G to the zero-homogeneous part of C(α, β) ⊗ (A1α ⊕ A1β ⊕ A2γ ) we proceed as follows.
On the Classification of Automorphic Lie Algebras
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5.1. The groups T, O, I. To obtain an homogeneous element we first of all need to choose an automorphic function to start with. This is equivalent to fixing the poles of our G–invariant elements. In other words, this choice corresponds to the choice of an orbit of the group G (see [LM05]). Let us choose α (the β-choice is treated in Appendix C) and consider j
Aφ =
β m2 j A αm1 φ
m 1 ∈ Z>0 , m 2 ∈ Z≥0 ,
where A01 ≡ A and the overline notation indicates division by α. Therefore, for each j Aφ , we need to solve the diophantine relation 2 − 2 j + ωφ + 2m 2 (ωα − 2) = m 1 ωα , m 1 ∈ Z>0 , m 2 ∈ Z≥0 , where we denote ωφ=1 = 0 and we recall that ωβ = 2ωα − 4, ωγ = 3ωα − 6. This allows us to define homogeneous elements. Suppose ωα = 0 mod 4, as is the case for G = T, I. Then for φ = 1, α, β we have j
2|Aφ | = 2 − 2 j = 0
mod 4, m 1 ∈ Z>0 , m 2 ∈ Z≥0 ,
j
This implies |Aφ | = 0, excluding A, A2α and A2β . To these equations we add the invariance requirement. On the covariants acts the abelianized group AG = G/[G, G]. Since the icosahedral group is perfect, AI is trivial and so is its action. Suppose α goes to χ α, then β goes to χ 2 β and γ to χ 3 γ . In the same manner Aiα goes to χ Aiα , i = 1, 2, Aiβ goes to χ 2 Aiβ , i = 1, 2 and A2γ to χ 3 A2γ . Furthermore χ should be compatible with the relation ωγ γ 2 = −ωα α 2 q(α) + ωβ β 2 p(α), which implies χ 4 = χ deg(qG ) . It follows that χ 3 = 1 when G = T, χ 2 = 1 when G = O and χ = 1 when G = I. And this in turn implies that I : α → α, β → β, γ → γ , O : α → χ α, β → β, γ → χ γ , T : α → χ α, β → χ 2 β, γ → γ ,
AI = 1; AO = Z/2; AT = Z/3.
This leads to T : 2m 2 + δφ = m 1 mod 3, δα = 1, δβ = 2, δγ = 0; O : δφ = m 1 mod 2, δα = 1, δβ = 0, δγ = 1. We now restrict to G = O: ωφ + 4m 2 = 3m 1 , m 1 ∈ Z>0 , m 2 ∈ Z≥0 , 2 O : δφ = m 1 mod 2, δα = 1, δβ = 0, δγ = 1. O : 1− j +
(16) (17)
Take φ = α and j = 2. Then m 1 is even by the first equation and odd by the second. This rules out A2α . With φ = β and j = 2, m 1 is odd by the first equation and even by the second. This rules out A2β . In the case j = 0 one has m 1 is odd by the first and even by the second equation. This rules out A.
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Remark 5.1. If one would have divided by β instead of α, the analysis leading to the excluded elements remains the same. j
j
Corollary 5.1. The matrices Aφ such that |Aφ | = 1 cannot be homogenized, neither by division by α nor β powers. i
5.2. The group Dn . As before, we consider expressions Aφ that are zero homogeneous in X, Y and invariant under the Dn group action. Let j
Aφ =
β m2 j A . αm1 φ
Then the homogeneity equation is m m 2 + 2 − 2 j + ωφ = 2m 1 m 2 ≥ 0, m 1 > 0, and the invariance equation is m m 2 + δφ = 2m 1
mod 4,
where δφ = ωφ for φ = 1, α, β, and δγ = ωβ + ωα . Recall that m = n if n is odd while m = 2 n if n is even. It follows that 2 − 2 j + ωφ = δφ
mod 4.
Since for φ = 1, α, β, δφ = ωφ , this rules out A, A2α and A2β . i
5.3. The homogeneous basis. Having defined the homogeneous elements Aφ in the previous section we can now define a basis over the ring C(IG ), where IG is defined below. Definition 5.1. By sl2 (C; φ)G we denote the G–Automorphic Lie Algebra based on sl2 (C) with homogeneous coefficients having poles at the zeros of φ. Theorem 5.1. A basis of sl2 (C; α)G is given by 1 A1 , ωα α α I 1 Aβ = G A1β , ωβ β 1
Aα =
2
Aγ = where IG =
ωβ β 2 pG (β) ωα α 2 qG (α)
(18) (19)
ωα α IG A2 , 4(ωγ − 1)β pG (β) γ
(20) 2
for G = I, O, T and Dn , n odd, and IG =
Dn , n even. In other words,
ωβ β 2 pG (β) ωα α 2 qG (α)
1+rG
= IG
1+rG
for G =
, where rG = 0 for G = I, O, T, Dn ,
n odd, and 1 for Dn , n even. If we, moreover, define JG = immediately that IG
ωβ β 2 pG (β) ωα α 2 qG (α)
= 1 + JG .
ωγ γ 2 ωα α 2 qG (α)
then it follows
On the Classification of Automorphic Lie Algebras
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Proof. The last identity follows from the definition of IG and JG and from Corollary 4.2 : 1+rG
IG
ωβ β 2 pG (β) ωβ β 2 pG (β) ωβ β 2 pG (β) ωγ γ 2 − = + 1 − = 1. ωα α 2 qG (α) ωα α 2 qG (α) ωα α 2 qG (α) ωα α 2 qG (α)
− JG = 1
1
That Aα and Aβ are well defined is clear from their definition once it is shown that IG is 2
indeed invariant (see Appendix B). We see that the degree of Aγ equals ωβ −ωα −deg(qG ) for G = I, O, T and Dn , n odd. By inspection we see that this indeed equals zero. For G = Dn , one has IDn = αβn and we see that indeed the degree of α n−1 is equal to the degree of γ − 2, that is, 2n − 2. The invariance equations for G = T, O are T : 2m 2 = m 1 mod 3, O : 1 = m 1 mod 2, and one easily verifies from Table 6 that they are satisfied. For Dn they are m m 2 + m + 2 = 2m 1
mod 4.
One verifies that the chosen scaling indeed satisfies these equations.
Corollary 5.2. If one computes the trace-form among the basis elements one finds 1
1
1
Aα , Aα = 2, 1
1
1
1
Aα , Aβ = 2IG ,
2
1
1−rG
2
Aβ , Aβ = 2IG , Aβ , Aγ = −IG
2
Aα , Aγ = −IG , 2
2
1−rG
, Aγ , Aγ = 21 IG
.
Remark 5.2. It is clear from Tables 6 and 7 that all homogeneous elements defined in (18)–(20) have poles at the zeros of α only. Table 6. α divisor: homogeneous elements of I, O, T G
I
IG
β3 1 24490059264 α 5 γ2 1 − 40479436800 α5 1 A1 12 α α β2 1 1 489801185280 α 5 Aβ β 2 1 − 586951833600 A α4 γ
JG 1
Aα 1
Aβ 2
Aγ
O β3 1 − 13500000 4 α
γ2 1 17280000 α 4 1 1 6 α Aα β2 1 1 − 108000000 A α4 β β 2 1 190080000 α 3 Aγ
T 1 √ β3 2654208 i 3 α 3 1 √ γ2 − 2359296 i 3 α 3 1 1 4 α Aα 1 √ β 2 A1 10616832 i 3 α 3 β 1 √ β A2 − 17694720 i 3 α 2 γ
Table 7. α divisor: homogeneous elements of Dn , for n even and odd G
Dn , n even
Dn , n odd
IG
β αn 1 γ2 (2n)2 α 2n 1 1 2 α Aα 1 1 1 2n α n Aβ 1 A2 1 2(2n)2 (2n−1) α n−1 γ
β2 αn 1 γ2 n2 αn 1 1 2 α Aα 1 β 1 n α n Aβ β 1 A2 2n 2 (n−1) α n−1 γ
JG 1
Aα 1
Aβ 2
Aγ
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Remark 5.3. The projective representation σ induces a linear representation on C(λ). This implies that mφˆ ∈ sl2 (C; φ)G , that is, it is invariant under the action of G. Remark 5.4. (Towards Lax pairs) Defining a Lax operator L ∈ sl2 (C; φ)G gives us a G–invariant (automorphic) Lax operator and therefore a G–invariant (automorphic) integrable systems of equations. Example 5.1. (Dn , with n = 2) With reference to Example 4.2 one has 1 0 −λ , ρ(Aα ) = 0 −λ−1 1−λ4 −1 −λ 1 2 2λ ρ(Aβ ) = , 4 −λ − 1−λ 2 2λ 1 0 λ−1 2 . ρ(Aγ ) = 0 2 λ The structure constants in the homogeneous case are given by the following theorem: Theorem 5.2. The commutation relations for the basis of sl2 (C; α)G are 1
1
1
1
[Aα , Aβ ] = 2 IG Aα − 2 Aβ , 1 2 [Aα , Aγ ] 1 2 [Aβ , Aγ ]
= =
(21)
1 2 Aβ + 2 Aγ , 1−r 1 2 I G G Aα + 2 I G Aγ .
Proof. This follows immediately from Theorems 4.4 and 5.1.
(22) (23)
6. Normal Form of the Lie Algebra In this section we derive the normal form of the algebra sl2 (C; α)G (diagonalising it 1 with respect to Aα ). Let ⎛ ⎞ x1 1 1 2 X = x1 Aα + x2 Aβ + x3 Aγ = ⎝ x2 ⎠ . x3 Then 1
1
1
1
2
ad(Aα )X = x2 (2 IG Aα − 2 Aβ ) + x3 (Aβ + 2 Aγ ) ⎛ ⎞ 2x2 IG = ⎝ −2x2 + x3 ⎠ 2x3 ⎞⎛ ⎞ ⎛ x1 0 2IG 0 = ⎝ 0 −2 1 ⎠ ⎝ x2 ⎠ . x3 0 0 2
On the Classification of Automorphic Lie Algebras
This leads to the transformation matrix ⎛ 1 ⎝ −IG 1 4 IG
813
0 1 1 4
⎞ 0 0⎠ 1
and suggests the definition of a new basis: 1
e0 = Aα ,
(24)
1 − IG Aα ,
(25)
1 1 1 2 1 e+ = Aγ + Aβ + IG Aα . 4 4
(26)
e− =
1 Aβ
Corollary 6.1. Computing the trace-form among the basis elements: e0 , e0 = 2, e− , e− = 0,
e0 , e− = 0, e0 , e+ = 0, 1−r e− , e+ = IG G JG , e+ , e+ = 0.
Example 6.1. (Dn , with n = 2) With reference to Example 5.1 we find 0 −λ , ρ(e0 ) = −λ−1 0 1 − λ4 1 −λ , ρ(e− ) = λ−1 −1 2λ2 1 − λ4 1 λ ρ(e+ ) = . −λ−1 −1 8λ2 See Sect. 7 for comparison with earlier results. It turns out that 1 1 1 1 2 1 1 1 [e+ , e− ] = [Aγ + Aβ + IG Aα , Aβ − IG Aα ] 4 4 4 1 1 2 1 2 1 1 = −[Aβ , Aγ ] + IG [Aα , Aγ ] + IG [Aα , Aβ ] 2 1−rG
= (−IG
1−rG
1+rG
= IG
(IG
= IG
JG e0 .
1−rG
1−rG
Notice that Q(IG ) = IG
2
1
+ IG )Aα 1
− 1)Aα
JG is a quadratic polynomial in IG . We check that [e0 , e− ] = −2e− , [e0 , e+ ] = 2e+ .
We can now formulate the following main theorems:
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Theorem 6.1. The G–Automorphic Lie algebras sl2 (C; α)G are isomorphic as modules to C[IG ] ⊗ sl2 (C). A basis for sl2 (C; α)G over C is given by el· = Iˆ lG e· , · = 0, ±, l ∈ Z≥0 , where Iˆ G = aIG + b for some a, b ∈ C. The commutation relations can be brought into the form [el+1 , el−2 ] = el01 +l2 + el01 +l2 +2 , [el01 , el±2 ] = ±2el±1 +l2 . The algebras are quasi–graded (e.g. [LM05]), with grading depth 2. Proof. In all cases one has [e0+ , e0− ] = Q G (IG )e00 , where Q G is a quadratic polynomial. Using the allowed complex scaling and affine transformations on IG and rescaling e0− , this can be normalized to [e0+ , e0− ] = (1 + Iˆ 2G )e00 . Theorem 6.2. Let G be either one of the groups T, O, I or Dn . Then, the Automorphic Lie Algebras sl2 (C; α)G are isomorphic as Lie algebras. Remark 6.1. This proves a conjecture by A. Mikhailov, made in 2008. This conjecture is proven in [BM09] by completely different methods, unpublished at the time of the writing of the present paper. Definition 6.1. A basis for sl2(C)G , is given by l
el· = IG e· , · = 0, ±, l ∈ Z. The commutation relations are [el+1 , el−2 ] = el01 +l2 +1 − el01 +l2 +2 , [el01 , el±2 ] = ±2el±1 +l2 . −1
G as a C[IG , IG ]Theorem 6.3. Let rG = 0. Then sl2(C)G = sl2(C)+G ⊕ sl2(C)− G module, with subalgebras sl2(C)± to be defined in the proof. G Proof. Let sl2(C)+G = el0,± l≥0 ⊕ e−1 − . The C[IG ]-module sl2(C)+ is a Lie algebra, 0 1 l −1 −1 G G since [e0+ , e−1 − ] = e0 − e0 ∈ sl2(C)+ . Next consider sl2(C)− = e0,± l≤−2 ⊕ e+ , e0 . −2 −1 −2 −1 G This is also a Lie algebra, since [e− , e+ ] = −e0 +e0 ∈ sl2(C)− (here we only treated G the worst commutators, the others are less critical). Then sl2(C)G = sl2(C)+G ⊕sl2(C)− −1
G as a C[IG , IG ]-module, with subalgebras sl2(C)± .
Corollary 6.2. The Automorphic Lie Algebras sl2(C)G are isomorphic as Lie algebras for the groups T, O, I and Dn , n odd.
On the Classification of Automorphic Lie Algebras
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7. The Example sl2 (C; α)D2 and Comparison with Averaging We discuss here the case of the Automorphic Lie Algebra sl2 (C; α)D2 ; this was first found in [LM05] via group average (and denoted as slD2 (2, C; 0)) . This simple example allows us to show the equivalence of the two methods in this case. Let x0 (λ) =
λ−1 0
0 λ
,
λ y0 (λ) = , 0 1 − λ4 1 0 h 0 (λ) = 0 −1 λ2 0 λ−1
(27)
be the generators of the algebra obeying the commutation relations [x0 , y0 ] = h 0 , [h 0 , x0 ] = (IG + 1) x0 − y0 , [h 0 , y0 ] = −(IG + 1) y0 + x0 , 4 where IG = 21 1+λ (the reader is referred to [LM05] for details). To compare the λ2 results let us first of all write this algebra in normal form; this is equivalent to diagonalise the algebra with respect to y0 following the scheme used in Sect. 6. This leads to the transformation matrix ⎛
1 ⎝ 2I G −2 IG
0 −2 2
⎞ 0 1⎠ 1
and suggests the definition of a new basis: ρ(e0 ) = −y0 , 1 −2 IG y0 + 2 x0 + h 0 , ρ(e− ) = 2 1 ρ(e+ ) = 2 IG y0 − 2 x0 + h 0 . 8
(28) (29) (30)
In this new basis the commutation relations read [e0 , e± ] = ±2e± , [e+ , e− ] = (I2G − 1)e0 . The expressions ρ(e0 ), ρ(e− ) and ρ(e+ ) in (28)–(30) are nothing but the generators ρ(e0 ), ρ(e− ), ρ(e+ ) in Example 6.1.
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8. Explicit Bases for the Automorphic Lie Algebras sl2 (C; α) G In this section we give explicit bases for sl2 (C; α)G , using concrete formulas for the covariants. We notice the remarkable likeness in all cases and remark upon the somewhat surprising fact that the determinant of all matrices is constant. We compute the Jordan normal form of the whole algebra and see that it is even more uniform in occurrence. If understood, this might lead to better and quicker insight in the general case. Theorem 8.1. Let A0 , A± be given by Remark 3.2. In λ–notation this reads as follows: α X = αλ and αY = ωα α − λαλ , where αλ = dα dλ . Then 1 A , ωα α 0 2 IG γ A , e− = − ωα ωβ αβ − e0 =
ωβ β JG (ωα − 1)2 IG γ A+ = − A , 2 ωα ωβ αβ(α, α) 2ωα αγ IrG + G λ −λ2 and A0 = A1α . for all Platonic groups G, where A− = A, ρ(A− ) = 1 −λ e+ =
(31) (32) (33)
Proof. Case by case inspection, using the results in the following sections. The second equality in (33) uses Lemma 4.5. In what follows we use the λ–notation (ωα α − λ αλ )2 αλ (ωα α − λ αλ ) , ρ(A+ ) = −αλ2 −αλ (ωα α − λ αλ ) ωα α − 2λ αλ −2λ(ωα α − λ αλ ) ρ(A0 ) = . −2αλ −ωα α + 2λ αλ 8.1. Explicit basis for sl2 (C; α)I . Let ωα be 12; it follows then that ωβ = 2ωα − 4 = 20 and ωγ = 3ωα − 6 = 30; let also α(λ) = λ(λ10 + 11λ5 − 1), dα(λ) = (11λ10 + 66λ5 − 1), αλ (λ) = dλ
β(λ) = −242 λ20 − 228λ15 + 494λ10 + 228λ5 + 1 , γ (λ) = −4840 λ30 + 522λ25 − 10005λ20 − 10005λ10 − 522λ5 + 1 , IG =
Then
β3 1 . 24490059264 α 5
11λ − 792λ6 + 4234λ11 + 792λ16 + 11λ21 121λ2 − 1452λ7 + 4334λ12 + 132λ17 + λ22 −1 + 132λ5 − 4334λ10 − 1452λ15 − 121λ20 −11λ + 792λ6 − 4234λ11 − 792λ16 − 11λ21 −10λ − 10λ11 22λ2 − 132λ7 − 2λ12 . ρ(A0 ) = 2 − 132λ5 − 22λ10 10λ + 10λ11
ρ(A+ ) =
,
On the Classification of Automorphic Lie Algebras
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8.2. Explicit basis for sl2 (C; α)O . Let ωα be 6; it follows then that ωβ = 2ωα − 4 = 8 and ωγ = 3ωα − 6 = 12; let also α(λ) = λ(λ4 − 1), dα(λ) = 5λ4 − 1, αλ (λ) = dλ β(λ) = −50 λ8 + 14λ4 + 1 , γ (λ) = −400 λ12 − 33λ8 − 33λ4 + 1 , IG = − Then
β3 1 . 13500000 α 4
5λ − 26λ5 + 5λ9 25λ2 − 10λ6 + λ10 −1 + 10λ4 − 25λ8 −5λ + 26λ5 − 5λ9 −4λ − 4λ5 10λ2 − 2λ6 ρ(A0 ) = . 2 − 10λ4 4λ + 4λ5
ρ(A+ ) =
,
8.3. Explicit basis for sl2 (C; α)T . Let ωα be 4; it follows then that ωβ = 2ωα − 4 = 4 and ωγ = 3ωα − 6 = 6; let also √ α(λ) = λ4 − 2 i 3 λ2 + 1, √ dα(λ) = 4λ(λ2 − i 3), αλ (λ) = dλ √ √ β(λ) = −96 i 3 λ4 + 2 i 3 λ2 + 1 , γ (λ) = −9216λ λ4 − 1 , IG = Then
β3 √ 3. 2654208 i 3 α 1
√ √ √ 3 − 16i 3λ5 4 −16i 3λ − 32λ 16 − 32i 3λ2 − 48λ √ √ √ , 48λ2 + 32i 3λ4 − 16λ6 16i 3λ + 32λ3 + 16i 3λ5 √ 4 − 4λ4 −8λ + 8i 3λ3 √ . ρ(A0 ) = 8i 3λ − 8λ3 −4 + 4λ4
ρ(A+ ) =
8.4. Explicit basis for sl2 (C; α)Dn . Let ωα be 2 and ωβ be m, then ωγ = ωα + ωβ − 2 = m; recall that in this case (α, α)2 = −2 = β. Let α(λ) = λ, αλ (λ) = 1, 1 β(λ) = (λm + 1), 2 m γ (λ) = (λm − 1), 2
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where m = n if n is odd while m = 2 n if n is even. Moreover, IG = while IG =
β2 αn
β αn
if n is even
if n is odd. It turns out then that λ λ2 , ρ(A+ ) = −1 −λ 0 λ2 . ρ(A0 ) = −2 1 0
Remark 8.1. One notices that det(ρ(e0 )) = −1 and det(ρ(e± )) = 0 in all cases. We find that we can undress the representation, losing the invariance of the elements but keeping the commutation relations, to the following form: 1 0 ν(e0 ) = , 0 −1 ωβ JG βαλ 0 1 , ν(e+ ) = 0 0 2 IrG γ G γ 2 0 0 ν(e− ) = . I ωβ G βαλ 1 0 The undressing is done by computing the Jordan normal form of ρ(e0 ) and applying the same conjugation to the other elements. That this is possible suggests that there must be a method to it. The conjugation is with the matrix
−λ 1
ωα α−λαλ αλ
1
.
9. Conclusions We have shown that the problem of reduction can be formulated in a uniform way using the theory of invariants. This gives us a powerful tool of analysis and it opens the road to new applications of Automorphic Lie Algebras, beyond the context of integrable systems. It turns out that in the explicit case we present here where the underlying Lie algebra is sl2 (C), we can compute Automorphic Lie Algebras only using geometric data. Moreover, we prove that Automorphic Lie Algebras associated to the groups T, O, I and Dn are isomorphic in the α case. This fact, i.e. that the Automorphic Lie Algebras are independent from the group is not quite what one would expect from the topological point of view. Indeed, if one divides out the group action one obtains usually an orbifold, but a manifold in the case of I. This distinction is not visible at the level of the algebra and therefore not on the level of the integrable systems that follow from the reduction procedure. It may turn up again when one looks for the actual solutions, since then the domain starts to play a role again. We leave this for further investigation. On the other hand, the treatment of the groups, including Z/n, in the McKay-correspondence (see [Nak09]) and the resolution of the singularities of the relation between the invariants using invariant quotients of the covariants α, β and γ is remarkably uniform. We notice that the corresponding Dynkin diagram (without its weights) can be easily read off from the degrees of α, β and γ in Corollary 4.2 for each group, as long as G = Z/n.
On the Classification of Automorphic Lie Algebras
819
Preliminary computations based on the icosahedral group I suggest that in the case of non equivalent σ and τ2 one finds an Automorphic Lie Algebra isomorphic to the previous ones. In the case of higher dimensional Lie algebras, say slk (C), one could proceed as follows. Let k be such that one of T, O, I has an irreducible projective representation τk . Fix a 2-dimensional irreducible representation σ . One can read off the existence of an invariant matrix Ai,0 of degree 2i from the corresponding Dynkin diagram. Here A1,0 is the A as used in this paper. We plan to investigate these matters further. Acknowledgement. The authors are grateful to A. V. Mikhailov for enlightening and fruitful discussions on various occasions. One of the authors, S. L., acknowledges financial support initially from EPSRC (EP/E044646/1) and then from NWO through the scheme VENI (016.073.026). Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
A. Z/n, α-Divisor The analysis for the group Z/n is different from the other groups and so we give it here a completely independent treatment. As before, let X Y −X 2 . ρ(A) = Y 2 −X Y Take α = X and β = Y . Let m = n if n is odd and m = 2n if n is even. Let ω be a m th root of unity. The action of Z/m is given by g X = ωX, gY = ω−1 Y. This induces an action of Z/n on λ = Y / X . We see that (α, β)1 = −1, so the previous setting does not apply. In fact, C[X, Y ]Z/m = C[α, β], so the algebra of covariants is polynomial. We can now compute the Poincaré function for C[α, β] ⊗ A and we see that it equals 3 , (1 − tα )(1 − tβ ) where 3 stands for the 3-dimensional space generated by (α 2 , A)2 , (αβ, A)2 and (β 2 , A)2 , in other words, for d n m n sl2 (C). Let m = 2d. Then we take IZ/2d = β d = βα n . Let m = 2d + 1. Then we take IZ/2d+1 = βα m = βα n . α We can thus draw the conclusion that sl2 (C; α)Z/n = C[IZ/n ] ⊗ sl2 (C).
B. Invariant, α-Divisor In this Appendix we derive the α-divisor invariants associated to each group. We consider expressions of the m2 form βα m 1 , m 1 ∈ Z>0 , m 2 ∈ Z≥0 .
B.1. T. To compute the invariants we have to solve the homogeneity equation m 2 = m 1 , m 1 ∈ Z>0 , m 2 ∈ Z≥0 , and the invariance equation 2m 2 = m 1 3 We take m 1 = 3. This leads to IT ≡ β 3 .
α
mod 3.
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S. Lombardo, J. A. Sanders
B.2. O. To compute the invariants we have to solve the homogeneity and invariance equations 4m 2 = 3m 1 , m 1 ∈ Z>0 , m 2 ∈ Z≥0 , 0 = m 1 mod 2. We let m 1 = 2k1 and m 2 = 3k2 . Then 2k2 = k1 , k1 ∈ Z>0 , k2 ∈ Z≥0 . 3 Let k1 = 2, that is, m 1 = 4 and k2 = 1, that is, m 2 = 3. Then IO ≡ β 4 .
α
B.3. I. To compute the invariants we have to solve the homogeneity equation 5m 2 = 3m 1 , m 1 ∈ Z>0 , m 2 ∈ Z≥0 . Let m 1 = 5k1 and m 2 = 3k2 . Then k1 = k2 and k1 > 0. We let k1 = k2 = 1, that is, m 1 = 5 and m 2 = 3. 3 We have found II ≡ β 5 . α
B.4. Dm . To compute the invariants we have to solve the homogeneity equation m m 2 = 2m 1 , m 2 ≥ 0, m 1 > 0, and the invariance equation m m 2 = 2m 1
mod 4.
The last one follows from the first. Let n = 2d + p, p = 0, 1. If n is even, we take m 1 = n and m 2 = 1. If n 1+ p is odd, we take m 1 = n and m 2 = 2. Thus the invariant is IDn ≡ βα n .
C. The β-Divisor We consider here the algebra sl2 (C; β)G , that is the G–Automorphic Lie Algebra based on sl2 (C) with homogeneous coefficients having poles at the zeros of β.
C.1. Invariant, β-divisor. We compute first of all the β-divisor invariants associated to each group; we
m2 use an underlined notation for this case. Let us consider expressions of the form βα m 1 , m 1 ∈ Z≥0 , m 2 ∈ Z>0 .
C.1.1. T To compute the invariants we have to solve the homogeneity and invariance equations m 2 = m 1 , m 1 ∈ Z>0 , m 2 ∈ Z≥0 , m 2 = 2m 1 mod 3. We take m 2 by β.
= 3. This leads to IT
≡
α 3 , where the underline notation indicates division β3
C.1.2. O To compute the invariants we have to solve the homogeneity and invariance equations 3m 2 = 4m 1 , m 1 ∈ Z>0 , m 2 ∈ Z≥0 , 0 = m 2 mod 2. We let m 2 = 2k2 and m 1 = 3k1 . Then k2 = 2k1 , k1 ∈ Z>0 , k2 ∈ Z≥0 . 4 Let k1 = 1, that is, m 1 = 3 and k2 = 2, that is, m 2 = 4. Then IO ≡ α 3 .
β
On the Classification of Automorphic Lie Algebras
821
C.1.3. I To compute the invariants we have to solve the homogeneity equation 3m 2 = 5m 1 , m 1 ∈ Z>0 , m 2 ∈ Z≥0 . Let m 1 = 3k1 and m 2 = 5k2 . Then k1 = k2 and k1 > 0. We let k1 = k2 = 1, that is, m 1 = 3 and m 2 = 5. 5 We have found II ≡ α 3 . β
C.2. Dm . To compute the invariants we have to solve the homogeneity equation 2m 2 = m m 1 , m 2 ≥ 0, m 1 > 0, and the invariance equation 2m 2 = m m 1
mod 4.
The last one follows from the first. Let n = 2d + p, p = 0, 1. If n is even, we take m 2 = n and m 1 = 1. If n n is odd, we take m 2 = n and m 1 = 2. Thus the invariant is ID ≡ α1+ p . n
β
C.3. The homogeneous basis. Lemma C.1. The homogeneous basis of sl2 (C; β)G is given by 1
A1β = IG Aβ ,
(34)
1 A1α = IG Aα , 2 A2γ = IG Aγ ,
(35) (36)
2 2 ωα α qG (α) ωα α qG (α) where IG = ω for G = T, O, I and Dn , n odd, and I2G = ω for G = Dn , n even. In 2 2 β β pG (β) β β pG (β) 2 1+rG ωα α qG (α) other words, ω 2 = IG , where rG = 0 for G = T, O, I, Dn , n odd, and 1 for Dn , n even. If we, β β pG (β)
moreover, define HG =
ωγ γ 2 then it follows immediately that ωβ β 2 pG (β) 1+rG
IG
= 1 − HG .
Corollary C.1. IG IG = 1 and JG = IG HG .
Table 8. β divisor: homogeneous elements of I, O, T β IG
I 5 24490059264 α 3
O 4 −13500000 α 3
2654208 i
HG
121 γ 2 − 200 3
25 γ 2 − 32 3
− 98 γ 3
A1α
2040838272 α 3 A1α
−2250000 α 3 A1α
663552 i
A1β
1 1 1 20 β Aβ 121 α A2 − 2900 β2 γ
1 1 1 8 β Aβ 25 α A2 − 352 β2 γ
1 1 1 4 β Aβ 3 α A2 − 20 β2 γ
A2γ
β
β
4
β
β
β
3
β
T
√ α3 3 3 β
2
β
√ α2 1 3 3 Aα β
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S. Lombardo, J. A. Sanders Table 9. β divisor: homogeneous elements of Dn , for n even and odd
β
Dn , n even
Dn , n odd
IG
αn β 1 γ2 (2n)2 β 2 1 α n−1 1 2 β Aα 1 1 1 2n β Aβ α A2 1 2(2n)2 (2n−1) β γ
αn β2 1 γ2 n2 β 2 1 α n−1 1 2 β 2 Aα 1 1 1 n β Aβ 1 α A2 2n 2 (n−1) β γ
HG A1α A1β A2γ
Example C.1 (Dn , with n = 2). In the case of D2 one finds ρ(A1α ) = ⎛
0 − 2 λ4
3 − 2λ4
1+λ
0
1+λ
, ⎞
1−λ4 − 2λ 1 1+λ4 1+λ4 ⎠ , ⎝ ρ(Aβ ) = 3 4 − 2λ 4 − 1−λ4 1+λ 1+λ λ 0 1+λ4 . ρ(A2γ ) = λ3 0 1+λ4
Theorem C.1. The commutation relations for the basis of sl2 (C; β)G are [A1β , A1α ] = 2 IG A1β − 2 A1α ,
r [A1β , A2γ ] = IGG A1α + 2 A2γ , [A1α , A2γ ] = IG A1β + 2IG A2γ .
Proof. This follows immediately from Theorem 5.2.
As for the case of the α-divisor one can now diaginalise the algebra to find its normal form; let ⎛ ⎞ x1 1 2 1 X = x1 Aβ + x2 Aα + x3 Aγ = ⎝ x2 ⎠ . x3 Then r
ad(A1β )X = x2 (2 IG A1β − 2 A1α ) + x3 (IGG A1α + 2 A2γ ) ⎛
⎞
2x2 IG
r ⎟ ⎜ = ⎝ −2x2 + x3 I G ⎠ G
2x3 ⎛
⎞⎛ ⎞ 0 2IG 0 x rG ⎠ ⎝ 1 ⎠ ⎝ x2 . = 0 −2 I G x3 0 0 2 This leads to the transformation matrix ⎛
1 ⎜ −I G ⎝ 1 rG +1 4 IG
0 1
1 rG 4 IG
⎞ 0 0⎟ ⎠ 1
(37) (38) (39)
On the Classification of Automorphic Lie Algebras
823
and suggests the definition of a new basis: e0 = A1β ,
(40)
e− = A1α − IG A1β ,
(41)
1 r 1 r +1 e+ = A2γ + IGG A1α + IGG A1β . 4 4
(42)
In the new basis the commutation relations read [e+ , e− ] = −IG HG e0 , [e0 , e± ] = ±2e± . The following theorems hold true: Theorem C.2. The G–Automorphic Lie algebras sl2 (C; β)G are isomorphic as modules to C[IG ] ⊗ sl2 (C). A basis for sl2 (C; β)G , is given by el· = IlG e· , · = 0, ±, l ∈ Z≥0 , The commutation relations can be brought into the form [el+ , el− ] = el0 +l +1 − el0 +l +2+r , 1 1 2 1 2 2 G [el0 , el± ] = ±2el±+l . 1 2 1 2 The algebras are quasi–graded (e.g. [LM05]), with grading depth 2 + rG . Theorem C.3. Let G be either one of the groups T, O, I or Dn , with n odd. Then, the Automorphic Lie Algebras sl2 (C; β)G are isomorphic as Lie algebras to the Automorphic Lie Algebras sl2 (C; α)G . Proof. In all cases one has [e+ , e− ] = Q G (IG )e0 , where Q G is a quadratic polynomial. Using the allowed complex scaling and affine transformations on IG and rescaling e− , this can be normalized to [e+ , e− ] = ˆ 2 )e0 . (1 + I G
Remark C.1. In [LM05, Sect. 3] it was remarked that automorphic algebras corresponding to different orbits are not isomorphic, i.e. elements of one algebra cannot be represented by a finite linear combination of the basis elements of the other algebra with complex constant coefficients. This seems to be in contradiction with Theorem C.3 for odd n.
References [Bel80]
Belavin, A.A.: Discrete groups and integrability of quantum systems. Funkt. Anal. i Pril. 14(4), 18–26, 95 (1980) [Bel81] Belavin, A.A.: Dynamical symmetry of integrable quantum systems. Nucl. Phys. B 180(2, FS 2), 189–200 (1981) [BM] Bury, R.T., Mikha˘ılov, A.V.: Solitons and wave fronts in periodic two dimensional Volterra system. In preparation (2009) [BM09] Bury, R.T., Mikha˘ılov, A.V.: Automorphic Lie algebras and corresponding integrable systems. 2009 [Dol] Dolgachev, I.V.: McKay Correspondence. Lecture notes, Winter 2006/07 [GGIK01] Gerdjikov, V.S., Grahovski, G.G., Ivanov, R.I., Kostov, N.A.: N -wave interactions related to simple Lie algebras. Z2 -reductions and soliton solutions. Inverse Problems 17(4), 999–1015, (2001). (Special issue to celebrate Pierre Sabatier’s 65th birthday (Montpellier, 2000)) [GGK01] Gerdjikov, V.S., Grahovski, G.G., Kostov, N.A.: Reductions of N -wave interactions related to low-rank simple Lie algebras. I. Z2 -reductions. J. Phys. A 34(44) 9425–9461 (2001) [GKV07a] Gerdjikov, V., Kostov, N., Valchev, T.: Soliton equations with deep reductions. Generalized Fourier transforms. In: Topics in Contemporary Differential Geometry, Complex Analysis and Mathematical Physics, Hackensack, NJ: World Sci. Publ., 2007, pp. 85–96
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[GKV07b] Gerdjikov, V.S., Kostov, N.A., Valchev, T.I.: N-wave Equations with Orthogonal Algebras: Z2 and Z2 × Z2 Reductions and Soliton Solutions. SIGMA Symmetry Integrability Geom. Methods Appl. 3, Paper 039, 19 pp. (electronic), 2007 [Gun86] Gundelfinger, S.: Zur Theorie der binären Formen. J. Reine Angew. Nath. 100(1), 413–424 (1886) [HSAS84] Harnad, J., Saint-Aubin, Y., Shnider, S.: Soliton solutions to Zakharov-Shabat systems by the reduction method. In: Wave Phenomena: Modern Theory and Applications (Toronto, 1983), Volume 97 of North-Holland Math. Stud., Amsterdam: North-Holland, 1984, pp. 423–432 [Kac90] Kac, V.G.: Infinite-Dimensional Lie Algebras. Cambridge: Cambridge University Press, third edition, 1990 [Kle56] Klein, F.: Lectures on the icosahedron and the solution of equations of the fifth degree. New York, N.Y.: Dover Publications Inc., revised edition, 1956, Translated into English by George Gavin Morrice [Kle93] Klein, F.: Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade. Basel: Birkhäuser Verlag, 1993, Reprint of the 1884 original, Edited, with an introduction and commentary by Peter Slodowy [Lam86] Lamotke, K.: Regular Solids and Isolated Singularities. Advanced Lectures in Mathematics. Braunschweig: Friedr. Vieweg & Sohn, 1986 [LM04] Lombardo, S., Mikha˘ılov, A.V.: Reductions of integrable equations: dihedral group. J. Phys. A 37(31), 7727–7742 (2004) [LM05] Lombardo, S., Mikha˘ılov, A.V.: Reduction groups and automorphic Lie algebras. Commun. Math. Phys. 258(1), 179–202 (2005) [Mik81] Mikha˘ılov, A.V.: The reduction problem and the inverse scattering method. Physica D 3(1&2), 73– 117 (1981) [Mol97] Molien, Th.: Über die Invarianten der linearen Substitutionsgruppen. Berlin: Sitz.-Ber. d. Preub. Akad. d. Wiss., 52, 1897 [MSY87] Mikha˘ılov, A.V., Shabat, A.B., Yamilov, R.I.: On an extension of the module of invertible transformations. Dokl. Akad. Nauk SSSR 295(2), 288–291 (1987) [Nak09] Nakamura, I.: McKay correspondence. In: Groups and Symmetries, Volume 47 of CRM Proc. Lecture Notes, Providence, RI: Amer. Math. Soc., 2009, pp. 267–298 [Olv99] Olver, P.J.: Classical Invariant Theory, Volume 44 of London Mathematical Society Student Texts. Cambridge: Cambridge University Press, 1999 [Ovs97] Ovsienko, V.: Exotic deformation quantization. J. Diff. Geom. 45(2), 390–406 (1997) [Smi95] Smith, L.: Polynomial Invariants of Finite Groups. Volume 6 of Research Notes in Mathematics. Wellesley, MA: A K Peters Ltd., 1995 [Spr87] Springer, T.A.: Poincaré series of binary polyhedral groups and McKay’s correspondence. Math. Ann. 278(1-4), 99–116 (1987) [SVM07] Sanders, J.A., Verhulst, F., Murdock, J.: Averaging Methods in Nonlinear Dynamical Systems, Volume 59 of Applied Mathematical Sciences. New York: Springer, second edition, 2007 [Ver00] Vermaseren, J.A.M.: New features of FORM. Technical report, Nikhef, Amsterdam, 2000. http:// arXiv.org/abs/Math-ph/0010025vz, 2000 [Zhm92] Zhmud , È.M.: Kernels of projective representations of finite groups. J. Soviet Math. 59(1), 607– 616, (1992), translation of Teor. Funkt. Anal. i Prilozhen. 55, 34–49 (1991) Communicated by Y. Kawahigashi
Commun. Math. Phys. 299, 825–866 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1051-6
Communications in
Mathematical Physics
Hidden Grassmann Structure in the XXZ Model IV: CFT Limit H. Boos1 , M. Jimbo2 , T. Miwa3 , F. Smirnov4,∗ 1 Physics Department, University of Wuppertal, D-42097, Wuppertal, Germany.
E-mail:
[email protected] 2 Department of Mathematics, Rikkyo University, Toshima-ku, Tokyo 171-8501, Japan.
E-mail:
[email protected] 3 Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan.
E-mail:
[email protected] 4 Laboratoire de Physique Théorique et Hautes Energies, Université Pierre et Marie Curie, Tour 16 1er étage,
4 Place Jussieu, 75252 Paris Cedex 05, France. E-mail:
[email protected] Received: 18 December 2009 / Accepted: 18 December 2009 Published online: 22 April 2010 – © Springer-Verlag 2010
Abstract: The Grassmann structure of the critical XXZ spin chain is studied in the limit to conformal field theory. A new description of Virasoro Verma modules is proposed in terms of Zamolodchikov’s integrals of motion and two families of fermionic creation operators. The exact relation to the usual Virasoro description is found up to level 6. 1. Introduction In the present paper we continue the series of works [1–3] on the XXZ model. In [3] we considered it in the presence of the Matsubara direction, or equivalently the six vertex model on a cylinder. We computed the normalised partition function with a defect localised between two horizontal lines, which corresponds to an insertion of a quasi-local operator: Tr S Tr M TS,M q 2κ S+2αS(0) O κ 2αS(0) . Z q O = (1.1) Tr S Tr M TS,M q 2κ S+2αS(0) Here q = eπiν is related to the coupling parameter (see (2.2) below), and TS,M stands for the monodromy matrix on the two tensor products of evaluation representations of Uq ( sl2 ): one for the horizontal (or ‘space’) direction S, and another for the vertical (or Matsubara) direction M. For more details see Sect. 2 below, in particular fig. 1. It was important in [3] to incorporate inhomogeneities in the Matsubara chain. This allows, for example, to consider the temperature expectation values in the spirit of [4,5], by adjusting inhomogeneities and taking the limit to the infinite chain in the Matsubara direction. The clue to our calculation was the introduction of operators t∗ (ζ ), b∗ (ζ ), c∗ (ζ ) [2] which, by acting on the primary field q 2αS(0) , create the space of quasi-local operators on Membre du CNRS.
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H. Boos, M. Jimbo, T. Miwa, F. Smirnov
the horizontal chain. More precisely, quasi-local operators are created by Taylor coefficients of t∗ (ζ ), b∗ (ζ ), c∗ (ζ ) at the point ζ 2 = 1. In this paper we change the definition of b∗ (ζ ), c∗ (ζ ) from those of [2,3] by applying a certain Bogolubov transformation. Compared with the original ones, they have better asymptotic properties. We shall explain this in Sect. 2. There is an obvious similarity with conformal field theory (CFT), where the descendants are created from the primary field by the action of the Virasoro algebra. Our aim in this paper is to examine the scaling limit of our construction in the critical regime, and to establish its precise relation with CFT. We shall consider the case of a homogeneous Matsubara chain. The functional (1.1) is non-trivial only for operators O of spin zero. As it turns out, for the study of the scaling limit, it is quite useful to relax this restriction. We shall first introduce the following generalisation of (1.1) which is of interest on its own right. For s > 0 we define (−s) Tr S Tr M YM TS,M q 2κ S b∗∞,s−1 · · · b∗∞,0 q 2αS(0) O Z κ,s q 2αS(0) O = , (−s) Tr S Tr M YM TS,M q 2κ S b∗∞,s−1 · · · b∗∞,0 q 2αS(0) where the b∗∞, j ’s denote the coefficients of the singular part of b∗ (ζ ) at ζ 2 = 0. When s < 0, a similar definition is in force using the expansion coefficients of c∗ (ζ ). As long (−s) as YM is taken generically, this definition is independent of its choice (see Sect. 2 for (−s) more details). In general the dependence on YM enters, but only in a “topological” way. Needless to say, what we we are dealing with is a lattice analogue of the screening operators à la Feigin-Fuchs-Dotsenko-Fateev [6,7]. It is interesting to see that quasilocal operators and screening operators both arise from the same operators b∗ (ζ ), c∗ (ζ ), as expansions either around ζ 2 = 1 or ζ 2 = 0. After these modifications the main formula of [3] remains valid. It reads Z κ,s t∗ (ζ10 ) · · · t∗ (ζ p0 )b∗ (ζ1+ ) · · · b∗ (ζr+ )c∗ (ζr− ) · · · c∗ (ζ1− ) q 2αS(0) =
p
2ρ(ζi0 |κ, κ + α, s) × det ω(ζi+ , ζ j− |κ, α, s)
i, j=1,··· ,r
i=1
,
(1.2)
where the functions ρ(ζi0 |κ, κ + α, s) and ω(ζi+ , ζ j− |κ, α, s) are defined by the data of the Matsubara direction. We refer to this formula as the determinant formula. Due to the Bogolubov transformation of b∗ and c∗ the function ω(ζ, ξ |κ, α, 0) is slightly different from ω(ζ, ξ |κ, α) used in [3,8]. Now let us turn to the scaling limit. We have two twisted transfer matrices TM (ζ, κ +α) and TM (ζ, κ) in the Matsubara direction. As the number of sites n becomes large, their Bethe roots tend to distribute densely on R+ . Fixing R > 0 and introducing the step of the lattice a, we consider the limit n → ∞, a → 0, na = 2π R fixed.
(1.3)
At the same time we rescale the spectral parameter as ζ = (Ca)ν λ, λ fixed,
(1.4)
Grassmann Structure in XXZ Model
827
so that the Bethe roots close to 0 stay finite in terms of the variable λ. Here C is a constant chosen for fine tuning (see Sect. 8, (8.3)). In this limit the twisted transfer matrices turn into the transfer matrices of chiral CFT on the cylinder C yl = C/2πi R Z, introduced and studied by Bazhanov, Lukyanov and Zamolodchikov [9,10]. We wish to mention here that the present work owes a great deal to these remarkable papers without which it would have been impossible. Details about the scaling limit can be found in Sect. 8 below. The relevant CFT has central charge c =1−
6ν 2 . 1−ν
We shall parametrise the conformal dimension as α =
ν2 (α − 1)2 − 1 , 4(1 − ν)
and write the action of the Virasoro algebra on a local field φ(y) as ln (φ)(y). In the following we set a¯ = Ca, and let limscaling indicate the scaling limit (1.3), (1.4). The functions entering the determinant formula (1.2) also have finite limits, ρ sc (λ|κ, κ ) = lim ρ(λa¯ ν |κ, κ + α, s) , scaling
4 ω (λ, μ|κ, κ , α) = lim ω(λa¯ ν , μa¯ ν |κ, α, s), sc
scaling
where κ = κ + α + 2 1−ν ν s.
(1.5)
So all these partition functions (1.2) have finite limits. They should have some definite meaning in the context of CFT. We contend that they are the three point functions of the descendants of the chiral primary field φα (0), computed in the presence of two other primary fields (or their descendants) inserted at the two ends of the cylinder. More specifically, we conjecture that the following picture holds true. First, the creation operators tend to a limit, 2τ ∗ (λ) = lim t∗ (λa¯ ν ), 2β ∗ (λ) = lim b∗ (λa¯ ν ), 2γ ∗ (λ) = lim c∗ (λa¯ ν ). scaling
scaling
scaling
As λ → ∞, these operators have asymptotic expansions of the form ∞
2 j−1 log τ ∗ (λ) τ ∗2 j−1 λ− ν , j=1
√
1 τ ∗ (λ)
∗
β (λ)
∞
j=1
2 j−1 β ∗2 j−1 λ− ν ,
√
1 τ ∗ (λ)
∗
γ (λ)
∞
(1.6) 2 j−1 γ ∗2 j−1 λ− ν .
j=1
In the limit, the quasi-local operator q 2αS(0) becomes the product of two chiral primary fields φα (0)⊗ φ¯ −α (0). The operators τ ∗2 j−1 and the quadratic combinations β ∗2 j−1 γ ∗2k−1 act only on the left component φα (0), and create the entire Verma module spanned by the Virasoro descendants l−m 1 · · · l−m s (φα )(0).
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H. Boos, M. Jimbo, T. Miwa, F. Smirnov (−s)
Furthermore, if YM is chosen to be generic then lim Z κ,s t∗ (ζ10 ) · · · t∗ (ζ p0 )b∗ (ζ1+ ) · · · b∗ (ζr+ )c∗ (ζr− ) · · · c∗ (ζ1− ) q 2αS(0) scaling ∗ 0 ∗ 0 ∗ + ∗ + ∗ − ∗ − τ = 2 p+2r Z κ,κ (λ ) · · · τ (λ )β (λ ) · · · β (λ )γ (λ ) · · · γ (λ ) (0)) . (φ α p r r 1 1 1 R (1.7)
In the right hand side, the symbol Z κ,κ R {X (0)} stands for the three point function norκ,κ malised as Z R {φα (0)} = 1, with X (0) inserted at x = 0 and the primary fields φκ+1 , φ−κ +1 being inserted at x = ∞ and x = −∞, respectively (see (6.8) below). (−s) Non-generic choice of YM corresponds to replacing the primary fields at x = ±∞ by (−s) their descendants. In this paper we discuss only the case of generic YM . The coefficients in (1.6) are homogeneous operators in the sense that l0 , τ ∗2 j−1 = (2 j − 1)τ ∗2 j−1 , l0 , β ∗2i−1 γ ∗2 j−1 = (2i + 2 j − 2)β ∗2i−1 γ ∗2 j−1 . Hence, for each degree, the descendants created by l−k ’s, and those created by τ ∗2 j−1 ’s, β ∗2 j−1 ’s and γ ∗2 j−1 ’s, must be finite linear combinations of each other. The main goal of this paper is to show, for low degrees, that this is indeed the case, and that the coefficients can be found explicitly. To determine the coefficients of the linear combination, we compare the values of Z κ,κ R . For the Virasoro descendants, they can be easily computed by the conformal Ward-Takahashi identities. For the descendants by τ ∗2 j−1 and others, we need the coefficients of the asymptotic expansion of the functions ρ R (λ|κ, κ ) and ω R (λ, μ|κ, κ , α). In Sect. 10 we develop a systematic method for computing them. We note that in both cases the results are polynomials in the conformal dimensions κ+1 , κ +1 . We may regard them as independent variables and compare the coefficients, since s in (1.5) can take any integer values. This was one of reasons for us to introduce the screening operators. We consider first τ ∗2 j−1 . CFT allows an integrable structure based on Zamolodchikov’s integrals of motion i2m−1 [11]. With the above procedure we are led to a result which should not be surprising, τ ∗2m−1 = Cm · i2m−1 , where Cm are some ν-dependent constants which can be found in [10]. We then consider the action of β ∗2 j−1 ’s and γ ∗2 j−1 ’s. Because of a technical difficulty we have not been able to compute the asymptotics of ω R (λ, μ|κ, κ , α) for κ = κ . Here we restrict to the case κ = κ . Since Z κ,κ R (i2n−1 (X )) = 0 for any X , restricting to κ = κ means that we consider the quotient space of the Verma module modulo the action of the integrals of motion. We assume that the vectors i2k1 −1 · · · i2kr −1 l−2m 1 · · · l−2m s (φα (0)) span the Verma module, so the quotient space is created by the l−2m ’s. With primary fields as asymptotical states, we can compare up to the level 6. It should be added, however, that up to this level the system of equations is overdetermined. So the very possibility of finding a solution is the strongest support of our fermionic picture.
Grassmann Structure in XXZ Model
829
We give one example on the level 4: β ∗1 γ ∗3 (φα (0)) =
1 2c − 32 − 4dα 2 D1 (α)D3 (2 − α) l−2 l−4 (φα (0)), + 2 9
where 1 (25 − c)(24 α + 1 − c) , 6 1 (2n − 1) α2 + 2ν 2n−1 i 1 − 2n−1 . D2n−1 (α) = (ν) ν (1 − ν) 2 · ν (n − 1)! α + (1−ν) (2n − 1)
dα =
2
2ν
In the right-hand side, we have a particular combination of the Virasoro descendants. This equation says that its three-point function remains of the same determinant form before and after integrable perturbation. The text is organised as follows. In Sect. 2 we review the results of [3] and describe the Bogolubov transformation mentioned above. In Sect. 3 we define the functions ρ and ω. In Sect. 4 we define screening operators on the lattice, and describe a generalisation of the previous results. In Sect. 5 we start discussing the scaling limit of the XXZ chain, examining the behaviour of the Bethe roots in the Matsubara direction as the length of the chain becomes infinite. Sections 6 and 7 are a review of the CFT integrals of motion on the cylinder, and the series of works of Bazhanov, Lukyanov and Zamolodchikov (BLZ). We explain in Sect. 8 how the Matsubara transfer matrix turns into that of BLZ in the continuous limit. In Sect. 9 we discuss the CFT interpretation of the scaling limit in the space direction. In Sect. 10 we study the asymptotics of Thermodynamic Bethe Ansatz (TBA) function a for CFT. In Sect. 11 we find the asymptotical expansion of ω for κ = κ . In Sect. 12 we compare descendants created by l−2m with those created by β ∗2 j−1 ’s and γ ∗2 j−1 and give some concluding remarks. In the Appendix we present general properties of asymptotics of ω which apply to the case κ = κ . 2. Review of Previous Results Let us start with a brief review of the papers [1–3]. Consider the XXZ spin chain in the infinite volume. The space of states of the model is HS =
∞
C2 ,
j=−∞
and the Hamiltonian is given by H=
∞ 1 1 1 1 2 3 σk σk+1 + σk2 σk+1 , = (q + q −1 ). + σk3 σk+1 2 2
(2.1)
k=−∞
We consider the critical XXZ model in the following range of the coupling: q = eπiν , 1/2 < ν < 1.
(2.2)
Together with HS we consider the Matsubara space HM . In [3] the most general case was treated: namely, HM was the tensor product of spaces of different dimensions, and to every site m an independent inhomogeneity parameter τm was attached. In the present paper we shall restrict ourselves to the case
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H. Boos, M. Jimbo, T. Miwa, F. Smirnov
HM =
n
C2 .
j=1
We consider the monodromy matrix TS,M . Mathematically this is nothing but the sl2 ) on the tensor product of evaluation repimage of the universal R matrix of Uq ( resentations corresponding to HS and HM . It has been said that we shall consider a homogeneous Matsubara chain only. In the notation of [3], this corresponds to setting 1 τm = q 2 for all m. We shall absorb this into redefinition of the L-operator comparing to [3]. Let us write the definition explicitly:
TS,M
∞ = T j,M , j=−∞
where
T j,M
n ≡ T j,M (1), T j,M (ζ ) = L j,m (ζ ) , m=1
with 1
1
− + + σ j− σm ). L j,m (ζ ) = q − 2 σ j σm − ζ 2 q 2 σ j σm − ζ (q − q −1 )(σ j+ σm 3 3
3 3
A local operator O on HS is by definition an operator which acts nontrivially only on a finite number of the tensor components C2 of HS . More generally we consider quasi-local operators with tail α, which are operators of the form q 2αS(0) O, S(k) =
1 2
k j=−∞
σ j3 ,
with O being local. In this notation S = S(∞) is the total spin. In [3] we computed the expectation values defined by Tr S Tr M TS,M q 2κ S+2αS(0) O κ 2αS(0) . Z q O = (2.3) Tr S Tr M TS,M q 2κ S+2αS(0) This is a linear functional on the space Wα,0 of spinless quasi-local operators with tail α. It is helpful to think of the functional Z κ as a ratio of partition functions of the six vertex model on the cylinder. For example, the numerator of (2.3) can be presented graphically as follows: On this picture the summation is performed over all edges except the broken ones in the middle. The arrows on the broken edges are fixed, representing the local operator O. The one dimensional sublattice going in the infinite space direction will be called the space chain, while the compact one dimensional sublattice in the Matsubara direction will be referred to as the Matsubara chain. We computed the expectation values (2.3) using the fermionic description of Wα,0 found in [2]. Let us briefly recall it. Consider the space
Grassmann Structure in XXZ Model
831
Space
M a t s u b a r a
= qκσ
=Lij
3
=q(α+κ) σ
3
Fig. 1. Partition function on the cylinder. The functional Z κ is a ratio of two partition functions on the cylinder. On each crossing of a row and a column, one associates the Boltzmann weights of the six vortex model. On κσ 3
(σ +κ)σ 3
j (marked by circles). The a particular row there are also twist fields q j (marked by crosses) and q numerator of Z κ corresponds to a lattice with defects representing an insertion of a local operator
W (α) =
∞
Wα−s,s ,
s=−∞
where Wα−s,s denotes the space of quasi-local operators of spin s with tail α − s. We have the creation operators t∗ (ζ ), b∗ (ζ ), c∗ (ζ ) and the annihilation operators b(ζ ), c(ζ ) which act on W (α) . To be precise the operators t∗ (ζ ), b∗ (ζ ), c∗ (ζ ) were defined in [2] as formal power series in ζ 2 − 1, the quasi-local operators in W (α) are created by coefficients of these series. However, when the series are substituted into Z κ the result allows analytical continuation. So, in the present paper we shall adopt another point of view which is similar to that of CFT. Namely, we shall consider the operators t∗ (ζ ), b∗ (ζ ), c∗ (ζ ) as analytical functions. Then the relation to real quasi-local operators is achieved by considering t∗ (ζ ), b∗ (ζ ), c∗ (ζ ) around the point ζ 2 = 1. The creation-annihilation operators have the block structures t∗ (ζ ) : Wα−s,s → Wα−s,s , b (ζ ), c(ζ ) : Wα−s+1,s−1 → Wα−s,s , c∗ (ζ ), b(ζ ) : Wα−s−1,s+1 → Wα−s,s . ∗
(2.4)
The operator τ = t1∗ /2 plays a special role: it is the right shift by one site along the space chain. We have b(ζ )(q 2αS(0) ) = 0, c(ζ )(q 2αS(0) ) = 0, ∗ c(ξ ), c (ζ ) + = ψ(ξ/ζ, α), b(ξ ), b∗ (ζ ) + = −ψ(ζ /ξ, α), where ψ(ζ, α) = ζ α
ζ2 + 1 . 2(ζ 2 − 1)
(2.5)
The operators in the space Wα,0 are created from the primary field q 2αS(0) by action of t∗ ’s and of equal number of b∗ ’s and c∗ ’s. The completeness [12] says that the entire
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H. Boos, M. Jimbo, T. Miwa, F. Smirnov
space Wα,0 is generated by coefficients of the creation operators considered as series in ζ 2 − 1. Certainly, this description is reminiscent of CFT. The main result of [3] is the following relations which allow for recursive computations of the expectation values: Z κ t∗ (ζ )(X ) = 2ρ(ζ |κ, κ + α)Z κ {X } , (2.6) 2 dξ 1 ω(ζ, ξ |κ, α)Z κ {c(ξ )(X )} 2 , (2.7) Z κ b∗ (ζ )(X ) = 2πi ξ dξ 2 1 Z κ c∗ (ζ )(X ) = − ω(ξ, ζ |κ, α)Z κ {b(ξ )(X )} 2 , (2.8) 2πi ξ where goes around all the singularities of the integrand except ξ 2 = ζ 2 . We think no further explanation is needed nowadays when the method of CFT is a part of common knowledge. The functions ρ(ζ |κ, κ + α) and ω(ζ, ξ |κ, α) will be defined in Sect. 3. We changed the notation for the former from [3]. The present notation agrees better with the explicit formula given below. The set of equations (2.6), (2.7), (2.8) implies a determinant representation for the expectation values [3], we shall discuss this later. Let us describe the modification of b∗ (ζ ), c∗ (ζ ) by a Bogolubov transformation which was mentioned in the Introduction. Denoting the operators used in [2,3] by ∗ (ζ ), define b∗rat (ζ ), crat 1 dξ 2 ∗ ∗ b (ζ ) = brat (ζ ) + Dζ Dξ −1 ζ ψ(ζ /ξ, α) · c(ξ ) 2 , 2πi ξ (2.9) 1 dξ 2 −1 ∗ ∗ Dζ Dξ ζ ψ(ξ/ζ, α) · b(ξ ) 2 ., c (ζ ) = crat (ζ ) − 2πi ξ where Dζ is the following finite difference operator of the second order: Dζ f (ζ ) = f (ζ q) + f (ζ q −1 ) − t∗ (ζ ) f (ζ ). The function −1 ζ ψ(ζ, α) is transcendental. For ζ 2 > 0, − we define it as −1 ζ ψ(ζ, α) = −V P
∞ 0
1 < Re α < 0 ν
1 dη2 ψ(η, α) , 1 2πiη2 2ν 1 + (ζ /η) ν
(2.10)
where the principal value is taken with regards to the pole at η2 = 1. In general we define it by analytic continuation with respect to both α and ζ 2 , obtaining a meromorphic function of log ζ . It is bounded at log ζ → ±∞, and its singularities closest to the real axis are the simple poles at log ζ = ±πiν with residues of opposite signs. The function Dζ Dξ −1 ζ ψ(ζ /ξ, α) is regular at ζ = ξ , so the Taylor series for ∗ ∗ 2 b (ζ ), c (ζ ) at ζ = 1 are well-defined. The function ω changes following the change of b∗ , c∗ : ω(ζ, ξ |κ, α) = ωrat (ζ, ξ |κ, α) + D ζ D ξ −1 ζ ψ(ζ /ξ, α) ,
(2.11)
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833
where D ζ f (ζ ) = f (ζ q) + f (ζ q −1 ) − 2ρ(ζ |κ, κ + α) f (ζ ).
(2.12)
Let us mention one marvellous property of our modified operators b∗ , c∗ . The main subject of our original study [1,2] was the following normalised matrix element Z ∞ {q 2αS(0) O} =
vac|q 2αS(0) O|vac , vac|q 2αS(0) |vac
(2.13)
where |vac denotes the ground state of the XXZ Hamiltonian in the infinite volume. Equivalently, the numerator of (2.13) is the partition function of the six vertex model on the plane with a defect localised between two horizontal lines. It is easy to see using the formulae from [1,2] that, if we create the quasi-local fields by the operators t∗ (ζ ), b∗ (ζ ), c∗ (ζ ), then Z ∞ vanishes on all of them with the sole exception of the descendants created by τ = t1∗ /2. For the latter we have Z ∞ {τ m (q 2αS(0) O)} = 1, m ∈ Z. We know from the algebraic construction [2] that the Taylor coefficients of the part ∗ (ζ ) in (2.9) produce only rational functions of q and q α . Hence all tranb∗rat (ζ ), crat scendental pieces of the expectation values (2.9) come from rewriting a given quasilocal operator in the fermionic basis and picking Taylor coefficients of the function D ζ D ξ −1 ζ ψ(ζ /ξ, α). The property of t∗ (ζ ) (except t1∗ ) and b∗ (ζ ), c∗ (ζ ) mentioned above is analogous to that of the Virasoro generators in CFT: on the Riemann sphere, all normalised one point functions of the descendants vanish due to the conformal invariance. Strictly speaking, the one point function of the primary field also vanishes, so a word of clarification is necessary. Take a massive model with a mass scale m and consider the conformal limit m → 0. While the normalised one point function of the primary field stays equal to 1, those of the descendants vanish in the limit, because for dimensional reasons they contain additional powers of m. At this point one may wonder why we did not do the Bogolubov transformation killing completely the function ω in (2.7), (2.8). The answer is that it is impossible to rewrite the left hand sides of (2.7), (2.8) because the function ω cannot be written as a function of ρ. 3. Functions ρ and ω Introduce the twisted Matsubara transfer matrix: 3 TM (ζ, κ) = Tr j T j,M (ζ )q κσ j .
(3.1)
Let |κ be the eigenvector of T (1, κ) whose eigenvalue is maximal in the absolute value. Similarly let κ + α| be the eigencovector of T (1, κ + α) whose eigenvalue is maximal in the absolute value. It is well-known that these eigenvectors have spin 0. We call them the maximal eigenvectors, and assume that they are not orthogonal. We denote the eigenvalues of TM (ζ, κ) (resp. TM (ζ, κ + α)) on |κ (resp. κ + α|) by T (ζ, κ) (resp. T (ζ, κ + α)).
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Then ρ(ζ |κ, κ + α) is defined by T (ζ, κ) and T (ζ, κ + α). We have ρ(ζ |κ, κ + α) =
T (ζ, κ + α) . T (ζ, κ)
(3.2)
The function ω(ζ, ξ |κ, α) is more complicated. In [3] it was shown that it is completely determined by two requirements the singular part and the normalisation condition. Then they were explicitly solved in terms of q-deformed Abelian integrals. In the present paper it is convenient to use an alternative, TBA-like, description used in [8]. Let us explain this simplifying a little the notation of [8]. Together with the transfer matrix TM (ζ, κ) we consider in [3] Baxter’s Q-operators − Q± M (ζ, κ). In this paper we use only one of them: Q M (ζ, κ), denoting it just as Q M (ζ, κ). It is defined by the trace over the highest weight representation of the q-oscillator algebra with generators a, a∗ , D (see [2] for notation), − (3.3) (ζ )q −2κ D A , Q M (ζ, κ) = ζ −κ+SM (1 − q −2(κ−SM ) )Tr − TOsc,M where n − TOsc,M (ζ ) = L− Osc,m (ζ ) , m=1
L− Osc,m (ζ )
1 3 − + − + = I − ζ 2 q 2D+1 σm σm − ζ q − 2 (a σm + a∗ σm ) q Dσm .
Its eigenvalue on the eigenvector discussed above will be denoted by Q(ζ, κ). The main role in TBA is played by the function: a(ζ, κ) =
d(ζ )Q(ζ q, κ) , a(ζ )Q(ζ q −1 , κ)
(3.4)
where a(ζ ) = (1 − qζ 2 )n , d(ζ ) = (1 − q −1 ζ 2 )n . It follows from the Baxter equation that the solutions to the equation a(ζ, κ) = −1 , are the zeros of either Q(ζ, κ) or T (ζ, κ). The function a(ζ, κ) satisfies the nonlinear integral equation [13,4,14]:
d(ζ ) dξ 2 − log a(ζ, κ) = −2πiνκ + log K (ζ /ξ ) log (1 + a(ξ, κ)) 2 , (3.5) a(ζ ) ξ γ where the cycle γ goes around the zeros of Q(ζ, κ) (Bethe roots) in the clockwise direction, as opposed to all other contours. We shall need a slightly more general kernel than K (ζ /ξ ), so, let us define them together. First, we introduce operations: ζ f (ζ ) = f (ζ q) − f (ζ q −1 ), δζ− f (ζ ) = f (ζ q) − ρ(ζ |κ, κ + α) f (ζ ).
(3.6)
Grassmann Structure in XXZ Model
835
Then 1 ζ ψ(ζ, α), 2πi
K (ζ, α) =
K (ζ ) = K (ζ, 0).
(3.7)
We shall use the the following notation: f g = f (η)g(η)dm(η) , γ
where the measure is given by dm(η) =
dη2 . η2 ρ(η|κ, κ + α) (1 + a(η, κ))
(3.8)
Now we introduce the resolvent of a certain integral operator Rdress − Rdress K α = K α ,
(3.9)
where K α stands for the integral operator with the kernel K (ζ /ξ, α). Introducing further the two kernels f left (ζ, ξ ) =
1 2πi
δζ− ψ(ζ /ξ, α),
f right (ζ, ξ ) = δξ− ψ(ζ /ξ, α),
(3.10)
we are ready to write the definition of [8] cleaning it from irrelevant auxiliary objects and taking into account the modification (2.11): 1 4 ω(ζ, ξ |κ, α) = f left f right + f left Rdress f right (ζ, ξ ) +δζ− δξ− −1 ζ ψ(ζ /ξ, α).
(3.11)
4. Introducing Screening Operators on the Lattice Now we want to describe an important generalisation of results of [3]. We have three constants: α which defines the tail of the operator and κ, κ + α which defines the twist of Matsubara transfer matrices at +∞ and −∞. As it has been said in the Introduction we need more freedom. Let us explain how an additional parameter s ∈ Z can be introduced into Z κ . We shall see later that in the scaling limit introduction of this parameter leads to emancipation of κ + α from κ and α. Let us consider the trace (−s) Tr S Tr M YM (4.1) TS,M q 2κ S+2(α−s)S(0) O(s) , (−s)
where YM carries spin −s. For definiteness we suppose s > 0. It follows from the ice condition that in this situation the operator O(s) must have spin s. We assume that among the eigenvectors of the transfer matrices TM (ζ, κ) and TM (ζ, κ + α − s) there are maximal ones |κ , |κ + α − s, s with eigenvalues T (ζ, κ) and T (ζ, κ + α − s, s) which are defined by the requirement that T (1, κ) · T (1, κ + α − s, s) is of maximal absolute value among all the pairs of eigenvectors. We assume further the generality condition: (−s)
κ|YM |κ + α − s, s = 0.
(4.2)
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H. Boos, M. Jimbo, T. Miwa, F. Smirnov
Obviously the difference between spins of |κ + α − s, s and |κ must be equal to s. We make the technical assumption that a spin of |κ remains equal to zero. The natural idea is to create the operators q 2(α−s)S(0) O(s) by b∗ , c∗ , t∗ having an excess of operators b∗ : q 2(α−s)S(0) O(s) = b∗ (ξ1 ) · · · b∗ (ξs )
×b∗ (ζm+ ) · · · b∗ (ζ1+ )c∗ (ζ1− ) · · · c∗ (ζm− )t∗ (ζ10 ) · · · t∗ (ζn0 ) q 2αS(0) .
However, the formulae (2.7) are not applicable in this case. Let us explain why it is so.
The method of [3] requires to start the consideration by the operator b∗ (ξ1 ) which is the closest to TS,M q 2κ S . The operator ξ1−α b∗rat (ξ1 )(X ) is a meromorphic function of ξ12 with singularities at the points (ζ j− )2 . It satisfies n + 1 normalisation conditions [3]. The additional term in (2.9) is of the form satisfying (2.7) from the very beginning. These singularities and normalisation conditions are studied algebraically, they do not change comparing to [3] where the spin of X was equal to −1. However, the behaviour at zero changes: in the present case the spin of X equals s − 1, and according to [2] ξ1−α b∗ (ξ1 )(X ) does not vanish at zero. Generally, ζ −α b∗ (ζ )(X ) =
s−1
ζ −2 j b∗∞, j (X ) + ζ −α b∗reg (ζ )(X ), spin(X ) = s − 1,
(4.3)
j=0
where ζ −α b∗reg (ζ )(X ) vanishes at 0. But in [3] the fact that ζ −α b∗ (ζ )(X ) vanishes at 0 for spin(X ) = −1 was important when deriving (2.7). As a result in the present case we do not have enough conditions to define ω. Let us turn this problem into advantage. The operators b∗∞, j , j = 0, . . . s − 1 constitute a finite Grassmann algebra. So, we just consider ξ j → 0 and replace b∗ (ξ1 ) . . . b∗ (ξs ) by b∗∞,s−1 · · · b∗∞,0 . Now move one of the remaining operators b∗ , namely, b∗ (ζ1+ ) to the left. Obviously, (ζ1+ )−α b∗ (ζ1+ ) vanishes as (ζ1+ )2 → 0 because the singular part disappears due to multiplication by b∗∞,s−1 · · · b∗∞,0 . Effectively, this operator reduces to b∗reg (ζ1+ ). It is not hard to see that we can introduce ω(ζ, ξ ) and obtain (2.6), (2.7), (2.8) for the functional on Wα,0 : (−s) Tr S Tr M YM TS,M q 2κ S b∗ ∞,s−1 · · · b∗ ∞,0 q 2αS(0) O Z κ,s q 2αS(0) O = . (4.4) (−s) Tr S Tr M YM TS,M q 2κ S b∗ ∞,s−1 · · · b∗ ∞,0 q 2αS(0) The function ρ changes in the most natural way to ρ(ζ |κ, κ + α, s) =
T (ζ, κ + α − s, s) . T (ζ, κ)
(4.5)
The function ω(ζ, ξ |κ, α, s) is defined by (3.11), replacing ρ(ζ |κ, κ + α) by ρ(ζ |κ, κ + α, s) but keeping the same a(ζ, κ). Let us discuss one important property of the Bethe vector of spin s. The basic objects in the theory are the transfer matrix TM (ζ, κ + α − s) and Baxter’s Q-operator Q M (ζ, κ + α − s). Their eigenvalues on the vector |κ + α − s, s have the following analytical properties: T (ζ, κ + α − s, s) is a polynomial of ζ 2 of degree n,
Grassmann Structure in XXZ Model
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Q(ζ, κ + α − s, s) = ζ −α−κ+2s A(ζ, κ + α − s, s) , where A(ζ, κ + α − s, s) is a polynomial in ζ 2 of degree n/2 − s. In terms of T and A the Baxter equation reads T (ζ, κ + α − s, s)A(ζ, κ + α − s, s)
= q −κ d(ζ )A(ζ q, κ + α − s, s) + q κ a(ζ )A(ζ q −1 , κ + α − s, s),
(4.6)
with κ = κ + α + 2 1−ν ν s.
(4.7)
Formally, κ is defined modulo 2Z/ν. However, the eigenvalues are multi-valued functions of κ and we have to be careful about the choice of the branch. The choice in (4.7) is consistent with the semi-classical limit ν → 1. We shall return to this point when we discuss the scaling limit. Notice that if we write the Baxter equation for twist κ and spin 0:
T (ζ, κ )A(ζ, κ ) = q −κ d(ζ )A(ζ q, κ ) + q κ a(ζ )A(ζ q −1 , κ ),
(4.8)
it looks exactly the same as (4.6) if we identify T (ζ, κ ) with T (ζ, κ + α − s, s) and A(ζ, κ ) with A(ζ, κ + α − s, s). There is, however an important difference: the polynomial A(ζ, κ ) in (4.8) is of degree n/2 while the polynomial A(ζ, κ + α − s, s) in (4.6) is of degree n/2 − s. Still, this similarity will be very important for us later when we discuss the scaling limit. (s) Similarly to the previous discussion we can consider an operator YM which carries positive spin s. Then the operator c∗ (ζ ) will have nontrivial behaviour at ζ 2 = 0: α ∗
ζ c (ζ )(X ) =
s−1
∗ α ∗ ζ −2 j c∞, j (X ) + ζ creg (ζ )(X ), spin(X ) = −s + 1,
(4.9)
j=0 ∗ ∗ and we can repeat the entire procedure using c∞, j instead of b∞, j . So, s in (4.7) can take any integer value. Obviously, what we are doing here is nothing else but introducing screening operators on the lattice. This is important for relating to the CFT. The screening operators b∗∞, j ∗ . Naively, one could say that anticommute among themselves, the same is true for c∞, j ∗ ∗ b∞, j anticommute with c∞,i , but this does not make sense because the product of these operators do not act nontrivially on any subspace of W (α) .
Remark 4.1. Let us mention one more, less dramatic, generalisation of the results of (s) [3]. Clearly the functional Z κ,s is independent of the choice of YM provided the condition (4.2) is satisfied. However different choices are also possible. For instance one can take any eigenvector |A and eigencovector B| of the Matsubara transfer matrices and consider the projector |A B|. The main formula is applicable in this more general setting.
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H. Boos, M. Jimbo, T. Miwa, F. Smirnov
5. Scaling Limit We take α, κ to be real, and restrict our consideration to the region κ < 1, κ < 1.
(5.1)
Then we continue analytically. The crucial data for our construction are the eigenvectors |κ + α − s, s , |κ . Consider the second of them. The corresponding eigenvalue T (ζ, κ) corresponds to the maximal eigenvector |κ of spin zero. Denote by Q(ζ, κ) the eigenvalue of the Q-operator on |κ . These eigenvalues have the form Q(ζ, κ) = ζ −κ
n/2
(1 − ζ 2 /ξ p2 ),
p=1
T (ζ, κ) = (q κ + q −κ )
n
(1 − ζ 2 /θ p2 ).
p=1
In the domain |κ| < 1, the vector |κ is uniquely characterised by the two requirements for the roots: ξk2 , θ 2j ∈ R, and ξk2 > 0 > θ 2j for all k, j. Let us study the behaviour of the Bethe roots ξ p as n → ∞. Suppose they are numbered in the order ξ12 < ξ22 < ξ32 < · · ·. In the limit n → ∞ they are subject to the Lieb distribution [15] : for 1 m we have
1 1 ξm+1 πν − ν1 ν +O . (5.2) ξm + ξm −1= ξm n n2 We are interested in the Bethe roots which are not very far from ζ 2 = 0. In other words 1
we assume 1 m n. Since ξm is small, we can drop the term ξmν in the (5.2), obtaining m ν ξm π . (5.3) n Similar power law is obeyed by θm . So far we have concentrated on the ground states in Matsubara direction. But according to Remark 4.1 the main formulae can be generalised to arbitrary Bethe states. There are low-lying excited states which satisfy (5.3), and to which the same analysis as for the ground states apply. Readers who are familiar with the papers by Bazhanov, Lukyanov and Zamolodchikov [9,10,16] would immediately say that in the limit we obtain the CFT transfer matrices treated by them. We shall come to this relation later in Sect. 7. The formulae (2.6), (2.7) and (2.8) imply the explicit expression Z κ,s t∗ (ζ10 ) · · · t∗ (ζ p0 )b∗ (ζ1+ ) · · · b∗ (ζr+ )c∗ (ζr− ) · · · c∗ (ζ1− ) q 2αS(0) =
p
2ρ(ζi0 |κ, κ + α, s) × det ω(ζi+ , ζ j− |κ, α, s)
i, j=1,...,r
i=1
.
(5.4)
We consider the scaling limit in the Matsubara direction, n → ∞, a → 0, na = 2π R fixed.
(5.5)
Grassmann Structure in XXZ Model
839
Let us introduce the following strange-looking notation: a¯ = Ca,
(5.6)
where C is some ν-dependent constant which will be needed for fine tuning comparing the scaling limit with CFT. The following limits exist for finite λ: T sc (λ, κ) = Q sc (λ, κ) =
lim
T (λa¯ ν , κ),
lim
a¯ νκ Q(λa¯ ν , κ).
n→∞, a→0, 2π R=na n→∞, a→0, 2π R=na
(5.7)
The eigenvalues of T sc (λ, κ), Q sc (λ, κ) are given by convergent infinite products due to (5.3). In particular, it is easy to see from (5.3) that the following asymptotics hold: log Q sc (λ, κ)λ2 →−∞ ∼ 2π R ·
1 C 2 2ν π (−λ ) . sin 2ν
(5.8)
Certainly, the limits exist for eigenvalues corresponding to any eigenvectors satisfying (5.3). Now we turn to the operators TM (ζ, κ + α − s), Q M (ζ, κ + α − s) for which the eigenvectors of spin s are considered. The definitions of T sc (λ, κ +α−s, s), Q sc (λ, κ +α−s, s) are the same as before. The important statement is that T sc (λ, κ + α − s, s) = T sc (λ, κ ) ,
Q sc (λ, κ + α − s, s) = Q sc (λ, κ ),
where κ is given by (4.7). The right hand sides are understood as “correct” analytical continuations from the spin 0 sector. We have to explain two points: first, what is the reason that in the scaling limit the eigenvalues in the spin s sector equals analytical continuations of those in the spin 0 sector; second, what we mean by “correct” analytic continuations. The first point is simple: recall the discussion concerning the similarity of Eqs. (4.6), (4.8). The only difference between them was the number of Bethe roots, but this number is infinite in the scaling limit, so, the difference disappears. On the other hand the eigenvalue T sc (λ, κ) is a multi-valued function of κ, so we have to explain the choice of its branch. At this point we refer to the semi-classical domain: ν close to 1. We take a good branch at this domain, and then continue analytically. Notice that T (0, κ) = 2 cos(π νκ). We require that introducing s does not deviate us far from this value. This was the reason for choosing definition (3.1) because with this definition we have T (0, κ − s, s) = 2 cos(π ν(κ + 2 1−ν ν s)), which stays close to 2 cos(π νκ) for all s if ν is close to 1. From now on we shall often consider κ and κ as arbitrary numbers implying the possibility of analytical continuation from values (4.7). Using Q sc (λ, κ), Q sc (λ, κ ) we obtain finite limits ρ sc (λ|κ, κ ) =
lim
n→∞, a→0, 2π R=na
4 ωsc (λ, μ|κ, κ , α) =
lim
n→∞, a→0, 2π R=na
ρ(λa¯ ν |κ, α, s), ω(λa¯ ν , μa¯ ν |κ, α, s),
(5.9) (5.10)
where κ is given by (4.7), and then for ρ sc (λ|κ, κ ), ωsc (λ, μ|κ, κ , α) the analytical continuation with respect to κ is used.
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According to Remark 4.1 (5.4) remains valid for any Bethe states in the Matsubara direction. It has been already said that to Bethe states close to the ground states the scaling procedure applies. We shall argue later on that these vectors span the Verma module of chiral CFT. So, we would like to use the right hand side of (5.4) in order to consider the scaling limit in the space direction ja = x finite. In our setting it amounts to considering the operators: 2τ ∗ (λ) = lim t∗ (λa¯ ν ),
(5.11)
a→0
2β ∗ (λ) = lim b∗ (λa¯ ν ), 2γ ∗ (λ) = lim c∗ (λa¯ ν ), a→0
a→0
and α (0) = lim q 2αS(0) , a→0
so that (5.4) gives in the scaling limit τ ∗ (λ01 ) · · · τ ∗ (λ0p )β ∗ (λ+1 ) · · · β ∗ (λr+ )γ ∗ (λr− ) · · · γ ∗ (λ− Z κ,κ 1 ) (α (0)) R =
p
ρ sc (λi0 |κ, κ ) × det ωsc (λi+ , λ−j |κ, κ , α)
i, j=1,··· ,r
i=1
.
(5.12)
We understand the formula (5.4) as giving the expectation values of certain non-local operators making contact with quasi-local ones near ζ 2 = 1. After introducing a this point moves to λ2 = a¯ −2ν , and in the scaling limit it goes further to λ2 = +∞. It should not be a surprise that this limit is described by CFT. To make this statement precise, we have to establish certain asymptotic properties of ρ sc (λ|κ, κ ), ωsc (λ, μ|κ, κ , α) for λ2 , μ2 → +∞. But first we shall need some information about the integrable structure of CFT. 6. CFT on a Cylinder and Three Point Functions In this section we introduce our notation concerning CFT, and collect a few facts which will be relevant to the subsequent sections. Consider chiral CFT on a cylinder Cyl = C/2πi RZ with circumference 2π R, the points x and x + 2πi R being identified. Along with the local coordinate x, we shall also use the global coordinate x
z = e− R . The two points x = −∞, ∞ on the boundary of Cyl correspond respectively to the points z = ∞, 0 on the Riemann sphere. Let T (x) =
∞
n=−∞
ln x −n−2
Grassmann Structure in XXZ Model
841
be the energy-momentum tensor in the coordinate x, where the ln ’s satisfy the commutation relations of the Virasoro algebra with the central charge c =1−
6 ν2 . 1−ν
(6.1)
The energy-momentum tensor in the coordinate z, T˜ (z) =
∞
L n z −n−2 ,
n=−∞
is related to T (x) via the transformation rule T˜ (z)(dz)2 = (T (x) − (c/12){z; x})(d x)2 . Here {z; x} denotes the Schwarzian derivative. In turn, T (x) is written as ∞
nx 1 c T (x) = 2 Lne R − . R n=−∞ 24 The Virasoro algebra acts on a local field O(y) by the contour integral dx (x − y)n+1 T (x)O(y), (ln O)(y) = 2πi Cy
(6.2)
where C y encircles the point y anticlockwise. From now on, we fix a primary field φ (y) with the scaling dimension : (l0 φ )(y) = φ (y), (ln φ )(y) = 0 (n > 0), and study the expectation values T (xk ) · · · T (x1 )φ (y) + , − .
(6.3)
The suffix indicates that we consider (6.3) in the presence of two other primary fields inserted at x = ±∞. More precisely, we impose the boundary conditions 1 c (6.4) lim T (x) = 2 ± − x→±∞ R 24 inside the expectation values (6.3), where ± are the conformal dimensions of the inserted primary fields. For readers who prefer the language of representation theory, we are considering a highest weight vector | + at z = 0 satisfying L n | + = δn,0 + | + (n ≥ 0), and a co-vector − | at z = ∞ satisfying − |L n = δn,0 − − | (n ≤ 0). The singular part of (6.3) is known from OPEs. In order to write them in the coordinate x, it is useful to introduce the function x B x 2n−1 1 2n coth = . 2 2R (2n)! R ∞
χ (x) =
n=0
Here B0 = 1, B2 = 1/6, B4 = −1/30, · · · are the Bernoulli numbers. The main OPEs then read, as x → y, T (x)T (y) = −
c 2T (y) T (y) χ (x − y) − χ (x − y) + χ (x − y) + O(1) , 12R R R (6.5)
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H. Boos, M. Jimbo, T. Miwa, F. Smirnov
T (x)φ (y) = −
φ (y) φ (y) χ (x − y) + χ (x − y) + O(1) , R R
(6.6)
where the prime stands for the derivative. The OPEs (6.5), (6.6), combined with (6.4) and the behaviour 1 χ (x) = ± + O(e∓x/R ) (x → ±∞), 2 allow us to write the conformal Ward-Takahashi identity which determines (6.3) recursively: T (xk ) · · · T (x1 )φ (y) + , − =−
k j c χ (x1 − x j ) T (xk ) · · · · · · T (x2 )φ (y) + , − 12R j=2
k
2 1 ∂ − χ (x1 − x j ) + (χ (x1 − x j ) − χ (x1 − y)) − χ (x1 − y) R R ∂x j R j=2 1 1 c T (xk ) · · · T (x2 )φ (y) + , − . +( + − − ) 2 χ (x1 − y)+ 2 ( + + − )− R 2R 24R 2
+
From these one can extract, for example, ⎧ Bn δn,2 c ⎪ ⎨ 2 ( + + − − ) − (l−n φ )(y) + , − 2R 12 n(n − 2)!R n = ( − )B − n−1 ⎪ φ (y) + , − ⎩ + (n − 1)!R n
(n : even); (n : odd).
In general, the normalised three point function of any particular descendant l−n k · · · l−n 1 φ (y) + , − φ (y) + , −
(6.7)
is determined as a polynomial in , + , − . In writing these formulas, we have tacitly assumed that φ (y) + , − is non-trivial. Actually, in the theory with c < 1, for a given generic value of there is a discrete but an infinite collection of such + , − . In view of the polynomial dependence mentioned above, one can regard (6.7) as a linear functional, defined for arbitrary , + , − , on the Verma module consisting of the descendants of φ (y). Henceforth we shall adopt this point of view. In later sections we shall use the parametrisation = α , + = κ+1 , − = −κ +1 , and write this functional as
Z κ,κ R {X (y)} =
X (y) κ+1 , −κ +1 φ α (y) κ+1 , −κ +1
,
(6.8)
where X (y) is a descendant of φ α (y). In [11], A. Zamolodchikov introduced the local integrals of motion which survive the φ1,3 -perturbation of CFT. They act on local operators as dx (i2n−1 O)(y) = h 2n (x)O(y) (n ≥ 1). (6.9) 2πi Cy
Grassmann Structure in XXZ Model
843
The densities h 2n (y) are certain descendants of the identity operator I . The simplest examples are h 2 (x) = (l−2 I )(x) = T (x), h 4 (x) = (l−2 T )(x),
(6.10)
for which we have (i1 O)(y) = (l−1 O)(y), (i3 O)(y) = 2
∞
(l−n−2 ln−1 O)(y). n=0
In general, for a descendant h(x) of the identity operator, the three point function dx h(x)O(y) + , − 2πi Cy is reducible to that of O(y). Indeed, it can be rewritten as dx dx h(x)O(y) + , − + O(y)h(x) + , − , − 2πi 2πi C(u) C(v) where C(u)(u ∈ R) is a circle with the real part u, starting from u − π Ri and ending at u + π Ri. Choosing u Re y v and using the boundary conditions at x → ±∞, one can show that each of them reduce to a constant. From the above remark it follows that − + i2n−1 (O(y)) + , − = (I2n−1 − I2n−1 ) · O(y) + , − ,
(6.11)
± denote the vacuum eigenvalues of the local integrals of motion on the Verwhere I2n−1 ma module with conformal dimension ± . Their explicit formulas for small n can be found in [9] (see also Sect. 10, (10.18)–(10.20) below). Notice that, in the special case + = − , the three point function vanishes on the image of the local integrals. As mentioned in the Introduction, we accept the conjectural statement that the Verma module is spanned by the elements
i2k1 −1 · · · i2k p −1 l−2m 1 · · · l−2m q (φα (0)). Formula (6.11) tells that for the computation of the linear functional (6.7) it suffices to consider the descendants by the even Virasoro generators {l−2n }n≥1 . Before closing this section, let us comment on a point which could be a source of confusion. The local integrals of motion arise in two different ways. At the boundary of the cylinder, they appear as the operators I2n−1 constructed from the modes L n of the energy-momentum tensor in the coordinate e−x/R . These operators preserve the subspace of the Verma module of a given degree and can be diagonalised. In the classical limit, the eigenvalues of I2n−1 correspond to the values of the integrals of motion on quasi-periodic solutions to the KdV hierarchy. In contrast, the action of the integrals of motion on local fields i2n−1 are constructed from the modes ln in the coordinate x. Unlike in the first case, they do not commute with l0 (for example, the first integral of motion is l−1 ). They act as a creation part of the Heisenberg algebra. In the classical limit, they correspond to the action of the Hamiltonian vector fields generated by the local integrals of motion. So it does not make sense to talk about their diagonalisation.
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H. Boos, M. Jimbo, T. Miwa, F. Smirnov
7. Brief Review of BLZ In a series of papers [9,10,16] Bazhanov, Lukyanov and Zamolodchikov (BLZ) studied the integrable structure of the chiral CFT on a circle. We shall recall these results briefly, since they are quite relevant to us. It is convenient to write the energy-momentum tensor in terms of the chiral boson x an nx ϕ(x) = i Q + P + eR. R n n =0
The operators P, Q are canonically conjugate, and the an ’s satisfy the Heisenberg algebra [P, Q] =
i(1 − ν) m(1 − ν) , [am , an ] = δm+n,0 . 2 2
The energy-momentum tensor is expressed as (1 − ν)T (x) =: ϕ (x)2 : + νϕ (x) −
1−ν . 24R 2
The chiral vertex operator ν 2 α2 x
ν
φα (x) = e− 4(1−ν) R : e 1−ν αϕ(x) :
(7.1)
is a primary field of scaling dimension α =
α(α − 2)ν 2 . 4(1 − ν)
(7.2)
We note that the parameter β 2 used in [10] is identified as β 2 = 1 − ν, hence their q = eπiβ is our −q −1 . We have also changed the sign of P, an and the normal ordering convention in [10] to 2
: eλϕ(x) := eλ
an nx/R n0 n e
,
ν2 a2 x − 4(1−ν) R
which results in the appearance of a scalar factor e in (7.1). The main object studied by BLZ is the universal monodromy matrix in CFT. It is an element of Uq (b+ ) ⊗ AH , where Uq (b+ ) is the Borel subalgebra of Uq ( sl2 ) generated by e0 , e1 , h 1 , and AH is the algebra generated by P, Q, an (the suffix H stands for Heisenberg). Set KH (x) = e0 ⊗ V+ (x) + e1 ⊗ V− (x) , HH = − x
V± (x) = e−(1−ν) R : e±2ϕ(x) :,
2P , 1−ν
(7.3) (7.4)
so that [HH , V± (x)] = ±2V± (x). The universal monodromy matrix is defined to be
2π R 1 TH (λ) = P exp λ KH (−i y)dy q − 2 h 1 ⊗HH . (7.5) 0
Grassmann Structure in XXZ Model
845
Here P exp stands for the path ordered exponential. Formula (7.5) is understood as a power series in λ. The integrals in each term converge in the domain 1/2 < ν < 1. Otherwise divergences occur and a regularisation is needed. We have considered two maps: Uq (b+ ) → End(Va ) with two-dimensional Va , and Uq (b+ ) → Osc A with Osc A being the q-oscillator algebra (see [2] for the notation). The images of TH (λ) under these maps are denoted by Ta,H (λ) and T A,H (λ). Then following [10] we define 3 THCFT (λ) = Tr a Ta,H (λ)e−2πi(σa ⊗P) , (7.6) 2P 2πi P ν Q CFT )Tr A T A,H (λ)e2πi(D A ⊗P) . H (λ) = λ (1 − e There is a slight difference with [10] due to different notation for the q-oscillator algebra. These operators satisfy the Baxter equation CFT −1 CFT THCFT (λ)Q CFT H (λ) = Q H (λq ) + Q H (λq).
(7.7)
An important property of these transfer matrices is that they commute with the local c integrals of motion. The first local integral of motion is nothing but L 0 − 24 , which commutes with the transfer matrices as mentioned above. Hence each of their eigenstates on a Verma module belongs to the subspace of a definite degree. In particular, the highest weight vector of the Verma module (primary field) is an eigenvector. Actually, the local integrals of motion are all encoded in the transfer matrix THCFT (λ). The latter is known [9] to be an entire function of λ2 . One of the main statements of [9] is that it has the following asymptotics for λ2 → ∞, λ2 ∈ / R 0 > μ2j for all i, j. 8. Conformal Limit in the Matsubara Direction Let us return to the XXZ model. Using the notation introduced in Sect. 5, we write the Baxter equation TM (λa¯ ν , κ)Q M (λa¯ ν , κ) = a(λa¯ ν )Q M (λa¯ ν q −1 , κ) + d(λa¯ ν )Q M (λa¯ ν q, κ), (8.1) where a(λa¯ ν ) = (1 − q a¯ 2ν λ2 )n , d(λa¯ ν ) = (1 − q −1 a¯ 2ν λ2 )n . We are interested in the maximal eigenvector |κ of TM (λa¯ ν , κ). We want to consider the limit n → ∞, a¯ → 0, while keeping na¯ = 2π R · C and λ fixed. In this limit, for 1/2 < ν < 1, a(λ) → 1, d(λ) → 1, so, if we identify νκ = −2P,
(8.2)
the Baxter equation (8.1) turns into (7.7). Now we want to fix the constant C in order to make the equivalence between Q sc and Q CFT exact. We had the asymptotics (5.8). On the other hand, it is known [10] that
1 1 1 1 1−ν CFT 1− (ν) ν (−λ2 ) 2ν . log Q (λ, κ) ∼ R · √ 2ν 2ν π So, comparing we see that the agreement is exact if 1−ν 1 C = √ 2ν 1 (ν) ν . 2 π 2ν So, we come to
Q sc (λ, κ) = Q CFT (λ)
P=− νκ 2
(8.3)
.
(8.4)
Let us argue that the vector |κ goes to the primary field with the dimension κ+1 =
ν2 (κ 2 − 1). 4(1 − ν)
On the lattice the maximal eigenvector |κ is defined by the requirement that the eigenvalue T (1, κ) is of maximal absolute value. In the scaling limit this corresponds to the requirement that THCFT (λ, κ) is maximal for λ2 large and positive. But the asymptotic behaviour of the BLZ transfer matrix is given by (7.8), so in the domain −
πν πν < arg λ < 2 2
(8.5)
Grassmann Structure in XXZ Model
847
the maximal eigenvalue corresponds to the primary field. Comparing with (7.2) and (7.9), we find that in the picture of Sect. 6 we have + = κ+1 . Hence the boundary conditions at +∞ is described by φκ+1 (+∞). Similarly, considering the left Matsubara transfer matrix we find that its ground state corresponds to the scaling dimension − = −κ +1 .
(8.6)
In other words, in the picture of Sect. 6 we have the left boundary condition described by the primary field φ−κ +1 (−∞). The argument above is not rigourous, because it involves two limits which are a priori non-commutative. Nevertheless we believe it makes sense because of integrability, which stipulates that |κ is an eigenvector of TM (λa¯ ν , κ) for all values of λ. As a supporting argument, we quote from [17] knowledge that the eigenvalue T (ζ, κ) of the lattice transfer matrix for τ = q 1/2 has maximal absolute value in the range |ζ | = 1, −π ν/2 < arg ζ < π ν/2. This agrees exactly with (8.5). We call the previous reasoning a macroscopic one. For completeness let us give a less formal, microscopic derivation providing at the same time a constant which is important for physics. Consider the Matsubara transfer matrix TM (λa¯ ν , κ), which is given explicitly by 3 TM (λa¯ ν , κ) = Tr j L j,n (λa¯ ν ) · · · L j,1 (λa¯ ν )q κσ j , (8.7) where
1 3 1 3 − q − 2 σm − a¯ 2ν λ2 q 2 σm −(q − q −1 )λa¯ ν σm L j,m (λa¯ ) = . 1 3 3 + q 21 σm −(q − q −1 )λa¯ ν σm − a¯ 2ν λ2 q − 2 σm j ν
Let us make the gauge transformation 1
3 m σ 3 i=1 i
L j,m (λa¯ ν ) = q − 2 σ j where
1
3 m−1 σ 3 i=1 i
L j,m (λa¯ ν )q 2 σ j
,
1
(8.8) m−1
1 − a¯ 2ν λ2 q σm −(q − q −1 )λa¯ ν q − 2 + i=1 L j,m (λa¯ ) = 3 3 − 12 − m−1 σ −1 ν + i=1 i σ −(q − q )λa¯ q 1 − a¯ 2ν λ2 q −σm m 3
ν
σi3
− σm
. j
Now we recall known formulae concerning the continuous limit of the XXZ chain [18]. A very accurate account of this matter is given in Lukyanov’s paper [19]. Notice, however, that the Hamiltonian in [19] differs from ours by a similarity transformation ! 3 . Having this in mind we rewrite the main order with the operator U = m: odd σm formulae for n → ∞, y = ma (formulae (2.19) in [19]) as follows: a ∂ y (ϕ(−i y) − ϕ(−i ¯ y)) , iπ(1 − ν) 1 ¯ y)) → (−1)m F/2 a 2 (1−ν) : e±(ϕ(−i y)+ϕ(−i :.
3 σm → ± σm
(8.9)
848
H. Boos, M. Jimbo, T. Miwa, F. Smirnov
Here ϕ(x), ϕ(x) ¯ are two chiral bosons with the same normalisation as in Sect. 7. The fractional power of a in the second formula is needed in order to compensate the anom¯ y)) :, and F is related to the one point function of alous dimension of : e±(ϕ(−i y)−ϕ(−i the latter [19]. From these formulae we see that m−1 3 √ ± Wm± = q ∓ i=1 σi σm → Z a 1−ν V± (−i y), (8.10) where V± (x) are the chiral vertex operators (7.4). The power of a changed due to a normal reordering, while the constant Z is obviously related to the asymptotical behaviour at m → ∞ of the following two-point function for XXZ model: Wm+ W0− =
Z m 2(1−ν)
.
We do not know a direct way to fix this constant, however, our construction allows an indirect one. Indeed, in order to have complete agreement with CFT on this microscopic level we need that L j,m (λa¯ ν ) = 1 + aλK j,H (−i y) + O(a 2ν ),
(8.11)
which would imply
ν ν L j,n (λa¯ ) · · · L j,1 (λa¯ ) → P exp λ
2π R
K j,H (−i y)dy ,
(8.12)
0
where we have set
1 K j,H (x) = iq − 2 σ j+ V− (x) + σ j− V+ (x) .
So the microscopic picture agrees with the macroscopic one if Z=
1 4 sin2 (π ν)C 2ν
Altogether we obtain from (8.7), "
ν πiνκ TM (λa¯ , κ) → Tr j e P exp λ
.
2π R
# K j,H (−i y)dy
,
(8.13)
0
giving rise to the BLZ transfer matrix (7.6) with the identification (8.2). In particular, the second chirality decouples. 9. Conformal Limit in the Space Direction Let us return to the formula ∗ 0 ∗ 0 ∗ + ∗ + ∗ − ∗ − Z κ,κ τ (λ ) · · · τ (λ )β (λ ) · · · β (λ )γ (λ ) · · · γ (λ ) (0)) ( α p r r 1 1 1 R =
p i=1
ρ sc (λi0 |κ, κ ) det ωsc (λi+ , λ−j |κ, κ , α)
i, j=1,··· ,r
.
(9.1)
We have seen that the functions in the right-hand side are defined through the eigenvalues of the BLZ transfer matrix on the primary fields φκ+1 , φ−κ +1 . Thus the right hand side of (9.1) is defined. We want to interpret the left-hand side of this equation. Our arguments are far from being mathematically rigourous, so, we formulate our statement as a conjecture.
Grassmann Structure in XXZ Model
849
Conjecture. Asymptotics of (9.1) for λi± , λi0 → ∞ describes the expectation values of ν descendants for chiral CFT with c = 1 − 6 1−ν of the primary field φα (0) inserted on the cylinder with the asymptotic conditions described by φ−κ +1 and φκ+1 . 2
Recall that we start with κ, κ , α satisfying (4.7), and then continue analytically. The three point function of the operators φ−κ +1 (−∞), φα (0), φκ+1 (∞) does not vanish because (4.7) can be rewritten as (−κ + 1) + α + (κ + 1) = 2 − 2 1−ν ν s, which coincides in our normalisation with the Dotsenko-Fateev condition [7] for one type of screening operators condition. We do not know if the second set of screening operators can be defined starting from the lattice model. In the present section we shall first present qualitative arguments in favour of this conjecture, and then explain how it can be verified quantitatively. The actual verification will be done in Sects. 11, 12. Consider the primary field q 2αS(0) on the lattice. Making the scaling limit in the horizontal direction on the cylinder in the same way as it was done in the vertical one we conclude that this operator turns into α (0) = φα (0)⊗ φ¯ α (0). Typical operators in the space Wα−s,s are of the form q 2(α−s)S(0) σk+1 · · · σk+s O(s ≥ 0) or q 2(α−s)S(0) σk−1 · · · σk−−s O(s < 0), where O is spinless. Then the same bosonisation formulae as (8.9) ν
¯ q 2αS(0) ∼ e 1−ν α(ϕ(0)−ϕ(0)) , q −2S(k−1) σk+ ∼ e2ϕ(x)
imply that Scaling limit Wα−s,s ⊂ Vα+2 1−ν s ⊗ V −α . ν
τ ∗ (λ), β ∗ (λ), γ ∗ (λ)
So, the operators do not change the Verma module for the second chirality. This is one reason to assume that they do not act on it at all. Let us give the first evidence for this claim. Consider the operator τ ∗ (λ). It originates from its counterpart on the lattice, t∗ (ζ ). We know that close to ζ 2 = 1 the operator t∗ (ζ ) describes the adjoint action of XXZ local integrals of motion. So, we expect the same in the CFT. Using the BLZ formulae (7.8) we see that log ρ sc (λ|κ, κ )
∞
λ−
2n−1 ν
Cn I2n−1 (κ) − I2n−1 (κ ) ,
(9.2)
n=1
which implies together with (6.11) that ∞
2n−1 τ ∗ (λ) exp λ− ν Cn i2n−1 . n=1
So, as it has been expected, the local operators are extracted from the action of τ ∗ (λ) 1 in the asymptotics at λ → ∞ as coefficients of fractional degrees λ ν which have the dimension of inverse length. Certainly, this exercise is quite tautological, and we would not write this paper if this were the only thing we can do. But it demonstrates the chiral nature of our operators.
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In the Appendix we prove the general statement that ωsc (λ, μ|κ, κ , α) has the following asymptotics for λ2 , μ2 → +∞: ωsc (λ, μ|κ, κ , α)
∞
2 j−1 2i−1 ρ sc (λ|κ, κ ) ρ sc (μ|κ, κ ) λ− ν μ− ν ωi, j (κ, κ , α). i, j=1
(9.3) The proof will be given for the primary fields φ−κ −1 , φκ+1 as asymptotical conditions, but it can be generalised to arbitrary descendants. So, in the weak sense the following operators are defined: ∞
2 j−1 τ ∗2 j−1 λ− ν , log τ ∗ (λ)
(9.4)
j=1
√ √
1 τ ∗ (λ) 1 τ ∗ (λ)
β ∗ (λ) γ ∗ (λ)
∞
j=1 ∞
β ∗2 j−1 λ−
2 j−1 ν
,
(9.5)
γ ∗2 j−1 λ−
2 j−1 ν
,
(9.6)
j=1
which act between different Verma modules: τ ∗2 j−1 : Vα+2 1−ν s β ∗2 j−1 γ ∗2 j−1
ν
→ Vα+2 1−ν s , ν
: Vα+2 1−ν (s−1) → Vα+2 1−ν s , ν
ν
: Vα+2 1−ν (s+1) → Vα+2 1−ν s . ν
ν
Consider the Verma module Vα . We obtain different elements of this module by operators τ ∗2 j−1 acting on the primary field φα , and by the same number of operators β ∗2 j−1 and γ ∗2 j−1 acting further. Due to the completeness in the lattice case [12], in this way we obtain linearly independent vectors from Vα . Counting the characters we see that the entire Verma module is created in this way. Indeed, from a combinatorial point of view, τ ∗ is one odd boson, and β ∗ , γ ∗ are Gross-Neveu fermions which in the uncharged sector produce one even boson. Now we shall proceed to computation of the coefficients ωi, j (κ, κ , α) and comparing them with the three-point functions of CFT. Since the operator τ ∗ (λ) is already settled we shall consider the case κ = κ which means to ignore the image of the actions by the local integrals of motion i2n−1 in the Verma module Vα . Acting on the primary field φα (0), the even generators of the Virasoro algebra, l−2k , create the quotient space of the Verma module by these descendants. 10. Asymptotics of log asc In this section we study the asymptotic behaviour of the function asc (λ, κ) =
Q sc (λq, κ) Q sc (λq −1 , κ)
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as λ2 → ∞. Following closely the analysis developed in [10], we give a recursive algorithm for determining the coefficients of the asymptotic expansion. It is known (see [10], (3.17) and (3.23)) that for large κ the smallest Bethe root behaves as λ21 ∼ c(ν)κ 2ν , where ν 2ν c(ν) = (ν)−2 eδ , 2R (10.1) δ = −ν log ν − (1 − ν) log(1 − ν). The main technical idea in [10] is to consider the limit λ2 , κ → ∞ ,
keeping t = c(ν)−1
λ2 fixed. κ 2ν
Henceforth we change the variable from λ to t and write F(t, κ) = log asc (λ, κ). This function is to be determined from the non-linear integral equation. In order to write the equation, it is convenient to redefine some functions in terms of the variables t, u. We use
1 1 tq 2 + 1 tq −2 + 1 K (t) = · − . 2πi 2 tq 2 − 1 tq −2 − 1 We also use R(t, u) to represent the following resolvent kernel. ∞ dv R(t, v)K (v/u) = K (t/u), (t, u > 1). R(t, u) − v 1 An explicit formula for R(t, u) will be given below. The non-linear integral equation for F(t, κ)(t > 1) reads ∞ du F(t, κ) − = −2πiνκ K (t/u)F(u, κ) u i 1 e−i ·∞ e ·∞ F(u,κ) du −F(u,κ) du − − K (t/u) log(1 + e ) K (t/u) log(1 + e ) , u u 1 1 (10.2) where is a small positive number. From Appendix A, we see that 1
1
1
1
F(t, κ) = −F+ (arg t, κ)|t| 2ν + O(|t|− 2ν ), 0 < arg t < π, F(t, κ) =
F− (arg t, κ)|t| 2ν + O(|t|− 2ν ), −π < arg t < 0,
where F± is positive. We seek for the solution of (10.2) in an asymptotic series in κ −1 , F(t, κ)
∞
κ −2n+1 Fn (t).
n=0
Consider first the leading coefficient F0 (t). For t > 1, from (9.2), ((I − K )F0 )(t) = −2πiν
(10.3)
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follows, where K denotes the integral operator on the interval [1, ∞), ∞ du K (t/u) f (u) . K f (t) = u 1 Equation (10.3) can be solved by the standard Wiener-Hopf technique. Quite generally, for a function f (t) let ∞ ∞ dt dk f (t)t −ik , fˆ(k)t ik f (t) = fˆ(k) = t 2π 0 −∞ denote the Mellin transform and its inverse transform. For the solution of (10.3) we shall need sinh(2ν − 1)π k Kˆ (k) = , sinh π k along with the Riemann-Hilbert factorisation 1 − Kˆ (k) = S(k)−1 S(−k)−1 , (1 + (1 − ν)ik)(1/2 + iνk) iδk S(k) = e , √ (1 + ik) 2π(1 − ν) where δ is defined in (10.1). The function S(k) is holomorphic on the lower half plane Im k < 1/2ν, and for k → ∞ behaves as S(k) = 1 + O(k −1 ). If we demand that 1 1 (t → ∞) , F0 (t) = const. t 2ν + O t − 2ν then (10.3) admits a unique solution given by −i f F0 (t) = , dl t il S(l) l(l + 2νi ) R− 2νi −i0
(t > 1) ,
(10.4)
where 1 f = √ . 2 2(1 − ν) The right-hand side of (9.4) gives a continuous function F˜0 (t) on the half line (0, ∞) such that F˜0 (t) = 0 for 0 < t ≤ 1. However, the function F0 (t) for t > 1 can be analytically continued to the sector | arg t| < 2(1 − ν)π , rewriting Eq. (9.3): F0 (t) = −2πiν + (K F0 )(t) , −i f . dl t il S(l) Kˆ (l) (K F0 )(t) = l(l + 2νi ) R− 2νi −i0
(10.5) (10.6)
It is also possible to check directly the consistency of the formulas (10.4) and (10.5). The difference of two integrals (10.4) and (10.6) has the only pole in the upper half plane 1 Im l + 2ν > 0 at l = 0, where we pick up the residue −2πiν.
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Now we turn to the higher order terms. Similarly as above, the Wiener-Hopf method allows us to find R(t, u) for t > 1, ∞ ∞ dl dm il im −i t u S(l)S(m) Kˆ (m) . R(t, u) = 2π 2π l + m − i0 −∞ −∞ The analytic continuation is given by ∞ ∞ dl dm il im −i R(t, u) = K (t/u) + t u S(l)S(m) Kˆ (l) Kˆ (m) . l + m − i0 −∞ −∞ 2π 2π From this
R(t, u) =
∞ −∞
dl il ˆ u) t S(l) Kˆ (l) R(l, 2π
follows, where
∞
dm im −i u S(m) l + m − i0 −∞ 2π 1 dm = u −il S(l)−1 + u im S(m) Kˆ (m). 2πi l + m − i0
ˆ u) = R(l,
(10.7)
ˆ e x ) is analytic near x = 0. Equation (10.2) can be converted The last line shows that R(l, into F(t, κ) = κ F0 (t) i e ·∞
− 1
du − R(t, u) log(1 + e F(u,κ) ) u
e−i ·∞ 1
du R(t, u) log(1 + e−F(u,κ) ) u
.
(10.8) Motivated by the formula (10.5), let us set ∞ dl t il S(l) Kˆ (l)( (l, κ) − κ 0 (l)), F(t, κ) = κ F0 (t) + −∞
(10.9)
where (l, κ) has an asymptotic expansion, (l, κ)
∞
n=0
κ −2n+1 n (l) , 0 (l) =
−i f l(l +
i 2ν )
.
(10.10)
We show below that each coefficient n (l) (n ≥ 1) can be determined as a polynomial in l by a purely algebraic procedure. With a change of integration variable, (10.8) is brought further into the form i −i∞+ d x ˆ i x/ f κ ,κ) ) (l, κ) − κ 0 (l) = − R(l, ei x/ f κ ) log(1 + e F(e fκ 0 2π i∞+ dx ˆ −i x/ f κ ,κ) + ) . (10.11) R(l, e−i x/ f κ ) log(1 + e−F(e 2π 0
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Since log(1 + e±F(u,κ) ) decays exponentially for ± Im u > 0, the asymptotics of the right hand side of (10.11) is completely determined from the behaviour of the integrand at x = 0. In order to develop a systematic expansion, let us first make a general remark. Consider a Fourier integral ∞ G(x) = eikx g(k)dk. −∞
We assume that g(k) is the boundary value of a holomorphic function on the lower half plane Im k < 0, satisfying the asymptotic expansion ∞
g(k)
gn (ik)−n (k → ∞, Im k < 0).
n=−n 0
Then integration by parts shows that for any N > 0 we have 0
G(x) =
gn 2π δ (n) (x) + 2π
n=−n 0
N
n=1
gn x n−1 + R N , (n − 1)! +
where R N = O(x N ) as x → 0. Suppose further that G(x) can be prolonged analytically around x = 0. In this situation its Taylor expansion can be computed from the right hand side, discarding the delta function terms. The result is summarised in a compact form ∞ eikx g(k)dk = 2πi resk [eikx g(k)] , −∞
where resk [· · · ] signifies the coefficient of k −1 in the expansion at k = ∞. The above consideration applies to (10.7), and we obtain the Taylor expansion at x = 0, " −hx/ f κ # e i x/ f κ ˆ S(h) . (10.12) ) = resh R(l, e l+h For the factor log(1 + e F(u,κ) ), we proceed as follows. Set ¯ F(ei x/ f κ , κ) = −2π x − F(x, κ) . ¯ Similarly as above, the Taylor expansion of F(x, κ) at x = 0 is calculated as ¯ F(x, κ) = x + resh e−hx/ f κ S(h)i (h, κ) .
(10.13)
¯ Actually the term x is cancelled by a term coming from κ 0 (h), so that F(x, κ) = O(κ −1 ). We rewrite the corresponding piece of the integrand as log(1 + e±F(e
±i x/ f κ ,κ)
)=
∞ ¯
∂ n F(±x, κ)n ∓ log(1 + e−2π x ). n! ∂x n=0
(10.14)
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Substituting (10.14), (10.12) into (10.11), we arrive at i (l, κ) − iκ 0 (l) " −hx/ f κ
$ # % ∞ e 2 1 ∞ dx ∂ n n ¯ log(1 + e−2π x ). resh S(h) F(x, κ) − fκ n! 0 2π l+h ∂x even n=0
(10.15) Here {· · · }even (resp. {· · · }odd ) means the even (resp. odd) part in x. To evaluate the integral in (10.15) we need only to develop the integrand into a Taylor series and apply the formula
∞ ∂ n dx m ζ (m − n + 2) − x log(1 + e−2π x ) = m!(1 − 2−m−1+n ) . 2π ∂x (2π )m−n+2 0 In summary, the asymptotic expansion (10.10) can be calculated order by order in κ −2 , from the set of equations (10.15) and (10.13). The first few terms of the expansion read i (l, κ) =
1 i 2ν )
fκ +
1 1 24 f κ
l(l +
i 2ν 2 − 6ν + 6 1 7 l− l −i + 6 + ··· . 2 · 90 2ν 7ν(1 − ν) ( f κ)3
In general, the coefficients have the structure n (l) =
n−1 j=1
i(2 j − 1) l− 2ν
× ( Polynomial in l of degree n − 1).
(10.16)
From the knowledge of log asc (λ, κ), it is straightforward to extract the asymptotic expansion of log T sc (λ, κ) [10]: √
δ−πiν 2 il (1−il)( 21 +iνl) e 1−ν λ (l, κ) log T sc (λ, κ) πiνκ + √ dl (1−i(1−ν)l) κ 2ν c(ν) 2π R− 2νi −i0
∞
Cn I2n−1 (κ)λ−
2n−1 ν
,
n=0
where
√ ( 2n−1 2n−1 π 1 2ν ) (1 − ν)n (ν)− ν , 1−ν ν n! (1 + 2ν (2n − 1)) (10.17)
i(2n − 1) 2 n−1 2n−1 −2n+1 I2n−1 (κ) = −i , κ × n(2n − 1)(2ν ) ( f κ) R , 2ν Cn = −
and we set by definition I−1 = R. Notice that C0 > 0 while Cn < 0 for n ≥ 1. The factors in (10.16) ensure that at these special values of l the asymptotic series (10.10) truncates. This has to be the case, because according to [10], I2n−1 (κ)(n ≥ 1) are the vacuum eigenvalues of the integrals of motion which are polynomials in c and κ+1 .
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For instance 1 c κ+1 − , R 24 1 1 c I3 (κ) = I1 (κ)2 − I1 (κ) + , 2 R 6R 1440R 3 1 1 c+5 c(5c + 28) I5 (κ) = I3 (κ)I1 (κ) − I3 (κ) + I1 (κ) − . R 3R 2 360R 4 181440R 5 I1 (κ) =
(10.18) (10.19) (10.20)
We have verified upto n = 4 that (10.17) matches perfectly the formulas for I2n−1 given in [9]. 11. Asymptotics of ω for κ = κ In this section, we restrict our consideration to the case κ = κ , so that ρ sc (λ|κ, κ ) = 1. Our goal is to give an algorithm for deriving the asymptotic expansion of the function ωsc (λ, μ|κ, κ, α). We start from the representation ωsc (λ, μ|κ, κ, α) = f left f right + f left Rdress f right (λ, μ) + ω0 (λ, μ|α) , (11.1) where 1 − δ ψ(λ/μ, α), f right (λ, μ, α) = δμ− ψ(λ/μ, α) , 2πi λ ω0 (λ, μ|α) = δλ− δμ− −1 λ ψ(λ/μ, α) ,
f left (λ, μ, α) =
and Rdress denotes the resolvent for the integral equation Rdress − Rdress K α = K α . Here we have set
f g = dm(λ) =
ei ∞
σ2
−
e−i ∞
(11.2)
σ2
f (λ)g(λ)dm(λ) ,
dλ2 , λ2 (1 + asc (λ, κ))
and σ 2 is a point lying between the smallest Bethe root and the largest zero of T sc (λ, κ). Strictly speaking, and dm(λ) have slightly different meaning than those used in Sect. 3. Since we use them here only locally, there should not be fear of confusion. From now until (11.5) below, we shall work with the variables t = c(ν)−1 λ2 /κ 2ν , u = c(ν)−1 μ2 /κ 2ν , and write K (t, α) =
1 tq 2 + 1 tq −2 + 1 1 · (tq 2 )α/2 2 − (tq −2 )α/2 −2 . 2πi 2 tq − 1 tq − 1
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857
Again we start with the leading order approximation as κ → ∞, where (11.2) becomes ∞ dv R(t, u, α) − R(v, u, α)K (t/v, α) = K (t/u, α). v 1 This can be solved in the same manner as before, using sinh π (2ν − 1)k − Kˆ (k, α) = sinh π k + iα 2
iα 2
.
The only point worth noting is that in writing the Riemann-Hilbert factorisation 1 − Kˆ (k, α) = S(k, α)−1 S(−k, 2 − α)−1 , 1 + (1 − ν)ik − α2 21 + iνk iδk S(k, α) = e , √ 1 + ik − α2 2π (1 − ν)(1−α)/2 we are naturally led to assume that 0 < α < 2. So the naïve symmetry (−k, −α) → (k, α) of Kˆ (k, α) is replaced by the symmetry (k, α) → (−k, 2 − α). With our normalisation of α, the reflection α → 2 − α is the usual one for CFT with c < 1. For the resolvent kernel we obtain the representation R(t, u, α) = K (t/u, α) ∞ ∞ dl dm il im −i t u S(l, α)S(m, 2 − α) Kˆ (l, α) Kˆ (m, 2 − α) . + 2π 2π l + m − i0 −∞ −∞ The ‘dressed’ resolvent kernel Rdress (t, u) satisfies ei ∞ 1 Rdress (t, u) − R(t, u, α) = − R(t, v, α)Rdress (v, u) −F(v,κ) 1 + e 1 e−i ∞ 1 dv dv − R(t, v, α)Rdress (v, u) . × v v 1 + e F(v,κ) 1 Setting Rdress (t, u) = K (t/u, α) ∞ ∞ dl dm il im t u S(l, α)S(m, 2 − α) Kˆ (l, α) Kˆ (m, 2 − α) (l, m|κ, α) , + −∞ −∞ 2π 2π (11.3) ∞
−i , (11.4) n (l, m|α)κ −2n , 0 (l, m) = (l, m|κ, α) l +m n=0
and repeating the analysis of the previous section, we arrive at the linear recursion relation for the n (l, m|α): ∞ e−l x/ f κ 2 1 ∞ (l, m|κ, α) − 0 (l, m) − d x resl S(l , 2 − α) fκ n! 0 l + l n=0 "
% ∂ n 1 ¯ κ)n − × . ×resm e−m x/ f κ S(m , α) (m , m|κ, α) F(x, ∂x 1 + e2π x odd
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The coefficients of the series (11.4) can be calculated by Taylor expanding the integrand and applying
∞
dxx
m
0
∂ − ∂x
n
1 ζ (m − n + 1) = m!(1 − 2−m+n ) . 1 + e2π x (2π )m−n+1
The first non-trivial term reads 1 1 i i (l, m|κ, α) + l + m 24ν ( f κ)2
1 1 −iν(l + m) − + α + O . 2 κ4
Returning to the original variables λ and μ, the formula for ωsc (λ, μ|κ, κ, α) can be obtained from (11.1). We have 1 ˜ α) S(m, ˜ dldm S(l, 2 − α) (l + i0, m|κ, α) 2πi
δ+πiν 2 il δ+πiν 2 im e λ μ e , (11.5) × 2ν 2ν κ c(ν) κ c(ν) −ik + α2 21 + iνk ˜ α) = S(k, , √ −i(1 − ν)k + α2 2π (1 − ν)(1−α)/2
ωsc (λ, μ|κ, κ, α)
where l + i0 is important only in 0 (l, m). Notice that originally in R(t, u, α) we had rather l − i0. The change appeared due to the addition of ω0 (λ, μ) which explains the importance of this term. Picking the residues at the poles in the upper half plane, its asymptotics as t, u → ∞ can be calculated: ωsc (λ, μ|κ, κ, α) −
∞
r,s=1
×λ−
1 D2r −1 (α)D2s−1 (2 − α) r +s−1
2r −1 ν
μ−
2s−1 ν
2r −1,2s−1 (κ, α) ,
(11.6)
where D2n−1 (α) =
1 (2n − 1) α2 + 2ν 2n−1 i 1 − 2n−1 , (11.7) (ν) ν (1 − ν) 2 · ν (n − 1)! α + (1−ν) (2n − 1) 2
2ν
and √ 2r +2s−2
r +s−1 2 f κν i(2r − 1) i(2s − 1) , . 2r −1,2s−1 (κ, α) = − κ, α × 2ν 2ν ν R The counterpart of the factorisation (10.16) for n (l) is the vanishing property
n
i(2r − 1) i(2s − 1) , α = 0 2ν 2ν
(n ≥ r + s).
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859
This ensures that the coefficients 2r −1,2s−1 (κ, α) are polynomials in κ+1 , α and c. For instance, 1 I1 (κ) − R 1 1,3 (κ, α) = I3 (κ) − 3,1 R 1 1,5 (κ, α) = I5 (κ) − 5,1 R 1,1 (κ, α) =
α , 12R 2 α 2α c+5 α I (κ) + + α ∓ dα , 1 6R 3 144R 4 1080R 4 360R 4
2 α α c + 11 I3 (κ) + + α I1 (κ) 4R 3 48R 5 360R 5
3α 13(c + 35) 2 2c2 + 21c + 70 − − α 6 1728R 90720R 6 α 60480R 6
α 1 c+7 ∓ I1 (κ) − 2α − α dα , 6 5 1440R 7560R 6 120R
2 α 1 α c+2 c+2 3,3 (κ, α) = I5 (κ) − I (κ) + + + I1 (κ) 3 α R 4R 3 48R 5 360R 5 1440R 5 −
−
1 5c − 14 2 10c2 + 37c + 70 1/2c2 + c 3α − α − α − . 6 6 6 1728R 18144R 362880R 36288R 6
Here I2n−1 (κ) are given in (10.18)–(10.20), and dα =
√ ν(ν − 2) (α − 1) = 16 (25 − c)(24 α + 1 − c). ν−1
(11.8)
These structures exhibit a remarkable consistency with our fermionic picture. 12. Final Results and Conclusions Now we clearly see the structure of our fermions in the CFT limit. They naturally split into two parts: β ∗2m−1 = D2m−1 (α)β CFT∗ 2m−1 ,
γ ∗2m−1 = D2m−1 (2 − α)γ CFT∗ 2m−1 ,
(12.1)
the multipliers D2m−1 (α), D2m−1 (2 − α) absorb all the transcendental dependence on CFT∗ α, the operators β CFT∗ 2m−1 , γ 2m−1 are purely CFT-objects. The fermions act between different Verma modules. In order to stay in one Verma module it is convenient to introduce the bilinear combinations of fermions 1 CFT∗ CFT∗ CFT∗ φ even β 2m−1 γ 2n−1 + β CFT∗ 2m−1,2n−1 = (m + n − 1) 2n−1 γ 2m−1 , 2 1 CFT∗ CFT∗ −1 CFT∗ φ odd β 2n−1 γ 2m−1 − β CFT∗ 2m−1,2n−1 = dα (m + n − 1) 2m−1 γ 2n−1 . 2 The Verma module has a basis consisting of the vectors i2k1 −1 · · · i2k p −1 l−2l1 , · · · l−2lq (φα ) .
(12.2)
Conjecturally the same space is also created by the action of the i2k−1 ’s and fermions: even odd odd i2k1 −1 · · · i2k p −1 φ even 2m 1 −1,2n 1 −1 · · · φ 2m r −1,2nr −1 φ 2m¯ 1 −1,2n¯ 1 −1 φ 2m¯ s −1,2n¯ s −1 (φα ) . (12.3)
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For small degrees, the transition coefficients between (12.2) and (12.3), modulo descendants of the i2k−1 , can be determined by taking the expectation values with κ = κ and equating like powers of κ. Abbreviating φα and writing α as , we find φ even 1,1 ≡ l−2 , 2 φ even 1,3 ≡ l−2 +
φ odd 1,3 ≡
2c − 32 l−4 , 9
2 l−4 , 3
c + 2 − 20 + 2c l−4 l−2 3( + 2) (12.4) −5600 +428c −6c2 +2352 2 −300c 2 +12c2 2 +896 3 − 32c 3 l−6 , + 60 ( +2) 2 56 − 52 − 2c + 4c l−4 l−2 + l−6 , φ odd 1,5 ≡ +2 5( + 2) 6 + 3c − 76 + 4c 3 l−2 l−4 φ even 3,3 ≡ l−2 + 6( + 2) −6544 +498c −5c2 +2152 2 −314c 2 +10c2 2 −448 3 +16c 3 l−6 . + 60 ( +2)
3 φ even 1,5 ≡ l−2 +
4 , l l2 , l2 , l l , l , At the next degree, there are 5 Virasoro descendants l−2 −4 −2 −4 −6 −2 −8 which are polynomials in κ+1 of degree 4, 2, 1, 1, 0, respectively. With the data at hand, obtained from the primary field φκ+1 , there remains one parameter undetermined. This can be fixed considering the first descendent L −1 φκ+1 which we hope to do in the future. Nevertheless we have checked that the determinant,
β CFT∗ γ CFT∗ γ CFT∗ , β CFT∗ 1 3 3 1 4 , has the correct degree 2 in after subtracting a suitable multiple of l−2 κ+1 . We regard it as further supporting evidence in favour of the fermionic structure. Let us pass to conclusions. We believe that the fermionic description will provide new results for the theory of integrable models. For example, there is an obvious similarity between our fermions and those introduced in [20]. With the formulae (12.4) at hand, it should be possible to upgrade the qualitative description of form factors of descendants in [20] to a quantitative level. We hope to explain this in future works. Here, however, we would like to emphasise that, even for CFT, the fermionic description must give something completely new. Let us explain that. Consider the functional Z κ,κ with κ = κ . It describes the three point function R for descendants of φα and two primary fields φ−κ+1 , φκ+1 of equal dimension −κ+1 = κ+1 . It was said several times that the construction generalises if we replace the asymptotic states described by φκ+1 by any other eigenstate of the integrals of motion I2n−1 . The only change is that the function ω is to be computed for the new asymptotic condition. It is assumed [9,10] that the joint spectrum of I2n−1 is simple, so, in this way we compute all the three-point functions for a descendant of φα and descendants of φ−κ+1 , φκ+1 , provided the latter are eigenstates of the integrals of motion. Notice that the descendant of φκ+1 can be very deep in the Verma module. In that case the usual CFT
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computation is rather hard to perform. Let us be more precise appealing to the classical limit. In the classical limit ν → 1, the eigenstates of I2n−1 are in correspondence with the periodic solutions of the KdV equation. Let us give some explanation about this point. Consider the classical KdV hierarchy with the second Poisson structure: {u(y1 ), u(y2 )} = 2(u(y1 ) + u(y2 ))δ (y1 − y2 ) + δ (y1 − y2 ). The integrability of the KdV equation is due to existence of the auxiliary linear problem: (∂ y2 + u(y))ψ(y, λ) = λ2 ψ(y, λ). We consider the periodic case u(y + 2π R) = u(y). In this case one defines the monodromy matrix M(λ) for the auxiliary linear problem in a standard way. Then the local integrals of motion in involution are found in the asymptotical expansion of T cl (λ) = TrM(α) for λ2 → +∞: log(T (λ)) 2π Rλ + cl
∞
cl C cl I2n−1 λ−(2n−1) ,
n=1
where Cncl
√ =− π
2n−1 2
n!
.
cl The integrals of motion I2n−1 are well-known functionals of u(y). Here we use for them the normalisation of [9]. So, the classical limit is
T (−i y) →
1 u(y). 1−ν
It brings the Virasoro commutation relations to the second Poisson structure of the KdV hierarchy, and ensures the finite limits cl (1 − ν)n I2n−1 → I2n−1 .
It is well known that periodic solutions of KdV are in correspondence with hyperelliptic Riemann surfaces which are a two-fold covering of the Riemann sphere of λ2 . In particular, the solution corresponding after the quantisation to the primary field φκ+1 corresponds to the Riemann surface of genus 0: μ2 = λ2 − κ 2 . From the point of view of classical theory, this is a completely trivial case which describes a constant solution of KdV. This case becomes non-trivial after the quantisation, because KdV is a theory with infinitely many degrees of freedom, and quantising the simplest classical solution one has to take into account infinitely many zero oscillations (see [21] for a relevant discussion). Still it is rather unpleasant to be able to quantise only trivial classical solutions. The consideration of the usual, low-lying descendants of φκ+1 does not change the situation seriously: they describe excitations for the same classical solution. What are really interesting solutions in the classical case? They correspond to other Riemann surfaces. The simplest one is described by the elliptic curve: μ2 = (λ2 − λ21 )(λ2 − λ22 )(λ2 − λ23 ).
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H. Boos, M. Jimbo, T. Miwa, F. Smirnov
At the quantum level, this solution corresponds to the following distribution of the Bethe roots over the real axis in the plane of λ2 . Going from λ2 = −∞ we first have no Bethe roots. Then there is a large interval where the Bethe roots are dense. Then there is a large interval without Bethe roots, wherein we find one or several zeros of T sc (λ, κ). Then starting from a certain point and up to λ2 = ∞, the Bethe roots are again dense. For a reader who is not familiar with periodic solutions of KdV, it is useful to think about this solution as a periodic analogue of one-soliton solution which we really obtain in the limit R → ∞. Everybody would agree that quantising only the trivial solutions when there are solitons around is a waste of such possibility. Our fermionic construction gives a possibility to treat this kind of asymptotic states. Moreover, we suppose that the function ω has a clear algebra-geometric meaning in the classical limit. We hope to return to all that in one of our future publications. A. General Results on the Asymptotics of ωsc (λ, μ|κ, κ , α) In this section we derive the asymptotic behaviour (9.3) of ωsc (λ, μ|κ, κ , α) when λ2 , μ2 → ∞. The main point of the argument is that in a certain domain, which we call the A-domain, the expansion (9.2) of log ρ sc (λ|κ, κ ) holds for both λ and λq −1 , and by the cancellation due to 1
1
λ ν = −(λq −1 ) ν , we have ρ sc (λ|κ, κ )ρ sc (λq −1 |κ, κ ) 1. We shall suppress the arguments κ, κ and α in ρ sc (λ|κ, κ ) and ωsc (λ, μ|κ, κ , α). We set ρ sc (λ) =
T sc (λ, κ ) , T sc (λ, κ)
asc (λ) =
Q sc (λq, κ) . Q sc (λq −1 , κ)
For ωsc (λ, μ), after simple computations we get: ωsc (λ, μ) = f left f right + f left Rdress f right (λ, μ) + δλ− δμ− −1 λ ψ(λ/μ, α). (A.1) The symbol stands for integration over the contour γ going clockwise around the zeros of Q sc (λ, κ) with the measure dm(θ ) =
dθ 2 θ 2 ρ sc (θ )(1 + asc (θ ))
.
Here again , dm(λ) and Rdress are slightly different than those used in Sect. 3 or Sect. 11, but this should not cause any confusion. The measure dm(λ) has simple poles at the zeros of Q sc (λ, κ) and T sc (λ, κ ). For simplicity of presentation, we assume κ and κ are close enough so that any zero of T sc (λ, κ ) is smaller than any zero of Q sc (λ, κ). In this section if we say f (λ) g(λ) on some half line of arg(λ2 ), it means f (λ) = g(λ) + O(|λ|−N ) for all N there. From [10] one concludes that 1
1
1 ν
− ν1
log asc (λ) = −F+ (arg(λ2 ))|λ| ν + O(|λ|− ν ), log asc (λ) = F− (arg(λ2 ))|λ| + O(|λ|
),
0 < arg(λ2 ) < π , −π < arg(λ2 ) < 0 ,
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863
where F± (arg(λ2 )) are some functions taking positive values in corresponding domains. Hence asc (λ) decays rapidly in the upper half plane and grows rapidly in the lower half plane. Following [10] we write in the integral over the upper bank in (A.1) using 1 1 1 =1− , a¯ sc (η) = sc , sc sc ¯ 1 + a (η) 1 + a (η) a (η) in order to separate the rapidly decreasing part. To formalise the story we introduce the notation f g = f ◦ g − f ∗ g, where f ◦g =
ei0 ∞
σ2
f (λ)g(λ)
dλ2 , λ2 ρ sc (λ)
f ∗g=
∞ σ2
f (λ)g(λ)d m &(λ).
Here σ 2 is an arbitrary point lying between the largest zero of T (α, κ ) and the smallest zero of Q sc (λ, κ), and the modified measure is
1 1 dλ2 + . dm &(λ) = 2 sc λ ρ (λ) 1 + a¯ sc (λei0 ) 1 + asc (λe−i0 ) Introduce the resolvent R by the equation R − Kα ◦ R = Kα , and two “dressed” kernels: Fleft = f left + f left ◦ R ,
Fright = f right + R ◦ f right ,
(A.2)
where the functions f left (λ, η), f right (η, λ) are defined in (3.10). They are singular at η2 = λ2 . According to our general prescription we understand real λ2 in them as λ2 e−i0 , and then continue analytically. The equation for the resolvent takes the form Rdress + R ∗ Rdress = R ∗ Rdress + Rdress = R , and the definition of ω can be rewritten as ωsc (λ, μ) = ω(1) (λ, μ) + ω(2) (λ, μ) , ω(1) (λ, μ) = (−Fleft ∗ Fright + Fleft ∗ Rdress ∗ Fright )(λ, μ) , ω(2) (λ, μ) = ( f left ◦ Fright )(λ, μ) + δλ− δμ− −1 λ ψ(λ/μ, α). Now we are ready to study the asymptotic behaviour. We shall consider λ2 and μ2 in the A-domain defined as follows: π(2ν − 1) < arg(λ2 ), arg(μ2 ) < π. We prove the correct asymptotic behaviour there, then assume that it is valid for all −π < arg(λ2 ), arg(μ2 ) < π . The latter assumption is not even necessary for our goals, but we do not see why it should not be true having in mind that the only infinite series of poles of ω(λ, μ) are the zeros of T sc (λ, κ) which accumulate to λ2 = −∞. The importance of the A-domain is due to the fact that in it ρ(λ)ρ(λq −1 ) 1.
(A.3)
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H. Boos, M. Jimbo, T. Miwa, F. Smirnov
Introduce the operation δλ+ f (λ) = f (λ) + ρ(λ) f (λq −1 ).
(A.4)
Using the definitions it is not hard to show that for λ2 , μ2 in the A-domain δλ+ Fleft (λ, η) 0, δμ+ Fright (η, μ) 0. These equations imply Fleft (λ, η)
Fright (η, μ)
ρ(λ)
∞
λ−
k=1 ∞
ρ(μ)
2k−1 ν
Fleft, k (η), (A.5)
− 2k−1 ν
μ
Fright, k (η).
k=1
It is easy to argue that Fleft, k (η), Fright, k (η) grow for η → ∞ as powers of η. This is enough to ensure that the “connected part” ω(1) (λ, μ) has the desired asymptotics: we just substitute the asymptotics (A.5) into the formula for ω(1) (λ, μ) and observe that all the integrals converge because of exponential in η decay of d m &(η). With the “disconnected part” ω(2) (λ, μ) the situation is far more delicate, studying it we shall understand the importance of the term δλ− δμ− −1 λ ψ(λ/μ, α) in ω(λ, μ). Let us evaluate the last term in ω(2) (λ, μ) considering λ2 , μ2 > σ 2 . Using the definition (2.10) it is easy to see that δμ− −1 λ ψ(λ/μ) = −
ei0 ∞ 0
dη2 1 1 f right (η, μ) . + δμ+ 1 1 2 2πiη 2ν 1 + (λ/η) ν 4ν 1 − (λ/μ) ν
The function Fright (η, μ) allows analytical continuation with respect to η, so, we shall use it for all η2 ∈ R+ . Substitute f right = Fright − K α ◦ Fright and compute the integral
∞ 0
K α (θ, η)
dθ 2 1 . = −ψ(λ/η, α) − 1 2 θ 2ν(1 + (λ/θ ) ) 2ν(1 − (λ/η) ν ) 1 ν
We get after some straightforward computations ω(2) (λ, μ) = ω(3) (λ, μ) + ω(4) (λ, μ) , where ω(3) (λ, μ) = −
σ2 0
ω(4) (λ, μ) = −V P
δλ−
1 dη2 (η, μ) , · F right 1 2πiη2 2ν 1 + (λ/η) ν
∞
σ2
δλ− δη+
1
· Fright (η, μ) 1
2ν 1 − (λ/η) ν
where the principal value refers to the pole at η2 = μ2 .
dη2 , 2πiη2 ρ(η)
(A.6)
(A.7)
Grassmann Structure in XXZ Model
865
For ω(3) (λ, μ) the asymptotics of the kind (9.3) follows immediately from (A.5) and δλ+ δλ−
1
1
2ν 1 + (λ/η) ν
0.
Consider ω(4) (λ, μ). We check the equations δλ+ ω(4) (λ, μ) 0,
δμ+ ω(4) (λ, μ) 0.
(A.8)
The first of them follows immediately from two facts. First, δλ+ δλ− δη+
1 0. 1 2ν 1 − (λ/η) ν
Second, writing explicitly δλ− δη+
1 ρ(η) − ρ(λ) 1 − ρ(λ)ρ(η) = + , 1 1 1 2ν 1 − (λ/η) ν 2ν 1 − (λ/η) ν 2ν 1 + (λ/η) ν
(A.9)
and recalling the asymptotic expansion for ρ(λ), ρ(η), we see that asymptotically for λ2 → +∞, η2 → +∞ the singularities in (A.9) disappear. Altogether we have for the asymptotics in both arguments, δλ− δη+
1
1
2ν 1 − (λ/η) ν
ρ(λ)
∞
λ−
2m−1 ν
n
η− ν Cm,n .
(A.10)
m,n=1
To prove the second equation in (A.8) it is not sufficient to use (A.5) because Fright (η, μ) has simple poles at η2 = μ2 and η2 = μ2 q 2 which contribute to the analytic continuation μ → μq −1 . However, it is easy to see that the corresponding contributions cancel. Using the first of Eqs. (A.8) we get ω
(4)
(λ, μ)
ρ(λ)
∞
λ−
2m−1 ν
(4) ωm (μ) ,
m=1 (4)
where due to (A.10) the functions ωm (μ) are given by convergent integrals. These (4) functions satisfy δμ+ ωm (μ) 0, and do not grow for μ2 → +∞. Hence ω(4) (λ, μ) has the asymptotics of the kind (9.3). Acknowledgements. HB is grateful to the Volkswagen Foundation for financial support. Research of MJ is supported by the Grant-in-Aid for Scientific Research B-20340027. Research of TM is supported by the Grant-in-Aid for Scientific Research B-17340038. Research of FS is supported by RFBR-CNRS grant 09-02-93106 and by EU-grant MEXT-CT-2006-042695 during his visit to DESY, Hamburg. FS thanks Masaki Kashiwara for inviting him to RIMS where the most important part of this research was carried out. This visit was supported by the Grant-in-Aid for Scientific Research B-18340007. HB would like to thank F. Göhmann, A. Klümper and K. Nirov for the stimulating discussions. FS would like to thank S. Lukyanov for many valuable discussions.
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