Communications in Mathematical Physics - Volume 302

Commun. Math. Phys. 302, 1–51 (2011) Digital Object Identifier (DOI) 10.1007/s00220-010-1175-8 Communications in Mathe...

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Commun. Math. Phys. 302, 1–51 (2011) Digital Object Identifier (DOI) 10.1007/s00220-010-1175-8

Communications in

Mathematical Physics

Transition to Longitudinal Instability of Detonation Waves is Generically Associated with Hopf Bifurcation to Time-Periodic Galloping Solutions Benjamin Texier1, , Kevin Zumbrun2, 1 Université Paris Diderot (Paris 7), Institut de Mathématiques de Jussieu, UMR CNRS 7586,

75205 Paris Cedex 13, France. E-mail: [email protected]

2 Indiana University, Bloomington, IN 47405, USA. E-mail: [email protected]

Received: 17 December 2008 / Revised: 25 June 2010 / Accepted: 4 October 2010 Published online: 9 January 2011 – © Springer-Verlag 2011

Abstract: We show that transition to longitudinal instability of strong detonation solutions of reactive compressible Navier–Stokes equations is generically associated with Hopf bifurcation to nearby time-periodic “galloping”, or “pulsating”, solutions, in agreement with physical and numerical observation. In the process, we determine readily numerically verifiable stability and bifurcation conditions in terms of an associated Evans function, and obtain the first complete nonlinear stability result for strong detonations of the reacting Navier–Stokes equations, in the limit as amplitude (hence also heat release) goes to zero. The analysis is by pointwise semigroup techniques introduced by the authors and collaborators in previous works. Contents 1.

Introduction . . . . . . . . . . . . . . . . . . . . 1.1 The reacting Navier-Stokes equations . . . . 1.2 Assumptions . . . . . . . . . . . . . . . . . . 1.3 Coordinatizations . . . . . . . . . . . . . . . 1.4 Strong detonations . . . . . . . . . . . . . . 1.5 Structure of the equations and the profiles . . 1.6 The Evans function . . . . . . . . . . . . . . 1.7 Results . . . . . . . . . . . . . . . . . . . . . 1.7.1 Stability. . . . . . . . . . . . . . . . . . 1.7.2 Transition from stability to instability. . . 1.7.3 Nonlinear instability. . . . . . . . . . . . 1.8 Verification of stability/bifurcation conditions 1.9 Discussion and open problems . . . . . . . .

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Research of B.T. was partially supported under NSF grant number DMS-0505780.

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2 3 4 4 5 7 8 9 9 10 11 11 12

Research of K.Z. was partially supported under NSF grants no. DMS-0300487 and DMS-0801745.

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B. Texier, K. Zumbrun

Strong Detonations . . . . . . . . . . . . . . . . . . . . Resolvent Kernel and Green Function Bounds . . . . . 3.1 Laplace transform . . . . . . . . . . . . . . . . . . 3.1.1 The limiting, constant-coefficient equations. . . 3.1.2 Low-frequency behaviour of the normal modes. 3.1.3 Description of the essential spectrum. . . . . . 3.1.4 Gap Lemma and dual basis. . . . . . . . . . . 3.1.5 Duality relation and forward basis. . . . . . . . 3.1.6 The resolvent kernel. . . . . . . . . . . . . . . 3.1.7 The Evans function. . . . . . . . . . . . . . . 3.2 Inverse Laplace transform . . . . . . . . . . . . . . 3.2.1 Pointwise Green function bounds. . . . . . . . 3.2.2 Convolution bounds. . . . . . . . . . . . . . . Stability: Proof of Theorem 1.14 . . . . . . . . . . . . 4.1 Linearized stability criterion . . . . . . . . . . . . 4.2 Auxiliary energy estimate . . . . . . . . . . . . . . 4.3 Nonlinear stability . . . . . . . . . . . . . . . . . . Bifurcation: Proof of Theorem 1.18 . . . . . . . . . . . 5.1 The perturbation equations . . . . . . . . . . . . . 5.2 Coordinatization . . . . . . . . . . . . . . . . . . . 5.3 Poincaré return map . . . . . . . . . . . . . . . . . 5.4 Lyapunov-Schmidt reduction . . . . . . . . . . . . 5.4.1 Pointwise cancellation estimate. . . . . . . . . 5.4.2 Reduction. . . . . . . . . . . . . . . . . . . . 5.4.3 Bifurcation. . . . . . . . . . . . . . . . . . . . Nonlinear Instability: Proof of Theorem 1.19 . . . . . .

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1. Introduction Motivated by physical and numerical observations of time-oscillatory “galloping” or “pulsating” instabilities of detonation waves [MT,BMR,FW,MT,AlT,AT,F1,F2,KS], we study stability and Hopf bifurcation of viscous detonation waves, or traveling-wave solutions of the reactive compressible Navier–Stokes equations. This extends a larger program begun in [Zl,LyZ1,LyZ2,JLW,LRTZ] toward the dynamical study of viscous combustion waves using Evans function/inverse Laplace transform techniques introduced in the context of viscous shock waves [GZ,ZH,ZS,Zl,MaZ3], continuing the line of investigation initiated in [TZ1,TZ2,SS,TZ3] on bifurcation/transition to instability. It has long been observed that transition to instability of detonation waves occurs in certain predictable ways, with the archetypal behavior in the case of longitudinal, or onedimensional instability being transition from a steady planar progressing wave U (x, t) = U¯ (x1 −st) to a galloping, or time-periodic planar progressing wave U˜ (x1 −st, t), where U˜ is periodic in the second coordinate, and in the case of transverse, or multi-dimensional instability, transition to more complicated “spinning” or “cellular behavior”; see [KS,TZ1,TZ2], and references therein. The purpose of this paper is, restricting to the one-dimensional case, to establish this principle rigorously, arguing from first principles from the physical equations that transition to longitudinal instability of detonation waves is generically associated with Hopf bifurcation to time-periodic galloping solutions, not only at the spectral but also at the full nonlinear level. In the process, we establish the first full nonlinear stability results for

Transition to Longitudinal Instability of Detonation Waves

3

strong detonations of the reacting Navier–Stokes equations, extending the sole previous result obtained by Tan–Tesei [TT] for the special class of initial perturbations with zero integral. 1.1. The reacting Navier-Stokes equations. The single-species reactive compressible Navier–Stokes equations, in Lagrangian coordinates, appear as [Ch] ⎧ ∂t τ − ∂x u = 0, ⎪ ⎪ ⎨ −1 ∂ u), ∂t u + ∂x p = ∂x (ντ x (1.1) −2 ∂ z + κτ −1 ∂ T + ντ −1 u∂ u , qdτ ∂ E + ∂ ( pu) = ∂ ⎪ t x x x x x ⎪ ⎩ −2 ∂t z + kφ(T )z = ∂x (dτ ∂x z), where τ > 0 denotes specific volume, u velocity, E > 0 total specific energy, and 0 ≤ z ≤ 1 mass fraction of the reactant. The variable U := (τ, u, E, z) ∈ R4 depend on time t ∈ R+ , position x ∈ R, and parameters ν, κ, d, k, q, where ν > 0 is a viscosity coefficient, κ > 0 and d > 0 are respectively coefficients of heat conduction and species diffusion, k > 0 represents the rate of the reaction, and q is the heat release parameter, with q > 0 corresponding to an exothermic reaction and q < 0 to an endothermic reaction. In (1.1), T = T (τ, e, z) > 0 represents temperature, p = p(τ, e, z) pressure, where the internal energy e > 0 is defined through the relation 1 E = e + u 2 + qz. 2 In (1.1), we assume a simple one-step, one-reactant, one-product reaction kφ(T )

A −→ B,

z := [A ],

[A ] + [B ] = 1,

where φ is an ignition function. More realistic reaction models are described in [GS2]. In the variable U, after the shift x → x − st,

s ∈ R,

the system (1.1) takes the form of a system of differential equations ∂t U + ∂x (F(U )) = ∂x (B(U )∂x U ) + G(U ), where

and

⎛

⎛

⎞ −u ⎜ p ⎟ F := ⎝ − s(ε)U, pu ⎠ 0

(1.2)

⎛

⎞ 0 0 ⎜ ⎟ G := ⎝ ⎠, 0 −kφ(T )z

⎞ 0 0 0 0 ⎜ ⎟ 0 ντ −1 0 0 ⎟ B := ⎜ ⎝ κτ −1 ∂τ T −κuτ −1 ∂e T + ντ −1 u κτ −1 ∂e T κτ −1 (∂z T − q∂e T ) + qdτ −2 ⎠. 0 0 0 dτ −2

4


The characteristic speeds of the first-order part of (1.1), i.e., the eigenvalues of ∂U F(U ), are {−s − σ, −s, −s + σ , −s }, reactive eigenvalue fluid eigenvalues

(1.3)

where σ, the sound speed of the gas, is 1

1

σ := ( p∂e p − ∂τ p) 2 = τ −1 ( ( + 1)e) 2 . 1.2. Assumptions. We make the following assumptions: Assumption 1.1. We assume a reaction-independent ideal gas equation of state, p = τ −1 e,

T = c−1 e,

(1.4)

where c > 0 is the specific heat constant and is the Gruneisen constant. Assumption 1.2. The ignition function φ is smooth; it vanishes identically for T ≤ Ti , and is strictly positive for T > Ti . Remark 1.3. A typical ignition function is given by the modified Arrhenius law E

φ(T ) = Ce T −Ti ,

(1.5)

where E is activation energy. Remark 1.4. The specific choice (1.4) is made for concreteness/clarity of exposition. Our results remain valid for any reaction–independent equation of state with pτ < 0, pe > 0, and Te > 0.1 With further effort, reaction-dependence should be treatable as well. 1.3. Coordinatizations. We let w := (u, E, z) ∈ R3 ,

v := (τ, u, E) ∈ R3 .

Then we have the coordinatizations U = (v, z) = (τ, w). In particular, Assumption 1.1 implies that in the (τ, w) coordinatization, B takes the block-diagonal form 0 0 , B= 0 b where b is full rank for all values of the parameters and U ; the system (1.2) in (τ, w) coordinates is ∂t τ − s∂x τ − J ∂x w = 0, ∂t w + ∂x f (τ, w) = ∂x (b(τ, w)∂x w) + g(w), 1 An obvious exception is Lemma 1.6, which depends on specific structure.


5

with the notation

⎛

⎞ ⎛ ⎞ p 0 ⎠. 0 f := ⎝ pu ⎠ − sw, g := ⎝ 0 −kφ(T )z

J := 1 0 0 ,

(1.6)

In the (v, z) coordinatization, the system (1.2) takes the form ∂t v + ∂x f (v, z) = ∂x (b1 (v)∂x v + b2 (v)∂x z) ∂t z − s∂x z + kφ(T )z = ∂x (dτ −2 ∂x z), where the flux is f = (−u − sτ, p − su, pu − s E), and, under Assumption 1.1, the diffusion matrices are ⎛ ⎛ ⎞ ⎞ 0 0 0 0 ⎠, ⎠. 0 ντ −1 0 b2 = ⎝ b1 = ⎝ 0 −2 −1 −1 −1 −1 −1 q(dτ − κτ ) 0 τ (ν − κc )u κτ c Note that, in the (v, z) coordinatization, the first component is a conservative variable, in the sense that ∂t v is a perfect derivative, hence (v(x, t) − v(x, 0)) d x ≡ 0, (1.7) R

for v(t) − v(0) ∈ W 2,1 (R). 1.4. Strong detonations. We prove in this article stability and bifurcation results for viscous strong detonations of (1.1), defined as follows: Definition 1.5. A one-parameter, right-going family of viscous strong detonations is a family {U¯ ε }ε∈R of smooth stationary solutions of (1.2), associated with speeds s(ε), s(ε) > 0, model parameters (ν, κ, d, k, q)(ε) and ignition function φ ε , with U¯ ε , φ ε , (s, ν, κ, d, k, q)(ε) depending smoothly on ε in L ∞ × L ∞ × R6 ,satisfying U¯ ε (x, t) = U¯ ε (x),

lim U¯ ε (x) = U±ε ,

x→±∞

(1.8)

connecting a burned state on the left to an unburned state on the right, ε z− ≡ 0, z +ε ≡ 1,

(1.9)

with a temperature on the burned side above ignition temperature T−ε > Ti ,

(1.10)

and satisfying the Lax characteristic conditions σ− := σ (U−ε ) > s > σ+ := σ (U+ε ), uniformly in ε.

(1.11)

6


Fig. 1. Characteristic speeds for strong detonations

Consider a standing wave (1.8), U = (τ, u, E, z), solution of (1.2), with endstates U± = (τ± , u ± , E ± , z ± ). It satisfies the linear constraint −s(τ − τ− ) = u − u − , the system of ordinary differential equations (Fig. 1) ⎧ ντ −1 u = p − su − ( p − su)− , ⎪ ⎪ ⎪ −1 −1 −1 ⎪ ⎨ κτ c E + τ (ν − κc−1 )uu = pu − s E − ( pu − s E)− + (κτ −1 c−1 − dτ −2 )qy, ⎪ = y, ⎪ z ⎪ ⎪ ⎩ dτ −2 y = −sy + kφ(T )z, and the Rankine-Hugoniot relations ⎧ −s(τ+ − τ− ) ⎪ ⎪ ⎪ ⎨ ( p − su)+ ( pu − s E)+ ⎪ ⎪ y± ⎪ ⎩ φ(T± )z ±

= u+ − u−, = ( p − su)− , = ( pu − s E)− , = 0, = 0,

(1.12)

(1.13)

expressing the fact that (u ± , E ± , 0, z ± ) are rest points of (1.12). From (1.11) and (1.13), we note that the right endstate of a strong detonation satisfies φ(T+ε ) = 0,

(1.14)

which, by Assumption 1.2, implies also φ (T+ε ) = 0.

(1.15)

Lemma 1.6. Under Assumptions 1.1, 1.2, if q > 0 and s is large enough with respect to q, then for any z + ∈ (0, 1], there exists an open subset O− in R3 , such that any left endstate U− = (v− , 0) with v− ∈ O− satisfies (1.10) and (1.11), and is associated with a right endstate U+ = (v+ , z + ) satisfying T+ < Ti , (1.11) and (1.13).


7

The existence of strong detonations was proved by Gasser and Szmolyan [GS1] for small dissipation coefficients ν, κ and d. We restrict throughout the article to strong detonations with left endstates as in the above lemma. Remark 1.7. In the small-heat-release limit q → 0, the equations in (y, z) (in system (1.12)) are decoupled from the fluid equations; in particular, strong detonations converge to ordinary nonreacting gas-dynamical shocks of standard Lax type, the existence of which has been established by Gilbarg [G]. A consequence of Lemma 1.6 is that strong detonations converge exponentially to their endstates, a key fact of the subsequent stability and bifurcation analysis. Corollary 1.8. Under Assumptions 1.1, 1.2, let {U¯ ε }ε be a family of viscous strong detonations. There exist C, η0 > 0, such that, for k ≥ 0 and j ∈ {0, 1}, j |∂ε ∂xk (U¯ ε − U−ε )(x)| ≤ Ce−η0 |x| , j |∂ε ∂xk (U¯ ε − U+ε )(x)| ≤ Ce−η0 |x| ,

x < 0, x > 0.

(1.16)

In particular, |(U¯ ε ) (x)| ≤ Ce−η0 |x| , for all x. Remark 1.9. In the ZND limit, strong detonations are transverse orbits of (1.12), a result proved in Sect. 3.6 of [LyZ2], following [GS1]. Lemma 1.6 and Corollary 1.8 are proved in Sect. 2. 1.5. Structure of the equations and the profiles. System (1.2), seen as a system in τ, w, satisfies (A1) the convection terms in the equation in τ are linear in (τ, w); (A2) the diffusion matrix b is positive definite. For strong detonation waves, the convection terms in (1.1) satisfy (H1) The convection coefficient s(ε) in the evolution equation in τ is nonzero, uniformly in ε. (H2) The spectrum of ∂U F, given in (1.3), is real, simple, and nonzero, uniformly in ε. System (1.2) satisfies the Kawashima dissipativity condition (H3) For all ε, for all ξ ∈ R, θξ2 , σ iξ ∂U F(U±ε ) − ξ 2 B(U±ε ) + ∂U G(U±ε ) ≤ − 1 + ξ2 at the endstates U±ε of a family of strong detonations. In (H3), σ denotes spectrum of a matrix, and θ > 0 is independent of ξ and ε. To verify (H3), it suffices, by a classical result of [ShK], to check that (1.2) has a symmetrizable hyperbolic-parabolic structure, and that the genuine coupling condition holds. These conditions are coordinates-independent, and easily checked in (τ, u, e) coordinates. Finally, the assumption (H4) Considered as connecting orbits of (1.12), U¯ ε lie in a smooth one-dimensional manifold of solutions of (1.12), obtained as a transversal intersection of the unstable manifold at U−ε and the stable manifold at U+ε ,

8


holds in the ZND limit, as stated in Remark 1.9. Under (H4), in a vicinity of U¯ ε , the set of stationary solutions of (1.2) with limits U±ε at ±∞ is a smooth one-dimensional manifold, given by {U¯ ε (· − c), c ∈ R}, and the associated speed ε → s(ε) is smooth. Conditions (A1)–(A2), (H0)–(H4) are the assumptions of [TZ3] (where G ≡ 0), themselves a strengthened version of the assumptions of [MaZ3]. 1.6. The Evans function. A central object in the study of stability of traveling waves is the Evans function D(ε, ·) (precisely defined in Sect. 3.1.7), a Wronskian of solutions of the eigenvalue equation (L(ε) − λ)U = 0 decaying at plus or minus spatial infinity [AGJ],2 where the linearized operator L is defined as L(ε) := −∂x (A ·) + ∂x (B(U¯ ε )∂x ·) + ∂U G(U¯ ε ),

(1.17)

A := −∂U F(U¯ ε ) + (∂U B(U¯ ε ) ·)(U¯ ε ) .

(1.18)

with the notation

Recall the important result of [LyZ2]: Proposition 1.10 ([LyZ2], Theorem 4). Under Assumptions 1.1 and 1.2, let {U¯ ε }ε be a one-parameter family of viscous strong detonation waves satisfying (H4). For all ε, the associated Evans function has a zero of multiplicity one at λ = 0: D(ε, 0) = 0,

and

D (ε, 0) = 0.

Proof. By translational invariance, D(ε, 0) = 0, for all ε. Generalizing similar results known for shock waves [GZ,ZS], there was established in [Zl,LyZ1,LyZ2] the fundamental relation D (ε, 0) = γ δ.

(1.19)

In (1.19), γ is a coefficient given as a Wronskian of solutions of the linearized travelingwave ODE about U¯ ; transversality corresponds to γ = 0. In (1.19), δ is the Lopatinski determinant δ := det r1− r2− r4− ( τ+ − τ− u + − u − E + − E − )tr , ε (where r − j denote the eigenvectors of ∂U F(U− ) associated with outgoing eigenvalues, F as in (1.2), and tr denotes transverse matrix3 ) determining hyperbolic stability of the Chapman–Jouget (square wave) approximation modeling the detonation as a shock discontinuity. Hyperbolic stability corresponds to δ = 0. See [Zl,LyZ1,LyZ2,JLW] for further discussion. By (H4), γ = 0, while δ = 0 by direct calculation comparing to the nonreactive (shock-wave) case.

Remark 1.11. The vectors r1− , r2− and r4− correspond to outgoing modes to the left of x = 0, see Sect. 3.1.2 and Fig. 4. (The fluid modes r − j , 1 ≤ j ≤ 3, are ordered as usual by increasing characteristic speeds: −s − σ− < −s < 0 < −s + σ− , so that r3− is incoming.) 2 For applications of the Evans function to stability of viscous shock and detonation waves, see, e.g., [AGJ,GZ,ZS,Zl,LyZ1,LyZ2,LRTZ]. 3 This notation will be used throughout the article.


9

1.7. Results. Let X and Y be two Banach spaces, and consider a traveling wave U¯ solution of a general evolution equation. Definition 1.12. A traveling wave U¯ is said to be X → Y linearly orbitally stable if, for any solution U˜ of the linearized equations about U¯ with initial data in X, there exists a phase shift δ, such that |U˜ (·, t) − δ(t)U¯ (·)|Y is bounded for 0 ≤ t ≤ ∞. It is said to be X → Y linearly asymptotically orbitally stable if it is X → Y linearly orbitally stable and if moreover |U˜ (·, t) − δ(t)U¯ (·)|Y → 0 as t → ∞. Definition 1.13. A traveling wave U¯ is said to be X → Y nonlinearly orbitally stable if, for each δ > 0, for any solution U˜ of the nonlinear equations with |U˜ (·, 0) − U¯ | X sufficiently small, there exists a phase shift δ, such that |U˜ (·, t) − U¯ (· − δ(t), t)|Y ≤ δ for 0 ≤ t ≤ ∞. It is said to be X → Y nonlinearly asymptotically orbitally stable if it is X → Y nonlinearly orbitally stable and if moreover |U˜ (·, t) − U¯ (· − δ(t), t)|Y → 0 as t → ∞. 1.7.1. Stability. Our first result, generalizing that of [LRTZ] in the artificial viscosity case, is a characterization of linearized stability and a sufficient condition for nonlinear stability, in terms of an Evans function condition. Theorem 1.14. Under Assumptions 1.1, 1.2, let {U¯ ε }ε be a one-parameter family of viscous strong detonation waves. For all ε, U¯ ε is L 1 ∩ L p → L p linearly orbitally stable if and only if, for all ε, the only zero of D(ε, ·) in λ ≥ 0 is a simple zero at the origin.

(1.20)

If (1.20) holds, U¯ ε is L 1 ∩ H 3 → L 1 ∩ H 3 linearly and nonlinearly orbitally stable, and L 1 ∩ H 3 → L p ∩ H 3 asymptotically orbitally stable, for p > 1, with |U˜ ε (·, t) − U¯ ε (· − δ(t))| L p ≤ C|U˜ 0ε − U¯ ε | L 1 ∩H 3 (1 + t)

− 21 (1− 1p )

,

(1.21)

where U˜ ε is the solution of (1.2) issued from U˜ 0ε , for some δ(·) satisfying |δ(t)| ≤ C|U˜ 0ε − U¯ ε | L 1 ∩H 3 , ˙ |δ(t)| ≤ C|U˜ 0ε − U¯ ε | L 1 ∩H 3 (1 + t)− 2 . 1

Remark 1.15. It is shown in [LyZ2] that in the small heat-release limit q → 0, strong detonations are Evans stable if and only if the limiting gas-dynamical profile (see Remark 1.7) is Evans stable: in particular, for shock (or equivalently detonation) amplitude sufficiently small [HuZ2]. Corollary 1.16. Under Assumptions 1.1, 1.2, strong detonation profiles are linearly and nonlinearly orbitally stable (in the strong sense of (1.21) in the limit as amplitude |U+ − U− | (hence also heat release q) goes to zero, with U− (or U+ ) held fixed. Corollary 1.16 is notable as the first complete nonlinear stability result for strong detonations of the reacting Navier–Stokes equations. The only previous result on this topic, a partial stability result applying to zero mass (i.e., total integral) perturbations, was obtained by Tan and Tesei under similar, but more restrictive assumptions (in particular, for nonphysical Heaviside-type ignition function) in 1997.

10


1.7.2. Transition from stability to instability. Theorem 1.17. Under Assumptions 1.1, 1.2, let {U¯ ε }ε be a one-parameter family of viscous strong detonation waves satisfying (H4). Assume that the family of Eqs. (1.2) and profiles U¯ ε undergoes transition to instability at ε = 0 in the sense that U¯ ε is linearly stable for ε < 0 and linearly unstable for ε > 0. Then, one or more pair of nonzero complex conjugate eigenvalues of L(ε) move from the stable (negative real part) to the neutral or unstable (nonnegative real part) half-plane as ε passes from negative to positive through ε = 0, while λ = 0 remains a simple root of D(ε, ·) for all ε. That is, transition to instability is associated with a Hopf-type bifurcation in the spectral configuration of the linearized operator about the wave. Proof of Theorem 1.17. By Theorem 1.14, transition from stability to instability must occur through the passage of a root of the Evans function from the stable half-plane to the neutral or unstable half-plane. However, Proposition 1.10 implies that D has a zero of multiplicity one at the origin, for all ε, and so no root can pass through the origin. It follows that transition to instability, if it occurs, must occur through the passage of one or more nonzero complex conjugate pairs λ = γ ± iτ, τ = 0, from the stable half-plane (γ < 0 for ε < 0) to the neutral or unstable half-plane (γ ≥ 0 for ε ≥ 0). Our third result and the main object of this paper is to establish, under appropriate nondegeneracy conditions, that the spectral Hopf bifurcation configuration described in Theorem 1.17 is realized at the nonlinear level as a genuine bifurcation to time-periodic solutions. Given k ∈ N and a weight function ω > 0, define the Sobolev space and associated norm 1

Hωk := { f ∈ S (R), ω 2 f ∈ H k (R)},

1

f Hωk := ω 2 f H k .

(1.22)

Let ω ∈ C 2 be a growing weight function such that, for some θ0 > 0, C > 0, for all x, y, ⎧ 1 2 ⎨ 1 ≤ ω(x) ≤ eθ0 (1+|x| ) 2 , (1.23) ⎩ |ω (x)| + |ω (x)| ≤ Cω(x), ω(x) ≤ Cω(x − y)ω(y). Theorem 1.18. Under Assumptions 1.1, 1.2, let {U¯ ε }ε be a family of viscous strong detonation waves satisfying (H4). Assume that the family of Eqs. (1.2) and profiles U¯ ε undergo transition from linear stability to linear instability at ε = 0. Moreover, assume that this transition is associated with passage of a single complex conjugate pair of eigenvalues of L(ε), λ± (ε) = γ (ε) + iτ (ε) through the imaginary axis, satisfying γ (0) = 0, τ (0) = 0, dγ /dε(0) = 0.

(1.24)

Then, given a growing weight ω satisfying (1.23) with θ0 sufficiently small, for r ≥ 0 sufficiently small and C > 0 sufficiently large, there are C 1 functions r → ε(r ), r →


11

T (r ), with ε(0) = 0, T (0) = 2π/τ (0), and a C 1 family of time-periodic solutions U˜ r (x, t) of (1.2) with ε = ε(r ), of period T (r ), with C −1 r ≤ U˜ r − U¯ ε Hω2 ≤ Cr.

(1.25)

Up to translation in x, t, these are the only time-periodic solutions nearby in · Hω2 with period T ∈ [T0 , T1 ] for any fixed 0 < T0 < T1 < +∞. That is, transition to linear instability of viscous strong detonation waves is “generically” (in the sense of (1.24)) associated with Hopf bifurcation to time-periodic galloping solutions, as asserted in the title of this paper. 1

The choices ω ≡ 1 and ω = eθ0 (1+|x| ) 2 are allowed in (1.23), as well as ω = (1 + |x|2 ) p , for any real p > 0. In Theorem 1.18, we need, in particular, θ0 < η0 , where η0 is as in Corollary 1.8, so that the spatial localization given by (1.25) is less precise than the spatial localization of the background profile U¯ ε . The smallness condition on θ0 is described in Remark 5.9. 2

1.7.3. Nonlinear instability. We complete our discussion with the following straightforward result verifying that the exchange of linear stability described in Theorem 1.18, as expected, corresponds to an exchange of nonlinear stability as well, the new assertion being nonlinear instability for ε > 0. Theorem 1.19. Under the assumptions of Theorem 1.18, the viscous strong detonation waves U¯ ε undergo a transition at ε = 0 from nonlinear orbital stability to instability; that is, U¯ ε is nonlinearly orbitally stable for ε < 0 and unstable for ε > 0. 1.8. Verification of stability/bifurcation conditions. The above theory not only describes the nature of possible bifurcation/exchange of stability but characterizes its occurrence in terms of corresponding spectral conditions involving zeros of the Evans function of the linearized operator about the wave. These may readily and efficiently be computed numerically [HuZ1,BHRZ], answering in a practical sense the question of whether or not such transitions which actually occur as parameters are varied in any given compact region. Much more can be said in certain interesting limiting cases. It is shown in [LyZ2] that in the small heat-release limit q → 0, strong detonations are Evans stable if and only if the limiting gas-dynamical profile (see Remark 1.7) is Evans stable. As noted in Corollary 1.16, this implies in particular that strong detonations are stable in the smallamplitude limit as the distance between endstates goes to zero with one endstate held fixed (forcing q → 0 as well). For an ideal gas law (1.4), stability of large-amplitude detonations in the small heat-release limit is strongly suggested by the recent asymptotic and numerical studies of [HLZ,HLyZ] indicating that viscous ideal gas shocks are stable for arbitrary amplitudes. A more interesting limit from the viewpoint of stability transitions is the smallviscosity, or ZND limit as ν, κ, d go to zero. Recall, [GS1,GS2], that in this limit, the viscous detonation profile approaches an invscid profile composed of a smooth reaction zone preceded by a shock discontinuity. In [Z4], it has recently been shown that strong detonations are stable in the ZND limit if and only if both the limiting ZND profile and the viscous shock profile associated with its component shock discontinuity satisfy spectral Evans stability conditions like those developed here for viscous detonations.

12


Since viscous shocks for ideal gas law (1.4) as just mentioned are uniformly stable, this means that Evans stability of rNS profiles reduces in the small viscosity limit to Evans stability of the limiting ZND profile. For ZND profiles, there is a wealth of numerical [Er1,Er2,FW,S2,KS,BMR,BM,KS] and asymptotic [F1,FD,B,BN,S1,Er4] literature indicating that stability transitions do, and do often, occur. Indeed, a classic benchmark problem of Fickett and Woods [FW] tests numerical code for parameters = 1.2, E = 50, q = 50 for which transition to stability is known to occur as overdrive is varied as a bifurcation parameter [BMR]. In multidimensions, a theorem of Erpenbeck [Er3] gives a rigorous proof of instability for certain detonation types, occurring through high-frequency transverse modes (the only such proof to our knowledge). In short, the evidence is overwhelming that spectral bifurcation occurs in the ZND context, whence (by the results of [Z4]) also for (1.1) for ν, κ, d sufficiently small. Together with these observations, the results of this paper answer definitively and positively the fundamental question whether the reacting Navier–Stokes equations are adequate to capture the bifurcation phenomena observed for more than half a century in physical experiments [FD,Er1]. A very interesting problem would be to establish in one dimension a rigorous spectral instability result for ZND analogous to that of Erpenbeck for multi-d, thus completing an entirely mathematical proof; in this regard, we mention that the analyses of [BN,S1] appear to come very close.

1.9. Discussion and open problems. This analysis in large part concludes the one-dimensional program set out in [TZ2]. However, a very interesting remaining open problem is to determine linearized and nonlinear stability of the bifurcating time-periodic solutions, in the spirit of Sect. 4.3. For a treatment in the shock wave case with semilinear viscosity, see [BeSZ]. Likewise, it would be very interesting to carry out a numerical investigation of the spectrum of the linearized operator about detonation waves with varying physical parameters, as done in [LS,KS] in the inviscid ZND setting, but using the viscous methods of [Br1,Br2,BrZ,BDG,HuZ1] to treat the full reacting Navier–Stokes equations, in order to determine the physical bifurcation boundaries. Other interesting open problems are the extension to multi-dimensional (spinning or cellular) bifurcations, as carried out for artificial viscosity systems in [TZ2], and to the case of weak detonations (analogous to the case of undercompressive viscous shocks; see [HZ,RZ,LRTZ]). The strong detonation structure considerably simplifies both stability and bifurcation arguments over what was done in [LRTZ]. We remark that, at the expense of further complication, nonlinear stability of general (time-independent) combustion waves, including also weak detonations and strong or weak deflagrations, may be treated by a combination of the pointwise arguments of [LRTZ] and [RZ]. We remark finally that the restriction to a scalar reaction variable is for simplicity only. Indeed, the results of this article (as well as the results of the article by Lyng and Zumbrun [LyZ2] from which it draws) are independent of the dimension of the reactive equation, so long as the reaction satisfies an assumption of exponential decay of space-independent states (with temperature at −∞ above the ignition temperature). Plan of the paper. Lemma 1.6 and Corollary 1.8 are proved in Sect. 2. We give a detailed description of the low-frequency behavior of the resolvent kernel for the linearized equations in Sect. 3, following [MaZ3]. In Sect. 4, we prove Theorem 1.14, while Sect. 5 is devoted to the proof of Theorem 1.18. Finally, in Sect. 6, we prove Theorem 1.19.


13

2. Strong Detonations Proof of Lemma 1.6. Let U− be a given left endstate, with z − = 0, satisfying (1.10) and (1.11). We look for a right endstate U+ , with z + ∈ (0, 1], that satisfies (1.13), (1.11), and T+ < Ti . We note that (1.13)(i) determines u + and that T+ < Ti entails (1.13)(v). The Rankine-Hugoniot relations in the (τ+ , p+ ) plane are

p = −s 2 τ + c1 p = (c0 − sτ (1 + −1 ))−1 (c2 + sqz + + 21 s 3 τ 2 − s 2 c0 τ )

(R), (H),

where (R) is the Rayleigh line, corresponding to (1.13)(ii), (H) the Hugoniot curve, corresponding to (1.13)(iii), and where c0 := u − + sτ− ,

c1 := p− + s 2 τ− ,

1 c2 := ( p− u − − s E − ) + c02 s 2

depend on parameters U− and s. The temperature and Lax constraints for both endstates are

τ+ p+ < c Ti < τ− p− τ+−1 p+ < ( + 1)−1 s 2 < τ−−1 p−

(T)± , (L)± .

We restrict to left endstates satisfying in the large s regime p− = 2s 2 −1 τ− + p˜ − ,

τ− = O(1),

u − = s u˜ − ,

(2.1)

with u˜ − = O(1) and p˜ − = O(1). Under (2.1), conditions (T)− and (L)− are satisfied as soon as s is large enough. The Hugoniot curve takes the form −1 pH = u˜ − +τ− −(1+ −1 )τ

1 3 s (τ −(1+2 −1 )τ− )(τ −(1−2 −1 )τ− −2u˜ − )+sqz + . 2

Assume that u˜ − is such that τ− < (1 − 2 −1 )τ− + 2u˜ − < (1 + 2 −1 )τ− . 1 + −1

(2.2)

For any such u˜ − , any given τ− and any q > 0, if s is large enough then, for any z + ∈ (0, 1], the Hugoniot curve has two zeros τ < τ , with asymptotic expansions τ = (1 − 2 −1 )τ− + 2u˜ − + O(s −2 ), p˜ − u˜ − + qz + τ = (1 + 2 −1 )τ− − s −2 −1 + O(s −3 ). 2 τ− − u˜ −

(2.3) (2.4)

If s is large, by (2.2), τ0 < τ < τ , where τ0 := c0 s −1 (1 + −1 )−1 is the pole of (H).

14


The Rayleigh line and the Hugoniot curve have at least one intersection point to the right of τ0 if pR (τ ) < 0 < pR (τ ). Under (2.2), the inequality 0 < pR (τ ) holds, and pR (τ ) < 0 holds as well if in addition p˜ − < −

p˜ − u˜ − + qz + . 2 −1 τ− − u˜ −

(2.5)

Let τ+ be an intersection point of (R) and (H) to the right of τ0 . Condition (T)+ is satisfied if τ+ = (1 + 2 −1 )τ− + s −2 τ˜+ + O(s −3 ),

(2.6)

(1 + 2 −1 )τ− ( p˜ − − τ˜+ ) < c Ti .

(2.7)

with

Condition (L)+ is satisfied if (1 + 2 −1 )τ− < (1 + (1 + )−1 )τ+ ,

(2.8)

which holds under (2.6), if s is large. We plug the ansatz (2.6) in the equation pH = pR , to find

p˜ −

qz + −1 −1 (1 +

)(1 + 2

) − 1 + . (2.9) τ˜+ = −1 (1 + 2 )τ− (1 + 2 −1 )τ− The intersection point τ+ is an admissible right specific volume if pH (τ+ ) > 0 and pR (τ+ ) > 0. These inequalities hold if τ < τ+ < (α + 1)τ− + s −2 p˜ − .

(2.10)

The inequalities (2.5), (2.7) and (2.10) are constraints on τ− , p˜ − , and u˜ − . The lower bound on τ+ in (2.10) is satisfied in the regime (2.1) if s is large. If we let p˜ − =

−2 qz + + O(s −1 ), τ−

then (2.5) holds. Finally, if τ− satisfies 1
0. This triangular structure of the matrix −μA± + μ2 B± + G ± allows a simple description of the solutions of (3.14). Indeed, (3.14), a polynomial, degree four equation in λ, splits into the linear equation μs + μ2 d − kφ± − λ = 0,

(3.20)

and the degree three equation

det(−μ∂v f |± + μ2 b1|± − λ) = 0.

(3.21)

By inspection, (3.20) is quadratic in μ, while (3.21) is degree five in μ. Thus, the four solutions λ(μ) of (3.14) correspond to seven eigenvalues μ(λ) of A(λ). 3.1.2. Low-frequency behaviour of the normal modes. We describe here the behaviour of the normal modes in a small ball B(0, r ) := {λ ∈ C, |λ| < r }. Definition 3.4 (Slow modes, fast modes). We call slow mode at ±∞ any family of normal modes {(λ, U (λ)}λ∈B(0,r ) ,

f or some r > 0,

such that, in (3.16), μ±j (0) = 0, for all j. Normal modes which are not slow are called fast modes. We define similarly slow dual modes and fast dual modes, using (3.19). The solutions of (3.20) are 1 1 (−s + (s 2 + 4d(λ + kφ± )) 2 ), 2d 1 1 2 2 μ± 5 = − 2d (s + (s + 4d(λ + kφ± )) );

μ± 4 =

(3.22) (3.23)

20


they depend analytically on λ (in the case of μ4+ and μ+5 , this is ensured by s > 0, assumed in Definition 1.5), and satisfy, for λ in a neighborhood of the origin, ± μ+4 = s −1 λ − s −3 dλ2 + O(λ3 ), μ− 4 > 0, μ5 < 0.

(3.24)

Note that the inequality μ− 4 > 0 is a consequence of φ− > 0. By (3.18), the eigenvector ˜ + that is associated with −μ+ is of A L +4

=

4

+4 + μ4 b+tr +4

∈ C4 × C3 , +4 (0) = +4 ,

(3.25)

where tr +4 := 0 0 0 1

(3.26)

is the reactive left eigenvector of A+ associated with the reactive eigenvalue of A+ . We ± − ± ˜ label L − 4 , L 5 the eigenvectors of A± associated with −μ4 and −μ5 . By the block − − 2 structure of −μA± + μ B± + G ± , spectral separation of μ4 and μ5 (and of μ+4 and ± μ+5 ), the eigenvectors L ± 4 and L 5 are analytic in λ, in a neighborhood of the origin (see for instance [Kat], II.1.4); in particular, +4 = +4 + O(λ), μ+4 b+tr +4 = O(λ).

(3.27)

The solutions of (3.21), seen as an equation in λ, are the eigenvalues of the matrix −μ∂v f |± + μ2 b1|± . By (1.3) and the block structure of A± , we find that the spectrum

of ∂v f |± is

σ (∂v f |± ) = {−s(ε) − σ± , −s(ε), −s(ε) + σ± }.

The eigenvalues of ∂v f |± are distinct, hence, by Rouché’s theorem, the eigenvalues of

−∂v f |± + μb1|± are analytic in μ, for small μ, with expansions λ1 = s + σ± + β1± μ + O(μ2 ), λ2 = s + β2± μ + O(μ2 ), λ3 = s − σ± + β3± μ + O(μ2 ).

(3.28)

By (H3) (Sect. 1.5), β ± j > 0 for all j. Inversion of these expansions yields analytic ± functions μ j , called fluid modes, and defined in a neighborhood of the origin in Cλ : −1 −3 ± 2 3 μ± 1 := (s + σ± ) λ − (s + σ± ) β1 λ + O(λ ), ± ± 2 −1 −3 3 μ2 := s λ − s β2 λ + O(λ ), −1 −3 ± 2 3 μ± 3 := (s − σ± ) λ − (s − σ± ) β3 λ + O(λ ).

˜ that are associated with these eigenvalues are By (3.18), the eigenvectors of A ±j ± L j (λ) = ∈ C4 × C3 , ±j (0) = ±j , 1 ≤ j ≤ 3, tr ± μ±j b± j

(3.29)

(3.30)


21

± ± where the vectors ± 1 , 2 and 3 are the left eigenvectors of A± associated with the fluid eigenvalues −s − σ± , −s, and −s + σ± ; they have the form

tr ±j := ∗ ∗ ∗ 0 ,

1 ≤ j ≤ 3.

(3.31)

The eigenvalues of −∂v f |± + μb1|± being distinct, the associated eigenvectors are analytic as well, so that the L ±j , 1 ≤ j ≤ 3, are analytic in λ; in particular, ±j = ±j + O(λ),

tr ± μ±j b± j = O(λ).

(3.32)

Finally, the equation det(−μ∂v f |± +μ2 b1|± ) = 0 has two non-zero solutions γ6± , γ7± , corresponding to the remaining two (fast) modes, solutions of κτ±−2 c−1 sνμ2 + (κc−1 (s 2 − τ±−2 e± ) + νs 2 )τ±−1 μ + s(s 2 − σ±2 ) = 0.

(3.33)

The Lax condition (1.11) implies that solutions of (3.33) are distinct and have small frequency expansions ± μ± 6 = γ6 + O(λ), ± μ7 = γ7± + O(λ),

γ6± < 0, γ7− > 0, γ7+ < 0.

(3.34)

± ± ± ˜ We label L ± 6 and L 7 the eigenvectors of A associated with −μ6 and −μ7 . Again, by ± ± spectral separation, L 6 and L 7 are analytic in λ.

Lemma 3.5. For some r > 0, Eqs. (3.13) have analytic bases of solutions in B(0, r ), ±

B˜ ± := {V˜ j± }1≤ j≤7 , V˜ j± := e−yμ j (λ) L ±j (λ),

(3.35)

where the eigenvalues μ±j are given in (3.22), (3.23), (3.29), and (3.34) and the eigenvectors associated with the slow modes are given in (3.25), (3.27), (3.30) and (3.32). Proof. The above discussion describes analytic families μ±j , L ±j , such that the vectors V j± defined in (3.35) are analytic solutions of (3.13). For λ = 0, the eigenvalues μ±j are simple, so that the families B˜ ± define bases of Eqs. (3.13). By inspection of the expansions at λ = 0, the families B˜ ± define bases of Eqs. (3.13) at λ = 0 as well. The above low-frequency expansions of the eigenvalues show that ˜ − (λ)W˜ has a 3-dimensional subspace of solutions associated (i) Equation W˜ = A with slow modes (μ−j , j = 1, 2, 3) and 4-dimensional subspace of solutions asso− − − ciated with fast modes (μ− 4 , μ5 , μ6 , μ7 ). ˜ + (λ)W˜ has a 4-dimensional subspace of solutions associated (ii) Equation W˜ = A with slow modes (μ+j , j = 1, 2, 3, and μ+4 ) and a 3-dimensional subspace of solutions associated with fast modes (μ+5 , μ+6 , μ+7 ).

22


3.1.3. Description of the essential spectrum. We adopt Henry’s definition of the essential spectrum [He]: Definition 3.6 (Essential spectrum). Let B be a Banach space and T : D(T ) ⊂ B → B a closed, densely defined operator. The essential spectrum of T, denoted by σess (T ), is defined as the complement of the set of all λ such that λ is either in the resolvent set of T, or is an eigenvalue with finite multiplicity that is isolated in the spectrum of T. By Lemma 3.3, the matrix A± (λ) has a non trivial center subspace if and only if λ ∈ C± , C± := {λ ∈ C, det(−iξ A± − ξ 2 B± + G ± − λ) = 0, for some ξ ∈ R}. The following lemma can be found in [He] (Theorem A.2, Chap. 5 of [He], based on Theorem 5.1, Chap. 1 of [GK]): Lemma 3.7. The connected component of C\ (C− ∪ C+ ) containing real +∞ is a connected component of the complement of the essential spectrum of L(ε). The reactive eigenvalues of −iξ A± − ξ 2 B± + G ± are λ = iξ s − ξ 2 d − kφ± . For small |ξ |, the fluid eigenvalues satisfy λ = iαξ − βξ 2 + O(ξ 3 ),

α ∈ R, β > 0,

as described in Sect. 3.1.2; for large |ξ |, they satisfy λ = −ξ 2 (α + O(ξ −1 ))

(parabolic eigenvalues),

(3.36)

with α ∈ {ντ±−1 , κc−1 τ±−1 }, or λ = isξ + O(1)

(hyperbolic eigenvalue).

(3.37)

This implies that the essential spectrum is confined to the shaded area in Fig. 3, the boundary of which is the union of an arc of parabola and two half-lines. (The origin λ = 0 is an eigenvalue, associated with eigenfunction (U¯ ε ) ; the existence of bifurcation eigenvalues γ (ε) ± iτ (ε) is assumed in Theorem 1.18, the proof of which is given in Sect. 5.) Remark 3.8. The essential spectrum, as given by Definition 3.6, is not stable under relatively compact perturbations (see [EE], Chap. 4, Ex. 2.2); namely, a domain of the complement of the essential spectrum of a (closed, densely defined) operator T is either a subset of the complement of the essential spectrum of T + S, or is filled with point spectrum of T + S, where S is a relatively compact perturbation of T. Remark 3.9. By the Fréchet-Kolmogorov theorem, L is a relatively compact perturbation of L ± . (This observation is the first step of the proof of Lemma 3.7, see Henry [He].) The pathology described in Remark 3.8 does not occur in the right half-plane here, as we know by an energy estimate that if λ is large and real, λ ∈ / σ p (L).


23

Fig. 3. Spectrum of L(ε)

3.1.4. Gap Lemma and dual basis. Let be the connected component of C \ (C− ∪ C+ ) containing real +∞. Definition 3.10 (Stable and unstable subspaces at ±∞). Given λ ∈ ∪ B(0, r ), r as ˜ ± (λ)) the stable subspace of A ˜ ± (λ) (i.e., the subspace of in Lemma 3.5, denote by S(A generalized eigenvectors associated with eigenvalues with negative real parts) and by ˜ ± (λ)) the unstable subspace of A ˜ ± (i.e., the subspace of generalized eigenvectors U (A associated with eigenvalues with positive real parts). We define similarly S(A± (λ)) and U (A± (λ)). By definition of C± , given λ ∈ , the matrices A± (λ) do not have purely imaginary ˜ ± (λ)) ⊕ U (A ˜ ± (λ)) = C7 , eigenvalues, so that S(A± (λ)) ⊕ U (A± (λ)) = C7 , and S(A for all λ ∈ . ˜ ± (λ)) have analytic bases in . ˜ ± (λ)) and U (A Lemma 3.11. The vector spaces S(A Proof. By simple-connectedness of , the lemma follows from a result of Kato ([Kat], II.4), that uses spectral separation in . Corollary 3.12. Equations (3.13) have analytic bases of solutions in . Proof. Basis elements of the stable and unstable spaces defined in Definition 3.10 are associated, through the flow of (3.13), with bases of solutions of (3.13). The matrices ˜ ± depending analytically on λ, the flow of (3.13) is analytic in λ. A Lemma 3.13. For λ real and large, dim S(A+ (λ)) = dim S(A− (λ)) = 3. Proof. From Lemma 3.3, μ is an eigenvalue of A± (λ) if and only if λ is an eigenvalue of −μA± +μ2 B± +G ± . As in Sect. 3.1.3, for large μ, the eigenvalues of −μA± +μ2 B± +G ± are sμ + O(1) (hyperbolic mode) and ντ±−1 μ2 + O(μ), κτ±−1 c−1 μ2 + O(μ), dμ2 + O(μ) (parabolic modes), c−1 as in Assumption 1.1. Inversion of these expansions gives three stable eigenvalues for both A− and A+ .

24


Fig. 4. Normal modes on the −∞ side

Remark 3.14. The above lemma implies in particular that is a domain of consistent splitting, as defined in [AGJ]. (See also Sect. 3.1 of [LyZ2].) ˜ + (λ)) with soluGiven λ ∈ , the flow of (3.13) associates basis elements of S(A tions of (3.13) which are exponentially decaying as t → +∞, and basis elements of ˜ − (λ) with solutions which are exponentially decaying as t → −∞. Similarly, the U (A ˜ − (λ)) and U (A ˜ + (λ) are associated with exponentially growing solutions, at spaces S(A −∞ and +∞ respectively. Definition 3.15 (Decaying and growing normal modes). We call decaying dual normal mode at ±∞ any continuous family of dual normal modes {λ, U˜ (λ)}, λ ∈ B(0, r ), r as in Lemma 3.5, such that for all λ ∈ ∩ B(0, r ), U˜ (λ) corresponds to a decaying solution of (3.13) at ±∞. Families of normal modes which are not decaying are growing. We define similarly decaying dual normal modes and growing dual normal modes. By continuity of the eigenvalues and spectral separation in , if for some λ ∈ a continuous family of normal modes corresponds to a decaying (resp. growing) solution, then it corresponds for all λ ∈ to a decaying (resp. growing) solution. By (1.11), (3.24) and (3.29), μ+1 , μ+2 , μ+3 and μ+4 are growing (in the sense of Definition 3.15) at +∞, while μ+5 , μ+6 and μ+7 are decaying. − − − − − − Similarly, μ− 3 , μ5 and μ6 are growing, while μ1 , μ2 , μ4 and μ7 are decaying. The normal modes with which the characteristics of (1.1) are associated are pictured on Figs. 5 and 4. In particular, slow normal modes associated with incoming characteristics are growing. Definition 3.16 (Normal residuals). A map (y, λ) → + (y, λ) ∈ C7 defined on [y0 , +∞) × B(0, r ), for some y0 > 0, r > 0, is said to belong to the class of normal


25

Fig. 5. Normal modes on the +∞ side

residuals if it satisfies the estimates |+ | ≤ C,

|∂ y + | ≤ C(|λ| + e−θ|y| )

for some θ > 0 and C > 0, uniformly in y ≥ y0 and λ ∈ B(0, r ). We define similarly the class of normal residuals on (−∞, −y0 ) × B(0, r ). Lemma 3.17 (Fast dual modes). Equations (3.8) has solutions W˜ 4− , W˜ 5+ , W˜ 6+ , W˜ 7+ (growing)

and W˜ 5− , W˜ 6− , W˜ 7− (decaying),

which for λ ∈ B(0, r ), r possibly smaller than in Lemma 3.5, satisfy −yμ±j (λ) ˜ ± + λ ˜± , L ±j (0) + e−θ|y| y ≷ ±y0 , W˜ ± j =e 1j 2j

(3.38)

for some y0 > 0 independent of λ, where the constant vectors L ±j (0) are defined in ˜ ± , ˜ ± are normal residuals in the sense of Definition 3.16. Sect. 3.1.2, and 1j

2j

Proof. With the description of the normal modes in Lemma 3.5, this is a direct application of the Gap Lemma (for instance in the form of Proposition 9.1 of [MaZ3]). Lemma 3.18 (Slow dual modes). Equations (3.8) has solutions W˜ 1− , W˜ 2− (gr owing)

and W˜ 3− , W˜ 1+ , W˜ 2+ , W˜ 3+ , W˜ 4+ (decaying),

which for λ ∈ B(0, r ), r possibly smaller than in Lemma 3.5, satisfy −yμ±j (λ) ± ˜ ± , y ≷ ±y0 , W˜ ± L = e (0) + λ j j j

(3.39)

26


for some y0 > 0 independent of λ, where the constant vectors L ±j (0) are defined in ˜ ± are normal residuals. Sect. 3.1.2, and j

Proof. The Conjugation Lemma ([MeZ]; Lemma 3.1 of [MaZ3]) implies that there ˜ + (·, λ)}λ∈B(0,r ) , for some r > 0 possiexists a family of matrix-valued applications { ˜ + is invertible for all λ and bly smaller than in Lemma 3.5, such that the matrix Id + + ˜ is smooth in y and analytic in λ, with exponential bounds y, the application +

˜ | ≤ C jk e−θ y , |∂λ ∂xk j

for some θ > 0, C jk > 0, for y ≥ y0 ,

for some y0 > 0, and such that any solution W˜ of (3.8) has the form ˜ + )V˜ + , W˜ = (Id +

for y ≥ y0 ,

(3.40)

where V˜ + is a dual normal mode, and, conversely, if V˜ + is a dual normal mode, then W˜ defined by (3.40) solves (3.8) on y ≥ y0 . Equation (3.8) at λ = 0 has a four-dimensional subspace of constant solutions; let {W˜ 0j }1≤ j≤4 be a generating family. The normal modes with which, through (3.40), the W˜ 0j are associated are slow normal modes. Hence, by Lemma 3.5, there exist coordinates c jk such that ˜ + (·, 0)) W˜ 0j = (Id + c jk L +k (0), y ≥ y0 , 1≤k≤4

which implies in particular that the matrix c := (c jk )1≤ j,k≤4 is invertible. Then, for 1 ≤ j ≤ 4, ˜ + (·, 0))L + (0) = (Id + (c−1 ) jk W˜ k0 , j 1≤k≤4 +

˜ (·, 0))L + (0) is constant, hence, by exponential decay of ˜ + , equal in particular, (Id + j to L +j (0). We can conclude that, for 1 ≤ j ≤ 4, ˜ + )V˜ + W˜ +j := (Id + j (where V˜ j+ is defined in Lemma 3.5) is a solution of (3.8) on y ≥ y0 , which can be put in the form (3.39). The proof on the −∞ side is based similarly on the decomposition of the fluid components of the W˜ 0j onto the (fluid) dual slow modes V˜ j− , for 1 ≤ j ≤ 3. 3.1.5. Duality relation and forward basis. We use the duality relation, introduced in [MaZ3], W˜ tr SW = 1

(3.41)

that relates solutions W of the forward Eq. (3.3) with solutions W˜ of the adjoint Eq. (3.8) through the conjugation matrix in (τ, w, bw ) coordinates ⎛ ⎞ −A11 −A12 0 S := ⎝ −A21 −A22 IdC3 ⎠, 0 −IdC3 0


27

where A is the convection matrix defined in (1.18). Namely, W is a solution of (3.3) if and only if it satisfies (3.41) for all solutions W˜ of (3.8), and conversely W˜ is a solution of (3.8) if and only if it satisfies (3.41) for all solutions W of (3.3). (See Lemma 4.2, [MaZ3]; note that the reactive term contains no derivative, hence does not play any role here.) Remark that there exist vectors rk± such that ±j A±rk± = −δ jk , Let Rk± be vectors of the form Rk±

:=

rk± ∗

1 ≤ j, k ≤ 4.

+ e−θ ± 1k ,

(3.42)

(3.43)

where for 1 ≤ k ≤ 4, rk± are given by (3.43), and where ± 1k are normal residuals. With the notation of Lemmas 3.17 and 3.18, let L ±j (0) if μ±j is slow, ¯L ± := (3.44) j ˜± L ±j (0) + e−θ|y| if μ±j is fast. 1j Lemma 3.19 (Forward and dual basis) For some r > 0 and y0 > 0, • Equation (3.3) has analytic bases of solutions {W1± , . . . , W7± }λ∈∪B(0,r ) , for y ≷ ±y0 ; • Equation (3.8) has analytic bases of solutions {W¯ 1± , . . . , W¯ 7± }λ∈∪B(0,r ) , for y ≷ ±y0 , such that for λ ∈ B(0, r ), ±

xμ j (λ) ± W± (R ± j =e j + λ j ),

W¯ ± j =e

−yμ±j (λ)

¯ ± ), ( L¯ ±j + λ j

y ≷ ±y0 , y ≷ ±y0 ,

(3.45) (3.46)

± ¯± ¯± where R ± j and L j are defined in (3.43) and (3.44), and j and j are normal residuals; the fast forward modes W4− and W7+ satisfy also ε (U¯ ) (x) ± + λ±j (x, λ), x ≷ ±y0 , (3.47) W j (x, λ) = ∗

where |±j | + |∂x ±j | ≤ Ce−θ|x| , for some C, θ > 0, uniformly in λ ∈ B(0, r ). 7 Proof. Given a family {F1, . . . , F7 } of vectors in C , let col(F j ) denote the 7 × 7 matrix col(F j ) := F1 . . . F7 . Let y0 , r, and W˜ ± j as in Lemma 3.17 and 3.18. For all λ ∈ ∪ B(0, r ), the families − − ˜ ˜ {W , . . . , Wn } and {W˜ + , . . . , W˜ n+ } are bases of solutions of (3.8), on y ≤ −y0 and 1

1

˜ 0± := col(W˜ ± ) are invertible y ≥ y0 respectively. In particular, the 7 × 7 matrices W j for all λ ∈ B(0, r ) and y ≷ ±y0 . Let ˜ 0± )tr S)−1 =: col(W 0± ). W0± := ((W k

(3.48)

28


For the forward modes W 0± j defined in (3.48) to satisfy the low-frequency description ±

xμ j (λ) −θ|x| 0± W 0± (R 0± 1 j + λ0± j =e j +e 2 j ),

y ≷ ±y0 ,

(3.49)

0± where R 0± j are constant vectors and j are normal residuals, it suffices, by (3.41), that

0± 0± the matrices R0± := col(R 0± j ) and := col( j ) satisfy

Ltr SR0 = IdC7 , 0 ˜ 1 )tr S01 = − ˜ tr (L + e−θ|x| 1 SR , ˜ 01 (L + e−θ|x|

˜ 2 )tr S02 = + λ

0 ˜ tr − 2 SR ,

(3.50) (3.51) (3.52)

±

˜ := col( ¯ ± ) appear in the low-frequency description where L± := col(L ±j (0)) and j of the W˜ ± . In (3.50)–(3.52), the ± exponents are omitted. The matrices L± being invertj ible, (3.50) (with + or −) has a unique solution, and, for y0 large enough and r small enough, Eqs. (3.51) and (3.52) have unique solutions in the class of normal residuals. Note that for 1 ≤ j, k ≤ 4, Eq. (3.50) reduces to (3.42), up to exponentially decaying terms, so that the vectors Rk0± have the form (3.43). Remark now that (U¯ ε ) satisfies L(ε)(U¯ ε ) = 0, and decays at both −∞ and +∞, hence (U¯ ε ) is associated with decaying fast normal modes; by Lemma 3.17, there exist constants c±j , such that ε (U¯ ) (y) = c4 − W40− |λ=0 = c+j W 0+ (3.53) j |λ=0 . ∗ 5≤ j≤7

We may assume, without loss of generality, that c7+ = 0. Let now W− := W10− W20− W30− c4− W40− W50− W60− W70− , W+ := W10+ W20+ W30+ W40+ W50+ W60+ 7j=5 c+j W 0+ , j and W± =: col(W ± j ). These forward modes satisfy (3.45) and (3.47). Let finally ±tr ± −1 ˜ 0± and ¯ := (SW ) =: col(W¯ ± W j ), so that, in particular, the slow modes of W ¯ ± coincide. We can prove as above that the low-frequency description (3.49) of the W forward modes carries over to the dual modes through the duality relation, so that (3.46) is satisfied. 3.1.6. The resolvent kernel. Let L 2 (, D (R)) := {φ ∈ D ( × R), for all ϕ ∈ D(R), φ, ϕ ∈ L 2 ()}. A linear continuous operator T : L 2 (R) → L 2 (R) operates on L 2 (R, D (R)), by T φ, ϕ := T φ, ϕ. Let τ(·) δ ∈ L 2 (R, D (R)) be defined by τx δ, ϕ = ϕ(x), for all x ∈ R. Definition 3.20 (Resolvent kernel). Given λ in the resolvent set of L(ε), define the resolvent kernel Gλ of L(ε) as an element of L 2 (Rx , D (R y )) by Gλ := (L(ε) − λ)−1 τ(·) δ.


29

Given y ∈ R, let s y = sgn(y), and s ˜ D(y) := { j, μ˜ jy slow and decaying},

so that ˜ D(y) = {3}, if y < 0,

˜ D(y) = {1, 2, 3, 4}, if y > 0.

Given x, y ∈ R, let D(x, y) be the set of all ( j, k) such that for all x, y, for λ > 0 and s |λ| small enough, (μsjx x − μky y) < 0, that is, s

D(x, y) = {( j, k), μsjx andμ˜ ky slow and decaying} {( j, j), sx = s y , |y| < |x|, μsjx slow and decaying} s {( j, j), sx = s y , |x| < |y|, μ˜ jy slow and decaying}, so that

⎧ {(1, 1), (2, 2), (1, 3), (2, 3)}, ⎪ ⎪ ⎪ {(1, 3), (2, 3), (3, 3)}, ⎪ ⎪ ⎨ ∅, D(x, y) := {( j, k), 1 ≤ j ≤ 4, 1 ≤ k ≤ 2}, ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ {( j, j), 1 ≤ j ≤ 4}, ∅,

x ≤ y ≤ 0, y ≤ x ≤ 0, y ≤ 0 ≤ x, x ≤ 0 ≤ y, 0 ≤ x ≤ y, 0 ≤ y ≤ x.

Define now the excited term Eλ (x, y) := λ−1 (U¯ ε ) (x)

s

s tr −yμ jy (λ)

[c0j,s y ] jy

e

,

˜ j∈D(y)

and the scattered term

Sλ (x, y) :=

( j,k)∈D(x,y)

s

s tr xμsjx (λ)−yμky (λ)

j,s

[ck,sxy ]r sj x ky

e

,

where the vectors ±j are defined in (3.26) and (3.31), the vectors r ± j are defined in (3.42), j,±

0 ] and [c and the transmission coefficients [ck,± k,± ] are constants.

Proposition 3.21. Under (1.20), for λ ∈ B(0, r ), the radius r being possibly smaller 0 ] and [c j,± ] such that the than in Lemma 3.5, there exist transmission coefficients [ck,± k,± resolvent kernel decomposes as Gλ = Eλ + Sλ + Rλ , where Rλ satisfies

|∂xα ∂ yα Rλ | ≤ Ce−θ|x−y| + Cλα e−θ|x|

+ λ

˜ j∈D(y) 1+min(α,α )

+ λα e−θ|x|

sy

e−yμ j

sx

e xμk

sy

−yμ j

,

( j,k)∈D(x,y)

for α ∈ {0, 1, 2, }, α ∈ {0, 1}, for some C, θ > 0, uniformly in x, y and λ ∈ B(0, r ).

30


Proof. The duality relation (3.41) allows to apply Proposition 4.6 of [MaZ3] (and its Corollary 4.7), which describes Gλ as sums of pairings of forward and dual modes, for λ in the intersection of and the resolvent set of L . By Lemma 3.19, Gλ extends as a meromorphic map on B(0, r ). The excited term Eλ comprises the pole terms, corresponding to pairings of a fast, decaying forward mode associated with the derivative of the background wave with a slow, decaying dual mode, i.e. W7+ /W¯ 3− for y ≤ 0 and W4− /W¯ +j for y ≥ 0, 1 ≤ j ≤ 4. The next-to-leading order term is the scattered term Sλ . It corresponds to pairings of a slow forward mode with a slow dual mode. For y ≤ 0, the scattered term comprises only fluid modes. For y ≤ 0 ≤ x and for 0 ≤ y ≤ x, the scattered term vanishes, as there are no outgoing modes to the right of the shock (see Figs. 1 and 4). By the Evans function condition (1.20) and Lemma 6.11 of [MaZ3], the residual Rλ does not contain any pole term; it comprises: (a) the contribution of the normal residuals to the fast forward/slow dual pairings involving the derivative of the background profile, (b) the fast forward/slow dual pairings not involving the derivative of the background profile, (c) the contribution of the normal residuals to the slow forward/slow dual pairings, and (d) the slow forward/fast dual pairings. Term (a) is bounded by the first two terms in the upper bound for Rλ . Term (b) is smaller than term (a) by a O(λ) factor. Term (c) is bounded by the third term in the upper bound. By the Lax condition (1.11), the Evans function condition (1.20) and Lemma 6.11 of [MaZ3], term (d) is also bounded by the third term. 3.1.7. The Evans function. By Lemma 3.13, for all λ ∈ , the dimensions of U (A− (λ)) and S(A+ (λ)), the vector spaces associated with decaying solutions of (3.3) at −∞ and +∞, add up to the full dimension of the ambient space: dim U (A− (λ)) + dim S(A+ (λ)) = 7. Definition 3.22 (Evans function). On ∪ B(0, r ), define the Evans function as D(ε, λ) := det(W1− , W2− , W4− , W7− , W5+ , W6+ , W7+ )|x=0 . The Evans function D satisfies Proposition 1.10; it has a zero at λ = 0, as reflected in equality (3.53). 3.2. Inverse Laplace transform. Similarly as in Sect. 3.1.6 (or Sect. 2 of [MaZ3]), define the Green function of L(ε) as G := et L(ε) τ(·) δ,

(3.54)

where {et L(ε) }t≥0 is the semi-group generated by L(ε). That is, the kernel of the integral operator et0 L(ε) is the Green function G evaluated at t = t0 . Assuming (1.20), the inverse Laplace transform representation of the semi-group by the resolvent operator (see for instance [Pa] Theorem 7.7; [Z3] Prop. 6.24) yields η0 +i∞ 1 G(ε, x, t; y) = P.V. eλt Gλ (ε, x, y) dλ, (3.55) 2πi η0 −i∞ for η0 > 0 sufficiently large.


31

3.2.1. Pointwise Green function bounds. Introduce the notations y 2 errfn(y) := e−z dz, −∞

and let, for y < 0 : e :=

⎛

⎞

⎛

⎛

⎞⎞

y − a−t y + a−t 0 ⎝errfn ⎝ 3 ⎠ − errfn ⎝ 3 ⎠⎠, ]−tr [c3,− 3 4β − 4β − j t j t

for y > 0 :

(3.56)

⎞ ⎛ ⎞⎞ +t +t y − a y + a ⎝errfn ⎝ j ⎠ − errfn ⎝ j ⎠⎠, e := [c0j,+ ]+tr j 4β +j t 4β +j t 1≤ j≤4

(3.57)

E(ε, x, t; y) := (U¯ ε ) (x)e(ε, t; y).

(3.58)

⎛

⎛

and In (3.56)–(3.57) and below, the {a ± j }1≤ j≤4 are the characteristic speeds, i.e. the limits at ±∞ of the eigenvalues of ∂U F(ε, U¯ ε ), ordered as in (1.3), the β ± j , 1 ≤ j ≤ 3, are the positive diffusion rates that were introduced in (3.28) (and which depend on ε, as do the characteristic speeds), and β4+ := d, the species diffusion coefficient. Let for y < 0 : − − −tr − 21 −(x−y−a j t)2 /4β j t r− (4πβ − e S := χ{t≥1} j j j t) 1≤ j≤2

⎛ − 1 e−x ⎝ − −tr 2 +χ{t≥1} −x r3 3 (4πβ3− t)− 2 e−(x−y+(s−σ− )t) /4β3 t x e +e ⎞ j,− j,− 2 j,− 1 j,− −tr + [c3,− ]r − (4πβ3,− t)− 2 e−(x−z 3,− ) /4β3,− t ⎠, j 3

(3.59)

1≤ j≤2

and for y > 0: S := χ{t≥1}

ex + − 21 −(x−y−a +j t)2 /4β +j t r +j +tr e j (4πβ j t) −x x e +e 1≤ j≤4

+χ{t≥1}

e−x e−x

+ ex

k,− 2 /4β k,− j,+ t

k,− − 2 −(x−z j,+ ) − +tr [ck,− e j,+ ]rk j (4πβ j,+ t) 1

,

1≤ j≤4 1≤k≤2

(3.60) where the indicator function χ{t≥1} is identically equal to 1 for t ≥ 1 and 0 otherwise, and ± 2 aj |x| ± |y| j,± k,± ± ± −1 z j,± := a j (t − |y||ak | ), βk,± := ± β j + ± βk± . |a j |t |ak |t ak±

32


Let H := h(ε, t, x, y)τx+st δ,

h+4 ≡ 0,

(3.61)

where the notation τ(·) δ was introduced at the beginning of Sect. 3.1.6. j,± Let finally S0 be the scattered term defined in (3.59)–(3.60) in which [ck,± ] = 1 for all j, k. j,±

Proposition 3.23. Under (1.20), there exists transmission coefficients [c0j,± ] and [ck,± ], satisfying ⎧ 0 ⎪ ⎨ [c4+ ] = 0, 0 ](v ε − v ε ) + [c1,− ]r − + [c2,− ]r − , rk = [ck, − + (3.62) k, 1 k, 2 ⎪ ⎩ c1,− = [c2,− ] = 0, 4,+ 4,+ ε , z ε ), such that the where 1 ≤ k ≤ 3 if = + and 1 ≤ k ≤ 2 if = −, U±ε =: (v± ± Green function G(ε, x, t; y) defined in (3.54) may be decomposed as a sum of hyperbolic, excited, scattered, and residual terms, as follows:

G = H + E + S + R,

(3.63)

where H, E and S are defined in (3.56)–(3.61), with the estimates

|∂tk ∂xα ∂ yα h| ≤ C e−θt ,

|∂xα ∂ yα R| ≤ C e−θ(|x−y|+t) 1 1 2 +C t − 2 (1+α+α ) (t + 1)− 2 + e−θt e−(x−y) /Mt 1 1 1 +C (t − 2 + e−θ|x| )t − 2 (α+α ) + α t − 2 e−θ|y| |S0 |.

(3.64)

uniformly in ε, for k ∈ {0, 1}, α ∈ {0, 1, 2}, α ∈ {0, 1}, for some θ, C, M > 0. Proof. We only check (3.62), as decomposition (3.63) and bounds (3.64) are easily deduced from Proposition 7.1 of [MaZ3] and Proposition 7.3 of [LRTZ]. (See also Proposition 3.7 of [TZ2], especially Eqs. (3.30)–(3.33) and (3.38).) The description of the residue of Gλ at λ = 0 for y < 0 and y > 0 implies 0 0 + [c3,− ]− = [c ] + [c0j,+ ]+j , 4,+ 4 3 1≤ j≤3

corresponding to Eq. (1.34) in [MaZ3]. The (reactive) left eigenvector vector +4 being orthogonal to the (fluid) left eigenspace span{±j }1≤ j≤3 (see (3.26) and (3.31)), this implies (3.62)(i). Given U0 ∈ L 1 , the estimates for H and R imply (H + R)U0 d y d x = 0. lim t→+∞ R2

Hence, by conservation of mass in the fluid variables, (1.7), for all U0 ∈ L 1 , π(E + S)U0 d y d x = πU0 dy, lim t→+∞ R2

R

(3.65)


33

where π : C4v,z → C3v is defined by π(v, z) := v. (Eq. (3.65) corresponds to (1.33) and (7.60) in [MaZ3].) Taking U0 ∈ span{±j }1≤ j≤3 , we find (3.62)(ii), and taking U0 parallel to +4 , we find (3.62)(iii). Remark 3.24. The terms E and S correspond to the low-frequency part of the representation of G by inverse Laplace transform of the resolvent kernel Gλ , while the term H corresponds to the high-frequency part. As observed in [MaZ3], for low frequencies, the resolvent kernel in the case of real (physical) viscosity obeys essentially the same description as in the artificial (Laplacian) viscosity case, hence the estimates on E and S follow by the analysis in [LRTZ] of the corresponding artificial viscosity system, specialized to the case of strong detonations (more general waves were treated in [LRTZ]). The estimate of the terms H and R follows exactly as for the nonreactive case treated in [MaZ3,Z2]. Remark 3.25. Bound (3.64)(ii) is implied by bounds (7.1)–(7.4) of Proposition 7.1 of [MaZ3] and bounds (3.30), (3.32) and (3.38) of Proposition 3.7 of [TZ2]. Here the contribution of the hyperbolic, delta-function terms to the upper bounds for the spatial derivatives of R is absorbed in H, and the short-time, t ≤ |ak± ||y|, contributions of the scattered terms are absorbed in the generic parabolic residual term 2 e−θt e−(x−y) /Mt . Corollary 3.26. The excited terms Eλ and E contain only fluid terms: Eλ +4 ≡ 0 and E +4 ≡ 0. Proof. The equality E +4 ≡ 0 follows from (3.62)(i). The resolvent kernel Gλ is the Laplace transform of the Green function G, so that the coefficients [c0j,± ] in Propositions (3.21) and (3.23) must agree. Hence, (3.62)(i) implies also Eλ +4 ≡ 0. Corollary 3.27. For all η > 0, for some C, M > 0, some θ1 (η, s) > 0, the following bounds hold, for α ∈ {0, 1, 2}: |e−ηy ∂xα S+4 | ≤ Ce−θ1 t e−η|x−y|/2 , +

|e−ηy ∂xα R+4 | ≤ Ce−θ1 t (e−η|x−y|/2 + e−(x−y) +

2 /Mt

).

(3.66)

Proof. By (3.62)(iii), the contribution of the reactive modes to S is χ{y>0} χ{t≥1}

ex

ex 2 r + +tr (4π dt)−1/2 e−(x−y+st) /4dt . + e−x 4 4

Given 0 ≤ x ≤ y, we can bound e−ηy e−(x−y+st) e−ηy e−(st/2)2/4t ≤ e−ηy e−s and, for |x − y| >

2 t/16

2 /4dt

(3.67)

, for |x − y| ≤ 21 st, by

≤ e−η|y−x| e−s

2 t/16

,

1 2 st, by −ηy

e

≤ e−η|y−x|/2 e−ηy/2 ≤ e−η|y−x|/2 e−ηst/4 ,

and this implies (3.66)(i). To prove (3.66)(ii), we note that the contribution of the para1 1 2 bolic terms t − 2 (t + 1)− 2 e−(x−y) /Mt and S0 to R+4 comes from Riemann saddle-point estimates of the sole scattered terms Sλ +4 (see the proof of Proposition 7.1 in [MaZ3] for more details). Hence (3.66)(i) implies (3.66)(ii). Remark 3.28. The proof of Proposition 7.1 of [MaZ3] shows that Proposition 3.23 applies more generally to linear operators of the form (1.17) that satisfy (1.20) and the conditions (A1)–(A2), (H1)–(H4) of Sect. 1.5.

34


3.2.2. Convolution bounds. From the pointwise bounds of Proposition 3.23 and Remarks 3.26 and 3.27, we obtain by standard convolution bounds the following L p → L q estimates, exactly as described in [MaZ1,MaZ2,MaZ3,MaZ4,Z2] for the viscous shock case. Corollary 3.29. Under (1.20), for all t ≥ 1, some C > 0, any η > 0, for any 1 ≤ q ≤ p, 1 ≤ p ≤ +∞, and f ∈ L q ∩ W 1, p , ! ! ! ! ! (S + R)(·, t; y) f (y) dy ! ! ! p ! R !L ! ! ! ∂ y (S + R)(·, t; y) f (y) dy ! ! ! p R ! !L ! ! + + −θ y ! (S + R)(·, t; y) f (y)e dy !! 4 ! p R ! !L ! ! ! H(·, t; y) f (y)dy ! ! ! p R

≤ Ct

− 21 ( q1 − 1p )

≤ Ct

− 21 ( q1 − 1p )− 21

| f | L q + Ce−ηt | f | L p ,

≤ Ct

− 21 ( q1 − 1p )− 21

| f | L q + Ce−ηt | f | L p ,

| f |L q ,

≤ Ce−ηt | f | L p ,

L

where y + := max(y, 0). Likewise, for all x and all t ≥ 0, |∂ y e(·, t)| L p + |∂t e(·, t)| L p ≤ Ct |∂t ∂ y e(·, t)| L p ≤ Ct

− 21 (1− 1p )

,

− 21 (1− 1p )− 21

.

4. Stability: Proof of Theorem 1.14 We often omit to indicate dependence on ε in the proof below. All the estimates are uniform in ε.

4.1. Linearized stability criterion. Proof of Theorem 1.14. Linear case. Sufficiency of (1.20) for linearized orbital stability follows immediately by the bounds of Corollary 3.29, exactly as in the viscous shock case, setting δ(t) :=

R

e(x, t; y)U0 (y) dy

so that U − δ(t)U¯ =

R

(H + S + R)(x, t; y)U0 (y) dy;

see [ZH,MaZ3,Z2] for further details. Necessity follows from more general spectral considerations not requiring the detailed bounds of Proposition 3.23; see the discussion of effective spectrum in [ZH,MaZ3,Z2]. The argument goes again exactly as in the viscous shock case.


35

4.2. Auxiliary energy estimate. Consider U˜ the solution of (1.2) issued from U˜ 0 , and let U (x, t) := U˜ (x + δ(t), t) − U¯ (x).

(4.1)

Then, the following auxiliary energy estimate holds. Lemma 4.1 (Proposition 4.15, [Z2]). Under the hypotheses of Theorem 1.14, assume ˙ and the H 3 norm that U˜ 0 ∈ H 3 , and suppose that, for 0 ≤ t ≤ T , the suprema of |δ| of U each remain bounded by a sufficiently small constant. Then, for all 0 ≤ t ≤ T , for some θ > 0, t ˙ 2 )(s) ds. |U (t)|2H 3 ≤ Ce−θt |U (0)|2H 3 + C e−θ(t−s) (|U |2L 2 + |δ| 0

4.3. Nonlinear stability. Proof of Theorem 1.14. Nonlinear case. Let U be the perturbation variable associated with solution U˜ as in (4.1); by a Taylor expansion, U solves the perturbation equation ˙ U¯ + ∂x U ), ∂t U − LU = ∂x Qf (U, ∂x U ) + Qr (U ) + δ(t)( where the linear operator L is defined in (1.17), and |Qf | ≤ C|U |(|U | + |∂x U |),

(4.2)

where C depends on U L ∞ and U¯ W 1,∞ . Lemma 4.2. Under Assumptions 1.1, 1.2, if the temperature T associated with solution U satisfies T L ∞ < Ti − T+ (by Lemma 1.6, 0 < Ti − T+ ), then the nonlinear reactive term Qr has the form Qr (U ) = +4 e−η0 x qr (U ), +

(4.3)

where x + := max(x, 0), η0 > 0 is as in Corollary 1.8, and qr (U ) = qr (w, z) is a scalar such that |qr (U )| ≤ C|U | 2,

(4.4)

where C depends on U L ∞ and U¯ L ∞ . Proof. We use the specific form −kφ(T )z+4 of the reactive source in (1.1), together with Taylor expansion (φ(T¯ + T )(¯z + z) − (φ(T¯ )¯z − (φ (T¯ )T z¯ + φ(T¯ )z) = φ (T¯ )T z + φ (T¯ + βT )T 2 z¯ , + for some 0 < β < 1, and the fact that φ (T¯ + T ) ≤ Ce−η0 x for |T | < Ti − T+ , for η > 0 as in Corollary 1.8, by φ(T+ ) = 0 together with the property that φ (T ) ≡ 0 for T ≤ Ti and exponential convergence of U¯ (x) to U+ as x → +∞.

36


Recalling the standard fact that U¯ is a stationary solution of the linearized Eqs. (3.1), ¯ L U = 0, or G(x, t; y)U¯ (y)dy = et L U¯ (x) = U¯ (x), R

we have by Duhamel’s principle: U (x, t) = δ(t)U¯ (x) + G(x, t; y)U0 (y) dy R t + + G(x, t − s; y)+4 e−ηy qr (U )(y, s) dy ds 0 R t ˙ )(y, s) dy ds. − ∂ y G(x, t − s; y)(Qf (U, ∂x U ) + δU R

0

Defining δ(t) = −

e(y, t)U0 (y) dy R t + − e(y, t − s)+4 e−ηy qr (U )(y, s) dy ds 0 R t + ∂ y e(y, t − s)(Qf (U, ∂x U ) + δ˙ U )(y, s)dyds, 0

R

(4.5)

following [Z3,MaZ1,MaZ2,MaZ4], and recalling Proposition 3.23, we obtain finally the reduced equations: U (x, t) = (H + S + R)(x, t; y)U0 (y) dy R t ˙ ) + +4 e−ηy + qr (U ) dy ds + H(x, t − s; y) ∂ y (Qf (U, ∂x U ) + δU 0 R t + + (S + R)(x, t − s; y)+4 e−ηy qr (U ) dy ds 0 R t ˙ )dy ds, − ∂ y (S + R)(x, t − s; y)(Qf (U, ∂x U ) + δU (4.6) 0

R

and, differentiating (4.5) with respect to t, and recalling Corollary 3.26: ˙δ(t) = − ∂t e(y, t)U0 (y) dy R t ˙ )(y, s) dy ds, + ∂ y ∂t e(y, t − s)(Qf (U, ∂x U ) + δU 0 R δ(t) = − e(y, t)U0 (y) dy R t + ∂ y e(y, t − s)(Qf (U, ∂x U ) + δ˙ U )(y, s) dy ds. 0

R

(4.7)

(4.8)


Define ζ (t) :=

sup

37

1

|U (·, s)| L p (1 + s) 2

(1− 1p )

1 ˙ + |δ(s)|(1 + s) 2 + |δ(s)| .

0≤s≤t, 2≤ p≤∞

We shall establish: Claim. There exists c0 > 0, such that, for all t ≥ 0 for which a solution exists with ζ uniformly bounded by some fixed, sufficiently small constant, there holds ζ (t) ≤ c0 (|U0 | L 1 ∩H 3 + ζ (t)2 ). From this result, it follows by continuous induction that, provided |U0 | L 1 ∩H 3
0. By Lemma 4.1, we have t ˙ 2 )(τ ) dτ |U (t)|2H 3 ≤ C|U (0)|2H 3 e−θt + C e−θ(t−τ ) (|U |2L 2 + |δ| 0 1 ≤ C |U (0)|2H 3 + ζ (t)2 (1 + t)− 2 , for some C, C , θ > 0, and so the solution continues so long as ζ remains small, with bound (4.9), at once yielding existence and the claimed sharp L p ∩ H 3 bounds, 2 ≤ p ≤ ∞. Proof of Claim. We must show that each of the quantities 1

|U | L p (1 + s) 2

(1− 1p )

˙ + s) 2 , and , |δ|(1 1

|δ|

is separately bounded by C(|U0 | L 1 ∩H 3 + ζ (t)2 ), for some C > 0, all 0 ≤ s ≤ t, so long as ζ remains sufficiently small. By (4.6)–(4.7) and the triangle inequality, we have |U | L p ≤ Ia + Ib + Ic + Id , ˙ |δ(t)| ≤ IIa + IIb , |δ(t)| ≤ IIIa + IIIb , where Ia is the L p norm of the first integral term in the right-hand side of (4.6), Ib the second term, etc., and similarly IIa is the modulus of the first term in the right-hand side of (4.7), etc. We estimate each term in turn, following the approach of [MaZ1,MaZ4].

38


The linear term Ia satisfies bound Ia ≤ C|U0 | L 1 ∩L p (1 + t)

− 21 (1− 1p )

,

by Proposition 3.23 and Corollary 3.29. Likewise, applying the bounds of Corollary 3.29, we have t 1 Ib ≤ Cζ (t)2 e−η(t−s) (1 + s)− 2 ds 0 1

≤ Cζ (t) (1 + t)− 2 , 2

and (taking q = 2 in the second estimate of Corollary 3.29) t ˙ Ic + I d ≤ C e−η(t−s) (|U | L ∞ + |∂x U | L ∞ + |δ|)|U | L p (s)ds 0 t −3+ 1 ˙ +C (t − s) 4 2 p (|U | L ∞ + |δ|)|U | H 1 (s)ds 0 t − 1 (1− 1p )− 21 ≤ Cζ (t)2 e−η(t−s) (1 + s) 2 ds 0 t 3 −3+ 1 +Cζ (t)2 (t − s) 4 2 p (1 + s)− 4 ds 0

≤ Cζ (t) (1 + t) 2

− 21 (1− 1p )

, 1

IIa ≤ |∂t e(t)| L ∞ |U0 | L 1 ≤ C|U0 | L 1 (1 + t)− 2 , and

t

˙ |∂ y ∂t e(t − s)| L 2 (|U | L ∞ + |δ|)|U | H 1 (s)ds t 3 3 ≤ Cζ (t)2 (t − s)− 4 (1 + s)− 4 ds

IIb ≤

0

0 1

≤ Cζ (t) (1 + t)− 2 , 2

while |U0 | L 1 ≤ C|U0 | L 1 , IIIa ≤ |e(t)| L ∞ y and

t

˙ |∂ y e(t − s)| L 2 (|U | L ∞ + |δ|)|U | H 1 (s)ds t 1 3 ≤ Cζ (t)2 (t − s)− 4 (1 + s)− 4 ds

IIIb ≤

0

0

≤ Cζ (t) . 2

This completes the proof of the claim, establishing (1.21) for p ≥ 2. The remaining bounds 1 ≤ p < 2 then follow by a bootstrap argument as described in [Z2]; we omit the details.


39

5. Bifurcation: Proof of Theorem 1.18 Given two Banach spaces X and Y, we denote by L(X, Y ) the space of linear continuous applications from X to Y, and let L(X ) := L(X, X ). We use (1.22) to denote weighted Sobolev spaces and norms. Let x + := max(0, x). Given a constant η > 0 and a weight function ω > 0, define subspaces of S (R) by L 1η+ := { f, eη(·) f ∈ L 1 }, +

L 1ω := { f, ω f ∈ L 1 },

L 1ω,η+ := { f, ω f ∈ L 1η+ }.

Definition 5.1. Given a constant η > 0 and a weight function ω satisfying (1.23), define the Banach spaces B1 , B2 , X 1 , X 2 ⊂ D (R; C3v × Cz ) by B1 := H 1 , B2 := H 1 ∩ (∂x L 1 × L 1η+ ), X 1 := Hω2 , X 2 := Hω2 ∩ (∂x L 1ω × L 1ω,η+ ), with norms (v, z)B1 := (v, z) H 1 , (∂x v, z)B2 := (∂x v, z) H 1 + v L 1 + eη(·) z L 1 , (v, z) X 1 := (v, z) Hω2 , +

(∂x v, z) X 2 := (∂x v, z) Hω2 + ωv L 1 + ωeη(·) z L 1 . +

In particular, X 2 → X 1 → B1 , with ·B1 ≤ · X 1 ≤ · X 2 , and X 2 → B2 → B1 , with · B1 ≤ · B2 ≤ · X 2 , and the unit ball in X 1 is closed in B1 . ε 5.1. The perturbation equations. If U˜ ε solves (1.2) with initial datum U˜ |t=0 = U¯ ε +U0ε , then the perturbation variable U (ε, x, t) := U˜ ε (x, t) − U¯ ε (x) satisfies

∂t U − L(ε)U = ∂x Qf (ε, U, ∂x U ) + Qr (ε, U ), ε(x) U (ε, x, 0) = U0 .

(5.1)

The nonlinear term Qf satisfies (4.2), while Qr satisfies Lemma 4.2. ε be the eigenfunctions of L(ε) associated with the bifur5.2. Coordinatization. Let ϕ± ε be the corresponding left eigenfunctions. cation eigenvalues γ (ε) ± iτ (ε), and let ϕ˜± ε decay exponentially We know from Sect. 3.1.3 that (γ ±iτ )(ε) ∈ C \ C− ∪ C+ , hence ϕ± ε ∈ H 2 . Let at both −∞ and +∞, in particular, if in (1.23) θ0 is small enough, then ϕ± ω ε ) parallel to span(ϕ˜ ε )⊥ . Decomposing be the L 2 -projection onto span(ϕ± ±

U = u 11 ϕ+ε + u 12 ϕ+ε + u 2 ,

U0ε = a1 ϕ+ε + a2 ϕ+ε + b,

40


ε ) (so that, in particular, where u 11 ϕ+ε + u 12 ϕ+ε and a1 ϕ+ε + a2 ϕ+ε belong to span(φ± u 1 j and a j are real), and coordinatizing as (u 1 , u 2 ), u 1 := (u 11 , u 12 ) ∈ R2 , we obtain after a brief calculation that U solves (5.1) if and only if its coordinates solve the system ⎧ γ (ε) τ (ε) ⎪ ⎪ ∂t u 1 = u + N (ε, u 1 , u 2 ), ⎪ −τ (ε) γ (ε) 1 ⎨ (5.2) ∂t u 2 = (1 − )L(ε)u 2 + (1 − )N (ε, u 1 , u 2 ), ⎪ ⎪ u = a, ⎪ 1|t=0 ⎩ u 2|t=0 = b,

where N (ε, u 1 , u 2 ) := (∂x Qf + Qr )(ε, U¯ ε , U ). Given T0 > 0, there exist ζ0 > 0 and C > 0, such that, if |a|+b Hω2 < ζ0 , the initial value problem (5.2) possesses a unique solution (u 1 , u 2 )(a, b, ε) ∈ C 0 ([0, T0 ], R2 ×Hω2 ) satisfying C −1 |a| − Cb2H 2 ≤ |u 1 (t)| ≤ C(|a| + b2H 2 ), ω

u 2 (t) Hω2 ≤ C(b Hω2 + |a|2 ), ∂(a,b) (u 1 , u 2 )(t)L(R2 ×H 1 ,H 1 ) ≤ C.

ω

(5.3)

(For more details on the initial value problem (5.2) and estimate (5.3), see [TZ2], Prop. 4.2.) 5.3. Poincaré return map. We express the period map (a, b, ε) → bˆ := u 2 (a, b, ε, T ) as a discrete dynamical system bˆ = S(ε, T )b + N˜ (a, b, ε, T ),

(5.4)

where S(ε, T ) := e T (1−)L(ε) is the linearized solution operator in v and T ˜ N (a, b, ε, T ) := S(ε, T − s)(1 − )N (ε, u 1 , u 2 )(s)ds 0

the difference between nonlinear and linear solution operators. Evidently, periodic solutions of (5.2) with period T are fixed points of the period map (equilibria of (5.4)) or, equivalently, zeros of the displacement map (a, b, ε, T ) := (S(ε, T ) − Id)b + N2 (a, b, ε, T ). 5.4. Lyapunov-Schmidt reduction. We now carry out a nonstandard Lyapunov–Schmidt reduction following the “inverse temporal dynamics” framework of [TZ2], tailored for the situation that Id − S(ε, T ) is not uniformly invertible, or, equivalently, the spectrum


41

of (1 − )L(ε) is not bounded away from { jπ/T } j∈Z . In the present situation, (1 − )L(ε) has both a 1-dimensional kernel (a consequence of (H4), see Sect. 1.5) and essential spectra accumulating at λ = 0, and no other purely imaginary spectra, so that Id − S(ε, T ) inherits the same properties; see [TZ2] for further discussion. Our goal, and the central point of the analysis, is to solve (a, b, ε, T ) = 0 for b as a function of (a, ε, T ), eliminating the transverse variable and reducing to a standard planar bifurcation problem in the oscillatory variable a. A “forward” temporal dynamics technique would be to rewrite = 0 as a fixed point map b = S(ε, T )b + N˜ (a, b, ε, T ),

(5.5)

then to substitute for T an arbitrarily large integer multiple j T . In the strictly stable case σ ((1 − )L) ≤ −θ < 0, S(ε, j T )L(X 1 ) < 21 for j sufficiently large. Noting that N˜ is quadratic in its dependency, we would have therefore contractivity of (5.5) with respect to b, yielding the desired reduction. However, in the absence of a spectral gap between σ ((1 − )L) and the imaginary axis, S(ε, j T )L(X 1 ) does not decay, and may be always greater than unity; thus, this naive approach does not succeed. The key idea in [TZ2] is to rewrite = 0 instead in “backward” form b = (Id − S(ε, T ))−1 N˜ (a, b, ε, T ),

(5.6)

then show that (Id − S(ε, T ))−1 is well-defined and bounded on Range N˜ , thus obtaining −1 ˜ ˜ contractivity by quadratic ∞ dependence of N . Since the right inverse (Id − S(ε, T )) N ˜ is formally given by j=0 S(ε, j T ) N this amounts to establishing convergence: a stability/cancellation estimate. Quite similar estimates appear in the nonlinear stability theory, where the interaction of linearized evolution S and nonlinear source N˜ are likewise crucial for decay. The formulation (5.6) can be viewed also as a “by-hand” version of the usual proof of the standard Implicit Function Theorem [TZ2]. Lemma 5.2. Under the assumptions of Theorem 1.18, if the constant η in Definition 5.1 satisfies η < η0 , where η0 was introduced in Corollary 1.8, then N˜ : (a, b, ε, T ) ∈ R2 × X 1 × R2 → N˜ (a, b, ε, T ) ∈ X 2 , is quadratic order, and C 1 as a map from R2 × B1 × R2 to B2 for b X 1 uniformly bounded, with N˜ X 2 ≤ C(|a| + b X 1 )2 , ∂(a,b) N˜ L(R2 ×B1 ,B2 ) ≤ C(|a| + b H 2 ), ∂(ε,T ) N˜ L(R2 ,B2 ) ≤ C(|a| + b H 2 )2 .

(5.7)

Proof. We use the variational bounds of [TZ3] (see Propositions 5 and 6, [TZ3]) and Lemma 4.2. Note that, in 1.23, only ω−1 ∈ L ∞ and 1.23(ii) were used at this point. 5.4.1. Pointwise cancellation estimate. We now develop the key cancellation estimates, adapting the pointwise semigroup methods of [ZH,MaZ3,Z2] to the present case. Our starting point is the inverse Laplace transform representation (3.55). Deforming the contour using analyticity of Gλ across oscillatory eigenvalues λ± (ε) we obtain G = G˜ + O, where ε ε O(x, t; y) := eλ+ (ε)t ϕ+ε (x)ϕ˜+ε (y)tr + eλ− (ε)t ϕ− (x)ϕ˜− (y)tr

42


ε is the sum of the residues of the integrand at λ± (the right- and left-eigenfunctions ϕ± ε are defined in Sect. 5.2). The Green function G˜ is the kernel of the integral operand ϕ˜± ator S(ε, t) defined in Sect. 5.3. Note that, under the assumptions of Theorem 1.18, the Evans function associated with (1 − )L(ε) satisfies (1.20), so that, by Remark 3.28, ˜ Proposition 3.23 applies to G. For ν, ν0 > 0, let be the counterclockwise arc of circle ∂ B(0, r ) (r as in Proposition 3.21) connecting −ν − iν0 and −ν + iν0 . If ν and ν0 > 0 are sufficiently small, then

is entirely contained in the resolvent set of (1 − )L(ε), and G˜ can be decomposed as GI + GII , with 1 GI (ε, x, t; y) := eλt Gλ (ε, x, y) dλ, 2πi

−ν−iν0 −ν+i∞ (5.8) 1 GII (ε, x, t; y) := eλt Gλ (ε, x, y) dλ. P.V. + 2πi −ν−i∞ −ν+iν0

Let SI and SII denote the integral operators with respective kernels GI and GII , so that S = SI + SII , and let := (−¯ε0 , ε¯ 0 ) × (0, +∞), for some ε¯ 0 > 0. Remark 5.3. The contour being contained in the resolvent set of L , the elementary bound holds: |∂ y Gλ | ≤ Ce−θν |x−y| ,

λ ∈ ,

for some θν > 0 depending on ν. See for instance Proposition 4.4, [MaZ3]. Our treatment of the high-frequency term follows [TZ3]: Lemma 5.4. Under the assumptions of Theorem 1.18, the sequence of operators with N kernel n=0 GII (ε, nT ) is absolutely convergent in L(H 1 ), uniformly in (ε, T ) ∈ . Proof. Starting from the description of the resolvent kernel given in Proposition 3.21, we find by the same inverse Laplace transform estimates that give terms H and R in Proposition 3.23, that the high-frequency resolvent kernel GII , defined in (5.8), may be expressed as GII = Ce−θ(|x−y|+t) + hτx+st δ,

(5.9)

where C and its space-time derivatives are bounded, θ > 0, and hτx+st δ is a generic hyperbolic term; in particular h has the form (3.61) and satisfies (3.64)(i). The lemma follows. ˜ Its fluid terms are handled as in Next we turn to the low-frequency component of G. [TZ3]: Lemma 5.5. Under the assumptions of Theorem 1.18, the sequence of operators with N kernel n=0 GI (ε, nT ) converges in L(∂x L 1 , H 1 ), uniformly with respect to (ε, T ) ∈ . Proof. We argue as in the proof of Proposition 3 of [TZ3]. Let f ∈ L 1 . By (5.8), N −1 n=0 ∂ y GI f decomposes into I − II N , where 1 1 ∂ y Gλ f dλ dy, I= 2iπ R 1 − eλT eN T λ 1 II N = ∂ y Gλ f dλ dy. 2iπ R 1 − eλT For small ν and λ ∈ , (1 − eλT )−1 = λ−1 T −1 (1 + O(λ)).


43

The boundary term I is independent of N and is seen to belong to H 1 by Remark 5.3. By (3.56)–(3.60), λ−1 ∂ y Eλ and λ−1 ∂ y Sλ have the same form as Eλ and Sλ . By Proposition 3.21, λ−1 ∂ y Rλ behaves like the sum of Rλ and a pole term of form λ−1 e−θ|x−y| . Hence, by the same Riemann saddle-point estimates used to bound G in Proposition 3.23, we find that eλN T ∂ G dλ = (E + S + R) (ε, N T ), (5.10) λT y λ

1−e up to a constant (independent of N ) term of the form Ce−θ|x−y| , where the space-time derivatives of C are uniformly bounded. This constant term satisfies the same bound as term I. In (5.10), E, S, R denote generic excited, scattered and residual terms of form (3.58), (3.59)–(3.60) and (3.64)(ii). By dominated convergence, H1 − lim E(ε, N T ) f (y) dy N →∞ R

exists and is equal to a sum of terms of the form f (y) dy. C(ε, T )(U¯ ε )

(5.11)

Besides, by (3.59)–(3.60) and (3.64), " " " " " (S + R)(ε, N T ) f (y) dy " " "

(5.12)

R

R

H1

≤ C(N T )− f rac14 f L 1 .

This proves convergence in H 1 of the sequence II N . We examine finally the contribution to the series n S(ε, nT ) of the new (not present in [TZ3]), reactive terms. Lemma 5.6. Under the assumptions of Theorem 1.18, the sequence of operators with N kernels n=0 GI (ε, nT )+4 is absolutely convergent in L(L 1η+ , H 1 ), uniformly with respect to (ε, T ) ∈ . Proof. Let f ∈ L 1 . By (5.8), Proposition 3.23 and Corollary 3.26, the low-frequency kernel GI satisfies + + e−ηy GI (ε, t)+4 f (y) dy = e−ηy (S + R) (ε, t)+4 f (y) dy, (5.13) R

R

and, by Corollary 3.27, " " " " " e−ηy + (S + R)(ε, t)+ f (y) dy " 4 " " R

1

H1

≤ C(1 + t 4 )e−θ1 t f L 1 ,

and the upper bound defines for t = N T an absolutely converging series in H 1 .

44


From Lemmas 5.4, 5.5 and 5.6 and the fact that S(ε, T ) ∈ L(B1 ), for all (ε, T ) ∈ , we can conclude that, under the assumptions of Theorem 1.18, the operator Id − S(ε, T ) has a right inverse S(ε, nT ) : B2 → B1 , (Id − S(ε, T ))−1 := n≥0

that belongs to L(B2 , B1 ), locally uniformly in (ε, T ) ∈ . We will need the following regularity result for the right inverse: Lemma 5.7. Under the assumptions of Theorem 1.18, the operator (Id − S(ε, T ))−1 is C 1 in (ε, T ) ∈ , with respect to the L(B2 , B1 ) norm on B2 . Proof. Note that, by (3.55), ∂t G has kernel λGλ ; in particular, the small λ (low-frequency) estimates of the proofs of Lemmas 5.5 and 5.6 imply the convergence of the N sequence n=0 ∂T SI (ε, nT ) in L(B2 , B1 ). The contribution of ∂T GII (ε, nT ) is handled as in Lemma 5.4, by (5.9) and (3.64)(i) with k = 1. Bounds for ε-derivatives are handled as in [TZ3], using either the variational equation (L − λ)∂ε Gλ = −(∂ε L)Gλ , or the ∂ε G bounds of Proposition 3.11 from [TZ2]. Note that the ε-derivative bound (5.7)(iii) is stated on a proper subspace of B1 , namely X 1 . In this respect, the following lemma, asserting boundedness of the right inverse on X 2 → B2 , in L(X 2 , X 1 ) norm, is key to the reduction procedure of the following Section. (See Remark 5.12.) Lemma 5.8. Under the assumptions of Theorem 1.18, (Id − S(ε, T ))−1 belongs to L(X 2 , X 1 ), for all (ε, T ) ∈ . Proof. The convolution bound " " " 1 " −θ|x−y| "ω 2 e f (y) dy " " " R

L2

≤ C min f L 2ω , f L 1ω ,

(5.14)

1

where C depends on ω 2 e−θ|·| L 1 ∩L 2 , holds by (1.23)(i) and (1.23)(iii). It implies that the contributions of GII , of I and of the constant pole term in II N (see the proofs of Lemmas 5.4 and 5.5) are all bounded in L(X 1 ). The scattered and residual terms in II contribute nothing to the limit, by (5.12). We use again Corollary 3.27 to handle the contribution of the reactive term. In (3.66), there are two terms in the upper bound for (S + R)+4 . The first term is handled by (5.14), and the second by " " " 1 " " 1 " 2 " −|x−y|2 /Mt " ≤ e−θt " 2 2 e −|x−y| /Mt " ω ω e−θt " e f (y) dy f L 1ω , (5.15) " " " 2 R

L2

noting that 1

e−θt ω 2 e−|x−y|

2 /Mt

L

1 1 2 2 L 2 ≤ Ce−θ1 t ω 2 e−|·| /Mt L 2 + ω 2 e−|·| /Mt L 2 |x| 0 small enough, for all 0 ≤ t ≤ T.

(5.19)


47

Likewise, any periodic solution of (5.2) originating in C, since it necessarily satisfies = 0, must originate from data (a, b) of the form (5.19). ˜ ε, T, α), Defining b ≡ β(a, ˆ and recalling invariance of C under flow (5.2), we may view v(t) as a multiple u 2 (x, t) = c(a, ε, T, α, ˆ x, t)u 1 (t)

(5.20)

of u 1 (t), where c is bounded, Lipschitz in all arguments, and C 1 away from a = 0. Substituting into (5.2)(i), we obtain a planar ODE, γ (ε) τ (ε) ∂t u 1 = u + M(u 1 , ε, T, t, α, ˆ a) −τ (ε) γ (ε) 1 in approximate Hopf normal form, with nonlinearity M := N now nonautonomous and depending on the additional parameters (T, α, ˆ a), but, by (4.2) and (4.4), still satisfying the key bounds |M|, |∂ε,T,αˆ M| ≤ C|u 1 |2 ; |∂a,w M| ≤ C|u 1 |

(5.21)

along with planar bifurcation criterion (1.24). From (5.21), we find that M is C 1 in all arguments, also at a = 0. By standard arguments (see, e.g., [HK,TZ1]), we thus obtain a classical Hopf bifurcation in the variable u 1 with regularity C 1 , yielding existence and uniqueness up to time-translates of a 1-parameter family of solutions originating in C, indexed by r and δ with r := a1 and (without loss of generality) a2 ≡ 0. Bound (1.25) is a consequence of (5.3)(i) and (5.20). Finally, in order to establish uniqueness up to spatial translates, we observe, first, that, by dimensional considerations, the one-parameter family constructed must agree with the one-parameter family of spatial translates, and second, we argue as in [TZ2] that any periodic solution has a spatial translate originating in C, yielding uniqueness up to translation among all solutions and not only those originating in C; see Proposition 2.20 and Corollary 2.21 of [TZ2] for further details. 6. Nonlinear Instability: Proof of Theorem 1.19 We describe a nonlinear instability result in general setting. Consider ∂t U = LU + ∂x N (U ) + R(U ),

(6.1)

well-posed in H s , where L = ∂x (B∂x U ) + ∂x (AU ) + GU, and |N (U )|, |R(U )| ≤ C|U |2 for |U | ≤ C. Suppose that L has a conjugate pair of simple unstable eigenvalues λ± = γ ± iτ, γ > 0, and the rest of the spectrum is neutrally stable, without loss of generality e(1−)Lt ≤ Ct, where is the projection onto the eigenspace associated with λ± . Coordinatizing similarly as in Sect. 5 by U (x, t) = u 11 ϕ1 (x) + u 12 ϕ2 (x) + u 2 (x, t),

48


where ϕ j = O(e−θ|x| ) are eigenfunctions of L, denote r (t) := |u 1 |(t). Then, so long as |U | H s ≤ C , we have existence (by variation of constants, standard continuation) of solutions of (6.1) in H s , with estimates r = γ r + O( )|U |, u 2 = (1 − )Lu 2 + O( )|U |

(6.2)

in L 2 . We shall argue by contradiction. That is, using (6.2), we shall show, for C > 0 fixed, > 0 sufficiently small, and |u 2 (0)| H 1 ≤ Cr (0), that eventually r (t) ≥ , no matter how small r (0) is, or equivalently |U | H s (0). This, of course entails nonlinear instability. Define α(x, t) := u 2 (x, t)/r (t). Then, α =

r u 2 − u 2 r u 2 u2 r − , = r2 r r r

yielding after some rearrangement the equation α = ((1 − )L − γ )α + O( (e−θ|x| + |α| + |α|2 )).

(6.3)

From (6.3) and standard variation of constants/contraction mapping argument, we find that |α(t)| H 1 remains less than or equal to C|α(t0 )| H 1 for t − t0 small. By variation of constants and the semigroup bound |e((1−)L−γ )t | H 1 →H 1 ≤ Ce−γ t (note: γ is scalar so commutes with (1 − )L), we obtain δ(t) ≤ C(|α(0)| H 1 + (1 + δ(t)2 )), for δ(t) := sup0≤τ ≤t |α(τ )| H 1 . So long as δ remains less than or equal to unity and C ≤ 21 , this yields δ(t) ≤ 2C(|α(0)| H 1 + ), and thus 1 δ(t) ≤ 2C|α(0)| H 1 + . 2 Substituting into the radial equation, we obtain r ≥ (γ − (1 + δ) )r, yielding exponential growth for sufficiently small. In particular, r ≥ C for some time, and thus |U | H 1 ≥ , a contradiction. We may conclude, therefore, that |U | L 2 eventually grows larger than any , no matter how small the initial size r (0), and thus we may conclude instability of the trivial solution U ≡ 0. Taking now (6.1) to be the perturbation equations about a strong detonation profile, we obtain the result of nonlinear instability of the background profile U¯ . Remark 6.1. In the easier case of a single, real eigenvalue, the scalar, w equation, would play the role of the radial equation here. This case is subsumed in our analysis as well. Acknowledgements. Thanks to Björn Sandstede and Arnd Scheel for their interest in this work and for stimulating discussions on spatial dynamics and bifurcation in the absence of a spectral gap. Thanks to Gregory Lyng for pointing out reference [Ch]. B.T. thanks Indiana University for their hospitality during the collaborative visit in which the analysis was carried out. B.T. and K.Z. separately thank the Ecole Polytechnique Fédérale de Lausanne for their hospitality during two visits in which a substantial part of the analysis was carried out.


49

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Communicated by P. Constantin


Communications in


Critical Measures, Quadratic Differentials, and Weak Limits of Zeros of Stieltjes Polynomials A. Martínez-Finkelshtein1,2 , E. A. Rakhmanov3 1 Department of Statistics and Applied Mathematics, University of Almería,

04120 Almeria, Spain. E-mail: [email protected]

2 Instituto Carlos I de Física Teórica y Computacional,

Granada University, 18071 Granada, Spain

3 Department of Mathematics, University of South Florida, Tampa, FL 33620, USA.

E-mail: [email protected] Received: 6 April 2009 / Accepted: 25 July 2010 Published online: 8 January 2011 – © Springer-Verlag 2011

Abstract: We investigate the asymptotic zero distribution of Heine-Stieltjes polynomials – polynomial solutions of second order differential equations with complex polynomial coefficients. In the case when all zeros of the leading coefficients are all real, zeros of the Heine-Stieltjes polynomials were interpreted by Stieltjes as discrete distributions minimizing an energy functional. In a general complex situation one deals instead with a critical point of the energy. We introduce the notion of discrete and continuous critical measures (saddle points of the weighted logarithmic energy on the plane), and prove that a weak-* limit of a sequence of discrete critical measures is a continuous critical measure. Thus, the limit zero distributions of the Heine-Stieltjes polynomials are given by continuous critical measures. We give a detailed description of such measures, showing their connections with quadratic differentials. In doing that, we obtain some results on the global structure of rational quadratic differentials on the Riemann sphere that have an independent interest. The problem has a rich variety of connections with other fields of analysis; some of them are briefly mentioned in the paper. Contents 1. Generalized Lamé Equation . . . . . . . . . . . . . . . . . . . . . . 2. Discrete and Continuous Extremal Measures . . . . . . . . . . . . . 3. Discrete and Continuous Critical Measures . . . . . . . . . . . . . . 4. Rational Quadratic Differentials on the Riemann Sphere in a Nutshell 5. Critical Measures in the Field of a Finite System of Fixed Charges . . 6. Critical Measures and Extremal Problems . . . . . . . . . . . . . . . 7. Weak Limit of Zeros of Heine-Stieltjes Polynomials . . . . . . . . . 8. Heun’s Differential Equation ( p = 2) . . . . . . . . . . . . . . . . . 9. General Families of A-Critical Measures . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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54 56 59 64 67 81 87 90 101 108

54

A. Martínez-Finkelshtein, E. A. Rakhmanov

1. Generalized Lamé Equation Let us start with a classical problem more than 125 years old. Given a set of pairwise distinct points fixed on the complex plane C, A = {a0 , a1 , . . . , a p },

(1.1)

( p ∈ N), and two polynomials, A(z) =

p

(z − ai ),

B(z) = αz p + lower degree terms ∈ P p , α ∈ C, (1.2)

i=0

where we denote by Pn the set of all algebraic polynomials of degree ≤ n, we are interested in the polynomial solutions of the generalized Lamé differential equation (in algebraic form), A(z) y (z) + B(z) y (z) − n(n + α − 1)Vn (z) y(z) = 0,

(1.3)

where Vn is a polynomial (in general, depending on n) of degree ≤ p − 1; if deg V = p − 1, then V is monic. An alternative perspective on the same problem can be stated in terms of the second order differential operator L[y](z) = A(z) y (z) + B(z) y (z), def

and the associated generalized spectral problem (or multiparameter eigenvalue problem, see [97]), L[y](z) = n(n + α − 1)Vn (z) y(z), n ∈ N,

(1.4)

where Vn ∈ P p−1 is the “spectral polynomial”. Special instances of Eq. (1.3) are well known. For instance, p = 1 corresponds to the hypergeometric differential equation. Case p = 2 was studied by Lamé in the 1830s in the special setting B = A /2, a j ∈ R, and a0 + a1 + a2 = 0, in connection with the separation of variables in the Laplace equation using elliptical coordinates (see e.g. [100, Ch. 23]). For the general situation of p = 2 we get Heun’s equation, which still attracts interest and poses open questions (see [76]). Recently, Eq. (1.3) has also found other applications in studies as diverse as the construction of ellipsoidal and sphero-conal h-harmonics of the Dunkl Laplacian [98,99], the quantum asymmetric top [1,17,37], or certain quantum completely integrable systems called generalized Gaudin spin chains [40], and their thermodynamic limits. Heine [41] proved that for every n ∈ N there exist at most n+ p−1 σ (n) = (1.5) n different polynomials Vn such that (1.3) (or (1.4)) admits a polynomial solution y = Q n ∈ Pn . These particular Vn are called Van Vleck polynomials, and the corresponding polynomial solutions y = Q n are known as Heine-Stieltjes (or simply Stieltjes) polynomials. Heine’s theorem states that if the polynomials A and B are algebraically independent (that is, they do not satisfy any algebraic equation with integer coefficients) then for any n ∈ N there exist exactly σ (n) Van Vleck polynomials Vn , their degree is exactly p − 1,

Asymptotic Zero Distribution of Heine-Stieltjes Polynomials

55

and for each Vn Eq. (1.3) has a unique (up to a constant factor) solution y of degree n. The condition of algebraic independence of A and B is sufficient but not necessary. It should be noted that the original argument of Heine is far from clear, and even Szeg˝o [92] cites his result in a rather ambiguous form. Recently significant research on the algebraic aspects of this theory has been carried out by B. Shapiro in [82], and we refer the reader to his work for further details. In particular, it has been proved in [82] that for any polynomials A and B like in (1.2) there exists N ∈ N such that for any n ≥ N , there exist σ (n) Van Vleck polynomials Vn of degree exactly p − 1 such that (1.3) has a polynomial solution of degree exactly n. Stieltjes discovered an electrostatic interpretation of zeros of the polynomials discussed in [41], which attracted common attention to the problem. He studied the problem (1.3) in a particular setting, assuming that A ⊂ R and that all residues ρk in B(x) ρk = A(x) x − ak p

(1.6)

k=0

are strictly positive (which is equivalent to the assumption that the zeros of A alternate with those of B and that the leading coefficient of B is positive). He proved in [90] (see also [92, Theorem 6.8]) that in this case for each n ∈ N there are exactly σ (n) different Van Vleck polynomials of degree p − 1 and the same number of corresponding HeineStieltjes polynomials y of degree n, given by all possible ways how the n zeros of y can be distributed in the p open intervals defined by A (see Sect. 2). Further generalizations of the work of Heine and Stieltjes followed several paths; we will mention only some of them. First, under Stieltjes’ assumptions (A ⊂ R and ρk > 0), Van Vleck [95] and Bôcher [15] proved that the zeros of each Vn belong to the convex hull of A (see also the work of Shah [78–81]). Pólya [70] showed that this is true for A ⊂ C if we keep the assumption of positivity of the residues ρk . Marden [58], and later, Al-Rashed, Alam and Zaheer (see [2,3,101,102]) established further results on location of the zeros of the Heine-Stieltjes polynomials under weaker conditions on the coefficients A and B of (1.3). An electrostatic interpretation of these zeros in cases when A ⊂ R and some residues ρk are negative has been studied by Grünbaum [39], and by Dimitrov and Van Assche [25]. For some interlacing properties, see e.g. [18]. We are interested in the asymptotic regime (so called semiclassical asymptotics) when n (the degree of the Heine-Stieltjes polynomials) tends to infinity. The first general result in this direction, based precisely on the Stieltjes model, is due to Martínez-Finkelshtein and Saff [61]. There the limit distribution of zeros of Heine-Stieltjes polynomials has been established in terms of the traditional extremal problem for the weighted logarithmic energy on a compact set of the plane. The main goal of this paper is to consider the weak-* asymptotics of the HeineStieltjes and Van Vleck polynomials in the general setting of A ⊂ C and ρk ∈ C, which leads to a very different electrostatic problem - equilibrium problem in the conducting plane (with a finite exceptional set of points). It is essentially known that zeros of Heine-Stieltjes polynomials present a discrete critical measure – saddle point of the discrete energy functional. A continuous analogue of this notion leads to a concept of “continuous” critical measure, i.e. critical point of the usual energy functional defined on Borel measures with respect to a certain class of local variations. We prove (Sect. 7) that the weak limit of discrete critical measures is a continuous critical measure (as the number of atoms or mass points tends to infinity). Thus, discrete critical measures are limit distributions of zeros of the Heine-Stieltjes polynomials.

56


To complete the description of the limit zero distributions of these polynomials we have to study more deeply the set of continuous critical measures. The problem, rather complex, is connected to many other classical problems of analysis, and has potentially a large circle of applications. In Sect. 6 we mention a few connections, in particular, to minimal capacity problem and its generalizations. In Sect. 5 we characterize critical measures in terms of trajectories of a (closed) rational quadratic differential on the Riemann sphere; for completeness of reading we summarize basic results on quadratic differentials in Sect. 4. Further investigation of such differentials in carried out in Sects. 8 (case p = 2) and 9 (general case). In the following two sections, 2 and 3, we discuss in some detail the concepts of the discrete and continuous equilibrium. 2. Discrete and Continuous Extremal Measures 2.1. Stieltjes electrostatic model: discrete equilibrium. We denote by Mn the class of uniform discrete measures on C, n def def δz k , z k ∈ C , and M = Mn , Mn = n≥1

k=1

where δx is a unit mass (Dirac delta) at x. With any polynomial P(z) = we associate its zero counting measure ν(P) =

n

n

j=1 (z

− ζj)

δζ j ∈ Mn ,

j=1

where the zeros

are counted according to their multiplicity. For μ = nk=1 δζk ∈ M we define its (discrete) energy 1 def , E(μ) = log |ζi − ζ j | i= j

(if two or more ζ j ’s coincide, then E(μ) = +∞). Additionally, given a real-valued function (external field) ϕ, finite at supp(μ), we consider the weighted energy Eϕ (μ) = E(μ) + 2 def

n

ϕ(ζk ).

(2.1)

k=1

In the above mentioned paper [90] Stieltjes introduced the following extremal problem. For fixed subset A = {a0 , . . . , a p } ⊂ R, a0 < · · · < a p , values ρk ≥ 0, k = p 0, 1, . . . , p, and an arbitrary vector n = (n 1 , . . . , n p ) ∈ Z+ (where Z+ = N∪{0}), define def p |n| = n 1 + · · · + n p , j = [a j−1 , a j ], j = 1, . . . , p, and = ∪ j=1 j = [a0 , a p ]. Consider the class of discrete measures def M|n| ( , n) = μ ∈ M|n| : supp(μ) ⊂ , μ( j ) = n j , j = 1, . . . , p , (2.2) and the external field ϕ(x) = Re ( (x)), (x) =

p ρj j=0

2

log

1 . x − aj

(2.3)


57

We seek a measure μ∗ = μ∗ (n) minimizing the weighted energy (2.1) in the class M|n| ( , n): (2.4) Eϕ (μ∗ ) = min Eϕ (μ) : μ ∈ M|n| ( , n) . In other words, we place n j unit electric charges on the conductor j and look for the equilibrium position of such a system of charges in the external field ϕ, if the interaction obeys the logarithmic law. Stieltjes proved that the global minimum (2.4) provides the only equilibrium position, and that the zeros of the solution y = Q n of (1.3) are exactly points of the support of the extremal measure μ∗ in (2.4): ν(Q n ) = μ∗ . Actually, μ∗ provides also the unique component-wise or point-wise minimum of Eϕ (“Nash-type” equilibrium). The Stieltjes equilibrium problem (2.4) is a constrained one: the constraints are embedded in the definition of the class M|n| ( , n). A classical non-constrained version of the same problem leads to the (weighted) Fekete points. Given a compact ⊂ C and def n ∈ N, we want to find μ∗ ∈ Mn ( ) = {μ ∈ Mn : supp(μ) ⊂ } with Eϕ (μ∗ ) = min Eϕ (μ) : μ ∈ Mn ( ) . Stieltjes’ model for the hypergeometric case ( p = 1) provides the well known electro(α,β) static interpretation of the Jacobi polynomials. Zeros of the Jacobi polynomials Pn are β+1 α+1 1 1 also weighted Fekete points for = [−1, 1] and ϕ(x) = 2 log |x−1| + 2 log |x+1| . Similarly, zeros of Laguerre and Hermite polynomials are weighted Fekete points for 1 x x2 = [0, +∞), ϕ(x) = α+1 2 log |x| + 2 and = R, ϕ(x) = 2 , respectively. It was pointed out in [42] that zeros of general orthogonal polynomials with respect to a measure on R may be interpreted as weighted Fekete points with an external field ϕ = ϕn in general depending on the degree n. Besides its elegance, the electrostatic model just described allows to establish monotonicity properties of the zeros of the Heine-Stieltjes polynomials as a function of the parameters ρk . Furthermore, the minimization problem for the discrete energy it is based upon, admits substantial generalizations (one of them is the subject of the present paper). The problem of the limit distribution of the discrete extremal points as n → ∞ leads to the corresponding continuous energy problems. 2.2. Extremal problem for Borel measures: continuous equilibrium. We denote by M (resp., MR ) the set of all finite positive (resp., real) Borel measures μ with compact support supp(μ) ⊂ C. Hereafter, |μ| stands for the total variation of μ ∈ MR , and def μ = |μ|(C). For n ∈ N, let Mn = {μ ⊂ M : μ = n} be the set of positive Borel measures with total mass n on C. With every measure μ ∈ MR we can associate its (continuous) logarithmic energy

1 def dμ(x)dμ(y). (2.5) E(μ) = log |x − y| Given the external field ϕ ∈ L 1 (|μ|), we consider also the weighted energy

def E ϕ (μ) = E(μ) + 2 ϕ dμ.

(2.6)

58


If is a subset of C, we denote by M( ) (resp., MR ( )) the restriction of the corresponding families to measures supported on . Again, a standard extremal problem of the potential theory is to seek for a global minimizer λ ,ϕ ∈ M1 ( ) such that def E ϕ (λ ,ϕ ) = ρ = min E ϕ (μ) : μ ∈ M1 ( ) .

(2.7)

It is well known that under certain conditions on ϕ this minimizer λ ,ϕ exists and is unique; it is called the equilibrium measure of in the external field ϕ, see e.g. [77] for further details. For ϕ ≡ 0, measure λ = λ ,0 is also known as the Robin measure of . In terms of the extremal constant ρ we can also define the weighted (logarithmic) capacity of , capϕ ( ) = e−ρ . For ϕ ≡ 0 we simplify notation writing cap( ) instead of cap0 ( ). If cap( ) = 0, then is a polar set. Observe that E(μ) = +∞ for any μ ∈ M, so that any finite set is polar. There is a number of properties characterizing the equilibrium measure λ ,ϕ . For instance, if we define the logarithmic potential of μ ∈ MC by

1 def dμ(t), U μ (z) = log |z − t| then up to a polar subset of , U

λ ,ϕ

(z) + ϕ(z)

= ρ ∗ , if z ∈ supp(λ ,ϕ ), ≥ ρ ∗ , if z ∈ ,

(2.8)

where ρ ∗ is a constant related to ρ and ϕ. Furthermore, if and ϕ are sufficiently regular, min U λ ,ϕ (z) + ϕ(z) = max min U μ (z) + ϕ(z) . (2.9) z∈

μ∈M1 ( ) z∈

This max- min property is a basis for applications of the equilibrium measure in the asymptotic theory of extremal (in particular, orthogonal) polynomials, see [34,66,73], and also the monograph [77]. Like for the discrete measures, we will consider general external fields of the form ϕ(z) = Re (z), where is analytic, but in general multivalued. What we require in the sequel is that is holomorphic in C\A, allowing further construction below. Remark 2.1. Further generalizations of this construction can be obtained either considering several measures on respective sets interacting according to a certain law (vector equilibrium) [35], or including additional constraints. For instance, prescribing an upper bound on the density of the extremal measure on in (2.7) we obtain the so-called constrained equilibrium [26,74], relevant for the asymptotic description of polynomials of discrete orthogonality. Another way is to impose in (2.7) the size of μ on each compodef nent of , such as it was done in [61]: if A = {a0 , . . . , a p } ⊂ R, a0 < · · · < a p , j = p [a j−1 , a j ], j = 1, . . . , p, = ∪ j=1 j = [a0 , a p ], and N is the standard simplex in R p−1 , p N = θ = (θ1 , . . . , θ p ) : θi ≥ 0, i = 1, . . . , p, and θi = 1 , i=1


59

then for each θ = (θ1 , . . . , θ p ) ∈ N we can consider the global minimum of the weighted energy E ϕ (·) restricted to the class def M1 ( , θ ) = μ ∈ M1 : supp(μ) ⊂ , μ( j ) = θ j , j = 1, . . . , p − 1 . Again, for ϕ like in (2.3) with ρ j ≥ 0 there exists a unique minimizing energy, λ ,ϕ (θ). Remark 2.2. It should be mentioned that a characterization of the weighted Fekete points on the real line and its continuous limit were used in [21] to prove new results on the support of an equilibrium (i.e. extremal) measure in an analytic external field on R. 2.3. Relation between discrete and continuous equilibria. The transfinite diameter of a compact set is defined by the limit process when the number of Fekete points tends to infinity. It was Pólya who proved the remarkable fact that the transfinite diameter of is equal to its capacity. Fekete observed further that the normalized counting measure of Fekete points converges to the equilibrium (Robin) measure of . For the weighted analogue of this result, see [77, Ch. III]. The connection between the discrete and continuous equilibria allowed to use the Stieltjes model in [61] in order to obtain in this situation the limit distribution of zeros p of Heine-Stieltjes polynomials. Namely, if for each vector n = (n 1 , . . . , n p ) ∈ Z+ ∗ we consider the discrete extremal measure μ (n) introduced in (2.4), and assume that |n| → ∞ in such a way that each fraction n k /|n| has a limit, nj = θj, |n|→∞ |n| lim

j = 1, . . . , p,

then μ∗ (n)/|n| weakly converges to the equilibrium measure λ ,0 (θ), = [a0 , a p ], θ = (θ1 , . . . , θ p ), defined in the previous section. In a certain sense, this can be regarded as a generalization of the just mentioned classical result of Fekete. 3. Discrete and Continuous Critical Measures According to a well-known result of Gauss, there are no stable equilibrium configurations (i.e. local minima of the energy) in a conducting open set under a harmonic external field. Unstable equilibria usually do not attract much attention from a point of view of physics. However, as we will show further, they constitute a rich and relevant object that appears naturally in many fields of analysis. We introduce now the concept that plays the leading role in this paper: the family of measures providing saddle points for the logarithmic energy on the plane, with a separate treatment of the discrete and continuous cases. 3.1. Discrete critical measures. We start with the following definition: Definition 3.1. Let be a domain on C, A ⊂ a subset of zero capacity, and ϕ be a C 1 real-valued function in \A. A measure μ=

n k=1

δζk ∈ Mn , ζi = ζ j

for i = j,

(3.1)

60


is a discrete (A, ϕ)-critical measure in , if supp(μ) ⊂ C\A, and for the weighted discrete energy Eϕ (μ) = Eϕ (ζ1 , . . . , ζn ) we have grad Eϕ (ζ1 , . . . , ζk ) = 0,

(3.2)

or equivalently, ∂ Eϕ (ζ1 , . . . , z, . . . ζn )|z=ζk = 0, k = 1, . . . , n, ∂z

1 ∂ = ∂z 2

∂ ∂ −i ∂x ∂y

More generally, if ϕ = Re , where is an analytic (in general, multivalued) function in with a single-valued derivative , then this definition does not need any modification. In the sequel we omit the mention to if = C. The following proposition is just a reformulation of Eq. (1.3) in this new terminology: Proposition 3.1. Assume that A = {a0 , a1 , . . . , a p }, p ∈ N, is a set of pairwise distinct points on C, and the external field ϕ is given by (2.3). Then μ=

n

δζk ∈ Mn , ζi = ζ j

for i = j,

(3.3)

k=1

supported on C\A, is a discrete (A, ϕ)-critical measure if and only if there exists a polynomial Vn ∈ P p−1 such that y(z) = yn (z) = nk=1 (z − ζk ) is a solution of the differential equation (1.3), with B(x) ρk = . A(x) x − ak p

k=0

In other words, discrete (A, ϕ)-critical measures with external field generated by complex charges fixed at A correspond precisely to zeros of Heine-Stieltjes polynomials. Proof. A straightforward computation shows that for z = w, 2

∂ 1 log |z − w| = . ∂z z−w

Hence, 2

∂ 1 ∂ E(ζ1 , . . . , ζn ) = −2 log |ζi − ζ j | = − . ∂ζk ∂ζk ζk − ζ j i= j

j=k

On the other hand, the multivalued function ϕ has a single-valued derivative given by (see (2.3)) ρj 1 ∂ ϕ(z) = − = (z). ∂z 2 z − aj p

2

j=0

Thus, using the notation from (1.6), we can rewrite condition (3.2) as ⎛ ⎞ 1 1 B 2⎝ − 2 (ζk )⎠ = 2 + (ζk ) = 0, k = 1, . . . , n, ζk − ζ j ζk − ζ j A j=k

j=k

(3.4)


and with y(z) = def

n

61

i=1 (z

− ζi ) this identity takes the form y B (ζk ) = 0, k = 1, . . . , n. + y A

(3.5)

As a consequence, polynomial A(z) y (z) + B(z)y (z) ∈ Pn+ p−1 , n ∈ P p−1 such that is divisible by y, so there exists a polynomial V n (z)y(z), A(z) y (z) + B(z)y (z) = V which concludes the proof. In the sequel we will make use of the following uniform boundedness of the supports of the discrete critical measures, corresponding to a sequence of external fields of the form ϕn = Re n , n (z) = −

p ρk (n) k=0

2

log(z − ak ),

(3.6)

where ρk (n) ∈ C. Proposition 3.2. Let μn ∈ Mn , n ∈ N, be a discrete (A, ϕn )-critical measure corresponding to an external field (3.6). If lim inf Re n

then

n

p ρk (n)

n

k=0

1 >− , 2

(3.7)

supp(μn ) is bounded in C.

In other words, if we assume that in (1.3) the coefficient B = Bn may depend on n, but Bn /n is bounded (in such a way that (3.7) holds), then the zeros of the Heine-Stieltjes polynomials are also uniformly bounded.

Proof. Let μn = nk=1 δζk (n) ∈ Mn , and assume that |ζ1 (n)| ≥ · · · ≥ |ζn (n)|. Since |ζ1 (n)| > 0, by (3.4), n j=2

1 ζ1 (n) =− ρk (n) . 1 − ζ j (n)/ζ1 (n) ζ1 (n) − ak p

k=0

But

|ζ j (n)/ζ1 (n)| ≤ 1

⇒

1 Re 1 − ζ j (n)/ζ1 (n)

so that 1 ζ1 (n) 1 Re ρk (n) ≤− . n−1 ζ1 (n) − ak 2 p

k=0

≥ 1/2,

62


Hence, if ζ1 (n) → ∞ along a subsequence of N, then lim inf Re n

p ρk (n) k=0

n

1 ≤− , 2

which contradicts our assumptions. Remark 3.1. It was proved in [82] that for a fixed ϕ of the form (3.6) (that is, ρk (n) ≡ ρk , k = 0, . . . , p), the zeros of the Heine-Stieltjes polynomials accumulate on the convex hull of A. Remark 3.2. Condition (3.7) is in general necessary for the assertion of Proposition 3.2. Indeed, for p = 0, a0 = 0, and ϕn (z) =

n−1 log |z|, 2

any discrete uniform measure supported at the scaled zeros of unity, that is, μn =

n

δζk (n) ∈ Mn , ζk (n) = ζn e2πik/n , ζn ∈ C\{0},

k=1

is (A, ϕn )-critical, which is easily established using (3.4) and (3.5). Obviously, for ζn → ∞ the support of μn is not uniformly bounded in n. 3.2. Continuous critical measures. Unlike in the discrete case, we provide now a variational definition for the continuous critical measure. Any smooth complex-valued function h in the closure of a domain generates a local variation of by z → z t = z + t h(z), t ∈ C. It is easy to see that z → z t is injective for small values of the parameter t. The transformation above induces a variation of def sets e → et = {z t : z ∈ e}, and (signed) measures: μ → μt , defined by μt (et ) = μ(e); in the differential form, the pullback measure μt can be written as dμt (x t ) = dμ(x). Definition 3.2. Let be a domain on C, A ⊂ a subset of zero capacity, and ϕ be a C 1 real-valued function in \A. We say that a signed measure μ ∈ MR () is a continuous (A, ϕ)-critical if for any h smooth in \A such that h |A ≡ 0 , E ϕ (μt ) − E ϕ (μ) d E ϕ (μt )|t=0 = lim = 0. t→0 dt t

(3.8)

Furthermore, if ϕ = Re , where is an analytic (in general, multivalued) function in with a single-valued derivative , then this definition does not need any modification. In what follows we will always mean by an (A, ϕ)-critical measure the continuous one, satisfying Definition 3.2. Furthermore, in order to simplify notation, we speak about an A-critical measure meaning a continuous (A, ϕ)-critical measure with the external field ϕ ≡ 0. Observe that if A = ∅, this notion is nontrivial. A particularly interesting case is treated in the following lemma:


63

Lemma 3.1. If ϕ = Re , and is analytic in a simply connected domain , then condition (3.8) is equivalent to f ϕ (μ; h) = 0, with

f ϕ (μ; h) = def

h(x) − h(y) dμ(x)dμ(y) − 2 x−y

(3.9)

(x) h(x) dμ(x).

(3.10)

Proof. It is sufficient to show that E ϕ (μt ) − E ϕ (μ) = − Re t f (μ; h) + O(t 2 ) . We have

1 dμt (x t )dμt (y t ) − yt |

1 dμ(x)dμ(y), = log |(x − y) + t (h(x) − h(y))|

E(μ ) = t

so that

log

|x t

h(x) − h(y) dμ(x)dμ(y) E(μ ) − E(μ) = − log 1 + t x−y

h(x) − h(y) log 1 + t = − Re dμ(x)dμ(y). x−y

t

On the other hand,

t t t ϕ(x ) dμ (x ) − ϕ(x) dμ(x) = ϕ(x + th(x)) dμ(x) − ϕ(x) dμ(x) t

= Re ( (x + th(x)) − (x)) dμ(x). Taking into account the behavior of log(1 + x) for small x, we conclude that as t → 0,

h(x) − h(y) t + O(t 2 ) dμ(x)dμ(y) E ϕ (μt ) − E ϕ (μ) = − Re x−y

+2 Re t (x) h(x) + O(t 2 ) dμ(x), and the statement follows. Remark 3.3. For a finite set A and the external field given by (2.3), the discrete (A, ϕ)critical measures fit into the same variational definition as their continuous counterparts, as long as we replace in (3.8) the continuous energy E ϕ (μ) by Eϕ (μ). Indeed, arguments similar to those used in the proof of Lemma 3.1 show that for μ in (3.3), the condition d Eϕ (μt )|t=0 = 0, dt

(3.11)

64


written for h(ζ ) =

A(ζ ) , z∈ / A, ζ −z

yields i= j

n B(z) 1 D(z) 1 + = , (ζi − z)(ζ j − z) A(z) ζi − z A(z) i=1

where D is a polynomial. In particular, the residue of the left hand side (as a function def n of z) is 0 at w = ζk , k = 1, . . . , n; setting y(z) = i=1 (z − ζi ), we arrive again at the system (3.5). And viceversa, using the chain rule it is easy to show that the condition (3.2) implies (3.11). Critical measures constitute an important object; for a finite set A the natural description of their structure is in terms of the trajectories of quadratic differentials. In the next section we give an abridged introduction to quadratic differentials on the Riemann sphere in the form needed for our purposes. For a comprehensive account on this theory see for instance [43,72,91,96]. 4. Rational Quadratic Differentials on the Riemann Sphere in a Nutshell Let A and V be monic polynomials of degree p + 1 and p − 1, respectively, with A given by (1.2) with all ak ’s pairwise distinct. The rational function V /A defines on the Riemann sphere C the quadratic differential (z) = −

V (z) (dz)2 . A(z)

(4.1)

The only singular points of (assuming that the zeros of V and A are disjoint) are: – the points ak ∈ A, where has simple poles (critical points of order −1); – the zeros of V of order k ≥ 1, where has zeros of the same order; – the infinity, where has a double pole (critical point of order −2) with the residue −1. The rest of the points in C are the regular points of , and their order is 0. All singular points of order ≥ −1 are called finite critical points of . In a neighborhood of any regular point z 0 we can introduce a local parameter

z

z √ V (t) dt, (4.2) ξ = ξ(z) = = − A(t) in terms of which the representation of is identically equal to one. This parameter is not uniquely determined: any other parameter ξ with this property satisfies ξ = ±ξ + const. The function ξ is called the distinguished or natural parameter near z 0 . Following [72] and [91], a smooth curve γ along which −V (z)/A(z) (dz)2 > 0

⇔

Im ξ(z) = const

is a horizontal arc of the quadratic differential . More precisely, if γ is given by a parametrization z(t), t ∈ (α, β), then


V − (z(t)) A

dz dt

65

2 > 0, t ∈ (α, β).

A maximal horizontal arc is called a horizontal trajectory (or simply a trajectory) of . Analogously, trajectories of − are called orthogonal or vertical trajectories of ; along these curves V (z)/A(z) (dz)2 > 0

⇔

Re ξ(z) = const.

Any simply connected domain D not containing singular points of and bounded by two vertical and two horizontal arcs is called a -rectangle. In other words, if ξ is any distinguished parameter in D, then ξ(D) is a (euclidean) rectangle, and D → ξ(D) is a one-to-one conformal mapping. Obviously, this definition is consistent with the freedom in the selection of the natural parameter ξ . We can define a conformal invariant metric associated with the quadratic differential √ , given by the length element |dξ | = | V /A|(z)|dz|; the -length of a curve γ is

V 1 (z) |dz|; γ = π γ A (observe that this definition differs by a normalization constant from Definition 5.3 in [91]). Furthermore, if D is a simply connected domain not containing singular points of , we can introduce the -distance by dist(z 1 , z 2 ; , D) = inf{ γ : z 1 , z 2 ∈ γ¯ , γ ⊂ D}. Trajectories and orthogonal trajectories are in fact geodesics (in the -metric) connecting any two of its points. Indeed, according to [72, Thm. 8.4], in any simply connected domain D not containing singular points of , a trajectory arc γ joining z 1 with z 2 is the shortest: if L 1 , L 2 are the orthogonal trajectories through z 1 and z 2 , respectively, then any rectifiable curve γ that connects L 1 with L 2 in D satisfies γ . γ ≤ The local structure of the trajectories is well known (see the references cited at the end of the previous section). For instance, at any regular point trajectories look locally as simple analytic arcs passing through this point, and through every regular point of passes a uniquely determined horizontal and uniquely determined vertical trajectory of , that are locally orthogonal at this point [91, Theorem 5.5]. If z is a finite critical point of of order k ≥ −1, then from z emanate k + 2 trajectories under equal angles 2π/(k + 2) (see Fig. 1). In the case of a double pole, the trajectories have either the radial, the circular or the spiral form, depending whether the residue at this point is negative, positive or non-real, see Fig. 2. In particular, with the assumptions on A and V above all trajectories of the quadratic differential (4.1) in a neighborhood of infinity are topologically identical to circles. The global structure of the trajectories is much less clear. The trajectories and orthogonal trajectories of a given differential produce a transversal foliation of the Riemann sphere C. The main source of troubles is the existence of the so-called recurrent trajectories, whose closure may have a non-zero plane Lebesgue measure. We refer the reader to [91] for further details. A trajectory γ is critical or short if it joins two (not necessarily different) finite critical points of . The set of critical trajectories of together with their endpoints (critical

66


Fig. 1. The local trajectory structure near a simple zero (left) or a simple pole

Fig. 2. The local trajectory structure near a double pole with a negative (left), positive (center) or non-real residue

Fig. 3. -rectangles intersecting the support of a positive (left) and sign-changing measure (right); for further details, see Sect. 8.4

points of ) is the critical graph of . Critical and closed trajectories are the only trajectories of with finite -length. The quadratic differential is called closed if all its trajectories are either critical or closed (i.e. all its trajectories have a finite -length). In this case the trajectories of that constitute closed Jordan curves cover the whole plane, except a set of critical trajectories of a plane Lebesgue measure zero; see e.g. Fig. 3 for a typical structure of such trajectories.


67

If the quadratic differential (4.1) with A given by (1.2) is closed, there exists a set of at √ most p critical trajectories of such that the complement to is connected, and V /A has a single-valued branch in C\ . 5. Critical Measures in the Field of a Finite System of Fixed Charges In what follows we fix the set of p+1 distinct points A = {a0 , . . . , a p } ⊂ C and consider the basic domain = C\A, A = {a0 , a1 , . . . , a p }, and an external field ϕ of the form ϕ = Re , (z) = −

p ρk k=0

2

log(z − ak ), (z) = −

p ρk /2 B(z) =− , z − ak 2 A(z) k=1

(5.1) where we have used notation from (1.6). If {ρ0 , . . . , ρ p } ⊂ R, then this external field corresponds to the potential of a discrete signed measure supported on A: ϕ(z) = U σ (z),

σ =

p ρk k=0

2

δak ∈ M p+1 .

(5.2)

However, if any ρk ∈ C\R, then ϕ is not single-valued in C\A; nevertheless, the notion of an (A, ϕ)-critical measures for this case has been discussed in Definition 3.2. In particular, Lemma 3.1 applies. In this section we state and prove the main structural theorem for (A, ϕ)-critical measures, which asserts that the support of any such a measure is a union of analytic curves made of trajectories of a rational quadratic differential. On each arc of its support the measure has an analytic density with respect to the arc-length measure. Finally, we describe the Cauchy transform and the logarithmic potential of an (A, ϕ)-critical measure. 5.1. The main theorem. According to (5.1), A is exactly the set of singularities of the external field ϕ, except for the case when ρk = 0 for some k ∈ {0, . . . , p}. In such a case we do not drop the corresponding ak from the set A; it remains as a fixed point of the class of variations (Definition 3.2). However, the status of the point ak ∈ A with ρk = 0 is different from the case ρk = 0, see the next theorem. Theorem 5.1. Let A = {a0 , a1 , . . . , a p } and ϕ given by (5.1). Then for any continuous (A, ϕ)-critical measure μ there exists a rational function R with poles at A and normalized by R(z) =

2 1 κ + O 3 , z → ∞, z z

κ = μ(C) + def

p 1 ρj, 2

(5.3)

j=0

such that the support supp(μ) consists of a union of trajectories of the quadratic differential (z) = −R(z)dz 2 . If all ρ j ∈ R, then is closed, and supp(μ) is made of a finite number of trajectories of . If in the representation (5.1), ρ j = 0, j ∈ {0, 1, . . . , p}, then a j is either a simple pole or a regular point of R; otherwise R has a double pole at a j .

68


The proof of this theorem reduces to two lemmas below. The first of them deals with the principal value of the Cauchy transform

1 def μ C (z) = lim dμ(x) (5.4) →0+ |z−x|> x − z of the (A, ϕ)-critical measure μ. Lemma 5.1. For any (A, ϕ)-critical measure μ there exists a rational function R with properties listed in Theorem 5.1 such that μ 2 C (z) + (z) = R(z)

mes2 − a.e.,

(5.5)

where mes2 is the plane Lebesgue measure on C. Remark 5.1. Formula (5.5) and its variations for equilibriums measures of compact sets of minimal capacity (see Sect. 6) are well-known, although occasionally written in terms of quadratic differentials, see e.g. the work of Nuttall [67], Stahl [86,88], Gonchar and Rakhmanov [36,75], Deift and collaborators [21]. Notice that in the situation considered here the support of the critical measure is not known a-priori. Remark 5.2. Formula (5.5) is also sufficient for μ being (A, ϕ)-critical, so that it in fact characterizes these critical measures. The proof of this statement lies beyond the scope of this already lengthy paper, and we do not go into further details. Proof. Assume that μ is an (A, ϕ)-critical measure for ϕ like in (5.1). We will actually show that (5.5) is valid at any point z ∈ C where the integral defining C μ is absolutely convergent. It is well known that at such a z,

dμ(x) lim+ = 0, (5.6) r →0 |x−z| 0 denote Dr = {ζ ∈ C : |ζ − z| < r }. def Function m(r ) = μ (Dr ) is continuous from the left and monotonically increasing, so that the subset m(r + ε) − m(r − ε) def = r ∈ (0, 1) : m (r ) = lim exists ε→0 2ε has the linear Lebesgue measure 1.


69

For r ∈ and ε ∈ (0, 1) define the “smooth step” function ⎧ if 0 ≤ x < 1 − ε, ⎨ 0, 2 def (x, ε) = (x−1−ε) (x−1+2ε) , if 1 − ε ≤ x < 1 + ε, 4ε3 ⎩ 1, if x ≥ 1 + ε. It is easy to see that (·, ε) ∈ C 1 (R+ ) and that | ddx (x, ε)| < 1/ε for all ε > 0. Using this function we define on C the C 1 function |ζ − z| def θ (ζ ) = θ (ζ, r, ε) = ,ε , r and consider the condition (3.9) with the following particular choice of h: h(ζ ) = h ε (ζ ; r ) =

A(ζ ) θ (ζ, r, ε). ζ −z

(5.7)

For the sake of brevity we use the notation Kr,ε = Dr (1+ε) \Dr (1−ε) , Fr,ε = C\Dr (1+ε) , def

def

so that Dr (1−ε) , Kr,ε and Fr,ε provide a partition of C. Furthermore, by construction 0, if ζ ∈ Dr (1−ε) , h(ζ ) = A(ζ ) (5.8) ζ −z , if ζ ∈ Fr,ε . Consider first

h ε (x; r ) − h ε (y; r ) dμ(x)dμ(y) x−y = I (Dr (1−ε) × Dr (1−ε) ) + I (Kr,ε × Kr,ε ) + I (Fr,ε × Fr,ε ) + 2I (Dr (1−ε) × Kr,ε ) + 2I (Dr (1−ε) × Fr,ε ) + 2I (Kr,ε × Fr,ε ),

where I () means the integral in the l.h.s. taken over the set . Observe that by (5.8), I (Dr (1−ε) × Dr (1−ε) ) = 0. Let ζ ∈ Kr,ε ; since |ζ − z| |ζ − z| ∂ ζ −z ∂ 1 1 ,ε ,ε , θ (ζ ) = |ζ − z| = r r r r |ζ − z| ∂ζ ∂ζ we have

∂ 1 1 grad θ (ζ ) = θ (ζ ) ≤ . 2 rε ∂ζ

In consequence, for x, y ∈ Kr,ε , h ε (x; r ) − h ε (y; r ) const ≤ , x−y rε

(5.9)

where the constant in the right hand side is independent of ε. Obviously, by definition of h we have that this inequality is valid (with a different constant) if x ∈ Kr,ε and y lies on a compact subset of C.

70


From (5.9) we conclude that h (x; r ) − h (y; r ) ε ε I (Kr,ε × Kr,ε ) = dμ(x)dμ(y) Kr,ε ×Kr,ε x−y 2 const ≤ μ Kr,ε . rε Taking into account that r ∈ , we have that μ Kr,ε = 2r m (r ), lim ε→0+ ε

(5.10)

(5.11)

so by (5.10), I (Kr,ε × Kr,ε ) = o(1) as ε → 0+. Consider now x ∈ Kr,ε and y ∈ Dr,ε . Then h ε (x; r ) − h ε (y; r ) h ε (x; r ) A(x) = = θ (x). x−y x−y (x − z)(x − y) Consider two cases. If |y − z| < r (1 − 2ε), then h ε (x; r ) − h ε (y; r ) const const ≤ ≤ . x−y r (1 − ε)|x − y| r (1 − ε)(|x| − |y|) Hence, with a different constant, h ε (x; r ) − h ε (y; r ) dμ(x)dμ(y) x∈Kr,ε , |y−z| 0 this system is still uniquely solvable. Equations Im w j (v) = 0 imply that differential in (9.1) is closed, the associated measure μ is A-critical, and supp(μ) = v = γ1,v ∪ · · · ∪ γ p,v . The homotopic type and signs of the components of μ depend on the behavior of trajectories of , which are originated at the points a ∗ ⊂ A∗ and close to trajectories δ j . Any such a trajectory will hit the corresponding point v j if h j = 0. If h j > 0, then it will pass from the left of v j or from the right of v j , see Fig. 14. A change from s j to −s j will change the direction of the turn. Therefore, there is a unique selection of vectors (s1 , . . . , s p−1 ) such that all turns are right. Then the branch

108


Fig. 14. Left and right turns

√ √ of R in C\ v will be close to the branch of R ∗ in C\ ∗ , and therefore the corresponding measure μ will be positive. In this sense, the cell we entered contains some positive measures. Therefore, they are positive, since supp(μ) are all homotopic. Acknowledgements. We are indebted to B. Shapiro for interesting discussions and for providing us with the early version of his manuscripts [82] and [83]; after the first version of this paper was made public in the arxiv, we learned about a work in preparation of B. Shapiro and collaborators, which has some overlappings with this paper. Fortunately, the methods and the paths we follow are very different. We also gratefully acknowledge many helpful conversations with H. Stahl and A. Vasil ev, as well as useful remarks from M. Yattselev concerning the first version of this manuscript. The software for computing the parameters of Chebotarev’s compacts, provided by the authors of [69] and freely available at their web site, was also useful for gaining some additional insight. AMF is partially supported by Junta de Andalucía, grants FQM-229, P06-FQM-01735, and P09-FQM4643, as well as by the research project MTM2008-06689-C02-01 from the Ministry of Science and Innovation of Spain and the European Regional Development Fund (ERDF). EAR is partially supported by the NSF grant DMS-9801677.

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Commun. Math. Phys. 302, 113–159 (2011) Digital Object Identifier (DOI) 10.1007/s00220-010-1180-y

Communications in


Gravitational Descendants in Symplectic Field Theory Oliver Fabert Mathematisches Institut, Ludwig-Maximilians-Universität München, Theresienstr. 39, 80333 München, Germany E-mail: [email protected] Received: 13 July 2009 / Accepted: 19 August 2010 Published online: 6 January 2011 – © Springer-Verlag 2011

Abstract: It was pointed out by Y. Eliashberg in his ICM 2006 plenary talk that the rich algebraic formalism of symplectic field theory leads to a natural appearance of quantum and classical integrable systems, at least in the case when the contact manifold is the prequantization space of a symplectic manifold. In this paper we generalize the definition of gravitational descendants in SFT from circle bundles in the Morse-Bott case to general contact manifolds. After we have shown using the ideas in Okounkov and Pandharipande (Ann Math 163(2):517–560, 2006) that for the basic examples of holomorphic curves in SFT, that is, branched covers of cylinders over closed Reeb orbits, the gravitational descendants have a geometric interpretation in terms of branching conditions, we follow the ideas in Cieliebak and Latschev (http://arixiv.org/abs/0706.3284v2 [math.s6], 2007) to compute the corresponding sequence of Poisson-commuting functions when the contact manifold is the unit cotangent bundle of a Riemannian manifold. Contents Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Symplectic Field Theory with Gravitational Descendants . . 1.1 Symplectic field theory . . . . . . . . . . . . . . . . . 1.2 Gravitational descendants . . . . . . . . . . . . . . . . 1.3 Invariance statement . . . . . . . . . . . . . . . . . . . 1.4 The circle bundle case . . . . . . . . . . . . . . . . . . 2. Example: Symplectic Field Theory of Closed Geodesics . . 2.1 Symplectic field theory of a single Reeb orbit . . . . . 2.2 Gravitational descendants = branching conditions . . . 2.3 Branched covers of trivial half-cylinders . . . . . . . . 2.4 Obstruction bundles and transversality . . . . . . . . . 2.5 Additional marked points and gravitational descendants References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Research supported by the German Research Foundation (DFG).

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Summary Symplectic field theory (SFT), introduced by H. Hofer, A. Givental and Y. Eliashberg in 2000 ([EGH]), is a very large project and can be viewed as a topological quantum field theory approach to Gromov-Witten theory. Besides providing a unified view on established pseudoholomorphic curve theories like symplectic Floer homology, contact homology and Gromov-Witten theory, it leads to numerous new applications and opens new routes yet to be explored. While symplectic field theory leads to algebraic invariants with very rich algebraic structures, which are currently studied by a large group of researchers, for all the geometric applications found so far it was sufficient to work with simpler invariants like cylindrical contact homology. Although cylindrical contact homology is not always defined, it is much easier to compute, not only since it involves just moduli spaces of holomorphic cylinders but also due to the simpler algebraic formalism. While the rich algebraic formalism of the higher invariants of symplectic field theory seems to be too complicated for concrete geometric applications, it was pointed out by Eliashberg in his ICM 2006 plenary talk ([E]) that the integrable systems of rational Gromov-Witten theory very naturally appear in rational symplectic field theory by using the link between the rational symplectic field theory of circle bundles in the Morse-Bott version and the rational Gromov-Witten potential of the underlying symplectic manifold. Indeed, after introducing gravitational descendants as in Gromov-Witten theory, it is precisely the rich algebraic formalism of SFT with its Weyl and Poisson structures that provides a natural link between symplectic field theory and (quantum) integrable systems. In particular, in the case where the contact manifold is a circle bundle over a closed symplectic manifold, the rich algebraic formalism of symplectic field theory seems to provide the right framework to understand the deep relation between Gromov-Witten theory and integrable systems, at least in the genus zero case. While in the Morse-Bott case in [E] it follows from the corresponding statements for the Gromov-Witten descendant potential that the sequences of commuting operators and Poisson-commuting functions are independent of auxiliary choices like almost complex structure and abstract perturbations, for the case of general contact manifolds it is well-known that the SFT Hamiltonian, however in general, explicitly depend on choices like contact form, cylindrical almost complex structure and coherent abstract perturbations, and hence is not an invariant for the contact manifold itself. But before we can come down to the question of invariance, we first need to give a rigorous definition of gravitational descendants in the context of symplectic field theory. While in Gromov-Witten theory the gravitational descendants were defined by integrating powers of the first Chern class of the tautological line bundle over the moduli space, which by Poincaré duality corresponds to counting common zeroes of sections in this bundle, in symplectic field theory, more generally every holomorphic curves theory, where curves with punctures and/or boundary are considered, we are faced with the problem that the moduli spaces generically have codimension-one boundary, so that the count of zeroes of sections in general depends on the chosen sections in the boundary. It follows that the integration of the first Chern class of the tautological line bundle over a single moduli space has to be replaced by a construction involving all moduli space at once. Note that this is similar to the choice of coherent abstract perturbations for the moduli spaces in symplectic field theory in order to achieve transversality for the Cauchy-Riemann operator. Keeping the interpretation of descendants as common zero sets of sections in powers of the tautological line bundles (which will turn out to be particularly useful when one studies the topological meaning of descendants by

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localizing on special divisors, see [FR]), we define in this paper the notion of coherent collections of sections in the tautological line bundles over all moduli spaces, which just formalizes how the sections chosen for the lower-dimensional moduli spaces should affect the section chosen for moduli spaces on its boundary. To be more precise, since the sections should be invariant under obvious symmetries like reordering of the punctures and the marked points, we actually need to work with multi-sections in order to meet both the symmetry and the transversality assumption. We will then define descendants j of moduli spaces M ⊂ M, which we obtain inductively as zero sets of these coherent collections of sections (s j ) in the tautological line bundles over the descendant moduli spaces M

j−1

⊂ M, and define descendant Hamiltonians Hi,1 j by integrating chosen j

closed differential forms θi over M . For these we prove the following theorem. Theorem. Counting holomorphic curves with one marked point after integrating differential forms and introducing gravitational descendants defines a sequence of distinguished elements Hi,1 j ∈ H∗ −1 W0 , D 0 −1 0 in the full SFT homology algebra with differential D 0 = [H0 , ·] : −1 W0 → W , −1 0 0 which commute with respect to the commutator bracket on H∗ W , D ,

Hi,1 j , H1k, = 0, (i, j), (k, ) ∈ {1, . . . , N } × N .

In contrast to the Morse-Bott case considered in [E] it follows that, when the differential in symplectic field theory counting holomorphic curves without additional marked points is no longer zero, the sequences of generating functions no longer commute with respect to the bracket, but only commute after passing to homology. On the other hand, in the same way as the rational symplectic field theory of a contact manifold is defined by counting only curves with genus zero, we immediately obtain a rational version of the above statement by expanding H0 and the Hi,1 j in powers of the formal variable for the genus. Corollary. Counting rational holomorphic curves with one marked point after integrating differential forms and introducing gravitational descendants defines a sequence of distinguished elements hi,1 j ∈ H∗ P0 , d 0 , 0 0 0 0 in the rational SFT homology algebra with differential 0d = {h , ·} : P → P , which 0 commute with respect to the Poisson bracket on H∗ P , d ,

hi,1 j , h1k, = 0, (i, j), (k, ) ∈ {1, . . . , N } × N .

As we already outlined above, in contrast to the circle bundle case we have to expect that the sequence of descendant Hamiltonians depends on the auxiliary choices like contact form, cylindrical almost complex structure and coherent abstract polyfold perturbations. Here we prove the following natural invariance statements.

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Theorem. For different choices of contact form λ± , cylindrical almost complex structure J± , abstract polyfold perturbations and sequences of coherent collections of sec −1 0,− 0,− 1,− ,D tions s ± j the resulting systems of commuting operators Hi, j on H∗ W −1 0,+ 0,+ 1,+ are isomorphic, i.e., there exists an isomorphism of and Hi, j on H∗ W , D −1 0,− 0,− the Weyl algebras H∗ W , D and H∗ −1 W0,+ , D 0,+ which maps Hi,1,− j ∈ −1 0,− 0,− −1 0,+ 0,+ 1,+ H∗ W , D to Hi, j ∈ H∗ W , D . Note that this theorem is an extension of the theorem in [EGH] stating that for different choices of auxiliary data the Weyl algebras H∗ −1 W0,− , D 0,− and H∗ −1 W0,+ , D 0,+ are isomorphic. As above we clearly also get a rational version of the invariance statement: Corollary. For different choices of contact form λ± , cylindrical almost complex structure J ± , abstract polyfold perturbations and sequences of coherent collecthe resulting system of Poisson-commuting functions hi,1,− tions of sections s ± j on j 0,− 0,− and hi,1,+j on H∗ P0,+ , d 0,+ are isomorphic, i.e., there exists an isoH∗ P , d morphism of the Poisson algebras H∗ P0,− , d 0,− and H∗ P0,+ , d 0,+ which maps 0,− 0,− hi,1,− to hi,1,+j ∈ H∗ P0,+ , d 0,+ . ,d j ∈ H∗ P As a concrete example beyond the case of circle bundles discussed in [E] we consider the symplectic field theory of a closed geodesic. For this recall that in [F2] the author introduces the symplectic field theory of a closed Reeb orbit γ , which is defined by counting only those holomorphic curves which are branched covers of the orbit cylinder R ×γ in R ×V . In [F2] we prove that these orbit curves do not contribute to the algebraic invariants of symplectic field theory as long as they do not carry additional marked points. Our proof explicitly uses that the subset of orbit curves over a fixed orbit is closed under taking boundaries and gluing, which follows from the fact that they are also trivial in the sense that they have trivial contact area and that this contact area is preserved under taking boundaries and gluing. It follows that every algebraic invariant of symplectic field theory has a natural analog defined by counting only orbit curves. In particular, in the same way as we define sequences of descendant Hamiltonians Hi,1 j and hi,1 j by counting general curves in the symplectization of a contact manifold, we can define sequences of descendant Hamiltonians H1γ ,i, j and h1γ ,i, j by just counting branched covers of the orbit cylinder over γ with signs (and weights), where the preservation of the contact area under splitting and gluing of curves proves that for every theorem from above we have a version for γ . For this let W0γ be the graded Weyl subalgebra of the Weyl algebra W0 , which is generated only by those p- and q-variables pn = pγ n , qn = qγ n corresponding to Reeb orbits which are multiple covers of the fixed orbit γ and which are good in the sense of [BM]. In the same way we further introduce the Poisson subalgebra P0γ of P0 . We further prove that for branched covers of orbit cylinders over any closed Reeb orbit the gravitational descendants indeed have a geometric interpretation in terms of branching conditions, which generalizes the work of [OP] used in [E] for the circle. Since all the considered holomorphic curves factor through the embedding of the closed Reeb orbit into the contact manifold, it follows that it only makes sense to consider differential forms of degree zero or one. While it follows from the result h0γ = 0 in [F2] that the sequences h1γ ,i, j indeed commute with respect to the Poisson bracket (before passing to homology), the same proof as in [F2] shows that every descendant

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Hamiltonian in the sequence vanishes if the differential form is of degree zero. For differential forms of degree one the strategy of the proof however no longer applies and it is indeed shown in [E] that for γ = V = S 1 and θ = dt we get nontrivial contributions from branched covers. In this paper we want to study the corresponding Poisson-commuting sequence in the special case where the contact manifold is the unit cotangent bundle S ∗ Q of a (m-dimensional) Riemannian manifold Q, so that every closed Reeb orbit γ on V = S ∗ Q corresponds to a closed geodesic γ¯ on Q. When the closed geodesic γ¯ represents a hyperbolic Reeb orbit in the unit cotangent bundle of a surface Q a simple computation shows that all moduli spaces with 2 j + 1 punctures possibly contribute to the descendant Hamiltonian h1γ , j . Since in this case the Fredholm index is 2 j − 1 and hence for j > 0 strictly smaller than the dimension of the underlying nonregular moduli space of branched covers, which is 4 j − 2, transversality cannot be satisfied but the cokernels of the linearized operators fit together to give an obstruction bundle

of rank 2 j − 1. While like for every closed Reeb orbit we have that h1γ ,0 = h1S 1 ,0 = n pn qn , the other Hamiltonians h1γ , j are not so easy to determine. While in the case of the circle we obtain a complete set of integrals, our following theorem shows that the Hamiltonian system with symmetries obtained for different choices of Reeb orbits does not need to be integrable.

Theorem. Assume that the closed geodesic γ¯ represents a hyperbolic Reeb orbit in the unit cotangent bundle of a surface Q. Then gγ1¯ , j = 0 and hence h1γ , j = 0 for all j > 0. Apart from the fact that this result shows that the resulting Hamiltonian systems with symmetries are in general not very interesting from the point of view of integrable systems, let us sketch how the Hamiltonian systems with symmetries assigned to a closed Reeb orbit can be applied to embedding problems in symplectic geometry. To this end the author is currently working on a local version of SFT, which generalizes local Gromov-Witten (GW) theory in the same way as the usual SFT generalizes usual GW theory: While in local GW theory we count multiple covers over a fixed super-rigid closed holomorphic curve, in local SFT we count multiple covers over super-rigid punctured holomorphic curves, where the technical assumption of super-ridity guarantees that multiple covers are isolated. In particular, instead of getting invariants for contact manifolds, we now get the above invariants for closed Reeb orbits by counting multiple covers over the corresponding orbit cylinder. On the other hand, in the very same way as Paolo Rossi was able to compute part of the GW potential of the sphere using the SFT of the circle in [R1], we can use these new SFT invariants for the closed Reeb orbits appearing in the splitting process to derive information about the local GW potential of the original closed holomorphic curve. On the other hand, it can also be used in order to derive a contradiction and hence should be applicable to embedding problems in symplectic geometry. We claim that our above theorem can be used to show that an exceptional sphere cannot split along a hyperbolic Reeb orbit in the unit cotangent bundle, which also gives an alternative proof of the fact that every oriented embedded Lagrangian in a closed symplectic four-manifold, which intersects an exceptional sphere in a homologically nontrivial way, must have genus zero or one: Since it follows from the above theorem that the descendant Hamiltonians of every hyperbolic orbit representing a closed geodesic are zero, and this then implies that there are no descendant contributions of degree-two classes in the local Gromov-Witten descendant potential, we can easily derive a contradiction using the topological recursion relation in rational Gromov-Witten theory.

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More precisely, we will show that the resulting system of Poisson-commuting functions h1γ , j , j ∈ N on P0γ is isomorphic to the system of Poisson-commuting functions g1γ¯ , j , j ∈ N on Pγ0¯ = P0γ , where for every j ∈ N the descendant Hamiltonian g1γ¯ , j is given by qn 1 · . . . · qn j+2 gγ1¯ , j = , ( n) ( j + 2)! where q−n = pn and the sum runs over all ordered monomials qn 1 · · · qn j+2 with n 1 + · · · + n j+2 = 0 and which are of degree 2(m + j − 3). Here ( n ) ∈ {−1, 0, +1} is fixed by a choice of coherent orientations in symplectic field theory and is zero if and only if one of the orbits γ n 1 , . . . , γ n j+2 is bad. For this recall from [BM] that in order to orient moduli spaces in symplectic field theory one additionally needs to choose orientations for all occuring Reeb orbits, while the resulting invariants are independent of these auxiliary choices. While it follows from our proof that when the degree is maximal we have an obstruction bundle of rank zero over a discrete non-regular moduli space, we show in Proposition 2.8 how (for j = 1) this obstruction bundle and hence its orientation is determined by the tangent spaces to the unstable manifolds of the multiply-covered geodesics. While the orientation of a closed Reeb orbit in SFT corresponds to an orientation of the (finite-dimensional) unstable manifold, the sign in front of pn 1 pn 2 qn k (n 1k + n 2k = n k ) in gγ1¯ ,1 is given by k

k

comparing the orientations of the finite-dimensional linear subspaces T W − (γ¯ 2 ) and (T W − (γ¯ ) ⊕ T W − (γ¯ )) ∩ = {(v1 , v2 ) ∈ T W − (γ¯ ) ⊕ T W − (γ¯ ) : v1 (0) = v2 (0)} of C ∞ ((γ¯ 2 )∗ N ) (N is the normal bundle to γ¯ in Q, see Proposition 2.8). For j > 1 the obstruction bundle gets much more complicated, but the idea is the same. Apart from the fact that the commutativity condition {gγ1¯ , j , gγ1¯ ,k } = 0 clearly leads to relations between the different ( n ), observe that a choice of orientation for γ does not lead to a canonical choice of orientations for its multiples γ k . While we expect that it is in general very hard to write down a set of signs ( n ) explicitly, for all the geometric applications we have in mind and the educational purposes as a test model beyond the Gromov-Witten case we are rather interested in proving vanishing results as the one above. Forgetting about the appearing sign issues, it follows that the sequence g1γ¯ , j is obtained from the sequence for the circle by removing all summands with the wrong, that is, not maximal degree, so that the system is completely determined by the KdV hierarchy and the Morse indices of the closed geodesic and its iterates. Indeed note that when the underlying Poisson algebra is graded so that the Poisson bracket is of pure degree, then one naturally gets from a Hamiltonian system with symmetries h1j a new Hamiltonian system with symmetries h1j , where h1j denotes the part of h1j with maximal degree, since [h1j ], [h1k ] = h1j , h1k . Note that with our grading conventions the Poisson bracket is indeed of pure degree since | pn | + |qn | = − CZ (γ n ) + (m − 3) + CZ (γ n ) + (m − 3) = 2(m − 3) is independent of the multiplicity n, where CZ (γ n ) denotes the index of γ n (Morse index of γ¯ n ).

Conley-Zehnder inx −inx On the other hand, since u(x) = n pn e + qn e is not of pure degree, our new Hamiltonian systems with symmetries have no good translation (using inverse Fourier transform) into the formal loop space {u : S 1 → Rk }, k = 1 which is the classical phase space of the integrable systems of Gromov-Witten theory, see [R2]. Note that in the case of the circle γ¯ = Q = S 1 the degree condition is automatically fulfilled and we just get back the sequence of descendant Hamiltonians for the

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circle in [E], which agrees with the sequence of Poisson-commuting integrals of the dispersionless KdV integrable hierarchy, while in the case of a hyperbolic geodesic on a surface it follows from the multiplicativity of the Conley-Zehnder index that none of the monomials qn 1 · . . . · qn j+2 has the right degree. Apart from using the geometric interpretation of gravitational descendants for branched covers of orbit cylinders over a closed Reeb orbit in terms of branching conditions mentioned above, the second main ingredient for the proof is the idea in [CL] to compute the symplectic field theory of V = S ∗ Q from the string topology of the underlying Riemannian manifold Q by studying holomorphic curves in the cotangent bundle T ∗ Q. More precisely, we compute the symplectic field theory of a closed Reeb orbit γ in S ∗ Q including differential forms and gravitational descendants by studying branched covers of the trivial half-cylinder connecting the closed Reeb orbit in the unit cotangent bundle with the underlying closed geodesic in the cotangent bundle T ∗ Q with special branching data, where the latter uses the geometric interpretation of gravitational descendants. In order to give a complete proof we also prove the neccessary transversality theorems using finite-dimensional obstruction bundles over the underlying nonregular moduli spaces. While on the SFT side one has very complicated obstruction bundles over nonregular moduli spaces of arbitrary large dimension, on the string side all relevant nonregular moduli spaces already turn out to be discrete, so that the obstruction bundles disappear if the Fredholm index is right. It follows that the system of Poisson-commuting function for a closed geodesic is completely determined by the KdV hierarchy and the Morse indices of the closed geodesic and its iterates. This paper is organized as follows. Section One is concerned with the definition and the basic results about gravitational descendants in symplectic field theory. After we recalled the basic definitions of symplectic field theory in Subsect. 1.1, we define gravitational descendants in Subsect. 1.2 using the coherent collections of sections and prove that the resulting sequences of descendant Hamiltonians commute after passing to homology. In Subsect. 1.3 we prove the desired invariance statement and discuss the important case of circle bundles in the Morse-Bott setup outlined in [E] in 1.4. After we treated the general case in Sect. One, Sect. Two is concerned with a concrete example beyond the case of circle bundles, the symplectic field theory of a closed geodesic, which naturally generalizes the case of the circle in [E]. After we have recalled the definition of symplectic field theory for a closed Reeb orbit including the results from [F2] in Subsect. 2.1, we show in Subsect. 2.2 that for branched covers of orbit cylinders the gravitational descendants have a geometric interpretation in terms of branching conditions. After outlining that there exists a version of the isomorphism in [CL] involving the symplectic field theory of a closed Reeb orbit in the unit cotangent bundle, we study the moduli space of branched covers of the corresponding trivial half-cylinder in the cotangent bundle in Subsect. 2.3. Since we meet the same transversality problems as in [F2], we study the neccessary obstruction bundle setup including Banach manifolds and Banach space bundles in Subsect. 2.3. In Subsect. 2.4 we finally prove the above theorem by studying branched covers of the trivial half-cylinder with special branching behavior. 1. Symplectic Field Theory with Gravitational Descendants 1.1. Symplectic field theory. Symplectic field theory (SFT) is a very large project, initiated by Eliashberg, Givental and Hofer in their paper [EGH], designed to describe in a

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unified way the theory of pseudoholomorphic curves in symplectic and contact topology. Besides providing a unified view on well-known theories like symplectic Floer homology and Gromov-Witten theory, it shows how to assign algebraic invariants to closed contact manifolds (V, ξ = {λ = 0}): Recall that a contact one-form λ defines a vector field R on V by R ∈ ker dλ and λ(R) = 1, which is called the Reeb vector field. We assume that the contact form is Morse in the sense that all closed orbits of the Reeb vector field are nondegenerate in the sense of [BEHWZ]; in particular, the set of closed Reeb orbits is discrete. The invariants are defined by counting J -holomorphic curves in R ×V which are asymptot ically cylindrical over chosen collections of Reeb orbits ± = γ1± , . . . , γn±± as the R-factor tends to ±∞, see [BEHWZ]. The almost complex structure J on the cylindrical manifold R ×V is required to be cylindrical in the sense that it is R-independent, links the two natural vector fields on R ×V , namely the Reeb vector field R and the R-direction ∂s , by J ∂s = R, and turns the distribution ξ on V into a complex subbundle of T V, ξ = T V ∩ J T V . We denote by Mg,r ( + , − ) the corresponding compactified moduli space of genus g curves with r additional marked points ([BEHWZ,EGH]). Possibly after choosing abstract perturbations using polyfolds (see [HWZ]), obstruction bundles ([F2]) or domain-dependent structures ([F1]) following the ideas in [CM] we get that Mg,r ( + , − ) is a branched-labelled orbifold with boundaries and corners of dimension equal to the Fredholm index of the Cauchy-Riemann operator for J . Note that in the same way as we will not discuss transversality for the general case but just refer to the upcoming papers on polyfolds by Hofer and his co-workers, in what follows we will for simplicity assume that every moduli space is indeed a manifold with boundaries and corners, since we expect that all the upcoming constructions can be generalized in an appropriate way. Let us now briefly introduce the algebraic formalism of SFT as described in [EGH]: Recall that a multiply-covered Reeb orbit γ k is called bad if CZ γ k = CZ(γ ) mod 2, where CZ(γ ) denotes the Conley-Zehnder index of γ . Calling a Reeb orbit γ good if it is not bad we assign to every good Reeb orbit γ two formal graded variables pγ , qγ with grading | pγ | = m − 3 − CZ(γ ), |qγ | = m − 3 + CZ(γ ) when dim V = 2m −1. In order to include higher-dimensional moduli spaces we further assume that a string of closed (homogeneous) differential forms = (θ1 , . . . , θ N ) on V is chosen and assign to every θi ∈ ∗ (V ) formal variables ti with grading |ti | = 2 − deg θi . Finally, let be another formal variable of degree || = 2(m − 3). Let W be the graded Weyl algebra over C of power series in the variables , pγ and ti with coefficients which are polynomials in the variables qγ , which is equipped with the associative product in which all variables super-commute according to their grading except for the variables pγ , qγ corresponding to the same Reeb orbit γ , [ pγ , qγ ] = pγ qγ − (−1)| pγ ||qγ | qγ pγ = κγ . (κγ denotes the multiplicity of γ .) Following [EGH] we further introduce the Poisson algebra P of formal power series in the variables pγ and ti with coefficients which are

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polynomials in the variables qγ with Poisson bracket given by ∂ f ∂g | f ||g| ∂g ∂ f . κγ − (−1) { f, g} = ∂ pγ ∂qγ ∂ pγ ∂qγ γ As in Gromov-Witten theory we want to organize all moduli spaces Mg,r ( + , − ) into a generating function H ∈ −1 W, called Hamiltonian. In order to include also higher-dimensional moduli spaces, in [EGH] the authors follow the approach in Gromov-Witten theory to integrate the chosen differential forms θ1 , . . . , θ N over the moduli spaces after pulling them back under the evaluation map from target manifold V . The Hamiltonian H is then defined by + − H= ev∗1 θi1 ∧ . . . ∧ evr∗ θir g−1 t I p q

+ , −

Mg,r ( + , − )/ R

−

with t I = ti1 . . . tir , p = pγ1+ . . . pγ ++ and q = qγ − . . . qγ − . Expanding +

n

H = −1

1

n−

H g g

g

we further get a rational Hamiltonian h = H0 ∈ P, which counts only curves with genus zero. While the Hamiltonian H explicitly depends on the chosen contact form, the cylindrical almost complex structure, the differential forms and abstract polyfold perturbations making all moduli spaces regular, it is outlined in [EGH] how to construct algebraic invariants, which just depend on the contact structure and the cohomology classes of the differential forms. 1.2. Gravitational descendants. For the relation to integrable systems it is outlined in [E] that, as in Gromov-Witten theory, symplectic field theory must be enriched by considering so-called gravitational descendants of the primary Hamiltonian H. Before we give a rigorous definition of gravitational descendants in SFT, we recall the definition from Gromov-Witten theory. Denote by Mr = Mg,r (X, J ) the compactified moduli space of closed J -holomorphic curves in the closed symplectic manifold X of genus g with r marked points (and fixed homology class). Following [MDSa] we introduce over Mr so-called tautological line bundles L1 , . . . , Lr , where the fibre of Li over a punctured curve (u, z 1 , . . . , zr ) ∈ Mr in the noncompactified moduli space is given by the cotangent line to the underlying, possibly unstable closed nodal Riemann surface S at the i th marked point, (Li )(u,z 1 ,...,zr ) = Tz∗i S, i = 1, . . . , r. To be more formal, observe that there exists a canonical map π : Mr +1 → Mr by forgetting the (r + 1)st marked point and stabilizing the map, where the fibre over the curve (u, z 1 , . . . , zr ) agrees with the curve itself. Then the tautological line bundle Li can be defined as the pull-back of the vertical cotangent line bundle of π : Mr +1 → Mr under the canonical section σi : Mr → Mr +1 mapping to the i th marked point in the fibre. Note that while the vertical cotangent line bundle is rather a sheaf than a true bundle

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since it becomes singular at the nodes in the fibres, the pull-backs under the canonical sections are indeed true line bundles as the marked points are different from the nodes and hence these sections avoid the singular loci. Denoting by c1 (Li ) the first Chern class of the complex line bundle Li , one then considers for the descendant potential of Gromov-Witten theory integrals of the form ev∗1 θi1 ∧ c1 (L1 ) j1 ∧ . . . ∧ evr∗ θir ∧ c1 (Lr ) jr , Mr

where (i k , jk ) ∈ {1, . . . , N } × N, which can again be organized into a generating function. Like pulling-back cohomology classes from the target manifold, the introduction of the tautological line bundles hence has the effect that the generating function also sees the higher-dimensional moduli spaces. On the other hand, in contrast to the former, the latter refers to partially fixing the complex structure on the underlying punctured Riemann surface. Before we can turn to the definition of gravitational descendants in SFT, it will turn out to be useful to give an alternative definition, where the integration of the powers of the first Chern classes is replaced by considering zero sets of sections. Restricting for notational simplicity to the case with one marked point, we can define by induction over j+1 j j ∈ N a nested sequence of moduli spaces M1 ⊂ M1 ⊂ M1 such that 1 ev∗ θi ∧ c1 (L) j = ev∗ θi . · j! M1j M1 For j = 1 observe that, since the first Chern class of a line bundle agrees with its Euler class, the homology class obtained by integrating c1 (L) over the compactified moduli space M1 can be represented by the zero set of a generic section s1 in L. Note that here we use that M1 represents a pseudo-cycle and hence has no codimension-one boundary strata. In other words, we find that ev∗ θi ∧ c1 (L) = ev∗ θi , 1 M1

M1

where M1 = s1−1 (0). 1

j−1

⊂ M1 . Now consider the restriction of the tautological line bundle L to M1 Instead of describing the integration of powers of the first Chern class in terms of common zero sets of sections in the same line bundle L, it turns out to be more geometric (see 2.2) to choose a section s j not in L but in its j-fold (complex) tensor product L⊗ j and define M1 = s −1 j (0) ⊂ M1 j

⊗j

Since c1 (L

.

) = j · c1 (L) it follows that ∗ ev θ = j · ev∗ θi ∧ c1 (L), i j j−1 M1

so that by induction

M1

as desired.

j−1

M1

ev∗ θi ∧ c1 (L) j =

1 ev∗ θi · j! M1j

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While the result of the integration is well-known to be independent of the choice of the almost complex structure and the abstract polyfold perturbations, it also follows that the result is independent of the precise choice of the sequence of sections s1 , . . . , s j . Like for the almost complex structure and the perturbations this results from the fact that the moduli spaces studied in Gromov-Witten theory have no codimension-one boundary. On the other hand, it is well-known that the moduli spaces in SFT typically have codimension-one boundary, so that now the result of the integration will not only depend on the chosen contact form, cylindrical almost complex structure and abstract polyfold perturbations, but also additionally explicitly depend on the chosen sequences of sections s1 , . . . , s j . While the Hamiltonian is hence known to depend on all extra choices, it is well-known from Floer theory that we can expect to find algebraic invariants independent of these choices. While the problem of dependency on contact form, cylindrical almost complex structure and abstract polyfold perturbations is sketched in [EGH], we will now show how to include gravitational descendants into their algebraic constructions. For this we will define descendants of moduli spaces, which we obtain as zero sets of coherent collections of sections in the tautological line bundles over all moduli spaces. From now on let Mr denote the moduli space Mg,r ( + , − )/ R studied in SFT for chosen collections of Reeb orbits + , − . In complete analogy to Gromov-Witten theory we can introduce r tautological line bundles L1 , . . . , Lr , where the fibre of Li over a punctured curve (u, z 1 , . . . , zr ) ∈ Mr is again given by the cotangent line to the underlying, possibly unstable nodal Riemann surface (without ghost components) at the i th marked point and which again formally can be defined as the pull-back of the vertical cotangent line bundle of π : Mr +1 → Mr under the canonical section σi : Mr → Mr +1 mapping to the i th marked point in the fibre. Note again that while the vertical cotangent line bundle is rather a sheaf than a true bundle since it becomes singular at the nodes in the fibres, the pull-backs under the canonical sections are still true line bundles as the marked points are different from the nodes and hence these sections avoid the singular loci. For notational simplicity let us again restrict to the case r = 1. Following the compactness statement in [BEHWZ], the codimension-one boundary of M1 consists of curves with two levels (in the sense of [BEHWZ]), whose moduli spaces can be represented as products M1,1 × M2,0 or M1,0 × M2,1 of moduli spaces of strictly lower dimension, where the marked point sits on the first or the second level. As we want to keep the notation as simple as possible, note that here and in what follows for product moduli spaces the first index refers to the level and not to the genus of the curve. To be more precise, after introducing asymptotic markers as in [EGH] for orientation issues, one obtains a fibre rather than a direct product, see also [F2]. However, since all the bundles and sections we will consider do or should not depend on these asymptotic markers, we will forget about this issue in order to keep the notation as simple as possible. On the other hand, it directly follows from the definition of the tautological line bundle L over M1 that over the boundary components M1,1 × M2,0 and M1,0 × M2,1 it is given by L |M1,1 ×M2,0 = π1∗ L1 , L |M1,0 ×M2,1 = π2∗ L2 , where L1 , L2 denotes the tautological line bundle over the moduli space M1,1 , M2,1 and π1 , π2 is the projection onto the first or second factor, respectively. With this we can now introduce the notion of coherent collections of sections in (tensor products of) tautological line bundles.

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Definition 1.1. Assume that we have chosen sections s in the tautological line bundles L over all moduli spaces M1 of J -holomorphic curves with one additional marked point. Then this collection of sections (s) is called coherent if for every section s in L over a moduli space M1 the following holds: Over every codimension-one boundary component M1,1 × M2,0 , M1,0 × M2,1 of M1 the section s agrees with the pull-back π1∗ s1 , π2∗ s2 of the chosen section s1 , s2 in the tautological line bundle L1 over M1,1 , L2 over M2,1 , respectively. Remark. Since in the end we will again be interested in the zero sets of these sections, we will assume that all occuring sections are transversal to the zero section. Furthermore, we want to assume that all the chosen sections are indeed invariant under the obvious symmetries like reordering of punctures and marked points. In order to meet both requirements, it follows that we actually need to employ multi-sections as in [CMS], which we however want to suppress for the rest of this exposition. The important observation is clearly that one can always find coherent collections of (transversal) sections (s) by using induction on the dimension of the underlying moduli space. While for the induction start it suffices to choose a non-vanishing section in the tautological line bundle over the moduli space of orbit cylinders with one marked point, for the induction step observe that the coherency condition fixes the section on the boundary of the moduli space. Here it is important to remark that the coherency condition further ensures that two different codimension-one boundary components actually agree on their common boundary strata of higher codimension. On the other hand, we can use our assumption that every moduli space is indeed a manifold with corners to obtain the desired section by simply extending the section from the boundary to the interior of the moduli space in an arbitrary way. For a given coherent collection of transversal sections (s) we will again define for every moduli space 1

M1 = s −1 (0) ⊂ M1 . 1

As an immediate consequence of the above definition we find that M1 is a neat submanifold (with corners) of M1 , i.e., the components of the codimension-one bound1 1 1 1 ary of M1 are given by products M1,1 × M2,0 and M1,0 × M2,1 , where M1,1 = s1−1 (0), M2,1 = s2−1 (0) for the section s1 in L1 over M1,1 , s2 in L2 over M2,1 , respectively. To be more precise, since we actually need to work with multi-sections rather than sections in the usual sense, the zero set is indeed a branched-labelled manifold. On the other hand, since we already suppressed the fact that our moduli spaces are indeed branched and labelled, we want to continue ignoring this technical aspect. On the other hand, we can use the above result as an induction start to obtain for every moduli space j j−1 M1 a sequence of nested subspaces M1 ⊂ M1 ⊂ M1 as in Gromov-Witten theory. 1

j−1

Definition 1.2. Let j ∈ N. Assume that for all moduli spaces we have chosen M1 ⊂ j−1 M1 such that the components of the codimension-one boundary of M1 are given by j−1 j−1 products of the form M1,1 × M2,0 and M1,0 × M2,0 . Then we again call a collection of transversal sections (s j ) in the j-fold tensor products L⊗ j of the tautological line j−1

bundles over M1

⊂ M1 coherent if for every section s j the following holds: Over

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125 j−1

j−1

j−1

every codimension-one boundary component M1,1 × M2,0 , M1,0 × M2,1 of M1 the section s j agrees with the pull-back π1∗ s1, j , π2∗ s2, j of the section s1, j , s2, j in the line ⊗j

j−1

⊗j

j−1

bundle L1 over M1,1 , L2 over M2,1 , respectively. With this we will now introduce (gravitational) descendants of moduli spaces. Definition 1.3. Assume that we have inductively defined a subsequence of nested subj j−1 j j−1 spaces M1 ⊂ M1 ⊂ M1 by requiring that M1 = s −1 for a coherent j (0) ⊂ M1 j−1

collection of sections s j in the line bundles L⊗ j over the moduli spaces M1 we call

j M1

the

j th

. Then

(gravitational) descendant of M1 .

Let W0 be the graded Weyl algebra over C of power series in the variables and pγ with coefficients which are polynomials in the variables qγ , which is obtained from the big Weyl algebra W by setting all variables ti equal to zero. In the same way define the subalgebra P0 of the Poisson algebra P. Apart from the Hamiltonian H0 ∈ −1 W0 counting only curves with no additional marked points, H0 =

−

#Mg,0 ( + , − )/ R g−1 p q , +

+ , −

we now want to use the chosen differential forms θi ∈ ∗ (V ), i = 1, . . . , N and the j j sequences M1 = Mg,1 ( + , − )/ R of gravitational descendants to define sequences of new SFT Hamiltonians Hi,1 j ∈ −1 W0 , (i, j) ∈ {1, . . . , N } × N, by Hi,1 j =

+ , −

j Mg,1 ( + , − )/ R

−

ev∗ θi g−1 p q . +

We want to emphasize that the following statement is not yet a theorem in the strict mathematical sense as the analytical foundations of symplectic field theory, in particular, the necessary transversality theorems for the Cauchy-Riemann operator, are not yet fully established. Since it can be expected that the polyfold project by Hofer and his collaborators sketched in [HWZ] will provide the required transversality theorems, we follow other papers in the field in proving everything up to transversality and state it nevertheless as a theorem. Theorem 1.4. Counting holomorphic curves with one marked point after integrating differential forms and introducing gravitational descendants defines a sequence of distinguished elements Hi,1 j ∈ H∗ −1 W0 , D 0 −1 0 −1 0 in the full SFT homology algebra with differential D 0 = [H0 , ·] : W → W , −1 0 0 which commute with respect to the bracket on H∗ W , D ,

Hi,1 j , H1k, = 0, (i, j), (k, ) ∈ {1, . . . , N } × N .

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Proof. While the boundary equation D 0 ◦ D 0 = 0 is well-known to follow from the identity [H0 , H0 ] = 0, the fact that every Hi,1 j , (i, j) ∈ {1, . . . , N } × N defines an element in the homology H∗ −1 W0 , D 0 follows from the identity H0 , Hi,1 j = 0, since this proves Hi,1 j ∈ ker D 0 . On the other hand, in order to see that any two Hi,1 j , H1k, commute after passing to homology it suffices to prove the identity Hi,1 j , H1k, ± H0 , H2(i, j),(k,) = 0 for any (i, j), (k, ) ∈ {1, . . . , N } × N, where the new Hamiltonian H2(i, j),(k,) is defined below using descendant moduli spaces with two additional marked points. The latter two identities directly follow from our definition of gravitational descendants of moduli spaces based on the definition of coherent sections in tautological line bundles and the compactness theorem in [BEHWZ]. Indeed, in the same way as the identity H0 , H0 = 0 follows from the fact that the codimension-one boundary of every moduli space M0 is formed by products of moduli spaces M1,0 × M2,0 , the second = 0 follows from the fact that the codimension-one boundary of

identity H0 , Hi,1 j

j

j

a descendant moduli space M1 is given by products of the form M1,1 × M2,0 and j

M1,0 × M2,1 .

In order to prove the third identity Hi,1 j , H1k, ± H0 , H2(i, j),(k,) = 0 for every (i, j), (k, ) ∈ {1, . . . , N }×N, we slightly have to enlarge our definition of gravitational descendants in order to include moduli spaces with two additional marked points. For ( j,k) of M2 by this observe that for every pair j, k ∈ N we can define decendants M2 ( j,k)

( j,0)

(0,k)

( j,0)

(0,k)

setting M2 = M2 ∩ M2 , where M2 , M2 ⊂ M2 are defined in the j k same way as M1 , M1 ⊂ M1 by simply forgetting the second or first additional marked point, respectively. Since the boundary of a moduli space of curves with two marked points consists of products of the form M1,1 × M2,1 and M1,0 × M2,2 , M1,2 × M2,0 , ( j,0) j j it follows that the boundary of M2 consists of products M1,1 × M2,1 , M1,1 × M2,1 ( j,0)

( j,0)

and M1,0 ×M2,2 , M1,2 ×M2,0 . Together with the similar result about the boundary (0,k)

of M2

and using the inclusions we hence obtain that the codimension-one boundary

( j,k) j k k j of M2 is given by products of the form M1,1 × M2,1 , M1,1 × M2,1 and M1,0 × ( j,k) ( j,k) M2,2 , M1,2 × M2,0 . While summing over the first two products (with signs) we obtain Hi,1 j , H1k, , summing over the latter two we get H0 , H2(i, j),(k,) , which hence

sum up to zero.

Remark. While the proof suggests that for the above algebraic relations one only has to care about the codimension-one boundary strata of the moduli spaces, it is actually even more important that the coherency condition further ensures that two different codimension-one boundary components can be glued along their common boundary strata of higher codimension.

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As above we further again obtain a rational version of the above statement by expanding H0 and the Hi,1 j in powers of . Corollary 1.5. Counting rational holomorphic curves with one marked point after integrating differential forms and introducing gravitational descendants defines a sequence of distinguished elements hi,1 j ∈ H∗ (P0 , d 0 ),

in the rational SFT homology algebra with differential d 0 = h0 , · : P0 → P0 , which 0 0 commute with respect to the Poisson bracket on H∗ P , d , hi,1 j , h1k, = 0, (i, j), (k, ) ∈ {1, . . . , N } × N . So far we have only considered the case with one additional marked point. On the other hand, the general case with r additional marked points is just notationally more involved. Indeed, as we did in the proof of the above theorem we can easily define for every moduli space Mr with r additional marked points and every r -tuple of natural ( j1 ,..., jr ) ⊂ Mr by setting numbers ( j1 , . . . , jr ) descendants Mr ( j1 ,..., jr )

Mr

( j1 ,0,...,0)

= Mr

(0,...,0, jr )

∩ . . . ∩ Mr

,

(0,...,0, j ,0,...,0)

k where the descendant moduli spaces Mr ⊂ Mr are defined in the same j way as the one-point descendant M1k ⊂ M1 by looking at the r tautological line bundles over the moduli space Mr = Mr ( + , − )/ R separately and forgetting about the other points. With this we can define the descendant Hamiltonian of SFT, which we will continue denoting by H, while the Hamiltonian defined in [EGH] will from now on be called primary. In order to keep track of the descendants we will assign to every chosen differential form θi now a sequence of formal variables ti, j with grading

|ti, j | = 2(1 − j) − deg θi . Then the descendant Hamiltonian H of SFT is defined by + − H= ev∗1 θi1 ∧ . . . ∧ evr∗ θir g−1 t I p q , ( j ,..., jr )

+ , − ,I

Mg,r1

( + , − )/ R −

where p = pγ1+ . . . pγ ++ , q = qγ − . . . qγ − and t I = ti1 , j1 . . . tir , jr for I = n 1 n− ((i 1 , j1 ), . . . , (ir , jr )). Note that expanding the Hamiltonian H in powers of the formal variables ti, j , H = H0 + ti, j Hi,1 j +o(t 2 ), +

i, j

we get back our Hamiltonians H0 and the sequences of descendant Hamiltonians Hi,1 j from above and it is easy to see that the primary Hamiltonian from [EGH] is recovered by setting all formal variables ti, j with j > 0 equal to zero.

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In the same way as it was shown for the primary Hamiltonian in [EGH], the descendant Hamiltonian continues to satisfy the master equation [H, H] = 0, which is just a generalization of the identities for H0 , Hi,1 j and hence can be shown along the same lines by studying the codimension-one boundaries of descendant moduli spaces. On the other hand, expanding H ∈ −1 W in terms of powers of , g−1 Hg , H= g

note that for the rational descendant Hamiltonian h = H0 ∈ P we still have {h, h} = 0. 1.3. Invariance statement. We now turn to the question of independence of these nice algebraic structures from the choices like contact form, cylindrical almost complex structure, abstract polyfold perturbations and, of course, the choice of the coherent collection of sections. This is the content of the following theorem, where we however again want to emphasize that the following statement is not yet a theorem in the strict mathematical sense as the analytical foundations of symplectic field theory, in particular, the neccessary transversality theorems for the Cauchy-Riemann operator, are not yet fully established. Theorem 1.6. For different choices of contact form λ± , cylindrical almost complex structure J ± , abstract polyfold perturbations and sequences of coherent collections of sections (s ± ) the resulting systems of commuting operators Hi,1,− j on −1 0,− 0,− j −1 0,+ 0,+ 1,+ H∗ W , D and Hi, j on H∗ W , D are isomorphic, i.e., there exists −1 0,− 0,− an isomorphism of the Weyl algebras H∗ W , D and H∗ −1 W0,+ , D 0,+ −1 0,− , D 0,− to H1,+ ∈ H −1 W0,+ , D 0,+ . which maps Hi,1,− ∗ j ∈ H∗ W i, j As above we clearly also get a rational version of the invariance statement: Corollary 1.7. For different choices of contact form λ± , cylindrical almost complex structure J ± , abstract polyfold perturbations and sequences of coherent collec1,− tions of sections (s ± j ) the resulting system of Poisson-commuting functions hi, j on H∗ (P0,− , d 0,− ) and hi,1,+j on H∗ (P0,+ , d 0,+ ) are isomorphic, i.e., there exists an isomorphism of the Poisson algebras H∗ (P0,− , d 0,− ) and H∗ (P0,+ , d 0,+ ) which maps 0,− 0,− hi,1,− , d ) to hi,1,+j ∈ H∗ (P0,+ , d 0,+ ). j ∈ H∗ (P This theorem is an extension of the theorem in [EGH] which states that for different choices of auxiliary data the small Weyl algebras H∗ −1 W0,− , D 0,− and H∗ −1 W0,+ , D 0,+ are isomorphic. On the other hand, assuming that the contact form, the cylindrical almost complex structure and also the abstract polyfold sections are fixed to have well-defined moduli spaces, the isomorphism of the homology algebras is the identity and hence the theorem states the sequence of commuting operators is indeed independent of the chosen sequences of coherent collections of sections (s ± j ), 1,+ −1 0 0 . Hi,1,− j = Hi, j ∈ H∗ W , D For the proof we have to extend the proof in [EGH] to include gravitational descendants. To this end we have to study sections in the tautological line bundles over moduli spaces of holomorphic curves in symplectic manifolds with cylindrical ends.

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ω) be a symplectic manifold with cylindrical ends R+ ×V + , λ+ and −Let (W, R ×V − , λ− in the sense of [BEHWZ] which is equipped with an almost complex structure J which agrees with the cylindrical almost complex structures J ± on R+ ×V + . Then we study J -holomorphic curves in W which are asymptotically cylindrical over chosen collections of orbits ± = {γ1± , . . . , γn±± } of the Reeb vector fields R ± in V ± as the R± -factor tends to ±∞, see [BEHWZ], and denote by Mg,r ( + , − ) the corresponding moduli space of genus g curves with r additional marked points ([BEHWZ,EGH]). Possibly after choosing abstract perturbations using polyfolds, obstruction bundles or domain-dependent structures, which agree with chosen abstract perturbations in the boundary as described above, we find that Mg,r ( + , − ) is a weighted branched manifold of dimension equal to the Fredholm index of the Cauchy-Riemann operator for J . Note that as remarked above we will for simplicity assume that moduli space is indeed a manifold with corners, since this will be sufficient for our example and we expect that all the upcoming constructions can be generalized in an appropriate way. We further extend the chosen differential forms θ1± , . . . θ N± on V ± to differential forms θ1 , . . . , θ N on W as described in [EGH]. From now on let Mr denote the moduli space Mg,r ( + , − ) of holomorphic curves in W for chosen collections of Reeb orbits + , − . Note in particular that there is no longer an R-action on the moduli space which we have to quotient out. In order to distinguish these moduli spaces in non-cylindrical manifolds from those of holomorphic ± curves in the cylindrical manifolds, we will use the short-hand notation Mr for mod+ − ± uli spaces Mg,r ( , )/ R of holomorphic curves in R ×V , respectively. Like in Gromov-Witten theory we can introduce r tautological line bundles L1 , . . . , Lr , where the fibre of Li over a punctured curve (u, z 1 , . . . , zr ) ∈ Mr in the noncompactified moduli space is again given by the cotangent line to the underlying closed Riemann surface at the i th marked point and which formally can be defined as the pull-back of the vertical cotangent line bundle under the canonical section σi of π : Mr +1 → Mr mapping to the i th marked point in the fibre. For notational simplicity let us again restrict to the case r = 1. Following the compactness statement in [BEHWZ] the codimension-one boundary of M1 now consists of curves with one non-cylindrical level and one cylindrical level (in the sense of [BEHWZ]), whose moduli spaces can now be represented as products M1,1 × + − + − M2,0 , M1,1 × M2,0 or M1,0 × M2,1 , M1,0 × M2,1 of moduli spaces of strictly lower dimension, where the marked point sits on the first or the second level. Again note that here and in what follows for product moduli spaces the first index refers to the level and not to the genus of the curve. Furthermore it follows from the definition of the tautological + − line bundle L over M1 that over the boundary components M1,1 ×M2,0 , M1,1 ×M2,0 +

−

and M1,0 × M2,1 , M1,0 × M2,1 it is given by L |M

+ 1,1 ×M2,0

= π1∗ L1 , L |M ×M+ = π2∗ L+2 , 1,0 2,1

= π2∗ L2 , L |M− ×M = π1∗ L− 1 , L |M− 2,0 1,1 1,0 ×M2,1 (−)

(+)

(−)

(+)

where L1 , L2 denotes the tautological line bundle over the moduli space M1,1 , M2,1 and π1 , π2 is the projection onto the first or second factor, respectively. With this we can now introduce collections of sections in (tensor products of) tautological line bundles coherently connecting two chosen coherent collections of sections.

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Definition 1.8. Let W be a symplectic manifold with cylindrical ends V ± and let (s± ) be two coherent collections of sections in the tautological line bundles L± over all ± moduli spaces M1 of J -holomorphic curves with one additional marked point in the cylindrical manifolds R ×V ± . Assume that we have chosen transversal sections s in the tautological line bundles L over all moduli spaces M1 of J -holomorphic curves in the non-cylindrical manifold W with one additional marked point. Then this collection of sections (s) is called coherently connecting (s−) and (s+ ) if for every section s in L over a moduli space M1 the following holds: Over every codimension-one boundary + − + − component M1,1 × M2,0 , M1,1 × M2,0 and M1,0 × M2,1 , M1,0 × M2,1 of M1 the section s agrees with the pull-back π1∗ s1 , π1∗ s1− or π2∗ s2+ , π2∗ s2 of the chosen sec(−)

tions s1,(−) , s2,(+) in the tautological line bundles L1 respectively.

(−)

(+)

over M1,1 , L2

(+)

over M2,1 ,

Note that one can always find collections of sections (s) coherently connecting given coherent collections of sections (s+ ) and (s− ) as before by using induction on the dimension of the underlying moduli space. Indeed, for the induction step observe that the coherency condition again fixes the section on the boundary of the moduli space, so that the desired section can be obtained by simply extending the section from the boundary to the interior of the moduli space in an arbitrary way. For a given coherently connecting collection of sections (s) we will again define for every moduli space 1

M1 = s −1 (0) ⊂ M1 . As an immediate consequence of the above definition we find that the components 1 1 + 1,− of the codimension-one boundary of M1 are given by products M1,1 × M2,0 , M1,1 × −

1,+

1

1,(−)

M2,0 and M1,0 × M2,1 , M1,0 × M2,1 , where M1,1 (−)

(−)

(+)

−1 −1 = s1,(−) (0), M2,1 = s2,(+) (0) 1,(+)

(+)

for the section s1,(−) in L1 over M1,1 , s2,(+) in L2 over M2,1 , respectively. As before we can use this result as an induction start to obtain for every moduli space M1 a sequence j j−1 of nested subspaces M1 ⊂ M1 ⊂ M1 . Definition 1.9. Let j ∈ N and let (s j,± ) be two coherent collections of sections in the j-fold tensor products L±,⊗ j of the tautological line bundles over the j −1st gravitational j−1,± ± ⊂ M1 of all moduli spaces of curves in the cylindrical manifolds descendants M1 R ×V ± . Assume that for all moduli spaces of curves in the non-cylindrical manifold W j−1 we have chosen M1 ⊂ M1 such that the components of the codimension-one boundj−1 j−1 + j−1,− ary of M1 are given by products of the form M1,1 × M2,0 , M1,1 × M2,0 and j−1,+

M1,0 ×M2,0

−

j−1

, M1,0 ×M2,0 . Then we again call a collection of transversal sections j−1

(s j ) in the j-fold tensor products L⊗ j of the tautological line bundles over M1 ⊂ M1 coherently connecting (s j,− ) and (s j,+ ) if for every section s j the following holds: Over j−1

+

j−1,−

every codimension-one boundary component M1,1 × M2,0 , M1,1

j−1,+ − j−1 j−1 M1,0 × M2,1 , M1,0 × M2,1 of M1 the section s j agrees ∗ ∗ ∗ ∗ π1 s1, j , π1 s1, j,− or π2 s2, j,+ , π2 s2, j of the section s1, j,(−) , s2, j,(+) j−1,(−) j−1,(+) (−),⊗ j (+),⊗ j L1 over M1,1 , L2 over M2,1 , respectively.

× M2,0 and

with the pull-back in the line bundle

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With this we can now introduce gravitational descendants of moduli spaces for symplectic manifolds with cylindrical ends. Definition 1.10. Assume that we have the inductively defined subsequence of nested j j−1 j j−1 for a colsubspaces M1 ⊂ M1 ⊂ M1 by requiring that M1 = s −1 j (0) ⊂ M1 j−1

lection of sections s j in the line bundles L⊗ j over the moduli spaces M1

coherently j

connecting the coherent collections of sections (s j,− ) and (s j,+ ). Then we call M1 the j th (gravitational) descendant of M1 . In order to prove the above invariance theorem we now recall the extension of the algebraic formalism of SFT from cylindrical manifolds to symplectic cobordisms with cylindrical ends as described in [EGH]. Let D0 be the space of formal power series in the variables , pγ+ with coefficients which are polynomials in the variables qγ− . Elements in W0,± then act as differential operators from the right/left on D0 via the replacements ←−− −−→ ∂ ∂ + − q γ → κγ + , p γ → κγ − . ∂ pγ ∂qγ Apart from the potential F0 ∈ −1 W0 counting only curves in W with no additional marked points, + − F0 = #Mg,0 ( + , − ) g−1 p q ,

+ , −

we now want to use the extensions θi , i = 1, . . . , N on W of the chosen differential j j forms θ1± , . . . θ N± on V ± and these sequences M1 = Mg,1 ( + , − ) of gravitational descendants to define sequences of new SFT potentials Fi,1 j , (i, j) ∈ {1, . . . , N } × N, by + − Fi,1 j = ev∗ θi g−1 p q . j

+ , −

Mg,1 ( + , − )

For the potential counting curves with no additional marked points we have the following identity, where we however again want to emphasize that the following statement should again be understood as a theorem up to the transversality problem in SFT. Theorem ([EGH]). The potential F0 ∈ −1 D satisfies the master equation −− −−→ 0 0← eF H0,+ − H0,− eF = 0. In [EGH] it is shown that this implies that 0 0 ←−− 0 0 0 −−→ 0 D F : −1 D0 → −1 D0 , D F g = e− F H0,− geF − (−1)|g| geF H0,+ e− F 0

0

satisfies D F ◦ DF = 0, and hence can be used to define the homology algebra 0 H∗ −1 D0 , D F . Furthermore it is shown that the maps F 0,− : −1 W0,− → −1 D0 , F 0,+ : −1 W0,+ → −1 D0 ,

0− → 0 f → e− F f e+ F , 0← 0 − f → e+ F f e− F

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commute with the boundary operators, 0

F 0,± ◦ D 0,± = D F ◦ F 0,± , and hence descend to maps between the homology algebras 0 F∗0,± : H∗ −1 W0,± , D 0,± → H∗ −1 D0 , D F . Now assume that the contact forms λ+ and λ− are chosen such that they define the same contact structure (V + , ξ + ) = (V − , ξ − ) =: (V, ξ ) and let W = R ×V be the topologically trivial cobordism. Then in [EGH] the authors prove (up to transversality) the following fundamental theorem. Theorem ([EGH]). The map −1 F∗0,+ ◦ F∗0,− : H∗ −1 W0,− , D 0,− → H∗ −1 W0,+ , D 0,+ is an isomorphism of graded Weyl algebras. For the proof of the invariance statement we want to show that this map identifies the −1 0,± , D 0,± ). In order to get the sequences Hi,1,± j , (i, j) ∈ {1, . . . , N } × N on H∗ ( W right idea for the proof, it turns out to be useful to even enlarge the picture as follows. Precisely in the same way as for cylindrical manifolds we can define for every tuple ( j1 ,..., jr ) ( j1 , . . . , jr ) of natural numbers gravitational descendants M ⊂ M1 of moduli spaces of curves in non-cylindrical manifolds with more than one additional marked point, which are collected in the descendant potential F ∈ −1 D, where D is again obtained from D0 by considering coefficients which are formal powers in the graded formal variables ti, j , (i, j) ∈ {1, . . . , N } × N. Assuming for the moment that we have proven the fundamental identity ← − −→ e F H+ − H− e F = 0 and expanding the potential F ∈ −1 D and the two Hamiltonians H± ∈ −1 W± in powers of the t-variables, 2 F = F0 + ti, j Fi,1 j +o(t 2 ), H± = H0,± + ti, j Hi,1,± j +o(t ), i, j

i, j

we can deduce besides the master equation for F0 , −− −−→ 0 0← eF H0,+ − H0,− eF = 0 and other identities also the identity 0 ←−− −− −−→ 0 −−→ 0 0← F 0,− F 1 F 1 0,+ eF Hi,1,+j − Hi,1,− e − e e = H F F i, j i, j H , j about F0 , Fi,1 j and H0,± , Hi,1,± j , where we used that ⎛ ⎞ 0 eF = eF · ⎝1 + ti, j Fi,1 j ⎠ + o t 2 . i, j

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Proof of the theorem. Instead of proving the master equation for the full descendant potential F, we first show that it suffices to prove 0 ←−− −− −−→ 0 −−→ 0 0← F = H0,− eF Fi,1 j − eF Fi,1 j H0,+ . eF Hi,1,+j − Hi,1,− j e Indeed, it is easy to see that the desired identity implies that −− 0← 0 0 −−→ + F0 = e+ F Hi,1,+j e− F − e− F Hi,1,− F 0,+ Hi,1,+j − F 0,− Hi,1,− j j e is equal to 0 0 ←−− 0 −−→ 0 0 e− F H0,− e+ F Fi,1 j − e+ F Fi,1 j H0,+ e− F = D F Fi,1 j , so that, after passing to homology, we have 0 F∗0,+ Hi,1,+j = F∗0,− Hi,1,− ∈ H∗ −1 D0 , D F j as desired. On the other hand, the above identity directly follows from our definition of gravitational descendants of moduli spaces based on the definition of coherently connecting sections in tautological line bundles and the compactness theorem in [BEHWZ]. Indeed, in the same way as it is shown in [EGH] that the master equation for F0 and H0,± follows from the fact that the codimension-one boundary of every moduli space M0 is + − formed by products of moduli spaces M1,0 × M2,0 and M1,0 × M2,0 , the desired identity relating F0 , Fi,1 j and H0,± , Hi,1,± j can be seen to follow from the fact that the j

codimension-one boundary of a descendant moduli space M1 is given by products of j + j,− j,+ − j the form M1,1 × M2,0 , M1,1 × M2,0 and M1,0 × M2,1 , M1,0 × M2,1 : While the two 1,+ summands involving F0 and Hi,1,− j , Hi, j on the left-hand-side of the equation collect all j,−

j,+

boundary components of the form M1,1 × M2,0 , M1,0 × M2,1 , the two summands involving Fi,1 j and H0,− , H0,+ on the right-hand-side of the equation collect all boundary −

j

j

+

components of the form M1,0 × M2,1 , M1,1 × M2,0 , respectively. Note that as for the master equation for F0 and H0,± the appearance of F0 in the exponential follows from the fact that there corresponding curves may appear with an arbitrary number of 1 connected components, while the curves counted for in H0,± , Hi,1,± j , Fi, j can only appear once due to index reasons or since there is just one additional marked point. 1,+ Finally, in order to see why we actually have Hi,1,− j = Hi, j on homology if we fixed − + − + λ = λ = λ, J = J = J and the abstract polyfold perturbations to have well0 defined moduli spaces, observe that in this case F just counts orbit cylinders, so that F 0,± and hence F 0,± ∗ is the identity.

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1.4. The circle bundle case. In this subsection we briefly want to discuss the important case of circle bundles over closed symplectic manifolds, which links our constructions to gravitational descendants in Gromov-Witten theory, see also [R]. For this recall that to any closed symplectic manifold (M, ω) with integral symplectic form [ω] ∈ H 2 (M, Z) one can canonically assign a principal circle bundle π : V → M over (M, ω) by requiring that c1 (V ) = [ω]. Furthermore, it is easy to see that an S 1 -connection form λ with curvature ω on π : V → M is a contact form on the total space V , where the underlying contact structure agrees with the corresponding horizontal plane field ξ = ker λ, while the Reeb vector field R agrees with the infinitesimal generator of the S 1 -action. Observe that a ω-compatible almost complex structure J on M naturally equips R ×V with a cylindrical almost complex structure by requiring that J maps the Reeb vector field to the R-direction and agrees with J on the horizontal plane field ξ , which is naturally identified with T M. Since every fibre of the circle bundle is hence a closed Reeb orbit for the contact form λ, it follows that the space of orbits is given by M × N, where the second factor just refers to the multiplicity of the orbit. Hence, while every contact form in this class is not Morse as long as the symplectic manifold is not a point, it is still of Morse-Bott type. Following [EGH] the Weyl algebra W0 in this Morse-Bott case is now generated by sequences of graded formal variables pα,k , qα,k , k ∈ N assigned to cohomology classes α forming a basis of H ∗ (M, Z). For circle bundles in the Morse-Bott setup we now show that the general theorem from above leads to the following stronger statement. Note that in the following theorem we do not assume that the sequences of coherent collections of sections are neccessarily S 1 -invariant. Theorem 1.11. For circle bundles over symplectic manifolds, which are equipped with S 1 -invariant contact forms, cylindrical almost complex structures (and abstract polyfold perturbations) as described above, the descendant Hamiltonians Hi,1 j define a sequence of commuting operators on W0 , which is independent of the auxiliary data. Proof. Observing that a map u˜ : (, j) → (R ×V, J ) from a punctured Riemann sphere to the cylindrical manifold R ×V , which is equipped with the canonical cylindrical almost complex structure J defined by the ω-compatible almost complex structure J on M, can be viewed as tuple (h, u), where u : (, j) → (M, J ) is a J -holomorphic curve in M and h is a holomorphic section in R ×u ∗ V → , it is easy to see that every moduli space studied in SFT for the contact manifold V carries a natural circle bundle structure after quotienting out the natural R-action. It follows that D 0 = 0, so that by our first theorem the H1j already commute as elements in W0 . On the other hand, as long as the two different collections of auxiliary structures for V are actually obtained as pull-backs of the corresponding auxiliary structures on M, it follows in the same way that the only rigid holomorphic curves in the resulting cobordisms are the orbit cylinders, so that the resulting automorphism is indeed the identity. For S 1 and S 3 Eliashberg already pointed out in his ICM 2006 talk, see [E], that the corresponding sequences h1j counting only genus zero curves lead to classical integrable systems, while the sequences of commuting operators H1j provide deformation quantizations for these hierarchies. This is based on the surprising fact that the sequence h1j of Poisson-commuting functions actually agrees with the integrable system for genus zero from Gromov-Witten theory obtained using the underlying Frobenius manifold

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structure. In particular, for V = S 1 it follows that that the resulting system of Poissoncommuting functions are precisely the commuting integrals of the dispersionless KdV hierarchy, u j+2 (x) d x, u(x) = h1j = pn e+2πinx + qn e−2πinx , 1 ( j + 2)! S n∈N

while in the case of the Hopf fibration V = S 3 over M = S 2 one arrives at the Poissoncommuting integrals of the continuous limit of the Toda lattice. In order to see why in genus zero the SFT of the circle bundle V is so closely related to the Gromov-Witten theory of its symplectic base M, we recall from the proof of the theorem that every J -holomorphic curve u˜ can be identified with a tuple (h, u), where u is a J -holomorphic curve in M and h is a holomorphic section in R ×u ∗ V → , whose poles and zeroes correspond to the positive and negative punctures with multiplicities. Since the zeroth Picard group of S 2 is trivial and hence every degree zero divisor is indeed a principal divisor, it follows that for every map u the space of sections is isomorphic to C and hence that the SFT moduli space of J -holomorphic curves in R ×V is indeed a circle bundle over the corresponding Gromov-Witten moduli space of J -holomorphic curves in M. While this explains the close relation of SFT of circle bundles and Gromov-Witten theory in the genus zero case, the non-triviality of the Picard group for nonzero genus implies that the relation gets much more obscure when we allow for curves of arbitrary genus. Indeed, while in the case of V = S 1 the sequence H1j defined by counting curves of arbitrary genus in R ×V leads to the deformation quantization of the dispersionless KdV hierarchy, in particular, a quantum integrable system, counting curves of all genera in the underlying symplectic manifold, that is, the point, leads by Witten’s conjecture to the classical integrable system given by the full KdV hierarchy as proven by Kontsevich. At the end of this subsection we again want to emphasize that the above statement crucially relies on the fact that V is equipped with a S 1 -invariant contact form, cylindrical almost complex structure and abstract polyfold perturbations. Assuming for the moment that the sequences of coherent collections of sections are also chosen to be S 1 -invariant, note that in this case the above invariance statement can directly be deduced from the independence of the descendant Gromov-Witten potential of the auxiliary data used to define it, which essentially relies on the fact that all moduli spaces have only boundary components of codimension greater than or equal to two, so that absolute rather than relative virtual classes are defined. In particular, the gravitational descendants can be defined by integrating powers of the first Chern class over the absolute moduli cycle. On the other hand, recall that for the above theorem we did not require that the sequences of coherent collections of sections are neccessarily S 1 -invariant. While our definition of coherent collections of sections seems to be very weak, our above theorem shows that the nice invariance property continues to hold even for a larger class of sections. 2. Example: Symplectic Field Theory of Closed Geodesics 2.1. Symplectic field theory of a single Reeb orbit. We are now going to consider a concrete example, which actually formed the starting point for the formal discussion from above. As above consider a closed contact manifold V with chosen contact form λ ∈ 1 (V ) and let J be a compatible cylindrical almost complex structure on R ×V . For any closed

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orbit γ of the corresponding Reeb vector field R on V the orbit cylinder R ×γ together with its branched covers are the basic examples of J -holomorphic curves in R ×V . In [F2] we prove that these orbit curves do not contribute to the algebraic invariants of symplectic field theory as long as they do not carry additional marked points. Our proof explicitly uses that the orbit curves (over a fixed orbit) are closed under taking boundaries and gluing, which follows from the fact that orbit curves are also trivial in the sense that they have trivial contact area and that this contact area is preserved under taking boundaries and gluing. In particular, it follows, see [F2], that every algebraic invariant of symplectic field theory has a natural analog defined by counting only orbit curves. Further specifying the underlying Reeb orbit let us hence introduce the symplectic field theory of the Reeb orbit γ : For this denote by W0γ the graded Weyl subalgebra of the Weyl algebra W, which is generated only by those p- and q-variables pn = pγ n , qn = qγ n corresponding to Reeb orbits which are multiple covers of the fixed orbit γ and which are good in the sense of [BM]. In the same way we further introduce the Poisson subalgebra P0γ of P0 . It will become important that the natural identification of the formal variables pn and qn does not lead to an isomorphism of the graded algebras W0γ and P0γ with the corresponding graded algebras W0S 1 and P0S 1 for γ = V = S 1 , not only since the gradings of pn and qn are different and hence even the commutation rules may change but also that variables pn and qn may not be there since they would correspond to bad orbits. In the same way as we introduced the (rational) Hamiltonian H0 and h0 as well as sequences of descendant Hamiltonians H1j and h1j by counting general curves in the symplectization of a contact manifold, we can define distinguished elements H0γ ∈ −1 W0γ and h0γ ∈ P0γ , as well as sequences of descendant Hamiltonians H1γ , j and h1γ , j by just counting branched covers of the orbit cylinder over γ with signs (and weights), where the preservation of the contact area under splitting and gluing of curves proves that for every theorem from above we have a version for γ . While for the general part described above we have already emphasized that the theorems are not yet theorems in the strict mathematical sense since the necessary transversality theorems for the Cauchy-Riemann operator are part of the on-going polyfold project by Hofer and his collaborators and we further used the assumption that all occurring moduli spaces are manifolds with corners, for the rest of this paper we will restrict to the rational case, i.e., we will only be interested in the Poisson-commuting sequences h1γ , j on H∗ (P0γ , dγ0 ), but in return solve the occurring analytical problems in all detail. In particular, we have already proven in the paper [F2] that for (rational) orbit curves the transversality problem can indeed be solved using finite-dimensional obstruction bundles instead of infinite-dimensional polybundles. In order to see why this is even neccessary, observe that while in the case when γ = V = S 1 the Fredholm index equals the dimension of the moduli space, for general γ ⊂ V the Fredholm index of a true branched cover is in general strictly smaller than the dimension of the moduli space of branched covers, so that transversality for the Cauchy-Riemann operator can in general not be satisfied. So let us recall the main results about obstruction bundle transversality for orbit curves, where we refer to [F2] for all details. The first observation for orbit curves is that the cokernels of the linearized Cauchy-Riemann operators indeed fit together to give a smooth vector bundle Coker ∂¯ J over the compactified (nonregular) moduli spaces M of orbit curves (of constant rank). It follows that every transveral section ν¯ of this cokernel bundle leads to a compact perturbation making the Cauchy-Riemann operator transversal to the zero section in the underlying polyfold setup.

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In Gromov-Witten theory we would hence obtain the contribution of the regular perturbed moduli space by integrating the Euler class of the finite-dimensional obstruction bundle over the compactified moduli space. On the other hand, passing from GromovWitten theory back to symplectic field theory again, we see that we just arrive at the same problem we had to face with when we wanted to define gravitational descendants in symplectic field theory. Indeed, as for the tautological line bundles, the presence of codimension-one boundary of the (nonregular) moduli spaces of branched covers implies that Euler numbers for sections in the cokernel bundles are not defined in general, since the count of zeroes depends on the compact perturbations chosen for the moduli spaces in the boundary. Instead of looking at a single moduli space, we hence again have to consider all moduli spaces at once. Replacing the tautological line bundle L by the cokernel bundle Coker ∂¯ J and considering the nonregular moduli space of branched covers instead of the regular moduli space itself, we hence now define coherent collections of sections in the obstruction bundles Coker ∂¯ J over all moduli spaces M as follows. Following the compactness statement in [BEHWZ] for the contact manifold S 1 the codimension-one boundary of every moduli space of branched covers M again consists of curves with two levels (in the sense of [BEHWZ]), whose moduli spaces can be represented as products M1 ×M2 of moduli spaces of strictly lower dimension, where the first index again refers to the level. On the other hand, it follows from the linear gluing result in [F2] that over the boundary component M1 × M2 the cokernel bundle Coker ∂¯ J is given by 1 2 Coker ∂¯ J |M1 ×M2 = π1∗ Coker ∂¯ J ⊕ π2∗ Coker ∂¯ J , 1 2 where Coker ∂¯ J , Coker ∂¯ J denotes the cokernel bundle over the moduli space M1 , M2 and π1 , π2 is the projection onto the first or second factor, respectively. Assuming that we have chosen sections ν¯ in the cokernel bundles Coker ∂¯ J over all moduli spaces M of branched covers, we again call this collection of sections (¯ν ) coherent if over every codimension-one boundary component M1 × M2 of a moduli space M the corresponding section ν¯ agrees with the pull-back π1∗ ν¯ 1 ⊕ π2∗ ν¯ 2 of the 1 2 chosen sections ν¯ 1 , ν¯ 2 in the cokernel bundles Coker ∂¯ J over M1 , Coker ∂¯ J over M2 , respectively. Since in the end we will again be interested in the zero sets of these sections, we will again assume that all occurring sections are transversal to the zero section. As before it is not hard to see that one can always find such coherent collections of (transversal) sections in the cokernel bundles by using induction on the dimension of the underlying nonregular moduli space of branched covers. Note that the latter is not equal to the Fredholm index. In [F2] we prove the following result about orbit curves with no additional marked points.

Theorem ([F2]). For the cokernel bundle Coker ∂¯ J over the compactification M of every moduli space of branched covers over an orbit cylinder with dim M − rank Coker ∂¯ J = 0 the following holds: • For every pair ν¯ 0 , ν¯ 1 of coherent and transversal sections in Coker ∂¯ J the algebraic count of zeroes of ν¯ 0 and ν¯ 1 are finite and agree, so that we can define an Euler

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number χ Coker ∂¯ J for coherent sections in Coker ∂¯ J by χ Coker ∂¯ J := (¯ν 0 )−1 (0) = (¯ν 1 )−1 (0). • This Euler number is χ Coker ∂¯ J = 0. This theorem in turn has the following consequence. Corollary 2.1. For every closed Reeb orbit γ the Hamiltonian h0γ vanishes independently of the chosen coherent collection of sections (¯ν ) in the cokernel bundles over all moduli spaces of branched covers, h0 = h0,¯ν = 0. In particular, the sequences of descendant Hamiltonians h1γ , j already Poisson-commute as elements in P0γ . Note that the latter statement is obvious in the case γ = V = S 1 . While it directly follows from index reasons that h1S 1 , j = 0 when the string of differential forms just con-

sists of the zero-form 1 on S 1 , it is shown in [E] using the results from Okounkov and Pandharipande in [OP] that for the one-form dt on S 1 the system of Poisson commuting functions on P0S 1 is given by h1S 1 , j =

S1

u j+2 (x) d x, u(x) = pn e+2πinx + qn e−2πinx , ( j + 2)! n∈N

i.e., hence agrees with the dispersionless KdV (or Burger) integrable hierarchy. Going back from γ = V = S 1 to the case of orbit curves over general Reeb orbits γ , observe that, since for the orbit curves the evaluation map to V factors through the inclusion map γ ⊂ V , it follows that it again only makes sense to consider zero- or one-forms, where we can assume without loss of generality that the zero-form agrees with 1 ∈ 0 (V ) and that the integral of the one-form θ ∈ 1 (V ) over the Reeb orbit is one, θ = 1. γ

For the case with no gravitational descendants, note that it follows from index reasons that the only curves to be considered are orbit cylinders with one marked point, since introducing an additional marked point adds two or one to the Fredholm index. Since orbit cylinders are always regular and their contribution hence just equals the integral of the form θ over the closed orbit γ , we hence get just like in the case of γ = V = S 1 that the zeroth descendant Hamiltonian h1γ ,0 vanishes if deg θ = 0 and h1γ ,0 =

S1

u 2 (x) dx = pn q n 2!

if deg θ = 1 with the normalization from above. For the sum note that we only assigned formal variables pn , qn to Reeb orbits which are good in the sense of [BM]. While the Hamiltonians h1γ ,0 hence agree with the Hamiltonian h1S 1 ,0 for γ = V = S 1 up to the problem of bad orbits, since no obstruction bundles have to be considered, it is

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easy to see that the argument breaks down when gravitational descendants are introduced, since the underlying orbit curve then has non-zero Fredholm index 1 + 2( j − 1) + deg θ and hence need not be an orbit cylinder anymore. While for the case of a one-form we can hence expect to find new integrals for the nontrivial Hamiltonian h1S 1 ,0 = h1γ , j , we first show that in the case of a zero-form not only the zeroth Hamiltonian but even the whole sequence of descendant Hamiltonians h1γ , j is trivial. Theorem 2.2. Let γ be a Reeb orbit in any contact manifold V and assume that the string of differential forms on V just consists of the zero-form 1 ∈ 0 (V ). Then the sequence of Poisson-commuting functions h1γ , j on P0γ is trivial, h1γ , j = 0,

j ∈N

just like in the case of γ = V = S 1 . Proof. Since the proof of this theorem follows from completely the same arguments as the proof of our theorem in [F2] about Euler numbers of coherent sections in obstruction bundles from above, we shortly give the main idea for the proof in [F2] about orbit curves without additional marked points and then discuss its generalization to orbit curves with zero-forms and gravitational descendants. After proving that we can work with finite-dimensional obstruction bundles instead of infinite-dimensional polybundles, recall that the main problem lies in the presence of codimension-one boundary of the (nonregular) moduli space, so that Euler numbers of Fredholm problems are not defined in general, since the count of zeroes in general depends on the compact perturbations chosen for the moduli spaces in the boundary. In [F2] we prove the existence of the Euler number for moduli spaces of orbit curves without additional marked points by induction on the number of punctures. For the induction step we do not only use that there exist Euler numbers for the moduli spaces in the boundary, but it is further important that all these Euler numbers are in fact trivial. The vanishing of the Euler number in turn is deduced from the different parities of the Fredholm index of the Cauchy-Riemann operator and the actual dimension of the moduli space of branched covers following the idea for the vanishing of the Euler characteristic for odd-dimensional manifolds. For the generalization to the case of additional marked points and gravitational descendants, it is clear that it still suffices to work with finite-dimensional obstruction bundles. On the other hand, recall that the only further ingredient to our proof in [F2] was that the Fredholm index and the dimension of the moduli spaces always have different parity. Hence it follows that the proof in [F2] also works for the case when θ is a zero-form as the actual dimension of the moduli spaces is still even, while it breaks down in the case when θ is a one-form. Observe that for one-forms it is indeed no longer clear that the every Euler number has to be zero, as we for γ = V = S 1 and θ = dt we get nontrivial contributions from true branched covers. While at first glance the major problem seems to be the truly complicated computation of the Euler number (see [HT1,HT2] for related results), we further have the problem that Euler numbers need no longer exist for all Fredholm problems. For the rest of this paper we will hence only be interested in the case where the chosen differential form has degree one, deg θ = 1. While for γ = V = S 1 we actually get a unique sequence of Poisson-commuting functions, observe that for general fixed Reeb orbits γ in contact manifolds V the

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ν descendant Hamiltonians h1γ , j = h1,¯ γ , j may indeed depend on the chosen collection of sections in the cokernel bundles Coker ∂¯ J . Hence the invariance statement is no longer trivial, but implies that for different choices of coherent abstract perturbations ν¯ ± for the moduli spaces the resulting system of commuting elements h1,− γ , j , j = 0, 1, 2, .. and

0 h1,+ γ , j , j = 0, 1, 2, .. on Pγ are just isomorphic, i.e., there exists an automorphism of the

1,+ 0 0 Poisson algebra P0γ which identifies h1,− γ , j ∈ Pγ with hγ , j ∈ Pγ for all j ∈ N. The above discussion hence shows that the computation of the symplectic field theory of a closed Reeb orbit gets much more difficult when gravitational descendants are considered. In what follows we want to determine it in the special case where the contact manifold is the unit cotangent bundle S ∗ Q of a (m-dimensional) Riemannian manifold Q, so that every closed Reeb orbit γ on V = S ∗ Q corresponds to a closed geodesic γ¯ on Q. Before we can state the theorem we first want to expand the descendant Hamiltonians h1S 1 , j in terms of the pn - and qn -variables, where set pn = q−n . Abbreviating

u n (x) = qn einx for every nonzero integer n it follows from u = n u n that u n 1 (x) · . . . · u n j+2 (x) u j+2 (x) h1S 1 , j = dx = d x. ( j + 2)! S 1 ( j + 2)! S1

On the other hand, note that the integration around the circle corresponds to selecting only those sequences of multiplicities (n 1 , . . . , n j+2 ), whose sum is equal to zero, so that qn 1 · . . . · qn j+2 . h1S 1 , j = ( j + 2)! n 1 +...+n j+2 =0

Apart from the sequence of Poisson-commuting functions for the circle, the grading of the functions given by the grading of pn - and qn -variables will play a central role for the upcoming theorem. For this observe that it follows from the grading conventions in symplectic theory that the grading of the full Hamiltonian H0 is −1, so that

field 0 g−1 by H = g H0g the grading for the rational Hamiltonian h0 = H00 is given by 0 0 | h | = | H | + || = −1 + 2(m − 2). Since this grading has to agree with the grading of t j h1j with |t j | = 2(1 − j) − deg θ = 1 − 2 j, it follows that for every Reeb orbit γ ⊂ V we have 1 hγ , j = −1 + 2(m − 2) − 1 + 2 j = 2(m + j − 3). We already mentioned that the natural identification of the formal variables pn and qn does not lead to an isomorphism of the graded algebras W0γ and P0γ with the corresponding graded algebras W0S 1 and P0S 1 for γ = V = S 1 , not only since the gradings of pn and qn are different and hence even the commutation rules may change but even that variables pn and qn may not be there since they would correspond to bad orbits. While for the grading of γ = V = S 1 given by | pn | = |qn | = −2 in the descendant Hamiltonians h1S 1 , j every summand indeed has the same degree 2(m + j − 3), passing over to a general Reeb orbit γ with the new grading given by | pn | = m − 3 − CZ(γ n ), |qn | = m − 3 + CZ(γ n ) the descendant Hamiltonian h1S 1 , j is no longer of pure degree, i.e., different summands of the same descendant Hamiltonian usually have different degree. While the Poissoncommuting sequence for the circle seems not to be related to the sequence of descendant

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Hamiltonians for general Reeb orbits γ , we prove the following result in the case when the Reeb orbit corresponds to a closed geodesic. Theorem 2.3. Assume that the contact manifold is the unit cotangent bundle V = S ∗ Q of a Riemannian manifold Q, so that the closed Reeb orbit γ corresponds to a closed geodesic γ¯ on Q, and that the string of differential forms just consists of a single one-form which integrates to one around the orbit. Then the resulting system of Poisson-commuting functions h1γ , j , j ∈ N on P0γ is isomorphic to the system of Poisson-commuting functions gγ1¯ , j , j ∈ N on Pγ0¯ = P0γ , where for every j ∈ N the descendant Hamiltonian gγ1¯ , j is given by gγ1¯ , j =

( n)

qn 1 · . . . · qn j+2 ( j + 2)!

,

where the sum runs over all ordered monomials qn 1 · . . . · qn j+2 with n 1 + · · · + n j+2 = 0 and which are of degree 2(m + j − 3). Further ( n ) ∈ {−1, 0, +1} is fixed by a choice of coherent orientations in symplectic field theory and is zero if and only if one of the orbits γ n 1 , . . . , γ n j+2 is bad. We have the following immediate corollary, which immediately follows from the behavior of the Conley-Zehnder index for multiple covers. Corollary 2.4. Assume that the closed geodesic γ¯ represents a hyperbolic Reeb orbit in the unit cotangent bundle and dim Q > 1. Then gγ1¯ , j = 0 and hence h1γ , j = 0 for all j > 0. Indeed, since for hyperbolic Reeb orbits the Conley-Zehnder index CZ(γ n ) of γ n is given by CZ(γ n ) = n · CZ(γ ), an easy computation shows that there are no products of the above form of the desired degree. On the other hand, note that without the degree condition we would just get back the sequence of descendant Hamiltonians for the circle. Forgetting about orientation issues, in simple words we can hence say that the sequence gγ1¯ , j is obtained from the sequence for γ¯ = Q = S 1 by removing all summands with the wrong, that is, not maximal degree, where the latter can explicitly be computed using the formulas in [Lo] but also follows from our proof. The proof relies on the observation that for orbit curves the gravitational descendants indeed have a geometric meaning in terms of branching conditions, which is a slight generalization of the result for the circle shown by Okounkov and Pandharipande in [OP]. Applying (and generalizing) the ideas of Cieliebak and Latschev in [CL] for relating the symplectic field theory of V = S ∗ Q to the string topology of the underlying Riemannian manifold Q, we then study branched covers of the corresponding trivial half-cylinders in the cotangent bundle connecting the Reeb orbit γ with the underlying geodesic γ¯ to prove that the sequence of Poisson-commuting functions h1γ , j is isomorphic to a sequence of Poisson-commuting functions gγ1¯ , j . While the descendant Hamiltonians h1γ , j on the SFT side are defined using very complicated obstruction bundles over (nonregular) moduli spaces of arbitary large dimension, the key observation is that for the descendant Hamiltonians gγ1¯ , j on the string side we indeed only have to study obstruction bundles over discrete sets, which clearly disappear if the Fredholm index is right. With this we get that the Poisson-commuting sequences for the closed geodesics can be computed from the sequences for the circle and the Morse indices of the geodesic and its iterates as stated in the theorem.

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2.2. Gravitational descendants = branching conditions. Recall that by the above theorem from the last subsection we only have to consider the case where θ is a one-form on V , where we still assume without loss of generality that the integral of θ over γ is one. It follows that integrating the pullback of θ under the evaluation map over the moduli space of orbit curves with one additional marked point and dividing out the natural R-action on the target R ×S 1 ∼ = R ×γ is equivalent to restricting to orbit curves where the additional marked point is mapped to a special point on R ×S 1 . In other words, in what follows we will view h 1γ , j no longer as part of the Hamiltonian for γ but as part of the potential for the cylinder over γ equipped with a non-translation-invariant two-form. In order to save notation, M1 = M1 ( + , − ) will from now on denote the corresponding moduli space. On the other hand, after introducing coherent collections (¯ν ) of obstruction ν¯ bundle sections, it is easy to see that the tautological line bundle Lν¯ over M1 is just the ν¯ restriction of the tautological line bundle L over M1 to M1 = ν¯ −1 (0) ⊂ M1 . For the orbit curves we now want to give a geometric interpretation of gravitational descendants in terms of branching conditions over the special point on R ×S 1 . Before we state the corresponding theorem and give a rigorous proof using the stretching-ofthe-neck procedure from SFT, we first informally describe a naive direct approach based on our definition of gravitational descendants from above, which should illuminate the underlying geometric ideas. ν¯ Recall that if (h, z) is an element in the non-compactified moduli space Mν1 ⊂ M1 ∗ the fibre of the canonical line bundle L over (h, z) is given by L(h,z) = Tz S. Identifying the tangent space to the cylinder at the special point with C it follows that ν ∗ s(h, z) = ∂h ∂z (z) ∈ Tz S is a section in the restriction of L to M1 . Since s is a transversal ν section in the tautological line bundle over M1 if and only if it extends to a section over M1 such that s ⊕ ν is transversal to the zero section in L ⊕ Coker ∂¯ J over M1 , we may assume after possibly perturbing ν that s is indeed transversal. On the other hand, since ∂h ∂z (z) = 0 is equivalent to saying that z ∈ S is a branch point of the holomorphic map h : S → CP1 , it follows that M11 := s −1 (0) ⊂ M1 indeed agrees with the space of all orbit curves (h, z) with one additional marked point, where z is a branch point of h. Further moving on to the case j = 2 observe that a natural candidate for a generic section s2 in the restriction of the product line bundle L⊗2 to M11 ⊂ M1 is given by 2 s2 (h, z) = ∂∂zh2 (z) ∈ (Tz∗ S)⊗2 , for which M21 = s2−1 (0) ⊂ M11 agrees with the space of holomorphic maps where z ∈ S is now a branch point of order at least two. For general j we can hence proceed by induction and define the section s j in L⊗ j over j−1 j ∂ jh M1 := s −1 j−1 (0) ⊂ M1 by s j (h, z) = ∂z j (z), so that M1 agrees with the space of holomorphic maps where z ∈ S is a branch point of order at least j. If the chosen sections s1 , . . . , s j over the non-compactified moduli spaces would extend in the same way to a coherent collection of sections in the tautological line bunν¯ dles over the compactified moduli spaces M1 , the above would show that in the case of orbit curves considering the j th descendant moduli space is equivalent after passing to homology to requiring that the underlying additional marked point is a branch point of order j. In [OP] it was however shown that already for the case of the circle γ = V = S 1 the latter assumption is not entirely true, but that one instead additionally obtains corrections from the boundary M1 − M1 . To this end, we define a branching condition to be a tuple of natural numbers μ = (μ1 , . . . , μ(μ) ) of length (μ) and total branching order |μ| = μ1 + · · · + μ(μ) . Then

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μ

the moduli space M = M ( + , − ) consists of orbit curves with (μ) connected components, where every connected component carries one additional marked point z i , which is mapped to the special point on R ×γ and is a branch point of order μi − 1 for i = 1, . . . , (μ). For every branching condition μ = (μ1 , . . . , μ(μ) ) we then define ν new Hamiltonians h1γ ,μ = h1,¯ γ ,μ by setting h1γ ,μ =

μ

−

#M1 ( + , − ) p q . +

+ , −

With the following theorem we will prove that the abstract descendants-branching correspondence from [OP] holds for every closed Reeb orbit γ ⊂ V . For every j ∈ N and every branching condition μ we let ρ 0j,μ be the number given by integrating the j th power of the first Chern class of the tautological line bundle over the moduli space of connected rational curves over CP1 with one marked point mapped to 0 and (μ) additional marked points z i mapped to ∞ which are branch points of order μi −1, i = 1, . . . , (μ). Lemma 2.5. Each of the descendant Hamiltonians h1γ , j can be written as a sum, h1γ , j =

1 · h1γ ,( j+1) + ρ 0j,μ · h1γ ,μ , j! |μ|< j

where h1γ ,μ ∈ P0γ counts branched covers of the orbit cylinder with (μ) connected components, where each component carries one additional marked point z i , which is mapped to the special point on R ×γ and is a branch point of order μi − 1 for i = 1, . . . , (μ). Note that the statement of the lemma can be rephrased by saying that the integration of the j th power of the first Chern class corresponds to a weighted sum of branching conditions, . 1 c1 (L) j = · ( j + 1) + ρ 0j,μ · μ, j! |μ|< j

which is the rational version of the abstract descendants-branching correspondence from 0 is nonzero and agrees with the coefficient [OP] for the circle, where the coefficient ρμ, j ρ j,μ from [OP] only if the genus g determined by the Fredholm index, j + 1 = 2g − 1 + |μ| + (μ) is zero. Proof. Recall that the result in [OP] for the circle relies on the degeneration formula from relative Gromov-Witten theory, where the target sphere with three special points x − = 0, x + = ∞ and x degenerates in such a way that the original sphere only carries the two special points x − = 0, x + = ∞ while the third special point sits on a second sphere connected to the original one by a node. Viewing a sphere with two special points as a cylinder, it is clear that a corresponding statement can be proven for a Reeb orbit γ in a general contact manifold if the standard cylinder is replaced by the orbit cylinder R ×γ in the symplectization of the contact manifold which degenerates to an orbit cylinder with a ghost bubble attached.

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Since the degeneration formula from relative Gromov-Witten theory is no longer applicable, we will have to use the neck-stretching process from symplectic field theory, which however agrees with the degeneration process from relative Gromov-Witten theory in the case of the circle. For this observe that performing a neck-stretching at a small circle around the special point on the standard cylinder we obtain a pair-of-pants together with a complex plane carrying the special point, which can be identified with spheres with three or two special points, respectively. Replacing the circle by a Reeb orbit γ in a general contact manifold the neck-stretching yields besides a complex plane with a special point a pair-of-pants with a positive and a negative cylindrical end over γ together with a cylindrical end over the circle. Note that in order to include infinitesimal deformations needed for the obstruction bundles, we identify the orbit cylinder R ×γ (together with an infinitesimal tubular neighborhood) with (an infinitesimal neighborhood of the zero section in) its normal bundle over R ×S 1 with fibre given by the contact distribution ξ and twist around the puncture given by the linearized Reeb flow along γ . Then the (infinitesimal) neckstretching is performed along the (infinitesimal) hypersurface given by the restriction of the normal bundle to the small circle in R ×S 1 . Before we make the proof rigorous by studying coherent collections of sections in the cokernel bundles and the tautological line bundles over the moduli space of branched covers for the circle, observe that Theorem 2.5.5 in [EGH] concerning composition of cobordisms suggests that h1γ , j , viewed as a potential on P0γ , is homotopic, and by h0γ = 0 hence agrees with a potential, which can directly be computed from the potential for the complex plane counting rational curves with one additional marked point mapped to the special point and the potential for the pair-of-pants with its cylindrical ends over γ and the circle counting rational curves with no additional marked points. Indeed it follows from the compactness statement in [BEHWZ] that under the neck-stretching procedure every branched cover of the orbit cylinder with one additional marked point mapped to the special point splits into a branched cover of the complex plane with one additional marked point mapped to the special point and a branched cover of the pair-of-pants with no additional marked points. While from S 1 -symmetry reasons the potential for the complex plane with one special point can only count connected curves, note that under the splitting process the connected curve may split into branched covers of the pair-of-pants with more than one connected component. On the other hand, since the glued curve has genus zero, it follows that the branched cover of the complex plane and any connected component of the branched cover of the pair-of-pants cannot be glued at more than one cylindrical end, so that the number of connected components of the branched cover of the pair-of-pants agrees with the number of cylindrical ends of the branched cover of the complex plane. Note that a collection of closed Reeb orbits in the contact manifold S 1 is naturally identified with a tuple μ = (μ1 , . . . , μ(μ) ) of multiplicities and a branched cover is asymptotically cylindrical over the μith iterate of the circle near the puncture z i precisely if z i is a branch point of order μi − 1. With this it follows that h1γ , j can be computed as desired by summing over all branching conditions μ = (μ1 , . . . , μ(μ) ), where for each μ the summand is given by the product of ρ 0j,μ , obtained by integrating the j th power of the first Chern class of the tautological line bundle over the moduli space of branched covers of CP1 with one marked point mapped to the special point and (μ) additional marked points z i mapped to ∞ which are branch points of order μi − 1, with the branching Hamiltonians h1γ ,μ ∈ P0γ , counting branched covers of the orbit cylinder with (μ) connected components, where each component carries one additional marked

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point z i , which is mapped to the special point on R ×γ and is again a branch point of order μi − 1 for i = 1, . . . , (μ). In order to make the proof rigorous it remains to understand the above statement on the level of coherent collections of sections in the cokernel bundles and the tautological line bundles over the moduli spaces of branched covers for the circle. For this observe that for every chosen collections of Reeb orbits + , − the neck-stretching procedure at a small circle around the special point on the standard cylinder leads 1 = M 1 ( + , − ). It is shown in [BEHWZ] that to a compactified moduli space M this compactified moduli space has the desired codimension-one boundary components M1 = M1 ( + , − ) counting branched covers of the original orbit cylinder with one special point and M1,1 × M2,0 with M1,1 = M1 ( ) and M2,0 = M0 ( + , , − ) counting branched covers of the complex plane with one additional marked point and of the pair-of-pants with possibly more than one connected component, respectively. On the other hand, in contrast to the degeneration process from relative Gromov-Witten theory, it follows from [BEHWZ] that one also has to consider codimension-one boundary 1,1 × M2,0 with M 1,1 = M 1 ( + , − ), M2,0 = M0 ( + , − )/ R strata of the form M 1 2 1 2 − + ˜ 2,1 with M1,0 = M0 ( , )/ R, M 2,1 = M 1 ( + , − ), which corand M1,0 × M 1 2 1 2 respond to a splitting of a curve into two levels during the stretching process and which are irrelevant for the case of the circle due to the S 1 -symmetry on M1,0 and M2,0 , respectively. First, since the coherent collections of sections in the cokernel bundles over the moduli spaces of branched covers by definition are not affected by the position of the additional marked point, it follows that one can use the same obstruction bundle perturbations ν¯ = ν¯ ( + , − ) throughout the stretching process. In particular, it follows that the reg 1 has codimension-one boundary components ν1¯ = ν¯ −1 (0) ⊂ M ular moduli space M ν¯ ν¯ ¯2 ν¯ 1 ν ¯ ¯2 1 1,1 × Mν2,0 ν2,1 M1 and M1,1 × M2,0 as well as M and M1,0 × M , where ν¯ 1 , ν¯ 2 are sec1 2 tions in the cokernel bundles Coker ∂¯ J , Coker ∂¯ J over M1,0 = M0 ( + , − ), M2,0 = 1

1

M0 ( 2+ , 2− ) and which are determined by ν¯ by the coherency condition. On the other hand, concerning the coherent collections of sections in the tautologν¯ ical line bundles, it can be shown as above that the tautological line bundle L = L ν¯ ν ¯ 1 agrees with the tautological line bundle L over M1 , with the pullback π ∗ L1 over M 1 ν¯ ν¯ 2 ¯1 ν1,1 L1 over M L2 over over M1,1 × M2,0 and with the pullbacks π ∗ × M2,0 and π ∗

1 2 ν¯ 1 ν¯ 2 M1,0 × M2,1 , respectively. Assuming that we have chosen coherent collections of secν¯ ν¯ tions (s) in the tautological line bundles L over all moduli spaces M1 = M1 ( + , − ) of branched covers of the orbit cylinder with one special point and (s1 ) in the tautolog-

ical line bundles L1 over all moduli spaces M1,1 = M1 ( ) of branched covers of the complex plane with one special point, we as above can choose coherent collection of ν1¯ the sections (˜s ) connecting (s) and (s1 ) by requiring that over every moduli space M ν¯ ν¯ section s˜ agrees with the section s over M1 , with the pullback π1∗ s1 over M1,1 × M2,0 ν¯ 2 ν¯ 1 ¯1 ν¯ 2 ν1,1 2,1 and with the pullbacks π1∗ s˜1 over M × M2,0 and π2∗ s˜2 over M1,0 × M , respectively. Proceeding by induction it then follows that the regular descendant moduli space ν¯ , j j ν¯ ν¯ , j M1 has codimension-one boundary components M1 and M1,1 × M2,0 as well as ν¯ 2 ν¯ 1 ν¯ 1 ν¯ 2 ¯1 , j ¯2 , j ν2,1 ν1,1 ×M2,0 and M1,0 ×M , respectively. Since we have that #M1,0 = #M2,0 = 0 M

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by the result in [F] it hence follows that ν¯ , j

#M1

ν¯

j

= #M1,1 · #M2,0 ,

which finally proves the decendants-branching correspondence on the level of coherent collections of sections in obstruction bundles and tautological line bundles. Note in particular that using this stretching process we were able to separate the transversality problem from the problem of defining gravitational descendants. Since the moduli space M1,1 = M1 ( ) is independent of the chosen Reeb orbit and agrees with the moduli space obtained from the degeneration process in relative GromovWitten theory, it follows precisely like in the circle bundle case described above that the j count of elements in the descendant moduli space M1,1 is independent of the chosen coherent collection of sections and agrees with the integral of the j th power of the first Chern class over M1,1 . 2.3. Branched covers of trivial half-cylinders. In the case when the contact manifold V is the unit cotangent bundle S ∗ Q of a Riemannian manifold Q, Cieliebak and Latschev have shown in [CL] that, when suitably interpreted, the symplectic field theory of V = S ∗ Q without differential forms and gravitational descendants agrees with the string topology of Q. The required isomorphism is established by studying punctured holomorphic curves in T ∗ Q with boundary on the Lagrangian Q ⊂ T ∗ Q. For this they equip T ∗ Q with an almost complex structure J such that (T ∗ Q, J ) is an almost complex manifold with one positive cylindrical end (R+ ×S ∗ Q, J ). After showing that the contact area of holomorphic curve is given as differences of the sums of the actions of the Reeb orbits in S ∗ Q and the sum of the lengths of the boundary components on Q, they use the natural filtration by action on symplectic field theory and by length on string topology to show that the morphism has the form of a unitriangular matrix. The entries on the diagonal count cylinders with zero contact area, which are precisely the trivial half-cylinders in T ∗ Q connecting the geodesic γ¯ on Q with the corresponding Reeb orbit γ in S ∗ Q. On the other hand, since orbit curves are characterized by the fact that they have zero contact area, it hence directly follows from their proof that there exists a version of their isomorphism statement for the symplectic field theory of a closed Reeb orbit γ by studying branched covers over the trivial half-cylinder connecting γ¯ and γ . For this let us first recall some definitions from [CL]. Let A0 be the graded commutative subalgebra of W of polynomials in the variables qγ , where, following our notation from before, the subscript 0 indicates that no t-variables are involved. The Hamiltonian − → H0 ∈ −1 W0 defines a differential operator 0SFT := H0 : A0 [[]] → A0 [[]] via the replacements p γ → κγ

−−→ ∂ . ∂qγ

The resulting pair (A0 [[]], 0SFT ) has then the structure of a BV∞ -algebra, in particular, 0SFT ◦ 0SFT = 0. On the contrary, given a BV∞ -algebra (A0 [[]], 0) where A0 is a space of polynomials in variables q, it follows, see [CL], that 0 : A0 [[]] → A0 [[]] is a true differential operator. In particular, we naturally get a Weyl algebra W0 with

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distinguished element H0 ∈ −1 W satisfying [H0 , H0 ] = 0 by introducing for each qvariable a dualizing p-variable, considering the natural commutator relation and using the replacement for p-variables from above. As already mentioned above, in [CL] it is shown that the BV∞ -algebra (A0 [[]], SFT) representing the symplectic field theory of S ∗ Q is isomorphic to a BV∞ -algebra (C0 [[]], 0string ) constructed from the string topology of Q, where C0 is a space of chains in the string space = Q of Q. The differential is given by 0string = ∂ + + ∇ : C0 [[]] → C0 [[]], where ∂ is the singular boundary operator and ∇, are defined using the string bracket and cobracket operations of Chas and Sullivan. The BV∞ -iso− → morphism L0 is defined using the potential of (T ∗ Q, Q), L0 = evg,s − ( ) p g−1 g,s − ,

− using the evaluation cycles evg,s − ( ) = ev = ev1 , . . . , evs : Mg,s − ( ) → Q × · · · × Q (s − -times) starting from the moduli space of holomorphic curves in T ∗ Q with positive asymptotics , genus g and s − boundary components on Q. Now for moving from the symplectic field theory of S ∗ Q to the symplectic field theory − → of a closed Reeb orbit γ in S ∗ Q, we obviously just have to replace (A0 [[]], 0SFT = H0 ) − → by the BV∞ -algebra (A0γ [[]], 0, γ SFT = H0γ ) generated only by the q-variables representing the multiples of the fixed orbit γ . Furthermore the potential L0 of (T ∗ Q, Q) is now replaced by the potential L0γ ,γ¯ counting branched covers of the trivial halfcylinder connecting γ¯ in Q and γ in S ∗ Q, which defines a BV∞ -isomorphism from (A0γ [[]], 0, γ SFT ) to a BV∞ -algebra (C0γ [[]], 0, γ¯ string ). Assigning as for the Reeb orbits formal q-variables to multiples of the underlying closed geodesic γ¯ , the potential L0γ ,γ¯ is defined by L0γ ,γ¯ =

¯ p q ¯ g−1 #Mg ( , )

¯ g, ,

¯ of branched covers of the trivial half-cylinsumming over all moduli spaces Mg ( , ) der with Fredholm index zero. Note that it follows from the area estimate from above for curves in T ∗ Q with boundary on Q in terms of action of the Reeb orbits and length of the ¯ boundary component that, assuming enough transversality, the moduli space Mg ( , ) agrees with the preimage of the product stable manifold ¯ = W + (γ¯ n 1 ) × · · · × W + (γ¯ n s − ) ⊂ Q × · · · × Q W + ( ) of the energy functional E : Q → R on the string space under the evaluation evg,s − : Mg,s − ( ) → Q × · · · × Q ¯ = evg,s − ( )−1 (W + ( )) ¯ ⊂ Mg,s − ( ). Mg ( , ) Now the BV∞ -algebra (C0 [[]], 0string ) is replaced by the BV∞ -algebra (Cγ0¯ [[]], 0, γ¯ string ) of polynomials in the q-variables assigned to multiples of γ¯ . Since this algebra is now indeed an algebra of polynomials, we have seen above that we assign to (Cγ0¯ [[]], 0, γ¯ string ) again a Weyl algebra Wγ0¯ with bracket [·, ·] generated by

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p- and q-variables assigned to multiples of γ¯ together with a distinguished element Gγ0¯ ∈ −1 Wγ0¯ satisfying [G0γ¯ , Gγ0¯ ] = 0. Since BV∞ -algebras (A0γ [[]], 0, γ SFT = − → − → H0γ ), (Cγ0¯ [[]], 0, γ¯ string = Gγ0¯ ) determine the Weyl algebras with Hamiltonians (W0γ , H0γ ), (W0γ¯ , Gγ0¯ ) and vice versa, it follows that the BV∞ -isomorphism given by the potential L0γ ,γ¯ indeed leads to an isomorphism of the structures defined by (W0γ , H0γ ) and (Wγ0¯ , Gγ0¯ ): Indeed, let D0γ ,γ¯ be the space of formal power series in the p-variables for multiples of γ and with coefficients which are polynomials in the q-variables assigned to multiples of γ¯ . Then it follows that L0γ ,γ¯ is an element of −1 D0γ ,γ¯ satisfying the master equation − − → 0 0 ← eLγ ,γ¯ H0γ − Gγ0¯ eLγ ,γ¯ = 0. In particular, it follows in the notation of Sect. 1 that the map

L0,+ γ ,γ¯

−1 0,γ¯ 0,γ ◦ L0,− : H∗ −1 Wγ0¯ , Dstring → H∗ −1 W0γ , DSFT γ ,γ¯ ∗

∗

is an isomorphism of Weyl algebras. 0,γ¯ In order to understand Dstring , recall that the differential in the string topology was given by 0string = ∂ + + ∇ : C0 [[]] → C0 [[]], where ∇ is defined using the string bracket and using the string cobracket operations defined by Chas and Sullivan. While the singular boundary ∂ does not appear as we restrict ourselves to zero-dimensional moduli spaces, we expect to get contributions of the string bracket and string cobracket to Gγ0¯ , where we claim that the string bracket restricts to the operation of concatenating two multiples γ¯ n 1 , γ¯ n 2 to the multiple γ¯ n 1 +n 2 of γ¯ , while the string cobracket corresponds to splitting up the multiple γ¯ n 1 +n 2 again into γ¯ n 1 , γ¯ n 2 . In order to see this note that the compactification of the moduli spaces of branched covers of the trivial half-cylinder counted in the potential L0γ ,γ¯ can be entirely understood in terms of branch points of the branched covering map. While branch points moving the infinite end lead to the appearance of H0γ in the master equation, the Hamiltonian Gγ0¯ describes what happens if branch points are moving through the boundary of the branched cover, which itself sits over the boundary of the half-cylinder. The important observation is now that for the codimension-one boundary of the moduli space we only have to consider the case where a single branch point is leaving the branched cover through the boundary. In order to see that this is described by the concatenation and splitting operations of the multiples of γ¯ , observe that the case when a branch point sits in the boundary of the branched cover is equivalent to the fact that the boundary of R+ ×S 1 is a critical level set of the branching map followed by the projection to the first factor. Observe that the branch point may leave the branched cover through any point of its boundary, which itself is diffeomorphic to (a number of copies of) the circle. Note that this corresponds to the fact that the concatenation and splitting operation may take place anywhere over any point on γ¯ . It follows that we always get an one-dimensional family of configurations. Before we continue, we want to restrict ourselves as before to the rational case. In particular, there exists a version of the above isomorphism, given by counting rational branched covers of the trivial half-cylinder, which relates the rational symplectic field

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theory H∗ P0γ , dγ0 of γ with H∗ Pγ0¯ , dγ0¯ , where dγ0¯ = gγ0¯ , · : P0γ¯ → Pγ0¯ and Gγ0¯ = −1 gγ0¯ +o () . Before we discuss the rational Hamiltonian gγ0¯ ∈ Pγ0¯ , recall that it was shown in [F2] that h0γ = 0. Note that we have indeed not considered additional marked points so far. In particular, it follows from the above isomorphism that also gγ0¯ has to vanish. Since we have seen above that for Gγ0¯ and hence for gγ0¯ we always get one-dimensional sets of configurations, the vanishing of gγ0¯ seems to follow from a stupid dimension argument. On the other hand, recall that we have shown in [F2] that the corresponding statement for h0γ does not simply follow from a symmetry argument but indeed requires a careful study of sections in obstruction bundles in order to find compact perturbations making the Cauchy-Riemann operator transversal to the zero section. With the work in [F2] it is clear that the same transversality problem should continue to hold for branched covers of trivial half-cylinders. In the next section it will turn out that, like on the symplectic field theory side, also on the string side we are working in a highly degenerate situation, so that the transversality requirement is usually not fulfilled. 2.4. Obstruction bundles and transversality. In order to solve the transversality problem we follow the author’s paper [F2] in employing finite-dimensional obstruction bundles over the nonregular configuration spaces. Here is a sketch of the main points. For this let S˙ denote a (possibly disconnected) punctured Riemann surface with − boundary of genus zero with s + punctures circles s boundary z+1 , . . . , z s ++ and − n− n n n + C1 , . . . , Cs − and fix two ordered sets = γ 1 , . . . , γ s , ¯ = γ¯ 1 , . . . , γ¯ s − of iterates of γ , γ¯ , respectively. Let ξ = T T ∗ Q/T (R+0 ×S 1 ), J ξ denote the complex normal bundle to the trivial half-cylinder R+0 ×S 1 , {0} × S 1 → (T ∗ Q, Q) as defined in [CL], which over the boundary {0} × S 1 ∼ = γ¯ ⊂ Q has the property that ξ ∩ T Q agrees with the normal bundle N to the geodesic γ¯ in Q. Note that the tangent space T W + (γ¯ n ) to the stable manifold of the energy functional in the critical point γ¯ n can be identified with a subspace of the space of normal deformations C 0 ((γ¯ n )∗ N ). ˙ ∂ S) ˙ → (R+ ×S 1 , {0} × S 1 ) of the trivial half-cylGiven a branched covering h : ( S, 0 inder, for p > 2 let H 1, p (h ∗ ξ ) ⊂ C 0 (h ∗ ξ ) denote the space of H 1, p -sections in h ∗ ξ − which over every boundary component Ck ⊂ ∂ S˙ restrict to a section in C 0 ((γ¯ n k )∗ N ). 1, p Furthermore we will consider the subspace H ¯ (h ∗ ξ ) ⊂ H 1, p (h ∗ ξ ) consisting of all sections in h ∗ ξ , which over every boundary circle Ck restrict to sections in the subspace − − T W + (γ¯ n k ) ⊂ C 0 ((γ¯ n k )∗ N ). While the latter Sobolev spaces describe the normal deformations of the branched covering, we introduce similar as in [F2] for sufficiently small 1, p,d ˙ d > 0 a Sobolev space with asymptotic weights Hconst ( S, C) in order to keep track of tangential deformations, where, additionally to the definitions in [F2], we impose the natural constraint that the function is real-valued over the boundary. In the same way we define the Banach spaces L p ((0,1) S˙ ⊗ j,J ξ h ∗ ξ ) and L p,d ((0,1) S˙ ⊗ j,i C). Further we denote by M0,s − ,s + the moduli space of Riemann surfaces with s − boundary circles, s + punctures and genus zero. Following [F2,BM] for the general case and [W] for the case with boundary, there exists a Banach space bundle E over a Banach manifold of maps B in which the Cauchy-Riemann operator ∂¯ J extends to a smooth section. In our special case it follows

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as in [F2] that the fibre is given by E h, j = L p,d (0,1 S˙ ⊗ j,i C) ⊕ L p (0,1 S˙ ⊗ j,J ξ h ∗ ξ ), while the tangent space to the Banach manifold of maps B = B 0,s − ( ) at (h, j) ∈ M = M0,s − ( ) is given by ˙ C) ⊕ H 1, p (h ∗ ξ ) ⊕ T j M0,n . Th, j B = Hconst ( S, 1, p,d

It follows that the linearization Dh, j of the Cauchy-Riemann operator ∂¯ J is a linear map from Th, j B to E h, j , which is surjective in the case when transversality for ∂¯ J is satisfied. In this case it follows from the implicit function theorem that ker Dh, j = Th, j M. ¯ = In order to prove that the dimension of the desired moduli space M ¯ = M( , ) ¯ ⊂ M( ) agrees with the virtual dimension expected by the Fredholm ev−1 (W + ( )) − index, it remains to prove that the evaluation map ev : M → Q s is transversal to + ¯ the product stable manifold W ( ). In order to deal with this additional transversality problem, we introduce the Banach ¯ ⊂ B with tangent space submanifold of maps B ¯ = ev−1 (W + ( )) 1, p,d ˙ 1, p Th, j B ¯ = Hconst ( S, C) ⊕ H ¯ (h ∗ ξ ) ⊕ T j M0,n

¯ = {v ∈ Th, j B : v|∂ S˙ ∈ T W + ( )}

and view the Cauchy-Riemann operator as a smooth section in E → B ¯ . Then we have the following nice transversality lemma. Lemma 2.6. Assume that Dh, j : Th, j B ¯ → E h, j is surjective. Then the linearization of ¯ = C 0 ( ¯ ∗ N )/T W + ( ) ¯ is surjective. the evaluation map dh, j ev : Th, j M → T W − ( ) ¯ choose v˜ ∈ Th, j B such that dh, j ev ·v˜ = v0 . On the other Proof. Given v0 ∈ T W − ( ), hand, since Dh, j : Th, j B ¯ → E h, j is onto, we can find v ∈ Th, j B ¯ with Dh, j v = ¯ Dh, j v, ˜ that is, v−v ˜ ∈ ker Dh, j = Th, j M. On the other hand, since dh, j ev ·v ∈ T W + ( ) for all v ∈ Th, j B ¯ by definition, we have dh, j ev ·(v˜ − v) = dh, j ev ·v˜ = v0 and the claim follows. We have seen that, instead of requiring transversality for the Cauchy-Riemann operator in the Banach space bundle over B and geometric transversality for the evaluation map, it suffices to require transversality for the Cauchy-Riemann operator in the Banach space bundle over the smaller Banach manifold B ¯ . Along the same lines as for Proposition 2.1 in [F2] it can be shown that the linearized Cauchy-Riemann operator is of the form ˙ C) ⊕ H (h ∗ ξ ) ⊕ T j M0,n Dh, j : Hconst ( S,

¯ 1, p,d

1, p

→ L p,d (0,1 S˙ ⊗ j,i C) ⊕ L p (0,1 S˙ ⊗ j,J ξ h ∗ ξ ), ¯ 1 + D j y, D ξ v2 ), Dh, j · (v1 , v2 , y) = (∂v h

1, p,d ˙ C) → L p,d (0,1 S˙ ⊗ j,i C) is the standard Cauchy-Riemann operwhere ∂¯ : Hconst ( S, ξ 1, p ∗ ator, Dh : H (h ξ ) → L p (0,1 S˙ ⊗ j,J ξ h ∗ ξ ) describes the linearization of ∂¯ J in the

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direction of ξ ⊂ T T ∗ Q and D j : T j M0,n → L p,d (T ∗ S˙ ⊗ j,i C) describes the variation of ∂¯ J with j ∈ M0,n . In [F2] we have shown that for branched covers of orbit cylinders the cokernels of the linearizations of the Cauchy-Riemann operator have the same dimension for every branched cover and hence fit together to give a smooth vector bundle over the nonregular moduli space of branched covers, so that we can prove transversality without waiting for the completion of the polyfold project of Hofer, Wysocki and Zehnder. The following proposition, proved in complete analogy, outlines that this still holds true for branched covers of trivial half-cylinders. Proposition 2.7. The cokernels of the linearizations of the Cauchy-Riemann operator fit together to give a smooth finite-dimensional vector bundle over the moduli space of branched covers of the half-cylinder. Proof. As in [F2] this result relies on the transversality of the standard Cauchy-Riemann operator and the super-rigidity of the trivial half-cylinder ξ coker ∂¯ = {0} and ker Dh = {0},

where the second statement is now just a linearized version of Lemma 7.2 in [CL] which states that, as for orbit cylinders in the symplectizations, the branched covers of the trivial half-cylinder are characterized by the fact that they carry no energy in the sense that the action of Reeb orbits above agrees with the lengths of the closed geodesics below. It remains to study the extension Coker ∂¯ J of the cokernel bundle Coker ∂¯ J to the compactified moduli space. For this recall that the components of the codimensionone-boundary of the nonregular moduli space M = M ¯ of branched covers of the half-cylinder are either of the form M1 × M2 , where M1 = M1 ( 1+ , 1− )/ R, M2 = M2 ( 2 , ¯ 2 ) are nonregular compactified moduli spaces of branched covers of the orbit cylinder or of the trivial half-cylinder, respectively, or of the form M0 × S 1 , where M0 = M0 ( , ¯ 0 ) is again a nonregular compactified moduli space of branched covers of the trivial half-cylinder while S 1 refers to the concatenation or splitting locus, which agrees with the locus where the single branch point is leaving the branched covering through the boundary. Note that for ¯ = (γ¯ n 1 , . . . , γ¯ n s − ) the ordered set ¯ 0 is either of the form 1 2 or

¯ 0 = γ¯ n 1 , . . . , γ¯ n k−1 , γ¯ n k , γ¯ n k , γ¯ n k+1 , . . . , γ¯ n s − n

¯ 0 = γ¯ 1 , . . . , γ¯ n k−1 , γ¯ n k +n k+1 , γ¯ n k+2 , . . . , γ¯ n s − , 1

1

corresponding to concatenating γ¯ n k and γ¯ n k to get γ¯ n k (n 1k + n 2k = n k ) or the splitting of γ¯ n k +n k+1 to get γ¯ n k and γ¯ n k+1 . Restricting to the concatenation case, recall that the chosen special point on the simple closed Reeb orbit determines a special point on the underlying simple geodesic and that we may assume that every holomorphic curve comes equipped with asymptotic markers in the sense of [EGH] not only on the cylindrical ends but also on the boundary circles. In particular, for the concatenation and splitting processes we may assume that all multiply-covered geodesics come equipped with a parametrization 1 1 by S 1 . Denoting by t1 , t2 ∈ S 1 the points on γ¯ n k , γ¯ n k , where we want to concatenate 1 2 the two multiply-covered geodesics to get the multiply-covered geodesic γ¯ n k +n k , we see

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that the coordinates must satisfy n 1k t1 = n 2k t2 in order to represent the same point on the underlying simple geodesic, so that the configuration space agrees with S 1 by setting t1 = n 2k t, t2 = n 1k t for t ∈ S 1 . While it directly follows from [F2] that over the boundary components M1 × M2 ⊂ M the extended cokernel bundle Coker ∂¯ J is of the form 1 2 Coker ∂¯ J |M1 ×M2 = π1∗ Coker ∂¯ J ⊕ π2∗ Coker ∂¯ J , 1 2 where Coker ∂¯ J , Coker ∂¯ J denote the (extended) cokernel bundles over M1 , M2 , respectively, it remains to study the cokernel bundle over the boundary components M0 × S 1 .

Proposition 2.8. Over the boundary components M0 × S 1 ⊂ M the extended cokernel bundle Coker ∂¯ J is also of product form, 0 Coker ∂¯ J |M0 ×S 1 = π1∗ Coker ∂¯ J ⊕ π2∗ ,

where Coker ∂¯ J denotes the (extended) cokernel bundle over the moduli space M0 and is a vector bundle over S 1 which is determined by the tangent spaces to the stable manifolds of the multiply-covered closed geodesics involved into the concatenation or splitting process. 0

Proof. Still restricting to the concatenation case, let S˙0 = S˙01 ∪ S˙02 denote the disconnected Riemann surface of genus zero with s + punctures and s − + 1 boundary circles C1 , . . . , Ck1 , Ck2 , . . . , Cs − , where we assume that ∂ S˙01 = C1 ∪ . . . ∪ Ck1 and ∂ S˙02 = Ck2 , . . . , Cs − . As before we know that the tangent spaces to the corresponding Banach manifolds of maps B 0 , B 0 ¯ at a branched covering (h 0 , j0 ) : ( S˙0 , ∂ S˙0 ) → (R+0 ×S 1 , {0} × S 1 ) are given by

0

Th 0 , j0 B 0 = Hconst ( S˙0 , C) ⊕ H 1, p (h ∗0 ξ ) ⊕ T j0 M0,n , 1, p,d

Th 0 , j0 B 0 ¯ = Hconst ( S˙0 , C) ⊕ H ¯ (h ∗0 ξ ) ⊕ T j0 M0,n 0 0 0 = v ∈ Th 0 , j0 B : v|∂ S˙0 ∈ T W + ( ¯ 0 ) , 1, p,d

1, p

while the fibre of the corresponding Banach space bundle is given by E 0h 0 , j0 = L p,d 0,1 S˙0 ⊗ j0 ,i C ⊕ L p 0,1 S˙0 ⊗ j0 ,J ξ h ∗0 ξ . For (h 0 , j0 , t) ∈ M0 × S 1 we further introduce the Banach manifold of maps B ∗ ¯ ⊂ B∗ ⊂ B0 which should consist of all branched covers of the trivial half-cylinder in B 0 for which the boundary circles Ck1 , Ck2 ∼ = S 1 are concatenated at (t1 , t2 ) = (n 2k t, n 1k t) ∈ 1 2 Ck ×Ck , to give the singular Riemann surface S˙∗ with s − boundary circles C1 , . . . , Ck1 ∪t Ck2 , . . . , Cs − and we have for v1,2 := v|C 1,2 , Th 0 , j0 ,t B ∗ = v ∈ Th 0 , j0 B 0 : v1 n 2k t = v2 n 1k t k Th 0 , j0 ,t B∗ ¯ = v ∈ Th 0 , j0 ,t B∗ : v|∂ S˙∗ ∈ T W + ¯ 0 .

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The proof of the general gluing theorem in [MDSa] suggests that over (h 0 , j0 , t) ∈ M0 × S 1 ⊂ M the extended cokernel bundle Coker ∂¯ J has fibre Coker ∂¯ J h , j ,t = coker Dh 0 , j0 ,t , Dh 0 , j0 ,t : Th 0 , j0 ,t B ∗ ¯ → E 0h 0 , j0 . 0

0

Before we describe the relation to the cokernel bundle Coker ∂¯ J over the first factor M0 with fibre 0 Coker ∂¯ J = coker Dh 0 , j0 , Dh 0 , j0 : Th 0 , j0 ,t B 0 ¯ → E 0h 0 , j0 , 0

h 0 , j0

0

observe that we still have ξ

ξ

coker Dh 0 , j0 = coker Dh 0 ,

0,ξ

0

ξ

coker Dh 0 , j0 ,t = coker Dh 0 ,t , ξ

ξ

Dh 0 : Th 0 , j0 B 0 ¯ → E h 0 , j0 , ξ

ξ

Dh 0 ,t : Th 0 , j0 ,t B ∗ ¯ → E h 0 , j0 ,

ξ

ξ

and ker Dh 0 = ker Dh 0 ,t = {0}, where Th 0 , j0 B 0 ¯

0

0,ξ

ξ

⊂ Th 0 , j0 B0 ¯ , Th 0 , j0 ,t B ∗ ¯ ⊂ 0

Th 0 , j0 ,t B∗ ¯ and E h 0 , j0 ⊂ E 0h 0 , j0 are the subspaces corresponding to normal deformations. Now assume without loss of generality that t = 0 and n 1k = n 2k = 1. Viewing γ¯ : S 1 → Q as a map starting from [0, 1] (without identifying 0 and 1), we introduce ∞ (γ¯ ∗ N ). With this auxiliary space it is not very hard to observe the space of sections C[0,1] that the space of deformations of γ¯ and γ¯ 2 can be expressed as linear subspaces 0,ξ

∞ C ∞ (γ¯ ∗ N ) = {v ∈ C[0,1] (γ¯ ∗ N ) : v(0) = v(1)}, ∞ ∞ C ∞ ((γ¯ 2 )∗ N ) = {(v1 , v2 ) ∈ C[0,1] (γ¯ ∗ N ) ⊕ C[0,1] (γ¯ ∗ N ) : v1 (0) = v2 (1), v1 (1) = v2 (0)}.

Observing for the tangent spaces to the stable manifolds W + (γ¯ ) at γ¯ that T W + (γ¯ ) ⊕ T W + (γ¯ ) ⊂ {(v1 , v2 ) ∈ T W + (γ¯ 2 ) : v1 (0) = v2 (0)} ⊂ C ∞ (γ¯ ∗ N ) ⊕ C ∞ (γ¯ ∗ N ),

we get from ξ ξ Th 0 , j0 B0 ¯ = {v ∈ Th 0 , j0 B 0 : v|∂ S˙0 ∈ T W + ( ¯ 0 )}, 0

ξ

ξ

¯ Th 0 , j0 ,0 B 0 ¯ = {v ∈ Th 0 , j0 ,0 B ∗ : v|∂ S˙0 ∈ T W + ( )} ξ

ξ

that Th 0 , j0 B0 ¯ ⊂ Th 0 , j0 ,0 B 0 ¯ with quotient space 0

ξ

Th 0 , j0 B 0 ¯

0

ξ Th 0 , j0 ,t

B0 ¯

=

T W + (γ¯ ) ⊕ T W + (γ¯ ) . {(v1 , v2 ) ∈ T W + (γ¯ ) : v1 (0) = v2 (0)} ξ

ξ

On the other hand, since ker Dh 0 = ker Dh 0 ,0 = {0} we find that ξ

coker Dh 0 ,0 ξ

coker Dh 0

ξ

ξ

=

im Dh 0 ξ

im Dh 0 ,0

=

Th 0 , j0 B 0 ¯ ξ

0

Th 0 , j0 ,0 B 0 ¯

=

T W + (γ¯ ) ⊕ T W + (γ¯ ) , T W + (γ¯ 2 ) ∩ (C ∞ (γ¯ ∗ N ) ⊕ C ∞ (γ¯ ∗ N )

154

O. Fabert ξ

ξ

where the first equality follows from the fact that Dh 0 and Dh 0 ,0 both map to the same 0,ξ

Banach space E h 0 . In order to finish the proof, it hence only remains to prove that T W + (γ¯ ) ⊕ T W + (γ¯ ) T W − (γ¯ 2 ) = . T W + (γ¯ 2 ) ∩ (C ∞ (γ¯ ∗ N ) ⊕ C ∞ (γ¯ ∗ N ) (T W − (γ¯ ) ⊕ T W − (γ¯ )) ∩ C ∞ ((γ¯ 2 )∗ N ) But this is an immediate consequence of T W + (γ¯ ) ⊕ T W + (γ¯ ) T W + (γ¯ 2 ) ∩ (C ∞ (γ¯ ∗ N ) ⊕ C ∞ (γ¯ ∗ N )) T W + (γ¯ ) ⊕ T W + (γ¯ ) ⊕ (C ∞ (γ¯ ∗ N ) ⊕ C ∞ (γ¯ ∗ N ))⊥ = T W + (γ¯ 2 ) (T W − (γ¯ ) ⊕ T W − (γ¯ ))⊥ = T W + (γ¯ 2 ) and T W − (γ¯ 2 ) (T W − (γ¯ ) ⊕ T W − (γ¯ )) ∩ C ∞ ((γ¯ 2 )∗ N ) T W − (γ¯ 2 ) ⊕ (C ∞ ((γ¯ 2 )∗ N ))⊥ = T W − (γ¯ ) ⊕ T W − (γ¯ ) (T W + (γ¯ 2 ))⊥ = , T W − (γ¯ ) ⊕ T W − (γ¯ ) ∞ (γ¯ ∗ N ) ⊕ where A⊥ denotes the complement of the linear subspace A in C[0,1] ∞ ∗ C[0,1] (γ¯ N ). Defining an obstruction bundle over S 1 by setting

t =

T W − (γ¯ n k ) n 1k

2

{(v1 , v2 ) ∈ T W − (γ¯ ) ⊕ T W − (γ¯ n k ) : v1 (n 2k t) = v2 (n 1k t)}

and putting everything together we hence found that 0 ∼ Coker ∂¯ J = Coker ∂¯ J h 0 , j0 ,t

as desired.

h 0 , j0

⊕ t ,

With this we can prove the desired statement about gγ0¯ . Corollary 2.9. We have gγ0¯ = 0. Proof. It follows that the obstruction bundle over the one-dimensional configuration space has rank 2 1 rank = Morse γ¯ n k − Morse γ¯ n k − Morse γ¯ n k + dim Q − 1 ≥ 0, where the latter inequality can be verified as in [F2] using the multiple cover index formulas in [Lo]. When by index reasons the configuration is expected to be discrete we get a rank-one obstruction bundle over the boundary of the branched cover, which by orientability reasons must indeed be trivial.

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On the other hand, we want to emphasize that the proof of gγ0¯ = 0 is much simpler than the proof of h0γ = 0 in [F2], which has to involve obstruction bundles of arbitrary large rank and uses induction. Besides that our proof in [F2] also holds for Reeb orbits in general contact manifolds, this does not come as surprise. Going back to the symplectic field theory of unit cotangent bundles S ∗ Q, it is already mentioned in − → [CL] that the SFT differential 0SFT = H0 : A0 [[]] → A0 [[]] involving all moduli spaces of holomorphic curves in R ×S ∗ Q is much larger than the string differential 0string = ∂ + + ∇ : C0 [[]] → C0 [[]], which just involves the singular boundary operator and the string bracket and cobracket operations.

2.5. Additional marked points and gravitational descendants. We now want to understand the system of commuting operators defined for Reeb orbits by studying moduli spaces of branched covers over the cylinder over γ in terms of operations defined for the underlying closed geodesic γ¯ . To this end we have to extend the picture of [CL] used for computing the symplectic field theory of Reeb orbits to include additional marked points on the moduli spaces, integration of differential forms and gravitational descendants. Reintroducing the sequence of formal variables t j , j ∈ N, we now consider the graded Weyl algebras Wγ , Wγ¯ of power series in , the p-variables corresponding to multiples of γ , γ¯ and t-variables with coefficients which are polynomials in the q-variables corresponding to multiples of γ , γ¯ . In the same way we can introduce the graded commutative algebras Aγ , Cγ¯ of power series in , the t-variables with coefficients which are polynomials

in the q-variables corresponding to multiples of γ , γ¯ . For the expansion Hγ = H0γ + j t j H1γ , j +o(t 2 ) of the Hamiltonian from before, we are hence looking for an extended potential Lγ ,γ¯ as well as extended string Hamiltonian Gγ¯ , Lγ ,γ¯ = L0γ ,γ¯ +

j

Gγ¯ =

Gγ0¯

+

t j L1γ ,γ¯ , j +o t 2 ,

t j Gγ1¯ , j +o t 2 ,

j

−−→ − → − → such that Lγ ,γ¯ : Aγ [[]], Hγ → Cγ¯ [[]], Gγ¯ is an isomorphism of BV∞ -algebras. For this we have to prove the extended master equation ← − − → eLγ ,γ¯ Hγ − Gγ¯ eLγ ,γ¯ = 0, while the isomorphism property again follows using the natural filtration given by the t-variables. Since we are only interested in the system of commuting operators H1γ , j , j ∈ N, which is defined by counting branched covers of orbit cylinders with at most one additional marked point, we again will only discuss the required compactness statements in the case of one additional marked point. Furthermore we will still just restrict to the rational case. In other words we will prove the following proposition, which is just a reformulation of our theorem from above. Proposition 2.10. The system of Poisson-commuting functions h1γ , j , j ∈ N on P0γ is isomorphic to a system of Poisson-commuting functions g1γ¯ , j , j ∈ N on Pγ0¯ = P0γ ,

156

O. Fabert

where for every j ∈ N the descendant Hamiltonian gγ1¯ , j given by gγ1¯ , j =

( n)

qn 1 · . . . · qn j+2 ( j + 2)!

,

where the sum runs over all ordered monomials qn 1 · . . . · qn j+2 with n 1 + · · · + n j+2 = 0 and which are of degree 2(m + j − 3). Further ( n ) ∈ {−1, 0, +1} is fixed by a choice of coherent orientations in symplectic field theory and is zero if and only if one of the orbits γ n 1 , . . . , γ n j+2 is bad. Proof. While the proof seems to require the definition of gravitational descendants for moduli spaces of holomorphic curves not only with punctures but also with boundary, instead of defining them, recall that we have shown in the previous Subsect. 2.2 that the gravitational descendants can be replaced by imposing branching conditions over the special marked point on the orbit cylinder. More precisely, recall the lemma in Subsect. 2.2 states that we can indeed write each of the Hamiltonians h1γ , j as a weighted sum, h1γ , j =

1 · h1γ ,( j) + ρ 0j,μ · h1γ ,μ , j! |μ|< j

where h1γ ,μ ∈ P0γ counts rational branched covers of the orbit cylinder with (μ) connected components carrying precisely one additional marked point z 1 , . . . , z (μ) , which are mapped to the special point on the orbit cylinder and z i is a branch point of order μi − 1 for all i = 1, . . . , (μ). While for the invariance statement for gravitational descendants we were studying the compactification of the moduli spaces of holomorphic curves with one additional marked point, it follows from the definition of h1γ ,μ that now it is natural to study the moduli spaces of branched covers of the trivial half-cylinder with (μ) connected components carrying precisely one additional marked point z 1 , . . . , z (μ) , which are mapped to the special point on the trivial half-cylinder and z i is a branch point of order μi − 1 for all i = 1, . . . , (μ). While for the orbit cylinder the natural R-action is used to fix not only the S 1 -coordinate but also the R-coordinate of the special point, note that, in order to find the branched covers of the orbit cylinder counted in h1γ ,μ in the boundary, for the trivial half-cylinder we still fix the S 1 -coordinate but allow the R-coordinate to vary in R+ = (0, ∞). It follows that besides the boundary phenomena of the moduli spaces of branched covers of the trivial half-cylinder already described above, which can be described as seen above as the moving of branch points to infinity or leaving the branched cover through the boundary, the new boundary phenomena are the moving of the additional marked points to infinity or leaving the branched cover through the boundary, which are equivalent to the moving of the special point to infinity or leaving the half-cylinder through the boundary. In particular, it follows from the latter equivalence that the additional marked points z 1 , . . . , z (μ) move to infinity or leave the branched cover all at once. While the moving of the additional marked points to infinity, possibly together with other branch points, is counted in h1γ ,μ , the corresponding string Hamiltonian g1γ¯ ,μ should describe what happens if the additional marked points leave the branched cover through the boundary. Provided that we have found gγ1¯ ,μ ∈ Pγ0¯ for all branching profiles

Descendants in SFT

157

μ, it then follows from linearity that we obtain the desired Poisson-commuting sequence gγ1¯ , j by setting gγ1¯ , j =

1 ρ 0j,μ · gγ1¯ ,μ . · g1γ¯ ,( j) + j! |μ|< j

On the other hand, recall that in the computation of gγ0¯ we were faced with a transversality problem. While we have shown that the set of configurations counted for gγ0¯ is always one-dimensional, one can compute using the Morse indices of the involved multiply-covered geodesics that it happens that the Fredholm index expects the same set to be discrete. In the case when the Fredholm index is right, we have shown that to get an obstruction bundle of rank one to cut down the dimension of the configuration space, which is however trivial by orientability. For gγ1¯ ,μ we now show that the situation is even nicer. Lemma 2.11. For every branching condition μ the set of configurations studied for gγ1¯ ,μ is already discrete before we add abstract perturbations to the Cauchy-Riemann operator. It follows that, if the Fredholm index is right, there is no obstruction bundle. Before we show why this lemma leads to a proof of the above proposition and hence of the theorem, note that when γ¯ = Q = S 1 transversality is always satisfied and hence there are no obstruction bundles at all. On the other hand, note that the above proposition is formulated such that it holds in this case, where we use that g1S 1 ,μ = h1S 1 ,μ , which follows from the fact that the (rational) potential L0S 1 ,S 1 l0S 1 ,S 1 only counts orbit cylinders. In order to see that for an arbitrary closed geodesic γ¯ ⊂ Q the lemma proves the proposition and hence the theorem, observe that the Fredholm index is right precisely when it leads to the maximal degree 2(m + j − 3) from the proposition. Since the configuration space is independent of γ¯ before perturbing, in this case the lemma tells us that the corresponding configurations counted for gγ1¯ ,μ indeed agree with the ones counted for g1S 1 ,μ , up to sign determined by a choice of coherent orientations for the moduli spaces as described in [BM]. On the other hand, the results in [BM] show that the bad orbits indeed cancel out. For both statements we refer to the work of Cieliebak and Latschev in order to show that the orientation choices for closed Reeb orbits have a natural translation into orientation choices for to the underlying closed geodesics, that is, their unstable manifolds for the energy functional. In particular, we have, see [CL], that the Reeb orbit γ is bad if and only if the unstable manifold of γ¯ is not orientable. On the other hand, when the Fredholm is not right and hence maximal, we do not get a contribution to gγ1¯ ,μ by definition. Hence it just remains to prove the lemma. Proof of the lemma. For simplicity we first prove the statement for μ = (2). Following 1 the above description of g1γ ,μ ¯ it follows that gγ¯ ,(2) describes what happens if the additional marked point, which is a simple branch point, leaves the branched cover through the boundary. While at first this sounds that gγ1¯ ,(2) agrees with gγ0¯ , note that now the branch point is required to sit over the special point on the boundary of the half-cylinder. Since the S 1 -coordinate of the special point is fixed, it follows that the branch point can

158

O. Fabert

no longer leave the branched cover through every point on the boundary. In particular, while for gγ0¯ we obtained a one-dimensional configuration space due to the obvious S 1 -symmetry, it follows that for the configurations counted in gγ1¯ ,(2) the S 1 -symmetry is no longer present. Due to the important observation (which we already used to compute g0γ¯ ) that for the codimension-one boundary we can assume that there are no other branch points leaving the boundary at the same time, it follows that the set of configurations is indeed discrete. On the other hand, it is clear that this argument immediately generalizes to all branching profiles μ, since all the (μ) additional marked points are mapped to the same fixed special point. Together with the observation that the additional marked points z 1 , . . . , z (μ) leave the branched cover through the boundary all at once when the special point leaves the half-cylinder through the boundary, but again no other branch points by codimension reasons, the corresponding set of configurations stays discrete. To finish the proof of the theorem, observe that the sign ( n ) ∈ {−1, 0, +1} is fixed by a choice of coherent orientations in symplectic field theory and is zero if and only if one of the orbits γ n 1 , . . . , γ n j+2 is bad. For this recall from [BM] that in order to orient moduli spaces in symplectic field theory one additionally needs to choose orientations for all occurring Reeb orbits, while the resulting invariants are independent of these auxiliary choices. Recall that we have shown in Proposition 2.8 how (for j = 1) this obstruction bundle and hence its orientation is determined by the tangent spaces to the unstable manifolds of the multiply-covered geodesics. While the orientation of a closed Reeb orbit in SFT corresponds to an orientation of the (finite-dimensional) unstable manifold, the sign in front of pn 1 pn 2 qn k (n 1k + n 2k = n k ) in gγ1¯ ,1 is given by k

k

comparing the orientations of the finite-dimensional linear subspaces T W − (γ¯ 2 ) and (T W − (γ¯ ) ⊕ T W − (γ¯ )) ∩ = {(v1 , v2 ) ∈ T W − (γ¯ ) ⊕ T W − (γ¯ ) : v1 (0) = v2 (0)} of C ∞ ((γ¯ 2 )∗ N ). For j > 1 the obstruction bundle gets much more complicated, but the 1 1 idea is the same. Apart from the fact that the commutativity condition gγ¯ , j , gγ¯ ,k = 0 clearly leads to relations between the different ( n ), observe that a choice of orientation for γ does not lead to a canonical choice of orientations for its multiples γ k . While we expect that it is in general very hard to write down a set of signs ( n ) explicitly, for all the geometric applications we have in mind and the educational purposes as a test model beyond the Gromov-Witten case we are rather interested in proving vanishing results as the one above.

Acknowledgements. This research was supported by the German Research Foundation (DFG). The author thanks K. Cieliebak, Y. Eliashberg, K. Fukaya, M. Hutchings and P. Rossi for useful discussions.

References [BEHWZ] [BM] [CL] [CM] [CMS]

Bourgeois, F., Eliashberg, Y., Hofer, H., Wysocki, K., Zehnder, E.: Compactness results in symplectic field theory. Geom. and Top. 7, 799–888 (2003) Bourgeois, F., Mohnke, K.: Coherent orientations in symplectic field theory. Math. Z. 248, (2003) Cieliebak, K., Latschev, J.: The role of string topology in symplectic field theory. http://arixiv. org/abs/0706.3284v2 [math.s6], 2007 Cieliebak, K., Mohnke, K.: Symplectic hypersurfaces and transversality for gromov-witten theory. J. Symp. Geom. 5, 281–356 (2007) Cieliebak, K., Mundet, I., Salamon, D.: Equivariant moduli problems, branched manifolds, and the euler class. Topology 42(3), 641–700 (2003)

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[E] [EGH] [F1] [F2] [FR] [HT1] [HT2] [HWZ] [L] [Lo] [MDSa] [OP] [R1] [R2] [Sch] [W]

159

Eliashberg, Y.: Symplectic field theory and its applications. Proceedings of the ICM 2006., available at http://math.stanford.edu/~eliash/Public/eliashberg.pdf, 2006 Eliashberg, Y., Givental, A., Hofer, H.: Introduction to symplectic field theory. GAFA 2000 Visions in Mathematics Special Volume, Part II, 560–673 (2000) Fabert, O.: Contact homology of hamiltonian mapping tori. Comm. Math. Helv. 85, 203– 241 (2010) Fabert, O.: Obstruction bundles over moduli spaces with boundary and the action filtration in symplectic field theory. http://arxiv.org/abs/0709.3312v3 [math.s6], 2010 Fabert, O., P. Rossi: String, dilaton and divisor equation in symplectic field theory. http://arxiv. org/abs/1001.3094v2 [math.s6], 2010 Hutchings, M., Taubes, C.: Gluing pseudoholomorphic curves along branched covered cylinders i. J. Symp. Geom. 5, 43–138 (2007) Hutchings, M., Taubes, C.: Gluing pseudoholomorphic curves along branched covered cylinders ii. J. Symp. Geom. 7, 29–133 (2009) Hofer, H., Wysocki, K., Zehnder, E.: A general fredholm theory i: a splicing-based differential geometry. J. Eur. Math. Soc. 9(4), 841–876 (2007) Li, J.: A degeneration formula of gw-invariants. J. Diff. Geom. 60(2), 199–293 (2002) Long, Y.: Index theory for symplectic paths with applications. Progress in Mathematics 207, Basel-Bostoni Birkhäuser, 2002 McDuff, D., Salamon, D.A.: J -holomorphic curves and symplectic topology. AMS Colloquium Publications, Providence RI; Amer. Math. Six., 2004 Okounkov, A., Pandharipande, R.: Gromov theory, hurwitz theory and completed cycles. Ann. of Math. 163(2), 517–560 (2006) Rossi, P.: Gromov-witten invariants of target curves via symplectic field theory. J. Geom. Phys. 58, 931–941 (2008) Rossi, P.: Integrable systems and holomorphic curves. http://arxiv.org/abs/0912.0451v2 [math.s6], 2010 Schwarz, M.: Cohomology operations from S 1 -cobordisms in Floer homology. Ph.D. thesis, Swiss Federal Inst. of Techn. Zurich, Diss. ETH No. 11182, 1995 Wendl, C.: Automatic Transversality and Orbifolds of Punctured Holomorphic Curves in Dimension Four. http://arxiv.org/abs/0802.3842v4 [math.s6], 2009

Communicated by N.A. Nekrasov


Communications in


A New Variational Approach to the Stability of Gravitational Systems Mohammed Lemou1 , Florian Méhats1 , Pierre Raphaël2 1 CNRS and IRMAR, Université de Rennes 1, Rennes, France.

E-mail: [email protected]; [email protected]

2 IMT, Université Paul Sabatier, Toulouse, France.

E-mail: [email protected] Received: 25 September 2009 / Accepted: 31 July 2010 Published online: 11 January 2011 – © Springer-Verlag 2011

Abstract: We consider the three dimensional gravitational Vlasov Poisson system which describes the mechanical state of a stellar system subject to its own gravity. A well-known conjecture in astrophysics is that the steady state solutions which are nonincreasing functions of their microscopic energy are nonlinearly stable by the flow. This was proved at the linear level by several authors based on the pioneering work by Antonov in 1961. Since then, standard variational techniques based on concentration compactness methods as introduced by P.-L. Lions in 1983 have led to the nonlinear stability of subclasses of stationary solutions of ground state type. In this paper, inspired by pioneering works from the physics litterature (MNRAS 241:15, 1989), (Mon. Not. R. Astr. Soc. 144:189–217, 1969), (Mon. Not. R. Ast. Soc. 223:623–646, 1988) we use the monotonicity of the Hamiltonian under generalized symmetric rearrangement transformations to prove that non increasing steady solutions are the local minimizer of the Hamiltonian under equimeasurable constraints, and extract compactness from suitable minimizing sequences. This implies the nonlinear stability of nonincreasing anisotropic steady states under radially symmetric perturbations. 1. Introduction and Main Results 1.1. Setting of the problem. We consider the three dimensional gravitational VlasovPoisson system

∂t f + v · ∇x f − ∇φ f · ∇v f = 0, f (t = 0, x, v) = f 0 (x, v) ≥ 0,

(t, x, v) ∈ R+ × R3 × R3 ,

where, throughout this paper, f (x, v) dv and φ f (x) = − ρ f (x) = R3

1 ∗ ρf 4π |x|

(1.1)

(1.2)

162

M. Lemou, F. Florian, P. Raphaël

are the density and the gravitational Poisson field associated to f . This nonlinear transport equation is a well known model in astrophysics for the description of the mechanical state of a stellar system subject to its own gravity and the dynamics of galaxies, see for instance [10,15]. Unique global classical solutions for initial data f 0 ∈ Cc1 , f 0 ≥ 0, where Cc1 denotes the space of compactly supported and continuously differentiable functions, have been shown to exist in [40,47,49] and to propagate the corresponding regularity. Two fundamental properties of the nonlinear transport flow (1.1) are then first the preservation of the total Hamiltonian 1 1 H( f (t)) = |v|2 f (t, x, v)d xdv − |∇φ f (t, x)|2 d x = H( f (0)), (1.3) 2 R6 2 R3 and second the preservation of all the so-called Casimir functions: ∀G ∈ C 1 ([0, +∞), R+ ) such that G(0) = 0, G( f (t, x, v)) d xdv = G( f 0 (x, v)) d xdv. (1.4) R6

R6

This last property induces a continuum of conservation laws and is the major difference between this kind of problem and other nonlinear dispersive problems like nonlinear wave or Schrödinger equations. 1.2. Nonlinear stability of steady state solutions. A classical problem which has attracted a considerable amount of work both in the astrophysical [2–4,25,26,41,42,54] and mathematical communities, is the question of the nonlinear stability of stationary states. If we restrict our study to radially symmetric stationary states –that is a priori depending on (|x|, |v|, x · v) only, Jean’s theorem [8] ensures that they can be described as functions of their own microscopic energy and their angular momentum: |v|2 + φ Q (x), (x, v) = |x × v|2 , 2 Q(x, v) = F (e(x, v), (x, v)) .

e(x, v) =

(1.5) (1.6)

The existence of such steady states has been discussed in [8] for a large class of smooth functions F. A well-known conjecture in astrophysics, [10], is now that among these stationary solutions, those who are nonincreasing functions of their microscopic energy e are nonlinearly stable by the Vlasov Poisson flow, explicitly: Conjecture. Non increasing anisotropic galaxies F = F(e, ) with ∂∂eF < 0 on the support of Q are stable by spherically symmetric perturbations for the flow (1.1). Non increasing isotropic spherical galaxies F = F(e) with ∂∂eF < 0 on the support of Q are orbitally stable against general perturbations for the flow (1.1). Remarkably enough, this conjecture has been proved at the linear level by Doremus, Baumann and Feix [14] (see also [17,25,52] for related works), following the pioneering work by Antonov in the 60’s [3,4]. These results are based on some coercivity properties of the linearized Hamiltonian under constraints formally arising from the linearization of the Casimir conservation laws (1.4), see Lynden-Bell [41]. At the nonlinear level, the general problem is open. However, the nonlinear stability of a large class of stationary solutions of so-called ground state type including

New Variational Approach to the Stability of Gravitational Systems

163

the polytropic states has been obtained using variational methods in [13,18,20–22,55], completed by [50]. In [28–30], see also [48], we observed that a direct application of Lion’s concentration compactness techniques [38,39], implies that or a large class of convex functions j, the two parameters –according to the scaling symmetry of (1.1)– minimization problem I (M1 , M j ) =

inf

| f | L 1 =M1 , | j ( f )| L 1 =M j

H( f ),

M1 , M j > 0

(1.7)

is attained up to symmetries on a steady state solution to (1.1) of the form (1.6), and all minimizing sequences to (1.7) are relatively compact up to a translation shift in the natural energy space E = { f ≥ 0 with | f |E = | f | L 1 + | f | L ∞ + ||v|2 f | L 1 < +∞}. The so-called Cazenave, Lions [11] theory of orbital stability then immediately implies the orbital stability of the corresponding ground state steady solution, [29]. In fact, this last step requires the knowledge of the uniqueness of the minimizer to (1.7) which is a delicate open problem in general, see [50], but this difficulty was overcome in [30]. Other non variational approaches based on linearization techniques have also been explored in [23,53]. Recently, Guo and Lin [19] proved the radial stability of the so called King model F(e) = (exp(e0 − e) − 1)+ which is not in the class of ground states as obtained in the framework of (1.7). Adapting a robust approach developed by Lin and Strauss in their study of the Vlasov Maxwell system, [35–37], the authors use the infinity of conservation laws provided by the nonlinear transport to construct a sufficient large approximation of the kernel of the linearized operator close to the steady state. This allows them to recover a coercivity statement of the linearized energy using Antonov’s coercivity property which after linearization and control of higher order terms for the King model yields the claimed stability in the radial class.

1.3. Additional conserved quantities in the radial setting. Our main purpose in this paper is to describe a generalized variational approach for the nonlinear stability of steady states which fully takes into account the nonlinear transport structure of the problem, and in particular the continuum of constraints at hand from (1.4). First recall that in general, the full set of invariant quantities conserved by the nonlinear transport flow (1.1) depends on the initial data and its possible symmetries. From now and for the rest of this paper, we shall restrict our attention to spherically symmetric solutions f (x, v) = f (|x|, |v|, x · v), where we will systematically abuse notations and identify f with its image through various diffeomorphisms. We then let Erad be the space of spherically symmetric distribution functions of finite energy Erad = { f ∈ E,

f spherically symmetric},

(1.8)

and recall that if f is spherically symmetric, then ρ f (x) = ρ f (|x|) and φ f (x) = φ f (|x|). This implies in particular from a direct computation that the momentum = |x × v|2 is conserved by the characteristic flow associated to (1.1), and hence a larger class of Casimir conservation laws (1.4) holds:

164


R6

G( f (t, x, v), |x × v|2 )d xdv =

R6

G( f 0 (x, v), |x × v|2 )d xdv

(1.9)

for all G ∈ C 1 ([0, +∞) × [0, +∞), R+ ) with G(0, ) = 0, ∀ ≥ 0. Let us reformulate (1.9) in terms of equimeasurability properties of f and f 0 . Performing the change of variables r = |x|, w = |v|, x · v = |x||v| cos θ, r, w > 0, θ ∈ ]0, π [, the Lebesgue measure is mapped onto: ∞ +∞ 2 f (x, v)d xdv = 8π R6

r =0 w=0

π θ=0

f (r, w, cos θ )r 2 w 2 sin θ dr dwdθ.

We then perform the second change of variables, r = r, u = w sign(cos θ ), = r 2 w 2 sin2 θ and get from Fubini: f (x, v)d xdv = R6

+∞

=0

(r,u)∈

f (r, u, )dν d

(1.10)

with = {(r, u) ∈ R+ × R with r 2 u 2 > }

(1.11)

dν = 4π 2 1r 2 u 2 > (r 2 u 2 − )−1/2 r |u|dr du.

(1.12)

and We then define the distribution function of f at given kinetic momentum : ∀ > 0, ∀s ≥ 0, μ f (s, ) = ν {(r, u) ∈ , f (r, u, ) > s}, or equivalently μ f (s, ) = 4π

2

+∞ +∞

r =0

u=−∞

1 f (r,u,)>s (r 2 u 2 − )−1/2 r |u|1r 2 u 2 > dr du.

(1.13)

(1.14)

We now define the set of distribution functions which are equimeasurable to f at given by: Eq( f ) = {g ≥ 0 spherically symmetric, ∀s > 0, μ f (s, ) = μg (s, ) a.e. }. (1.15) We then have from standard arguments: Lemma 1.1 (Characterization of Eq( f )). Let f ∈ L 1 ∩L ∞ , nonnegative and spherically symmetric, then the following are equivalent: (i) g ∈ Eq( f ); (ii) ∀G(h, ) ≥ 0, C 1 with G(0, ) = 0, G( f (x, v), |x × v|2 )d xdv = G(g(x, v), |x × v|2 )d xdv holds . R6

R6

Lemma 1.1 allows us to reformulate the conservation laws of the full Casimir class (1.9) in the radial setting as follows: ∀t ≥ 0,

f (t) ∈ Eq( f 0 ).

(1.16)


165

1.4. Assumption (A) on the steady state. Before stating the results, let us fix our assumptions on the steady state Q. (i) Q is a continuous, nonnegative, non zero, compactly supported steady state solution of the Vlasov-Poisson system (1.1). (ii) There exists a continuous function F : R × R+ → R+ such that ∀(x, v) ∈ R6 ,

Q(x, v) = F

|v|2 + φ Q (x), |x × v|2 . 2

(1.17)

(iii) There exists e0 < 0 such that: O = {(e, ) ∈ R × R+ : F(e, ) > 0} ⊂] − ∞, e0 [×R+ , F is C 1 on O,

with

∂F ∂e

< 0.

Remark 1.2. Note that ∂∂eF may be infinite at the boundary of O, as is the case for polyq tropic ground states F(e, ) = (e0 − e)+ κ , for some 0 < q < 1 and κ ≥ 0. Below we list a number of physically relevant models for which our non linear stability result applies. All these examples are extracted from [10] to which we refer for a detailed physical description of various gravitational models. Examples. – Polytropes and double-power models: The polytropes correspond to the following form of F: F(e, ) = (e0 − e)+ κ , q

0 < q < 7/2,

κ ≥ 0,

where e0 < 0 is a constant threshold energy. A generalization of these polytropes is provided by the so-called double-power model [10]: F(e, ) =

αi j (e0 − e)+i κ j , q

0≤i, j≤N

where αi j are nonnegative constants. – Michie-King models: F(e, ) = exp(−/2ra2 ) (exp(e0 − e) − 1)+ , where e0 < 0 and the constant ra > 0 is the anisotropy radius [10]. When ra goes to infinity, this model reduces to the King model. – Osipkov-Merritt models: F(e, ) = G e0 − e + 2 , 2ra where e0 < 0, ra > 0 are constants, and G is a nonincreasing C 1 function such that G(t) = 0 for all t ≤ 0.

166


1.5. Statement of the results. From (1.16), a natural generalization of (1.7) in the radial setting is to minimize the Hamiltonian under constraints of given equimeasurability. This is a very natural strategy to prove stability in a nonlinear transport setting which goes back in fluid mechanics to the celebrated works of Arnold, see e.g [5–7], Marchioro and Pulvirenti [43,45], Wolansky and Ghil [56], and references therein, and is also very much present in the physics litterature, see in particular Lynden-bell [41], Gardner [16], Wiechen, Ziegler, Schindler [54], Aly [2] and references therein. The mathematical implementation of the corresponding variational problem is however confronted to the description of bounded sequences in Eq( f 0 ) and a possible lack of compactness in general, see for example Alvino, Trombetti and Lions [1] for an introduction to this kind of problem. Our first result is the characterization of non increasing states as local minimizers of the Hamiltonian in Erad under a constraint of equimeasurability: Theorem 1.3 (Local variational characterization of Q). There exists a constant C0 > 0 such that the following holds. For all R > 0, there exists δ0 (R) > 0 such that, for all f ∈ Erad ∩ Eq(Q) satisfying | f − Q|E ≤ R,

|∇φ f − ∇φ Q | L 2 ≤ δ0 (R),

(1.18)

we have H( f ) − H(Q) ≥ C0 |∇φ f − ∇φ Q |2L 2 .

(1.19)

If in addition H( f ) = H(Q), then f = Q. Theorem 1.3 was first obtained by Guo, Rein [23] for a perturbation f near Q 1 in the specific case of the isotropic King model, and for isotropic relativistic models F(e, ) = F(e) with locally bounded derivative F (e) in [24], and this excludes any singularity at the boundary –as many polytropic models would have. Let us stress onto the fact that Theorem 1.3 by itself alone is too weak to yield a stability statement including the full set of radial pertubations. Hence the importance of Theorem 1.3 relies in fact mostly on its proof. Indeed, a new important feature of our analysis is to use a monotonicity property of the Hamiltonian under a generalized Schwarz symmetrization which is not the standard radial rearrangement but a rearrange2 ment with respect to a given microscopic energy |v|2 + φ(x), at fixed angular momentum |x × v|2 , see Proposition 2.8 for a precise definition and Proposition 3.1 for the monotonicity statement. This monotonicity is very much a consequence of the “bathtub” principle for symmetric rearrangements, see Lieb and Loss [33], and was already observed in the physics literature, see Gardner [16], Aly [2]. It produces a reduced functional J (φ f ) which depends on the Poisson field φ f only and not the full distribution function. The outcome is a lower bound H( f ) − H(Q) ≥ J (φ f ) − J (φ Q ).

(1.20)

Interestingly enough, the reduced functional J was first introduced on physical ground as a generalized potential energy in the pioneering works by Lynden-Bell [41], see also Wiechen, Ziegler, Schindler [54]. It now turns out from explicit computation that the critical points of J are the Poisson field of steady states, and that the Hessian of J near the Poisson field of a nondecreasing steady state can be directly connected to 1 And not only φ near φ , which is an issue for the proof of Theorem 1.4. f Q


167

the Hartree-Fock exchange operator [41], which is coercive from Antonov’s stability criterion, see Sect. 4, and hence φ Q itself is a local minimizer of J . The important outcome of the structure (1.20) is that by reducing the problem to a problem on the Poisson field only, we are able to extract compactness in the radial setting from any minimizing sequence whose Hamiltonian converges to Q without the assumption of equimeasurability, thanks to the smoothing and compactness provided by the radial Poisson equation. This allows us to prove the following compactness result which is the heart of our analysis. Given f ∈ Erad , we consider the family of its Schwarz symmetrizations f ∗ (·, ), > 0, as defined in Proposition 2.6. We then claim: Theorem 1.4 (Compactness of local minimizing sequences). There exists δ > 0 such that the following holds. Let f n be a sequence of functions of Erad , bounded in L ∞ , such that |∇φ fn − ∇φ Q | L 2 < δ,

(1.21)

and lim sup H( f n ) ≤ H(Q), n→+∞

f n∗ → Q ∗ in L 1 (R+ × R+ ) as n → +∞

(1.22)

then f n → Q in L 1 (R6 ),

|v|2 f n → |v|2 Q in L 1 (R6 ).

(1.23)

Theorem 1.4 is the key to the radial Cazenave-Lions’ theory of orbital stability [11] and implies that any compactly supported non increasing steady state Q as defined by (1.17), is nonlinearly stable under the action of the Vlasov-Poisson flow with respect to spherical perturbations. We thus obtain the main result of this paper: Theorem 1.5 (Nonlinear stability of Q under the nonlinear flow (1.1)). For all M large enough and for all ε > 0, there exists η > 0 such that the following holds true. Let f 0 ∈ Erad ∩ Cc1 with | f 0 − Q| L 1 < η, | f 0 | L ∞ < M, H( f 0 ) < H(Q) + η,

(1.24)

then the corresponding global strong solution f (t) to (1.1) satisfies: ∀t ≥ 0, | f (t) − Q| L 1 + ||v|2 ( f (t) − Q)| L 1 < ε, | f (t)| L ∞ < M.

(1.25)

Comments on Theorem 1.5. 1. Linear versus nonlinear stability. A natural strategy to pass from linear to nonlinear stability is to try to linearize the problem and estimate higher order terms as perturbations. This turns out to be quite delicate in general and the control of higher order terms may be challenging, see [19,53] for a treatment of the King model, [32] for the polytropic case. Our analysis avoids this classical difficulty using two facts. We first derive a global monotonicity property which is fundamentally a nonlinear property and does not rely on any linearization procedure, Proposition 3.1, and which reduces the problem to understanding a simpler functional on the Poisson field φ only. For this functional, we do apply a linearization procedure that is a Taylor expansion near φ Q , but we avoid the computation of higher order terms thanks to compactness properties of the Hessian, see (4.45), (4.61).

168


2. Comparison with previous nonlinear stability results. In view of the nonlinear stability result obtained for ground state type minimizers of (1.7) which are not restricted to the radial class, one may ask whether a generic steady solution of the form (1.17) can in fact be obtained as a ground state for (1.7). This is a nontrivial issue which is connected to the notion of equivalence of ensemble in statistical physics. In a forthcoming work [31] and following pioneering ideas from Lieb and Yau [34], we will exhibit a large class of monotonic functions F for which the equivalence of ensemble actually holds. There are however of course many well known examples where this equivalence of ensembles fails. Note also that physical investigations around these minimization problems can be found in [12] and the references therein. 3. Comparison with 2D incompressible Euler. The conservation of equimeasurability properties by the nonlinear transport flow has also been used in the literature to prove the stability of steady states for the 2D incompressible Euler flow, see for example Marchioro, Pulvirenti [45] and references therein. For a discussion on variational problems with equimeasurability constraints in fluid dynamics, one can also refer to Serre [51]. Our result generalizes this approach to the Vlasov-Poisson system which is however more delicate due to the non-trivial structure of both the Hamiltonian and the steady states solutions. The conjecture of stability of nonincreasing radially symmetric steady states is hence proved for radial perturbations. Note that the result is expected to be optimal for anisotropic galaxies with a non-trivial dependence on as some numerical simulations suggest the possible instability of anisotropic models against general perturbations, see [10]. One important open problem after this work is certainly the general setting of nonradial perturbations for spherical models. 1.6. Strategy of the proof. Let us give a brief insight into the proof of the variational characterization of Q given by Theorem 1.3 and the lower bound (1.20) which are key features of our analysis. It follows in three main steps. Step 1. Rearrangement with respect to a given Poisson field. Let a Poisson field φ and a radially symmetric distribution function f ∈ Erad ; we aim at defining the Schwarz 2 symmetrization of f with respect to the microscopic energy e = |v|2 + φ(x) at each 2 given kinetic momentum . In other words,

given = |x × v| > 0, we are looking for |v|2 ∗φ a function f (x, v) = G 2 + φ(x), which is a nonincreasing function of e and which is equimeasurable to f in the sense of (1.13), (1.15) i.e.: ∀t > 0, μ f (t, ) = μ f ∗φ (t, ) a.e > 0. As a simple change of variables formula similar to (1.10) reveals, the choice of f ∗φ is essentially unique and given by: 2 |v| + φ(x), |x × v|2 , |x × v|2 1 |v|2 f ∗φ (x, v) = f ∗ aφ , (1.26) 2 2 +φ(x) 0, consider the effective potential ψφ, (r ) = φ(r ) +

, 2r 2

r > 0.

(i) Structure of ψφ, : ψφ, ∈ C 1 (R3 \{0}) and

eφ, = inf ψφ, (r ) , r ≥0

(2.2)

(2.3)

is attained at a unique r0 (φ, ). ψφ, is strictly decreasing on (0, r0 (φ, )) and strictly increasing on (r0 (φ, ), +∞) with lim ψφ, (r ) = +∞,

r →0

lim ψφ, (r ) = 0.

r →+∞

(2.4)

Moreover, the function → eφ, is continuous on R∗+ , with the uniform bound: | f |2L 1 ∀ > 0, max φ(0), − (2.5) ≤ eφ, < 0. 2 (ii) Level sets of ψφ, : for eφ, < e < 0, let r1 (φ, e, ) = inf r ≥ 0 st. e − ψφ, (r ) > 0 , r2 (φ, e, ) = sup r ≥ 0 st. e − ψφ, (r ) > 0 .

(2.6) (2.7)

Then r1 (φ, e, ), r2 (φ, e, ) are C 1 functions of e with uniform bounds: ∀eφ, < e < 0: 0
0, such that φ(r ) ≤ −

| f |L 1 , ∀ r ≥ r˜ . 2r

(2.14)

172


Thus (2.13), (2.14) imply (2.4). From (2.14), eφ, given by (2.3) satisfies | f |L 1 + 2 < 0, eφ, ≤ inf − 2r 2r r ≥˜r since by assumption f = 0, and hence eφ, is attained at some r0 = r0 (e, φ, ). Thus from (2.13): | f |2L 1 | f |L 1 eφ, = φ(r0 ) + 2 ≥ max φ(0), − + 2 ≥ max φ(0), − ∀ > 0, r0 2 2r0 2r0 and (2.5) is proved. Observe now from (2.11) again that: ψφ, (r ) = φ (r ) −

, and (r 2 ψφ, (r )) = r 2 ρ f + 2 > 0, 3 r r

and hence from ψ (r0 (e, φ, )) = 0: ∀r > 0, r

2

ψφ, (r )

=

r

r0

2 r ρ f (r ) + 2 dr, r

(2.15)

(2.16)

which yields the uniqueness of the minimum r0 > 0 and the claimed monotonicity properties of ψφ, . Together with (2.4), we conclude from (2.16) that r1 , r2 given by (2.6), (2.7) are well defined for eφ, < e < 0, and are C 1 functions of e from the implicit function theorem. To prove the uniform bound (2.8), we observe from (2.13): | f |L 1 − 2 >0 , r ≥ 0; st. e − φ(r ) − 2 > 0 ⊂ r ≥ 0; st. e + 2r r 2r and hence using from (2.5) that | f |2L 1 + 2e > 0 for e > eφ, : ⎤ ⎡ | f |L 1 ⎦. r ≥ 0; st. e+ − 2 >0 ⊂⎣ , r 2r 2 2 | f | L 1 + | f | L 1 +2e | f | L 1 − | f | L 1 +2e We then use the definitions (2.6) and (2.7) to get 0
0. > 0, α2 = 2| f | L 1 |eφ,2 |


173

Hence, (r, ) → ψφ, (r ) being continuous, the function ∈ [1 , 2 ] → eφ, = min ψφ, (r ) r ∈[α,α2 ]

is continuous. It remains to prove the concavity bound (2.9). Let w(r ) = e − ψφ, (r ) − then

2r 2 r

1 r2

(r − r1 )(r2 − r ),

1 2 −(r ψφ, (r )) + 2 = −rρ f (r ) ≤ 0, (r w(r )) = r r

where we used (2.15). Hence the function r → r w(r ) is concave. Since it vanishes at r1 and r2 , we conclude that w(r ) ≥ 0 for all r ∈ [r1 , r2 ] and (2.9) is proved. This concludes the proof of Lemma 2.1. Let us now define the Jacobian function aφ (e, ) and examine its differentiability properties in e: Lemma 2.3 (Definition and differentiability properties in e of the Jacobian aφ ). For φ = φ f ∈ rad non zero and > 0, we define: 2 ν (r, u) ∈ (R+ )2 : u2 + φ(r ) < e for e < 0 and > 0, aφ (e, ) = (2.17) +∞ for e ≥ 0, and > 0, where ν is the measure given by (1.12), equivalently: ∀ > 0, ∀e < 0, √ r2 1/2 e − ψφ, (r ) dr. aφ (e, ) = 8π 2 2

(2.18)

r1

Then: (i) Behavior of aφ : aφ (e, ) = 0 for e < eφ, and ∀ > 0, aφ (eφ, , ) = 0,

lim aφ (e, ) = +∞.

e→0−

(2.19)

(ii) Uniform bounds on aφ : let 0 < m φ := inf (r + 1)|φ(r )| < +∞,

(2.20)

∀e < 0, aφ (e, ) ≤ 16π 2 |e|−1/2 | f | L 1 ,

(2.21)

r ≥0

then the bounds

and ∀e ∈

−

m 2φ 4(2m φ + )

,0 ,

aφ (e, ) ≥

4π 2 −1/2 |e| m φ hold. 3

(2.22)

174


(iii) Differentiability in e: the map e → aφ (e, ) is a C 1 -diffeomorphims from (eφ, , 0) to (0, +∞) with: √ ∂aφ (e, ) = 4π 2 2 ∀e ∈]eφ, , 0[, ∂e

r2

r1

−1/2 e − ψφ, (r ) dr > 0.

(2.23)

Abusing notations, we shall denote in the sequel aφ−1 (·, ) : (0, +∞) → (eφ, , 0) its inverse function. Proof. Step 1. Bounds on aφ . First compute from the definitions (2.17) and (1.12): ∀e < 0, ∀ > 0 : aφ (e, ) = 8π 2

r >0 u>0

1 u2

= 8π 2

r >0

2

1e−φ(r )−

√ 2 = 8π 2

r2

r1

1 2 2 (r 2 u 2 +φ(r )<e r u > √ 2(e−φ(r )) >0 2r 2

√

u=

e − φ(r ) − 2 2r

r

− )−1/2 r udr du

(r 2 u 2 − )−1/2 udu r dr

1/2 dr, +

this is (2.18) or, equivalently, (1.27). Then aφ (e, ) = 0 for e ≤ eφ, and aφ (e, ) > 0 on (eφ, , 0) from Lemma 2.1. We now estimate aφ from above for e < 0 using (2.13) and (2.8) as follows: 1/2 e − φ(r ) − 2 dr 2r r1 (φ,e,) √ r2 (φ,e,) | f | L 1 1/2 2 ≤ 8π 2 dr r r1 (φ,e,)

√ 1/2 ≤ 16π 2 2| f | L 1 r2 (φ, e, )1/2 − r1 (φ, e, )1/2

√ aφ (e, ) = 8π 2 2

r2 (φ,e,)

≤ 16π 2 | f | L 1 |e|−1/2 , and (2.21) is proved. To estimate aφ (e, ) from below, first observe that (2.20) follows from (2.11). We then write: √ aφ (e, ) ≥ 8π 2 2 √ ≥ 8π 2 2

+∞

e+ 0

mφ − 2 r + 1 2r

1/2 dr +

mφ 1/2 e+ − dr r + 1 2r 2 + 1+/m φ +∞

and observe that for r ≥ 1 + /m φ , we have mφ mφ mφ mφ − 2 ≥ − ≥ 1− ≥ . 2 r + 1 2r r + 1 2(r − 1) r +1 2m φ (r − 1) 2(r + 1)


Thus:

√ aφ (e, ) ≥ 8π 2 2 √ ≥ 8π 2 2 ≥ 8π

e+

mφ 2(r + 1)

1+/m φ |e| 1/2 +∞

√ 2

+∞

mφ

1+/m φ

175

1/2 dr +

1/2 m φ − 2|e|(r + 1) + dr

2m φ + 2 −1/2 |e| m φ 1 − 2|e| 3 m 2φ m2

3/2 . +

2

This means that for |e| ≤ 4(2m φφ +) , aφ (e, ) ≥ 4π3 |e|−1/2 m φ , and (2.19) and (2.22) are proved. The continuity and the monotonicity of the application e → aφ (e, ) is a consequence of (2.8) and of the dominated convergence theorem, since 1/2 e − φ(r ) − 2 ≤ (−φ(0))1/2 , for all r ∈]eφ, , 0[. 2r + Step 2. Differentiability of aφ . We are now in position to prove the differentiability of the function e → aφ (e, ) which follows from the version of Lebesgue’s derivation theorem given by Lemma A.1. Let us fix > 0 and write +∞ g(e, r )dr aφ (e, ) = 0

with

√ √ 1/2 1/2 g(e, r ) = 8π 2 2 e − ψφ, (r ) 1r1 (φ,e,) 0. Now using the monotonicity of the function e → aφn (e, ) at fixed and applying the second Dini’s theorem, we get the desired uniform convergence. Step 6. Proof of (2.31). Let (s, ) ∈ (R∗+ )2 . Denote (s, ), en = aφ−1 n

e0 = aφ−1 (s, ).

We will prove that en → e0 . From (2.21), we get |en | ≤ C

| fn |L 1 aφn (en , )

2

=C

| fn |L 1 s

2 .

(2.40)

m2 . 4( + 2m)

(2.41)

Now we claim that |en | ≥ C

m 2 s

> 0 if |en | ≤

Indeed, we first get from (2.22), |en | ≥ C provided that |e| ≤

m 2φn 4(+2m φn ) ,

m φn aφn (en , )

2 =C

m

φn

s

2

> 0,

(2.42)

with m φn defined by (2.20). From (2.29), we have m φn ≥

m > 0. Therefore, (2.42) implies (2.41) since the function t →

t2 +2t

is increasing.


179

We then deduce from (2.40) and (2.41) that the sequence en belongs to a compact interval of R∗− thus, up to a subsequence, we have en → e∞ ∈ R∗− as n → +∞. Using (2.30), we have s = aφn (en , ) → aφ (e∞ , ) as n → +∞. Hence, aφ (e∞ , ) = aφ (e0 , ) = s ∈ (0, ∞). Since e → aφ (e, ) is invertible from (eφ, , 0) onto (0, ∞), we deduce that e0 = aφ−1 (s, ) = e∞ , which means that en → e0 as n → +∞. The proof of (2.31) is complete. This concludes the proof of Lemma 2.4. Let us now examine the differentiability of aφ and aφ−1 with respect to φ. To shorten the statement of the next lemma, we introduce a few notations. We consider two nonzero = φ potentials φ = φ f ∈ rad and φ f ∈ rad and set: − φ. h=φ

(2.43)

For all > 0 and λ ∈ [0, 1], we recall the notation eφ+λh, = inf ψφ, (r ) + λh(r ) ,

(2.44)

where ψφ Q , (r ) is defined by (2.2), and denote , ) = (λ, e) : λ ∈ [0, 1] and e ∈]eφ+λh, , 0[ . (φ, φ

(2.45)

r ≥0

Let s ∈ R∗+ and λ ∈ [0, 1]. Recall that, by Lemma 2.8, there exists a unique e ∈ −1 (s, ), such that aφ+λh (e, ) = s. Finally, we set ]eφ+λh, , 0[, denoted by aφ+λh f | L 1 ). M = max(| f | L 1 , |

(2.46)

Lemma 2.5 (Differentiability of aφ (e, ) with respect to φ). Let > 0 be fixed. Consider . Then, with the notations (2.43)–(2.46), ∈ rad both nonzero and let h = φ − φ φ, φ the following holds: (i) The function (λ, e) → aφ+λh (e, ) , ). Moreover, we have is a C 1 function on (φ, φ √ r2 (φ+λh,e,) −1/2 ∂ aφ+λh (e, ) = −4π 2 2 e − ψφ, (r ) − λh(r ) h(r )dr, ∂λ r1 (φ+λh,e,) (2.47) with the bound: 2 ∂aφ+λh ≤ C M√ , ∀(λ, e) ∈ (φ, φ , ), (e, ) ∂λ e2 for some universal constant C > 0.

(2.48)

180


−1 (ii) Let s ∈ R∗+ . Then the function λ → aφ+λh (s, ) is differentiable on [0, 1] and we have

−1/2 r2 −1 h(r )dr r1 aφ+λh (s, ) − ψφ, (r ) − λh(r ) ∂ −1 aφ+λh (s, ) = , (2.49)

−1/2 ∂λ r2 −1 a (s, ) − ψ (r ) − λh(r ) dr φ, φ+λh r1 −1 (s, ), ). where (ri )i=1,2 shortly denotes ri (φ + λh, aφ+λh

Proof. Recall from Lemma 2.4 that the functions eφ+λh, , r1 (φ + λh, e, ) and r2 (φ + λh, e, ) are continuous functions of λ (for fixed e and ). Step 1. Proof of (i). This proof of (i) will be done with Lemma A.1, exactly in the same manner as the regularity of aφ+λh (e, ) with respect to e in Lemma 2.8. We fix > 0 and introduce the following function: √ 1/2 g(λ, e, r ) = 8π 2 2 e − ψφ, (r ) − λh(r ) + , so that aφ+λh (e, ) =

r2 (φ+λh,e,)

r1 (φ+λh,e,)

g(λ, e, r )dr.

By (2.13) and (2.8), we have the following uniform bound: g(λ, e, r ) ≤ C

M r1 (φ + λh, e, )

1/2

M ≤ C√ ,

where M is defined by (2.46). Hence, one deduces from standard dominated convergence that (λ, e) → aφ+λh (e, ) is a C 0 function on [0, 1] × R− and satisfies , ). aφ+λh (e, ) > 0 ⇔ (λ, e) ∈ (φ, φ Let us now prove the differentiability of aφ+λh (e, ) with respect to λ. Let λ0 ∈ [0, 1], e0 = eφ+λ0 h, , and e ∈]e0 , 0[ be fixed. From the continuity of eφ+λh, with respect to λ, we have e ∈]eφ+λh, , 0[ for λ in a neighborhood I0 of λ0 . Hence, for λ ∈ I0 , the distributional partial derivative of g is given by √ −1/2 ∂g (λ, e, r ) = −4π 2 2 e − ψφ, (r ) − λh(r ) 1r1 (φ+λh,e,) s} +∞ +∞ 2 1 f (r,u,)>s (r 2 u 2 − )−1/2 r |u|1r 2 u 2 > dr du. = 4π r =0

u=−∞

We then have the following elementary lemma:


183

Lemma 2.6 (Properties of μ f ). Let f ∈ L 1 ∩ L ∞ (R6 ), nonnegative and spherically symmetric, and let μ f (s, ) be the distribution function of f at given as defined by (1.14). Then there exists a set A with |A|R+ = 0 such that ∀ ∈ R+ \A, ∀s > 0, μ f (s, ) < +∞, ∀ ∈ R+ \A, ∀s ≥ | f | L ∞ , μ f (s, ) = 0.

(2.57) (2.58)

Moreover, ∀ ∈ R+ \A, the map s → μ f (s, ) is right continuous on R∗+ . We may now introduce the generalized Schwarz symmetrization: Proposition 2.7 (Schwarz symmetrization at fixed > 0). Let f ∈ L 1 ∩ L ∞ (R6 ), nonnegative and spherically symmetric, let μ f (t, ) given by (1.14) and let A be the zero measure set given by Lemma 2.6. We define the Schwarz symmetrization f ∗ (·, ) of f at fixed as being the pseudo inverse of μ f (·, ): sup{s ≥ 0 : μ f (s, ) > t} for t < μ f (0, ) ∗ ∗ ∀t ≥ 0, ∀ ∈ R+ \A, f (t, ) = 0 for t ≥ μ (0, ) f (2.59) f ∗ (·, )

is a nonincreasing function on [0, ∞) and (2.60) ∀t ≥ 0, ∀ ∈ R∗+ \A, μ f (t, ) = {s > 0; f ∗ (s, ) > t |R+

with μ f given by (1.14). Then

In particular | f ∗ | L p (R+ ×R+ ) = | f | L p (R6 ) , ∀ p ∈ [1, +∞].

(2.61)

Moreover, the contractivity relation | f ∗ − g ∗ | L 1 ≤ | f − g| L 1 holds.

(2.62)

Lemma 2.6 and Proposition 2.7 can be derived from standard arguments by adapting for example the arguments in [44]; this is left to the reader. Given f ∈ Erad be and φ ∈ rad , we now define the rearrangement of f with respect 2 to the microscopic energy |v|2 + φ(x). Proposition 2.8 (Symmetric rearrangement with respect to a given microscopic energy). Let f ∈ Erad and φ ∈ rad non zero. Let f ∗ be its symmetric rearrangement defined by (2.59). We define the rearrangement f ∗φ of f with respect to the microscopic 2 energy |v|2 + φ(x) by: 2 |v| f ∗φ (x, v) = f ∗ aφ + φ(x), |x × v|2 , |x × v|2 1 |v|2 , (2.63) 2 2 +φ(x) 0, ∀e ∈ [eφ Q , , 0[, (2.67) where eφ Q , is defined in Lemma 2.1. In particular, for all > 0, Q ∗ (·, ) is a C 1 function on ]0, μ Q (0, )[, where μ Q is defined by (1.13). Proof of Corollary 2.9. Let > 0 be fixed and recall the function F defined in Assumption (A). Assume ) = 0 for all e < 0. From definition (1.13) we have first that F(e, 2 μ Q (s, ) = ν (r, u) : F |u|2 + φ Q (r ), > s = 0 for all s ≥ 0. This implies from (2.59) that Q ∗ (·, ) = 0, and then identity (2.67) is satisfied. Assume now that F(·, ) is not zero on R∗− and let e0 () = sup {e < 0 : F(e, ) > 0} .

(2.68)

By Assumption (A), we have e0 () ≤ e0 < 0 and the function e → F(e, ) is continuous, strictly decreasing on ]−∞, e0 ()] and vanishes for e ≥ e0 (). As F is nonnegative, we have from (1.13): 2 |u| + φ Q (r ), > F(e, ) , ∀e ∈ R, μ Q (F(e, ), ) = ν (r, u) : F 2 and, F(·, ) being strictly decreasing on ] − ∞, e0 ()], this identity implies |u|2 + φ Q (r ) < e , ∀e ≤ e0 () μ Q (F(e, ), ) = ν (r, u) : 2 = aφ Q (e, ), ∀e ≤ e0 ().

(2.69)

Assume that μ Q (0, ) = 0, then μ Q (·, ) = 0 since it is a nonincreasing function. Hence, from definition (2.59) we get Q ∗ (·, ) = 0. Now, we write (2.69) for e = e0 () and deduce from the structure of aφ Q that e0 () ≤ eφ Q , . This means that F(e, ) = 0 for e ∈ [eφ Q , , 0[, and identity (2.67) is satisfied. We now assume μ Q (0, ) > 0, which implies from (2.69) that e0 () > eφ Q , . We know that aφ Q (·, ) (resp. F(·, )) is continuous and one-to-one from [eφ Q , , e0 ()] to [0, aφ Q (e0 (), )] (resp. [0, F(eφ Q , , )]). Hence, identity (2.69) ensures that μ Q (·, ) is invertible from [0, F(eφ Q , , )] to [0, aφ Q (e0 (), )] and Q ∗ (which is by definition its pseudoinverse) is its inverse in this case. Therefore, (2.69) implies Q ∗ aφ Q (e, ), = F(e, ), ∀e ∈ [eφ Q , , e0 ()]. Now (2.69) implies that aφ Q (e, ) ≥ aφ Q (e0 (), ) = μ Q (0, ) for e ∈ [e0 (), 0[, which together with the definition of Q ∗ ensure that both terms in (2.67) vanish for e ∈ [e0 (), 0[. This ends the proof of (2.67). Finally, using (2.67), we conclude that the stated C 1 regularity of Q ∗ on ]0, aφ Q (e0 (), )[ is an immediate consequence of the C 1 regularity and the non vanishing derivatives of F and aφ Q on ]eφ Q , , e0 ()[. To end the proof of Corollary 2.9, it remains to identify Q and Q ∗φ Q for a.e. x, v. Let 2 (x, v) ∈ R6 such that = |x × v|2 > 0 and let e(x, v) = |v|2 + φ Q (r ) ≥ ψφ Q , (r ) ≥


185

eφ Q , , where we used that |v|2 ≥ r2 . If e(x, v) < 0, then (2.67) gives directly Q(x, v) = F(e(x, v), ) = Q ∗φ Q (x, v), by Assumption (A) and (2.63). If e(x, v) ≥ 0, then we have Q(x, v) = F(e(x, v), ) = Q ∗φ Q (x, v) = 0, using again (2.63). This concludes the proof of Corollary 2.9. Proof of Proposition 2.8. We first notice that the formula (2.63) is well-defined for a.e. (x, v) ∈ R6 by Proposition 2.7. Indeed, from (1.10) we have that (x, v) ∈ R6 : |x × v|2 ∈ A 6 = 0, R

where A is the measure zero exceptional set given in Lemma 2.6. Step 1. The change of variables formula. The equimeasurability of f and f ∗φ relies on the following elementary change of variables formula: let two nonnegative functions α ∈ C 0 (R) ∩ L ∞ (R), β ∈ L 1 (R+ × R+ ), then ∀ > 0, 2 +∞ +∞ 2 u u + φ(r ) β aφ + φ(r ), , 1 u 2 2 α dν 2 2 2 +φ(r ) 0, we use the definition of μ f (·, ) (1.14), of f ∗φ (2.63) and the formula (2.70) with α = 1 and β(s, ) = 1 f ∗ (s,)>t to get: +∞ +∞ +∞ 1 f ∗φ (r,u,)>t dν = 1 f ∗ (s,)>t ds, μ f ∗φ (t, ) = 2 0

0

0

and hence from (2.60): ∀t ≥ 0, a.e. > 0, μ f ∗φ (t, ) = μ f (t, ), which implies the equimeasurability of f and f ∗φ according to the definition (1.15) It remains to control the kinetic energy of f ∗φ according to (2.66). Indeed: 2 |v| + φ f ∗φ (x, v)d xdv + 2 ∇φ(x) · ∇φ f ∗φ d x ||v|2 f ∗φ | L 1 = 2 2 1/4 7/12 1/6 ≤ 2 ∇φ(x) · ∇φ f ∗φ d x |∇φ| L 2 ||v|2 f ∗φ | L 1 | f ∗φ | L 1 | f ∗φ | L ∞ , where we used (2.63) and the interpolation inequality (2.10). This together with a straightforward localization argument concludes the proof of (2.66). This concludes the proof of Proposition 2.8. Let us conclude this section with an elementary lemma which will be useful in the sequel. Lemma 2.10 (Pseudo inverse of f ∗ (aφ (·, ), )). Let f ∈ Erad and φ ∈ rad be given nonzero functions, and let > 0 such that f ∗ (0, ) > 0. The function e → f ∗ (aφ (e, ), ) is nonincreasing from [eφ, , 0[ to [0, f ∗ (0, )]. We define its pseudo inverse, which we denote (with abuse of notation) s → ( f ∗ ◦ aφ )−1 (s, ), as follows: ( f ∗ ◦ aφ )−1 (s, ) = sup{e ∈ [eφ, , 0[: f ∗ (aφ (e, ), ) > s},

(2.72)

for all s ∈]0, f ∗ (0, )[. Then s → ( f ∗ ◦ aφ )−1 (s, ) is a nonincreasing function and ∀(x, v) ∈ (R3 )2 such that |x × v|2 = , ∀s ∈]0, f ∗ (0, )[, |v|2 + φ(x) ≤ ( f ∗ ◦ aφ )−1 (s, ), 2 |v|2 + φ(x) ≥ ( f ∗ ◦ aφ )−1 (s, ). f ∗φ (x, v) ≤ s ⇒ 2

f ∗φ (x, v) > s ⇒

(2.73) (2.74)

Proof. Let > 0 and s ∈ (0, f ∗ (0, )), then f ∗ (aφ (eφ, , ), ) = f ∗ (0, ) > s and hence {e ∈ [eφ, , 0) : f ∗ (aφ (e, ), ) > s} is not empty. This means that ( f ∗ ◦aφ )−1 (s, ) is well defined for s ∈ (0, f ∗ (0, )). The monotonicity of ( f ∗ ◦ aφ )−1 follows from the monotonicity of f ∗ and aφ . Let now (x, v) ∈ R6 be such that |x × v|2 = > 0. Assume f ∗φ (x, v) > s, 2 2 then f ∗ (aφ ( |v|2 + φ(x), ), ) > s and thus |v|2 + φ(x) < 0. Thus we have either |v|2 2

+ φ(x) < eφ, , and in this case (2.73) is trivial, or

|v|2 2

+ φ(x) ∈ [eφ, , 0), and


this implies

|v|2 2

+ φ(x) ≤ ( f ∗ ◦ aφ )−1 (s, )) from the definition (2.72). Thus (2.73) |v|2 2 + 2 that |v|2

is then proved. Assume now f ∗φ (x, v) ≤ s. If otherwise

187

f ∗φ (x, v)

≤s < f ∗ (a

e ∈ {e ∈ [eφ, , 0) : and (2.74) follows.

f ∗ (0, )

φ (e, ), )

implies

φ(x) ≥ 0 then (2.74) is trivial, + φ(x) ∈ (eφ, , 0). Thus for all

> s} which is a non empty set,

|v|2 2

+ φ(x) ≥ e,

3. Nonlinear Stability of the Vlasov Poisson System This section is devoted to the proof of the main results of this paper. We first exhibit the key monotonicity formula involving the generalized symmetric rearrangement with respect to the Poisson field (2.63), Proposition 3.1, which allows us to reduce the study of the minimization problem of Theorem 1.3 to the one of an unconstrained minimization problem on the Poisson field only. The study of this new problem, that is the proof of Proposition 3.2, is postponed to Sect. 4, and immediately yields Theorem 1.3. We then show how to extract compactness from minimizing sequences to prove Theorem 1.4 which now implies Theorem 1.5 from standard arguments. 3.1. The monotonicity formula. Given f ∈ Erad , we will note to ease notation: f = f ∗φ f ,

(3.1)

f ∈ Erad ∩ Eq( f ).

(3.2)

and recall from Proposition 2.8 that:

We introduce the functional of φ ∈ rad : 1 J f ∗ (φ) = H( f ∗φ ) + |∇φ − ∇φ f ∗φ |2 , 2

(3.3)

and claim the following monotonicity formula which is a fundamental key for our analysis -see also [2] for related statements: Proposition 3.1 (Monotonicity of the Hamiltonian under the f ∗φ f rearrangement). Let f ∈ Erad , non zero, and f given by (3.1), then: H( f ) ≥ J f ∗ (φ f ) ≥ H( f ).

(3.4)

Moreover, H( f ) = H( f ) if and only if f = f. Proof. Let f, g ∈ Erad , then: 1 1 2 H( f ) = |v| f − |∇φ f |2 2 R6 2 R3 2 |v| 1 1 2 + φ f ( f − g) + = |v| g + φf g + |∇φ f |2 2 2 R6 2 R6 R3 2 |v| 1 1 +φ f ( f − g)+ = H(g)+ |∇φg |2 − ∇φ f · ∇φg + |∇φ f |2 , 2 2 2 R6

188


and hence the general formula: ∀ f, g ∈ Erad , 2 |v| 1 2 + φ f (x) ( f − g) d xdv. H( f ) = H(g) + |∇φ f − ∇φg | L 2 + 2 2 R6

(3.5)

We apply this formula with g = f = f ∗φ f and rewrite the result using (3.3): 2 |v| + φ f (x) ( f − H( f ) = J f ∗ (φ f ) + f ) d xdv. 2 R6 We now claim:

R6

|v|2 f ) d xdv ≥ 0, + φ f (x) ( f − 2

(3.6)

with equality if and only if f = f , which immediately implies (3.4). The proof of (3.6) is reminiscent from the standard inequality for symmetric rearrangement known as the "bathtub" principle ∗ |x| f ≥ |x| f, R6

+∞

R6

see [33]. Indeed, use f (x, v) = t=0 1t< f (x,v) dt and Fubini to derive: 2 +∞ 2 |v| |v| + φf ( f − + φf dt f ) d xdv = 2 2 R6 R6 t=0

× 1t< f (x,v) − 1t< f (x,v) d xdv +∞ 2

|v| 1 + φf = dt − 1 f (x,v)≤t< f (x,v) f (x,v)≤t< f (x,v) d xdv 2 R6 t=0 ∞ f ∗ (0,) u2 + φ f (r ) dν , = d dt − (3.7) 2 S1, (t) S2, (t) =0 t=0 where dν is given by (1.12), and f (r, u, ) ≤ t < f (r, u, )}, S1, (t) = {(r, u) ∈ , S2, (t) = {(r, u) ∈ , f (r, u, ) ≤ t < f (r, u, )}. We now use (2.73) in Lemma 2.10 to obtain: ∀t ∈ (0, f ∗ (0, )), 2 u + φ f (r ) dν ≤ ( f ∗ ◦ aφ )−1 (t, )ν (S2, (t)), 2 S2, (t) where we recall that

ν (S2, (t)) = 4π 2

S2, (t)

1r 2 u 2 > (r 2 u 2 − )−1/2 r |u|dr du.

We then observe from f ∈ Eq( f ) that: for a.e. t > 0, ν (S1, (t)) = ν (S2, (t)),


189

and deduce

S2, (t)

u2 ∗ −1 + φ f (r ) dν ≤ ( f ◦ aφ ) (t, ) dν . 2 S1, (t)

Injecting this into (3.7) and using (2.74) yields: 2 |v| f ) d xdv + φf ( f − 2 R6 2 ∞ f ∗ (0,) u ≥ + φ f (r ) − ( f ∗ ◦ aφ )−1 (t, ) dν ≥ 0 d dt 2 S1, (t) =0 t=0 and the analogous inequality for S2, (t):

|v|2 f ) d xdv + φf ( f − 2 R6 ∞ f ∗ (0,) u2 ≥ ( f ∗ ◦ aφ )−1 (t, ) − − φ f (r ) dν ≥ 0. d dt 2 S2, (t) =0 t=0

2 Moreover, assume that R6 |v|2 + φ f (x) ( f − f ) d xdv = 0. Recalling that ν (S1, (t)) = ν (S2, (t)) = 0 for t > f ∗ (0, ), the above two chains of equalities imply that for a.e t, > 0, either ν (S1, (t)) = ν (S2, (t)) = 0 or ν (S1, (t)) = ν (S2, (t)) > 0 with: u 21 u2 + φ f (r1 ) = ( f ∗ ◦ aφ )−1 (t, ) = 2 + φ f (r2 ), 2 2 for a.e (r1 , u 1 ) ∈ S1, (t), a.e (r2 , u 2 ) ∈ S2, (t), which contradicts the fact that f (r1 , u 1 , ) ≤ t < f (r2 , u 2 , ). We conclude that a.e t, > 0, ν (S1, (t)) = ν (S2, (t)) = 0 which implies f = f . This concludes the proof of (3.6) and of Proposition 3.1. 3.2. Reduction to a variational problem on φ and proof of Theorem 1.3. We now claim the following local coercivity property of the functional of φ given by (3.3). To ease notations, we let for φ ∈ rad : 1 ∗φ |∇φ − ∇φ Q ∗φ |2 . (3.8) J (φ) = J Q ∗ (φ) = H(Q ) + 2 R3 Proposition 3.2 (φ Q is a local strict minimizer of J ). There exist a constant C0 > 0 such that the following holds. For all R > 0, there exists δ0 (R) ∈]0, 21 |∇φ Q | L 2 ] such that, for all f ∈ Erad satisfying | f − Q|E ≤ R,

|∇φ f − ∇φ Q | L 2 ≤ δ0 (R),

we have J (φ f ) − J (φ Q ) ≥ C0 |∇φ f − ∇φ Q |2L 2 .

(3.9)

190


The proof of this proposition essentially relies on Antonov’s coercivity property and is postponed to Sect. 4. Theorem 1.3 is now a straightforward consequence of Propositions 3.1 and 3.2. Proof of Theorem 1.3. Let R > 0 and f ∈ Erad ∩ Eq(Q) satisfying (1.18), where δ0 (R) is as in Proposition 3.2. In particular, note that |∇φ f − ∇φ Q | L 2 ≤

1 |∇φ Q | L 2 2

implies that φ f = 0 and f = 0. Then the monotonicity property (3.4), f ∗ = Q ∗ and (3.3) yield: H( f ) − H(Q) ≥ J f ∗ (φ f ) − H(Q) = J (φ f ) − H(Q).

(3.10)

On the other hand, recall from Corollary 2.9 that our assumption on the ground state Q ensures = Q ∗φ Q = Q and thus H(Q) = J (φ Q ). Q Injecting this together with (3.9) into (3.10) yields: H( f ) − H(Q) ≥ J (φ f ) − J (φ Q ) ≥ C0 |∇φ f − ∇φ Q |2L 2 ,

(3.11)

this is (1.19). If in addition H( f ) = H(Q), then φ f = φ Q and hence using f ∗ = Q ∗ : H( f ∗φ f ) = H(Q ∗φ f ) = H(Q ∗φ Q ) = H(Q) = H( f ). We thus are in the case of equality of Proposition 3.1 from which: f = f ∗φ f = f ∗φ Q = Q ∗φ Q = Q. This concludes the proof of Theorem 1.3.

3.3. Compactness of minimizing sequences. We are now in position to prove Theorem 1.4. Proof of Theorem 1.4. The key to extract compactness is the monotonicity formula (3.11) which yields a lower bound on the Hamiltonian involving the Poisson field φ f only, while standard Sobolev embeddings ensure that φ f enjoys nice compactness properties in the radial setting. Step 1. Weak convergence in L p , p > 1. Let 7/9

2/9

R = |Q|E + C(1 + |∇φ Q | L 2 )4/3 |Q| L 1 |Q| L ∞ + |Q| L 1 + |Q| L ∞ ,

(3.12)

where C is the constant in the interpolation inequality (2.66). Let f n ∈ Erad be a sequence satisfying (1.21), (1.22), where δ will be fixed further, satisfying in particular 1 δ ≤ min 1, |∇φ Q | L 2 . (3.13) 2


191

Observe that (1.21) and (3.13) imply φ fn = 0. The sequence f n∗ is bounded in L 1 by (1.22), so f n is itself bounded in L 1 . Moreover, from H( f n ) < C, the L ∞ bound of f n and the interpolation inequality (2.10), |v|2 f n is uniformly bounded in L 1 . Hence f n is bounded in the energy space Erad . We then get: f n f ∈ Erad in L p for all 1 < p < +∞,

(3.14)

up to a subsequence. Moreover, by a standard consequence of interpolation, Sobolev embeddings and elliptic regularity, we have |∇φ fn − ∇φ f | L 2 → 0 and |φ fn − φ f | L ∞ → 0 as n → +∞.

(3.15)

From assumptions (1.21) and (1.22): |∇φ f − ∇φ Q | L 2 ≤ δ.

(3.16)

In particular, φ f = 0, since δ < |∇φ Q | L 2 from (3.13). Hence, by Proposition 2.8, we have Q ∗φ f ∈ Eq(Q).

(3.17)

Step 2. Strong convergence in E of the sequence Q ∗φ fn . We now aim at extracting a preliminary compactness from f n . Let f n = Q ∗φ fn ,

f = Q ∗φ f ,

(3.18)

and observe that f n is in fact a function of φ fn . We then claim that the strong convergence (3.15) automatically implies some strong compactness in E for fn : (1 + |v|2 ) f n → (1 + |v|2 ) f in L 1 (R6 ).

(3.19)

We claim also that there exists δ1 (R) such that, for 0 < δ ≤ δ1 (R) we have |∇φ f − ∇φ Q | L 2 ≤

δ0 (R) , 2

(3.20)

where R is defined by (3.12) and δ0 (R) is defined in Theorem 1.3. We are now ready to fix the constant δ of Theorem 1.4 as follows: 1 δ = min 1, |∇φ Q | L 2 , δ1 (R) . 2 Proof of (3.19), (3.20). We first claim the a.e convergence: f as n → +∞ for a.e (x, v) ∈ R6 . fn → Indeed, let (x, v) ∈ R6 such that |x × v|2 = > 0. If e = (3.15),

|v|2 2

|v|2 2

(3.21)

+ φ f (x) < 0, then from

+ φ fn (x) < e/2 for n large enough and 2 2 |v| = − |v| − φ f (x) ≤ −φ f (0) ≤ C. + φ (x) fn n n 2 2

(3.22)

We now recall from Lemma 2.4 that for all > 0: aφ fn (e, ) → aφ f (e, ),

(3.23)

192


uniformly with respect to e lying in a compact subset of ] − ∞, 0[. Therefore, from |v|2 2 + φn (x) < e/2 < 0 and from (3.22), 2 2 |v| |v| aφ f n as n → +∞. + φ fn (x), |x × v|2 → aφ + φ f (x), |x × v|2 2 2 Since, by Corollary 2.9, Lemma 2.3 and Assumption (A), the function Q ∗ (·, ) is con2 tinuous, this implies Q ∗φ fn (x, v) → Q ∗φ (x, v). Similarly, |v|2 + φ f (x) > 0 implies |v|2 2 + φn (x) > Q ∗φ fn → Q ∗φ f

0 for n large enough and thus Q ∗φ fn (x, v) = Q ∗φ f (x, v) = 0. Hence a.e in R6 and (3.21) is proved. Now recall from Proposition 2.8 and from φ fn = 0, φ f = 0, that f n ∈ Eq(Q) and f ∈ Eq(Q) so that Q= fn = f. ∀n ≥ 1, R6

R6

R6

to f and the fact that | fn |L 1 = | f |L 1 The almost everywhere convergence of fn = allows us to apply the Brézis-Lieb Lemma (see [33], Theorem 1.9) and get the strong L 1 convergence, Q ∗φ fn

fn → f in L 1 as n → +∞.

(3.24)

It remains to prove the strong convergence of the kinetic energy. Let us decompose f n = 1|v|2 ≤R f n + 1|v|2 >R f n = gn,R + h n,R . The L 1 convergence (3.24) implies: ∀R > 0, |v|2 gn,R → |v|2 1|v|2 ≤R f n in L 1 . Consider 2 the other term. We recall that f n = Q ∗φ fn is supported in the set |v|2 + φ fn (x) < 0. Hence, by interpolation, ||v|2 h n,R | L 1 = |v|2 h n,R (x, v) d xdv ≤ −2 φ fn (x)h n,R (x, v) d xdv 1/4

7/12

1/6

|∇φ fn | L 2 ||v|2 h n,R | L 1 |h n,R | L 1 |Q| L ∞ , which yields 7/9

||v|2 h n,R | L 1 ≤ C |h n,R | L 1 . By writing |h n,R | L 1 ≤ |Q ∗φ fn − Q ∗φ f | L 1 +

|v|2 >R

Q ∗φ f (x, v) d xdv,

we obtain that ||v|2 h n,R | L 1 converges to 0 when R → +∞ and n → +∞ independently. This together with the convergence of |v|2 gn,R concludes the proof of (3.19). We now turn to the proof of (3.20) and claim that it follows directly from (3.16) and the definition f = Q ∗φ f . Indeed, arguing by contradiction, we extract a subsequence δ0 (R) ∇φn → ∇φ Q in L 2 and gn = Q ∗φn such that |∇φ gn − ∇φ Q | L 2 ≥ 2 . From (2.66), gn is a bounded sequence in Erad and then the same proof like for (3.19) yields gn → (1 + |v|2 )Q ∗φ Q = (1 + |v|2 )Q in L 1 (1 + |v|2 ) 2 and hence ∇φ gn → ∇φ Q in L , a contradiction. This concludes the proof of (3.20).


193

Step 3. Identification of the limit. Following (3.1), we let: f n = f n ∗φ fn .

(3.25)

We now claim that the variational characterization of Q given by Theorem 1.3 and the monotonicity of Proposition 3.1 allow us to identify the limit: f = Q and φ f˜ = φ f = φ Q ,

(3.26)

and to obtain the additional convergence: 2 |v| + φ fn (x) f n − f n d xdv → 0 as n → +∞. 2 R6

(3.27)

Proof of (3.26), (3.27). First observe from (3.19), | f n | L ∞ = |Q ∗ | L ∞ and (2.10) that: H( f n ) → H( f ),

∇φ in L 2 . f n → ∇φ f

(3.28)

From (2.71), | fn − fn |L 1 2 |v| ∗ ∗ 2 2 + φ fn , |x × v| , |x × v| 1 |v|2 = | f n − Q | aφ f n d xdv 2 2 +φ f n (x) 0, > 0. This implies that ∗φ Q

fn

(x, v) → Q(x, v), for ae (x, v) ∈ R6 .

As a consequence of the Brézis-Lieb Lemma (see [33], Theorem 1.9), we then get ∗φ Q

| fn

− Q| L 1 → 0.

(3.42)

We now write 2

|v| ∗φ + φ fn (x) Tn − f n − f n fn d xdv 2 R6

|v|2

∗φ ∗φ Q ∗φ + φ f n (x) = + φ Q − φ fn f n − f n f n fn − f n Q d xdv 2 R6 R6 2 |v| ∗φ Q ∗φ fn φ f − φ Q f n − f n∗φ Q + f − ≤ (x) − f − φ d xdv f n n n n 2 R6 R6 + ∗φ Q

≤ |φ f n − φ Q | L ∞ | f n − f n

∗φ fn

| L 1 + |φ f n (0)|| f n

∗φ Q

− fn

|L 1

→ 0,

where we have used the definition (2.63) of f ∗φ , the uniform convergence of the potential ∗φ φ fn , the boundedness of f n and f n Q in the energy space, and the L 1 convergences (3.40) and (3.42). Using Tn ≥ 0 and the convergence (3.27), we finally deduce that Tn → 0, and (3.41) is proved. Arguing as in the proof of (3.6), we write (3.41) in the following equivalent form 2 2 +∞ |v| |v| Tn = + φ Q (x) d xdv − + φ Q (x) d xdv → 0, dt 2 2 S1n (t) S2n (t) t=0 (3.43) where ∗φ Q

S1n (t) = {(x, v) ∈ R6 , f n S2n (t)

(x, v) ≤ t < f n (x, v)}, ∗φ Q

= {(x, v) ∈ R , f n (x, v) ≤ t < f n 6

((x, v)}.


197

Now from (2.74), we have |v|2 + φ Q (x) ≥ ( f n∗ ◦ aφ Q )−1 (t, |x × v|2 ), 2 if (x, v) ∈ S1n (t). Thus +∞

Tn ≥

dt

S1n (t)

t=0

( f n∗ ◦ aφ Q )−1 (t, |x ×v|2 )d xdv−

S2n (t)

|v|2 +φ Q (x) d xdv . 2 (3.44)

∗φ Q

As a consequence of the equimeasurability of f n S2n (t)

−

and f n , we claim that

( f n∗ ◦ aφ Q )−1 (t, |x × v|2 )d xdv = 0.

S1n (t)

(3.45)

Indeed, we first use the change of variables r = |x|, u = |v|, = |x × v|2 , to get S1n (t)

( f n∗

◦ aφ Q )

−1

(t, |x × v| )d xdv = 2

=

∞

=0 ∞ =0

n (t) S1,

( f n∗ ◦ aφ Q )−1 (t, )dν (r, u)d,

n ( f n∗ ◦ aφ Q )−1 (t, )ν (S1, )(t)d,

and the same identity holds for S2n (t), where ν is given by (1.12), and ∗φ Q

n S1, (t) = {(r, u) ∈ , f n

(r, u, ) ≤ t < f n (r, u, )}, ∗φ Q

n (t) = {(r, u) ∈ , f n (r, u, ) ≤ t < f n S2, ∗φ Q

Since f n

∈ Eq( f n ), we have:

n ν (S1, (t))

=

n ν (S2, (t))

= 4π

2 n (t) S2,

(r, u, )}.

1r 2 u 2 > (r 2 u 2 − )−1/2 r |u|dr du.

This implies (3.45) and then (3.44) gives: 2 +∞ |v| ( f n∗ ◦ aφ Q )−1 (t, |x × v|2 ) − Tn ≥ + φ Q (x) d xdv. dt 2 S2n (t) t=0

(3.46)

Now from (2.73), we have ( f n∗ ◦ aφ Q )−1 (t, |x × v|2 ) ≥

|v|2 + φ Q (x), 2

for (x, v) ∈ S2n (t). Thus, from (3.41) and (3.46), we get 2 |v| + φ Q (x) 1 S2n (t) (x, v) → 0, An = ( f n∗ ◦ aφ Q )−1 (t, |x × v|2 ) − 2

(3.47)

198


as n → +∞, for almost every (t, x, v) ∈ R+ × R3 × R3 . We now claim that this implies 2 |v| ∗ −1 2 Bn = (Q ◦ aφ Q ) (t, |x × v| ) − (3.48) + φ Q (x) 1 S n (t) (x, v) → 0, 2 2 as n → +∞, for almost every (t, x, v) ∈ R+ × R3 × R3 , where n

S 2 (t) = {(x, v) ∈ R6 ), f n (x, v) ≤ t < Q(x, v)}. To prove (3.48), we write

n n S2n = S2n \S 2 ∪ S2n ∩ S 2 ,

n n n S 2 = S 2 \S2n ∪ S2n ∩ S 2 ,

and get An − Bn =

|v|2 + φ Q (x) − (Q ∗ ◦ aφ Q )−1 (t, |x × v|2 ) 1 S n (t)\S n (t) 2 2 2 2 |v| − φ Q (x) 1 S n (t)\S n (t) + ( f n∗ ◦ aφ Q )−1 (t, |x × v|2 ) − 2 2 2 ! + ( f n∗ ◦ aφ Q )−1 (t, |x × v|2 ) − (Q ∗ ◦ aφ Q )−1 (t, |x × v|2 ) 1 S n (t)∩S n (t) . 2

2

(3.49) We shall now examine the behavior of each of these terms when n → ∞. We first observe from (3.39) and (3.42) that +∞ n ∗φ meas(S2n (t)\S 2 (t))dt ≤ | f n Q − Q| L 1 → 0, 0

which implies (up to a subsequence extraction) 1 S n (t)\S n (t) −→ 0, 2

2

for ae (t, x, v) ∈ R+ × R3 × R3 .

Using in addition the estimate ∗ ( f n ◦ aφ Q )(−1) (t, |x × v|2 ) ≤ |eφ Q ,|x×v|2 | ≤ |φ Q (0)|, we deduce that the first two terms of the decomposition (3.49) go to 0 when n goes to infinity, for almost every (t, x, v) ∈ R+ × R3 × R3 . We now treat the third term and show that ! q0 = lim inf ( f n∗ ◦ aφ Q )−1 (t, |x × v|2 ) − (Q ∗ ◦ aφ Q )−1 (t, |x × v|2 ) n→∞

×1 S n (t)∩S n (t) ≥ 0, 2

(3.50)

2

for almost every (t, x, v). To prove (3.50), one may assume that 1 S n (t)∩S n (t) (x, v) = 1 2 2 for n large enough, (t, x, v) being fixed, otherwise q0 = 0 and (3.50) is proved. Let us also recall from standard argument that the strong L 1 convergence (1.22) together with the monotonicity of f n∗ in e and the continuity of Q ∗ in e ensure: a.e. > 0, ∀e ∈ (eφ Q , , 0),

f n∗ (aφ Q (e, ), ) → Q ∗ (aφ Q (e, ), ).


199

Hence, from (1.10), we deduce that for a.e. (x, v) ∈ R6 , we have ∀e ∈ (eφ Q , , 0), f n∗ (aφ Q (e, ), ) → Q ∗ (aφ Q (e, ), ), where = |x × v|2 > 0. (3.51) Let then (t, x, v) being fixed such that 1 S n (t)∩S n (t) (x, v) = 1 for n large enough and 2 2 (3.51) holds. From 2 |v| Q(x, v) = Q ∗ aφ Q , + φ Q (x), , > t 2 and from the continuity of Q ∗ (·, ), we deduce that (Q ∗ ◦ aφ Q )−1 (t, ) = sup{e ∈ ]eφ Q , , 0[: Q ∗ (aφ Q (e, ), ) > t}.

(3.52)

Take now any e such that eφ Q , < e < 0,

and Q ∗ (aφ Q (e, ), ) > t,

(3.53)

then from (3.51): f n∗ (aφ Q (e, ), ) > t, for n large enough. Using the definition of the pseudo-inverse given in Lemma 2.10, we then get e ≤ ( f n∗ ◦ aφ Q )−1 (t, |x × v|2 ) for n large enough, and hence e ≤ lim inf ( f n∗ ◦ aφ Q )−1 (t, |x × v|2 ). n→∞

Since this equality holds for all e satisfying (3.53), we conclude from (3.52) that lim inf ( f n∗ ◦ aφ Q )−1 (t, |x × v|2 ) ≥ (Q ∗ ◦ aφ Q )−1 (t, |x × v|2 ), n→∞

and (3.50) is proved. We now turn to the decomposition (3.49) and get from (3.50), lim inf(An − Bn ) ≥ 0, for a.e. (t, x, v). Finally, observing that Bn ≥ 0 and using (3.47), we conclude that (3.48) holds true: |v|2 − φ Q (x) 1{ fn ≤t t.

By Assumption (A) and Corollary 2.9, e → F(e, |x × v|2 ) is continuous and strictly 2 decreasing with respect to e = |v|2 + φ Q (x) for (x, v) ∈ {Q > 0}, and thus: t < Q(x, v) implies (Q ∗ ◦ aφ Q )(−1) (t, |x × v|2 ) −

|v|2 − φ Q (x) > 0. 2

200


We then deduce from (3.54) that 1{ fn ≤t 0 .

(4.2)

At any (x, v) ∈ , we have ( |v|2 + φ Q (x), |x × v|2 ) ∈ O, where O is defined in Assumption (A), hence Fe (x, v) < 0. Moreover, the function (x, v) → Fe (x, v) is continuous on . We now consider the L 2 weighted Hilbert space: 2

L 2,r |Fe |

=

f ∈

1 L loc ()

spherically symmetric with

f2 d xdv < +∞ |Fe |

and introduce an orthogonal decomposition: 2,even ⊕ L 2,odd L 2,r |F | = L |F | |F | , e

e

e

where f ∈ L 2,r with f (x, −v) = f (x, v) , |Fe | = f ∈ L 2,r |F | with f (x, −v) = − f (x, v) .

L 2,even = |F | e

L 2,odd |Fe |

e

We then consider the unbounded transport operator: 2,r T f = v · ∇x f − ∇φ Q · ∇v f, D(T ) = f ∈ L 2,r , T f ∈ L |F | |F | . e

e

2,r Note that Cc∞ () ⊂ D(T ) is dense in L 2,r |Fe | and hence D(T ) is dense in L |Fe | . We claim the following properties of T :

Proposition 4.1 (Properties of T ). (i) Structure of the kernel: iT is a self adjoint operator with kernel: N (T ) = f

∈ L 2,r |Fe |

2 |v| 2 ˜ . +φ Q (x), |x ×v| of the form f (x, v) = f 2

(4.3)


203

(ii) Coercivity of the Antonov functional: The Antonov functional g2 d xdv − |∇φg |2L 2 A (g, g) := |Fe |

(4.4)

is continuous on L 2,r |F | . Moreover, e

∀ξ ∈ D(T ) ∩

L 2,odd |Fe | ,

A (T ξ, T ξ ) ≥

(ξ )2 φ Q (r ) d xdv. |Fe | r

(4.5)

(iii) Let g ∈ [N (T )]⊥ ∩ L 2,even |Fe | . Then A(g, g) ≥ 0 and we have A(g, g) = 0 if and only if g = 0. Proof. Step 1: Description of the kernel. Property (i) relies on the integration of the characteristic equations associated with T f = 0 and is a standard consequence of the integrability of Newton’s equation with central force field in radial symmetry. The proof follows similarly like for the proof of Jean’s theorem in [8], see also [19]. Step 2. Proof of (ii). Let g ∈ Cc0 (). We integrate by parts to get: |∇φg |2L 2

=−

1/2 2 g(x, v)φg (x) d xdv ≤ |g| L 2,r (φg (x)) Fe d xdv 6 |F | R

e

|g| L 2,r |∇φg | L 2 , |Fe |

where we used (C.1) proved in the Appendix. The density of Cc0 () into L 2,r |Fe | allows us to extend this estimate: ∀g ∈ L 2,r |F | , |∇φg | L 2 |g| L 2,r , |Fe |

e

and the continuity of (4.4) onto L 2,r |F | follows. e

Antonov’s coercivity property is now the following claim: ∀ξ ∈ Cc∞ () ∩ L 2,odd |F | , e

A (T ξ, T ξ ) ≥

(ξ )2 φ Q (r ) d xdv. |Fe | r

(4.6)

In the case where the function F depends only on e = |v|2 /2 + φ(x), a proof of this inequality can be found in [25,23,46,52]. In our context F depends on e = |v|2 /2+φ(x) and = |x × v|2 , and for the sake of clarity and completeness, we give a proof of this inequality in Appendix B which is a simple extension of the proof in [23]. Let us extend this estimate to all ξ ∈ D(T ) ∩ L 2,odd |F | using standard regularization e

arguments. Let ξ ∈ D(T ) ∩ L 2,odd |Fe | and assume first that Supp(ξ ) ⊂ . From the continuity of Fe , we deduce that Fe (x, v) ≤ δ < 0 for all (x, v) ∈ Supp(ξ ). Let a mol|v| ∞ 6 lifying sequence ζn (x, v) = n16 ζ ( |x| n , n ) ∈ Cc (R ) with ζ ≥ 0, then from standard regularization arguments: ζn ξ → ξ, ζn (T ξ ) → T ξ in L 2|F | as n → +∞, e

204


and T (ζn ξ ) → T ξ in L 2|F | as n → +∞. e

Antonov’s coercivity property applied to ζn ξ ∈ Cc∞ () ∩ L 2,odd |F | , the continuity of A on L 2|F | and the boundedness of e

φ Q (r ) r

e

yield the claim. Consider now a general

n a C ∞ function such that ξ ∈ D(T ) ∩ L 2,odd |F | . We let χ e

⎧ 1 (s) = 0 for s ≤ 2n , ⎪ ⎨χ

1 1 χ increasing on 2n , n , ⎪ ⎩ χ (s) = 1 for s ≥ n1 ,

(4.7)

n (Q(x, v)). χn (x, v) = χ

(4.8)

and we set C1

function with a compact support in , satisfying T χn = 0. Therefore Then χn is a 2,odd χn ξ ∈ L |F | , has compact support in and e

T (χn ξ ) = χn T ξ → T ξ in L 2|F | , e

and hence the previous step and the continuity of A on L 2|F | yield (4.5). e

Step 3. Proof of (iii). We first observe that the transport operator exchanges parity in v: ∀ξ ∈ D(T ),

2,even ξ ∈ L 2,odd , ξ ∈ L 2,even . ⇒ T ξ ∈ L 2,odd |F | ⇒ T ξ ∈ L |F | |F | |F | e

e

e

e

This implies: R(T |

L 2,odd |Fe |

2,even ) = R(T ) ∩ L |F . |

(4.9)

e

On the other hand, iT being self-adjoint, there holds –see Cor. II.17, p. 28 in [9]: R(T ) = N (T )⊥ . Let g sequence

∈ [N (T )]⊥ ∩ L 2,even |Fe | . From 2,odd ξn ∈ D(T ) ∩ L |F | such that e

(4.10)

(4.9) and (4.10), we infer the existence of a

T ξn → g in L 2,r |F |

(4.11)

e

as n → +∞. Hence, from the continuity of the Antonov functional on L 2,r |F | , we have e

A (T ξn , T ξn ) → A (g, g).

(4.12)

Moreover, by (4.5), we have A (T ξn , T ξn ) ≥

(ξn )2 φ Q (r ) d xdv ≥ 0. |Fe | r

Thus (4.12) and (4.13) imply A (g, g) ≥ 0.

(4.13)


205

Assume now that A (g, g) = 0. Then (4.12) and (4.13) imply that (ξn )2 φ Q (r ) d xdv → 0 r |Fe |

(4.14)

as n → +∞. Solving the Poisson equation in radial coordinates yields: r r 2 φ Q (r ) = 4π ρ Q (s)s 2 ds. 0

Denote r0 = inf (x,v)∈ |x|. From the definition (4.2) of and the continuity of Q, we have a sequence r j → r0 , r j > r0 , such that ρ Q (r j ) > 0. Hence, for all r > r0 , we have φ Q (r ) |Fe |r

r 2 φ Q (r ) ≥ r 2j φ Q (r j ) > 0, for j large enough. Thus, the function and strictly positive on and (4.14) implies that

is continuous

2 (). ξn → 0 in L loc

Therefore, T ξn 0 in the distribution sense D () and, by (4.11), g = 0. This concludes the proof of Proposition 4.1. A standard consequence of the explicit description of the kernel of T given by (4.3) is that we can compute the projection onto N (T )–see [19] for related statements. For later use, we introduce the following homogeneous Sobolev space: H˙ r1 = h ∈ H˙ 1 (R3 ) s.t. h is radially symmetric . Lemma 4.2 (Projection onto the kernel of T ). Let D = (e, ) ∈ R∗− × R∗+ : e > eφ Q , ,

(4.15)

where eφ Q , is defined by (2.3). Given h ∈ H˙ r1 , we define the projection operator:

−1/2 r2 (x,v) e(x, v) − φ (r ) − h(r )dr Q r1 2r 2 Ph(x, v) = 1(e(x,v),(x,v))∈D , (4.16)

−1/2 r2 (x,v) e(x, v) − φ (r ) − dr Q 2 r1 2r where r1 = r1 (φ Q , e(x, v), (x, v)), r2 = r2 (φ Q , e(x, v), (x, v)) are defined by (2.6), (2.7), and where e(x, v) =

|v|2 + φ Q (x), 2

(x, v) = |x × v|2 .

Then: h Fe ∈ L 2,r |F | , e

(Ph)Fe ∈ L 2,r |F |

(4.17)

e

and (Ph)|Fe | ∈ N (T ), (h − Ph)Fe ∈ [N (T )]⊥ ∩ L 2,even |F | e

with Fe given by (4.1). The proof is given in Appendix C.

(4.18)

206


4.2. Differentiability of J . Our aim in this section is to prove the differentiability of J at φ Q and to compute the first two derivatives. We shall in particular exhibit an intimate link between the Hessian of J and the projection operator (4.16). Proposition 4.3 (Differentiability of J ). The functional J defined by (3.8) on rad satisfies the following properties. = φ (i) Differentiability of J . Let φ = φ f ∈ rad and φ f ∈ rad , both nonzero. Then, the functional − φ)) λ → J (φ + λ(φ is twice differentiable on [0, 1]. (ii) Taylor expansion of J near φ Q . Let R > 0 and f ∈ B R := {g ∈ Erad such that |g − Q|E < R} .

(4.19)

Then we have the following Taylor expansion near φ Q : J (φ f ) − J (φ Q ) =

1 2 D J (φ Q )(φ f − φ Q , φ f − φ Q ) 2 +ε R (φ f ) |∇φ f − ∇φ Q |2L 2 ,

(4.20)

where ε R (φ f ) → 0 as |∇φ f − ∇φ Q | L 2 → 0 with f ∈ B R , and where the second derivative of J in the direction h is given by D 2 J (φ Q )(h, h) = |∇h|2 d x + h(x)(h(x) − Ph(e, ))Fe (e, )d xdv R3

R6

(4.21) with Ph given by (4.16) and e =

|v|2 2

+ φ Q (x), = |x × v|2 .

Proof. Let us decompose J into a kinetic part and a potential part: 1 1 J (φ) = J Q ∗ (φ) = H(Q ∗φ ) + |∇φ − ∇φ Q ∗φ |2 = |∇φ|2 d x + J0 (φ) 2 2 with

(4.22)

2 |v|2 |v| ∗ 2 2 + φ(x) Q aφ + φ(x), |x × v| , |x × v| d xdv J0 (φ) = 2 2 R6 2 |v| = + φ(x) Q ∗φ (x, v) d xdv. (4.23) 6 2 R

involve two derivObserve that (4.23) seems to suggest that two derivatives of J0 should√ atives of Q ∗ and aφ which are not available in particular from the · regularity only of the integral (2.17) defining aφ . We claim that is in fact not the case and that suitable integration by parts and change of variables and a careful track of the dependence on (e, φ, ) of the various estimates on aφ and its derivatives given by Lemmas 2.3, 2.4, 2.5 will yield the result.


207

Step 1. Bounds for the support of Q ∗ . In Corollary 2.9, we have identified the function Q∗:

(s, ), , ∀ > 0, ∀s ≥ 0. (4.24) Q ∗ (s, ) = F aφ−1 Q Recall that, by Assumption (A), for all ≥ 0 the function e → F(e, ) is nonincreasing. Let us define L = > 0 : F(eφ Q , , ) > 0 , (4.25) where eφ Q , is defined in Lemma 2.1. By Lemma 2.1 (i) and by the continuity of F, the function → F(eφ Q , , ) is continuous on R∗+ , thus L is an open set. (s, ) ≥ eφ Q , implies If ∈ R∗+ \L, then aφ−1 Q F(aφ−1 (s, ), ) ≤ F(eφ Q , , ) = 0 Q for all s ≥ 0, thus ∀ ∈ R∗+ \L ,

Q ∗ (·, ) = 0.

(4.26)

In particular, since Q = Q ∗φ Q is not zero, the measure of L cannot be zero. Let now ∈ L and let s0 () = aφ Q (e0 (), ), where we recall the definition (2.68) of e0 (). From Assumption (A), Lemma 2.3 and (4.24), we infer that the function Q ∗ (·, ) is continuous on R+ , that its support is [0, s0 ()] and that this function is strictly decreasing and C 1 on ]0, s0 ()[. Furthermore, from (2.21), we deduce that ∀ ∈ L , 0 < s0 () ≤ s0 := 16π 2 |Q| L 1 |e0 |−1/2 .

(4.27)

Finally, let us prove that the set L is bounded. From Assumption (A), (x, v) → Q(x, v) is compactly supported, thus there exist r0 , u 0 > 0 such that Q(x, v) = 0 for all (x, v) such that |x| ≥ r0 or |v| ≥ u 0 . Hence, we have Q(x, v) = 0 for all (x, v) such that |x × v|2 ≥ r0 u 0 and then, by definition of Q ∗ , Q ∗ (·, ) = 0 for all ≥ 0 := r02 u 20 . Therefore, we have L ⊂]0, 0 [.

(4.28)

Step 2. First derivative of J0 . We first transform the expression (4.23) of J0 . Using the change of variable (2.71) and the bounds (4.26), (4.27) for the support of Q ∗ , we get s0 () J0 (φ) = aφ−1 (s, )Q ∗ (s, )dds, (4.29) ∀φ ∈ rad \{0}, ∈L

0

where we recall that aφ−1 (·, ) is defined as the inverse function of e → aφ (·, ) at given φ ∈ rad \{0}, and > 0. as in Proposition 4.3 (i) and h = φ − φ. Let us differentiate the following Let φ and φ function with respect to λ ∈ [0, 1]: s0 () −1 J0 (φ + λh) = aφ+λh (s, )Q ∗ (s, )dds. (4.30) L

0

208


Let −1 (s, )Q ∗ (s, ). g(λ, s, ) = aφ+λh

According to (2.49), we have

−1/2 r2 −1 a (s, ) − ψ (r ) − λh(r ) h(r )dr φ, φ+λh r1 ∂g Q ∗ (s, ), (λ, s, ) =

−1/2 ∂λ r2 −1 dr r1 aφ+λh (s, ) − ψφ, (r ) − λh(r ) −1 where ri , i = 1, 2, shortly denotes ri (φ + λh, aφ+λh (s, ), ) defined by (2.6), (2.7), and ψφ, (r ) is defined by (2.2). Therefore,

∂g (λ, s, ) ≤ |h| L ∞ Q ∗ (s, ) ∈ L 1 (R+ , R+ ), ∂λ

0≤

and we deduce from dominated convergence that J0 is differentiable at φ in the direction h with:

−1/2 s0 () r2 a −1 (s, ) − ψφ, (r ) h(r )dr φ r1 DJ0 (φ)(h) = Q ∗ (s, ) dds.

−1/2 r2 −1 L s=0 dr r1 aφ (s, ) − ψφ, (r ) Using the change of variable s → e = aφ−1 (s, ) and (2.23), we now get the following equivalent expression: √ 0 r2 ∗ −1/2 2 DJ0 (φ)(h) = 4π 2 Q aφ (e, ), e − ψφ, (r ) h(r )dr ded. L

r1

eφ,

(4.31) Step 3. Second derivative of J0 . Let us now compute the second derivative of J0 (φ + λh) with respect to λ. First, we write the first derivative in a more convenient form. Let ∗ ∗ Dφ, = (r, e) ∈ R+ × R− s.t. e − φ(r ) − 2 > 0 2r = (r, e) ∈ R∗+ ×]eφ, , 0[ s.t. r1 (φ, e, ) < r < r2 (φ, e, ) . An integration by parts gives ∂ J0 (φ + λh) ∂λ √ = 8π 2 2 & ×

L

0 e=eφ+λh,

∂ Q ∗ aφ+λh (e, ), ∂e

' 1/2 e − ψφ, (r ) − λh(r ) h(r )dr ded

r2 (φ+λh,e,)

r1 (φ+λh,e,)

√ = −8π 2

∂aφ+λh ∂ Q∗ aφ+λh (e, ), (e, ) ∂e L Dφ+λh, ∂s 1/2 h(r )dr ded, × e − ψφ, (r ) − λh(r ) 2


209

where the boundary terms of the integration by parts vanish. Now, we perform the change of variable e → s = aφ+λh (e, ) and get √ s0 () +∞ ∂ J0 (φ + λh) = −8π 2 2 G(λ, s, , r )dr dsd, (4.32) ∂λ r =0 L 0 with G(λ, s, , r ) = We have

1/2 ∂ Q∗ −1 (s, ) aφ+λh (s, ) − ψφ, (r ) − λh(r ) h(r ). + ∂s

−1 ∂aφ+λh ∂G 1 ∂ Q∗ = (s, ) (s, ) − h(r ) h(r ) ∂λ 2 ∂s ∂λ

−1/2 −1 × aφ+λh (s, ) − ψφ, (r ) − λh(r ) 1r1 0. m φ+λh = inf (r + 1)((1 − λ)|φ(r )| + λ|φ r >0

210


From (4.28), (4.36) and (4.37), we get ∂G ∂λ ≤ C|h| L ∞ |r h| L ∞ qλ (s, , r ),

(4.38)

for ∈ L, s ≤ s0 (), with 0 ≤ qλ (s, , r ) = −

1r1 dν d DJ0 (φ Q )(h) = 2 F 2 0 0 0 = Q(x, v)h(x) d xdv comes, R6

where we used (1.10), Assumption (A), and recall that h is radially symmetric. Hence, from the Poisson equation, we deduce after an integration by parts that DJ0 (φ Q )(h) = − ∇φ Q · ∇h d x, R3

which together with (4.42) implies (4.41). Let us now identify the right second derivative of J at φ Q . We have 2 2 |∇h|2 d x D J (φ Q )(h, h) = D J0 (φ Q )(h, h) + R3

(4.43)

and, by (4.39), D 2 J0 (φ Q )(h, h) √ s () = 4π 2 2 L 0 0

−1/2 r2 −1 ∂ Q∗ a (s, ) (s, ) − ψ (r ) (h(r ))2 dr dsd φ , Q φQ r1 ∂s 2

−1/2 r 2 a −1 (s,)−ψ h(r )dr φ Q , (r ) √ φ r ∗ Q 1 s () Q (s, ) r dsd. −4π 2 2 L 0 0 ∂∂s

−1/2 2 a −1 (s,)−ψ dr φ Q , (r ) φ r Q

1

(s, ), (2.23) and (2.67), we get Using first the change of variable s → e = aφ−1 Q D 2 J0 (φ Q )(h, h) √ 2 = 4π 2 L

e0 () eφ Q ,

√ − 4π 2 2 L

Fe (e, )

e0 ()

eφ Q ,

−1/2 e − ψφ Q , (r ) (h(r ))2 dr ded

r2

r1

Fe (e, )

2 −1/2 e − ψφ Q , (r ) h(r )dr ded. r2 −1/2 dr r1 e − ψφ Q , (r )

r2 r1

212


( We next apply the change of variable e → u = 2(e − φ Q (x)) and use (1.10) to get: Fe (e, )(h(x))2 d xdv − Fe (e, )h(x)Ph(e, )d xdv, D 2 J0 (φ Q )(h, h) = R6

R6

where we used the definition (4.16) and where we shortly denoted e=

|v|2 + φ Q (x), 2

= |x × v|2 .

This together with (4.43) concludes the proof of (4.21). Step 5. Proof of the Taylor expansion (4.20). We are now ready to prove the Taylor expansion (4.20). We first deduce from (4.41) and from the fact that J (φ Q + λh) twice differentiable with respect to λ that 1 ∂2 J (φ Q + h) − J (φ Q ) = (1 − λ) 2 J (φ Q + λh) dλ. ∂λ 0 Hence, for h = 0, 1 J (φ Q + h) − J (φ Q ) − D 2 J (φ Q )(h, h) 2 1

= (1 − λ) D 2 J (φ Q + λh) − D 2 J (φ Q ) (h, h) dλ 0 1

2 2 2 = |∇h| L 2 (1 − λ) D J0 (φ Q + λh) − D J0 (φ Q ) 0

h h , |∇h| L 2 |∇h| L 2

dλ. (4.44)

We now claim the following continuity property:

sup sup D 2 J0 (φ Q + λ(φ f − φ Q ) − D 2 J0 (φ Q ) ( h, h) → 0 λ∈[0,1] |∇ h|

(4.45)

L 2 =1

as |∇φ f − ∇φ Q | L 2 → 0, f satisfying (4.19). Assume (4.45). Then: 1

2 2 (1 − λ) D J0 (φ Q + λh) − D J0 (φ Q ) 0

h h , |∇h| L 2 |∇h| L 2

dλ → 0

and (4.44) now yields (4.20). hn Proof of (4.45). We argue by contradiction and consider ε > 0, f n satisfying (4.19), and λn ∈ [0, 1] such that |∇φ fn − ∇φ Q | L 2 < and

1 , n

|∇ h n | L 2 = 1,

2 hn , h n ) − D 2 J0 (φ Q )( hn , h n ) > ε. D J0 (φ Q + λn (φ fn − φ Q ))(

(4.46)

(4.47)


213

We denote h n = λn (φ fn − φ Q ). Recall from (4.39): D 2 J0 (φ Q + h n )( hn , hn ) = rn s0 () ∂ Q ∗ √ 2 = 4π 2 2 (s, ) ∂s L 0 r1n

−1/2 × aφ−1 (s, ) − ψ (r ) − h (r ) ( h n (r ))2 dr dsd φ , n Q +h n Q √ s0 () ∂ Q ∗ −4π 2 2 (s, ) ∂s L 0 n 2

−1/2 r2 −1 a (s, ) − ψ (r ) − h (r ) (r )dr h φ Q , n n φ Q +h n r1n × dsd,

−1/2 r2n −1 a (s, ) − ψ (r ) − h (r ) dr n φ , n Q φ Q +h n r

(4.48)

1

where we have denoted, for i = 1, 2,

(s, ), . rin = ri φ Q + h n , aφ−1 Q +h n By (4.46) and standard radial Sobolev embeddings, the sequence of radially symmetric functions h n is compact in L ∞ ([a, b]) for all 0 < a < b. By diagonal extraction, we deduce the pointwise convergence of h n (up to a subsequence) to a function h: h(r ) as n → +∞. h n (r ) →

∀r ∈ R∗+

(4.49)

Moreover, r

1/2

| h n (r )| ≤

+∞

s ( h n (s))2 ds

1/2

2

r

≤ |∇ h n | L 2 = 1,

(4.50)

thus, in particular, r 1/2 h belongs to L ∞ (R+ ). Let us analyze the convergence of (4.48). In a first step, recalling (4.26) and (4.27), we fix ∈ L and s ∈ ]0, s0 ()] and set (s, ), en = aφ−1 Q +h n

e∞ = aφ−1 (s, ) < 0. Q

From (4.46), the uniform bound of f n in Erad and Lemma 2.4, we have: en → e∞ as n → +∞. For k = 0, 1 or 2, we introduce the functions −1/2 gk (n, s, , r ) = en − ψφ Q , (r ) − h n (r ) ( h n (r ))k 1ψ (r )+h n (r )<en and

(4.51)

(4.52)

−1/2 ( h(r ))k 1ψ (r )<e∞ . gk (∞, s, , r ) = e∞ − ψφ Q , (r )

We claim: ∀ ∈ L, ∀s ∈]0, s0 ()], +∞ gk (n, s, , r ) dr → 0

0

+∞

gk (∞, s, , r ) dr for k = 0, 1 or 2.

(4.53)

214


Indeed, from (4.49), (4.51), (4.46) and the bound of f n which imply |h n | L ∞ → 0, we first deduce that, for all s > 0, > 0, the function gk (n, s, , r ) converges pointwise in r ∈ R∗+ to the function gk (∞, s, , r ), for k = 0, 1 or 2, as n → +∞. Moreover, by applying (2.9) to the function φ Q + h n , we get ( r | h n (r )|k r1n r2n 1r1n 0.

(4.62)

Indeed, let h ∈ H˙ r1 \{0} and consider the projection Ph given by (4.16). From (4.17), 2,r the functions h Fe and (Ph)Fe belong to L 2,r |Fe | and hence g = (h − Ph)Fe ∈ L |Fe | . By the orthogonality property (4.18), we have (h − Ph)2 Fe d xdv = − (h − Ph)2 Fe d xdv + 2 h (h − Ph) Fe d xdv g2 d xdv − 2 ∇h · ∇φg d x, = |Fe | where we used the Poisson equation. We may thus rewrite the Hessian (4.21): 2 2 D J (φ Q )(h, h) = |∇h| d x + (h − Ph)2 Fe d xdv =

R6

g2 |Fe |

d xdv − |∇φg |2L 2 + |∇h − ∇φg |2L 2

= A (g, g) + |∇h − ∇φg |2L 2 . Now, from (4.18) and Proposition 4.1 (iii), we deduce that A (g, g) ≥ 0. Therefore, D 2 J (φ Q )(h, h) is nonnegative. Moreover, if D 2 J (φ Q )(h, h) = 0, then A (g, g) = |∇h − ∇φg | L 2 = 0 and using again Proposition 4.1 (iii) enables to conclude that g = φg = h = 0. This ends the proof of (4.62). Step 2. Coercivity of the Hessian and conclusion. In Remark 4.4, we have seen that the quadratic form D 2 J0 (φ Q ) is compact on H˙ r1 . Hence from (4.22), the Fredholm alternative can be applied to the quadratic form D 2 J (φ Q ). Together with the strict positivity property (4.62), this implies the coercivity of this quadratic form: ∀h ∈ H˙ r1

D 2 J (φ Q )(h, h) ≥ c|∇h|2L 2 ,

(4.63)

for some universal constant c > 0. We now may conclude the proof of (3.9). Let R > 0 be fixed. From Proposition 4.3 (ii), there exists δ0 (R) – chosen in ]0, 21 |∇φ Q | L 2 ] – such that, for all f ∈ Erad satisfying | f − Q|E ≤ R,

|∇φ f − ∇φ Q | L 2 ≤ δ0 (R),

we have c , 4 where c is the constant in (4.63) and ε R is defined in (4.20). Hence, for such f , we deduce from (4.63) and (4.20) that c J (φ Q + h) − J (φ Q ) ≥ |∇h|2L 2 . 4 The proof of Proposition 3.2 is complete. ε R (φ f ) ≤


217

Acknowledgements. The authors would like to thank P.-E. Jabin for stimulating discussions about this work, and are endebted to J.-J. Aly for having kindly guided them through the physics reference on the subject and in particular the pioneering important works [2,41,54]. M. Lemou was supported by the Agence Nationale de la Recherche, ANR Jeunes Chercheurs MNEC. F. Méhats was supported by the Agence Nationale de la Recherche, ANR project QUATRAIN. P. Raphaël was supported by the Agence Nationale de la Recherche, ANR Projet Blanc OndeNonLin and ANR Jeune Chercheur SWAP.

Appendix A. Dominated Convergence Lemma Lemma A.1. Let I be an interval of R and let g(λ, r ) be a real-valued function in C 0 (I, L 1 (R+ )). Let g(λ, r )dr, G(λ) = R+

∂g the weak partial derivative of g with respect to λ. Assume that g and denote by ∂λ satisfies the following assumptions: ∂g ∂g ∈ L 1 (I × R + ) and for all λ0 ∈ I , limλ→λ0 ∂λ (λ, r ) = ∂λ (λ0 , r ) for a.e. r ; ∂g (ii) for all λ ∈ I , ∂λ (λ, r ) ≤ qλ (r ) a.e., where qλ ∈ L 1 (R+ ), and for all λ0 ∈ I ,

(i)

∂g ∂λ

qλ → qλ0 in L 1 (R+ ) as λ → λ0 . Then, G is C 1 on I and G (λ) =

∂g (λ, r )dr. R+ ∂λ

∂g Proof. Let λ0 , λ ∈ I . Since g ∈ C 0 (I, L 1 (R+ )) and ∂λ ∈ L 1 (I × R+ ), we have λ ∂g g(λ, r ) − g(λ0 , r ) = (μ, r )dμ. λ0 ∂λ

Hence, by Fubini,

G(λ) − G(λ0 ) g(λ, r ) − g(λ0 , r ) = dr λ − λ0 λ − λ0 R+ λ 1 ∂g = (μ, r )dr dμ. λ − λ0 λ0 R+ ∂λ

(A.1)

Now, we use a generalized version of the dominated convergence as stated in [33] (see the Remark after Theorem 1.8) and deduce from Assumptions (i) and (ii) that ∂g ∂g (μ, r )dr = (λ0 , r )dr. (A.2) lim μ→λ0 R+ ∂λ ∂λ R+ Hence, by using (A.2), we pass to the limit in (A.1) and obtain G(λ) − G(λ0 ) ∂g (λ0 , r )dr, lim = λ→0 λ − λ0 R+ ∂λ which proves the differentiability of G. In fact, we observe that the same Assumptions (i) and (ii) associated to the same generalized version of the dominated convergence theorem provide the continuity of G . This ends the proof of the lemma.

218


Appendix B. Proof of the Antonov Inequality (4.6) Let be defined by (4.2) and let ξ ∈ Cc∞ () ∩ L 2,odd |Fe | . We recall that the linear transport operator T is defined by T ξ = v · ∇x ξ − ∇x φ Q · ∇v ξ. Our aim is to prove the coercivity property (4.6). Let g = T ξ ; we have from the Poisson equation r 1 r 2 φg (r ) = s 2 ρg (s)ds = ρg (x)d x 4π |x|≤r 0 1 v · ∇x ξ − ∇x φ Q · ∇v ξ dvd x = 4π |x|≤r R3 1 = ∇x · vξ dv d x 4π |x|≤r R3 x 1 = · vξ dv dσ (x). 4π |x|=r R3 r Now we observe from the spherical symmetry of ξ that the quantity R3 rx · vξ dv only depends on r = |x|. Therefore (x · v)ξ dv. (B.1) r φg (r ) = R3

We then use the Cauchy-Schwarz inequality and Supp(ξ ) ⊂ to estimate: ξ2 r 2 φg (r )2 ≤ (x · v)2 |Fe |dv dv , R3 R3 |Fe | where we recall that Fe is given by (4.1). Now we claim that (x · v)2 |Fe |dv = r 2 ρ Q (r ). R3

(B.2)

(B.3)

Indeed, we first pass to spherical coordinates in v, u = |v|,

|x × v|2 = r 2 u 2 sin2 θ, θ ∈ [0, π [, with r = |x|,

(B.4)

and get (recall that Fe ≤ 0) (x · v)2 |Fe |dv R3

= −4π 0

π/2 +∞ r =0

∂F r u cos θ sin θ ∂e 2 4

2

u2 2 2 2 + φ Q (r ), r u sin θ dudθ. 2

Now we perform the change of variable u2 (u, θ ) → e = + φ Q (r ), = r 2 u 2 sin2 θ , 2

(B.5)


and obtain R3

(x · v)

2

|Fe |dv

√ = −2π 2 ×

+∞ +∞ e=φ Q (r )+

=0

∂F (e, )ded. ∂e

2r 2

219

1/2 e − φ Q (r ) − 2 2r

We then integrate by parts with respect to the variable e: R3

√ (x · v)2 |Fe |dv = π 2

+∞ +∞

=0

e=φ Q (r )+

2r 2

−1/2 e − φ Q (r ) − 2 F(e, )ded. 2r

Using the same changes of variables (B.4) and (B.5), we get 2 |v| r 2 ρ Q (x) = r 2 F + φ Q (x), |x × v|2 dv 2 R3 √ +∞ +∞ −1/2 e − φ Q (r ) − 2 =π 2 F(e, )ded, 2r =0 e=φ Q (r )+ 2 2r

and (B.3) follows. Now, we integrate the inequality (B.2) with respect to r and use (B.3) to get +∞ |∇x φg |2 d x = 4π r 2 φg (r )2 dr R3

≤

0

R3 0

+∞

4πr 2 ρ Q (r )

ξ2 ξ2 dr dv = ρ Q (x) d xdv. |Fe | |Fe | R 3 ×R 3

From the definition (4.4) of the Antonov functional, we then deduce

1 A (T ξ, T ξ ) ≥ d xdv. (T ξ )2 − ρ Q (x)ξ 2 |Fe |

(B.6)

Let now ξ = (x · v)q(x, v) and write from the definition of T , (T ξ )2 = (qT (x · v) + (x · v)T q)2 = (x · v)2 (T q)2 + (x · v)T (x · v)T (q 2 ) + q 2 (T (x · v))2

= (x · v)2 (T q)2 + T (x · v)q 2 T (x · v) − (x · v)q 2 T (T (x · v)) . We observe, from the Poisson equation φ Q = ρ Q , that T (T (x · v)) = −(x · v)φ Q − v · ∇x φ Q = −(x · v) ρ Q (r ) +

φ Q (r ) r

.

Thus

φ Q (r ) . (T ξ )2 − ρ Q (r )ξ 2 = (x · v)2 (T q)2 + T (x · v)q 2 T (x · v) + ξ 2 r

(B.7)

220


We now insert this expression into (B.6) and directly get the desired Antonov’s inequality (4.6), provided the following claim is proved:

T (x · v)q 2 T (x · v) d xdv → 0 as ε → 0. (B.8) |x · v| > ε 3 |x| > ε

Proof of (B.8). We shall in fact deal with the singularity at x · v = 0 in the integral (B.8), recalling that q(x, v) = ξ(x, v)/(x · v). We observe that the function |x|q(x, v) is bounded. To see this, let x = 0 and Rx be the orthogonal transformation of R3 such x that Rx |x| = e1 , where e1 = (1, 0, 0)T . Then, due to the spherical symmetry, ξ(|x|e1 , Rx v) = ω(|x|, Rx v), with e1 · Rx v ξ(r e1 , v) ξ˜ (r, |v|, e1 · v) ω(r, v) = = . e1 · v e1 · v

|x|q(x, v) =

Now, we recall that ξ is odd in v and hence ξ˜ is odd with respect to the last coordinate, thus ω is bounded and so is rq. Note also that q is smooth on |x · v| > δ, for all δ > 0. Let ε > 0. We have T (x · v)q 2 T (x · v) d xdv |x · v| > ε 3 |x| > ε

= −2ε 3 x · v = ε3

dσ1 (x, v) 2 q 2 (T (x · v))2 ( − ε |x|2 + |v|2

|x| > ε

(x · v)2 q 2 T (x · v)dσ2 (x)dv, x · v > ε3

|x| = ε

where dσ1 (x, v) is the measure on the set {(x, v) s.t. x · v = ε3 and |x| > ε} induced by the Lebesgue measure of R6 , and dσ2 (x) is the usual measure on the sphere {x ∈ R3 ; |x| = ε}. Now let R > 0 such that Supp(ξ ) ⊂ {(x, v), |x|2 + |v|2 ≤ R 2 }, then 2 T (x · v)q T (x · v) d xdv |x · v| > ε3 |x| > ε 2 2 2 2 dσ1 (x, v) 2 2 ≤ 2ε r q (T (x · v)) ( + ξ |T (x · v)| dv dσ2 (x), |x|2 + |v|2 ε |x|=ε R3 x · v = ε3 |x| > ε

≤ 2ε|rq|2L ∞ I (ε, R) +

C 2 ε , ε

where we have set

I (ε, R) = x · v = ε3 |x|2 + |v|2 < R 2

dσ1 (x, v) , (T (x · v))2 ( |x|2 + |v|2

and where we have used in the last estimate that rq is bounded and that ξ is compactly supported. We claim that I (ε, R) ≤ C R ,

(B.9)


221

where C R is independent of ε, which concludes the proof of (B.8). Indeed, we integrate by parts to get: T (T (x · v)) d xdv I (ε, R) = −

x · v > ε3 |x|2 + |v|2 < R 2

dσ3 (x, v) T (x · v) x · v − v · ∇x φ Q ( |x|2 + |v|2

+ x · v > ε3 |x|2 + |v|2 = R 2

≤

|x|2 +|v|2 0. Amongst them there are soliton-like solutions which travel with constant velocity, ψ(x, t) = a tanh γ

x − vt − q , √ 2

√ where γ = 1/ 1 − v 2 is the Lorentz contraction. The solitons are related by a Lorentz boost, since Eq. (1.2) is relativistically invariant. We will consider more general double well potentials for which U (±a) = U (±a) = 0,

U (±a) > 0,

(1.4)

and U (ψ) > 0 for ψ ∈ (−a, a),

(1.5)

similarly to the quartic potential. In this case the soliton-like solutions also exist, ψ(x, t) = s(γ (x − vt − q)),

v, q ∈ R,

|v| < 1,

(1.6)

where s(·) is a “kink” solution to the corresponding stationary equation s (x) − U (s(x)) = 0,

s(±∞) = ±a.

(1.7)

In general our goal is to clarify the special role of the soliton-like solutions (1.6) as long time asymptotics for any finite energy topological excitations satisfying (1.3). Namely, if one chooses some arbitrary finite energy initial state satisfying (1.3), one would expect that for t → ∞ the solution separates into two pieces: one piece is a finite collection of travelling solitons of the form (1.6) and their negatives with some velocities v j ∈ (−1, 1) and the shifts q j depending in a complicated way on the initial data, and the second radiative piece which is a dispersive solution to the free Klein-Gordon equation which propagates to infinity with the velocity 1. Our aim here is to elucidate this general picture by mathematical arguments for initial data sufficiently close to a soliton (1.6). Let us discuss our choice of the smooth potentials U . The condition (1.5) is necessary and sufficient for the existence of a finite energy static solution s(x) to (1.7) when (1.4) holds. Indeed, the condition is obviously sufficient. On the other hand, the “energy conservation” (s (x))2 /2 − U (s(x)) = E

(1.8)

and s(±∞) = ±a imply that E = 0. Therefore, U (ψ) > 0 for ψ ∈ (−a, a) since otherwise the boundary conditions s(±∞) = ±a would fail. As a byproduct, our kink solution is monotone increasing, and s (x) > 0,

x ∈ R.

(1.9)

Asymptotic Stability of Moving Kink for Relativistic Ginzburg-Landau Equation

227

Let us note that only the behavior of U near the interval [−a, a] is of importance since the solution is expected to be close to a soliton. However, we will assume additionally the potential to be bounded from below inf U (ψ) > −∞

(1.10)

ψ∈R

to have a well posed Cauchy problem for all finite energy initial states. Summarising, we formulate our first basic condition on the potential, for technical reasons adding a flatness condition. Condition U1. The potential U is a real smooth function which satisfies (1.4), (1.5), (1.10), and the following condition holds with some m > 0, U (ψ) =

m2 (ψ ∓ a)2 + O(|ψ ∓ a|14 ), 2

ψ → ±a.

(1.11)

Let us comment on the condition (1.11) (see also Remark 4.10). First, the condition means that U (−a) = U (a), though we do not need the potential to be reflection symmetric. We consider the solutions close to the kink, ψ(x, t) = s(γ (x −vt −q))+φ(x, t), with small perturbations φ(x, t). For such solution the condition (1.11) and the asymptotics (1.3) mean that Eq. (1.2) is almost linear Klein-Gordon equation for large |x| which is helpful for application of the dispersive properties. Finally, we expect that the degree 14 in (1.11) is technical, and a smaller degree should be sufficent. Let us note that a similar condition has been introduced in [4,5] in the context of the Schrödinger equation. Further we need some assumptions on the spectrum of the linearised equation. Let us rewrite Eq. (1.2) in the vector form, ˙ ψ(x, t) = π(x, t) x ∈ R. (1.12) π˙ (x, t) = ψ (x, t) + F(ψ(x, t)) Now the soliton-like solutions (1.6) become Yq,v (t) = (ψv (x − vt − q), πv (x − vt − q))

(1.13)

for q, v ∈ R with |v| < 1, where ψv (x) = s(γ x), πv (x) = −vψv (x).

(1.14)

The states Sq,v := Yq,v (0) form the solitary manifold S := {Sq,v : q, v ∈ R, |v| < 1}.

(1.15)

The linearized operator near the soliton solution Yq,v (t) is (see Sect. 4, formula (4.20)) Av =

v∇ 1 − m 2 − Vv (y) v∇

, ∇=

d2 d , = , dx dx2

where Vv (x) = −F (ψv (x)) − m 2 = U (ψv (x)) − m 2 .

(1.16)

228

E. A. Kopylova, A. I. Komech

By (1.7) and condition U1, we have Vv (x) ∼ C(s(γ x) ∓ a)12 ∼ Ce−12mγ |x| , x → ±∞,

(1.17)

s(x) ∓ a ∼ Ce−m|x| , x → ±∞.

(1.18)

since

In Sect. 4 we show that the spectral properties of the operator Av are determined by the corresponding properties of its determinant, which is the Schrödinger operator Hv = −(1 − v 2 ) + m 2 + Vv .

(1.19)

The spectral properties of Hv are identical for all v ∈ (−1, 1) since the relation Vv (x) = V0 (γ x) implies Hv = Tv−1 H0 Tv , where Tv : ψ(x) → ψ(x/γ ).

(1.20)

This equivalence manifests the relativistic invariance of Eq. (1.12). The continuous spectrum of the operator Hv coincides with [m 2 , ∞). The point 0 belongs to the discrete spectrum with corresponding eigenfunction ψv . By (1.14) and (1.9) we have ψv (x) = γ s (γ x) > 0 for x ∈ R. Hence, ψv is the groundstate, and all remaining discrete spectrum is contained in (0, m 2 ]. l, p For α ∈ R, p ≥ 1, and l = 0, 1, 2, ... let us denote by Wα , the weighted Sobolev space of the functions with the finite norm ψ W l, p = α

l

(1 + |x|)α ψ (k) L p < ∞.

k=0

Denote Hαl := Wαl,2 , so L 2α := Hα0 are the Agmon’s weighted spaces. Definition 1.1 (cf. [9,16]). A nonzero solution ψ ∈ L 2−1/2−0 (R)\L 2 (R) to Hv ψ = m 2 ψ is called a resonance. Now we can formulate our second basic condition on the potential. Condition U2. For any v ∈ (−1, 1), i) 0 is only eigenvalue of Hv . ii) m 2 is not a resonance of Hv . We show that Condition U2 implies the boundedness of the resolvent of the operator Av in the corresponding weighted Agmon spaces at the edge points ±im/γ of its continuous spectrum. Both conditions U1, U2 can be satisfied though it is non-obvious. Let us note that the quartic Ginzburg-Landau potential does not satisfy (1.11) and condition U2. We will prove elsewhere that the corresponding examples of potentials satisfying both U1 and U2 can be constructed as smoothened piece-wise quadratic potentials. We now can formulate the main result of our paper. Namely, we will prove the following asymptotics: (ψ(x, t), π(x, t)) ∼ (ψv± (x − v± t − q± ), πv± (x −v± t −q± ))+W0 (t) ± , t → ±∞ (1.21)


229

for solutions to (1.12) with initial states close to a soliton-like solution (1.13). Here W0 (t) is the dynamical group of the free Klein-Gordon equation, ± are the corresponding asymptotic states, and the remainder converges to zero ∼ t −1/2 in the global energy norm of the Sobolev space H 1 (R) ⊕ L 2 (R). Let us comment on previous results in this field. • Orbital stability of the kinks. For 1D relativistic nonlinear Ginzburg-Landau equations (1.2) the orbital stability of the kinks has been proved in [10]. • The Schrödinger equation. The asymptotics of type (1.21) were established for the first time by Soffer and Weinstein [23,24] (see also [19]) for nonlinear U (1)-invariant Schrödinger equation with a potential for small initial states and sufficiently small nonlinear coupling constant. The results have been extended by Buslaev and Perelman [4] to the translation invariant 1D nonlinear U (1)-invariant Schrödinger equation. The novel techniques [4] are based on the “separation of variables” along the solitary manifold and in the transversal directions. The symplectic projection allows to exclude the unstable directions corresponding to the zero discrete spectrum of the linearized dynamics. Similar techniques were developed by Miller, Pego and Weinstein for the 1D modified KdV and RLW equations, [17,18].The extensions to higher dimensions were obtained in [6,12,22,27]. • Nonrelativistic Klein-Gordon equations. The asymptotics of type (1.21) were extended to the nonlinear 3D Klein-Gordon equations with a potential [25], and for translation invariant system of the 3D Klein-Gordon equation coupled to a particle [11]. • Wave front of 3D Ginzburg-Landau equation. The asymptotic stability of wave front was proved for 3D relativistic Ginzburg-Landau equation with initial data which differ from the wave front on a compact set [7]. The wave front is the solution which depends on one space variable only, so it is not a soliton. The equation differs from the 1D equation (1.2) by the additional 2D Laplacian which improves the dispersive decay for the corresponding linearized Klein-Gordon equation in the continuous spectral space. The proving of the asymptotic stability of the solitons and kinks for relativistic equations remained an open problem till now. The investigation crucially depends on the spectral properties for the linearized equation which are completely unknown for higher dimensions. For the 1D case the main obstacle was the slow decay ∼ t −1/2 for the free 1D Klein-Gordon equation (see the discussion in [7, Introduction]). Let us comment on our approach. We follow general strategy of [4–7,11,25]: symplectic projection onto the solitary manifold, modulation equations, linearization of the transversal equations and further Taylor expansion of the nonlinearity, etc. We develop for relativistic equations a general scheme which is common in almost all papers in this area: dispersive estimates for the solutions to the linearized equation, virial and L 1 − L ∞ estimates and the method of majorants. However, the corresponding statements and their proofs in the context of relativistic equations are completely new. Let us comment on our novel techniques. I. The decay ∼ t −3/2 from Theorem 4.7 for the linearized transversal dynamics relies on our novel approach [13,14] to the 1D Klein-Gordon equation. II. The novel “virial type” estimate (4.42) is the relativistic version of the bound [5, (1.2.5)] used in [5] in the context of the nonlinear Schrödinger equation (see Remark 4.10).

230


III. We establish an appropriate relativistic version (4.31) of L 1 → L ∞ estimates. Both estimates (4.42) and (4.31) play a crucial role in obtaining the bounds for the majorants. IV. Finally, we give the complete proof of the soliton asymptotics (1.21). In the context of the Schrödinger equation, the proof of the corresponding asymptotics were sketched in [5]. Our paper is organized as follows. In Sect. 2 we formulate the main theorem. In Sect. 3 we introduce the symplectic projection onto the solitary manifold. The linearized equation is defined in Sect. 4. In Sect. 5 we split the dynamics in two components: along the solitary manifold and in the transversal directions. In Sect. 6 the modulation equations for the parameters of the soliton are displayed. The time decay of the transversal component is established in Sects. 7-11. Finally, in Sect. 12 we obtain the soliton asymptotics (1.21). 2. Main Results 2.1. Existence of dynamics. We consider the Cauchy problem for the Hamilton system (1.12) which we write as Y˙ (t) = F(Y (t)), t ∈ R :

Y (0) = Y0 .

(2.1)

Here Y (t) = (ψ(t), π(t)), Y0 = (ψ0 , π0 ), and all derivatives are understood in the sense of distributions. To formulate our results precisely, let us introduce a suitable phase space for the Cauchy problem (2.1). Definition 2.1. norm

i) E α := Hα1 ⊕ L 2α is the space of the states Y = (ψ, π ) with finite Y E α = ψ Hα1 + π L 2α < ∞.

(2.2)

ii) The phase space E := S + E, where E = E 0 and S is defined in (1.15). The metric in E is defined as ρE (Y1 , Y2 ) = Y1 − Y2 E , Y1 , Y2 ∈ E.

(2.3)

iii) W := W02,1 ⊕ W01,1 is the space of the states Y = (ψ, π ) with the finite norm Y W = ψ W 2,1 + π W 1,1 < ∞. 0

(2.4)

0

Obviously, the Hamilton functional (1.1) is continuous on the phase space E. The existence and uniqueness of the solutions to the Cauchy problem (2.1) follows by methods [15,20,26]: Proposition 2.2. (i) For any initial data Y0 ∈ E there exists the unique solution Y (t) ∈ C(R, E) to the problem (2.1). (ii) For every t ∈ R, the map U (t) : Y0 → Y (t) is continuous in E. (iii) The energy is conserved, i.e. H(Y (t)) = H(Y0 ), t ∈ R.

(2.5)


231

2.2. Solitary manifold and main result. Let us consider the solitons (1.14). The substitution to (1.12) gives the following stationary equations: −vψv (y) = πv (y), (2.6) −vπv (y) = ψv (y) + F(ψv (y)). Definition 2.3. A soliton state is S(σ ) := (ψv (x − b), πv (x − b)), where σ := (b, v) with b ∈ R and v ∈ (−1, 1). Obviously, the soliton solution (1.13) admits the representation S(σ (t)), where σ (t) = (b(t), v(t)) = (vt + q, v).

(2.7)

Definition 2.4. A solitary manifold is the set S := {S(σ ) : σ ∈ := R × (−1, 1)}. The main result of our paper is the following theorem Theorem 2.5. Let the conditions U1 and U2 hold, and Y (t) be the solution to the Cauchy problem (2.1) with an initial state Y0 ∈ E which is close to a kink S(σ0 ) = Sq0 ,v0 : Y0 = S(σ0 ) + X 0 , d0 := X 0 E β ∩W 1,

(2.8)

where β > 5/2. Then for d0 sufficiently small the solution admits the asymptotics: Y (x, t) = (ψv± (x − v± t − q± ), πv± (x − v± t −q± )) + W0 (t) ±r± (x, t), t → ± ∞, (2.9) where v± and q± are constants, ± ∈ E, and W0 (t) is the dynamical group of the free Klein-Gordon equation, while r± (t) E = O(|t|−1/2 ).

(2.10)

It suffices to prove the asymptotics (2.9) for t → +∞ since the system (1.12) is time reversible. 3. Symplectic Projection 3.1. Symplectic structure and hamiltonian form. The system (2.1) reads as the Hamilton system 0 1 ˙ Y = J DH(Y ), J := , Y = (ψ, π ) ∈ E, (3.1) −1 0 where DH is the Fréchet derivative of the Hamilton functional (1.1). Let us identify the tangent space of E, at every point, with the space E. Consider the symplectic form on E defined by (Y1 , Y2 ) = Y1 , J Y2 , Y1 , Y2 ∈ E,

(3.2)

where and ψ1 , ψ2 =

Y1 , Y2 := ψ1 , ψ2 + π1 , π2 ψ1 (x)ψ2 (x)d x, etc. It is clear that the form is non-degenerate, i.e. (Y1 , Y2 ) = 0 for every Y2 ∈ E ⇒ Y1 = 0.

Definition 3.1. i) The symbol Y1 Y2 means that Y1 ∈ E, Y2 ∈ E, and Y1 is symplectic orthogonal to Y2 , i.e. (Y1 , Y2 ) = 0. ii) A projection operator P : E → E is said to be symplectic orthogonal if Y1 Y2 for Y1 ∈ Ker P and Y2 ∈ Range P.

232


3.2. Symplectic projection onto solitary manifold. Let us consider the tangent space T S(σ ) S of the manifold S at a point S(σ ). The vectors τ1 = τ1 (v) := ∂b S(σ ) = (−ψv (y), −πv (y)), τ2 = τ2 (v) := ∂v S(σ ) = (∂v ψv (y), ∂v πv (y))

(3.3)

form a basis in T S(σ ) S. Here y := x − b is the “moving frame coordinate”. Let us stress that the functions τ j are always regarded as functions of y rather than those of x. Formula (1.14) implies that τ j (v) ∈ E α ,

v ∈ (−1, 1),

j = 1, 2,

∀α ∈ R.

(3.4)

Lemma 3.2. The symplectic form is nondegenerate on the tangent space T S(σ ) S, i.e. T S(σ ) S is a symplectic subspace. Proof. Let us compute the vectors τ1√and τ2 . Recall that ψv (y) = s(γ y) and πv = −vψv (y) = −vγ s (γ y) with γ = 1/ 1 − v 2 . Then τ1 = (τ11 , τ12 ) = −γ s (γ y), vγ 2 s (γ y) , τ2 = (τ21 , τ22 ) = vyγ 3 s (γ y), − γ 3 s (γ y) − v 2 yγ 4 s (γ y) . Therefore (τ1 , τ2 ) = τ11 , τ22 − τ12 , τ21 = γ 4 s (γ y), s (γ y) > 0.

(3.5)

Now we show that in a small neighborhood of the soliton manifold S a “symplectic orthogonal projection” onto S is well-defined. Let us introduce the translations Tq : (ψ(x), π(x)) → (ψ(x − q), π(x − q)), q ∈ R. Note that the manifold S is invariant with respect to the translations. Definition 3.3. For any v < 1 denote by (v) = {σ = (b, v) : b ∈ R, |v| ≤ v}. Let us note that S ⊂ E α with α < −1/2. Lemma 3.4. Let α < −1/2 and v < 1. Then i) there exists a neighborhood Oα (S) of S in E α and a mapping : Oα (S) → S such that is uniformly continuous on Oα (S) in the metric of E α , Y = Y for Y ∈ S,

and

Y − S T S S, where S = Y.

(3.6)

ii) Oα (S) is invariant with respect to the translations Tq , and Tq Y = Tq Y,

for Y ∈ Oα (S) and q ∈ R.

(3.7)

iii) For any v < 1 there exists an rα (v) > 0 s.t. S(σ ) + X ∈ Oα (S) if σ ∈ (v) and X E α < rα (v).


233

Proof. We have to find σ = σ (Y ) such that S(σ ) = Y and (Y − S(σ ), ∂σ j S(σ )) = 0,

j = 1, 2.

(3.8)

Let us fix an arbitrary σ 0 ∈ and note that the system (3.8) involves two smooth scalar functions of Y . Then for Y close to S(σ 0 ), the existence of σ follows by the standard finite dimensional implicit function theorem if we show that the 2 × 2 Jacobian matrix with elements Ml j (Y ) = ∂σl (Y − S(σ 0 ), ∂σ j S(σ 0 )) is non-degenerate at Y = S(σ 0 ). First note that all the derivatives exist by (3.4). The non-degeneracy holds by Lemma 3.2 and the definition (3.3) since Ml j (S(σ 0 )) = −(∂σl S(σ 0 ), ∂σ j S(σ 0 )). Thus, there exists some neighborhood Oα (S(σ 0 )) of S(σ 0 ), where is well defined and satisfies (3.6), and the same is true in the union Oα (S) = ∪σ 0 ∈ Oα (S(σ 0 )). The identity (3.7) holds for Y, Tq Y ∈ Oα (S), since the form and the manifold S are invariant with respect to the translations. It remains to modify Oα (S) by the translations: we set Oα (S) = ∪b∈R Tb Oα (S). Then the second statement obviously holds. The last two statements and the uniform continuity in the first statement follow by translation invariance and the compactness arguments. We refer to as the symplectic orthogonal projection onto S. 4. Linearization on the Solitary Manifold Let us consider a solution to the system (1.12), and split it as the sum Y (t) = S(σ (t)) + X (t),

(4.1)

where σ (t) = (b(t), v(t)) ∈ is an arbitrary smooth function of t ∈ R. In detail, denote Y = (ψ, π ) and X = (, ). Then (4.1) means that ψ(x, t) = ψv(t) (x − b(t)) + (x − b(t), t), (4.2) π(x, t) = πv(t) (x − b(t)) + (x − b(t), t). Let us substitute (4.2) to (1.12), and linearize the equations in X . Setting y = x − b(t) which is the “moving frame coordinate”, we obtain that ˙ v (y) + (y, ˙ (y, t) = πv (y) + (y, t), ˙ t) − b ψ˙ = v∂ ˙ v ψv (y) − bψ ˙ v (y)+ (y, ˙ (y, t) = ψv (y)+ (y, t)+ F(ψv (y) + (y, t)). ˙ t)− b π˙ = v∂ ˙ v πv (y)− bπ (4.3) Using Eq. (2.6), we obtain from (4.3) the following equations for the components of the vector X (t): ˙ (y, t) + (b˙ − v)ψv (y) − v∂ ˙ (y, t) = (y, t) + b ˙ v ψv (y), ˙ (y, t)+(b−v)π ˙ ˙ (y, t) = (y, t)+ b (y)− v∂ ˙ π (y)+ F(ψ (y)+(y, t))− F(ψ (y)). v v v v v

(4.4) We can write Eq. (4.4) as X˙ (t) = A(t)X (t) + T (t) + N (t), t ∈ R,

(4.5)

234


where T (t) is the sum of terms which do not depend on X , and N (t) is at least quadratic in X . The linear operator A(t) = Av,w depends on two parameters, v = v(t), and ˙ and can be written in the form w = b(t) w∇ 1 w∇ 1 = := , Av,w + F (ψv ) w∇ − m 2 − Vv (y) w∇ (4.6) where Vv (y) = −F (ψv ) − m 2 .

(4.7)

Furthermore, T (t) and N (t) = N (σ, X ) are given by (w − v)ψv − v∂ ˙ v ψv 0 T = , N (σ, X ) = , N (v, ) (w − v)πv − v∂ ˙ v πv

(4.8)

where v = v(t), w = w(t), σ = σ (t) = (b(t), v(t)), X = X (t), and N (v, ) = F(ψv + ) − F(ψv ) − F (ψv ),

(4.9)

Remark 4.1. Formulas (3.3) and (4.8) imply: T (t) = −(w − v)τ1 − vτ ˙ 2,

(4.10)

and hence T (t) ∈ T S(σ (t)) S, t ∈ R. This fact suggests an unstable character of the nonlinear dynamics along the solitary manifold. 4.1. Linearized equation. Here we collect some Hamiltonian and spectral properties of the operator Av,w . First, let us consider the linear equation X˙ (t) = Av,w X (t),

t ∈R

(4.11)

with arbitrary fixed v ∈ (−1, 1) and w ∈ R. Let us define the space E + := H 2 (R) ⊕ H 1 (R). Lemma 4.2. i) For any v ∈ (−1, 1) and w ∈ R, Eq. (4.11) can be represented as the Hamiltonian system, X˙ (t) = J DHv,w (X (t)),

t ∈ R,

where DHv,w is the Fréchet derivative of the Hamiltonian functional 1 2 Hv,w (X ) = || + | |2 + (m 2 + Vv )||2 dy + w dy. 2

(4.12)

(4.13)

ii) The energy conservation law holds for the solutions X (t) ∈ C 1 (R, E + ), Hv,w (X (t)) = const,

t ∈ R.

(4.14)

iii) The skew-symmetry relation holds: (Av,w X 1 , X 2 ) = −(X 1 , Av,w X 2 ),

X 1 , X 2 ∈ E.

(4.15)


Proof.

i) The equation (4.11) reads as follows: d + w . = − (m 2 + Vv ) + w dt

235

(4.16)

The equations correspond to the Hamilton form since + w = D Hv,w , − (m 2 + Vv ) + w = −D Hv,w . ii) The energy conservation law follows by (4.12) and the chain rule for the Fréchet derivatives: d Hv,w (X (t)) = DHv,w (X (t)), X˙ (t) = DHv,w (X (t)), J DHv,w (X (t)) = 0, dt t ∈ R, (4.17) since the operator J is skew-symmetric by (3.1), and DHv,w (X (t)) ∈ E for X (t) ∈ E + . iii) The skew-symmetry holds since Av,w X = J DHv,w (X ), and the linear operator X → DHv,w (X ) is symmetric as the Fréchet derivative of a real quadratic form. Lemma 4.3. The operator Av,w acts on the tangent vectors τ = τ j (v) to the solitary manifold as follows: Av,w [τ1 ] = (v − w)τ1 , Av,w [τ2 ] = (w − v)τ2 + τ1 .

(4.18)

Proof. In detail, we have to show that −ψv (v − w)ψv ∂v ψv (w − v)∂v ψv −ψv = , A = + . Av,w v,w −πv (v − w)πv ∂v πv (w − v)∂v πv −πv Indeed, differentiate Eqs. (2.6) in b and v, and obtain that the derivatives of the soliton state in parameters satisfy the following equations: −vψv = πv , −vπv = ψv + F (ψv )ψv ,

−ψv − v∂v ψv = ∂v πv , −πv − v∂v πv = ∂v ψv + F (ψv )∂v ψv . Then (4.18) follows from (4.19) by definition of Av,w in (4.6)

(4.19)

Now we consider the operator Av = Av,v corresponding to w = v: v∇ 1 . Av := − m 2 − Vv v∇

(4.20)

In that case the linearized equation has the following additional specific features. The continuous spectrum of the operator Av coincides with := (−i∞, −im/γ ] ∪ [im/γ , i∞).

(4.21)

From (4.18) it follows that the tangent vector τ1 (v) is the zero eigenvector, and τ2 (v) is the corresponding root vector of the operator Av , i.e. Av [τ1 (v)] = 0, Av [τ2 (v)] = τ1 (v).

(4.22)

236


Lemma 4.4. Zero root space of operator Av is two-dimensional for any v ∈ (−1, 1). Proof. It suffices to check that the equation Av u = τ2 (v) has no solution in L 2 ⊕ L 2 . Indeed, the equation reads v∇ 1 vγ 2 yψv u1 = . (4.23) u2 − m 2 − Vv v∇ −γ 2 ψv − v 2 γ 2 yψv From the first equation we get u 2 = vγ 2 yψv − vu 1 . Then the second equation implies that Hv u 1 = γ 2 (1 + v 2 )ψv + 2v 2 γ 2 yψv ,

(4.24)

where Hv is the Schrödinger operator defined in (1.19). Setting u 1 = − 21 v 2 γ 4 y 2 ψv + u˜ 1 , we reduce the equation to Hv u˜ 1 = −γ 2 ψv ,

(4.25)

since ψv = γ 2 (m 2 + Vv )ψv by the first line of (4.19). Hence, u˜ 1 is the root function of the operator Hv since ψv is an eigenfunction. However, this is impossible since Hv is a selfadjoint operator. Lemma 4.5. The operator Av has only eigenvalue λ = 0. Proof. Let us consider the eigenvalues problem for operator Av : v∇ 1 u1 u1 = λ . u2 u2 − m 2 − Vv v∇ From the first equation we have u 2 = −(v∇ − λ)u 1 . Then the second equation implies that (Hv + λ2 − 2vλ∇)u 1 = 0.

(4.26)

Hence, for v = 0 the operator A0 has only eigenvalue λ = 0 by Condition U2 i). Further, let us consider the case v = 0. Taking the scalar product with u 1 , we obtain Hv u 1 , u 1 + λ2 u 1 , u 1 = 0. Hence, λ2 is real since the operator Hv is selfadjoint. The nonzero eigenvalues can bifurcate either from the point λ = 0 or from the edge points ±im/γ of the continuous spectrum of the operator Av . Let us consider each case separately. i) The point λ = 0 cannot bifurcate since it is isolated, and the zero root space is two dimensional by Lemma 4.4. ii) The bifurcation from the edge points also is impossible. Indeed, the bifurcated eigenvalue λ ∈ (−im/γ , im/γ ) is pure imaginary because λ2 is real. Hence, (4.26) is equivalent to Hv + γ 2 λ2 p = 0, (4.27) where p(x) = eγ vλx u 1 (x) ∈ L 2 that is forbidden by Condition U2 i) since −γ 2 λ2 ∈ (0, m 2 ). 2


237

4.2. Decay for the linearized dynamics. Let us consider the linearized equation X˙ (t) = Av X (t), t ∈ R,

(4.28)

where Av = Av,v is given in (4.20) with Vv is defined in (4.7). Definition 4.6. For |v| < 1, denote by Pvd the symplectic orthogonal projection of E onto the tangent space T S(σ ) S, and Pvc = I − Pvd . Note that by the linearity, Pvd X =

p jl (v)τ j (v)(τl (v), X ),

X∈E

(4.29)

with some smooth coefficients p jl (v). Hence, the projector Pvd , in the variable y = x −b, does not depend on b. Next decay estimates will play the key role in our proofs. The first estimate follows from our assumption U2 by Theorem 3.15 of [14] since the condition of type [14, (1.3)] holds in our case (see also [13]). Theorem 4.7. Let the condition U2 hold, and β > 5/2. Then for any X ∈ E β , the weighted energy decay holds: e Av t Pvc X E −β ≤ C(v)(1 + t)−3/2 X E β , t ∈ R,

(4.30)

Corollary 4.8. For β > 5/2 and for X ∈ E β ∩ W , (e Av t Pvc X )1 L ∞ ≤ C(v)(1 + t)−1/2 ( X W + X E β ), t ∈ R.

(4.31)

Here (·)1 stands for the first component of the vector function. Proof. Let us apply the projector Pvc to both sides of (4.28): Pvc X˙ = Av Pvc X = A0v Pvc X + Vv Pvc X,

(4.32)

where A0v =

v∇ 1 − m 2 v∇

, V=

0 0 . −Vv 0

Hence, the Duhamel representation gives, e

Av t

Y =e

A0v t

Y+ 0

t

e Av (t−τ ) Ve Av τ Y dτ, Y = Pvc X, t > 0. 0

(4.33)

0

0

Let us note that e Av t Z = e A0 t Tvt Z , where Tvt Z (x, t) = Z (x + vt, t). Then (4.33) reads 0

e Av t Y = e A0 t Tvt Y +

0

t

e A0 (t−τ ) Tvt [Ve Av τ Y ]dτ, t > 0. 0

(4.34)

238


Applying estimate (265) from [21], the Hölder inequality and Theorem 4.7 we obtain t (e Av t Y )1 L ∞ ≤ C(1 + t)−1/2 Tvt Y W + C (1 + t − τ )−1/2 Tvt [V (e Av τ Y )1 ] W 1,1 dτ 0 0 t (1 + t − τ )−1/2 V (e Av τ Y )1 W 1,1 dτ = C(1 + t)−1/2 Y W + C 0 0 t (1 + t − τ )−1/2 e Av τ Pvc X E −β dτ ≤ C(1 + t)−1/2 X W + C 0 t (1 + t − τ )−1/2 (1 + τ )−3/2 X E β dτ ≤ C(1 + t)−1/2 X W + C 0

−1/2

≤ C(1 + t)

( X W + X E β ).

4.3. Taylor expansion for nonlinear term. Now let us expand N (v, ) from (4.9) in the Taylor series N (v, ) = N2 (v, ) + N3 (v, ) + · · · + N12 (v, ) + N R (v, ) = N I (v, ) + N R (v, ),

(4.35)

where N j (v, ) =

F ( j) (ψv ) j , j!

j = 2, . . . , 12

(4.36)

and N R is the remainder. By condition U1 we have F(ψ) = −m 2 (ψ ∓ a) + O(|ψ ∓ a|13 ), ψ → ±a. Hence, the functions F ( j) (ψv (y)), 2 ≤ j ≤ 12 decrease exponentially as |y| → ∞ by (1.18) and (1.14). Therefore, N I L 2 ∩W 1,1 = R( L ∞ ) L ∞ H 1 = R( L ∞ ) L ∞ X E −β . β

−β

0

(4.37)

For the remainder N R we have |N R | = R( L ∞ )||13 ,

(4.38)

where R(A) is a general notation for a positive function which remains bounded as A is sufficiently small. Lemma 4.9. The bounds hold: N R W 1,1 = R( L ∞ ) 11 L∞ ,

(4.39)

0

N R L 2

5/2+ν

= R( L ∞ )(1 + t)

4+ν

12 L∞ ,

0 < ν < 1/2.

(4.40)


239

Proof. Step i) By the Cauchy formula, 13 (x, t) N R (x, t) = (13)!

0

1

(1 − ρ)12 F (13) (ψv + ρ(x, t))dρ,

(4.41)

Therefore, N R L 1 = R( L ∞ )

2 ||13 d x = R( L ∞ ) 11 L ∞ 2

= R( L ∞ ) ) 11 L∞ , since L 2 ≤ C(d0 ) by the results of [10]. Differentiating (4.41) in x, we obtain N R =

1 13 (1 − ρ)12 (ψv + ρ )F (14) (ψv + ρ)dρ (13)! 0 12 1 + (1 − ρ)12 F (13) (ψv + ρ)dρ, (12)! 0

Hence, 11 N R L 1 = R( L ∞ ) 13 + (x)|d x |(x) ∞ ∞ L L ≤ R( L ∞ ) 11 L∞ , since

|(x) (x)|d x ≤ L 2 L 2 ≤ C(d0 ). Then (4.39) follows.

Step ii) The bound (4.38) implies N R L 2

5/2+ν

= R( L ∞ ) 12 L ∞ L 2

5/2+ν

.

We will prove in Appendix B that (t) L 2

5/2+ν

Then (4.40) follows.

≤ C(d0 )(1 + t)4+ν .

(4.42)

Remark 4.10. Our choice of the degree 14 in the condition (1.11) is due to the competition between the factors in the estimate (4.40) for the remainder. Namely, the factor (1 + t)4+ν with ν < 1/2 comes from the virial type estimate (4.42) describing the expan−6 by sion of the support for the perturbation of the kink. On the other hand, 12 L∞ ∼ t the crucial decay estimate (7.1). Hence, the right-hand side (4.40) decays like ∼ t −2+ν , where −2 + ν < −3/2 which is sufficient for the method of majorants (in integral inequalities (9.2) and (9.3)).

240


5. Symplectic Decomposition of the Dynamics Here we decompose the dynamics in two components: along the manifold S and in transversal directions. Equation (4.5) is obtained without any assumption on σ (t) in (4.1). We are going to choose S(σ (t)) := Y (t), but then we need to know that Y (t) ∈ Oα (S),

t ∈R

(5.1)

with some Oα (S) defined in Lemma 3.4. It is true for t = 0 by our main assumption (2.8) with sufficiently small d0 > 0. Then S(σ (0)) = Y (0) and X (0) = Y (0) − S(σ (0)) are well defined. We will prove below that (5.1) holds with α = −β if d0 is sufficiently small. First, we choose v < 1 such that |v(0)| ≤ v.

(5.2)

Denote by r−β (v) the positive number from Lemma 3.4 iii) which corresponds to α = −β. Then S(σ ) + X ∈ O−β (S) if σ = (b, v) with |v| < v and X E −β < r−β (v). Therefore, S(σ (t)) = Y (t) and X (t) = Y (t) − S(σ (t)) are well defined for t ≥ 0 so small that X (t) E −β < r−β (v). This is formalized by the standard definition of the “exit time”. First, we introduce the “majorants” m 1 (t) := sup (1 + s)3/2 X (s) E −β ,

m 2 (t) := sup (1 + s)1/2 (s) L ∞ .

s∈[0,t]

(5.3)

s∈[0,t]

Here X = (X 1 , X 2 ) = (, ). Let us denote by ε ∈ (0, r−β (v)) a fixed number which we will specify below. Definition 5.1. t∗ is the exit time t∗ = sup{t ≥ 0 : m j (s) < ε,

j = 1, 2, 0 ≤ s ≤ t}.

(5.4)

Let us note that m j (0) < ε for sufficiently small d0 . One of our main goals is to prove that t∗ = ∞ if d0 is sufficiently small. This would follow if we show that m j (t) < ε/2,

0 ≤ t < t∗ .

(5.5)

6. Modulation Equations In this section we present the modulation equations which allow to construct the solutions Y (t) of Eq. (2.1) close at each time t to a kink, i.e. to one of the functions described in Definition 2.3 with time varying (“modulating”) parameters (b, v) = (b(t), v(t)). We look for a solution to (2.1) in the form Y (t) = S(σ (t))+ X (t) by setting S(σ (t)) = Y (t) which is equivalent to the symplectic orthogonality condition of type (3.7), X (t) T S(σ (t)) S, t < t∗ ,

(6.1)

The projection Y (t) is well defined for t < t∗ by Lemma 3.4 iii). Now we derive the “modulation equations” for the parameters σ (t) = (b(t), v(t)). For this purpose, let us write (6.1) in the form (X (t), τ j (t)) = 0,

j = 1, 2,

(6.2)

where the vectors τ j (t) = τ j (σ (t)) span the tangent space T S(σ (t)) S. It would be convenient for us to use some other parameters (c, v) instead of σ = (b, v), where c(t) = t b(t) − 0 v(τ )dτ and ˙ − v(t) = w(t) − v(t) c(t) ˙ = b(t)

(6.3)


241

Lemma 6.1. Let Y (t) be a solution to the Cauchy problem (2.1), and (6.2) hold. Then the parameters c(t) and v(t) satisfy the equations (τ1 , τ2 )(N , τ2 ) + (X, ∂v τ1 )(N , τ2 ) − (X, ∂v τ2 )(N , τ1 ) (6.4) D −(τ1 , τ2 )(N , τ1 ) − (X, τ2 )(N , τ1 ) − (X, τ1 )(N , τ2 ) , (6.5) v˙ = D c˙ =

where D = 2 (τ1 , τ2 ) + O( X E −β ). Proof. Differentiating the orthogonality conditions (6.2) in t we obtain 0 = ( X˙ , τ j ) + (X, τ˙ j ) = (Av,w X + T + N , τ j ) + (X, τ˙ j ),

j = 1, 2.

(6.6)

First, let us compute the principal (i.e. non-vanishing at X = 0) term (T, τ j ). By (4.10), (T, τ1 ) = −v(τ ˙ ˙ ˙ 2 , τ1 ) = v(τ 1 , τ2 ); (T, τ2 ) = −c(τ 1 , τ2 ).

(6.7)

Second, let us compute (Av,w X, τ j ). The skew-symmetry (4.15) implies that (Av,w X, τ j ) = −(X, Av,w τ j ). Then by (4.18) we have (Av,w X, τ1 ) = (X, cτ ˙ 1 ), (Av,w X, τ2 ) = −(X, cτ ˙ 2 + τ1 ) = −(X, cτ ˙ 2 ),

(6.8) (6.9)

since (X, τ1 ) = 0. Finally, let us compute the last term (X, τ˙ j ) in (6.6). For j = 1, 2 one has τ˙ j = ˙ b τ j + v∂ ˙ v τ j = v∂ ˙ v τ j since the vectors τ j do not depend on b according to (3.3). b∂ Hence, (X, τ˙ j ) = (X, v∂ ˙ v τ j ).

(6.10)

As the result, by (6.7)–(6.10), Eq. (6.6) becomes 0 = c(X, ˙ τ1 ) + v˙ ((τ1 , τ2 ) + (X, ∂v τ1 )) + (N , τ1 ), 0 = −c˙ (X, τ2 ) + ((τ1 , τ2 ) + v(X, ˙ ∂v τ2 ) + (N , τ2 ). Since (τ1 , τ2 ) = 0 by (3.5) then the determinant D of the system does not vanish for small X E −β and we obtain (6.4)–(6.5). Corollary 6.2. Formulas (6.4)–(6.5) imply |c(t)|, ˙ |v(t)| ˙ ≤ C(v) (t) 2L 2 ≤ C(v) X (t) 2E −β , −β

0 ≤ t < t∗ .

(6.11)

242


7. Decay for the Transversal Dynamics In Sect. 12 we will show that our main Theorem 2.5 can be derived from the following time decay of the transversal component X (t): Proposition 7.1. Let all conditions of Theorem 2.5 hold. Then t∗ = ∞, and X (t) E −β ≤

C(v, d0 ) C(v, d0 ) , (t) L ∞ ≤ , 3/2 (1 + |t|) (1 + |t|)1/2

t ≥ 0.

(7.1)

We will derive (7.1) in Sects. 11 from our Eq. (4.5) for the transversal component X (t). This equation can be specified using Corollary 6.2. Indeed, (4.10) implies that T (t) E β ∩W ≤ C(v) X 2E −β ,

0 ≤ t < t∗

(7.2)

by (6.11) since w − v = c. ˙ Thus (4.5) becomes the equation X˙ (t) = A(t)X (t) + T (t) + N I (t) + N R (t),

0 ≤ t < t∗ ,

(7.3)

where A(t) = Av(t),w(t) , T (t) satisfies (7.2), and

12 4+ν N R E 5/2+ν ≤ C(v)(1 + t) L ∞ , 0 < ν < 1/2, 0 ≤ t < t∗ , N R W ≤ C(v) 11 ∞ L

N I (t) E β ∩W ≤ C(v) L ∞ X E −β ,

(7.4)

by (4.37), (4.39–(4.40)). In remaining part of our paper we will analyze mainly Eq. (7.3) to establish the decay (7.1). We are going to derive the decay using the bounds (7.2) and (7.4), and the orthogonality condition (6.1). Let us comment on two main difficulties in proving (7.1). The difficulties are common for the problems studied in [4]. First, the linear part of the equation is nonautonomous, hence we cannot apply directly the methods of scattering theory. Similarly to the approach of [4], we reduce the problem to the analysis of the frozen linear equation, X˙ (t) = A1 X (t), t ∈ R,

(7.5)

where A1 is the operator Av1 defined by (4.6) with v1 = v(t1 ) for a fixed t1 ∈ [0, t∗ ). Then we estimate the error by the method of majorants. Second, even for the frozen equation (7.5), the decay of type (7.1) for all solutions does not hold without the orthogonality condition of type (6.1). Namely, by (4.22) Eq. (7.5) admits the secular solutions X (t) = C1 τ1 (v) + C2 [τ1 (v)t + τ2 (v)]

(7.6)

which arise also by differentiation of the soliton (1.13) in the parameters q and v in the moving coordinate y = x − v1 t. Hence, we have to take into account the orthogonality condition (6.1) in order to avoid the secular solutions. For this purpose we will apply the corresponding symplectic orthogonal projection which kills the “runaway solutions” (7.6). Remark 7.2. The solution (7.6) lies in the tangent space T S(σ1 ) S with σ1 = (b1 , v1 ) (for an arbitrary b1 ∈ R) that suggests an unstable character of the nonlinear dynamics along the solitary manifold (cf. Remark 4.1 ii)).


243

Definition 7.3. Denote by Xv = Pvc E the space symplectic orthogonal to T S(σ ) S with σ = (b, v) (for an arbitrary b ∈ R). Now we have the symplectic orthogonal decomposition E = T S(σ ) S + Xv ,

σ = (b, v)

(7.7)

and the symplectic orthogonality (6.1) can be written in the following equivalent forms: d Pv(t) X (t) = 0,

c Pv(t) X (t) = X (t),

0 ≤ t < t∗ .

(7.8)

Remark 7.4. The tangent space T S(σ ) S is invariant under the operator Av by (4.22), hence the space Xv is also invariant by (4.15): Av X ∈ Xv on a dense domain of X ∈ Xv . 8. Frozen Form of Transversal Dynamics Now let us fix an arbitrary t1 ∈ [0, t∗ ), and rewrite Eq. (7.3) in a “frozen form” X˙ (t) = A1 X (t) + (A(t) − A1 )X (t) + T (t) + N I (t) + N R (t), 0 ≤ t < t∗ , (8.1) where A1 = Av(t1 ),v(t1 ) and A(t) − A1 =

(w(t) − v(t1 ))∇ 0 . 0 (w(t) − v(t1 ))∇

The next trick is important since it allows us to kill the “bad terms” (w(t) − v(t1 ))∇ in the operator A(t) − A1 . Let us change the variables (y, t) → (y1 , t) = (y + d1 (t), t), where t (w(s) − v(t1 ))ds, 0 ≤ t ≤ t1 . (8.2) d1 (t) := t1

Next define X˜ (t) = ((y1 − d1 (t), t), (y1 − d1 (t), t)).

(8.3)

Then we obtain the final form of the “frozen equation” for the transversal dynamics X˙˜ (t) = A1 X˜ (t) + T˜ (t) + N˜ I (t) + N˜ R (t), 0 ≤ t ≤ t1 ,

(8.4)

where T˜ (t), N˜ I (t) and N˜ R (t) are T (t), N I (t) and N R (t) expressed in terms of y1 = y + d1 (t). Now we derive appropriate bounds for the “remainder terms” in (8.4). Let us recall the following well-known inequality: for any α ∈ R, (1 + |y + x|)α ≤ (1 + |y|)α (1 + |x|)|α| ,

x, y ∈ R.

(8.5)

Lemma 8.1. For f ∈ L 2α with any α ∈ R the following bound holds: f (y1 − d1 ) L 2α ≤ f L 2α (1 + |d1 |)|α| , d1 ∈ R.

(8.6)

244


Proof. The bound (8.6) follows from (8.5) since 2 2 2α f (y1 − d1 ) L 2 = | f (y1 − d1 )| (1 + |y1 |) dy1 = | f (y)|2 (1 + |y + d1 |)2α dy α ≤ | f (y)|2 (1 + |y|)2α (1 + |d1 |)2|α| dy ≤ (1 + |d1 |)2|α| f 2L 2 . α

Corollary 8.2. The following bounds hold for 0 ≤ t ≤ t1 by (7.2) and (7.4): T˜ (t) E β N˜ I (t) E β N˜ R E 5/2+ν N˜ R W

, T˜ (t) W ≤ C(v) X 2E , −β β ˜ ≤ C(v)(1 + |d1 (t)|) L ∞ X E −β , N I (t) W ≤ C(v) L ∞ X E −β , 12 5/2+ν 4+ν ≤ C(v)(1 + |d1 (t)|) (1 + t) L ∞ , 0 < ν < 1/2, ≤ C(v) 11 . L∞ ≤ C(v)(1 + |d1 (t)|)β X 2E

−β

(8.7) 9. Integral Inequality Equation (8.4) can be written in the integral form: t A1 t ˜ ˜ X (t) = e X (0) + e A1 (t−s) [T˜ (s) + N˜ I (s) + N˜ R (s)]ds, 0 ≤ t ≤ t1 .

(9.1)

0 c We apply the symplectic orthogonal projection P1c := Pv(t to both sides, and get 1) t e A1 (t−s) P1c [T˜ (s) + N˜ I (s) + N˜ R (s)] ds. P1c X˜ (t) = e A1 t P1c X˜ (0) + 0

We have used here that P1c commutes with the group e A1 t since the space X1 := P1c E is invariant with respect to e A1 t by Remark 7.4. Applying (4.30) we obtain that t ˜ C X˜ (0) E β T (s) + N˜ I (s) + N˜ R (s) E β + C ds. P1c X˜ (t) E −β ≤ (1 + t)3/2 (1 + |t − s|)3/2 0 Then for 5/2 < β < 3 and 0 ≤ t ≤ t1 the bounds (8.7) imply P1c X˜ (t) E −β ≤ + C(d 1 (t)) 0

t

C(d 1 (0)) X (0) E β (1 + t)3/2 X (s) 2E −β + (s) L ∞ X (s) E −β + (1 + s)3/2+β (s) 12 L∞ (1 + |t − s|)3/2

ds, (9.2)

where d 1 (t) := sup0≤s≤t |d1 (s)|. Similarly, (4.31) and (8.7) imply t ˜ C X˜ (0) E β ∩W T (s) + N˜ I (s) + N˜ R (s) E β ∩W c ˜ + C ds (P1 X (t))1 L ∞ ≤ (1 + t)1/2 (1 + |t − s|)1/2 0 C(d 1 (0)) ≤ X (0) E β ∩W + C(d 1 (t)) (1 + t)1/2 t X (s) 2 + (s) ∞ X (s) 3/2+β (s) 12 + (s) 11 L E −β +(1 + s) E −β L∞ L∞ ds. × 1/2 (1 + |t − s|) 0 (9.3)


245

Lemma 9.1. For t1 < t∗ we have |d1 (t)| ≤ Cε2 , 0 ≤ t ≤ t1 .

(9.4)

Proof. To estimate d1 (t), we note that

t1

w(s) − v(t1 ) = w(s) − v(s) + v(s) − v(t1 ) = c(s) ˙ +

v(τ ˙ )dτ

(9.5)

s

by (6.3). Hence, the definitions (8.2), (5.3), and Corollary 6.2 imply that t t1 t1 |d1 (t)| = | (w(s) − v(t1 ))ds| ≤ |v(τ ˙ )|dτ ds |c(s)| ˙ + t1

≤ Cm 21 (t1 )

t1 t

1 + (1 + s)3

t t1 s

dτ (1 + τ )3

s

ds ≤ Cm 21 (t1 ) ≤ Cε2 ,

0 ≤ t ≤ t1 . (9.6)

Now (9.2) and (9.3) imply that for t1 < t∗ and 0 ≤ t ≤ t1 , P1c X˜ (t) E −β ≤

C X (0) E β (1 + t)3/2 + (s) L ∞ X (s) E −β + (1 + s)3/2+β (s) 12 L∞

t X (s) 2 E −β

+C

(1 + |t − s|)3/2

0

(P1c X˜ (t))1 L ∞ ≤ +C 0

ds,

(9.7)

C X (0) E β ∩W (1 + t)1/2

3/2+β (s) 12 + (s) 11 t X (s) 2 ∞ E −β + (s) L X (s) E −β +(1 + s) L∞ L∞ ds. (1 + |t − s|)1/2

(9.8) 10. Symplectic Orthogonality Finally, we are going to change P1c X˜ (t) by X (t) in the left-hand side of (9.7) and (9.8). We will prove that it is possible since d0 1 in (2.8). Lemma 10.1. For sufficiently small ε > 0, we have for t1 < t∗ , X (t) E −β ≤ C P1c X˜ (t) E −β , 0 ≤ t ≤ t1 , (t) L ∞ ≤ 2 (P1c X˜ (t))1 L ∞ , 0 ≤ t ≤ t1 , where the constant C does not depend on t1 . Proof. The proof is based on the symplectic orthogonality (7.8), i.e. d Pv(t) X (t) = 0,

t ∈ [0, t1 ]

c E are almost parallel for all t. and on the fact that all the spaces X (t) := Pv(t)

(10.1)

246


˜ (t) E −β ˜ Namely, we first note that (t) L ∞ = (t) L ∞ , and X (t) E −β ≤ C X by Lemma 8.1, since |d1 (t)| ≤ const for t ≤ t1 < t∗ by (9.4). Therefore, it suffices to prove that c ˜ ˜ (t) (t))1 L ∞ , X˜ (t) E −β ≤ 2 P1c X˜ (t) E −β , 0 ≤ t ≤ t1 . L ∞ ≤ 2 (P1 X (10.2)

This estimate will follow from 1 1 d ˜ ˜ (P1d X˜ (t))1 L ∞ ≤ (t) (t) E −β ≤ X˜ (t) E −β , 0 ≤ t ≤ t1 . L ∞ , P1 X 2 2 (10.3) since Pc X˜ (t) = X˜ (t) − Pd X˜ (t). To prove (10.3), we write (10.1) as, 1

1

d P˜ v(t) X˜ (t) = 0,

t ∈ [0, t1 ]

(10.4)

d X ˜ (t) is Pd X (t) expressed in terms of the variable y1 = y + d1 (t). Hence, where P˜ v(t) v(t) (10.3) follows from (10.4) if the difference Pd − P˜ d is small uniformly in t, i.e. 1

P1d

d − P˜ v(t)

< 1/2,

v(t)

0 ≤ t ≤ t1 .

(10.5)

It remains to justify (10.5) for small enough ε > 0. In order to prove the bound (10.5), we will need the formula (4.29) and the following relation which follows from (4.29): d P˜ v(t) (10.6) X˜ (t) = p jl (v(t))τ˜ j (v(t))(τ˜l (v(t)), X˜ (t)), where τ˜ j (v(t)) are the vectors τ j (v(t)) expressed in the variables y1 . In detail (cf. (3.3)), τ˜1 (v) := (−ψv (y1 − d1 (t)), −πv (y1 − d1 (t))), τ˜2 (v) := (∂v ψv (y1 − d1 (t)), ∂v πv (y1 − d1 (t))),

(10.7)

where v = v(t). Since τ j are smooth and rapidly decaying at infinity functions, then Lemma 9.1 implies τ˜ j (v(t)) − τ j (v(t)) E β ≤ Cε2 , Furthermore,

τ j (v(t)) − τ j (v(t1 )) = t

t1

0 ≤ t ≤ t1 ,

j = 1, 2.

(10.8)

v(s)∂ ˙ v τ j (v(s))ds,

and therefore

t1 |v(s)|ds, ˙ τ j (v(t)) − τ j (v(t1 )) E β ≤ C t t1 v(s)∂ ˙ p (v(s))ds| ≤ C | p jl (v(t)) − p jl (v(t1 ))| = | v jl t

0 ≤ t ≤ t1 , t1

|v(s)|ds, ˙

(10.9) 0 ≤ t ≤ t1 ,

t

(10.10) since |∂v p jl (v(s))| is uniformly bounded by (5.2). Further, t1 t1 ds |v(s)|ds ˙ ≤ Cm 21 (t1 ) ≤ Cε2 , 3 (1 + s) t t

0 ≤ t ≤ t1 .

(10.11)

Hence, the bounds (10.5) will follow from (4.29), (10.6) and (10.8)–(10.10) if we choose ε > 0 small enough. The proof is completed.


247

11. Decay of Transversal Component Here we prove Proposition 7.1. Step i) We fix ε > 0 and t∗ = t∗ (ε) for which Lemma 10.1 holds. Then the bounds of type (9.7) and (9.8) holds with P1c X˜ (t) E −β and (P1c X˜ (t))1 L ∞ in the left-hand sides replaced by X (t) E −β and (t) L ∞ : X (t) −β ≤

t

+C

(1 + t)3/2 X (s) 2E −β + (s) L ∞ X (s) E −β + (1 + s)3/2+β (s) 12 L∞

0

(t) L ∞ ≤

t

+C

C X (0) E β

C X (0) E β ∩W

(1 + |t − s|)3/2

ds,

(11.1)

(1 + t)1/2 11 X (s) 2E −β + (s) L ∞ X (s) E −β +(1 + s)3/2+β (s) 12 L ∞ + (s) L ∞ (1 + |t − s|)1/2

0

ds

(11.2) for 0 ≤ t ≤ t1 and t1 < t∗ . This implies an integral inequality for the majorants m 1 and m 2 . Namely, multiplying both sides of (11.1) by (1 + t)3/2 , and taking the supremum in t ∈ [0, t1 ], we obtain t (1 + t)3/2 ds m 1 (t1 ) ≤ C X (0) E β + C sup 3/2 t∈[0,t1 ] 0 (1 + |t − s|) 3/2+β m 21 (s) m 1 (s)m 2 (s) m 12 2 (s)(1 + s) × + + (1 + s)3 (1 + s)2 (1 + s)6 for t1 < t∗ . Taking into account that m(t) is a monotone increasing function, we get m 1 (t1 ) ≤ C X (0) E β + C[m 21 (t1 ) + m 1 (t1 )m 2 (t1 ) + m 12 2 (t1 )]I1 (t1 ), t1 < t∗ , (11.3) where

I1 (t1 ) = sup

t

t∈[0,t1 ] 0

(1 + t)3/2 ds ≤ I 1 < ∞, (1 + |t − s|)3/2 (1 + s)9/2−β

t1 ≥ 0, 5/2 < β < 3.

Therefore, (11.3) becomes m 1 (t1 ) ≤ C X (0) E β + C I 1 [m 21 (t) + m 1 (t1 )m 2 (t1 ) + m 12 2 (t1 )],

t1 < t∗ . (11.4)

Similarly, multiplying both sides of (11.2) by (1 + t)1/2 , and taking the supremum in t ∈ [0, t1 ], we get m 2 (t1 ) ≤ C X (0) E β ∩W + C[m 21 (t1 ) + m 1 (t1 )m 2 (t1 ) 11 + m 12 2 (t1 ) + m 2 (t1 )]I2 (t1 ), t1 < t∗ ,

where

I2 (t1 ) = sup

t∈[0,t1 ] 0

t

(1 + t)1/2 ds ≤ I 2 < ∞, (1 + |t − s|)1/2 (1 + s)9/2−β

(11.5)

t1 ≥ 0, 5/2 < β < 3.

248


Therefore, (11.5) becomes m 2 (t1 ) ≤ C X (0) E β ∩W + C I 2 [m 21 (t1 ) 11 + m 1 (t1 )m 2 (t1 ) + m 12 2 (t1 ) + m 2 (t1 )] t1 < t∗ ,

(11.6)

Inequalities (11.4) and (11.6) imply that m 1 (t1 ) and m 2 (t1 ) are bounded for t1 < t∗ , and moreover, m 1 (t1 ), m 2 (t1 ) ≤ C X (0) E β ∩W , t1 < t∗

(11.7)

since m 1 (0) = X (0) E −β and m 2 (0) = (0) L ∞ are sufficiently small by (2.8). Step ii) The constant C in the estimate (11.7) does not depend on t∗ by Lemma 10.1. We choose d0 in (2.8) so small that X (0) E β ∩W < ε/(2C). It is possible due to (2.8). Finally, this implies that t∗ = ∞, and (11.7) holds for all t1 > 0 if d0 is small enough. 12. Soliton Asymptotics Here we prove our main Theorem 2.5 using the decay (7.1). The estimates (6.11) and (7.1) imply that |c(t)| ˙ + |v(t)| ˙ ≤

C1 (v, d0 ) , (1 + t)3

t ≥ 0.

Therefore, c(t) = c+ + O(t −2 ) and v(t) = v+ + O(t −2 ), t → ∞. Similarly, t v(s)ds = v+ t + q+ + α(t), α(t) = O(t −1 ). b(t) = c(t) +

(12.1)

(12.2)

0

We have obtained the solution Y (x, t) = (ψ(x, t), π(x, t)) to (1.12) in the form Y (x, t) = Yv(t) (x − b(t), t) + X (x − b(t), t),

(12.3)

˙ = v+ + α(t). where we define now v(t) = b(t) ˙ Since Yv(t) (x − b(t), t) − Yv+ (x − v+ t − q+ , t) E = O(t −1 ), it remains to extract the dispersive wave W0 (t) + from the term X (x − b(t), t). Substituting (12.3) into (1.12) we obtain by (2.6) the inhomogeneous Klein-Gordon equation for the X (x − b(t), t): X˙ (y, t) = A0v X (y, t) + R(y, t), 0 ≤ t ≤ ∞, where y = x − b(t), and v∇ 1 , A0v = − m 2 v∇

R(t) =

(12.4)

v∂ ˙ v ψv , v∂ ˙ v πv + F( + ψv ) − F(ψv ) + m 2

Now we change the variable y → y1 = y + α(t) + q+ . Then we obtain the “frozen” equation ˜ X˙˜ (t) = A+ X˜ (t) + R(t), 0 ≤ t ≤ ∞,

(12.5)


249

˜ are X (t) and R(t) of y = y1 − α(t) − q+ , and where X˜ (t) and R(t) 1 v+ ∇ , A+ = − m 2 v+ ∇ Equation (12.5) implies

X˜ (t) = W+ (t) X˜ (0) +

t

˜ W+ (t − s) R(s)ds,

(12.6)

(12.7)

0

where W+ (t) = e A+ t is the integral operator with integral kernel W+ (y1 − z, t) = W0 (y1 − z + v+ t, t) = W0 (x − z, t), since by (12.2) y1 + v+ t = y + α(t) + q+ + v+ t = x − b(t) + α(t) + q+ + v+ t = x. Hence, Eq. (12.7) implies X (x − b(t), t) = W0 (t) X˜ (0) +

t

˜ W0 (t − s) R(s)ds.

(12.8)

0

Let us rewrite (12.8) as

X (x − b(t), t) = W0 (t) X˜ (0) +

∞

˜ W0 (−s) R(s)ds −

0

∞

˜ W0 (t − s) R(s)ds

t

= W0 (t) + + r+ (t). To establish the asymptotics (2.9), it suffices to prove that ∞ ˜ ˜ + = X (0) + W0 (−s) R(s)ds ∈ E and r+ (t) E = O(t −1/2 ).

(12.9)

0

˜ Assumption (2.8) implies that X˜ (0) ∈ E. Let us split R(s) as the sum 0 v∂ ˙ v ψ˜ v ˜ ˜ ˜ + R(s) = ˜ = R (s) + R (s). ˜ + ψ˜ v ) − F(ψ˜ v ) + m 2 F( v∂ ˙ v π˜ v By (12.1), we obtain R˜ (s) E = O(s −3 ).

(12.10)

Let us consider R˜ = (0, R˜ 2 ). We have ˜ = (F (ψ˜ v ) + m 2 ) ˜ + N˜ (v, ), ˜ ˜ + ψ˜ v ) − F(ψ˜ v )+m 2 ˜ + N˜ (v, ) ˜ = −V˜v R˜ 2 = F( By (1.17) and (7.1), we obtain −3/2 ˜ ˜ V˜v (s) , L 2 ≤ C (s) L 2 ≤ C(v, d0 )(1 + |s|) −β

(12.11)

since |q+ + α(s)| ≤ C. Finally, (7.1), (7.4), and (8.6) imply −3/2 ˜ N˜ (v, (s)) . L 2 ≤ C(v, d0 )(1 + |s|)

(12.12)

Hence, (12.11)–(12.12) imply R˜ (s) E = O(s −3/2 ), and (12.9) follows by (12.10) and (12.13).

(12.13)

250


A. Virial Type Estimates Here we prove the weighted estimate (4.42). Let us recall that we split the solution Y (t) = (ψ(·, t), π(·, t)) = S(σ (t)) + X (t), and denote X (t) = ((t), (t)), (0 , 0 ) := ((0), (0)). Our basic condition (2.8) implies that for some ν > 0, X 0 E 5/2+ν ≤ d0 < ∞.

(A.1)

Proposition A.1. Let the potential U satisfy conditions U1, and X 0 satisfy (A.1). Then the bounds hold (t) L 2

5/2+ν

≤ C(v, d0 )(1 + t)4+ν ,

t > 0.

(A.2)

We will deduce the proposition from the following two lemmas. The first lemma is well known. Denote |π(x, t)|2 |ψ (x, t)|2 + + U (ψ(x, t)). 2 2 Lemma A.2. For the solution ψ(x, t) of Klein-Gordon equation (1.2) the local energy estimate holds a2 a2 +t e(x, t) d x ≤ e(x, 0) d x, a1 < a2 , t > 0. (A.3) e(x, t) =

a1 −t

a1

Proof. The estimate follows by standard arguments: multiplication of Eq. (1.2) by ˙ ψ(x, t) and integration over the trapezium ABC D, where A = (a1 − t, 0), B = (a1 , t), C = (a2 , t), D = (a2 + t, 0). Then (A.3) is obtained after partial integration using that U (ψ) ≥ 0. Lemma A.3. For any σ ≥ 0 and b ∈ R, σ σ +1 (1 + |x|σ )e(x, 0)d x. (A.4) (1 + |x − b| )e(x, t)d x ≤ C(σ )(1 + t + |b|) Proof. By (A.3) (1 + |y|σ )

e(x, t)d x dy ≤ (1 + |y|σ )

y+b y+b−1

Hence, e(x, t)

x−b+1

e(x, 0)d x dy.

y+b+t y+b−1−t

(1 + |y|σ )dy d x ≤ e(x, 0)

x−b

x−b+1+t

(1 + |y|σ )dy d x.

x−b−t

(A.5) Obviously,

x−b+1

(1 + |y|σ )dy ≥ c(σ )(1 + |x − b|σ )

(A.6)

x−b

with some c(σ ) > 0. On the other hand, x−b+1+t (1 + |y|σ )dy ≤ (2t + 1)(1 + t + |b| + |x|)σ ≤ C(1 + t + |b|)σ +1 (1 + |x|σ ), x−b−t

(A.7) since σ ≥ 0. Finally, (A.5)–(A.7) imply (A.4).


Proof of Proposition A.1. First, we verify that U0 = (1 + |x|5+2ν )U (ψ0 (x))d x < C(d0 ), ψ0 (x) = ψ(x, 0).

251

(A.8)

Indeed, ψ0 (x) = ψv0 (x − q0 ) + 0 (x) is bounded since 0 ∈ H 1 (R). Hence U1 implies that |U (ψ0 (x))| ≤ C(d0 )(ψ0 (x) ± a)2 ≤ C(d0 ) (ψv0 (x − q0 ) ± a)2 + 02 (x) and then (A.8) follows by (1.14), (1.18) and (A.1). Further, we have

2 ˙ (y, s)ds − 0 (y) dy = (1 + |y| ) 0 t ˙ 2 (y, s)ds. ≤ 2d02 + 2t (1 + |y|5+2ν )dy

(t) 2L 2 5/2+ν

5+2ν

t

(A.9)

0

Due to (4.2) and (12.1)–(12.2) we have 2 ˙ ˙ 2 (y, s) = b(s)ψ (y + b(s), s) + π(y + b(s), s) − v∂ ˙ v ψv (y) ≤ C(v, d0 ) (ψ (y + b(s), s))2 + π 2 (y + b(s), s) + (∂v ψv (y))2 ≤ C(v, d0 ) e(y + b(s), s) + (∂v ψv (y))2 . (A.10) Substituting (A.10) into (A.9) and changing variables we obtain by (A.4) and (A.8) that t 2 2 5+2ν (t) L 2 ≤ 2d0 + C(v, d0 )t )e(x, s)d x + C(v) ds (1 + |x − b(s)| 5/2+ν 0 ≤ 2d02 + C(v, d0 )t 2 + C(v, d0 )t (1 + |x|5+2ν )e(x, 0)d x t × (1 + s + |b(s)|)6+2ν ds 0

2 ≤ 2d0 + C(v, d0 )t 2 + C(v, d0 )(1 + t)8+2ν X 0 2E 5/2+ν + U0 ≤ C(v, d0 )(1 + t)8+2ν . References 1. Agmon, S.: Spectral properties of Schrödinger operator and scattering theory. Ann. Scuola Norm. Sup. Pisa, Ser. IV 2, 151–218 (1975) 2. Bais, F.A.: Topological excitations in gauge theories; An introduction from the physical point of view. Springer Lecture Notes in Mathematics, Vol. 926, Berlin-Heidelberg-New York: Springer, 1982 3. Bjørn, F.: Geometry, Particles, and Fields. New York: Springer, NY, 1998 4. Buslaev, V.S., Perelman, G.S.: Scattering for the nonlinear Schrödinger equations: states close to a soliton. St. Petersburg Math. J. 4(6), 1111–1142 (1993) 5. Buslaev, V.S., Sulem, C.: On asymptotic stability of solitary waves for nonlinear Schrödinger equations. Ann. Inst. Henri Poincaré. Anal. Non Linéaire 20(3), 419–475 (2003)

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6. Cuccagna, S.: Stabilization of solutions to nonlinear Schrödinger equations. Comm. Pure Appl. Math. 54, 1110–1145 (2001) 7. Cuccagna, S.: On asymptotic stability in 3D of kinks for the φ 4 model. Transactions of AMS 360(5), 2581– 2614 (2008) 8. Jensen, A., Kato, T.: Spectral properties of Schrödinger operators and time-decay of the wave functions. Duke Math. J. 46, 583–611 (1979) 9. Jensen, A., Nenciu, G.: A unified approach to resolvent expansions at thresholds. Rev. Math. Phys. 13(6), 717–754 (2001) 10. Henry, D.B., Perez, J.F., Wreszinski, W.F.: Stability theory for solitary-wave solutions of scalar field equations. Commun. Math. Phys. 85, 351–361 (1982) 11. Imaikin, V., Komech, A.I., Vainberg, B.: On scattering of solitons for the Klein-Gordon equation coupled to a particle. Commun. Math. Phys. 268(2), 321–367 (2006) 12. Kirr, E., Zarnesku, A.: On the asymptotic stability of bound states in 2D cubic Schrödinger equation. Commun. Math. Phys. 272(2), 443–468 (2007) 13. Komech, A., Kopylova, E.: Weighted energy decay for 1D Klein-Gordon equation. Comm. PDE 35(2), 353–374 (2010) 14. Kopylova, E.: On long-time decay for Klein-Gordon equation. Comm. Math. Anal. Conference 03, 137– 152 (2011). http://arriv.org/abs/1009.2649vz [math-ph]; 2010 15. Lions, J.L.: Quelques Mèthodes de Rèsolution des Problémes aux Limites non Linéaires. Paris: Dunod, 1969 16. Murata, M.: Asymptotic expansions in time for solutions of Schrödinger-type equations. J. Funct. Anal. 49, 10–56 (1982) 17. Miller, J., Weinstein, M.: Asymptotic stability of solitary waves for the regularized long-wave equation Comm. Pure Appl. Math. 49(4), 399–441 (1996) 18. Pego, R.L., Weinstein, M.I.: Asymptotic stability of solitary waves, Commun. Math. Phys. 164, 305–349 (1994) 19. Pillet, C.A., Wayne, C.E.: Invariant manifolds for a class of dispersive, Hamiltonian, partial differential equations. J. Differ. Eq. 141(2), 310–326 (1997) 20. Reed, M.: Abstract Non-Linear Wave Equations. Lecture Notes in Mathematics 507, Berlin: Springer, 1976 21. Reed, M., Simon, B.: Methods of Modern Mathematical Physics, III. New York: Academic Press, 1979 22. Rodnianski, I., Schlag, W., Soffer, A.: Dispersive analysis of charge transfer models. Commun. Pure Appl. Math. 58(2), 149–216 (2005) 23. Soffer, A., Weinstein, M.I.: Multichannel nonlinear scattering for nonintegrable equations. Commun. Math. Phys. 133, 119–146 (1990) 24. Soffer, A., Weinstein, M.I.: Multichannel nonlinear scattering for nonintegrable equations. II. The case of anisotropic potentials and data. J. Diff. Eq. 98(2), 376–390 (1992) 25. Soffer, A., Weinstein, M.I.: Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations. Invent. Math. 136, 9–74 (1999) 26. Strauss, W.A.: Nonlinear invariant wave equations. Lecture Notes in Physics 73, Berlin: Springer, 1978, pp. 197–249 27. Tsai, T.-P., Yau, H.-T.: Asymptotic dynamics of nonlinear Schrödinger equations: resonance-dominated and dispersion-dominated solutions. Commun. Pure Appl. Math. 55(2), 153–216 (2002) Communicated by H. Spohn


Communications in


The Interaction of a Gap with a Free Boundary in a Two Dimensional Dimer System M. Ciucu1, , C. Krattenthaler2, 1 Department of Mathematics, Indiana University, Bloomington, IN 47405-5701, USA.

E-mail: [email protected]

2 Fakultät für Mathematik der Universität Wien, Nordbergstraße 15, A-1090 Wien, Austria

Received: 11 December 2009 / Accepted: 23 August 2010 Published online: 14 January 2011 – © Springer-Verlag 2011

Abstract: Let be a fixed vertical lattice line of the unit triangular lattice in the plane, and let H be the half plane to the left of . We consider lozenge tilings of H that have a triangular gap of side-length two and in which is a free boundary — i.e., tiles are allowed to protrude out half-way across . We prove that the correlation function of this 1 gap near the free boundary has asymptotics 4πr , r → ∞, where r is the distance from the gap to the free boundary. This parallels the electrostatic phenomenon by which the field of an electric charge near a conductor can be obtained by the method of images.

1. Introduction The study of the interaction of gaps in dimer coverings was introduced in the literature by Fisher and Stephenson [15]. This pioneering work contains three different types of gap interaction in dimer systems on the square lattice: (i) interaction of two dimer-gaps (equivalently, interaction of two fixed dimers required to be contained in the dimer coverings); (ii) interaction of two non-dimer-gaps (specifically, two monomers), and (iii) the interaction of a dimer-gap with a constrained boundary (edge or corner). The first of these types of interactions was later generalized by Kenyon [20] to an arbitrary number of dimer-gaps on the square and hexagonal lattices, and recently by Kenyon, Okounkov and Sheffield [22] to general planar bipartite lattices. Interactions of the second type were studied by the first author of the present paper in [5–9], where close analogies to two dimensional electrostatics were established. Two instances of interaction of non-dimer-gaps with constrained boundaries can be found in [21, Sect. 7.5] (interaction of a monomer with a constrained straight line Research partially supported by NSF grant DMS-0500616.

Research partially supported by the Austrian Science Foundation FWF, grants Z130-N13 and S9607-

N13, the latter in the framework of the National Research Network “Analytic Combinatorics and Probabilistic Number Theory.”

254

M. Ciucu, C. Krattenthaler

boundary on the square lattice), and respectively [6, Theorem 2.2] (interaction of a family of triangular gaps with a constrained straight line boundary on the hexagonal lattice). In this paper we determine the interaction of a triangular gap with a free straight line boundary (i.e., dimers are allowed to protrude out across it) on the hexagonal lattice. This type of interaction has not been treated before in the literature. (We are aware of one other paper, namely [12], addressing the asymptotic behavior of lozenge tilings under the presence of a free boundary, but the regions considered there contain no gaps.) We find that the gap is attracted to the free boundary in precise analogy to the (two dimensional) electrostatic phenomenon in which an electric charge is attracted by a straight line conductor when placed near it. This develops further the analogy between dimer systems with gaps and electrostatics that the first author has described in [6–9]. More generally, our result shows that in any physical system that can be modeled by dimer coverings, a gap will tend to be attracted to an interface corresponding to a free boundary. This effect, purely entropic in origin, is reminiscent of the Cheerios effect by which an air bubble at the surface of a liquid in a container is attracted to the walls [35] (note that the Cheerios effect is not entropic in origin). 2. Set-up and Results There seem to be no methods in the literature for finding the interaction of a gap “in a sea of dimers” with a free boundary. However, as V. I. Arnold said, “mathematics is a part of physics where experiments are cheap.” We now design such an experiment in order to determine the interaction of a gap in a dimer system on the hexagonal lattice with a free boundary. Consider the triangular lattice in the plane consisting of unit equilateral triangles, drawn so that one family of lattice lines is vertical. Note that the hexagonal lattice is the dual of the triangular lattice. (To be precise, the hexagonal graph arises as the graph whose vertices are the unit triangles, and whose edges connect precisely those unit triangles that share an edge.) Dimers on the hexagonal lattice then correspond to lozenges (i.e., unit rhombi) consisting of pairs of adjacent unit triangles. The free boundary we choose is a lattice line — say vertical — on the triangular lattice, to the left of which the plane is covered completely and without overlapping by lozenges, except for a gap 2 in the shape of a triangle of side-length 2, pointing to the left; the lozenges are allowed to protrude halfway across the free boundary, to its right. (Figure 1 pictures a portion of such a tiling.) We define the correlation function (or simply correlation) of the hole 2 with the free boundary as follows. Choose a rectangular system of coordinates in which is the y-axis, the origin is some lattice point on , and the unit is the lattice spacing. Let√2 (k) be the placement of 2 so that the center C of its right side has coordinates (−k 3, 0) (i.e., C and the origin are the endpoints of a string of k contiguous horizontal lozenges; Fig. 1 illustrates 2 (2), the origin being denoted by O there). Let Hn,x be the lattice hexagon of side-lengths 2n, 2n, 2x, 2n, 2n, 2x (in counter-clockwise order, starting with the southwestern side) centered at the origin (thus Hn,x is vertically symmetric about , and its horizontal symmetry axis cuts 2 (k) into two equal parts; for example, Fig. 2 shows the hexagon H4,4 with the triangular hole 2 (2)). Let Fn,x be the region obtained from the left half of Hn,x by regarding its boundary along as free (i.e., lozenges in a tiling of Fn,x are allowed to protrude outward across ). Figure 3 shows the region F3,3 together with such a lozenge tiling; the origin is labelled by O.

Interaction of a Gap with a Free Boundary in a Dimer System

255

Fig. 1. A partial lozenge tiling of the left half plane with a gap

Following [15] and [6], for any fixed integer k ≥ 0 we define the correlation of 2 (k) with the free boundary , denoted ω f (k), by ω f (k) := lim

n→∞

M(Fn,n \ 2 (k)) , M(Fn,n )

(2.1)

where M(R) stands for the number of lozenge tilings of the region R (if R has portions of the boundary that are free — as in our case — then it is understood that what we count is tilings in which lozenges are allowed to protrude out across the free portions). A tiling of F4,4 \ 2 (2) of this type is illustrated in Fig. 1. We note that, by [11], in a regular hexagon lozenges have maximum entropy statistics (in the scaling limit) at the center. According to this, (2.1) is a natural definition for the correlation function. An analogous definition was used in [6]. In fact, it is worth generalizing the definition of correlation (2.1) to the situation when the side-lengths 2n and 2x of the half hexagon Fn,x go to infinity at different rates. More precisely, for any real number ξ > 0, define ω f (k; ξ ) by ω f (k; ξ ) := lim

n→∞

M(Fn,ξn n \ 2 (k)) , M(Fn,ξn n )

(2.2)

256


Fig. 2. The hexagon Hn,x with n = x = 4

Fig. 3. A lozenge tiling of the region Fn,n with n = 3

where (ξn )n≥1 is a suitable sequence of rational numbers approaching ξ . (“Suitable” here means that we have to choose ξn in such a way that ξn n is integral.) The number ω f (k; ξ ) is the correlation of the triangular gap 2 (k) with the free boundary, obtained when the large regions used in the definition are the left halfs of hexagons that are not


257

necessarily regular, but have their left vertical side ξ times as long as the two oblique sides. In Lemma 13 we obtain an exact expression for ω f (k; ξ ) in terms of an integral. What affords this is an exact formula for M(Fn,x \ 2 (k)), which we present in Theorem 4. We then deduce the asymptotics of ω f (k; ξ ) as k → ∞ using Laplace’s method (see Lemma 14 and the proof of Theorem 1 in sect. 7). The result is the following. Theorem 1. As k → ∞, the correlation ω f (k; ξ ) is asymptotically ω f (k; ξ ) ∼

1 1 · √ π(1 + ξ )2 ξ(2 + ξ ) k

2 1+ξ

4k .

(2.3)

Remark 1. Note that, by the results of [11], we should expect distorted dimer statistics around the gap for ξ = 1. As the theorem above shows, the distortion is quite radical. Indeed, for ξ = 1, Theorem 1 gives ω f (k) = ω f (k; 1) ∼

1 1 1 1 , √ · = 4π d(2 (k), ) 4π 3 k

(2.4)

where d is the Euclidean distance. However, for ξ = 1, ω f (k; ξ ) decays exponentially to 0 or blows up exponentially, according as ξ > 1 or ξ < 1. Remark 2. The exponential behavior of ω f (k; ξ ) for ξ = 1 is in fact closely mirrored also in the setting of [6], where the correlation of holes was defined by including them at the center of large hexagons. Indeed, using arguments of [6], it follows that the correlation of two 2 ’s on the symmetry axis of the hexagons is exponential for ξ = 1. This has the interesting consequence that the limits of entries of the inverse Kasteleyn matrices of hexagons on the one hand and of “corresponding” tori on the other do not agree, not even at the center of the hexagons. To be more precise, note that the center of large hexagons is in the liquid regime of [22]. By [32, Ch. 8 and 9] (see also [22, Theorem 2.1]), there exists an ergodic Gibbs measure coming from (weighted) lozenge tilings on a large torus whose slope is the same as the slope of the limit shape at the center of the large hexagon. One might be tempted to guess that, in this situation, the entries of the inverse Kasteleyn matrix (which determine correlation) behave similarly in the limit. However, using [22, Theorem 4.3, (7), with P(z, w) = a + bz + cw and Q(z, w) = 1] and appropriate asymptotic expansions in the special case that we are interested in, it turns out that the correlation of two 2 ’s under any torus measure in the liquid regime behaves polynomially in the distance of the 2 ’s, in contrast to the exponential behavior for the hexagon mentioned above. This provides a concrete example highlighting the subtlety of taking limits of entries of the inverse Kasteleyn matrix: they depend quite sensitively on how the infinite plane is achieved as a limit of graphs. In [7] the first author described how a distribution of fixed holes on the triangular lattice defines in a natural way two vector fields. The F-field is a discrete vector field defined at the center of each left-pointing unit triangle e, and equal to the expected orientation of the lozenge covering e (under the uniform measure on the set of tilings). To define the T-field, one introduces an extra “test-hole” t and measures the relative change in the correlation function under small displacements of it, as the other holes are kept fixed. One can prove (details will appear elsewhere) that in the scaling limit of the lattice spacing approaching zero, this relative change is given by the scalar product of

258


the displacement vector with a certain vector T(z), where z is the point to which the test hole t shrinks when the lattice spacing approaches zero. This defines the second field. When these fields are generated by lozenge tilings that cover the entire plane with the exception of a finite collection of fixed-size holes (the case treated in [7] and [9]), both the T-field and the scaling limit of the F-field turn out to be equal, up to a constant multiple, to the electrostatic field of the two dimensional physical system obtained by viewing the holes as electrical charges. But what if we do not tile the entire plane, but only the half-plane to the left of the free boundary , and we have no holes? The above definitions for the F-field and T-field would still work, provided (i) the scaling limit of the discrete field defining F exists, and (ii) the scaling limit of the relative changes in the correlation function under small displacements of a test hole exists and is given by taking scalar products of the displacement vector with the vectors of a certain field. Our exact determination of ω f (k; ξ ) (see Lemma 13) allows us to verify (ii) for displacements along the horizontal direction. 2 (k) plays now the role of a test charge. The ω (k+1;ξ ) expression that measures the relative change in correlation in this case is ωf f (k;ξ ) − 1. What we have to do is to determine the leading term in the asymptotics of this expression as k → ∞. We obtain the following result, whose proof is given in Sect. 7. Theorem 2. We have

⎧ ⎨

4

2 − 1 + O k1 if ξ = 1, ω f (k + 1; ξ ) 1+ξ −1= ⎩−1 + O 1 ω f (k; ξ ) if ξ = 1, k k2

(2.5)

as k → ∞. Remark 3. In order to indicate the dependence on ξ (the asymptotic ratio of the sides of our half hexagon), we write Tξ for the T-field. By symmetry, displacements of 2 (k) parallel to leave ω f unchanged, so the relative change in ω f corresponding to such displacements is zero. Suppose that ξ = 1. Then, provided the field Tξ exists, it follows from Theorem 2 that its value at z is 4 2 e1 Tξ (z) = √ −1 , (2.6) 1+ξ 2 3 √ where e1 is the unit vector in the positive direction of the x-axis (the 2 3 at the denominator comes from the fact that Tξ arises from the√expression on the left-hand side of (2.5) divided by the product of the displacement, 3 in this case, and the “charge” of the hole 2 (k), which is 2; see [7] for details). In particular, the field Tξ is constant. On the other hand, if ξ = 1, then the second assertion in Theorem 2 yields e1 e1 . (2.7) T1 (z) = − √ =− 2 d(z, ) 2 3k Note that by [7] we would obtain (up to a multiplicative constant of 2) the same T-field at z if we look at tilings of the entire plane, with the mirror image of our test-hole 2 (k) being a fixed hole. This is analogous to the phenomenon in electrostatics by which the field created by an electric charge placed near a conductor can be obtained by the method of images (see e.g. [13, Chap. 6]).


259

The F-field could be determined by an “experiment” analogous to the one we described at the beginning of this section: simply replace 2 (k) by L(k), the horizontal lozenge contained in 2 (k). Recall that, by definition, the F-field at a left-pointing unit triangle is determined by the probabilities p1 , p2 , p3 that is occupied by a horizontal, northwest-pointing, or southwest-pointing lozenge, respectively. More precisely, (cf. [9]) F() = p1 e1 + p2 e2 + p3 e3 , where e1 , e2 , e3 are unit vectors parallel√ to the long diagonals of the above three loz√ enges, that is, e1 = (1, 0), e2 = (− 21 , 23 ), e3 = (− 21 , − 23 ). Hence, since p2 = p3 by symmetry and p1 + p2 + p3 = 1, it suffices to determine p1 , that is, the limit of the proportion of the number of lozenge tilings of Fn,x that contain L(k), as n and x go to infinity so that x/n approaches a fixed positive real number ξ . It turns out that, for fixed n, x, and k, the number of lozenge tilings of Fn,x \L(k) is given by a formula similar to (3.2), namely by the formula in Theorem 15. By lemmas that are analogous to Lemmas 12–14 (see Lemmas 16 and 17, and the text in between), one can then derive that the probability p1 is given by (see Corollary 18) p1 =

1 2 arctan √ . π ξ(2 + ξ )

(2.8)

Using standard formulas for trigonometric functions, it can be seen that this value agrees with the probability of finding a lozenge in the center of a random tiling of a hexagon with side-lengths 2n, 2n, 2x, 2n, 2n, 2x in the limit n, x → ∞ so that x/n approaches ξ , as given in [11, Conjecture 6.1] (with x = y = 0, α = γ = 1, β = ξ ), proved in [3, Theorem 3.12]. (In fact, in the special case that is relevant here, Conjecture 6.1 of [11] was proved earlier in [10, Cor. 4].) Thus, the free boundary has no disturbing effect at all on the lozenge statistics. Let Fξ denote the F-field for the above situation. Then the definition of the F-field and the above considerations imply the following result. Corollary 3. Let e(k) be the leftmost left-pointing unit triangle of 2 (k). Then Fξ (e(k)) =

3 p1 − 1 e1 , 2

(2.9)

where p1 is given by (2.8). Remark 4. In the case where ξ = 1, Eqs. (2.7) and (2.9) imply that, in sharp contrast to the case of lozenge tilings of the plane with a finite number of fixed size holes, where the T- and F-field are the same up to a constant multiple (cf. the second paragraph after Remark 2), for the half-plane with free boundary the fields T and F have radically different behavior: while in the scaling limit the former behaves as the electrostatic field near a conductor, the latter is zero. It is amusing that, aside from ξ = 1, there is precisely one other value of ξ where the field intensities in (2.6) and (2.9) agree. Figure 4 shows a plot of the two functions, with the intensity in (2.9) being the one approaching −1/2 for ξ → ∞. Numerically, this other value of ξ is 3.28262 . . .. Our approach to proving Theorems 1 and 2 consists of solving first the counting problem exactly, see Theorem 4. This result generalizes Andrews’ theorem [1] (which proved

260


0.6

0.4

0.2

1

2

3

4

5

0.2

0.4 Fig. 4. Plot of the field intensities Tξ and Fξ

MacMahon’s conjecture on symmetric plane partitions) in the case q = 1. Its proof is given in Sects. 4 and 5, with some auxiliary results proved separately in Sect. 6. It is based on the “exhaustion/identification of factors” method described in [25, Sect. 2.4]. In Sect. 7, we perform the asymptotic calculations needed to derive Theorems 1 and 2 from the exact counting results. The final section, Sect. 8, presents the results that are needed for the determination of the F-field Fξ reported in Corollary 3. 3. An Exact Tiling Enumeration Formula Tilings of the region Fn,x are clearly equivalent to tilings of the hexagon Hn,x that are invariant under reflection across its symmetry axis . Counting such tilings was a problem considered (in the equivalent form of symmetric plane partitions) by MacMahon in the early twentieth century (see [28, p. 270]). MacMahon conjectured that the number of vertically symmetric lozenge tilings of a hexagon with side-lengths 2n, 2n, 2x, 2n, 2n, 2x is equal to

n x + 21 2n (2x + 2s)4n−4s+1 , (3.1) 1

(2s)4n−4s+1 2 2n

s=1


261

Fig. 5. A symmetric lozenge tiling of the hexagon Hn,x with two holes

where (α)m is the Pochhammer symbol, defined by (α)m := α(α + 1) · · · (α + m − 1) for m ≥ 1, and (α)0 := 1. This was first proved by Andrews [1]. Other proofs, and refinements, were later found by e.g. Gordon [17], Macdonald [27, pp. 83–85], Proctor [31, Prop. 7.3], Fischer [14], and the second author of the present paper [23, Theorem 13]. Our “experiment” — counting M(Fn,x \ 2 (k)) — is by the same token equivalent to counting vertically symmetric lozenge tilings of Hn,x with two missing triangles (compare Figs. 1 and 5). This is in fact a generalization of MacMahon’s symmetric plane partitions problem (see Remark 5). The key result that allows deducing Theorems 1 and 2 is the following. Theorem 4. For all positive integers n, x and nonnegative integers k ≤ n − 1, we have M(Fn,x \ 2 (k)) n

4k + 1 (2x + 2s)4n−4s+1 (n + k)! = 2k (x + n − k)2k+1 (2s)4n−4s+1 s=1

×

n−k−1

( 21 )i

i=0

i! (n − k − i − 1)!2 (n + k − i + 1)n−k (n + k − i + 1)i (2n − i + 21 )i

· ((x)i (x + i + 1)n−k−i−1 (x + n + k + 1)n−k −(x)n−k (x + n + k + 1)n−k−i−1 (x + 2n − i + 1)i ) .

(3.2)

262


Fig. 6. Forced lozenges when the hole touches the left border

Remark 5. Replacing x by x − 1, n by n + 1, and k by n, one can see that the above formula specializes to MacMahon’s formula (3.1). More precisely, because of forced lozenges (see Fig. 6), the enumeration problem in the statement of Theorem 4 reduces to the problem of enumerating vertically symmetric lozenge tilings of a hexagon with side-lengths 2n, 2n, 2x, 2n, 2n, 2x. The proof of Theorem 4 is given in the next two sections. In Sect. 4, we show that M(Fn,x \ 2 (k)) can be expressed in terms of a certain Pfaffian. This Pfaffian is then evaluated in Sect. 5. 4. Lozenge Tilings and Nonintersecting Lattice Paths The purpose of this section is to find a manageable expression for M(Fn,x \ 2 (k)) (see Lemma 6 at the end of this section). In this context, we will find it more convenient to think of the tilings of Fn,x \ 2 (k) directly as tilings of a half hexagon with an open boundary (cf. Fig. 7) as opposed to symmetric tilings of a hexagon with two holes


263

Fig. 7. A lozenge tiling of the region Fn,x \ 2 (k); the right boundary is free. The dotted lines mark paths of lozenges. They determine the tiling uniquely

(cf. Fig. 5). There is a well known bijection between lozenge tilings of lattice regions and families of “paths of lozenges” (see Fig. 7), which in turn are equivalent to families of non-intersecting lattice paths (see Fig. 8). Its application to our situation is illustrated in Figs. 7 and 8. The origin of the system of coordinates indicated in Fig. 8 corresponds to the point O in Fig. 7 (note that the bottommost path of lozenges in Fig. 7 is empty for the illustrated tiling; the corresponding lattice path in Fig. 8 has no steps). By this bijection, lozenge tilings of Fn,x \ 2 (k) are seen to be equinumerous with families (P1 , P2 , . . . , P2n ) of non-intersecting lattice paths consisting of unit horizontal and vertical steps in the positive direction, where Pi runs from Ai = (−i, i) to some point from the set I ∪ {S1 , S2 }, i = 1, 2, . . . , 2n, with I = {(−1, s) : s = 1, 2, . . . , 2x + 2n}, S1 = (−2k − 1, x + n + k), S2 = (−2k − 2, x + n + k + 1),

(4.1)

and the additional condition that S1 and S2 must be ending points of some paths. At this point, we need a slight extension of Stembridge’s Theorem 3.2 in [34] (which is, in fact, derivable from the minor summation formula of Ishikawa and Wakayama [19, Theorem 2]). The reader should recall that the Pfaffian of a skew-symmetric 2n × 2n

264


Fig. 8. The paths of lozenges of Fig. 7 drawn as non-intersecting lattice paths on Z2

matrix A can be defined by (see e.g. [34, p. 102]) Pf A := sgn π π ∈M[1,...,2n]

Ai, j ,

(4.2)

i< j

i, j matched in π

where M[1, 2, . . . , 2n] denotes the set of all perfect matchings (1-factors) of (the complete graph on) {1, 2, . . . , 2n}, and where sgn π = (−1)cr(π ) , with cr(π ) denoting the number of “crossings” of π , that is, the number of quadruples i < j < k < l such that, under π , i is paired with k, and j is paired with l. It is a well-known fact (see e.g. [34, Prop. 2.2]) that (Pf A)2 = det A.

(4.3)

Theorem 5. Let {A1 , A2 , . . . , A p , S1 , S2 , . . . , Sq } and I = {I1 , I2 , . . . } be finite sets of lattice points in the integer lattice Z2 , with p + q even. Then q Q H = (−1) 2 Pf (sgn π ) · P nonint (Aπ → S ∪ I ), (4.4) −H t 0 π ∈Sp

where S p denotes the symmetric group on {1, 2, . . . , p}, Aπ = (Aπ(1) , Aπ(2) , . . . , Aπ( p) ), and P nonint (Aπ → S ∪ I ) is the number of families (P1 , P2 , . . . , Pp ) of nonintersecting lattice paths consisting of unit horizontal and vertical steps in the positive direction, with Pk running from Aπ(k) to Sk , for k = 1, 2, . . . , q, and to I jk , for k = q + 1, q + 2, . . . , p, the indices being required to satisfy jq+1 < jq+2 < · · · < j p . The matrix Q = (Q i, j )1≤i, j≤ p is defined by Q i, j =

P(Ai → Is ) · P(A j → It ) − P(A j → Is ) · P(Ai → It ) , (4.5) 1≤s q + |I |, there are more starting points than available ending points. However, Theorem 5 still holds: then the right-hand side of (4.4) is clearly zero, and the Pfaffian on the left-hand side follows to be zero by the above indicated arguments that prove Theorem 5. We now apply Theorem 5 to our situation, that is, p = 2n, q = 2, Ai = (−i, i), for i = 1, 2, . . . , 2n, and S1 , S2 , and I are given by (4.1). It is not difficult to convince oneself that, for this choice of starting and ending points, all families of nonintersecting lattice paths counted on the right-hand side of (4.4) give rise to even permutations π . Hence, the right-hand side of (4.4) indeed counts the families of nonintersecting lattice paths that we need to count. By Theorem 5, their number is equal to the negative value of the Pfaffian of Q H , (4.6) Mn (x) := −H t 0 where Q is a (2n) × (2n) skew-symmetric matrix with (i, j)-entry Q i, j given by (4.5), and where H is a (2n) × 2 matrix, in which the (i, j)-entry Hi, j is equal to the number of paths from Ai to S j , i = 1,2, . . . , 2n, j = 1, 2. (It is the negative value of the Pfaffian q

because of the sign (−1) 2 on the right-hand side of (4.4), as we have q = 2.) In particular, using the fact that the number of lattice paths on the integer lattice Z2 between two given lattice points is given by a binomial coefficient, we have x +n−k−1 , (4.7) Hi,1 = i − 2k − 1 x +n−k−1 Hi,2 = . (4.8) i − 2k − 2 On the other hand, substituting Ai = (−i, i) and Is = (−1, s) in (4.5), we have Q i, j = (P(Ai → Is ) · P(A j → It ) − P(A j → Is ) · P(Ai → It )) 1≤s i − 2 ≥ 2k + 1 ≥ 2n − 2s + 1 > 2n − 2s, by our assumptions. To see that the left-hand side of (5.20) is as well zero when it is specialized to the (2n + 1)st or (2n + 2)nd column amounts to a routine verification using the expressions (4.7) and (4.8) for the corresponding matrix entries. We now assume that s < n − k and turn our attention to (5.16). The reader should notice that the relations (5.16) and (5.18) are relatively similar, the essential difference being the two extra terms in (5.16) corresponding to the (2n −2s)th and the (2n −2s +1)st row, respectively. If 1 ≤ j ≤ 2n, the proof of relation (5.16) specialized to column j is therefore identical with the proof of relation (5.18) specialized to column j, because the entries in the first 2n columns of the (2n − 2s)th and the (2n − 2s + 1)st row evaluated at x = −s − 21 are all zero. (The reader should recall formula (4.10).) To show the relation (5.16) specialized to the (2n +1)st respectively to the (2n +2)nd column requires however more work. We have to prove 1 n − s − k − 23 n − s − k − 23 + (n − s − k) 2n − 2s − 2k − ε + 1 2 2n − 2s − 2k − ε 2n−2s−1 (−1)i n − s − k − 23 = 0, + 2n−2s−i+2 i − 2k − ε 2 i=1

where ε = 1, 2, respectively, after simplification, (n − s − k)(ε − 2) (−n + s + k + ε − 21 )2n−2s−2k−ε · 2 (2n − 2s − 2k − ε + 1)! 2n−2s−1 (−1)i n − s − k − 23 = 0. + i − 2k − ε 22n−2s−i+2

(5.22)

i=1

We reverse the order of summation in the sum over i (that is, we replace i by 2n − 2s − i − 1), and subsequently we write the (new) sum over i in standard hypergeometric notation ∞ (a1 )m · · · (a p )m m a1 , . . . , a p ;z = z . (5.23) p Fq b1 , . . . , bq m! (b1 )m · · · (bq )m m=0


275

Thereby we obtain (n − s − k)(ε − 2) (−n + s + k + ε − 21 )2n−2s−2k−ε · 2 (2n − 2s − 2k − ε + 1)! (−n + k + s + ε + 21 )2n−2s−2k−ε−1 1, −2n + 2k + 2s + ε + 1 1 (5.24) − F ; 2 1 −n + k + s + ε + 21 8 (2n − 2s − 2k − ε − 1)! 2 for the left-hand side of (5.22). If ε = 2, then the 2 F1 -series in (5.24) can be evaluated using Gauß’ second 2 F1 -summation (cf. [33, (1.7.1.9); App. (III.6)]) ⎧ if N is an odd nonnegative integer, ⎪ ⎨0 1 1 a, −N = 2 N/2 (5.25) 2 F1 1 a N ; ifN is an even nonnegative integer. ⎪ 2 + 2 − 2 2 ⎩ 1−a 2

2 N /2

As a result, in this case, the expression (5.24) vanishes, whence (5.22) with ε = 2 is satisfied, and thus relation (5.16) specialized to the (2n + 2)nd column. If ε = 1, the 2 F1 -series in (5.24) cannot be directly evaluated by means of Gauß’ formula. However, we may in a first stage apply the contiguous relation az a, b a, b − 1 a + 1, b ; z = 2 F1 ;z + ;z 2 F1 2 F1 c c c+1 c to transform (5.24) into (n − s − k) (−n + s + k + 21 )2n−2s−2k−1 · 2 (2n − 2s − 2k)! 3 (−n + k + s + 2 )2n−2s−2k−2 1, −2n + 2k + 2s + 1 1 − ; 2 F1 −n + k + s + 23 8 (2n − 2s − 2k − 2)! 2 1 2, −2n + 2k + 2s + 2 1 . − ; 2 F1 −n + k + s + 25 2n − 2k − 2s − 3 2

−

Both 2 F1 -series in the last expression can now be evaluated by means of Gauß’ formula (5.25). The first series simply vanishes, while the second series evaluates to a non-zero expression. If this is substituted, after some simplifications we obtain −

(−n + k + s + 23 )2n−2s−2k−2 (−n + s + k + 21 )2n−2s−2k−1 − = 0. 4 (2n − 2s − 2k − 1)! 8 (2n − 2s − 2k − 2)!

This shows that for ε = 1 the expression (5.24) vanishes as well, whence (5.22) with ε = 1 is satisfied, and thus also relation (5.16) specialized to the (2n + 1)st column. The verification of (5.17) is completely analogous to that of (5.20) and is left to the reader. Step 4. Pf Mn (x) is a polynomial in x of degree at most 2n 2 + n − 4k − 3. By (4.13), Q i, j is a polynomial in x of degree i + j. On the other hand, by recalling the definitions (4.7) and (4.8) of Hi,1 and Hi,2 , respectively, one sees that the degree of Hi,1 in x is i − 2k − 1,

276


while the degree of Hi,2 is i − 2k − 2. It follows that, in the defining expansion of the determinant det Mn (x), each nonzero term has degree 2n

i+

2n

i=1

j − 2(2k + 1) − 2(2k + 2) = 4n 2 + 2n − 8k − 6.

j=1

The Pfaffian being the square root of the determinant (cf. (4.3)), the claim follows. Step 5. Evaluation of Pn (x) at x = 0, −1, . . . , −n + k + 1. The polynomial Pn (x) is defined by means of (5.4). So, what we would like to do is to set x = −σ in (5.4), σ being one of 0, 1, . . . , n − k − 1, evaluate Pf Mn (−σ ), divide both sides of (5.4) by the products on the right-hand side of (5.4), and get the evaluation of Pn (x) at x = −σ . However, the first product on the right-hand side of (5.4) unfortunately is zero for x = −σ , 1 ≤ σ ≤ n − k − 1. (It is not zero for σ = 0.) Therefore we have to find a way around this difficulty. Fix a σ with 1 ≤ σ ≤ n − k − 1. Before setting x = −σ in (5.4), we have to cancel (x + σ )σ (see (5.13)) on both sides of (5.4). That is, we should write (5.4) in the form Pn (x) = − ×

1 Pf Mn (x) (x + σ )σ n−k−1

n

s=1

s=n−k+1

(x + s)−s

s =σ

×

n

s=1

1 x +s+ 2

−1

2n−1

(x + s)−s+1

(x + s)−2n+s−χ (s>n+k)+1

s=n+1

,

(5.26)

2n−2s

and subsequently specialize x = −σ . However, in order to be able to perform this step, we need to evaluate 1 − Pf Mn (x) . x=−σ (x + σ )σ In order to accomplish this, we apply Lemma 11 with N = 2n +2, a = 2n −2σ , b = 2n, and A = Mn (x). Indeed, (x + σ ) is a factor of each entry in the i th row in matrix Mn (x), for i = 2n − 2σ + 1, 2n − 2σ + 2, . . . , 2n. We obtain 1 Pf(S), Pf M (x) = − Pf( Q) (5.27) − n (x + σ )σ x=−σ where = Q with Q being given by

and H by

Q t H

H 0

,

Q = Q i, j |x=−σ 1≤i, j≤2n−2σ

H = Hi, j |x=−σ 1≤i≤2n−2σ, 1≤ j≤2 ,

(5.28)


and where

S=

277

1 . Q i+2n−2σ, j+2n−2σ x +σ x=−σ 1≤i, j≤2σ

We point out that (5.27) also holds for σ = 0 once we interpret the Pfaffian of an empty matrix (namely the Pfaffian of S) as 1. In particular, under that convention, the arguments below can be used for 0 ≤ σ ≤ n − k − 1, that is, including σ = 0. and Pf(S). We start with the evaluation of Pf(S). It We must now compute Pf( Q) follows from (4.13) that the (i, j)-entry of S is given by Si, j =

i+2n−2σ −1

(−1)l+ j+1

l=0

j −i i + 2n − 2σ

j + 2n − 2σ − 1 i + 2n − 2σ − l − 1

l + j + 2n − 2σ l

(2n − 2σ + 1)! (l + j − 1)! . · (l + j + 2n − 2σ + 1)! If we write this using hypergeometric notation, we obtain the alternative expression Si, j =

(−1) j+1 ( j − i)i+2n−2σ 1 − i − 2n + 2σ, 1 + j + 2n − 2σ, j ; 1 . F 3 2 1 − i + j, 2 + j + 2n − 2σ (2n − 2σ + j + 1)! ( j)i− j+2n−2σ +1

Rewrite this expression as the limit (−1) j+1 ( j − i)i+2n−2σ 1 − i − 2n + 2σ, 1 + j + 2n − 2σ, j ; 1 . F 3 2 1 − i + j, 2 + ε + j + 2n − 2σ ε→0 (2n − 2σ + j + 1)! ( j)i− j+2n−2σ +1

Si, j = lim

Now we apply one of Thomae’s 3 F2 -transformation formulas (cf. [4, Ex. 7, p. 98]) (e) (d + e − a − b − c) a, b, c a, −b + d, −c + d ;1 = ;1 . 3 F2 3 F2 d, e d, −b − c + d + e (e − a) (d + e − b − c) Thus, we obtain (−1) j+1 (2n − 2σ + ε + 1) (2n − 2σ + j + ε + 2) ( j − i)i+2n−2σ ε→0 (ε − i + 2) (4n − 4σ + i + j + ε + 1) (2n − 2σ + j + 1)! ( j)i− j+2n−2σ +1 1 − i − 2n + 2σ, −i − 2n + 2σ, 1 − i ×3 F2 ;1 , 1 − i + j, 2 + ε − i

Si, j = lim

or, in usual sum notation, i−1 (−1) j+1 ( j − i) (2n − 2σ + ε + 1) (2n − 2σ + j + ε + 2) ε→0 (l − i + ε + 2) (4n − 4σ + i + j + ε + 1)

Si, j = lim

l=0

·

(1 − i)l (l − i + j + 1)2n−2σ +i−l−1 (2n − 2σ + i − l)l . l! (2n − 2σ + j + 1)! ( j)2n−2σ +i− j−l+1

Because of the term (l − i + ε + 2) in the denominator, in the limit only the summand for l = i − 1 does not vanish. After simplification, this leads to Si, j =

(−1)i+ j ( j − i) (2n − 2σ + i − 1)! (2n − 2σ + j − 1)! . (4n − 4σ + i + j)! (2n − 2σ + 1)!

278


a

b

Fig. 9. a A lozenge tiling for the degenerate region. b Forced lozenges in case x = 0

We must evaluate the Pfaffian Pf

1≤i, j≤2σ

(Si, j ).

By factoring some terms out of rows and columns, we see that Pf

1≤i, j≤2σ

(Si, j ) = (−1)σ (2n − 2σ + 1)!−σ ×

2σ

(2n − 2σ + i − 1)!

i=1

Pf

1≤i, j≤2σ

j −i . (4n − 4σ + i + j)!

This Pfaffian can be evaluated in closed form by Corollary 10 in the next section. The result is that 2σ

σ −σ Pf(S) = (−1) (2n − 2σ + 1)! (2n − 2σ + i − 1)! ×

σ −1

i=0

i=1

(2i + 1)! . (4n − 2σ + 2i + 1)!

(5.29)

If we compare (5.28) with (4.6), then We finally turn to the evaluation of det( Q). we see that Q = Mn−σ (0). Hence, using Lemma 6 with n replaced by n − σ and with is equal to M(Fn−σ,0 \ 2 (k)). (The reader should recall the x = 0, we see that − Pf( Q) definitions of the region Fn,x and of the triangular hole 2 (k) given in the Introduction, see again Fig. 7.) Fig. 9a shows a typical example where n − σ = 5 and k = 2. Since this region is degenerate, there are many forced lozenges, see Fig. 9b. The enumeration problem therefore reduces to the problem of determining the number of symmetric lozenge tilings of a hexagon with side-lengths 2k, 2k, 2, 2k, 2k, 2. This number is given by formula (3.1) with n = k and x = 1. If we substitute this in (5.27), together with the evaluation (5.29), then, after some manipulation, we obtain


279

(2σ )! 1 4k + 1 σ Pf M (x) = (−1) n σ 2k (x + σ ) (2n − 2σ + 1)!σ 2σ σ ! x=−σ σ −1 2σ

(2i)! (2n − 2σ + i − 1)! × . (4n − 2σ + 2i + 1)!

−

i=1

i=0

Hence, by inserting this in (5.26), we have (2σ )! 4k + 1 σ Pn (−σ ) = (−1) 2k (2n − 2σ + 1)!σ 2σ σ ! 2σ σ −1

(2i)! × (2n − 2σ + i − 1)! (4n − 2σ + 2i + 1)! i=1

×

i=0

n−k−1

(−σ + s)−s

s=1 n

s=1

(−σ + s)−s+1

2n−1

(−σ + s)−2n+s−χ (s>n+k)+1

s=n+1

s=n−k+1

s =σ

×

n

1 (−σ + s + )−1 . 2 2n−2s

(5.30)

This completes the proof of Theorem 4. 6. An Auxiliary Determinant Evaluation, and an Auxiliary Pfaffian Factorization Mehta and Wang proved the following determinant evaluation in [29]. (There is a typo in the formula stated in [29, Eq. (7)] in that the binomial coefficient nk is missing there.) Theorem 7 ([29, Eq. (7)]). For all real numbers a, b and positive integers n, we have det

0≤i, j≤n−1

=

n−1

i=0

((a + j − i) (b + i + j)) i! (b + i)

n k=0

(−1)k

n ((b − a)/2)k ((b + a)/2)n−k , k

(6.1)

as long as the arguments occurring in the gamma functions avoid their singularities. The sum on the right-hand side of (6.1) can be alternatively expressed as the coefficient of z n /n! in (1 + z)(a−b)/2 (1 − z)(−a−b)/2 . Therefore, in the case a = 0 we obtain the following simpler determinant evaluation. Corollary 8. For all real numbers b and positive integers n, we have n−1

n! (b/2)n/2 , i! (b + i) det (( j − i) (b + i + j)) = χ (n is even) 0≤i, j≤n−1 (n/2)! i=0

as long as the arguments occurring in the gamma functions avoid their singularities. Here, as before, χ (A) = 1 if A is true and χ (A) = 0 otherwise.

280


One can obtain the following slightly (but, for our purposes, essentially) stronger statement. It is stated as Eq. (4) in [29], with the argument how to obtain it hinted at at the bottom of p. 231 of [29]. Since, from there, it is not completely obvious how to actually complete the argument, we provide a proof. Proposition 9. For all real numbers b and positive even integers n, we have n 2 −1

Pf

0≤i, j≤n−1

(( j − i) (b + i + j)) =

(2i + 1)! (b + 2i + 1),

(6.2)

i=0

as long as the arguments occurring in the gamma functions avoid their singularities. Proof. Since the Pfaffian of a skew-symmetric matrix equals the square root of its determinant (cf. (4.3)), the formula given by Theorem 8 yields, after a little manipulation, that n 2 −1

Pf

0≤i, j≤n−1

(( j − i) (b + i + j)) = ε

(2i + 1)! (b + 2i + 1),

(6.3)

i=0

where ε = +1 or ε = −1. In order to determine the sign ε, we argue by induction on (even) n. Let us suppose that we have already proved (6.2) up to n − 2. We now multiply both sides of (6.3) by b + 1 and then let b tend to −1. Thus, on the right-hand side we obtain the expression ⎛n ⎞ ⎛n ⎞ −1 −1 2 2 ε⎝ (2i + 1)!⎠ ⎝ (2i)⎠ . (6.4) i=0

i=1

On the other hand, by the definition of the Pfaffian, on the left-hand side we obtain ⎛ ⎞ π ∈M[0,...,n−1]

⎜ (b + 1) sgn π lim ⎜ b→−1 ⎝

i< j

⎟ ( j − i) (b + i + j)⎟ ⎠

(6.5)

i, j matched in π

(with the obvious meaning of M[0, . . . , n −1]; cf. the sentence containing (4.2)). In this sum, matchings π for which all matched pairs i, j satisfy i + j > 1 do not contribute, because the corresponding summands vanish. However, there is only one possible pair i, j with 0 ≤ i < j for which i + j ≤ 1, namely (i, j) = (0, 1). Therefore, the sum in (6.5) reduces to

sgn π

π ∈M[2,...,n−1]

= =

lim (b + 1)(1 − 0) (b + 1)

b→−1

Pf

(( j − i) (i + j − 1))

Pf

(( j − i) (i + j + 3)) ,

2≤i, j≤n−1 0≤i, j≤n−3

( j −i) (i + j − 1)

i< j

i, j matched in π

where the next-to-last equality holds by the definition (4.2) of the Pfaffian. Now we can use the induction hypothesis to evaluate the last Pfaffian. Comparison with (6.4) yields that ε = +1.


281

By using the reflection formula (cf. [2, Theorem 1.2.1]) (x) (1 − x) =

π sin π x

for the gamma function, and the substitutions i → n − i − 1 and j → n − j − 1, it is not difficult to see that Proposition 9 is equivalent to the following. Corollary 10. For all positive even integers n, we have Pf

0≤i, j≤n−1

j −i (b + i + j)

n 2 −1

=

i=0

(2i + 1)! . (b + n + 2i − 1)

We close this section by proving a factorization of a certain specialization of a Pfaffian that we need in Step 5 in Sect. 5. Lemma 11. Let N , a, b be positive integers with a < b ≤ N , where N and b − a are even. Let A = (Ai, j )1≤i, j≤N be a skew-symmetric matrix with the following properties: (1) The entries of A are polynomials in x. (2) The entries in rows a + 1, a + 2, . . . , b (and, hence, also in the corresponding columns) are divisible by x + σ . Then

1 · Pf S, Pf A = Pf A (b−a)/2 (x + σ ) x=−σ

(6.6)

is the matrix which arises from A by deleting rows and columns a + 1, a + where A 2, . . . , b and subsequently specializing x = −σ , and S=

1 Ai, j . x +σ x=−σ a+1≤i, j≤b

Proof. By the definition (4.2) of the Pfaffian, we have

1 Pf A x=−σ (x + σ )(b−a)/2 ⎛ ⎜ =⎜ ⎝

1 (x + σ )(b−a)/2

π ∈M[1,...,N ]

sgn π

i< j

i, j matched in π

⎞ ⎟ ⎟ Ai, j ⎠

. x=−σ

Let M1 denote the subset of M[1, . . . , N ] consisting of those matchings that pair all the elements from {a + 1, a + 2, . . . , b} among themselves (and, hence, all the elements of the complement {1, 2, . . . , a, b + 1, b + 2, . . . , N } among themselves). Let M2 be the complement M[1, . . . , N ]\M1 . Then

282


1 Pf A (b−a)/2 (x + σ ) x=−σ ⎛

⎞

⎟ ⎜ 1 ⎟ =⎜ sgn π A i, j ⎠ ⎝ (x + σ )(b−a)/2 i< j π ∈M1 i, j matched in π x=−σ ⎞ ⎛

⎟ ⎜ 1 ⎟ ⎜ +⎝ sgn π Ai, j ⎠ . (x + σ )(b−a)/2 i< j π ∈M2 i, j matched in π

(6.7)

x=−σ

Each term in the third line of (6.7) vanishes, since the product contains more than (b − a)/2 factors Ai, j that are divisible by x + σ . On the other hand, every matching π in M1 is the disjoint union of a matching π ∈ M[1, 2, . . . , a, b + 1, b + 2, . . . , N ] and a matching π

∈ M[a + 1, a + 2, . . . , b]. If we also use the simple fact that sgn π = sgn π · sgn π

(as there are no crossings between paired elements of π and paired elements of π

), then we obtain 1 Pf A (b−a)/2 (x + σ ) x=−σ ⎛ ⎜ 1 =⎜ ⎝ (x + σ )(b−a)/2

π ∈M[1,...,a,b+1,...,N ] π

∈M[a+1,...,b]

i< j

i, j matched in π

=

⎞⎞

⎟⎜ ⎟⎟ ⎜ ⎟ ⎟ Ai, j ⎟ A i, j ⎠⎠ ⎠⎝ i< j i, j matched in π

x=−σ

sgn π

Ai, j

π ∈M[1,...,a,b+1,...,N ]

·

sgn π · sgn π

⎞⎛

⎛ ⎜ ·⎜ ⎝

π

∈M[a+1,...,b]

sgn π

x=−σ

i< j

i, j matched inπ

i< j

1 Ai, j . x +σ x=−σ

i, j matched in π

By the definition (4.2) of the Pfaffian, the last expression is exactly the right-hand side of (6.6).

7. Proofs of Theorems 1 and 2 In our proofs we make use of the following lemmas. Lemma 12. Let β be a real number with either β > 0 or β < −1. Then, for fixed positive k and all sequences (βn )n≥1 with βn → β as n → ∞, we have


283

1 −2n, 21 , −n + k + 1, −n + k + 1, βn n ; 1 lim √ 5 F4 n→∞ n −2n + 21 , −n − k, −n − k, βn n + 1 √ 1 2 (1 − α)4k+2 dα, = α √ 0 (1 + β ) π α(2 − α)

(7.1)

where, on the left-hand side, we used again the standard notation (5.23) for hypergeometric series. Proof. We write the 5 F4 -series in (7.1) explicitly as a sum over l: n−k−1 l=0

(2n + 1) (l + 21 ) (2n − l + 21 ) (n − k)2 (n + k − l + 1)2

βn n

(2n − l + 1) ( 21 ) (l + 1) (2n + 21 ) (n − k − l)2 (n + k + 1)2 (βn n + l)

.

(7.2) Let us denote the summand in this sum by F(n, l). We have ∂ 1 1 F(n, l) = F(n, l) ψ(l + ) − ψ(l + 1) + ψ(2n − l + 1) − ψ(2n − l + ) ∂l 2 2 1 , + 2ψ(n − k − l) − 2ψ(n + k − l + 1) − βn n + l where ψ(x) := ( ddx (x))/ (x) is the digamma function. Because of the functional equation ψ(x + 1) = ψ(x) + x1 (cf. [2, Eq. (1.2.15) with n = 1]), we have 1 1 ψ(l + 1) − ψ(l + ) ≥ ψ(2n − l + 1) − ψ(2n − l + ) 2 2 for 0 ≤ l ≤ n. Moreover, since either β > 0 or β < −1, for large enough n we have ψ(n + k − l + 1) ≥ ψ(n − k − l) +

1 1 > ψ(n − k − l) − . n +k −l βn n + l

Altogether, this implies that ∂l∂ F(n, l) < 0 for 0 ≤ l ≤ n − k − 1, that is, for fixed large enough n, the summand F(n, l) is monotone decreasing as a function in l. In particular, for 0 ≤ l ≤ n − k − 1 we have 0 < F(n, l) ≤ F(n, 0) = 1.

(7.3)

The sum (7.2) may therefore be approximated by an integral: n−k−1 l=0

log n−1

F(n, l) =

n−k−log n−1

F(n, l) +

l=0

= O(log n) +

F(n, l) +

l=log n n−k−log n−1

log n−1

F(n, l) d l,

n−k−1

F(n, l)

l=n−k−log n

as n → ∞.

The next step is to apply Stirling’s approximation 1 1 1 log(z) − z + log(2π ) + O log (z) = z − 2 2 z

(7.4)

(7.5)

284


for the gamma function, in the form l c 1 log a +b +log(n)+log 1+ log (an+bl +c) = an+bl +c − 2 n an+bl 1 1 −(an + bl + c) + log(2π ) + O 2 an + bl l 1 (log(a + b ) + log(n)) = an + bl + c − 2 n 1 1 , −(an + bl) + log(2π ) + O 2 an + bl where a, b, c are real numbers with a ≥ 0. If this is used in the defining expression for F(n, l), then after cancellations we obtain 1 l 1 l log(2) + (4k + 2) log 1 − − log 2 − 2 n 2 n l 1 1 1 l − log − log(n) − log(π ) − log 1 + 2 n 2 2 βn n 1 1 1 1 +O +O +O +O l n −l 2n − l n ⎛ ⎞ √ 2 (1 − nl )4k+2 1 ⎝ ⎠ = log √ , +O log n n(1 + βnl n ) π nl (2 − nl )

log F(n, l) =

as long as log n ≤ l ≤ n − k − log n. Substitution of this approximation in (7.4) yields n−k−1 l=0

F(n, l)

⎛

=⎝

n−k−log n−1

log n−1

⎞ 2 (1 − nl )4k+2 1 ⎠ 1+O + O (log n) , dl √ log n n(1 + βnl n ) π nl (2 − nl ) √

or, after the substitution l = αn, n−k−1

F(n, l)

l=0

√

= n

(n−k−log n−1)/n (log n−1)/n

√ 1 2 (1 − α)4k+2 1+ O + O (log n). dα √ log n (1 + β1n α) π α(2 − α)

The assertion of the lemma follows now immediately. We can now get an exact formula for the correlation ω f (k; ξ ) defined in (2.2).


285

Lemma 13. For any ξ > 0 and 0 ≤ k ∈ Z, we have 1 1 4k + 1 ω f (k; ξ ) = √ 4k+2 2k π (1 + ξ ) ξ(2 + ξ ) 1 1 4k+2 (1 − α) (1 − α)4k+2 dα − ξ dα × (2 + ξ ) α √ α √ 0 (1 + ξ ) α(2 − α) 0 (1 − 2+ξ ) α(2 − α) 1 1 1 4k + 1 2 (1 − α)4k+3 = dα. (7.6) √ α α √ 2k π (1 + ξ )4k+2 ξ(2 + ξ ) 0 (1 + ξ ) (1 − 2+ξ ) α(2 − α) Proof. By Theorem 4 and formula (3.1), the ratio between M(Fn,x \ 2(k)) and M(Fn,x ) is, when written in hypergeometric notation, (n + k)! ( 21 )2n (x + 1)n−k−1 (x + n + k + 1)n−k−1 4k + 1 1 2k (x + n − k)2k+1 (x + 2 )2n (n − k − 1)!2 (n + k + 1)n−k −2n, 21 , −n + k + 1, −n + k + 1, x ; 1 × (x + 2n) 5 F4 −2n + 21 , −n − k, −n − k, x + 1 −2n, 21 , −n + k + 1, −n + k + 1, −2n − x ;1 . × x 5 F4 −2n + 21 , −n − k, −n − k, −2n − x + 1 We now substitute x = ξn n in this expression. Use of Lemma 12 (which applies, as ξ > 0), together with Stirling’s formula (7.5), yields the assertion. Lemma 14. For any β = 0 we have 1 (1 − α)4k+2 π , k → ∞. dα ∼ α √ 8k 0 (1 + β ) α(2 − α)

(7.7)

Proof. Let Iβ (k) be the integral on the left-hand side of (7.7). The asymptotics of Iβ (k) as k → ∞ can be readily found using Laplace’s method as presented for instance in [30]. Conditions (i)–(v) of [30, pp. 121–122] are readily checked. By [30, Theorem 6.1, b p. 125], the large z asymptotics of a e−zp(t) q(t) dt is determined by the quantities λ, μ, p0 and q0 in the series expansions p(t) − p(a) = p0 (t − a)μ + p1 (t − a)μ+1 + · · · and q(t) = q0 (t − a)λ + q1 (t − a)λ+1 + · · · . Namely, under the above assumptions one has e

zp(a)

b

e a

−zp(t)

λ/μ λ q0 /(μp0 ) 1 q(t) dt = + O λ/μ+1 . μ z λ/μ z

(7.8)

In the case of Iβ (k) we have p(t) = − ln(1 − t), q(t) = (1−t/β)1√t (2−t) , a = 0, and b = 1. √ These yield parameters λ = 1/2, μ = 1, p0 = 1, and q0 = 1/ 2. In addition, p(a) = 0. As in our case z = 4k + 2, under these specializations (7.8) becomes (7.7).

286


Proof of Theorem 1. Combine Lemmas 13 (first expression) and 14 with Stirling’s

(4k+2) approximation (7.5) for the binomial coefficient 4k+1 =

2k (2k+1) (2k+2) in (7.6). Proof of Theorem 2. The case ξ = 1 follows directly from Theorem 1. From now on, let ξ = 1. Set Dk := 3I1 (k) − I−3 (k), where Iβ (k) denotes the integral on the left-hand side of (7.7) with ξ = 1. Using the earlier notation ω f (k) = ω f (k; 1), we have by Lemma 13 that ω f (k + 1) − ω f (k) ! (4k + 3)(4k + 5) 1 1 4k + 1 = − 1 Dk+1 + (Dk+1 − Dk ) , √ 2k π 24k+2 3 4(2k + 2)(2k + 3) and thus ω f (k + 1) − ω f (k) (4k + 3)(4k + 5) Dk+1 Dk+1 − Dk = −1 + . ω f (k) 4(2k + 2)(2k + 3) Dk Dk

(7.9)

By two applications of Lemma 14 it follows that √ π Dk ∼ √ , k → ∞. 2k

(7.10)

Thus Dk+1 /Dk → 1 as k → ∞, and elementary arithmetics implies that the first term on the right-hand side of (7.9) is asymptotically −1/(2k) as k → ∞. To determine the asymptotics of the second term, write by Lemma 13 " # Dk+1 − Dk = 3 [I1 (k + 1) − I1 (k)] − I−3 (k + 1) − I−3 (k) . (7.11) As Iβ (k) is the integral on the left-hand side of (7.7), we have Iβ (k + 1) − Iβ (k) =

1 0

$ % (1 − α)4k+2 √ (1 − α)4 − 1 dα. 1 + βα α(2 − α)

(7.12)

The asymptotics of the integral in (7.12) follows by Laplace’s method, in the same manner as the proof of Lemma 14. In this case λ = 3/2, μ = 1, and Eqs. (7.8) and (7.12) impliy that √ π Iβ (k + 1) − Iβ (k) ∼ √ , k → ∞. 4 2k 3/2

(7.13)

Equations (7.11) and (7.13) determine the asymptotics of Dk+1 − Dk , and combining this with the asymptotics of Dk given by (7.10) we obtain that the second term on the right-hand side of (7.9) has asymptotics −1/(2k) as k → ∞. The two terms on the righthand side of (7.9) thus have a sum that is asymptotically −1/(2k) − 1/(2k) = −1/k, and Theorem 2 is proved.


287

8. Lozenge Occupation Probability for a Free Boundary This section contains the results that are relevant for the calculation of the F-field Fξ reported in Sect. 2. Since the proofs are very similar to those of Theorem 4 and Lemmas 12–14, here we only give a brief outline of how to derive these results. By a method completely analogous to the one used in the proof of Theorem 4 given in Sect. 5, one can derive the following theorem. Recall that Fn,x is the half hexagon with side-lengths√2n, 2x, 2n and that L(k) is the horizontal lozenge on its symmetry axis at distance k 3 from the free boundary. Theorem 15. For all positive integers n, x and nonnegative integers k ≤ n − 1, we have M(Fn,x \L(k)) =

n

(2x + 2s)4n−4s+1 (2s)4n−4s+1

s=1

×

n−k−1

( 21 )i

i=0

i! (2n − i)! (2n − i + 21 )i

((x)i (x + i + 1)2n−i − (x)2n−i (x + 2n − i + 1)i ) . (8.1)

The sum in (8.1) can be written as a difference of two hypergeometric series, which turn out to be 3 F2 -series. For the asymptotic analysis of these 3 F2 -series, we need the following counterpart of Lemma 12. Lemma 16. Let β be a real number with either β > 0 or β < −1. Then, for fixed positive k and all sequences (βn )n≥1 with βn → β as n → ∞, we have √ 1 1 2 −2n, 21 , βn n lim √ 3 F2 ;1 = dα, (8.2) √ 1 α n→∞ n −2n + 2 , βn n + 1 0 (1 + β ) π α(2 − α) where, on the left-hand side, we used again the standard notation (5.23) for hypergeometric series. Using Stirling’s formula (7.5) and the above lemma, it is straightforward to determine the asymptotics of (8.1) as n and x tend to infinity so that x/n approaches ξ . We obtain the following counterpart of Lemma 13. Lemma 17. For any ξ > 0, any sequence (ξn )n≥1 with limn→∞ ξn = ξ and ξn n ∈ Z, and 0 ≤ k ∈ Z, we have M(Fn,ξn n \L(k)) M(Fn,ξn n ) 1 1 1 dα = √ −ξ (2 + ξ ) α √ π ξ(2 + ξ ) 0 (1 + ξ ) α(2 − α) 0 (1 − 1 2(1 − α) 1 dα. = √ α α √ π ξ(2 + ξ ) 0 (1 + ξ ) (1 − 2+ξ ) α(2 − α)

lim

n→∞

dα α √ 2+ξ ) α(2 − α)

It is a routine matter to check that & & dα ξ α(2 + ξ ) arctan . =2 α √ π(2 + ξ ) (2 − α)ξ (1 + ξ ) α(2 − α)

(8.3)

(8.4)

288


Corollary 18. For any ξ > 0, any sequence (ξn )n≥1 with limn→∞ ξn = ξ and ξn n ∈ Z, and 0 ≤ k ∈ Z, we have lim

n→∞

M(Fn,ξn n \L(k)) 2 1 = arctan √ . M(Fn,ξn n ) π ξ(2 + ξ )

(8.5)

Proof. From Lemma 17 (first expression) and (8.4), we obtain & & M(Fn,ξn n \L(k)) 2 2+ξ ξ = − arctan arctan . lim n→∞ M(Fn,ξn n ) π ξ 2+ξ The expession in (8.5) then follows by standard formulas for the arctangent function.

Acknowledgements. We are grateful to the referee for a very careful reading of the original manuscript and many helpful suggestions on the presentation of the material.

References 1. Andrews, G.E.: Plane partitions I: The MacMahon conjecture. In: Studies in foundations and combinatorics, G.-C. Rota, ed., Adv. in Math. Suppl. Studies, Vol. 1, New York London: Academic Press, pp. 131–150, 1978 2. Andrews, G.E., Askey, R.A., Roy, R.: Special functions. In: Encyclopedia of Math. And Its Applications 71, Cambridge: Cambridge University Press, 1999 3. Baik, J., Kriecherbauer, T., McLaughlin, K.T.-R., Miller, P.D.: Discrete orthogonal polynomials. In: Asymptotics and applications Ann. Math. Studies, Princeton, NJ: Princeton University Press, 2007 4. Bailey, W.N.: Generalized hypergeometric series. Cambridge: Cambridge University Press, 1935 5. Ciucu, M.: Rotational invariance of quadromer correlations on the hexagonal lattice. Adv. in Math. 191, 46–77 (2005) 6. Ciucu, M.: A random tiling model for two dimensional electrostatics. Mem. Amer. Math. Soc. 178(839), 1–106 (2005) 7. Ciucu, M.: Dimer packings with gaps and electrostatics. Proc. Natl. Acad. Sci. USA 105, 2766–2772 (2008) 8. Ciucu, M.: The scaling limit of the correlation of holes on the triangular lattice with periodic boundary conditions. Mem. Amer. Math. Soc. 199(935), 1–100 (2009) 9. Ciucu, M.: The emergence of the electrostatic field as a Feynman sum in random tilings with holes. Trans. Amer. Math. Soc. 362, 4921–4954 (2010) 10. Ciucu, M., Krattenthaler, C.: The number of centered lozenge tilings of a symmetric hexagon. J. Combin. Theory Ser. A 86, 103–126 (1999) 11. Cohn, H., Larsen, M., Propp, J.: The shape of a typical boxed plane partition. New York J. of Math. 4, 137– 165 (1998) 12. Di Francesco, P., Reshetikhin, N.: Asymptotic shapes with free boundaries. preprint; http://arxiv.org/abs/ 0908.1630v1 [mathph], 2009 13. Feynman, R.P.: The Feynman Lectures on Physics, vol. II, Reading, MA: Addison-Wesley, 1963 14. Fischer, I.: Another refinement of the Bender–Knuth (ex-)conjecture. Eur. J. Combin. 27, 290–321 (2006) 15. Fisher, M.E., Stephenson, J.: Statistical mechanics of dimers on a plane lattice. II. Dimer correlations and monomers. Phys. Rev. 132(2 ), 1411–1431 (1963) 16. Gessel, I.M., Viennot, X.: Determinants, paths, and plane partitions. Preprint, 1989, available at: http:// people.brandeis.edu/~gessel/homepage/papers/pp.pdf (1989) 17. Gordon, B.: A proof of the Bender–Knuth conjecture. Pac. J. Math. 108, 99–113 (1983) 18. Graham, R.L. Knuth, D.E., Patashnik, O.: Concrete Mathematics. Reading, MA: Addison-Wesley, 1989 19. Ishikawa, M., Wakayama, M.: Minor summation formula for pfaffians. Linear and Multilinear Algebra 39, 285–305 (1995) 20. Kenyon, R.: Local statistics of lattice dimers. Ann. Inst. H. Poincaré Probab. Statist. 33, 591–618 (1997) 21. Kenyon, R.: The asymptotic determinant of the discrete Laplacian. Acta Math. 185, 239–286 (2000) 22. Kenyon, R., Okounkov, A., Sheffield, S.: Dimers and amoebae. Ann. of Math. 163, 1019–1056 (2006)


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Communications in


Spectral Simplicity and Asymptotic Separation of Variables Luc Hillairet1 , Chris Judge2 1 Laboratoire de Mathématiques Jean Leray, UMR 6629, Université de Nantes, 2, rue de la Houssinière,

44322 Nantes Cedex 3, France. E-mail: [email protected]

2 Department of Mathematics, Indiana University, Bloomington, IN 47405, USA.

E-mail: [email protected] Received: 18 January 2010 / Accepted: 4 July 2010 Published online: 15 January 2011 – © Springer-Verlag 2011

Abstract: We describe a method for comparing the spectra of two real-analytic families, (at ) and (qt ), of quadratic forms that both degenerate as a positive parameter t tends to zero. We suppose that the family (at ) is amenable to ‘separation of variables’ and that each eigenspace of at is 1-dimensional for some t. We show that if (qt ) is asymptotic to (at ) at first order as t → 0, then the eigenspaces of (qt ) are also 1-dimensional for all but countably many t. As an application, we prove that for the generic triangle (simplex) in Euclidean space (constant curvature space form) each eigenspace of the Laplacian acting on Dirichlet functions is 1-dimensional. 1. Introduction In this paper we continue a study of generic spectral simplicity that began with [HlrJdg09] and [HlrJdg10]. In particular, we develop a method that allows us to prove the following. Theorem 1.1. For almost every Euclidean triangle T ⊂ R2 , each eigenspace of the Dirichlet Laplacian associated to T is one-dimensional. Although we establish the existence of triangles with simple Laplace spectrum, we do not know the exact geometry of a single triangle that has simple spectrum. Up to homothety and isometry, there are only two Euclidean triangles whose Laplace spectrum has been explicitly computed, the equilateral triangle and the right isoceles triangle, and in both of these cases the Laplace spectrum has multiplicities [Lame,Pinsky80, Berard79,Harmer08]. Numerical results indicate that other triangles might have spectra with multiplicities [BryWlk84]. Non-isometric triangles have different spectra [Durso88, Hillairet05]. More generally, we prove that almost every simplex in Euclidean space has simple Laplace spectrum. Our method applies to other settings as well. For example, we have the following.

292

L. Hillairet, C. Judge

Theorem 1.2. For all but countably many α, each eigenspace of the Dirichlet Laplacian associated to the geodesic triangle Tα in the hyperbolic plane with angles 0, α, and α, is one-dimensional. If α = π/3, then Tα is isometric to a fundamental domain for the group S L 2 (Z) acting on the upper half-plane as linear fractional transformations. P. Cartier [Cartier71] conjectured that Tπ/3 has simple spectrum. This conjecture remains open (see [Sarnak03]). Until now, the only extant methods for proving that a domain has simple Laplace spectrum consisted of either explicit computation of the spectrum, a perturbation of a sufficiently well-understood domain, or a perturbation within an infinite dimensional space of domains. As an example of the first approach, using separation of variables one can compute the Laplace spectrum of each rectangle exactly and find that this spectrum is simple iff the ratio of the squares of the sidelengths is not a rational number. In [HlrJdg09] we used this fact and an analytic perturbation to show that almost every polygon with at least four sides has simple spectrum. The method for proving spectral simplicity by making perturbations in an infinite dimensional space originates with J. Albert [Albert78] and K. Uhlenbeck [Uhlenbeck72]. In particular, it is shown in [Uhlenbeck72] that the generic compact domain with smooth boundary has simple spectrum. In the case of Euclidean triangles, the last method does not apply since the space of triangles is finite dimensional. We also do not know how to compute the Laplace spectrum of a triangle other than the right-isoceles and equilateral ones. One does know the eigenfunctions of these two triangles sufficiently well to apply the perturbation method, but unfortunately the eigenvalues do not split at first order and it is not clear to us what happens at second order. As a first step towards describing our approach, we consider the following example. Let Tt be the family of Euclidean right triangles with vertices (0,0), (1,0), and (1, t) and let qt denote the associated Dirichlet energy form ∇u2 d x d y. qt (u) = Tt

For each u, v ∈ C0∞ (Tt ), we have qt (u, v) = t u, v , where t is the Laplacian, and hence the spectrum of t equals the spectrum of qt on the domain H01 (Tt ) with respect to the L 2 -inner product on Tt . As t tends to zero, the triangle Tt degenerates to the segment that joins (0,0) and (1,0). The spectrum of an interval is simple and hence one can hope to use this to show that Tt has simple spectrum for some small t > 0 (Fig. 1). Indeed, the spectral study of domains that degenerate to a one-dimensional object is quite well developed. In particular, the asymptotic behaviour of the spectrum of ordered

Fig. 1. The triangle Tt and the sector St .

Spectral Simplicity

293

eigenvalues involves a limiting one-dimensional Schrödinger operator (see, for example, [ExnPst05,FrdSlm09 and Grieser]). Using these kinds of results it can be proved that for each n ∈ N, there exists tn > 0 so that the first n eigenvalues of Ttn are simple (as in [LuRowl]). Unfortunately, this does not imply the existence of a triangle all of whose eigenvalues are simple. This subtle point is perhaps best illustrated by a different example whose spectrum can be explicitly calculated: Let Ct be the cylinder [0, 1] × R/tZ. The spectrum of the Dirichlet Laplacian on Ct is π 2 · k 2 + 4 · 2 /t 2 | (k, ) ∈ N × (N ∪ {0}) . Moreover, for each t > 0 and (k, ) ∈ N × N, each eigenspace is 2-dimensional. On the 1 other hand, the first n eigenvalues of the cylinder Ct are simple iff t < 2(n 2 − 1)− 2 . The example indicates that the degeneration approach to proving spectral simplicity does not work at the ‘zeroth order’ approximation. The method that we describe here is at the next order. In the case of the degenerating triangles Tt , there is a second quadratic form at to which qt is asymptotic in the sense that at − qt is controlled by t · at . Geometrically, the quadratic form at corresponds to the Dirichlet energy form on the sector, St , of the unit disc with angle arctan(t) and it is quite a standard idea to analyse the spectra of thin right triangles using thin sectors (see for example [BryWlk84]). The spectrum of the sectorial form at can be analyzed using polar coordinates and separation of variables. In particular, we obtain the Dirichlet quadratic form b associated to the interval of angles [0, arctan(t)], and, asssociated to each eigenvalue (·π/ arctan(t))2 of b, we have a quadratic √ form at on the radial interval [0, 1]. Each eigenfunction of at is of the form r → Jν ( λ · r ), where Jν is a Bessel function of order ν = π/ arctan(t) and where the eigenvalue, λ, is determined by the condition that this function vanish at r = 1. The spectrum of at is the union of the spectra of at over ∈ N. Figure 2 presents the main qualitative features of the spectrum of at after renormalization by multiplying by t 2 . For each ∈ N, the (renormalized) real-analytic eigenvalue

Fig. 2. The spectrum of the family at

294


branches of a coming from at converge to the threshold ( · π )2 . The eigenvalues of at are simple for all t, and for all but countably many t, the spectrum of at is simple. From the asymptotics of the zeroes of the Bessel function, one can show that the distance between any two (renormalized) real-analytic eigenbranches of at is of order 2 at least t 3 . This ‘super-separation’ of eigenvalues is central to our method. Indeed, simplicity would then follow if one were to prove that each real-analytic eigenvalue branch of qt lies in an O(t) neighborhood of a real-analytic eigenvalue branch of at and that at most one eigenfunction branch of qt has its eigenvalue branch lying in this neighborhood. In fact, as sets, the distance between the spectrum of at and the spectrum of qt is O(t), and, with some work, one can prove that each (renormalized) real-analytic eigenvalue branch of qt converges to a threshold in {( · π )2 | ∈ N} (Theorem 13.1). Nonetheless, infinitely many real-analytic eigenbranches of at converge to each threshold and the crossing pattern of these branches and the branches of qt can be quite complicated.

Semiclassical analysis predicts that the eigenvalues of at become separated at order t away from the threshold ( · π )2 (see Remark 10.5). On the other hand, two real-analytic 2 eigenbranches that converge to the same threshold stay separated at order t 3 . In order to use the super-separation of eigenvalues, we will need to show that each eigenvector branch of qt whose eigenvalue branch converges to a particular threshold does not interact with eigenvector branches of at that converge to another threshold (see Lemmas 12.3 and 12.4). In this sense, we will asymptotically separate variables. One somewhat novel feature of this work is the melding of techniques from semiclassical analysis and techniques from analytic perturbation theory. We apply quasimode and concentration estimates to make comparative estimates of the eigenvalues and eigenfunctions of at and qt . We then feed these estimates into the variational formulae of analytic perturbation theory in order to track the real-analytic branches. So far, our description of the method has been limited to the special case of degenerating right triangles. In §15 we make a change of variables that places the problem for right triangles into the following more general context. We suppose that there exists a positive abstract quadratic from b with simple discrete spectrum, and define ∞ ∞ at (u ⊗ ϕ) = t 2 · (ϕ, ϕ) |u (x)|2 d x + b(ϕ) |u(x)|2 d x. (1) 0

0

We consider this family of quadratic forms relative to the weighted L 2 -inner product defined by ∞ u ⊗ ϕ, v ⊗ ψ = (ϕ, ψ) u · v σ d x, 0

σ

where σ is a smooth positive function with < 0 and lim x→∞ σ (x) = 0. See §11. The spectrum of at decomposes into the joint spectra of ∞ ∞ μ at (u) = t 2 |u (x)|2 d x + μ |u(x)|2 d x, (2) 0

0

where μ is an eigenvalue of b (and hence is positive). Because σ is a decreasing function, μ an eigenfunction of at with eigenvalue E oscillates for x > x E . Since σ < 0, one can approximate the eigenfunction (or a quasimode at energy E) with Airy functions in a neighborhood of x E . A good

Spectral Simplicity

295

deal of the present work is based on this approximation by Airy functions. For example, the asymptotics of the zeroes of Airy functions underlies the super-separation of eigenvalues. The following is the general result. Theorem 1.3 (Theorem 14.1). If qt is a real-analytic family of positive quadratic forms that is asymptotic to at at first order (see Definition 3.1), then for all but countably many t, the spectrum of qt is simple. Using an induction argument that begins with the triangle, we obtain the following: Corollary 1.4. For almost every simplex in Euclidean space, each eigenspace of the associated Dirichlet Laplacian is one-dimensional. Dirichlet boundary conditions can be replaced by any boundary condition that corresponds to a positive quadratic form b. In particular, one can choose any mixed Dirichlet-Neumann condition on the faces of the simplex except for all Neumann. Using the ‘pulling a vertex’ technique of [HlrJdg09], we can extend generic simplicity to certain classes of polyhedra. For example, a d-dimensional polytope P is called k-stacked if P can be triangulated by introducing only faces of dimension d − 1 [Grünbaum]. Corollary 1.5. Almost every d − 1-stacked convex polytope P ⊂ Rd with n vertices has simple Dirichlet spectrum. Finally, we note that by perturbing the curvature of Euclidean space as in §4 of [HlrJdg09] we obtain the following: Corollary 1.6. Almost every simplex in a constant curvature space form has simple Dirichlet Laplace spectrum.

Organization of the paper. In §2 we use standard resolvent estimates to quantify the assertion that if two quadratic forms are close, then their spectra are close. In particular, we consider the projection, PaI (u), of an eigenfunction q with eigenvalue E onto the eigenspaces of a whose eigenvalues lie in an interval I E. We show that this projection is essentially a quasimode at energy E for a. In §3 we specialize these estimates to the case of two real analytic families of quadratic forms at and qt . We define what it means for qt to be asymptotic to at at first order. We show that if the first order variation, a˙ t , of at is nonnegative, then each real-analytic eigenbranch of at converges as t tends to zero, and if qt is asymptotic to at at first order, then the eigenbranches of qt also converge. In Sect. 4, we use the variational formula along a real-analytic eigenfunction branch u t to derive an estimate on the projection PaIt (u t ). This results in the assertion that the function t →

a˙ t (PaIt (u t )) PaIt (u t )2

is integrable (Theorem 4.2). The integrability will be used several times in the sequel to control the projection PaIt (u t ), and in particular, it will be used to prove that the

296


eigenspaces essentially become one-dimensional in the limit. This result depends on both analytic perturbation theory and resolvent estimates. Sections 5 through 10 are devoted to the study of the one dimensional quadratic forms μ at in (2). Most of the material in these sections is based on asymptotics of solutions to second order ordinary differential equations (see, for example, [Olver]). In §6 we provide uniform estimates on the L 2 -norm of quasimodes and on the exponential decay of eigenfunctions for large x. In §7 we make a well-known change of variables to transform μ the second order ordinary differential equation associated to at into the inhomogeneous Airy equation. In §8 we use elementary estimates of the Airy kernel to estimate both quasimodes and eigenfunctions near the turning point x E . In §9 we use the preceding estimates to prove Proposition 9.1 which essentially says μ that the L 2 -mass of both eigenfunctions and quasimodes of at does not concentrate at x E as t tends to zero. This proposition is an essential ingredient in proving the projection estimates of §12. But first we use it in §10 to prove that each real-analytic eigenvalue μ branch of at converges to a threshold μ/σ (0). μ In §10 we also establish the ‘super-separation’ of eigenvalue branches for at . In the case of degenerating right-triangles, we may use the uniform asymptotics of the Bessel function (see [Olver]) to obtain the ‘super-separation’ near the threshold. We prove it directly in Proposition 10.4 for general σ . In §11 we establish some basic properties of the quadratic form defined in (1). In §12 we combine results of §2, §4, and §9 to derive estimates on PaIt (u t ), where u t is a real-analytic eigenfunction branch of qt with eigenvalue branch E t converging to a point E 0 belonging to the interior of an interval I . In §13 we show that each eigenvalue branch of qt converges to some threshold μ/σ (0) (Theorem 13.1). This leads to the following natural question: which thresholds μ/σ (0) are limits of some real-analytic eigenvalue branch of qt ? Strangely enough, we do not answer it here. In §14 we prove the generic simplicity of qt . In §15, we show how simplices and other domains in Euclidean space fit into the general framework presented here. Finally, in §16 we prove a generalization of Theorem 1.2. 2. Quasimode Estimates for Quadratic Forms Let H be a real Hilbert space with inner product ·, · . Let a be a real-valued, densely defined, closed quadratic form on H. Let dom(a) ⊂ H denote the domain of a. In the sequel, we will assume that the spectrum spec(a) of a with respect to ·, · is discrete. Moreover, we will assume that for each λ ∈ spec(a), the associated eigenspace Vλ is finite dimensional, and we will assume that there exists an orthonormal collection, {ψ }∈N , of eigenfunctions such that the span of {ψ } is dense in H. The following estimate is standard: Lemma 2.1 (Resolvent estimate). Suppose that the distance, δ, from E to the spectrum of a is positive. If |a(w, v) − E · w, v | ≤ · v then w ≤

. δ

Spectral Simplicity

297

Given a closed interval I ⊂ [0, ∞), define PaI to be the orthogonal projection onto ⊕λ∈I Vλ . Definition 2.2. Let q be a real-valued, closed quadratic form defined on dom(a). We will say that q is ε-close to a if and only if for each v, w ∈ dom(a), we have 1

1

|q(v, w) − a(v, w)| ≤ ε · a(v) 2 · a(w) 2.

(3)

For each quadratic form q defined on dom(a), define 1 2 n q (u) = u2 + q(u) . If q is ε-close to a, then the norms n q and n a are equivalent on dom(a). Thus, the form domains of q and a with respect to · coincide. We will denote this common form domain by D. Lemma 2.3. Let q and a be quadratic forms such that q is ε-close to a. If u is an eigenfunction of q with eigenvalue E contained in the open interval I ⊂ R, then E 2 I 2 a u − Pa (u) ≤ ε · a(u) · 1 + , (4) δ where δ is the distance from E to the complement R\I . Proof. Let v ∈ D. Since q(u, v) = E · u, v , from (3) we have 1

1

|E · u, v − a(u, v)| ≤ ε · a(u) 2 · a(v) 2 .

(5)

There exists a linear functional f such that for all v ∈ D we have Write f =

E · u, v − a(u, v) = f, v . f · ψ and define vtest = λ−1 f · ψ . Observe that

E · u, vtest − a(u, vtest ) = f, vtest =

(6)

| f l |2 = a(vtest ). λl

By substituting v = vtest into (5), we find that

| f l |2 ≤ ε2 · a(u). λl

(7)

−1 · f for each ∈ N. Let u = u · ψ . From (6) we find that u = (E − λl ) Therefore, a(u − PaI (u)) =

λl ∈I /

λl ·

λ2 | f l |2 2 ≤ ε · a(u) · sup , 2 |E − λl |2 λl ∈I / |E − λ |

where the inequality follows from (7). We have λ2 x2 ≤ sup = sup 2 2 λl ∈I / |E − λ | |1−x|>δ/E |1 − x| The desired bound follows.

2 E +1 . δ

298


The preceding lemma provides control of the norm of PaI (u). In particular, we have the following: Corollary 2.4. Let q and a be quadratic forms such that q is ε-close to a. If u is an eigenfunction of q with eigenvalue E contained in the open interval I ⊂ R, then E 2 a(u) I 2 2 , (8) Pa (u) ≥ 1 − ε · 1 + · δ sup(I ) where δ is the distance from E to the complement R\I . Proof. Since a(u − PaI (u), PaI (u)) = 0, we have a(PaI (u)) = a(u) − a u − PaI (u) . Thus, it follows from Lemma 2.3 that

a(PaI (u))

E 2 ≥ 1−ε 1+ · a(u). δ 2

Since, on the other hand, a(PaI (u)) ≤ sup(I ) · PaI (u)2 , the claim follows.

We use the preceding to prove the following. Lemma 2.5. Let I be an interval, let E ∈ I , and let δ denote the distance from E to the complement R\I . Let u be an eigenfunction of q with eigenvalue E. If ε < (1 + E/δ)−1 and q is ε-close to a, then for each v ∈ D, we have ε · sup(I ) · PaI (u) · v. (9) a PaI (u), v − E · PaI (u), v ≤ 1 E 2 2 2 1−ε 1+ δ Proof. Let u˜ = PaI (u), and v˜ = PaI (v). Since PaI is an orthogonal projection that commutes with a, we have a(u, ˜ v) = a(u, ˜ v) ˜ = a(u, v) ˜ and u, ˜ v = u, ˜ v ˜ = u, v . ˜ Therefore, by replacing v with v˜ in (5) we obtain 1

1

|a(u, ˜ v) − E · u, ˜ v | ≤ ε · a(u) 2 · a(v) ˜ 2. Since v˜ ∈ PaI (H), we have a(v) ˜ ≤ sup(I ) · v ˜ 2 ≤ sup(I ) · v2. By the hypothesis and Corollary 2.4, we have −1 E 2 2 a(u) ≤ 1 − ε 1 + · sup(I ) · PaI (u)2 . δ By combining these estimates, we obtain the claim.

Spectral Simplicity

299

Let {an }n∈N and {qn }n∈N be sequences of quadratic forms defined on D. For each n, let E n be an eigenvalue of qn . Proposition 2.6. Suppose that limn→∞ E n exists and is finite. If the quadratic form qn is 1/n-close to an for each n, then there exist N > 0 and C > 0 such that for each n > N and each eigenfunction u of qn with eigenvalue E n , we have 1 (10) an PaIn (u), v − E n · PaIn (u), v ≤ C · · PaIn (u) · v. n Proof. Let E 0 = limn→∞ E n and let I be an open interval that contains E 0 . Let δn be the distance from E n to R\I . Since E n converges to E 0 and I is open, there exists δ0 > 0 and N0 so that if n > N0 , then δn > δ0 . Choose N ≥ max{N0 , 1 + 2E 0 /δ0 } so that if n > N , then E n < 2E 0 . Then for each n > N we have n −1 (1 + E n /δn ) ≤ 1, and we can apply Lemma 2.5 to obtain the claim. 3. Asymptotic Families and Eigenvalue Convergence Given a mapping of the form t → f t , we will use f˙t to denote its first derivative. More precisely, we define d f˙t := fs . ds s=t Let at and qt be real-analytic families of closed quadratic forms densely defined on D ⊂ H for t > 0.1 In this section, we show that the nonnegativity of both at and a˙ t implies that each real-analytic eigenvalue branch of at converges as t tends to zero. We then show that if q is asymptotic to a in the following sense then the eigenvalue branches of qt also converge (Proposition 3.4). Definition 3.1. We will say that qt is asymptotic to a t at first order iff there exists C > 0 such that for each t > 0 and u, v ∈ D, 1

1

|qt (u, v) − at (u, v)| ≤ C · t · at (u) 2 · at (v) 2 ,

(11)

|q˙t (v) − a˙ t (v)| ≤ C · at (v).

(12)

and

Remark 3.2. By reparameterizing the family—replacing t by say t/C—one may assume, without loss of generality, that C = 1. We will do so in what follows. In what follows, we will assume that the eigenvalues and eigenfunctions of at and qt vary real-analytically. To be precise, we will suppose for each t > 0, there exists an orthonormal collection {ψ (t)}∈N of eigenvectors whose span is dense in H such that t → ψ (t) is real-analytic for each ∈ N. This assumption is satisfied if the operators that represent at and qt with respect to ·, · have a compact resolvent for each t > 0. See, for example, Remark 4.22 in §VII.4 of [Kato]. The following proposition is well-known: 1 For notational simplicity, we will often drop the index t, but note that each object related to a or q will, in general, depend on t.

300


Proposition 3.3. If at ≥ 0 and a˙ t ≥ 0 for all small t, then each real-analytic eigenvalue branch of at converges to a finite limit as t tends to zero. Proof. Let λt be a real-analytic eigenvalue branch of at . By standard perturbation theory (see [Kato]) λ˙ t · u t 2 = a(u ˙ t ).

(13)

Thus, since a˙ t ≥ 0, the function t → λt is increasing in t. Since λt is bounded below, the limit limt→0 λt exists. If qt is asymptotic to at , then the eigenvalues of qt also converge. Proposition 3.4. Suppose that for each t > 0, the quadratic forms at and a˙ t are nonnegative. If qt is asymptotic to at at first order, then each real-analytic eigenvalue branch of qt converges to a finite limit. Proof. Let (E t , u t ) be a real-analytic eigenbranch of qt with respect to ·, · . We have q˙t (u t ) = E˙ t · u t 2.

(14)

Using (11), we have qt (v) ≥

1 · at (v) 2

(15)

for all t sufficiently small. Since at ≥ 0, we have qt ≥ 0 and hence E t ≥ 0 for small t. From (12) and Remark 3.2 we have q˙t (u t ) ≥ a˙ t (u t ) − at (u t ) and hence, since a˙ t ≥ 0, we have q˙t (u t ) ≥ −at (u t ). By combining this fact with (14) and (15), we find that E˙ t + 2 · E t ≥ 0

(16)

for sufficiently small t. To finish the proof, define the function f by f (t) = E t · exp(2t). By (16) we have f (t) ≥ 0 for t < t0 and, since qt is non-negative, f is obviously bounded from below. Therefore limt→0 f (t) exists and is finite and so does limt→0 E t . 4. An Integrability Condition Let qt be a real-analytic family that is asymptotic to at at first order. In this section, we use the estimates of §2 to derive an integrability condition (Theorem 4.2) that will be used in §14 to prove that the spectrum of qt is simple for most t under certain additional conditions. Let E t be a real-analytic eigenvalue branch of qt that converges to E 0 as t tends to zero. Let Vt be the associated real-analytic family of eigenspaces. Let I be a compact interval whose interior contains E 0 . Remark 4.1. The definition of Vt implies that, for each t > 0, the vector space Vt is a subspace of ker(At − E t · I ). If a distinct real-analytic eigenvalue branch crosses the branch E t at t = t0 , then Vt0 is a proper subspace of ker(At0 − E t0 · I ).

Spectral Simplicity

301

Theorem 4.2. Let qt be asymptotic to at at first order, and suppose that for each t > 0, we have 0 ≤ a˙ t (v) ≤ t −1 · at (v).

(17)

If t → u t ∈ Vt is continuous on the complement of a countable set, then the function a˙ t PaIt (u t ) (18) t → P I (u t )2 at is integrable on each interval of the form (0, t ∗ ]. Proof. Let χt = PaIt (u t ). Since the spectrum of at is discrete and E t is real-analytic, the operator family t → PaIt is real-analytic on the complement of a countable set. By combining this with the hypothesis, we find that the function a(P ˙ aIt (u t ))/Pat (u t )2 is ∗ ∗ locally integrable on (0, t ] for each t > 0. By Lemma 4.3 below, there exists a constant C > 0 such that a˙ t (χt ) − C. E˙ t ≥ C · χt 2 Integration then gives

t∗

Et ∗ − Et ≥ C t

a˙ s (χs ) ds − C(t ∗ − t). χs 2

Since E t ≥ 0 and the integrand is nonnegative, the integral on the right-hand side converges as t tends to zero. Lemma 4.3. Suppose that for each t > 0, we have 0 ≤ a˙ t (v) ≤ t −1 · at (v).

(19)

If qt is asymptotic to at at first order, then there exists t > 0 and a constant C > 0 such that for each t ≤ t and each eigenvector u ∈ Vt we have ˙ (20) E t · u2 − a˙ t PaIt (u) ≤ C · u2 and PaI (u) ≥

1 · u. C

(21)

Proof. Since Vt is the real-analytic family of eigenspaces associated to E t , for each t > 0 and u ∈ Vt we have q(u) ˙ = E˙ · u2 (see Remark 4.1). Since E t converges to E 0 , we find using (11) that there exists t0 so that for t < t0 , at (u) ≤ 2qt (u) = 2E t · u2 ≤ 2(E 0 + 1) · u2. Thus, from (12) we find that ˙ E · u2 − a˙ t (u) ≤ 2(E 0 + 1) · u2 for t < t0 .

(22)

(23)

302


Let χt = PaIt (u). Since a˙ t is a nonnegative quadratic form, we have 1

1

a˙ t (u) ≤ a˙ t (χt ) + 2a˙ t (χt ) 2 · a˙ t (u − χt ) 2 + a˙ t (u − χt ) and 1

1

a˙ t (χt ) ≤ a˙ t (u) + 2a˙ t (u) 2 · a˙ t (χt − u) 2 + a˙ t (χt − u). The former estimate provides a bound on a˙ t (u) − a˙ t (χt ) and the latter one gives a bound on its negation. In particular, we find that 1 1 1 |a˙ t (u) − a˙ t (χt )| ≤ 2 · max a˙ t (u) 2 , a˙ t (χt ) 2 · a˙ t (u − χt ) 2 + a˙ t (u − χt ). Thus, by (19), we have 1 1 1 2 at (u − χt ) · max at (χt ) 2 , at (u) 2 at (u − χt ) 2 + . (24) t t Let δt be the distance from E t to the complement R\I . Since E 0 belongs to the interior of I and E t → E 0 , there exists δ > 0 and 0 < t1 ≤ t0 so that if t < t1 , then δt ≥ δ. Hence we may apply Lemma 2.3 to find that 2E 0 2 2 at (u − χt ) ≤ t · at (u) · 1 + δ |a˙ t (u) − a˙ t (χt )| ≤

for t < t1 . Since at is non-negative, from (22) we have at (χt ) ≤ at (u) ≤ 2(E 0 + 1) · u2 for t ≤ t0 . By combining these estimates with (24) we find that for t ≤ t1 , 2E 0 2E 0 2 · 2+t · 1+ . |a˙ t (u) − a˙ t (χt )| ≤ 2(E 0 + 1) · u · 1 + δ δ

(25)

Estimate (20) then follows from (23), (25) and the triangle inequality. If E 0 > 0, then there exists 0 < t2 ≤ t1 such that if t < t2 , then 1 1 1 · qt (u) = · E t · u2 ≥ · E 0 · u2. 2 2 4 Thus, if E 0 > 0, then (21) follows from Corollary 2.4. On the other hand, if E 0 = 0, then let t1 and δ be as above. Since PaI is a spectral projection and the eigenspaces are orthogonal, we have 2 a u − PaI (u) ≥ δ · u − PaI (u) . at (u) ≥

Thus, by Lemma 2.3 and (22) we have 2 E0 + 1 · u2 ≥ δ · u − PaI (u) . 2t 2 · (E 0 + 1) · 1 + δ In particular, if t 2 < (δ/8) · (E 0 + 1)−1 · (1 + (E 0 + 1)/δ)−1 , then 2 1 u2 ≥ · u − PaI (u) . 4 Estimate (21) then follows from the triangle inequality.

Spectral Simplicity

303

5. Definition and Basic Properties In the sequel σ : [0, ∞) → R+ will be a smooth positive function such that • lim x→∞ σ (x) = 0, • σ (x) < 0 for all x ≥ 0, • |σ

| has at most polynomial growth on [0, ∞). For u, v ∈ C0∞ ((0, ∞)), define u, v σ =

∞

u(x) · v(x) · σ (x) d x.

0

Let Hσ denote the Hilbert space obtained by completing C0∞ ((0, ∞)) with respect to √ the norm uσ := u, u σ . Let H 1 (0, ∞) and H01 (0, ∞) denote, respectively, the classical Sobolev spaces with respect to Lebesgue measure on (0, ∞). For each t > 0 and u in H 1 (0, ∞), we define ∞ μ t 2 · |u (x)|2 + μ · |u(x)|2 d x. at (u) = 0

Remark 5.1. If μ > 0, then since σ is decreasing, we have ∞ σ (0) μ 2 uσ ≤ σ (0) |u(x)|2 d x ≤ a (u). μ t 0 Let μ

dom D (at ) = H01 (0, ∞) ∩ Hσ and let μ

dom N (at ) = H 1 (0, ∞) ∩ Hσ . μ

μ

Both dom D (at ) and dom N (at ) are closed form domains for a that are dense in Hσ . μ

μ

Definition 5.2. The spectrum of the quadratic form at restricted to dom D (at ) (resp. μ dom N (at )) with respect to ·, · σ will be called the Dirichlet (resp. Neumann) specμ trum of at . μ

In the sequel, we will drop the subscript ‘D’ from dom D (at ) and the subscript ‘N ’ μ from dom N (at ). In particular, unless stated otherwise, all of the results below hold for both the Neumann and Dirichlet boundary conditions. When we refer to the ‘spectrum’ μ of at , we will mean either the Dirichlet or the Neumann spectrum. μ

Proposition 5.3. If μ > 0 and t > 0, then the quadratic form at has discrete spectrum with respect to ·, · σ . Proof. By a standard result in spectral theory—see, for example, Theorem XIII.64 [Reed-Simon]—it suffices to prove that for each r > 0 the set μ μ Ar = u ∈ dom(at )|at (u) ≤ r, uσ ≤ 1 is compact with respect to · σ . To verify this, one uses Rellich’s Lemma on compact sets. The decay of σ prevents the escape of mass at infinity.

304


6. Estimates of Quasimodes and Eigenfunctions In the sequel, unless otherwise stated, we assume that μ > 0. Let r ∈ Hσ and let E ≥ 0. In this section, we begin our analysis of functions w in μ dom(at ) that satisfy μ

at (w, v) − E · w, v σ = r, v σ

(26)

μ

for all v ∈ dom(at ). In applications, the function r in (26) will be negligible. For example, if r = 0, then w is an eigenfunction with eigenvalue E. More generally, if μ

atn (wn , v) − E n · wn , v σ = rn , v σ ,

(27)

μ

ρ

where tn → 0, wn ∈ dom(atn ), lim E n = E 0 and rn = O(tn ) · wtn , then the sequence wn is called a quasimode of order ρ at energy E 0 . (See also Proposition 2.6 and Remark 9.2.) Our goal is to understand the behavior of both eigenfunctions and quasimodes. Of course, in most situations, either the eigenfunction estimate will be stronger than the quasimode estimate and/or the proof will be simpler. In the following, we will first provide a general estimate—valid for any quasimode—and then, as needed, we will state and prove the stronger result for eigenfunctions. By unwinding the definitions, Eq. (26) may be rewritten as

t 2 · w (x) · v (x) + f E (x) · w(x) · v(x) d x =

∞ 0

∞

r (x) · v(x) · σ (x) d x,

0

(28) where f E (x) = μ − E · σ (x). By integrating (28) by parts, we find that w satisfies (28) for all v ∈ C0∞ ((0, ∞)) if and only if for each x ∈ (0, ∞), − t 2 · w

(x) + f E (x) · w(x) = r (x) · σ (x).

(29) μ

The function w is a Dirichlet (resp. Neumann) eigenfunction of at if and only if w is in Hσ ∩ H 1 , satisfies Eq. (29) and w(0) = 0 (resp. w (0) = 0). μ Let E ≥ μ/σ (0). For instance, we may choose E to be an eigenvalue of at . Since σ is strictly decreasing, there exists a unique point x E ∈ [0, ∞) such that f E (x E ) = 0. In particular, if x > x E , then f E (x) > 0 and if x < x E , then f E (x) < 0. If w is an eigenfunction (r = 0), then one expects w to behave like an exponential function when x >> x E and to oscillate for x 0 such that for each E ∈ K , x ∈ [0, ∞), and M ∈ R, we have |ρ M By applying this estimate to (32) and applying the Cauchy-Schwarz inequality, we obtain ∞ C · t2 ∞ 2 f E · w2 · ρ M d x ≤ w d x + r σ · wσ . 2 x Es xE

For x ≥ x Es , we have ρ M (x) = χ (M + 1 − x). Thus, since f E · w 2 is integrable, by the Lebesgue dominated convergence theorem, we may let M tend to ∞ and obtain ∞ C · t2 ∞ 2 f E · w2 d x ≤ w d x + r σ · wσ . 2 x Es xE Since σ is decreasing, the function f E (x) is increasing and ∞ w2σ ≤ σ (0) w 2 (x) d x. 0

Therefore, we find that ∞ 2 ∞ C ·t r σ f E (x Es ) · w2 d x ≤ w 2 d x. + σ (0) · 2 wσ x E+ 0

Since f E (x Es ) = s, the lemma follows by choosing C to be s −1 max( C2 , σ (0)).

6.2. An estimate of the L 2 mass of an eigenfunction. If w is an eigenfunction, then the bound given in Lemma 6.2 can be greatly improved. In particular, an eigenfunction is exponentially small in the classically forbidden region, and hence one can make L 2 estimates with polynomial weights. See Lemma 6.4. First, we quantify the exponential decay of each eigenfunction. μ

Lemma 6.3. Let w be an eigenfunction of at with eigenvalue λ ≤ E. If x ≥ y ≥ x Es , then

√ 2s 2 2 w (x) ≤ w (y) · exp − · (x − y) . (33) t Proof. The proof is a straightforward convexity estimate using the maximum principle. This estimate allows us to prove the following.

Spectral Simplicity

307

Lemma 6.4. For each ν > 0, there exists a function βν : (μ/σ (0), ∞) × (0, μ) → R μ such that if w is an eigenfunction of at with eigenvalue λ ≤ E, and t ≤ 1, then ∞ ∞ 2 ν w (x) · (1 + x ) d x ≤ βν (E, s) · t · w 2 (x) d x. 3x Es

Proof. Let α = x ∈ [x Es , y],

x Es

√ 2s/t. By exchanging the roles of x and y in (33), we find that for all w 2 (x) ≥ w 2 (y) · exp (α · (y − x)).

(34)

Integrating with respect to x, we obtain y 1 w 2 (x) d x ≥ · w 2 (y) · exp(α · (y − x Es ) − 1) , α x Es and thus α · w (y) ≤ exp(α · (y − x Es )) − 1 2

y x Es

w 2 (x) d x.

(35)

If u ≥ 0, then u ν ≤ cν · eu , where cν = sup{x ν e−u | u > 0}. Hence, we have ν 2 ν · eα·x/2 . x ≤ cν · α By combining this with (33), we find that for x ≥ y, ν x 2 −y . w 2 (x) · x ν ≤ cν · · w 2 (y) · exp −α · α 2 By integrating, we find that

∞ y

ν+1 2 w (x) · x d x ≤ cν · · w 2 (y) · exp(α · y/2). α 2

ν

Putting this together with (35) gives y ν ∞ 2 exp(α · y/2) w 2 (x) · x ν d x ≤ 2 · cν · · w 2 (x) d x. s )) − 1 s α exp(α · y − α · x y xE E (36) If we let

exp(3x/2) x > 0 cν = sup x · exp(2x) − 1

and set y = 3 · x Es , then we have cν exp(α · y/2) ≤ . exp(α · y − α · x Es )) − 1 α · x Es

308


By substituting this into (36) we obtain ∞ 2cν · cν ν+1 − ν+1 ∞ 2 2 w 2 (x) · x ν d x ≤ · t · s w (x) d x. x Es 3x Es x Es

(37)

The claim then follows by specializing (37) to the case ν = 0 and adding the resulting estimate to (37). In particular, we may define βν (E, s) = 2 ·

c0 · c0 + cν · cν x Es · s

ν+1 2

.

6.3. Comparing weighted L 2 inner products on eigenfunctions. Let p : [0, ∞) → R be a positive continuous function of (at most) polynomial growth. That is, there exist constants C p and ν p such that if x ≥ 0, then 0 < p(x) ≤ C p · 1 + x ν p . We will regard p as a weight for an L 2 -inner product. Proposition 6.5. Let p be as above. There exists a function α : [μ/σ (0), ∞)×(0, μ) → R such that if s ∈ (0, μ), then lim

E→μ/σ (0)

α(E, s) = 0

(38)

and a function β : (μ/σ (0), ∞) × (0, μ) → R such that if w± is an eigenfunction of μ at with eigenvalue λ± ≤ E, then ∞ ∞ ∞ ≤ s) + β(E, s) · t) w · w · p d x − p(0) w · w d x w 2 d x. (α(E, + − + − 0

0

0

The functions α and β depend only on p, E, σ , and μ. Proof. Set α(E, s) = sup | p(x) − p(0)| | 0 ≤ x ≤ 3x Es . Since p is continuous and lim E→μ/σ (0) x Es = 0 we have (38). Using the Cauchy-Schwarz inequality we find that s 3x s 3x E E w+ · w− · p d x − p(0) w+ · w− d x ≤ α(E, s) · w+ · w− . 0 0 We also have

∞

3x Es

w+ · w− · p d x

2

≤

∞ 3x Es

|w+ | · p d x · 2

∞

3x Es

|w− | · p d x . 2

Spectral Simplicity

309

By Lemma 6.4 we have ∞ |w± |2 · p d x ≤ C p · βν p (E, s) · t 3x Es

∞ 0

|w± |2 d x

and also p(0)

∞ 3x Es

|w± | d x ≤ p(0) · β0 (E, s) · t

∞

2

0

|w± |2 d x.

The claim then follows from combining these estimates and using the triangle inequality. 7. The Langer-Cherry Transform We wish to analyse the behavior of the solutions to (29) for x near x E and for t small. To do this, we will use a transform to put the solution into a normal form. The transform that we will use was first considered by Langer [Langer31] and Cherry [Cherry50] and is a variant of the Liouville-Green transformation. See Chapter 11 in [Olver]. As above, let f E = μ− E · σ, where σ is smooth with σ < 0 and lim x→∞ σ (x) = 0. For E ≥ μ/σ (0), there exists a unique x E ∈ [0, ∞) such that f E (x E ) = 0. In the present context, the Langer-Cherry transform is based on the function φ E : [0, ∞) → R defined by x 2 3 3 1 2 φ E (x) = sign(x − x E ) · | f E (u)| du . 2 xE

(39)

Before defining the Langer-Cherry transform, we collect some facts concerning φ E . Lemma 7.1. Let U = σ μ(0) , ∞ × [0, ∞). (1) (2) (3) (4)

The map (E, x) → φ E (x) is smooth on U. φ E (x) > 0 for each (E, x) ∈ U. (φ E )2 · φ E = f E . The map (E, x) → f E (x)/φ E (x) defined for x = x E extends to a smooth map from U to R+ . (5) The limit 2

2

1

lim x − 3 · φ E (x) = (3/2) 3 · μ 3

x→∞

holds uniformly for E in each compact subset of

μ σ (0) , ∞

.

Proof. These properties follow directly from the definition (39) or from the alternative expression (41) below that we now prove. Since σ (x) < 0 for all x ∈ [μ/σ (0), ∞), the map I : U → R, I (E, u) = 0

1

−E · σ (E, s · u + (1 − s) · x E ) ds,

310


is smooth and positive on U. The map π : U → R defined by 1 1 1 π(E, x) = s 2 · I 2 (E, s · x + (1 − s) · x E ) ds

(40)

0

is also smooth and positive. Since f E (x E ) = 0 and f E (x) = −Eσ (x), the fundamental theorem of calculus gives that μ − E · σ (u) = (u − x E ) · I (E, u). Direct computation shows that φ E (x) = (x − x E ) ·

3 · π(E, x) 2

2

3

.

(41)

Definition 7.2. Let w : [0, ∞) → R and let E ≥ μ/σ (0). Define the Langer-Cherry transform of w at energy E to be the function 1 (42) W E = (φ E ) 2 · w ◦ φ −1 E . It follows from Lemma 7.1 that the Langer-Cherry transform maps C k ([0, ∞)) to C k ([φ E (0), ∞)). The importance of this transform is due to its effect on solutions to the ordinary differential equation (29). In what follows we let 1

ρ E = (φ E )− 2 .

(43)

Proposition 7.3. Let r : [0, ∞) → R and let w ∈ C 2 ([0, ∞)). Let W E be the LangerCherry transform of w at energy E. Then w satisfies t 2 · w

− f E · w = −r · σ if and only if W E satisfies

3 t 2 · W E

− y · W E = −t 2 · (ρ E3 · ρ E

) ◦ φ −1 · W − ρ · r · σ ◦ φ −1 E E E E .

(44)

The proof is a straightforward but lengthy computation. See also, for example, §11.3 in [Olver], where the function fˆ is related to h E by fˆ = h 4 . In the analysis that follows, we will treat the right-hand side of (44) as an error term for t and r small. The following estimates will help justify this treatment. Lemma 7.4. Let K ⊂ [μ/σ (0), ∞) be compact. There exists C > 0 such that if x ≥ 0 and E ∈ K , then 1 −1 ρ (x) ≤ C · x 6 , E

and

1 |ρ E (x)| ≤ C · 1 + x 6 .

Moreover, there exists ν such that

ρ (x) ≤ C · 1 + x ν . E The exponent ν depends only on σ . The constant C depends only on μ, σ , and K .

Spectral Simplicity

311 1

Proof. By part (3) of Lemma 7.1, we have ρ = (φ/ f ) 4 . Hence since lim x→∞ f E (x) = μ, we find from part (5) that lim ρ E · x

− 16

x→∞

=

2 3μ

1 6

uniformly for E ∈ K . The first two estimates follow. To prove the last estimate, one computes using f = (φ )2 · φ that 3

1

3

5 f4 1 φ 4 · f

5 φ 4 · ( f )2 − + . ρ =− 9 16 φ 114 4 f 45 16 f4

By part (5) of Lemma 7.1, both φ and 1/φ have polynomial growth that is uniform for E ∈ K . By assumption, σ

has at most polynomial growth, and hence, by integration, the function σ also has at most polynomial growth. Therefore, f

and f both have polynomial growth that is uniform over K . Therefore, since lim x→∞ f E (x) = μ > 0, we find that ρ

has uniform polynomial growth. Lemma 7.5. Let I ⊂ [0, ∞) be a compact interval and let K ⊂ [μ/σ (0), ∞) be a compact set. There exists a constant C such that for each E ∈ K such that if w is a solution to (29) and W E is the Langer-Cherry transform of w at energy E, then we have ∞ 2 2

2 4 2 |w| d x . t · W E (y) − y · W E (y) dy ≤ C · r σ + t φ E (I )

0

The constant C depends only on μ, σ, I , and K . Proof. For each continuous function F : I → R, let |F|∞ = sup{|F(x)| | x ∈ I }. We perform the change of variables y = φ E (x). Since φ E = ρ E−2 , we have dy = ρ E−2 · d x, and thus by (42) and (43), |W |2 dy = ρ E−4 · |w|2 d x. Therefore φ E (I )

and

(45)

2 3 −1

2 2

2 · ρ ) ◦ φ · |W | dy ≤ |ρ | · |ρ | · |w|2 d x, (ρ E E E ∞ E ∞ E I

φ E (I )

2 3 2 2 (ρ E · r · σ ) ◦ φ −1 E dy ≤ |ρ E · σ |∞ · r σ .

The claim then follows from squaring and integrating (44) and applying the above estimates. μ

Suppose w is an eigenfunction of at , and denote by λ its eigenvalue. If we perform the Cherry-Langer transform at energy E = λ then r = 0 and hence the conclusion of Lemma 7.5 is stronger. Actually, we will need the following strengthening of Lemma 7.5 μ which treats the case when w is an eigenfunction of at but E is close to but not necessarily exactly the corresponding eigenvalue.

312


Lemma 7.6. Let K ⊂ [μ/σ (0), ∞) be compact. There exists a constant C K such that if μ t < 1, w is an eigenfunction of at with eigenvalue λ ∈ K , and W is the Cherry-Langer transform of w at energy E ∈ K , then ∞ 2 ∞ 2

2 4 w 2 d x. (46) t · W − y · W dy ≤ C K · |λ − E| + t φ E (0)

0

Proof. Since −t 2 · w

+ (μ − λ · σ ) · w = 0, the function w satisfies −t 2 · w

+ f E · w = r. with r = (E − λ) · σ · w. Therefore we may apply Proposition 7.3. In particular, it suffices to bound the integrals of the squares of the terms appearing on the right-hand side of (44). By Lemma 7.4 there exists ν1 and C1 (depending only on K ) such that |ρ E (x)|−4 · |ρ E3 · ρ E

(x)|2 ≤ C1 · (1 + x ν1 ). Hence by changing variables (recall that W 2 dy = ρ E−4 w 2 d x) we find that

2 3 −1

2 (ρ E · ρ E ) ◦ φ E · |W (y)| dy ≤ C1

∞

φ E (0)

∞

|w(x)|2 · (1 + x ν1 ) d x.

0

Since w is an eigenfunction, we can apply Lemma 6.4. By fixing s = μ/2, we obtain a constant C2 —depending only on K —such that ∞ ∞ 2 ν1 |w(x)| · (1 + x ) d x ≤ C2 · t |w(x)|2 d x. 3x Es

x Es

Let x ∗ = sup{x Es | E ∈ K , s = μ/2}. Then

3x Es

|w(x)|2 · (1 + (3x)ν1 ) d x ≤ 1 + (3x ∗ )ν1

0

∞

|w(x)|2 d x.

0

In sum, if t ≥ 1, then we have a constant C3 such that ∞ 3

−1 2 2 |ρ E · ρ E ◦ φ (y)| · |W (y)| dy ≤ C3 φ E (0)

∞

|w(x)|2 d x.

0

A similar argument shows that there exists C4 —depending only on K —such that ∞ ∞ 2 3 (y) · W (y) dy ≤ C |w(x)|2 d x. (ρ E · σ 2 ) ◦ φ −1 4 E φ E (0)

By putting these estimates together we obtain the claim.

0

The following lemma will allow us to control scalar products in w when they are expressed on the Cherry-Langer side, in the limit as E tends to μ/σ (0) and t tends to 0. It will be used in the proof of Theorem 10.4.

Spectral Simplicity

313

Lemma 7.7. Let q : [0, ∞) → R be a positive continuous function of at most polynomial growth. Given > 0, there exists δ > 0 such that if t < δ, E < μ/σ (0) + δ, and w± is an eigenfunction of aμt with eigenvalue λ± ≤ E, then ∞ ∞ 1 W+ · W− dy − 4 w+ · w− · q d x ≤ · w+ · w− , (47) φ E (0) ρ E (0) · q(0) 0 where W± is the Langer-Cherry transform of w± at energy E, and · is the standard (unweighted) L 2 norm. Proof. Changing variables gives ∞ W+ · W− dy = φ E (0)

0

∞

w+ · w− · ρ E−4 (x) d x.

ρ E−4

is bounded, and hence we can apply Proposition 6.5. By Lemma 7.4, the function In particular, choose δ1 > 0 so that if E < μ/σ (0) + δ1 , then α p (E, μ/2) < /4 and choose δ2 ≤ δ1 so that if t < δ2 , then β(δ1 , μ/2) · t < /4. Thus, if E < μ/σ (0) + δ2 and t < δ2 , then ∞ ∞ −4 ≤ · w+ · w− . W · W dy − ρ (0) w · w d x + − + − E 2 φ E (0)

0

In a similar fashion we can apply Lemma 6.5 to find δ ≤ δ2 so that if E < μ/σ (0) + δ and t < δ, then ∞ ∞ w+ · w− · q dy − q(0) w+ · w− d x ≤ · ρ E4 (0) · q(0) · w+ · w− . 2 0

The claim follows.

0

8. Airy Approximations In this section we analyse solutions to the inhomogeneous equation t 2 · W

(y) − y · W (y) = g(y).

(48)

To do this, we will use a solution operator, K˜ t , for the associated homogeneous equation t 2 W0

− y · W0 = 0.

(49) 2

The function W0 is a solution to (49) if and only if A(u) = W0 (t 3 · u) is a solution to the Airy equation A

− u · A = 0.

(50)

Using, for example, the method of variation of constants, one can construct an integral kernel K for an ‘inverse’ of the operator A(u) → A

(u) − u · A(u) in terms of Airy functions. We give the construction of K as well as its basic properties in Appendix A. By rescaling (or by direct construction) we obtain an integral kernel for the operator A(x) → t 2 · A

(x) − x · A(x). To be precise, define 2 4 2 K˜ t (y, z) = t − 3 · K t − 3 · y, t − 3 · z , where K is the integral kernel constructed in Appendix A.

314


Lemma 8.1. Let −∞ < a ≤ b ≤ ∞. For each locally integrable g : [a, b] → R of at most polynomial growth, the function b (51) K˜ t (y, z) · g(z) dz y → a

is a solution to (48). Proof. This follows from Lemma A.1 or directly from the variation of constants construction. The following estimate is crucial to the proof of Proposition 9.1. Lemma 8.2. Let g : R → R be continuous. For each −∞ < a < 0 < b, there exist constants C and t0 > 0 such that if t < t0 and W satisfies (48), then

a 0 b 2 2 2 − 53 W ≤C· W +t g2 , (52) a

a

and

b 2

W ≤C· t 2

0

1 3

a 2

a

b

2

W +

a

b 2

2

W +t

− 53

b

g

2

.

(53)

a

The constants C and t0 can be chosen to depend continuously upon a and b. Proof. Define W0 on [a, b] by W0 (y) = W (y) −

b

K˜ t (y, z) · g(z) dz.

a

Using Lemma 8.1 and linearity, W0 is a solution to (49). Using the Cauchy-Schwarz-Bunyakovsky inequality, we find that b 2 b ˜ 2 2 |g(z)| dz . |W (y) − W0 (y)| ≤ K t (y, z) dz a

(54)

a

A change of variables gives a

b

a

b

t − 23 b t − 23 b 2 4 ˜ |K (u, v)|2 du dv. K t (y, z) dy dz = t − 3 − 2 −2 t

3a

t

(55)

3a

√ 1 By Lemma A.3 in Appendix A, the latter integral is less than CAiry · δ · t − 3 , where CAiry is a universal constant and δ = max{|a|, b}. Therefore, by integrating (54) over an interval I ⊂ [a, b] and substituting (55), we find that 5

W − W0 2I ≤ C0 · t − 3 · g2[a,b] ,

(56)

where C0 = CAiry · δ and · J denotes the L 2 -norm over the interval J . In particular, by the triangle inequality we have 1

5

W I ≤ W0 I + C02 · t − 6 · g[a,b] ,

Spectral Simplicity

315

and hence 5

W 2I ≤ 2 · W0 2I + 2 · C0 · t − 3 · g2[a,b] .

(57)

Similarly, 5

W0 2I ≤ 2 · W 2I + 2 · C0 · t − 3 · g2[a,b] .

(58)

2

The function u → W0 (t 3 · u) satisfies the Airy equation (131). Hence, it follows 2 from Lemma A.4 (in which s is replaced by t − 3 ) that there exist constants M and t0 > 0—depending continuously on a and b—such that if t ≤ t0 , then 0 a 2 W02 dy ≤ M W02 dy (59) a

a

and

b 2

W02

0

dy ≤ M t

1 3

a

a 2

W02

dy +

b b 2

W02

dy .

(60)

By combining (60) with (57) and (58), we obtain (52). By combining (59) with (57) and (58), we obtain (53). 9. A Non-Concentration Estimate μ

Fix μ and σ and let at be the family of quadratic forms defined as in §5. The purpose of this section is to prove the following non-concentration estimate—see Remark 9.3—that is crucial to our proof of generic spectral simplicity. Proposition 9.1. Let K be a compact subset of (μ · σ (0)−1 , ∞), and C > 0. There exist μ constants t0 > 0 and κ > 0 such that if E ∈ K , if t < t0 , and if for each v ∈ dom(at ), the function w satisfies μ at (w, v) − E · w, v σ ≤ C · t · wσ · vσ , (61) then

∞ 0

(E · σ (x) − μ) · |w(x)|2 d x ≥ κ · w2σ .

(62)

The constants t0 and κ depend only upon K , C, μ, and σ . In contrast to previous estimates, Proposition 9.1 is concerned with so-called noncritical energies, those values of E that are strictly greater than the threshold μ/σ (0). Remark 9.2. Estimate (61) is a special case of an estimate of the following form: For all v ∈ dom(aμ,t ), μ at (w, v) − E t · w, v σ ≤ t ρ · wσ · vσ . (63) By the Riesz representation theorem, estimate (63) is equivalent to Eq. (26) with r such that r ≤ t ρ · w. In other words, a sequence wn satisfying (63) is what we have called a quasimode of order ρ at energy E 0 .

316

L. Hillairet, C. Judge μ

Remark 9.3. Suppose that wn is a sequence of eigenfunctions of atn with tn tending to zero as n tends to infinity. Then, by Lemma 6.3, each wn decays exponentially in the region {x | E · σ (x) − μ < 0} and the rate of decay increases as n increases. In particular, we can use Proposition 6.4 to prove that the measure |wn (x)|2 d x concentrates in the ‘classically allowed region’ {x | E · σ (x) − μ ≥ 0}. Proposition 9.1 is a twofold strengthening of this latter statement: We prove that if E is not critical then |wn (x)|2 d x does not concentrate solely on {x | E · σ (x) − μ = 0}, and we prove that this also holds true for a quasimode of order 1. Estimate (62) for eigenfunctions could be obtained using a contradiction argument which is standard in the study of semiclassical measures. (See [Hillairet10] for closely related topics.) However, we believe that this method fails for first order quasimodes. Proof of Proposition 9.1. Let E > μ/σ (0). Then f E (0) < 0 and since f E = μ − E · σ is strictly increasing with lim x→∞ f E (x) = μ, there exists a unique x E > 0 such that f E (x E ) = 0. Since f E changes sign at x E , we have

∞

xE

(− f E ) · w 2 d x =

| f E | · w2 d x −

0

0

∞

| f E | · w 2 d x.

(64)

xE

Thus, by Lemmas 9.5 and 9.4 below, there exist constants C + , c− > 0 and t ∗ > 0 such that if t < t ∗ , then

∞

(− f E ) · w 2 d x ≥ c−

0

∞

1

∞

w2 d x − C + · t 3

0

w 2 d x.

(65)

0

Thus, if t < t0 = (c− /2C + )3 , then we have (62) with κ = c− /(2 · σ (0)).

Lemma 9.4. There exist constants C + and t + > 0 so that if t < t + , then

∞

1

| f E | · w2 d x ≤ C + · t 3

xE

w 2 d x.

(66)

0

xE

Lemma 9.5. There exist constants c− > 0, and t − > 0 so that if t < t − , then

xE 0

| fE| · w dx ≥ c 2

−

∞

w 2 d x.

(67)

0

The proofs of Lemma 9.5 and 9.4 are based on estimates provided in Sects. 6, 7, and 8. In preparation for these proofs we provide the common context. First note that the Riesz representation theorem provides r ∈ Hσ so that for all v ∈ dom(at ), |at (w, v) − E · w, v σ | = r, v σ , where r σ ≤ C0 · t · wσ .

Spectral Simplicity

317

Let W denote the Langer-Cherry transform of w at energy E (see §7). In particular, 1 W = (φ E ) 2 · w ◦ φ −1 E , where φ E is defined by (39). By Proposition 7.3, the function W satisfies (48) with g equal to the right-hand side of Eq. (44). As a last preparation for the proofs, we define the endpoints of the intervals over which we will apply the estimates from the preceding sections. Let x E+ be defined by μ/2 f E (x E+ ) = μ/2. In other words, x E+ = x E , where x Es is defined in (30). Define y E+ = 2 · φ E (x E+ ) and y E− = φ E (0). Since σ is decreasing, we have 0 < x E < x E+ , and hence since φ E is strictly increasing, we have y E− < 0 < y E+ . It follows from Lemma 7.1 and Remark 6.1 that y E+ and y E− depend smoothly on E. Proof of Lemma 9.4. Since σ is decreasing, we have sup {| f E | | x ≥ x E } = μ, and thus ∞ ∞ | f E | · w2 d x ≤ μ w 2 d x. (68) xE

Since φ E [x E , x E+ ] = 0,

∞

+

yE 2

xE

and W 2 · dy = (φ E )2 · w 2 · d x, we have

y+ E 2

w d x ≤ C1 2

W 2 dy +

x E+

0

xE

∞

w 2 d x,

(69)

where C1 = max{(φ E (x))−2 | E ∈ K , x ∈ [x E , x E+ ]}. By Lemma 8.2 there exist constants C E and t E > 0 so that if t < t E , then ⎞ ⎛ − y+ yE y+ y +E E E 2 2 1 5 W 2 dy ≤ C E ⎝t 3 W 2 dy + y + W 2 dy + t − 3 g 2 d x ⎠. y E−

0

y E−

E 2

(70)

The constants C E and t E depend continuously on E and hence C2 = sup{C E | E ∈ K } is finite and t2 = inf{t E | E ∈ K } is positive. Since W 2 · dy = (φ E )2 · w 2 · d x, we have t

1 3

y− E 2

y E−

2

W dy +

y E+ y+ E 2

W dy ≤ C3 · t 2

1 3

∞

w dx + 2

0

∞ x E+

w dx , 2

(71)

where C3 := sup (φ E (x))2 | E ∈ K , x ∈ 0, φ −1 (y E+ ) . By Lemma 7.5, there exists a constant C ∗ so that

y E+ y E−

∗

g dy ≤ C · t 2

2 0

∞

w 2 d x.

(72)

318


By substituting (71) and (72) into (70) we find that if t < t2 , then

y+ E 2

1

W 2 dy ≤ C4 · t 3

0

∞

w2 d x + C5

0

∞ x E+

w 2 d x,

(73)

where C4 = C2 · (C3 + C ∗ ) and C5 = C2 · C3 . By Lemma 6.2, there exists a constant C6 so that if t < 1, then ∞ ∞ 1 w2 d x ≤ C6 · t 3 w 2 d x. (74) x E+

0

By combining (69), (73), and (74), we find that if t < t3 := min{1, t2 }, then ∞ ∞ 1 w2 d x ≤ C7 · t 3 w 2 d x, xE

(75)

0

where C7 = C1 · C4 + C1 · C5 · C6 + C6 . Finally, split the integral on the right-hand side of (75) into the integral over [0, x E ] and the integral over [x E , ∞). Then subtract the latter integral from both sides of (75). It follows that if t < min{t3 , (2C7 )−3 }, then xE 1 1 ∞ 2 3 w d x ≤ C7 · t w 2 d x. 2 xE 0

The claim then follows by combining this with (68). We have the following corollary of the proof.

Corollary 9.6. There exist constants C and t > 0 such that if t < t , ∞ xE 1 w2 d x ≤ C · t 3 w 2 d x. 0

xE

Proof of Lemma 9.5. Since W 2 · dy = (φ E )2 · w 2 · d x we have xE 0 2 2 | f E | · w d x ≥ c1 f E ◦ φ −1 E · W dy, y E−

0

where c1 = inf φ E (x)−2 | E ∈ K , x ∈ [0, x E ] . Since f E ◦ φ −1 E is negative and − − increasing on [y E , y E /2], we have

y− E 2

y E−

2 · W dy ≥ c f E ◦ φ −1 2 E

y− E 2

y E−

W 2 dy,

− E ∈ K . Putting these two estimates together y /2 where c2 = inf f E ◦ φ −1 E E we have

xE 0

| f E | · w d x ≥ c1 · c2 2

y− E 2

y E−

W 2 dy.

It follows from Lemma 7.1 that c1 and c2 are both positive.

(76)

Spectral Simplicity

319

By Lemma 8.2, there exist constants C E and t E > 0 so that if t < t E , then

⎛ 0 y E−

W ≤ CE · ⎝

y− E 2

2

y E−

2

W +t

− 53

y E+ y E−

⎞ g ⎠. 2

(77)

Moreover, C E and t E depend continuously on E, and hence the constants c3 = sup{1/C E | E ∈ K } and t1 = inf{t E | E ∈ K } are both positive. By manipulating (77) we find that

y− E 2

y E−

W 2 ≥ c3

0 y E−

5

W 2 − t− 3

y E+ y E−

g2

(78)

for each t < t1 . By combining (76), (78), and (72) we find that for t < t1 ,

xE

| f E | · w d x ≥ c4 2

0

0 y E−

W dy − C · t 2

1 3

∞

w 2 d x,

(79)

0

where c4 = c1 · c2 · c3 and C = c1 · c2 · C ∗ . Since W 2 · dy = (φ E )2 · w 2 · d x, we have

0 y E−

xE

W 2 dy ≥ c5

w 2 d x,

(80)

0

where c5 = inf{(φ E (x))2 | E ∈ K , x ∈ [0, x E ]} is positive by Lemma 7.1. By substituting (80) into (79) and applying Corollary 9.6, we find that if t < t2 = min{t1 , t }, then

xE 0

| f E | · w 2 d x ≥ c4 · c5

∞

1 w 2 d x − C + c4 · c5 · C · t 3

0

∞

w 2 d x. (81)

0

3 If t < t − = min t2 , c4 · c5 /2(C + c4 · c5 · C ) , then (67) holds with c− = c4 ·c5 /2.

10. Convergence, Estimation, and Separation of Eigenvalues μ

Let at be the family of quadratic forms defined as in §5. In this section, we will evaluate the limit to which each real-analytic eigenvalue branch converges (Proposition 10.1), estimate the asymptotic behavior of eigenvalues (Proposition 10.3), and show that if both t and E − μ/σ (0) are sufficiently small, then eigenvalues near energy E must be ‘super-separated’ at order t (Theorem 10.4).

320

L. Hillairet, C. Judge μ

10.1. Convergence. Let t → λt be a real-analytic eigenvalue branch of at with respect to ·, · σ . Since |w |2 ≥ 0 and σ is decreasing, we have ∞ μ 0 |wt |2 d x μ . (82) ≥ λt ≥ ∞ 2 · σ dx σ (0) |w | t 0 μ

The first derivative of at , μ a˙ t (u)

w (x)2 d x, t

∞

= 2t 0

is nonnegative, and hence by Proposition 3.3, the eigenbranch λt converges as t tends to zero. Proposition 10.1. We have lim λt =

t→0

μ . σ (0)

Proof. Let wt be an eigenfunction branch associated to E t . The variational formula (13) becomes ∞ 2 w (x) d x. (83) λ˙ · wt 2σ = 2t t 0

μ

Using the eigenvalue equation for at with respect to ·, · σ we find that t

2

w (x)2 d x = t

∞

0

∞

(λt · σ (x) − μ) · |wt (x)|2 d x.

0

By combining this with (83) and (82) we find that ∞ 2 ˙λ · wt 2σ ≥ · (λt · σ (x) − μ) · |wt (x)|2 d x. t 0

(84)

Suppose to the contrary that λ0 := limt→0 λt = μ/σ (0). Then by (82), we have λ0 > μ/σ (0). Let K be the compact interval [λ0 , λ0 + 1]. Then for all t sufficiently small, λt ∈ K . Hence we can apply Proposition 9.1, with E = λt , and obtain a constant κ > 0 such that ∞ (λt · σ (x) − μ) · |wt (x)|2 d x ≥ κ · wt (x)2σ . 0

By combining this with (84) we find that d 2·κ λt ≥ . dt t The left-hand side is integrable on an interval of the form [0, t0 ), but the right-hand side is not integrable on such an interval. The claim follows.

Spectral Simplicity

321

10.2. Airy eigenvalues. The remainder of this section concerns quantitative estimates μ on the eigenvalues of at for t small. In particular, we will use the Langer-Cherry transμ form to compare the eigenvalues of at to the eigenvalues of the operator associated to the Airy equation. We first define and study the eigenvalue problem for the model operator. For each z ∈ R and u ∈ C0∞ [z, ∞) define Az (u)(y) = −u

(y) + y · u(y). The operator Az is symmetric with respect to the L 2 ([z, ∞), dy) inner product, and we have Az (u), u ≥ z · u2 . Thus, by the method of Friedrichs, we may extend Az to a densely defined, self-adjoint operator on L 2 ([z, ∞), dy) with either Dirichlet or Neumann conditions at y = z. Let A± be the solutions to the Airy equation defined in Appendix A. Proposition 10.2. The real number ν is a Dirichlet (resp. Neumann) eigenvalue of Az with respect to the L 2 -norm if and only if z − ν is a zero of A− (resp. A − ). Moreover, each eigenspace of Az is 1-dimensional and each eigenvalue of Az is strictly greater than z. Proof. If ψ is an eigenfunction with eigenvalue ν, then x → ψ(x + ν) is solution to the Airy equation that decays as x tends to infinity. Sturm-Liouville theory ensures that the associated eigenspaces are one-dimensional. 10.3. Estimation. Proposition 10.3. There exists δ0 and C such that for any t ≤ δ0 , if λ ∈ [ σ μ(0) , σ μ(0) + δ0 ] μ is a Dirichlet (resp. Neumann) eigenvalue of at , then there exists a zero, z, of A− (resp.

A− ) such that 2 (85) φλ (0) − t 3 · z ≤ C · t 2 . Proof. We set = 21 min{ρ E−4 (0) | E ∈ [ σ μ(0) , σ μ(0) + 1]}, and we choose δ0 to be the minimum of 1 and the δ provided by Lemma 7.7 that is associated with this and q identically 1. Let K be the compact [ σ μ(0) , σ μ(0) + δ0 ]. Let w be an eigenfunction with eigenvalue λ ∈ K and t ≤ δ0 . Let W the Cherry-Langer transform of w at energy λ. According to Lemma 7.7 and to the choice we made of , we have ∞ ∞ ∞ 1 3 2 2 w d x ≤ W dy ≤ w 2 d x. 2ρλ (0)4 0 2ρλ (0)4 0 φλ (0) Combining with Lemma 7.6, (and using that ρ E (0) is uniformly bounded away from 0 over the compact K ), there exists a constant C such that ∞ ∞ |t 2 · W

− y · W |2 dy ≤ C · t 4 |W (y)|2 dy. (86) φλ (0)

φλ (0)

2 3

Setting U (x) = W (t · x), we have ∞ 8

2 |U − x · U | d x ≤ C · t 3 2 t − 3 ·φλ (0)

2

t − 3 ·φλ (0)

|U (x)|2 d x.

(87)

322

L. Hillairet, C. Judge 2

Let z t = t − 3 · φλ (0). Then U (z) = 0 (resp. U (z) = 0) if λ is a Dirichlet (resp. Neumann) eigenvalue. In particular, U belongs to the domain of Az t . Moreover, from (87) we have that Az (U )2 ≤ C · t 83 · U 2. (88) t Thus, since Az t is self-adjoint,

! 8 A2z t (U ), U ≤ C · t 3 · U 2.

(89) 8

Thus, by the minimax principle, A2z t has an eigenvalue in the interval [0, Ct 3 ]. Hence 1

4

1

4

Az t has an eigenvalue in the interval [−C 2 t 3 , C 2 t 3 ], and the claim follows from Proposition 10.2. μ

10.4. Separation. We next show that, as t tends to zero, the eigenvalues of at with respect ·, · σ are separated at order greater than t. More precisely, we have the following. Theorem 10.4. Let t1 , t2 , t3 , . . . be a sequence of positive real numbers such that limn→∞ tn = 0. For each n ∈ Z+ , let λ+n and λ− n be distinct eigenvalues of the quadratic μ form atn . If limn→∞ λ± n = μ/σ (0), then lim

n→∞

1 + · λn − λ− n = ∞. tn

This fact may be understood by using the following semiclassical heuristics: The threshold σ μ(0) is the bottom of the potential, and the eigenvalues near it are driven by the shape of this minimum. Since σ (0) = 0, the asymptotics are given by the eigenvalues of the model problem Pt u = −t 2 · u

+ x · u = 0 on (0, ∞). Denote by en (t) the n th 2 eigenvalue of the model operator. Using homogeneity, en (t) behaves like en (1) · t 3 (and en (1) actually is some zero of the Airy function see Proposition 10.2). For fixed n, the 2 separation between two eigenvalues is thus of order t 3 . It would be relatively straightforward to make the preceding reasoning rigorous in the case of a finite number of real-analytic eigenvalue branches. (For instance we could use [FrdSlm09]). Unfortunately, this is not enough for our purposes. In Sect. 14 we will need the result for a sequence of eigenvalues that may belong to an infinite number of distinct branches. Remark 10.5. The same semiclassical heuristics show that this super-separation does not hold near an energy strictly greater than σ μ(0) . Indeed, near a non-critical energy, the spectrum is separated at order t. Proof of Theorem 10.4. Suppose to the contrary that there exists a subsequence—that + ± we will abusively call tn —such that |λ− n − λn |/tn is bounded. Let wn denote a sequence ± ± + of eigenfunctions associated to λn with wn σ = 1. Since λ− n = λn , we have − + wn , wn σ = 0. + Let Wn± denote the Langer-Cherry transform of wn± at the energy E n = sup{λ− n , λn }. By hypothesis limn→∞ = μ/σ (0). By Lemma 7.6 and Lemma 7.7, we find that there exist N1 and C such that if n > N1 , then 2 2 (90) −tn2 · ∂ y2 − y Wn± ≤ C · tn2 · Wn± .

Spectral Simplicity

323

Since wn− , wn+ σ = 0 and wn± σ = 1, it follows from Lemma 7.7 that there exists N2 > N1 such that if n > N2 , then − W , W + ≤ 1 · W − · W + . n n n n 2 This implies that for any linear combination of Wn+ and Wn− we have |α+ |2 Wn+ 2 + |α− |2 Wn− 2 ≤ 2α+ Wn+ + α− Wn− 2. Therefore, it follows from (90) that if W belongs to the span, Wn , of {Wn− , Wn+ }, then 2 −tn2 · ∂ y2 − y W ≤ 4 · C · tn2 · W 2 . 2

Let U (x) = W (t 3 · x) and let Un denote the vector space corresponding to Wn . If U ∈ Un , then 2 2 2 (91) ∂x − x U ≤ 4 · C · tn3 · U 2 . Since wn± satisfies the boundary condition at 0, the Langer-Cherry transform Wn± at energy E n satisfies the boundary condition at φ E n (0). It follows that Un ⊂ dom(Az n ), − 23

where z n = tn

· φ E n (0). By (91) we have 2

A2z (U ), U ≤ 4 · C · tn3 · U 2 for each U ∈ Un . Hence, by the minimax principle, A2z n has at least two independent 2

eigenvectors with eigenvalues in the interval [0, 4C · tn3 ]. Thus, Az n has at least two 1 1 √ √ independent eigenvectors with eigenvalues in the interval [−2 C · tn3 , 2 C · tn3 ]. By Proposition 10.2, the eigenvalues of Az n are simple, and hence Az n has at least two 1 1 √ √ distinct eigenvalues, νn+ < νn− lying in [−2 C · tn3 , 2 C · tn3 ]. By Proposition 10.2, the number an± = z n − νn± is a zero of the funtion A− . Note that 1 √ |an+ − an− | ≤ 4 C · tn3 .

(92)

Since A− is real-analytic and A− (x) = 0 for x nonnegative, the zeroes Z of A− are a countable discrete subset of (−∞, 0). In particular, there is a unique bijection : Z → Z+ such that a < a implies (a) > (a ) and limk→∞ −1 (k) = −∞. From the asymptotics of A− —see Appendix A—one finds that there exists a constant c > 0 so that 2

lim k − 3 · −1 (k) = −c, 2 1 lim k 3 · −1 (k) − −1 (k + 1) = · c. k→∞ 3 k→∞

(93) (94)

Since limn→∞ tn = 0, estimate (92) implies that limn→∞ an± = −∞, and hence limn→∞ (an± ) = ∞. Therefore, since an+ = an− for all n, we have from (94) that there exists N such that if n > N then c 1 lim an+ − an− ≥ · (an+ )− 3 . k→∞ 2

324


By combining this with (92) we find that 1 c ((an ) · tn ) 3 ≥ √ . 4 C

(95)

But since limn→∞ E n = μ/σ (0), we have limn→∞ φ E n (0) = 0. Therefore, by Propo2 sition 10.3 we have limn→∞ t 3 · an± = 0. By (93) we have lim

n→∞ 2

an 2

(an ) 3

= −c.

2

Thus, limn→∞ tn3 · (an ) 3 = 0. This contradicts (95).

11. Separation of Variables in the Abstract Recall that the first step in our method for proving generic simplicity consists of finding a family at such that qt is asymptotic to at and such that at decomposes as a direct sum μ of ‘1-dimensional’ quadratic forms at of the type considered in the previous sections. μ In the present section we discuss the decomposition of at into forms at . Although the content is very well-known, we include it here for the purpose of establishing notation and context. Let ·, · σ be the inner product on Hσ defined in §5. Let H be Hilbert space "a real with inner product (·, ·). Consider the tensor product H := Hσ H completed with respect to the inner product ·, · determined by u 1 ⊗ ϕ1 , u 2 ⊗ ϕ2 := u 1 , u 2 σ · (ϕ1 , ϕ2 ).

(96)

Let b be a positive, closed, densely defined quadratic form on H . We will assume that the spectrum of b with respect to (·, ·) is discrete " and the eigenspaces are finite dimensional. For each t > 0 and u ⊗ ϕ ∈ C0∞ ([0, ∞)) dom(b), define ∞ ∞ 2

2 at (u ⊗ ϕ) = t · (ϕ, ϕ) |u (x)| d x + b(ϕ) |u(x)|2 d x. (97) 0

0

C0∞ ([0, ∞))

Let Y ⊂ be a subspace. The restriction of at to Y ⊗ H is a nonnegative real quadratic form. By Theorem 1.17 in Chap. VI of [Kato], this restriction has a unique minimal closed extension. In particular, let dom(at ) be the collection of u ∈ Hσ ⊗ H such that there exists a sequence u n ∈ Y ⊗ dom(b) such that limn→∞ u n − u = 0 and u n is Cauchy in the norm [u]t := at (u) + uH . For each u ∈ dom(at ) define at (u) := lim at (u n ), n→∞

where u n is a sequence as above. For t, t > 0 the norms [·]t and [·]t are equivalent, and hence dom(at ) does not depend on t. Remark 11.1. In applications, either Y = C0 ([0, ∞)) or Y consists of smooth functions whose support is compact and does not include zero. In the former case, eigenfunctions of at will satisfy a Neumann condition at x = 0 and in the latter case they will satisfy a Dirichlet condition at x = 0.

Spectral Simplicity

325

Proposition 11.2. The family t → at is a real-analytic family of type (a) in the sense of Kato.2 Proof. For each t the form at is closed with respect to ·, · , the domain dom(at ) is constant in t, and for each u ∈ dom(at ), the function t → at (u) is analytic in t. on a compact LipsExample 11.3. Let H be the space of square integrable functions " chitz domain U ⊂ Rn with the usual inner product. Then H H is isomorphic to the completion of C0∞ ((0, ∞) × U ) with respect to the inner product ∞ f, g = f (x, y) · g(x, y) · σ (x) d x d y. U

0

Let b˜ be the quadratic form defined on H 1 (U ) by ˜ b(φ) = |∇φ|2 d x d y.

(98)

U

We define b to be the restriction of b˜ to any closed subset of H 1 (U ) on which it defines a positive quadratic form. In this case the quadratic form at is equivalent to the form t 2 · |∂x u|2 + |∇ y u|2 d x d y. (99) a t (u) = R+ ×U

μ

μ

For each μ > 0 and t > 0, we define the quadratic form at as in §5. The form at is equivalent to the construction above with H = R with its standard inner product μ μ and b(s) = μ · s 2 . The norms [·]t,μ and [·]t ,μ that are used to extend at and at are μ equivalent. Hence dom(at ) is independent of t and μ. We will denote this common domain by D. Proposition 11.4. If φ is a μ-eigenvector for b with respect to ·, · H , and v is a μ λ-eigenvector of at with respect to ·, · σ , then v ⊗ φ is a λ-eigenvector of at with respect to ·, · H . Conversely, if u is a λ-eigenvector of at with respect to ·, · H , then u μ is a finite sum vμ ⊗ φμ , where vμ is a λ-eigenvector of at with respect to ·, · σ and φμ is a μ-eigenvector of b with respect to ·, · H . Proof. Straightforward.

Proposition 11.5. For each analytic eigenvalue branch λt of at , there exists a unique μ μ ∈ spec(b) such that λt is an analytic eigenvalue branch of at . In particular, λt μ decreases to σ (0) as t tends to 0. Proof. Let t0 > 0. For each μ ∈ spec(b), consider # the set Aμ of t ∈ (0, t0 ) such that μ λt ∈ spec(at ). By Proposition 11.4, the union μ Aμ equals (0, t0 ). Since spec(b) is countable, the Baire Category Theorem implies that there exists μ ∈ spec(b) such μ that Aμ has nonempty interior A0μ . For each real-analytic eigenvalue branch νt of at , let Bν ⊂ A0μ be the set of t such that νt = λt . Since there are only countably many eigenvalue branches, the Baire Category Theorem implies that there exists an eigenvalue μ branch νt of at such that Bμ has nonempty interior Bμ0 . Since λt and νt are real-analytic functions that coincide on a nonempty open set, they agree for all t. The latter statement then follows from Proposition 10.1. 2 See Chap. VII §4.2 in [Kato].

326


Corollary 11.6. If each eigenspace of b is 1-dimensional, then for each t belonging to the complement of a countable set, each eigenspace of at with respect to ·, · H is 1-dimensional. Proof. Use the assumption that b is simple and the fact that the eigenbranches are analytic. We end this section by establishing some notation that will be useful in the sections that follow. For each eigenvalue μ of b, let Vμ denote the associated eigenspace " and let Pμ : H → Vμ denote the associated orthogonal projection. Define μ : Hσ H by μ (v ⊗ w) = v ⊗ Pμ (w). If M is a collection of eigenvalues μ of b, then we define M to be the orthogonal projection onto the direct sum of μ-eigenspaces. That is,

M = μ . μ∈M

The subscript for may represent either an eigenvalue or a set of eigenvalues. Assumption 11.7. In what follows we assume that each eigenspace of b with respect to ·, · is 1-dimensional. One convenient consequence of this assumption is that for each w ∈ H, there exists w˜ μ ∈ Hσ and a unit norm eigenvector φμ of b such that μ (w) = w˜ μ ⊗ φμ .

(100)

Indeed, for each μ ∈ spec(b), let φμ ∈ Vμ . Since dim(Vμ ) = 1, each vector in Hσ ⊗ Vμ is of the form v ⊗ φμ . In particular, there exists w˜ μ so that (100) holds. Note that

w=

μ∈spec(b)

μ (w) =

w˜ μ ⊗ φμ .

μ∈spec(b)

12. Projection Estimates In this section, qt will denote a family of quadratic forms densely defined on H that is asymptotic at first order3 to the family at defined in the preceding section. Let PaIt be the orthogonal projection onto the direct sum of eigenspaces of at associated to the eigenvalues of at that belong to the interval I (see §2). We will provide some basic estimates on w := PaIt (u)

(101)

We begin with the following quasimode type estimate. In the sequel φμ will denote a unit norm eigenvector of b with eigenvalue μ. By Assumption 11.7, φμ is unique up to sign. 3 See Definition 3.1.

Spectral Simplicity

327

Lemma 12.1. Let J ⊂ I be a proper closed subinterval of a compact interval I . There exist constants C > 0 and t0 > 0 such that if μ ∈ spec(b), t < t0 , u is an eigenfunction of qt with eigenvalue E ∈ J, z ∈ D, then the projection w = PaIt (u) satisfies $ % at μ w, z ⊗ φμ − E · μ w, z ⊗ φμ ≤ C · t · zσ · w. (102) Proof. Since qt and at are asymptotic at first order, Lemma 2.5 applies. In particular, 1 −1 by letting δ = dist(J, " ∂ I ), t0 = 2 (1 + E/δ) , and C = (4/3) · sup(I ), we have for t < t0 and v ∈ D dom(b), |at (w, v) − E · w, v | ≤ C · t · v · w. For each μ ∈ spec(b), there exists w˜ μ ∈ D so that

w= w˜ μ ⊗ φμ

(103)

(104)

μ ∈spec(b)

and v = v˜μ ⊗ φμ . If μ = μ, then b(φμ , φμ ) = 0 and φμ , φμ = 0, and hence using (96) and (97) we find that % $ at (w˜ μ ⊗ φμ , z ⊗ φμ ) − E · w˜ μ ⊗ φμ , z ⊗ φμ = 0. Thus,

$ % at (w, v) − E · w, v = at μ w, v − E · μ w, v .

The claim then follows from substituting this into (103).

Lemma 12.2. Let J ⊂ I be a proper closed subinterval of a compact interval I . Let μ ∈ spec(b) with μ < σ (0) · inf(I ) and let > 0. There exist constants κ > 0 and t0 > 0 such that if t < t0 , u is an eigenfunction of qt with eigenvalue E ∈ J , and μ w ≥ · w, where w = PaIt (u), then we have 2 κ a˙ t μ (w) ≥ · μ (w)σ . t

(105)

Proof. We have μ w = w˜ μ ⊗ φμ for some w˜ μ ∈ D. Since, by assumption, φμ = 1, we have μ (w) = w˜ μ and hence the assumption becomes w˜ μ σ ≥ · w. Therefore, Lemma 12.1 gives μ at w˜ μ , z − E t · w˜ μ , z σ ≤ C · t · zσ · w˜ μ σ

(106)

for all sufficiently small t. Since μ/σ (0) < inf(I ), the compact set I is a subset of (μ/σ (0), ∞). Hence we may apply Proposition 9.1 to obtain κ > 0 and t1 > 0 so that if t < t1 , then ∞ (107) (E t · σ (x) − μ) · |w˜ μ |2 d x ≥ κ · w˜ μ 2σ . 0

328


Inspection of (97) gives that

a˙ t (w˜ μ ⊗ φμ , w˜ μ ⊗ φμ ) = 2t · φμ , φμ In particular a˙ w˜ μ ⊗ φμ = 2t

∞ 0

∂x w˜ μ · ∂x w˜ μ .

(108)

∞

0

∂x w˜ μ 2 d x.

μ at

Thus, by using the definition of and estimates (106) and (107) we find that ∞ 2 μ at (w˜ μ ) − μ |w˜ μ |2 d x a˙ μ (w) = t 0 ∞ 2C 2 ≥ · w˜ μ 2σ (E t · σ − μ) |w˜ μ |2 d x − t 0 C κ − · w˜ μ 2σ . ≥2 t By choosing t0 = min{t1 , C/( · κ)} we obtain the claim.

Remark 12.3. In the preceding lemma the constants t0 and κ a priori depend on the chosen μ. However, since there is only a finite number of eigenvalues of b that satisfy μ ≤ σ (0) inf I , we can choose t0 and κ depending only on I and not on the eigenvalue μ. It will be convenient to introduce the following notation. Given μ ∈ spec(b), define μ˜ =

μ , σ (0)

where σ is as in §11. For each compact interval I , define M I = {μ ∈ spec(b)| μ˜ ∈ I }, M− ˜ < inf I }, I = {μ ∈ spec(b)| μ M+I = {μ ∈ spec(b)| μ˜ > sup I }. + The spectrum spec(b) equals the disjoint union of M− I , M I , and M I , and in particular, each v ∈ H can be orthogonally decomposed as

v = M− (v) + M I (v) + M+I (v). I

The following lemma is crucial to our proof of generic simplicity. The proof uses both Theorem 4.2 and—by way of Lemma 12.2—Proposition 9.1. Lemma 12.4. Let E t be a real-analytic eigenvalue branch qt , and let Vt be the associated family of eigenspaces. Let t → u t be a map from (0, t0 ] to Vt that is continuous on the complement of a countable set. If wt = PaIt (u t ), then lim inf t→0

M− (wt ) I

wt

= 0.

Here if wt = 0, then we interpret the ratio to be equal to 1.

(109)

Spectral Simplicity

329

Proof. Suppose that (109) is false. We have the orthogonal decomposition M− (wt ) = I

μ (wt ),

μ∈M− I

and hence there exists > 0 and t0 > 0 such that for each t < t0 there exists μt ∈ M− I such that μt (wt ) ≥ · wt .

(110)

Using the orthogonal decomposition of w as in (104) we find that a˙ t (wt ) =

a˙ t μt (wt ) .

μ∈spec(b)

(See also (108).) In particular, since the quadratic form a˙ t is nonnegative, we have that a˙ t (wt ) ≥ a˙ t (μt (wt )). Thus we may apply Lemma 12.2 with J = E((0, t0 ]) as well as (110) to find that a˙ t (wt ) ≥

·κ · wt 2 t

for all t sufficiently small with some κ independent of t (according to Remark 12.3). Thus, it follows from Theorem 4.2 that the function 1/t is integrable on an interval whose left endpoint is zero. This is absurd. Lemma 12.5. Let I be a compact interval. If w belongs to the range of PaIt , then M+I (w) = 0. In particular, 2 2 w2 = M I (w) + M− (w) . I

Proof. By definition, w is a linear combination of eigenfunctions of at whose eigenvalues belong to I . Hence by Proposition 11.4, we have w=

vλ,μ ⊗ φμ ,

μ∈spec(b) λ∈I ∩spec(atμ ) μ

where vλ,μ belongs to the λ-eigenspace of at and φμ belongs to the μ-eigenspace of b. Hence

M+I (w) = vλ,μ ⊗ φμ . (111) μ∈M+I λ∈I ∩spec(atμ )

μ

˜ If μ ∈ M+I , According to Proposition 10.1, each eigenvalue λ of at satisfies λ ≥ μ. then μ˜ ≥ sup(I ). Hence each term in (111) vanishes.

330


13. The Limits of the Eigenvalue Branches of q t Proposition 3.4 implies that each real-analytic eigenvalue branch E t of qt converges as t tends to zero. In this section we use the results of the previous section to show that each limit belongs to the set & = {μ˜ | μ ∈ spec(b)}. spec(b) Theorem 13.1. For each real-analytic eigenvalue branch E t of qt , we have & lim E t ∈ spec(b).

t→0

& Since Proof. Suppose to the contrary that the limit, E 0 , does not belong to spec(b). & is discrete, there exists a nontrivial compact interval I such that E 0 ∈ J , such spec(b) that & = ∅. J ∩ spec(b)

(112)

Since J is nontrivial and E t is continuous, there exists t0 such that if t < t0 , then E t ∈ J . & . Let I be a compact interval such that J ⊂ I ⊂ R\spec(b) Let u t be a real-analytic eigenfunction branch associated to E t and let wt = PaIt (u t ). We have chosen I so that M I = ∅. Thus, by Lemma 12.5, 2 M− (wt ) = wt 2. I

This contradicts Lemma 12.4.

14. Generic Simplicity of q t In this section, we prove that the spectrum of qt is generically simple. We will make crucial use of the ‘super-separation’ of the eigenvalues of at for small t (see Theorem 10.4). Before providing the details of the proof, we first illustrate how super-separation can be useful in proving simplicity. Suppose that there exists an eigenvalue branch E t of qt such that E t → μ˜ and the associated real-analytic family of eigenspaces Vt is at least two dimensional. If for each u t ∈ Vt we knew that μ u t were uniformly bounded away from 0, then, arguing as in the beginning of the proof of Lemma 12.2, we would μ find that μ u t is a first order quasimode of at at energy μ. ˜ Then, since dim(Vt ) ≥ 2, μ we would have a sequence tn tending to zero and two distinct eigenvalues λ, λ of atn

such that λ − λ /tn is bounded. This would contradict super-separation. Theorem 14.1. Let E t be a real-analytic eigenvalue branch E t of qt , and let Vt be the associated real-analytic family of eigenspaces (see Remark 4.1). For each t ∈ (0, t0 ] we have dim(Vt ) = 1. Since each eigenvalue branch of qt is real-analytic and the spectrum of each qt is discrete with finite dimensional eigenspaces, we have the following corollary. Corollary 14.2. Let E t be a real-analytic eigenbranch, then E t is a simple eigenvalue of qt for all t in the complement of a discrete subset of (0, t0 ].

Spectral Simplicity

331

Proof of Theorem 14.1. Suppose that the conclusion does not hold. Since Vt is a realanalytic family of vector spaces, its dimension is constant and so for each t ∈ (0, t0 ], we have dim(Vt ) > 1. By Theorem 13.1 there exists μ ∈ spec(b) such that E t tends to μ˜ = μ/σ (0) as t & = {μ}. tends to zero. Let I be a compact interval so that I ∩ spec(b) ˜ By Lemma 14.3 below, there exists t3 ≤ t0 and a map t → u t from (0, t3 ] into Vt that is continuous on the complement of a discrete set so that if t ∈ (0, t3 ]\Z , then μ (wt )
0 we have dim(Vt ) > 1. Let μ ∈ spec(b) be such that limt→0 E t = μ, ˜ and let I be a compact interval such that & = μ. I ∩ spec(b) ˜ There exists t0 > 0 and a function t → u t that maps (0, t0 ] to Vt , is continuous on the complement of a discrete set, and satisfies μ (wt ) ≤ 1 · wt (113) 2 where wt = PaIt (u t ). To prove Lemma 14.3, we will use the following well-known fact. Lemma 14.4. Let {gk : (a, b) → R | k ∈ N} be a collection of real-analytic functions. If for each k ∈ N and t ∈ (a, b) we have gk+1 (t) > gk (t) then the set {t ∈ (a, b) | gk (t) = 0, k ∈ N} is a discrete subset of (a, b). Proof. Suppose that gk (t) = 0 for some k ∈ N and t ∈ (a, b). Since gk is real-analytic there exists an open set U t such that if t ∈ U \{t}, then gk (t) = 0. Since k > k

implies gk (t) > gk

(t) we have ' −1 (0, ∞) = gk−1 t ∈ gk+1

(0, ∞) k >k

and −1 (−∞, 0) = t ∈ gk−1

' k 0 such that if t ≤ t1 , z ∈ D, and u is an eigenfunction with eigenvalue E t , then μ $ % at w˜ μ , z − E t · w˜ μ , z ≤ C · t · w · zσ , (114) σ where w = PaIt (u) and w˜ ⊗ ϕμ = μ w. μ Since at is a real-analytic family of type (a) in the sense of [Kato], for each k ∈ N, there exists a real-analytic function λk : (0, t1 ] → R so that for each t ∈ (0, t1 ], we μ μ have spec(at ) = {λk (t) | k ∈ N}. Since each eigenspace of at is 1-dimensional, we

may assume that k > k implies λk (t) > λk (t) for all t ∈ (0, t1 ]. By Theorem 10.4, there exists t0 ∈ (0, t1 ] such that if t < t0 , then k = k , then |λk (t) − λk (t)| > 4C · t.

(115)

For each k ∈ N and t ∈ (0, t0 ), define gk± (t) = λk (t) − E t ± 2C · t. Thus, by Lemma 14.4, the set Z=

' ' (gk+ )−1 {0} (gk− )−1 {0} k∈N

is discrete in (0, t0 ]. On each component J of the complement (0, t0 ]\Z , we have either μ • for all t ∈ J , we have dist E t , spec(at ) ≥ 2C · t, or μ • for all t ∈ J , we have dist E t , spec(at ) < 2C · t. It suffices to construct in each of these cases a continuous map t → u t from J to Vt that satisfies (113). Without loss of generality, each interval J is precompact in (0, t0 ], for otherwise we may, for example, add the discrete set {1/n | n ∈ N} to Z . We consider the first case. Let u t be a real-analytic eigenfunction branch of qt associated to E t . By estimate (114), we may apply Lemma 2.1 with = C · t · wt and find that w˜ t σ ≤

1 · wt . 2

(116)

Since μ w = w˜ μ σ , the desired (113) follows. We consider the second case. By (115) and since J ⊂ (0, t0 ) there exists a unique k such that if t ∈ J , then |E t − λk (t)| < 2C · t. μ

(117)

Let t → v˜t be the unique eigenfunction branch of at associated to the eigenvalue branch λk . Since dim(Vt ) > 1 and Vt is an analytic family of vector spaces, there exist analytic eigenfunction branches xt , xt ∈ Vt so that for each t, the eigenvectors xt and xt are independent. The function t → xt , v˜t ⊗ φμ is real-analytic, and thus it vanishes on at most a finite subset Z J ⊂ J . Away from Z J , set c(t) = −

xt , v˜t ⊗ φμ . xt , v˜t ⊗ φμ

Spectral Simplicity

333

Then u t = c(t) · xt + xt depends real-analytically on t and satisfies u t , v˜t ⊗ φμ = 0. μ

For each t ∈ J \Z J , let rt denote the " restriction of the quadratic form at to the orthogonal complement of v˜t ⊗ φμ in D dom(b). Let wt = PaIt (u t ) and let w˜ μ,t ∈ D such that μ wt = w˜ μ,t ⊗ φμ . From (114), we have $ % rt w˜ μ,t , z − E t · w˜ μ,t , z ≤ C · t · wt · zσ . σ It follows from (115) that dist(E t , spec(rt )) ≥ 2C · t. Hence Lemma 2.1 applies with = 2C · t · w to give (113). # Therefore, on the complement of Z ∪ J Z J , we have constructed a real-analytic function t → Vt so that (113) holds. 15. Stretching Along an Axis In this section, we consider a family of quadratic forms qt obtained by ‘stretching’ certain domains in Euclidean space Rn+1 that fiber over an interval. To be precise, let I = [0, c] be an interval, let Y ⊂ Rn be a compact domain with Lipschitz boundary, and let ρ : [0, c] → R be a smooth nonnegative function. For t > 0, define φt : I × Y → Rn+1 by φt (x, y) = (x/t, ρ(x) · y).

(118)

We will consider the Dirichlet Laplacian associated to the domain t = φt (I × Y ). Example 15.1 (Triangles and simplices). Let Y = [0, a] and ρ(x) = x. Then t is the right triangle with vertices (0, 0), (c/t, 0), (c/t, c). More generally, if ρ(x) = x and Y is a n-simplex, then t is a n + 1-simplex. Theorem 15.2. If ρ : [0, a] → R is smooth, ρ(0) = 0, ρ > 0, c dx = ∞, lim →0 ρ(x) and each eigenspace of the Dirichlet Laplacian acting on L 2 (Y ) is 1-dimensional, then for all but countably many t, each eigenspace of the Dirichlet Laplacian acting on L 2 (t ) is 1-dimensional. Proof. In order to apply Theorem 14.1, we make the following change of variables. Define ψ : (0, c] → [0, ∞) by c dx . ψ(x) = x ρ(x) By hypothesis, ψ is an orientation reversing homeomorphism. Define t : C ∞ ([0, ∞)× Y ) → C ∞ (t ) by n−1 t (u) = ρ 2 · u ◦ (ψ × Id) ◦ φt , where φt is defined by (118). We will use t to pull-back the L 2 inner product and the Dirichlet energy form.

334


First note that the Jacobian matrix of φt is 1/t 0 , Jφ = ∂x ρ · y ρ · Id

(119)

where Id is the n × n identity matrix, and hence the Jacobian determinant |J φt | equals t −1 · ρ n . The Jacobian determinant of ψ × Id is ρ −1 . It follows that 1 ∞ u · v σ (x) d x d y, (120) (t (u) · t (v)) d V = t 0 t Y where σ = ρ 2 ◦ ψ −1 and where dy denotes Lebesgue measure on Y ⊂ Rn . In order to have an inner product that does not depend on t, we rescale by t. Define ∞ u, v = u · v σ (x) d x d y. Y

0

Define a family of quadratic forms on C ∞ ([0, ∞) × Y ) by |∇ (t (u))|2 d x d y. qt (u) = t · t

The map t defines an isomorphism from each eigenspace of qt with respect to ·, · to the eigenspaces of the Dirichlet energy form on t with respect to the L 2 -inner product on t . In particular, it suffices to show that each eigenspace of qt with respect to ·, · is 1-dimensional. Define ∞ 2 at (u) = t 2 · |∂x u|2 + ∇ y u d x d y. 0

Y

By Theorem 14.1, it suffices to show that qt is asymptotic to at at first order. Let τ = ρ ◦ ψ −1 . A straightforward calculation of moderate length shows that qt (u, v) − at (u, v) = t · (I1 (u, v) + I2 (u, v) + I3 (u, v) + I4 (u, v) + I5 (u, v) + I3 (v, u) + I4 (v, u) + I5 (v, u)), where I1 (u, v) = I2 (u, v) = I3 (u, v) = I4 (u, v) = I5 (u, v) =

(n − 1)2 ∞ t· τ 2 · u · v d x d y, 4 Y 0 ∞ t τ 2 · y · ∇ y u · y · ∇ y v d x d y, Y 0 n−1 ∞ t· τ 2 · u · y · ∇ y v d x d y, 2 Y ∞ 0 t τ · ∂x u · y · ∇ y v d x d y, Y 0 ∞ n−1 t· τ · u · ∂x v d x d y. · 2 0 Y

To get (11), it suffices to show that for each k = 1, . . . , 5, there exists a constant Ck 1 1 such that |Ik (u, v)| ≤ Ck · at (u) 2 · at (v) 2 for t < 1.

Spectral Simplicity

335

First note that by assumption |ρ |—and hence |τ |—is bounded by a constant C. Second, note that if λ0 > 0 is the smallest eigenvalue of the Dirichlet Laplacian on L 2 (Y ), then for each u ∈ C ∞ ([0, ∞) × Y ) we have

∞ 0

u2 d x d y ≤ Y

1 λ0

∞

∇ y u 2 d x d y.

(121)

Y

0

If n = 1, then |I1 (u, v)| is trivial. Otherwise, apply the Cauchy-Schwarz inequality and estimate (121). More precisely 4 · |I1 (u, v)| ≤ t · C 2 (n − 1)2

∞ 0

u2 d x d y Y

∞

1 2 ·

1

∞

0

v2 d x d y Y

1 2 ∇ y u 2 d x d y ·

t ≤ · λ0 0 Y 1 1 t ≤ · at (u) 2 · at (v) 2 . λ0

0

∞

2

∇ y v 2 d x d y

1 2

Y

To bound |I2 (u, v)|, note that |y · ∇ y u|2 ≤ |y|2 · |∇ y u|2 and that |y|2 is bounded since Y is compact. The desired bound of |I2 (u, v)| then follows from an application of the Cauchy-Schwarz inequality. If n = 1, then |I3 (u, v)| is trivial. Otherwise, we apply the Cauchy-Schwarz inequality and estimate (121) as in the bound of |I1 (u, v)|. To bound |I4 (x, y)| we apply Cauchy-Schwarz as follows:

|t · ∂x u| · y · ∇ y v ≤

1 |t · ∂x u|2

2

y · ∇ y v 2

1 2

.

It then follows that 1

1

|I4 (u, v)| ≤ C · at (u) 2 · at (u) 2 . To bound |I5 (u, v)| apply Cauchy-Schwarz and argue in a fashion similar to the above. Condition (12) also follows using that (t Ik ) = 2Ik .

15.1. Changing the boundary condition. Theorem 15.2 extends to a more general boundary condition that we describe here. Inspecting the proof, the only thing we have used from the Laplace operator on Y is that it satisfies the Poincaré inequality (121). This fact is true for any mixed Dirichlet-Neumann boundary condition except Neumann on all faces. As a consequence we may take on the faces of t of the form I × ∂Y any kind of boundary condition except full Neumann. On the face {1} × Y we may take Dirichlet or Neumann as we want since we have μ allowed Dirichlet or Neumann at 0 for the one-dimensional model operators at .

336


16. Domains in the Hyperbolic Plane with a Cusp Recall that the hyperbolic metric on the upper half-plane R × R+ is defined by (d x 2 + dy 2 )/y 2 . The associated Riemannian measure is given by dμ = y −2 d x d y and the gradient is given by ∇ f = y 2 (∂x f · ∂x + ∂ y f · ∂ y ). Let h : (−η, η) → R be a positive real-analytic function such that h (0) = 0. For each t < η, define t by t = (x, y) ∈ R × R+ | − t ≤ x ≤ t and y ≥ h(x) . The domain t is unbounded but has finite hyperbolic area. It is known that the hyperbolic Dirichlet Laplacian acting on L 2 (t , dμ) is compactly resolved and hence has discrete spectrum (see e.g. [LaxPhl]).4 √ Example 16.1. Let h : (−1, 1) → R be defined by h(x) = 1 − x 2 . For each t < 1, the domain t is a hyperbolic triangle with one ideal vertex. In particular, 1/2 is a fundamental domain for the modular group S L(2, Z) acting on R × R+ ⊂ C as linear fractional transformations. Theorem 16.2. For all but countably many t, each eigenspace of the Dirichlet Laplacian acting on L 2 (t , dμ) is 1-dimensional. The remainder of this section is devoted to the proof of Theorem 16.2. The spectrum of the hyperbolic Laplacian on t coincides with the spectrum of the Dirichlet energy form |∂x u|2 + |∂ y u|2 d x d y, (122) E(u) = t

with respect to the inner product

u, v μ =

t

u·v

dx dy . y2

(123)

In order to study the variational behavior of the eigenvalues, we first adjust the domains by constructing a family of diffeomorphisms φt from the fixed set U = [−1, 1] × [h(0), ∞[ onto t . In particular, define t ·a a . = φt b + h(t · a) − h(0) b For each u ∈ C0∞ (U), we define u˜ = ψ · u ◦ φt−1 , where ψ(x, y) =

y . y − h(x) + h(0)

Since φt is a smooth diffeomorphism from U onto t and ψ is smooth on t , the mapping u → u˜ is a bijection from C0∞ (U) onto C0∞ (t ). 4 The Neumann Laplacian is not compactly resolved, and in fact, has essential spectrum.

Spectral Simplicity

337

Since the Jacobian of φt is J (φt )

t 0 a = t · h (t · a) 1 b

(124)

and ψ ◦ φt = (y ◦ φt )/b, we find that, for any smooth u and v compactly supported in U, d xd y da db u˜ · v˜ = u·v . (125) t −1 2 y b2 t U In particular, the mapping u → u˜ extends to an isometry of H := L 2 (U, da · db/b2 ) onto L 2 (t , t −1 dμ). We now pull-back the Dirichlet energy form from t to U. In particular, we define qt : C0∞ (U) → R by ˜ qt (u) = t · E(u). The form extends to a closed densely defined form on H. By construction, λ belongs to the spectrum of qt if and only if t −2 ·λ belongs to the Laplace spectrum of the hyperbolic triangle t . Because h is real-analytic, t → φt is a real-analytic family of bi-Lipschitz homeomorphisms. It follows that qt is a real-analytic family of quadratic forms of type (a) in the sense of Kato [Kato]. On C0∞ (U), we also define t 2 · |∂b u|2 + |∂a u|2 da db. at (u) = U

Theorem 16.2 follows from Theorem 14.1 and the following proposition. Proposition 16.3. qt is asymptotic to at at first order. Proof. Let u¯ = (ψ ◦ φt ) · u. One computes that ¯ ∂ y u˜ ◦ φt = ∂b u, 1 ˜ ◦ φt = · ∂a u¯ − h (ta) · ∂b u. ¯ (∂x u) t Thus, by making a change of variables in the integral that defines E, we find that |∂a u| qt (u) = ¯ 2 − 2t · h (ta) · ∂a u¯ · ∂b u¯ + t 2 · (1 + h (ta)2 ) |∂b u| ¯ 2 da db, (126) U

where u¯ = ψ¯ · u. To aid in computation we define a weighted gradient ¯ = [∂a w, t · ∂b w], ∇w and we define

At =

1 −h (t · a) . −h (t · a) 1 + h (t · a)2

Thus, (126) becomes

qt (u, v) =

U

∇¯ u¯ · At · ∇¯ v¯ da db

338


and

at (u, v) =

U

¯ · ∇v ¯ da db. ∇u

Letting ψ¯ = ψ ◦ φ, we have ¯ + w · ∇ψ, ¯ ∇¯ w¯ = ψ¯ · ∇w and hence qt (u, v) − at (u, v) is the sum of four terms: ¯ da db, ¯ · (ψ¯ 2 · At − I ) · ∇v ∇u U ¯ da db, ψ¯ · v · (∇¯ ψ¯ · At · ∇u) U ¯ da db, ψ¯ · u · (∇¯ ψ¯ · At · ∇v) U ¯ · u · v da db, (∇ ψ¯ · At · ∇¯ ψ) U

(127) (128) (129) (130)

where I denotes the 2×2 identity matrix. To finish the proof, it suffices to show that each 1 1 of these terms is bounded by O(t) · at (u) 2 · at (v) 2 , where O(t) represents a function that is bounded by a constant times t for t small. ¯ In order to estimate these terms, we use elementary estimates of h(t · a), h (t · a), ψ,

¯ ¯ and ∇ ψ. In particular, since h (0) = 0 we have that |h(t · a) − h(0)| = O(t) and |h (t · a)| = O(t) uniformly for a ∈ [−1, 1]. Thus, since ¯ ψ(a, b) = 1 −

h(t · a) − h(0) , b

¯ = O(t) uniformly for (a, b) ∈ U. we find that |ψ¯ 2 (a, b) − 1| = O(t) and |∇ ψ| To bound (127), note that tr(ψ¯ 2 · A − I ) = 2(ψ¯ 2 − 1) + ψ¯ 2 · h (t · a)2 and det(ψ¯ 2 · A − I ) = (ψ¯ 2 − 1)2 − h (t · a)2. Hence tr(ψ¯ 2 · A− I ) = O(t) and det(ψ¯ 2 · A− I ) = O(t 2 ). It follows that the eigenvalues of ψ¯ 2 · A − I are O(t). Therefore, 2 ¯ ¯ ¯ · ∇v ¯ da db. ¯ ∇u · (ψ · At − I ) · ∇v da db = O(t) · ∇u U

U

To estimate (128) we first note that the eigenvalues of At are O(1). Then we apply Cauchy-Schwarz ¯ ≤ |∇¯ ψ| ¯ ¯ · |∇u|, |∇¯ ψ¯ · ∇u| ¯ to find that and then the elementary estimate on |∇¯ ψ| ¯ da db ≤ O(t) ¯ da db. ¯ · |v| · |∇¯ ψ¯ · ∇u| |ψ| v · |∇u| U

U

Spectral Simplicity

339

Cauchy-Schwarz applied to the latter integral gives U

¯ da db ≤ |v| · |∇u|

U

|v|2 da db

1 1 2 2 ¯ 2 da db . · |∇u| U

From a Poincaré inquality—Lemma 16.4 below—we find that 2 2 ¯ 2 da db. |v| da db ≤ π |∇v| U

U

1

1

In sum we find that the expression in (128) is bounded by O(t) · at (u) 2 · at (v) 2 . Switching the rôles of u and v, we obtain the same bound for the expression in (129). To estimate (130) we use the fact that the norm of the eigenvalues of At are O(1) ¯ 2 = O(t 2 ) to find that and the fact that |∇¯ ψ| ¯ · |u| · |v| da db = O(t) · |∇ ψ¯ · At · ∇¯ ψ| |u| · |v| da db. U

U

By applying Cauchy-Schwarz and the Poincaré inequality of Lemma 16.4 below we obtain the claim. Condition (12) follows using the same kind of arguments. Lemma 16.4. Any u ∈ C0∞ (U) satisfies: 2 2 |u| da db ≤ π |∂a u|2 da db. U

U

Proof. We decompose u = k u k (b) sin(kπa). Then we have ∞

|∂a u|2 = k2π 2 u k (b)2 db U

k

≥ π2

k

h(0) ∞

u k (b)2 db = π 2

h(0)

U

|u|2 da db.

Acknowledgements. L.H. would like to thank Indiana University for its invitation and hospitality and the ANR programs ‘Teichmüller’ and ‘Résonances et chaos quantiques’ for their support. C.J. thanks the Université de Nantes, MATPYL program, L’Institut Fourier, and the Max Planck Institut für Mathematik-Bonn for hospitality and support.

Appendix A. Solutions to the Airy Equation Here we consider solutions to Airy’s differential equation A

(u) = u · A(u)

(131)

for u ∈ R. It is well-known that there exist unique solutions A+ and A− that satisfy5 1 3 3 u− 4 2 1 + O u− 2 (132) A± (u) = 1±1 · exp ± · u 2 3 2 2 1

5 The functions π − 2 · A are the classical Airy functions Ai and Bi. See, for example, [Olver] Chap. 11. ±

340


and A± (−u) = u 3

− 14

3 2 π cos ·u2 ∓ 3 4

3 + O u− 2 ,

(133)

3

where u 2 · O(u − 2 ) is bounded on [1, ∞). Let W denote the Wronskian of {A+ , A− }. Define K : R × R → R by

K (u, v) = W

−1

⎧ ⎪ ⎨

A+ (u) · A− (v) A− (u) · A+ (v) · ⎪ ⎩ A+ (u) · A− (v) − A− (u) · A+ (v) 0

if v ≥ u ≥ 0 or v ≥ 0 ≥ u if u ≥ v ≥ 0 if u ≤ v ≤ 0 otherwise.

Lemma A.1. Let −∞ < α ≤ 0 ≤ β ≤ ∞. For each locally integrable function g : [α, β] → R of at most polynomial growth, we have (∂u2 − u)

β α

K (u, v) · g(v) dv = g(u),

(134)

Proof. The Wronskian W is constant and hence by, for example, variation of parameters we find that the function P(u) = W −1 · A+ (u)

β u

A− (v) · g(v) dv + W −1 · A− (u)

u

A+ (v) · g(v) dv

0

is a solution to P

(u) − u · P(u) = g(u). Hence K satisfies (134).

Lemma A.2. There exists a constant CAiry so that ⎧ if u, v ≥ 0 ⎨ exp (− |v −1 u|) −4 |u · v| if u≤v≤0 |K (u, v)| ≤ CAiry · ⎩ −1 |u| 4 · exp (−v) if u ≤ 0 ≤ v

(135)

⎧ u|) if u, v ≥ 0 ⎨ exp (− |v − 1 4 |u · v| if u ≤ v ≤ 0 |∂u K (u, v)| ≤ CAiry · ⎩ 1 |u| 4 · exp (−v) if u ≤ 0 ≤ v.

(136)

and

Proof. Straightforward using definition of K and the asymptotic behavior of the Airy functions [Olver]. Lemma A.3. There exists a constant C so that α α √ |K (u, v)|2 du dv ≤ C · α. −α

−α

Proof. This follows directly from Lemma A.2.

(137)

Spectral Simplicity

341

Lemma A.4. Let b− < a − < 0 < b+ < a + . There exist constants C and s0 such that if s > s0 and A is a solution to (131), then

0 s·a −

and

A2 du ≤ C

s·b+

A2 du ≤ C s

0

− 21

s·a −

s·b−

s·b− s·a −

A2 du,

A2 du +

s·2b+

s·b+

(138)

A2 du .

(139)

The constants C and s0 may be chosen to depend continuously on a − , b− , a + , and b+ . Proof. Let 0 < α < β. By using (133) and the identity cos2 (ξ ) = 2−1 · (1 + cos(2ξ )), we have β −α 1 β −1 1 β −1 A2± du = u 2 du + u 2 · cos(2ξ ) du + O (1 + u)−2 du, 2 α 2 α −β α 3

where ξ = (2/3) · u 2 ∓ π/4. Integration by parts gives β β 1 1 1 β −2 u − 2 · cos(2ξ ) du = · u −1 · sin(2ξ ) + u · sin(2ξ ) du, 2 2 α α α and hence we have

−α −β

1 1 A2± du = β 2 − α 2 + O β −1 + α −1 .

Since A± is bounded on [−1, 0] we also have 0 1 A2± du = β 2 + O(1). −β

(140)

(141)

Using (133) and the fact that 2 cos(ξ + π4 ) cos(ξ − π4 ) = cos(2ξ ), we find that for 0 < α < β, we have −α (142) A+ · A− du = O β −1 + α −1 . −β

Since A± is bounded on [−1, 0], it follows that 0 A+ · A− du = O(1). −β

(143)

We now specialize to the case α = −s · a − and β = −s · b− . By (140) and (141), there exists s1 —depending continuously on b− < a − < 0—such that for s > s1 ,

s·a − s·b−

A2± du ≥ m

0

s·a −

A2± du,

(144)

342


where

⎛ m=

1 ⎝ · 1− 2

a− 2

1

b−

⎞ ⎠.

By (141) and (142), there exists a constant s2 —depending continuously on b− , a − < 0— such that if s > s2 , then − s·a m 0 · A+ · A− du ≤ A2± du. (145) s·b− 2 s·a − If A is a general solution to (131), then there exist c+ , c− ∈ R such that A = c+ · A+ + c− · A− . Using (145) we find that − 0 s·a m 0 2 · c+2 A+ · A− du ≤ A2+ du + c− A2− du . 2|c+ · c− | · s·b− 2 s·a − s·a − By combining this with (144) we find that if s > max{s1 , s2 }, then s·a − m 0 A2 du ≥ A2 du. − − 4 s·b s·a

(146)

This finishes the proof of the first estimate. 3 To prove the second estimate, first define f (u) = exp((2/3) · u 2 ) and let 0 < α < β. By using (132) and integrating by parts we find that, for β large, β 1 A2+ du = · β −1 · f (β) · 1 + O(β −1 ) . 4 0 It follows that there exists s3 so that for s > s3 , s·b+ s·2b+ 1 A2+ du ≥ · A2+ du. 2 0 s·b+

(147)

Equation (132) also implies that β 1 1 A+ · A− du = β 2 − α 2 + O β −1 + α −1 . α

In particular, there exists s4 > 0 so that if s > s4 , then s·2b+ A+ · A− du ≥ 0.

(148)

s·b+

By (132), the function A2− is integrable on [0, ∞). Let I be the value of this integral. Using (140) we find that there exists s5 such that if s > s5 , then s·a − s·b+ 2 − 21 A− du ≤ M · s A2− du, (149) 0

s·b−

Spectral Simplicity

343

1 1 where M = 2I / (b− ) 2 − (a − ) 2 . From (140) and (142) we find that there exists s6 such that if s > s6 , then − s·a 1 s·a − A+ · A− du ≤ A2 du. (150) s·b− 2 s·b− ± Let A = c+ A+ + c− A− be a general solution to the Airy equation. From (147) and (148) it follows that if s > max{s3 , s4 }, then 2s·b+ s·b+ c+2 A2+ du ≤ 2 A2 du. (151) s·b+

0

From (150) we have that if s > s6 , then −

− s·a − s·a 1 s·a 2 2 2 2 A+ · A− du ≤ · c+ A+ du + c− A− du . 2|c+ · c− | · s·b− 2 s·b− s·b− It follows that for s > s6 , 2 c−

s·a −

s·b−

A2−

du ≤ 2

s·a −

s·b−

A2 du.

Putting this together with (149) gives 2 c−

s·b+ 0

1

A2− du ≤ 2M · s − 2

s·a − s·b−

A2 du.

(152)

By combining (151) and (152) we find that 1 2

s·b+ 0

1

A2 du ≤ 2M · s − 2

s·a − s·b−

A2 du + 2

This completes the proof of the second estimate.

2s·b+

A2 du.

s·b+

References [Albert78] [Berard79] [BryWlk84] [Cartier71] [Courant-Hilbert] [Cherry50] [Durso88] [ExnPst05]

Albert, J.H.: Generic properties of eigenfunctions of elliptic partial differential operators. Trans. Amer. Math. Soc. 238, 341–354 (1978) Bérard, P.: Spectres et groupes cristallographiques. C.R. Acad. Sci. Paris Sér. A-B 288(23), A1059–A1060 (1979) Berry, M.V., Wilkinson, M.: Diabolical points in the spectra of triangles. Proc. Roy. Soc. London Ser. A 392(1802), 15–43 (1984) Cartier, P.: Some numerical computations relating to automorphic functions in Computers in number theory. Proceedings of the Science Research Council Atlas Symposium No. 2. Edited by A.O.L. Atkin, B.J. Birch. London-New York: Academic Press, 1971 Courant, R., Hilbert, D.: Methods of Mathematical Physics. Volume 1. New York: Wiley Classics, 1989 Cherry, T.M.: Uniform asymptotic formulae for functions with transition points. Trans. Amer. Math. Soc. 68, 224–257 (1950) Durso, C.: Inverse spectral problem for triangular domains. Thesis, MIT, 1988 Exner, P., Post, O.: Convergence of spectra of graph-like thin manifolds. J. Geom. Phys. 54(1), 77–115 (2005)

344

[FrdSlm09] [Grieser]

[Grünbaum] [Harmer08] [Hillairet05] [Hillairet10] [HlrJdg09] [HlrJdg10] [Kato] [Lame] [Langer31] [LaxPhl] [LuRowl] [Olver] [Pinsky80] [Reed-Simon] [Sarnak03] [Uhlenbeck72]


Friedlander, L., Solomyak, M.: On the spectrum of the Dirichlet laplacian in a narrow strip. Israel J. Math. 170, 337–354 (2009) Grieser, D.: Thin tubes in mathematical physics, global analysis and spectral geometry in Analysis on Graphs and its Applications. Proceedings of Symposia in Pure Mathematics, Edited by P. Exner, J. Keating, P. Kuchment, T. Sunada, A. Teplyaev, Providence, RI: Amer. Math. Soc, 2008 Grünbaum, B.: Convex polytopes. 2nd ed. Graduate Texts in Mathematics 221. New York, Springer-Verlag, 2003 Harmer, M.: The spectra of the spherical and euclidean triangle groups. J. Aust. Math. Soc. 84(2), 217–227 (2008) Hillairet, L.: Contribution of periodic diffractive geodesics. J. Funct. Anal. 226(1), 48– 89 (2005) Hillairet, L.: Eigenvalue variations and semiclassical concentration in Spectrum and Dynamics: Proceedings of the Workshop Held in Montral, QC, April 7–11, 2008, Edited by D. Jakobson, S. Nonnenmacher, I. Polterovich, Montral, QC: Amer. Math. Soc, 2010 Hillairet, L., Judge, C.: Generic spectral simplicity of polygons. Proc. Amer. Math. Soc. 137(6), 2139–2145 (2009) Hillairet, L., Judge, C.: The eigenvalues of the Laplacian on domains with small slits. Trans. Amer. Math. Soc. 362(12), 6231–6259, (2010) Kato, T.: Perturbation Theory for Linear Operators, Springer-Verlag Classics in Mathematics, Berlin, Heidelberg-New York: Springer Verlag, 1995 Lamé, G.: Leçons sur la théorie mathématique de l’élasticité des corps solides. Paris: Bachelier, 1852 Langer, R.E.: On the asymptotic solutions of ordinary differential equations with an application to the Bessel functions of large order. Trans. of the Amer. Math. Soc. 33(1), 23– 64 (1931) Lax, P., Phillips, R.: Scattering theory for automorphic forms. Princeton, NJ: Princeton U. Press, 1976 Lu, Z., Rowlett, J.: The fundamental gap, http://arxiv.org/abs/1003.0191v1 [math.sp], 2010 Olver, F.W.J.: Asymptotics and Special Functions. AKP Classics. Wellesley, MA: A K Peters, Ltd., 1997 Pinsky, M.A.: The eigenvalues of an equilateral triangle. SIAM J. Math. Anal. 11(5), 819– 827 (1980) Reed, M., Simon, B.: Methods of modern mathematical physics. IV. Analysis of operators. New York-London: Academic Press, 1978 Sarnak, P.: Spectra of hyperbolic surfaces. Bull. Amer. Math. Soc. (N.S.) 40(4), 441– 478 (2003) Uhlenbeck, K.: Eigenfunctions of Laplace operators. Bull. Amer. Math. Soc. 78, 1073– 1076 (1972)

Communicated by S. Zelditch


Communications in


Existence and Uniqueness of SRB Measure on C 1 Generic Hyperbolic Attractors Hao Qiu School of Mathematical Sciences, Peking University, Beijing 100871, China. E-mail: [email protected] Received: 26 February 2010 / Accepted: 25 June 2010 Published online: 17 December 2010 – © Springer-Verlag 2010

Abstract: Let M be a smooth Riemannian manifold. We show that for C 1 generic f ∈ Diff1 (M), if f has a hyperbolic attractor f , then there exists a unique SRB measure supported on f . Moreover, the SRB measure happens to be the unique equilibrium state of potential function ψ f ∈ C 0 ( f ) defined by ψ f (x) = − log | det(D f |E xu )|, x ∈ f , where E xu is the unstable space of Tx M. 1. Preliminary Let M be a smooth Riemannian manifold. Assume m is the volume measure of M induced by Riemann metric. Denote by δx the probability atomic measure supported on x ∈ M. For any C 1 diffeomorphism f and ergodic measure μ, the statistical basin of μ is defined as n−1 1 k B(μ) = {x ∈ M : lim ϕ( f x) = ϕdμ, ∀ϕ ∈ C 0 (M)} n→∞ n k=0

n−1 1 δ f k x = μ}, = {x ∈ M : lim n→∞ n k=0

and its elements are called generic points of μ. If m(B(μ)) > 0, we call μ an SRB measure. The theory of SRB measure has been extensively studied since it was introduced by Sinai, Ruelle and Bowen in the early 1970’s. The classical SRB theory says that, if dynamical systems admit sufficient differentiability and hyperbolicity, then they do have SRB measures. A particular example will be C k,α hyperbolic attractors, where k = 1, 2, 3, . . . and 0 < α ≤ 1. In this situation, we have both existence and uniqueness of the SRB measure that is supported on such an attractor (see, for instance, [1,8]).

346

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With abundance of results in the case of high differentiability, people are curious whether the theory maintains for “most” f ∈ Diff1 (M). Towards this question, Campbell and Quas obtained the following C 1 generic result for circle expanding maps (see [2]). Theorem (Campbell, Quas). Let E 1 denote the set of C 1 expanding maps of the unit circle S 1 onto itself. Assume m is the normalized Lebesgue measure over S 1 . Then for generic T ∈ E 1 , there is a unique SRB measure μT , with m(B(μT )) = 1. In this paper, we push forward the above result to the setting of C 1 hyperbolic attractors: Let f 0 be a C 1 diffeomorphism of M. Assume there exists a compact invariant transitive hyperbolic set f0 , and an open neighborhood ⊃ f0 , so that f 0 () ⊂ and i≥0 f 0i () = f0 . By stability theory of an isolated hyperbolic set (see [7]), there exists a C 1 neighborhood U of f 0 , so that for any f ∈ U, the f -maximal invariant set of , denoted by f , is also hyperbolic. Moreover, for each f ∈ U there is a unique homeomorphism r f : f0 → f that is C 0 close to id| f0 , with f | f ◦ r f = r f ◦ f 0 | f0 . The main result of the paper is Theorem A. There exists a generic set U in U with the following property: for any f ∈ U , there is a unique SRB measure μ f supported on f , with m(B(μ f ) ∩ ) = m(). Moreover, μ f depends continuously in weak*-topology on f ∈ U . The proof of Theorem A is formulated through Sects. 3, 4. It basically follows Bowen’s convention of equilibrium state thermodynamical formalism developed in [1]. Thus we give in Sect. 2 a partial review on related concepts and results of this topic. Notation Hypotheses:

u ⊕ Es = u s 1) For any f ∈ U, denote by E x∈ f E x ⊕ E x the hyperbolic splitting f f u for T f M, and u = dim E f . 2) For compact metric space X and continuous map T over it, denote by M(X ) the set of Borel probability measures on X , by M(X ; T ) the set of T -invariant Borel probability measures on X , and by E(X ; T ) the set of T -ergodic Borel probability measures on X . 3) For any compact C 1 submanifold ⊂ M, denote by Tx the tangent space of at x, by T the tangent bundle of , and by m the volume measure induced by submanifold immersion. 4) For any finite set A, denote by A the cardinality of A.

2. A Partial Review on Thermodynamical Formalism Most contents of this section can be found in [1,3 and 9]. Let X be a compact metric space, and T be a continuous map over it. We call such a pair as (X, T ) a topological dynamical system. For any φ ∈ C 0 (X ) (φ is usually called a potential function), the topological pressure of φ (w.r.t T ) is defined by P(T ; φ) = sup {h μ (T ) + φdμ}, μ∈M(X ;T )

X

Existence and Uniqueness of SRB Measure on C 1 Generic Hyperbolic Attractors

347

where h μ (T ) is the measure theoretical entropy of T with respect to μ. If the topological entropy h(T )

sup

μ∈M(X ;T )

h μ (T ) < ∞,

then |P(T ; φ)| < ∞ for any φ ∈ C 0 (X ). In this situation, P(T ; ·) : C 0 (X ) → R has the following elementary properties (see Theorem 9.7 of [9]): 1. (Continuity) For any φ , φ ∈ C 0 (X ), P(T ; φ ) − P(T ; φ )| ≤ φ − φ C 0 (X ) .

(2.1)

2. (Convexity) For any φ , φ ∈ C 0 (X ) and 0 ≤ t ≤ 1 P(T ; tφ + (1 − t)φ ) ≤ t P(T ; φ ) + (1 − t)P(T ; φ ). As a consequence of convexity, for any φ, ϕ ∈ C 0 (X ), and t1 < t2 < t3 , we have P(T ; φ + t3 ϕ) − P(T ; φ + t1 ϕ) P(T ; φ + t2 ϕ) − P(T ; φ + t1 ϕ) ≤ , t2 − t1 t3 − t1

(2.2)

P(T ; φ + t2 ϕ) − P(T ; φ + t1 ϕ) P(T ; φ + t3 ϕ) − P(T ; φ + t2 ϕ) ≤ . t2 − t1 t3 − t2

(2.3)

and

In particular, taking t1 = 0, (2.2) implies that (P(T ; φ + tϕ) − P(T ; φ))/t monotonically decreases as t → 0+ . Moreover, taking t2 = 0, (2.3) implies that (P(T ; φ + tϕ) − P(T ; φ))/t, t > 0 is bounded from below. Thus limt→0+ (P(T ; φ + tϕ) − P(T ; φ))/t exists, and equals inf t>0 (P(T ; φ + tϕ) − P(T ; φ)/t. We denote the limit by τ (T ; φ, ϕ), i.e., τ (T ; φ, ϕ) = inf

t>0

P(T ; φ + tϕ) − P(T ; φ) P(T ; φ + tϕ) − P(T ; φ) = lim+ . t→0 t t

(2.4)

Lemma 2.1. Assume h(T ) < ∞ and φ ∈ C 0 (X ). Then 1. For any ϕ ∈ C 0 (X ), τ (T ; φ, ϕ) ≥ −τ (T ; φ, −ϕ).

(2.5)

2. τ (T ; φ, ·) : C 0 (X ) → R is continuous. More precisely, for any ϕ , ϕ ∈ C 0 (X ), |τ (T ; φ, ϕ ) − τ (T ; φ, ϕ )| ≤ ϕ − ϕ C 0 (X ) . Proof. Let t2 = 0 and take the limit as t1 → 0− , respectively t3 → 0+ in (2.3). Then the first statement is clear by definition of τ (T ; φ, ϕ). The second statement is straightforward by (2.1) and direct computation. Lemma 2.2. Assume h(T ) < ∞ and ϕ ∈ C 0 (X ). Then 1. τ (T ; ·, ϕ) : C 0 (X ) → R is upper semicontinuous. 2. For any φ ∈ C 0 (X ), if τ (T ; φ, ϕ) = −τ (T ; φ, −ϕ), then τ (T ; ·, ϕ) is continuous at φ.

348

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Proof. Upper semicontinuity of τ (T ; ·, ϕ) is clear from the first “=” of (2.4). For the C0

second statement, let φk → φ, then upper semicontinuity of τ (T ; ·, ϕ) gives lim sup τ (T ; φk , ϕ) ≤ τ (T ; φ, ϕ), k→∞

and lim sup τ (T ; φk , −ϕ) ≤ τ (T ; φ, −ϕ). k→∞

Therefore, if τ (T ; φ, ϕ) = −τ (T ; φ, −ϕ), we have τ (T ; φ, ϕ) = −τ (T ; φ, −ϕ) ≤ − lim sup τ (T ; φk , −ϕ) = lim inf −τ (T ; φk , −ϕ) k→∞

k→∞ (2.5)

≤ lim inf τ (T ; φk , ϕ) ≤ lim sup τ (T ; φk , ϕ) ≤ τ (T ; φ, ϕ). k→∞

k→∞

Then the above “≤” must all be “=”. In particular, lim inf τ (T ; φk , ϕ) = lim sup τ (T ; φk , ϕ) = τ (T ; φ, ϕ), k→∞

k→∞

thus limk→∞ τ (T ; φk , ϕ) = τ (T ; φ, ϕ).

An equilibrium state of φ (w.r.t. T ) is a T -invariant probability measure ν satisfying P(T ; φ) = h ν (T ) +

φdν. X

A tangent functional to P(T ; ·) at φ is a finite signed measure μ on X such that P(T ; φ + ϕ) − P(T ; φ) ≥

ϕdμ, ∀ϕ ∈ C 0 (X ) X

Let Eq(T ; φ) be the collection of equilibrium states of φ w.r.t. T, t (T ; φ) be the collection of tangent functionals to P(T ; ·) at φ. Lemma 2.3. Assume h · (T ) : M(X ; T ) → R is upper semicontinuous. Then for any φ ∈ C 0 (X ), Eq(T ; φ) = t (T ; φ). Proof. See Theorem 9.15 of [9]. Lemma 2.4. Assume h · (T ) : M(X ; T ) → R is upper semicontinuous and φ ∈ C 0 (X ). Then the following statements are equivalent: 1) Eq(T ; φ) = 1. 2) τ (T ; φ, ϕ) = −τ (T ; φ, −ϕ), ∀ϕ ∈ C 0 (X ). 3) For any ν ∈ Eq(T, φ), we have ϕdν = τ (T ; φ, ϕ), ∀ϕ ∈ C 0 (X ).


349

Proof. Consider “1) ⇒ 2)” first. Assume Eq(T ; φ) = 1. Suppose ∃ϕ ∈ C 0 (X ) so that τ (T ; φ, ϕ ) = −τ (T ; φ, −ϕ ). Then by (2.5) τ (T ; φ, ϕ ) > −τ (T ; φ, −ϕ ). We claim −ϕ ), τ (T ; φ, ϕ )], there exist ν ∈ Eq(T ; φ) so that that for any a ∈ [−τ (T ; φ, ϕ dν = a. In fact, consider ϕ the one-dimensional linear space generated by ϕ . We ˜ ) = at. Then the first “=” of (2.4) define the linear functional A˜ : ϕ → R by A(tϕ yields ˜ ), P(T ; φ + tϕ ) − P(T ; φ) ≥ tτ (T ; φ, ϕ ) ≥ at = A(tϕ (2.6) ), ˜ P(T ; φ − tϕ ) − P(T ; φ) ≥ tτ (T ; φ, −ϕ ) ≥ −at = A(−tϕ for t ≥ 0. This implies that the graph of A˜ is under the graph of P(T ; φ+·)− P(T ; φ)|ϕ . Applying the Hahn-Banach theorem and due to convexity of P(T ; φ + ·) − P(T ; φ), we can extend A˜ to A ∈ C 0 (X )∗ , so that A(tϕ ) = at, and the graph of A is under the graph of P(T ; φ + ·) − P(T ; φ), i.e. P(T ; φ + ϕ) − P(T ; φ) ≥ A(ϕ), ∀ϕ ∈ C 0 (X ). Let ν be the signed measure associated to A by the Rieszrepresentation theorem, then ν ∈ t (T ; φ), and by Lemma 2.3, ν ∈ Eq(T ; φ). Clearly, ϕ dν = a. Therefore, for arbitrary −τ (T ; φ, −ϕ ) ≤ a1 0, respectively t < 0, and taking limit as t → 0+ , respectively 0− , we obtain τ (T ; φ, ϕ) ≥ ϕdν ≥ −τ (T ; φ, −ϕ), ∀ϕ ∈ C 0 (X ). (2.8) Then if τ (T ; φ, ϕ) = −τ (T ; φ, −ϕ) for any ϕ ∈ C 0 (X ), (2.8) yields ϕdν = τ (T ; φ, ϕ), ∀ϕ ∈ C 0 (X ). “3) ⇒ 1)” is trivial. Corollary 2.5. Assume h · (T ) : M(X ; T ) → R is upper semicontinuous. Denote by R ⊂ C 0 (X ) the set of potential functions that have unique equilibrium state. Then R is a G δ set in C 0 (X ). Proof. Let {ϕi }i be a countable and dense subset of C 0 (X ). By Lemma 2.4 and 2) of Lemma 2.1, R can be represented as {φ ∈ C 0 (X ) | τ (T ; φ, ϕi ) = −τ (T ; φ, −ϕi )}. R= i

350

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Since τ (T ; φ, ϕi ) ≥ −τ (T ; φ, −ϕi ), {φ ∈ C 0 (X ) | τ (T ; φ, ϕi ) = −τ (T ; φ, −ϕi )} = {φ ∈ C 0 (X ) | τ (T ; φ, ϕi ) + τ (T ; φ, −ϕi ) < ε} ε>0

=

{φ ∈ C 0 (X ) : inf

t>0

ε>0

=

P(T ; φ + tϕi ) + P(T ; φ − tϕi ) − 2P(T ; φ) < ε} t

{φ ∈ C 0 (X ) : P(T ; φ + tϕi ) + P(T ; φ − tϕi ) − 2P(T ; φ) < tε}.

ε>0 t>0

This implies that R is G δ .

Remark 2.6. In fact, one may go further to prove that R is a dense G δ set in C 0 (X ), see Corollary 9.15.1 of [9]. Under the condition of Corollary 2.5, for any φ ∈ R we denote by μφ the unique equilibrium state of φ. Corollary 2.7. μφ depends continuously in weak*-topology on φ ∈ R. Proof. By 3) of Lemma 2.4, we have ϕdμφ = τ (T ; φ, ϕ) for any ϕ ∈ C 0 (X ). Thus it is sufficient to prove for any ϕ ∈ C 0 (X ), τ (T ; ·, ϕ) is continuous at φ, and this is derived from 2) of Lemma 2.2. Denote by d : X × X → R the distance function of X . Call E ⊂ X , (n, ε) separated, if whenever x, y are two distinct points in E, one can find 0 ≤ i ≤ n − 1 with d(T i x, T i y) > ε. Lemma 2.8. Given ε > 0 and ψ ∈ C 0 (X ), for each n ∈ N, let E n ⊂ X be an (n, ε) separated set, and μn ∈ M(X ) be defined by: μn =

x∈E n

where Sn ψ = and

n−1 i=0

n−1 e Sn ψ(x) 1 · δT i x , Sn ψ(x) n x∈E n e i=0

ψ ◦ T i . Assume μn i → μ in weak*-topology, then μ ∈ M(X ; T )

h μ (T ) +

ψdμ ≥ lim sup X

i→∞

1 log e Sni ψ(x) . ni x∈E ni

Proof. See part (2) of proof of Theorem 9.10 in [9]. 3. Generic Properties of P( f | f ; ψ f ) and Eq( f | f ; ψ f ) for f ∈ U For any f ∈ U, we define ψ f ∈ C 0 ( f ) by ψ f (x) = − log | det(D f |E xu )|, x ∈ f . With preparations in the previous section, we are going to study P( f | f ; ψ f ) and Eq( f | f ; ψ f ) for generic f ∈ U. Indeed, since f | f is expansive, the entropy map h · ( f | f ) : M( f ; f | f ) → R


351

is upper semicontinuous, thus h( f | f ) < ∞ (see Theorem 8.2 of [9]). Then all the results presented in the previous section hold in this situation. Recall that by classical SRB theory, if f ∈ U ∩ Diff2 (M), we have P( f | f ; ψ f ) = 0, Eq( f | f ; ψ f ) = 1.

(3.9)

Indeed, this is another presentation of the Ruelle-Pesin formula (see [4]). The next proposition says that this property holds for “most” f ∈ U. Proposition 3.1. 1. For any f ∈ U, P( f | f ; ψ f ) = 0. 2. There exists a generic subset U ⊂ U, so that for any f ∈ U , Eq( f | f ; ψ f ) = 1. Proof. We introduce a continuous map : U → C 0 ( f0 ) defined by ( f ) = ψ f ◦ r f . By invariance of topological pressure under conjugation, we have P( f | f ; ψ f ) = P( f 0 | f0 ; ( f )),

Eq( f | f ; ψ f ) = r f ∗ Eq( f 0 | f0 ; ( f )). (3.10)

For the first statement, let f be an arbitrary diffeomorphism in U, and { f k }k be C 2 C1

diffeomorphisms so that f k → f . Therefore P( f | f ; ψ f ) = P( f 0 | f0 ; ( f )) = lim P( f 0 | f0 ; ( f k )) k→∞ (3.9)

= lim P( f k | fk ; ψ fk ) = 0. k→∞

For the second statement, abusing the notations in Corollary 2.5, we denote by R the set of potentials in C 0 ( f0 ) that have unique equilibrium state w.r.t. f 0 | f0 . Let U = −1 (R). Clearly, for any f ∈ U , Eq( f | f , ψ f ) = 1. By Corollary 2.5, R is a G δ set, thus U is a G δ set in U. Moreover, by (3.9), U ∩ Diff2 (M) ⊂ U . This implies that U is dense in U. In the sequel, for any f ∈ U , we denote by μ f the unique equilibrium state for ψ f w.r.t. f | f . Derived directly from (3.10) and Corollary 2.7, we have: Corollary 3.2. μ f depends continuously in weak*-topology on f ∈ U . Corollary 3.3. μ f is ergodic (w.r.t f ). Proof. Let μ f = E ( f ; f | f ) μdη(μ) be the ergodic decomposition of μ f , where η ∈ M(M( f ; f | f )) with η(E( f ; f | f )) = 1. Therefore by Theorem 8.4 of [9], 0 = h μ f ( f | f ) +

f

ψ f dμ f =

E ( f ; f | f )

{h μ ( f | f ) +

f

ψ f dμ}dη(μ). (3.11)

By (3.9), h μ ( f | f ) + f ψ f dμ ≤ 0, and “=” holds if and only if μ = μ f . Then (3.11) implies that μ = μ f for η a.e. μ. Thus μ f is ergodic.

352

H. Qiu

4. Volume Estimate of B(μ f ) ∩ for f ∈ U Now we carry on to compute, for any fixed f ∈ U , the volume of B(μ f ) ∩ . Our aim is to derive estimate m(B(μ f ) ∩ ) = m()

(4.12)

through the thermodynamical properties P( f | f ; ψ f ) = 0,

Eq( f | f ; ψ f ) = {μ f }.

(4.13)

Recall that if we consider a local unstable manifold , by Bowen’s standard technique developed in [1], one can obtain the following estimate: m (B(μ f ) ∩ ) = m ()

(4.14)

C2

from (4.13). Then, when f is of class, by an absolutely continuous holonomy map derived by stable foliation of f , one can transfer (4.14) to every u-dimensional C 1 compact submanifold that is transverse to stable foliation (in the sequel, we call them u-transversal C 1 compact submanifold or u-TCSM in abbreviation). Observe that can be foliated by a smooth family of u-TCSM’s. Thus applying Fubini’s Theorem, one can integrate (4.14) over this family to obtain estimate (4.12). However, for f ∈ Diff1 (M), the above holonomy map is, in general, not absolutely continuous (see [6]). Our strategy in this situation is to generalize Bowen’s technique for every u-TCSM in to obtain (4.14). More specifically, we will prove: Lemma 4.1. Let ⊂ be a u-TCSM. Then m (B(μ f ) ∩ ) = m (). As an immediate consequence of Lemma 4.1 and Fubini’s Theorem, we have: Proposition 4.2. m(B(μ f ) ∩ ) = m(). Then Proposition 4.2, Proposition 3.1, Corollary 3.2 and Corollary 3.3 jointly accomplish the proof of Theorem A. Now we only need to prove Lemma 4.1. To illustrate the argument in a simple case, we first prove the lemma for those ’s so that: case *) for any i ∈ N, f i ∩ = ∅ and f i ∩ f = ∅. The proof of the general case is very similar. Before the formal argument, we need the following preparative lemma: Lemma 4.3. Let ⊂ be a u-TCSM. Then, 1. Given C1 > 0, there exist δ1 > 0, so that for any i, j ∈ N and any compact disk D ⊂ f i , diam(D) ≤ δ1 ⇒ m f i (D) ≤ C1 . 2. Given C2 > 1, there exist δ2 > 0, so that for any i ∈ N and any x ∈ f i , y ∈ f j , d(x, y) ≤ δ2 ⇒ C2−1 ≤ | det(D f |Tx f i )| · | det(D f |Ty f j )|−1 ≤ C2 , where d(·, ·) is the distance function of M. Proof. The detail of the proof is omitted. The key observation is that, due to λ-lemma (see p. 82 of [5]), f i “C 1 -converges” to f as i → ∞. Thus we can apply the argument of compactness over f ∪ i≥0 f i .


353

4.1. Proof of Lemma 4.1 for Case *). For of case *), we consider the positive invariant set = f ∪ i≥0 f i and potential ψ ∈ C 0 () defined by if x ∈ f ; ψ f (x), ψ(x) = i − log | det(D f |Tx f )|, if x ∈ f i , i = 0, 1, 2, . . . . By definition of case *), one sees that ψ is well defined. By λ-lemma, is a compact set, thus (, f |) is a topological dynamical system. Furthermore, i≥0 f i = f , which implies that any invariant measure on must be supported on f . Then the thermodynamical properties for f | f with potential ψ f can be handed to f | with potential ψ. Therefore, by (4.13) and upper-semicontinuity of h · ( f | f ) : M( f , f | f ) → R, we have 1) P( f |; ψ) = 0, Eq( f |; ψ) = {μ f }, (4.15) 2) h · ( f |) : M(, f |) → R is upper semicontinuous. For any r > 0, let Kr ⊂ M(; f |) be defined by {ν ∈ M(; f |) : h ν ( f |) + ψdν ≥ −r }.

Then by 1) of (4.15), r >0 Kr = {μ f }. Furthermore, by 2) of (4.15), Kr is closed in M(; f |), thus closed in M(). This implies M()\Kr is open in M(). Therefore by local compactness and local convexity of M(), M()\Kr can be covered by a countable family of open sets {Vi }i in M(), so that each Vi is convex, and the closure of Vi is contained in M()\Kr . For any W ⊂ M(), let (W, n) and (W) be defined by (W, n) = {x ∈ :

n−1 1 δ f i x ∈ W}, n i=0

1 i→∞ n i

(W) = {x ∈ : lim It is easy to see that (W) ⊂

n≥0

n i −1

δ f ni x ∈ W, for some{n i }i }.

i=0

i≥n

(W, i) whenever W is open.

Claim. For any V ∈ {Vi }i , m ((V)) = 0. Proof. We choose arbitrary C1 > 0, C2 > 1, and determine δ1 = ε1 (C1 , ), δ2 = ε2 (C2 , ) by Lemma 4.3. Let δ = min{δ1 , δ2 }. Moreover, choose 0 < ε < δ so that for any x, y ∈ M, d( f x, f y) < δ whenever d(x, y) < ε. For each n ∈ N, select E n an (n, ε) separated set that is maximal in (V, n). For each x ∈ E n , let B n,ε (x) = {y ∈ : d( f i x, f i y) ≤ ε, 0 ≤ i ≤ n − 1}. Due to maximality, (V, n) ⊂ x∈E n Bn,ε (x). Then m ((V, n)) ≤ m (Bn,ε (x)) = dm (y) x∈E n

=

x∈E n

x∈E n n−1 f n (Bn,ε (x)) i=0

Bn,ε (x)

| det(D f |T f −n+i y f i )|−1 dm f n (y )

354

H. Qiu

≤ C2n

e Sn ψ(x) m f n ( f n (Bn,ε (x)))

x∈E n

≤ C1 C2n

e Sn ψ(x) ,

(4.16)

x∈E n

n−1 where Sn ψ = i=0 ψ ◦ f i. Now we apply Lemma 2.8 to (, f |), ψ and {E n }n . For each n ∈ N, let νn =

x∈E n

n e Sn ψ(x) 1 · δ f ix, Sn ψ(x) n x∈E n e i=0

and {νn i }i be a subsequence converging to some ν in weak*-sense. Then Lemma 2.8 gives 1 Sni ψ(x) lim sup log e ≤ h ν ( f |) + ψdν. (4.17) i→∞ n i x∈E ni

n−1

n−1 Observe that νn is a convex combination of { n1 i=0 δ f i x , x ∈ E n }, and n1 i=0 δfix ∈ V for any x ∈ E n . By convexity of V we have νn ∈ V, thus ν = limi→∞ νn i ∈ V ⊂ M()\Kr . Then by definition of Kr , h ν ( f |) + ψdν < −r . Therefore lim sup i→∞

1 log e Sni ψ(x) < −r. ni

(4.18)

x∈E ni

Clearly, (4.18) holds for any {n i }i such that νn i converges. Substituting n i in (4.18) by n, lim sup n→∞

1 log m ((V, n)) < −r. n

(4.19)

Combine (4.19) with (4.16), lim sup n→∞

1 1 log m ((V, n)) ≤ lim sup log e Sn ψ(x) + log C2 n n→∞ n x∈E n

< −r + log C2 .

(4.20)

1 log m ((V, n)) ≤ −r. n

(4.21)

Let C2 → 1, we have lim sup n→∞

Then, given 0 < σ < r , there exist C > 1, so that

m ((V, n)) ≤ Ce−(r −σ )n .

(4.22)

(V, i) because V is open, m ((V)) ≤ lim sup m ((V, i)) ≤ lim sup Ce−(r −σ )i = 0.

Note that (V) ⊂

n≥0

i≥n

n→∞

n→∞

i≥n

This ends the proof of the claim.

i≥n

(4.23)


As a consequence of the claim, m ((M()\Kr )) ≤ then by

355

m ((Vi )) = 0,

i

r >0 Kr

= {μ f },

m ((M()\{μ f })) = lim m ((M()\Kr )) = 0. r →0

Clearly, we have = (M()\{μ f })∪(B(μ f )∩) and (M()\{μ f })∩(B(μ f )∩ ) = ∅. Thus m (B(μ f ) ∩ ) = m () − m ((M()\{μ f })) = m (). This completes the proof of Lemma 4.1 for case *).

(4.24)

4.2. Proof of Lemma 4.1. Now we are going to apply the above argument in the general case. Note that the crucial point in the previous proof is that we “naturally” extend ψ f to ψ, in the sense that ψ| is “compatible” with volume measure on . However, without the assumption in case *), such an extension may be unrealizable. For example, assume there exists x ∈ ∩ f so that Tx = E xu , ψ(x) should equal − log | det(D f |E xu )| if x is referred to a point in f , while ψ(x) must be − log | det(D f |Tx )| if x is considered contained in , and | det(D f |E xu )| = | det(D f |Tx )| in general. Similar problem happens when there exists y ∈ ∩ f i with Ty = Ty f i . To overcome this problem, we introduce the framework of the Grassmann bundle, in which the previously mentioned Tx and E xu (respectively, Ty and Ty f i ) are forced apart. In precise words, let π : G(M, u) → M be the u-dimensional Grassmann bundle over M. For any V ⊂ T M a u-dimensional linear subspace, we write [V ] to denote the corresponding element in G(M, u). The topology of G(M, u) is determined by the distance function ˆ d([V ], [V ]) = min{l(γ ) + ∠π([V ]) (V, Pγ V )|γ : [0, 1] → M is piecewise smooth with γ (0) = π([V ]), γ (1) = π([V ])}, where l(γ ) is the length of γ , Pγ the parallel translation along γ , and ∠π([V ]) (V, Pγ V ) sup{v − v | v ∈ V, v ∈ Pγ V , v = v = 1}. Under this topology π : G(M, u) → M is a continuous map. Let fˆ : G(M, u) → G(M, u) be a homeomorphism defined by fˆ[V ] = [D f (V )]. Then f ◦ π = π ◦ fˆ. Let potential ψˆ ∈ C 0 (G(M, u)) be defined by ˆ ψ([V ]) = − log | det(D f |V )|.

Proof of Lemma 4.1. Still as in case *), we consider = f ∪ i≥0 f i . Moreover, define the following sets of G(M, u) that are related to : ˆ ˆf = ˆ = ˆ = ˆf ∪ [E xu ], [Tx ], fî . x∈ f

x∈

i≥0

356

H. Qiu

ˆ f, ˆ and ˆ respectively onto f , and . In particular, π | ˆf : Clearly, π maps ˆ ˆ ˆ f) f → f is a homeomorphism. Then by upper semicontinuity of h · ( f | f ), h · ( f | ˆ is upper semicontinuous. Moreover, since (ψ ◦ π )| f = ψ f , by (4.13) and invariance of topological pressure, ˆ = 0, ˆ f ; ψ) P( fˆ|

ˆ f ; ψ) ˆ = {μˆ f }, Eq( fˆ|

(4.25)

ˆ f )−1 where μˆ f (π | ∗ μf. ˆ is a compact set, thus (, ˆ fˆ|) ˆ is a topological dynamical system. By λ-lemma, i ˆ = ˆ f . Then for a similar reason mentioned before (4.15), we Furthermore, i≥0 fˆ have ˆ ψ) ˆ = 0, Eq( fˆ|; ˆ ψ) ˆ = {μˆ f }, 1) P( fˆ|; (4.26) ˆ : M(; ˆ fˆ|) ˆ → R is upper semicontinuous. 2) h · ( fˆ|) ˆ fˆ|) ˆ be defined by For any r > 0, let Kˆ r ⊂ M(; ˆ νˆ ≥ −r }. ˆ + ˆ fˆ|) ˆ : h νˆ ( fˆ|) ψd {ˆν ∈ M(; ˆ

Then by 1) of (4.26), r >0 Kˆ r = {μˆ f }. Furthermore, by 2) of (4.26) Kˆ r is closed in ˆ fˆ|), ˆ thus closed in M(). ˆ This implies M()\ ˆ Kˆ r is open in M(). ˆ ThereM(; ˆ ˆ ˆ fore by local compactness and local convexity of M(), M()\Kr can be covered by a countable family of open sets {Vˆ i }i , so that each Vˆ i is convex, and the closure of Vˆ i is ˆ Kˆ r . contained in M()\ In the sequel, for any x ∈ , we write xˆ to represent [Tx ] for simplicity. For any ˆ ⊂ M(), ˆ n) and ( ˆ be defined by ˆ let ( ˆ W, ˆ W) W n−1 1 ˆ n) = {xˆ ∈ ˆ ˆ W, ˆ : ( δ fî xˆ ∈ W}, n i=0

1 i→∞ n i

ˆ = {xˆ ∈ ˆ W) ˆ : lim (

n i −1

ˆ for some {n i }i }. δ fˆni xˆ ∈ W,

i=0

ˆ ˆ V))) Claim. For any Vˆ ∈ {Vˆ i }i , m (π(( = 0. Proof. Again, we choose arbitrary C1 > 0, C2 > 1, determine δ1 = ε1 (C1 , ), δ2 = δ2 (C2 , ) and δ = min{δ1 , δ2 } by Lemma 4.3, and choose 0 < ε < δ as in case *). For ˆ n)). ˆ V, each n ∈ N, select E n an (n, ε) separated set (w.r.t. f ) that is maximal in π(( ˆ ˆ ˆ We write E n = {xˆ : x ∈ E n }, then E n is (n, ε) separated (w.r.t. f ). For each x ∈ E n , let Bn,ε (x) = {y ∈ : d( f i x, f i y) ≤ ε, 0 ≤ i ≤ n − 1}. Due to maximality, ˆ n)) ⊂ ˆ V, π(( x∈E n Bn,ε (x). Then similar to (4.16), ˆ n))) ≤ ˆ V, m (π((

x∈E n

where Sn ψˆ =

n−1 i=0

ψˆ ◦ fî .

m (Bn,ε (x)) ≤ C1 C2n

x∈ ˆ Eˆ n

ˆ

ˆ e Sn ψ(x) ,

(4.27)


357

ˆ fˆ|), ˆ ψˆ and Eˆ n . With same argument as in between Now we apply Lemma 2.8 to (, (4.17) and (4.21), we have lim sup n→∞

1 ˆ n))) ≤ −r, ˆ V, log m (π(( n

(4.28)

which implies, as in (4.23), that ˆ ˆ V))) m (π(( = 0.

(4.29)

This ends the proof of claim. As a consequence of the claim, ˆ ˆ Kˆ r ))) ≤ )\ m (π((M(

ˆ Vˆ i ))) = 0, m (π((

i

then by

ˆ = {μˆ f },

r >0 Kr

ˆ ˆ μˆ f }))) = lim m (π((M( ˆ ˆ Kˆ r ))) = 0. m (π((M( )\{ )\ r →0

Moreover, it is easy to check that ˆ ˆ μˆ f })) = (M()\{μ f }). π((M( )\{ Then similar to (4.24), m (B(μ f ) ∩ ) = m () − m ((M()\{μ f }) ˆ ˆ μˆ f }))) = m (). )\{ = m () − m (π((M(

(4.30)

This completes the proof of Lemma 4.1. Acknowledgements. We sincerely thank Professor HU Huyi and Professor GAN Shaobo for posing to him the problem addressed in this paper, and helpful discussion with them. We also thank Professor WEN Lan, Professor SUN Wenxiang and Professor CAO Yongluo for their useful comments.

References 1. Bowen, R.: Equilibrium states and ergodic theory of Anosov diffeomorphisms. Lecture Note in Mathematics 470. New York: Springer Verlag, 1975 2. Campbell, J., Quas, A.: A Generic C 1 Expanding Map has a Singular SRB Measure. Commun. Math. Phys. 221, 335–349 (2001) 3. Keller, G.: Equilibrium states in ergodic theory. Cambridge: Cambridge University Press, 1998 4. Ledrappier, F., Young, L-S.: The metric entropy of diffeomorphisms, Part I: Characterization of measures satisfying Pesin’s entropy formula. Annals Math. 122, 509–539 (1985) 5. Palis, J., de Melo, W.: Geometric theory of dynamic systems: an introduction. New York: Springer Verlag, 1982 6. Robinson, C., Young, L-S.: Nonabsolutely continuous foliations for an Anosov diffeomorphism. Invent. Math. 61, 159–176 (1980) 7. Shub, M.: Global stability of dynamical systems. New York: Springer-Verlag, 1987 8. Viana, M.: Stochastic dynamics of deterministic systems. Lecture Notes 21st Braz. Math. Colloq. Rio de Janeiro: IMPA, 1997 9. Walters, P.: An introduction to ergodic theory. Graduate Texts in Mathematics 79, New York: Springer Verlag, 1982 Communicated by G. Gallavotti


Communications in


KAM Theory in Configuration Space and Cancellations in the Lindstedt Series Livia Corsi1 , Guido Gentile1 , Michela Procesi2 1 Dipartimento di Matematica, Università di Roma Tre, Roma, I-00146, Italy.


2 Dipartimento di Matematica, Università di Napoli “Federico II”, Napoli, I-80126, Italy.

E-mail: [email protected] Received: 8 March 2010 / Accepted: 18 May 2010 Published online: 29 September 2010 – © Springer-Verlag 2010

Abstract: The KAM theorem for analytic quasi-integrable anisochronous Hamiltonian systems yields that the perturbation expansion (Lindstedt series) for any quasi-periodic solution with Diophantine frequency vector converges. If one studies the Lindstedt series by following a perturbation theory approach, one finds that convergence is ultimately related to the presence of cancellations between contributions of the same perturbation order. In turn, this is due to symmetries in the problem. Such symmetries are easily visualised in action-angle coordinates, where the KAM theorem is usually formulated by exploiting the analogy between Lindstedt series and perturbation expansions in quantum field theory and, in particular, the possibility of expressing the solutions in terms of tree graphs, which are the analogue of Feynman diagrams. If the unperturbed system is isochronous, Moser’s modifying terms theorem ensures that an analytic quasi-periodic solution with the same Diophantine frequency vector as the unperturbed Hamiltonian exists for the system obtained by adding a suitable constant (counterterm) to the vector field. Also in this case, one can follow the alternative approach of studying the perturbation expansion for both the solution and the counterterm, and again convergence of the two series is obtained as a consequence of deep cancellations between contributions of the same order. In this paper, we revisit Moser’s theorem, by studying the perturbation expansion one obtains by working in Cartesian coordinates. We investigate the symmetries giving rise to the cancellations which makes possible the convergence of the series. We find that the cancellation mechanism works in a completely different way in Cartesian coordinates, and the interpretation of the underlying symmetries in terms of tree graphs is much more subtle than in the case of action-angle coordinates.

1. Introduction Consider an isochronous Hamiltonian system, described by the Hamiltonian H (α, A) = ω · A + ε f (α, A), with f real analytic in Td × A and A an open subset of Rd .

360

L. Corsi, G. Gentile, M. Procesi

The corresponding Hamilton equations are α˙ = ω + ε∂ A f (α, A),

˙ = −ε∂α f (α, A). A

(1.1)

Let (α 0 (t), A0 (t)) = (α 0 + ωt, A0 ) be a solution of (1.1) for ε = 0. For ε = 0, in general, there is no quasi-periodic solution to (1.1) with frequency vector ω which reduces to (α 0 (t), A0 (t)) as ε → 0. However, one can prove that, if ε is small enough and ω satisfies some Diophantine condition, then there is a ‘correction’ μ(ε, A0 ), analytic in both ε and A0 , such that the modified equations α˙ = ω + ε∂ A f (α, A) + μ(ε, A0 ),

˙ = −ε∂α f (α, A) A

(1.2)

admit a quasi-periodic solution with frequency vector ω which reduces to (α 0 (t), A0 (t)) as ε → 0. This is a well known result, called the modifying terms theorem, or translated torus theorem, first proved by Moser [20]. By writing the solution as a power series in ε (Lindstedt series), the existence of an analytic solution means that the series converges. This is ultimately related to some deep cancellations in the series; see [1] for a review. Equations like (1.1) naturally arise when studying the stability of an elliptic equilibrium point. For instance, one can think of a mechanical system near a minimum point for the potential energy, where the Hamiltonian describing the system looks like 1 2 H (x1 , . . . , xn , y1 , . . . , yn ) = y j + ω2j x 2j + εF(x1 , . . . , xn , ε), 2 d

(1.3)

j=1

where F is a real analytic function at least of third order in its arguments, the vector ω = (ω1 , . . . , ωd ) satisfies some Diophantine condition, and the factor ε can be assumed to be obtained after a rescaling of the original coordinates – such rescaling makes sense if one wants to study the behaviour of the system near the origin. Indeed, the corresponding Hamilton equations, written in action-angle variables, are of the form (1.1). Unfortunately, the action-angle variables are singular near the equilibrium, and hence there are problems in the region where one of the actions is much smaller than the others. Thus, it can be worthwhile to work directly in the original Cartesian coordinates. In fact, there has been a lot of interest for KAM theory in configuration space, that is, without action-angle variables; see for instance [6,19,22]. 1.1. Set up of the problem. In this paper we consider the ordinary differential equations x¨ j + ω2j x j + f j (x1 , . . . , xd , ε) + η j x j = 0,

j = 1, . . . , d,

(1.4)

where x = (x1 , . . . , xd ) ∈ Rd , ε is real parameter (perturbation parameter), the function f (x, ε) = ( f 1 (x, ε), . . . , f d (x, ε)) is real analytic in x and ε at (x, ε) = (0, 0) and at least quadratic in x, f j (x, ε) =

∞ p=1

εp

f j,s1 ,...,sd x1s1 . . . xdsd ,

(1.5)

s1 ,...,sd ≥0 s1 +···+sd = p+1

(by taking f j (x, ε) = −ε∂x j F(x, ε) one recovers the Hamilton equations corresponding to the Hamiltonian (1.3)), η = (η1 , . . . , ηd ) is a vector of parameters, and the frequency vector (or rotation vector) ω = (ω1 , . . . , ωd ) satisfies the Diophantine condition |ω · ν| > γ0 |ν|−τ

∀ν ∈ Zd∗ ,

(1.6)

KAM Theory in Configuration Space and Cancellations in the Lindstedt Series

361

with Zd∗ = Zd \{0}, τ > d − 1 and γ0 > 0. Here and henceforth · denotes the standard scalar product in Rd , and |ν| = |ν1 | + . . . + |νd |. In light of Moser’s theorem of the modifying terms, one expects that, by taking the (arbitrary) unperturbed solution x0, j (t) = C j cos ω j t + S j sin ω j t = c j eiω j t + c∗j e−iω j t , j = 1, . . . , d, there exists a function η(ε, c), analytic both in ε and c = (c1 , . . . , cd ), such that, by fixing η j = η j (ε, c), there exists a quasi-periodic solution to (1.4) with frequency vector ω, which reduces to the unperturbed one as ε → 0. In fact, this is what happens: the result is just a rephrasing of Moser’s modifying terms theorem, with the advantage that it extends to the regions of phase space where the action-angle variables cannot be defined, and hence is not surprising; see also [6]. What is less obvious is the cancellation mechanism which is behind the convergence of the perturbation series. The problem can be described as follows. One can try to write again – as in action-angle variables – the solution as a power series in ε, and study directly the convergence of the series. In general, when considering the Lindstedt series of some KAM problem, first of all one identifies the terms of the series which are an obstruction to convergence: such terms are usually called resonances (or self-energy clusters, by analogy to what happens in quantum field theory). Crudely speaking, the series is given by the sum of infinitely many terms (finitely many for each perturbation order), and each term looks like a product of ‘small divisors’ times some harmless factors: a resonance is a particular structure in the product which allows a dangerous accumulation of small divisors. This phenomenon is very easily visualised when each term of the series is graphically represented as a tree graph (tree tout court in the following), that is, a set of points and lines connecting them in such a way that no loop arises; we refer to [10,13,15] for an introduction to the tree formalism. Shortly, in any tree, each line carries a label j ∈ {1, . . . , d} and a label ν ∈ Zd (that one calls momentum, again inspired by the terminology of quantum field theory) and with each such line a small divisor δ j (ω · ν ) is associated; here u → δ j (u) is a smooth function, which depends on both the model under study and the coordinates one is working with, for instance δ j (u) = u for (1.2), while δ j (u) = u 2 − ω2j for (1.4). Then a resonance becomes a subgraph which is between two lines 1 and 2 with the same small divisors, i.e. δ j 1 (ω · ν 1 ) = δ j 2 (ω · ν 2 ). A tree with a chain of resonances represents a term of the series containing a factor δ j (ω · ν) to a very large power, and this produces a factorial k! to some positive power when bounding some terms contributing to the k th order in ε of the Lindstedt series, so preventing a proof of convergence. However, a careful analysis of the resonances shows that there are cancellations to all perturbation orders. This is what can be proved in the case of the standard anisochronous KAM theorem, as first pointed out by Eliasson [8]; see also [9,10], for a proof which more deeply exploits the similarity with the techniques of quantum field theory. More precisely the cancellation mechanism works in the following way. Given a tree θ and two lines 1 and 2 of θ with the same small divisor, consider all possible resonances which can be inserted between 1 and 2 . For each possible resonance one obtains a different tree, which represents a term of the perturbation series, and each term can be written as the product of a numerical value corresponding to the resonance times a numerical value associated to the points and lines of θ which are outside the resonance: this second numerical value is the same for all such trees, and hence factorises out. When summing together the numerical values corresponding to all resonances, there are compensations and the sum is in fact much smaller than each summand (for more details we refer to [10,13]).

362


For the isochronous case, already in action-angle variables [1], there are some kinds of resonances which do not cancel each other. Nevertheless there are other kinds of resonances for which the gain factor due to the cancellation is more than what is needed (that is, one has a second order instead of a first order cancellation). Thus, the hope naturally arises that one can use the extra gain factors to compensate the lack of gain factors for the first kind of resonances, and in fact this happens. Indeed, the resonances for which there is no cancellation cannot accumulate too much without entailing the presence of as many resonances with the extra gain factors, in such a way that the overall number of gain factors is, in average, one per resonance (this is essentially the meaning of Lemma 5.4 in [1]). When working in Cartesian coordinates, one immediately meets a difficulty. If one writes down the lowest order resonances, there is no cancellation at all. This is slightly surprising because a cancellation is expected somewhere: if the resonances do not cancel each other, in principle one can construct trees containing chains of arbitrarily many resonances, and these trees represent terms of the formal power series expansion for which a bound proportional to some factorial seems unavoidable. However, we shall show that there are cancellations, as soon as one has at least two resonances. So, one has the curious phenomenon that resonances which do not cancel each other are allowed, but they cannot accumulate too much. Moreover, the cancellation mechanism is more involved than in other cases (including the same problem in action-angle variables). First of all, the resonances are no longer diagonal in the momenta, that is, the lines 1 and 2 considered above can have different momenta ν 1 and ν 2 . Second, the cancellation does not operate simply by collecting together all resonances to a given order and then summing the corresponding numerical values. As we mentioned, in this way no cancellation is produced: to obtain a cancellation one has to consider all possible ways to connect two resonances to each other. Thus, there is a cancellation only if there is a chain of at least two resonances. What emerges eventually is that working in Cartesian coordinates rather complicates the analysis. On the other hand, as remarked above, it can be worthwhile to investigate the problem in Cartesian coordinates. Moreover, the cancellations are due to remarkable symmetries in the problem, which can be of interest on their own; in this regard we mention the problem of the reducibility of the skew-product flows with Bryuno base [11], where the convergence of the corresponding Lindstedt series is also due to some cancellation mechanism and hence to some deep symmetry of the system. In this paper we shall assume the standard Diophantine condition on the frequency vector ω; see (1.6) below. Of course one could consider more general Diophantine conditions than the standard one (for instance a Bryuno condition [5]; see also [12] for a discussion using the Lindstedt series expansion). This would make the analysis slightly more complicated, without shedding further light on the problem. An important feature of the Lindstedt series method is that, from a conceptual point of view, the general strategy is exactly the same independent of the kind of coordinates one uses (and independent of the fact that the system is a discrete map or a continuous flow; see [2,10,15]). What is really important for the analysis is the form of the unperturbed solution: the simpler such a solution is the easier the analysis. Of course, an essential issue is that the system one wants to study is a perturbation of one which is exactly soluble. This is certainly true in the case of quasi-integrable Hamiltonian systems, but of course the range of applicability is much wider, and includes also non-Hamiltonian systems; see for instance [14,16]. Moreover an assumption of this kind is more or less always implicit in whatever method one can envisage to deal with small divisor problems of this kind; see also [6].


363

In the anisochronous case, the cancellations are due to symmetry properties of the model – essentially the symplectic character of the problem, as first pointed out by Eliasson [8]. The cancellation mechanism for the resonances is deeply related to that assuring the formal solubility of the equations of motions, which in turn is due to a symmetry property as already shown by Poincaré [21]. We refer to [17] for a detailed comparison between Eliasson’s method and the tree formalism that we are using here. Note that, despite what is sometimes claimed in the literature, Eliasson did not study how the resonances have to be regrouped in order to exhibit the cancellation; on the contrary, he proved that, because of aforementioned symmetry properties, the sum of (the leading parts of) all possible resonances must cancel out; a proof of the cancellation through a careful regrouping of the resonances was first given by Gallavotti [9]. Subsequently, stressing further the analogy with quantum field theory, Bricmont et al. showed that the cancellations can be interpreted as a consequence of suitable Ward identities of the corresponding field theory [4] (see also [7]): the symmetry property corresponds to the translation invariance of the field theoy. In the isochronous case, in terms of Cartesian coordinates the cancellation mechanism works in a completely different way with respect to action-angle coordinates. However, as we shall see, the cancellation is still related to underlying symmetry properties: it would be interesting to relate the symmetry properties that we find to invariance properties of the corresponding quantum field model, as done in [4] for the KAM theorem. 1.2. Statement of the results. Now, we give a formal statement of our results. As stressed above, the main point of the paper is not in the results themselves, but in the method used to prove them, in particular on the analysis of the perturbation series and of the cancellation mechanism which is at the base of the convergence of the series. We look for quasi-periodic solutions x(t) of (1.4) with frequency vector ω. Therefore we expand the function x(t) by writing x(t) = e i ν ·ω t x ν , (1.7) ν ∈Zd

ν th

and we denote by f ν (x, ε) the Fourier coefficient of the function that we obtain by Taylor-expanding f (x, ε) in powers of x and Fourier-expanding x according to (1.7). Thus, in Fourier space (1.4) becomes (ω · ν)2 − ω2j x j,ν = f j,ν (x, ε) + η j x j,ν . (1.8) For ε = 0, η = 0, the vector x (0) (t) with components (0)

x j (t) = c j eiω j t + c∗j e−iω j t ,

j = 1, . . . , d,

(1.9)

is a solution of (1.4) for any choice of the complex constant c = (c1 , . . . , cd ). Here and henceforth ∗ denotes complex conjugation. Define e j as the vector with components δi j (Kronecker delta). Then we can split (1.8) into two sets of equations, called respectively the bifurcation equation and the range equation,

f j,σ e j (x, ε) + η j x j,σ e j = 0, j = 1, . . . , d, σ = ±1, (1.10a) (ω · ν)2 − ω2j x j,ν = f j,ν (x, ε) + η j x j,ν , j = 1, . . . , d, ν = ±e j . (1.10b)

364


We shall study both Eqs. (1.10) simultaneously, by showing that for all choices of the parameters c there exist suitable counterterms η, depending analytically on ε and c, such that (1.10) admits a quasi-periodic solution with frequency vector ω, which is analytic in ε, c, and t. Moreover, with the choice x j,e j = c j for all j = 1, . . . , d, the counterterms are uniquely determined. We formulate the following result. Theorem 1.1. Consider the system described by Eqs. (1.4) and let (1.9) be a solution at ε = 0, η = 0. Set (c) = max{|c1 |, . . . , |cd |, 1}. There exist a positive constant η0 , small enough and independent of ε, c, and a unique function η(ε, c), holomorphic in the domain |ε| 3 (c) ≤ η0 and real for real ε, such that the system x¨ j + ω2j x j + f j (x1 , . . . , xd , ε) + η j (ε, c) x j = 0,

j = 1, . . . , d,

admits a solution x(t) = x(t, ε, c) of the form (1.7), holomorphic in the domain |ε| 3 (c)e3|ω| |Im t| ≤ η0 and real for real ε, t, with Fourier coefficients x j,e j = c j and x j,ν = O(ε) if ν = ±e j for j = 1, . . . , d. The proof is organised as follows. After introducing the small divisors and proving some simple preliminary properties in Sect. 2, we develop in Sect. 3 a graphical representation for the power series of the counterterms and the solution (tree expansion). In particular we perform a multiscale analysis which allows us to single out the contributions (self-energy clusters) which give problems when trying to bound the coefficients of the series. In Sect. 4 we show that, as far as such contributions are neglected, there is no difficulty in obtaining power-like estimates on the coefficients: these estimates, which are generalisations of the Siegel-Bryuno bounds holding for anisochronous systems [9,10], would imply the convergence of the series and hence analyticity. In Sect. 5 we discuss how to deal with the self-energy clusters: in particular we single out the leading part of their contributions (localised values), which are proved in Sect. 6 to satisfy some deep symmetry properties. Finally, in Sect. 7 we show how the symmetry properties can be exploited in order to obtain cancellations involving the localised parts, in such a way that the remaining contributions can still be bounded in a summable way. This will yield the convergence of the full series and hence the analyticity of both the solution and the counterterms. Note that the system dealt with in Theorem 1.1 can be non-Hamiltonian. On the other hand the most general case for a Hamiltonian system near a stable equilibrium allows for Hamiltonians of the form 1 2 y j + ω2j x 2j + εF(x1 , . . . , xn , y1 , . . . , yn , ε), 2 d

H (x1 , . . . , xn , y1 , . . . , yn ) =

j=1

(1.11) which lead to the equations x˙ j = y j + ε∂ yi F(x, y, ε), y˙ j = −ω2j x j − ε∂xi F(x, y, ε). Also in this case one can consider the modified equations x˙ j = y j + ε∂ yi F(x, y, ε), y˙ j = −ω2j x j − ε∂xi F(x, y, ε) + η j x j ,

(1.12)

(1.13)


365

which are not of the form considered in Theorem 1.1. However, a result in the same spirit as Theorem 1.1 still holds. Theorem 1.2. Consider the system described by Eqs. (1.13) and let (x (0) (t), y(0) (t)) be a solution at ε = 0, η = 0, with x (0) (t) given by (1.9) and y(0) (t) = x˙ (0) (t). Set

(c) = max{|c1 |, . . . , |cd |, 1}. Then there exist a positive constant η0 , small enough and independent of ε, c, and a unique function η(ε, c), holomorphic in the domain |ε| 3 (c) ≤ η0 and real for real ε, such that the system x˙ j = y j + ε∂ yi F(x, y, ε), y˙ j = −ω2j x j − ε∂xi F(x, y, ε) + η j (ε, c) x j admits a solution (x(t, ε, c), y(t, ε, c)), holomorphic in the domain |ε| 3 (c)e3|ω| |Im t| ≤ η0 and real for real ε, t, with Fourier coefficients x j,e j = y j,e j /iω j = c j and x j,ν = y j,ν = O(ε) if ν = ±e j for j = 1, . . . , d. The proof follows the same lines as that of Theorem 1.1, and it is discussed in Appendices A and B. Finally in Appendix C we briefly sketch an alternative approach based on the resummation of the perturbation series. 2. Preliminary Results We shall denote by N the set of (strictly) positive integers, and set Z+ = N ∪ {0}. For any j = 1, . . . , d and ν ∈ Zd define the small divisors (2.1) δ j (ω · ν) := min{ω · ν − ω j , ω · ν + ω j } = |ω · (ν − σ (ν, j) e j )|, where σ (ν, j) is the minimizer. Note that the Diophantine condition (1.6) implies that δ j (ω · ν) ≥ γ |ν|−τ

∀ j = 1, . . . , d, ∀ν = 0, σ (ν, j) e j ,

(2.2a)

δ j (ω · ν) + δ j (ω · ν ) ≥ γ |ν − ν |−τ ∀ j, j =1, . . . , d, ∀ν = ν , ν−ν = σ (ν, j) e j −σ (ν , j ) e j ,

(2.2b)

for a suitable positive γ > 0. We can (and shall) assume that γ is sufficiently smaller than γ0 , and hence than δ(0) = min{|ω1 |, . . . , |ωd |} and ω := min{||ωi | − |ω j || : 1 ≤ i < j ≤ d}. Lemma 2.1. Given ν, ν ∈ Zd , with ν = ν , and δ j (ω · ν) = δ j (ω · ν ) for some j, j ∈ {1, . . . , d}, then either |ν − ν | ≥ |ν| + |ν | − 2 or |ν − ν | = 2. Proof. One has δ j (ω · ν) = |ω · ν − σ ω j | and δ j (ω · ν ) = |ω · ν − σ ω j |, with σ = σ (ν, j) and σ = σ (ν , j ). Set ν¯ = ν − σ e j and ν¯ = ν − σ e j . By the Diophantine condition (1.6) one can have δ j (ω · ν) = δ j (ω · ν ), and hence |ω · ν¯ | = |ω · ν¯ |, if and only if ν¯ = ±¯ν . If ν¯ = −¯ν then for σ = −σ one has |ν − ν | = |ν| + |ν |, while for σ = σ one obtains |ν − ν | ≥ |ν| + |ν | − 2. If ν¯ = ν¯ and j = j one has νi = νi for all i = j and ν j − σ = ν j − σ , and hence |ν j − ν j | = 2. If ν¯ = ν¯ and j = j then νi = νi for all i = j, j , while ν j − σ = ν j and ν j = ν j − σ , and hence |ν j − ν j | = |ν j − ν j | = 1.

366


Lemma 2.2. Let ν, ν ∈ Zd be such that ν = ν and, for some n ∈ Z+ , j, j ∈ {1, . . . , d}, both δ j (ω · ν) ≤ 2−n γ and δ j (ω · ν ) ≤ 2−n γ hold. Then either |ν − ν | > 2(n−2)/τ or |ν − ν | = 2 and δ j (ω · ν) = δ j (ω · ν ). Proof. Write δ j (ω·ν) = |ω·ν −σ ω j | and δ j (ω·ν ) = |ω·ν −σ ω j |, with σ = σ (ν, j) and σ = σ (ν , j ), and set ν¯ = ν − σ e j and ν¯ = ν − σ e j as above. If ν¯ = ν¯ , by the Diophantine condition (2.2b), one has −τ γ ν¯ − ν¯ < ω · (¯ν − ν¯ ) ≤ |ω · ν¯ | + ω · ν¯ < 2−(n−1) γ , which implies |¯ν − ν¯ | > 2(n−1)/τ , and hence we have |ν − ν | > 2(n−2)/τ in such a case. If ν¯ = ν¯ then, as in Lemma 2.1, one has |ν − ν | = 2 and δ j (ω · ν) = δ j (ω · ν ).

Remark 2.3. Note that |ν − ν | ≤ 2 and δ j (ω · ν) = δ j (ω · ν ) if and only if ν − ν = σ (ν, j)e j − σ (ν , j )e j . Lemma 2.4. Let ν 1 , . . . , ν p ∈ Zd and j1 , . . . , j p ∈ {1, . . . , d}, with p ≥ 2, be such that |ν i − ν i−1 | ≤ 2 and δ ji (ω · ν i ) = δ j1 (ω · ν 1 ) ≤ γ for i = 2, . . . , p. Then |ν 1 − ν p | ≤ 2. Proof. Set σi = σ (ν i , ji ) and ν¯ i = ν i − σi e ji for i = 1, . . . , p. For all i = 2, . . . , p, the assumption δ ji (ω · ν i ) = δ ji−1 (ω · ν i−1 ) implies ν¯ i = ±¯ν i−1 , which in turn yields ν¯ i = ν¯ i−1 , since |ν i − ν i−1 | ≤ 2. In particular ν¯ 1 = ν¯ p , and hence |ν 1 − ν p | ≤ 2.

3. Multiscale Analysis and Diagrammatic Rules As we are looking for x(t, ε, c) and η(ε, c) analytic in ε, we formally write x j,ν =

∞

(k)

εk x j,ν ,

k=0

ηj =

∞

(k)

εk η j .

(3.1)

k=1

It is not difficult to see that using (3.1) in (1.10) one can recursively compute (at least (k) formally) the coefficients x (k) j,ν , η j to all orders. Here we introduce a graphical represen(k)

(k)

tation for each contribution to x j,ν , η j , which will allow us to study the convergence of the series. 3.1. Trees. A graph is a set of points and lines connecting them. A tree θ is a graph with no cycle, such that all the lines are oriented toward a unique point (root) which has only one incident line (root line). All the points in a tree except the root are called nodes. The orientation of the lines in a tree induces a partial ordering relation () between the nodes and the lines: we can imagine that each line carries an arrow pointing toward the root; see Fig. 1. Given two nodes v and w, we shall write w ≺ v every time v is along the path (of lines) which connects w to the root. We call E(θ ) the set of end nodes in θ , that is, the nodes which have no entering line, and V (θ ) the set of internal nodes in θ , that is, the set of nodes which have at least one entering line. Set N (θ ) = E(θ ) V (θ ). For all v ∈ N (θ ) denote by sv the number of lines entering the node v.


367

Fig. 1. An unlabelled tree: the arrows on the lines all point toward the root, according to the tree partial ordering

Remark 3.1. One has

v∈V (θ) sv

= |N (θ )| − 1.

We denote by L(θ ) the set of lines in θ . We call an internal line a line exiting an internal node and an end line a line exiting an end node. Since a line ∈ L(θ ) is uniquely identified with the node v which it leaves, we may write = v . We write w ≺ v if w ≺ v; we say that a node w precedes a line , and write w ≺ , if w . Notation 3.2. (1) If and are two comparable lines, i.e., ≺ , we denote by P( , ) the (unique) path of lines connecting to , the lines and being excluded. (2) Each internal line ∈ L(θ ) can be seen as the root line of the tree θ whose nodes and lines are those of θ which precede , that is, N (θ ) = {v ∈ N (θ ) : v ≺ } and L(θ ) = { ∈ L(θ ) : }. 3.2. Tree labels. With each end node v ∈ E(θ ) we associate a mode label ν v ∈ Zd , a component label jv ∈ {1, . . . , d}, and a sign label σv ∈ {±}; see Fig. 2. We call E σj (θ ) the set of end nodes v ∈ E(θ ) such that jv = j and σv = σ . With each internal node v ∈ V (θ ) we associate a component label jv ∈ {1, . . . , d}, and an order label kv ∈ Z+ . Set V0 (θ ) = {v ∈ V (θ ) : kv = 0} and N0 (θ ) = E(θ ) V0 (θ ). We also associate a sign label σv ∈ {±} with each v ∈ V0 (θ ). The internal nodes v with kv ≥ 1 will be drawn as black bullets, while the end nodes and the internal nodes with kv = 0 will be drawn as white bullets and white squares, respectively; see Fig. 2. With each line we associate a momentum label ν ∈ Zd , a component label j ∈ {1, . . . , d}, a sign label σ ∈ {±}, and scale label n ∈ Z+ ∪ {−1}; see Fig. 3. Denote by sv, j the number of lines with component label j = j entering the node v, and with rv, j,σ the number of end lines with component label j and sign label σ which enter the node v. Of course sv = sv,1 + · · · + sv,d and sv, j ≥ rv, j,+ + rv, j,− for all j = 1, . . . , d.

368


(a)

(b)

(c)

Fig. 2. Nodes and labels associated with the nodes: (a) end node v with sv = 0, jv ∈ {1, . . . , d}, σv ∈ {±}, and ν v = σv e jv (cf. Sect. 3.3); (b) internal node v with sv ≥ 2, jv ∈ {1, . . . , d}, and kv = sv − 1 (cf. Sect. 3.3); (c) internal node v with sv = 2, jv ∈ {1, . . . , d} kv = 0, σv ∈ {±} (cf. Sect. 3.3)

Fig. 3. Labels associated with a line. One has σ = σ (ν , j ) (cf. Sect. 3.3) Moreover if = v then j = jv ; if v ∈ V0 (θ ) one has also σ = σv ; if ν = σ e j then n = −1, otherwise n ≥ 0 (cf. Sect. 3.3)

Finally call k(θ ) :=

kv

v∈V (θ)

the order of the tree θ . In the following we shall call trees tout court the trees with labels, and we shall use the term unlabelled trees for the trees without labels. 3.3. Constraints on the tree labels. Constraint 3.3. We have the following constraints on the labels of the nodes (see Fig. 2): (1) if v ∈ V (θ ) one has sv ≥ 2; (2) if v ∈ E(θ ) one has ν v = σv e jv ; (3) if v ∈ V (θ ) then kv = sv − 1, except for sv = 2, where both kv = 1 and kv = 0 are allowed. Constraint 3.4. The following constraints will be imposed on the labels of the lines: (1) j = jv , ν = ν v , and σ = σv if exits v ∈ E(θ ); (2) j = jv if exits v ∈ V (θ ); (3) if is an internal line then σ = σ (ν , j ), i.e., δ j (ω · ν ) = |ω · ν − σ ω j | (see (2.1) for notations); (4) if v ∈ V0 (θ ) then (see Fig. 4) 1. sv = 2; 2. both lines 1 and 2 entering v are internal and have σ 1 = σ 2 = σv and j 1 = j 2 = jv ; 3. either ν 1 = σv e jv and ν 2 = σv e jv or ν 1 = σv e jv and ν 2 = σv e jv ; 4. σ v = σv ; (5) if is an internal line and ν = σ e j , then enters a node v ∈ V0 (θ ); (6) n ≥ 0 if ν = σ e j and n = −1 otherwise.


369

Fig. 4. If there is an internal node v with kv = 0 then sv = 2 and the following constraints are imposed on the other labels: σ v = σ 1 = σ 2 = σv ; j v = j 1 = j 2 = jv ; either ν 1 = σv e jv and ν 2 = σv e jv (as in the figure) or ν 2 = σv e jv and ν 1 = σv e jv . (The scale labels are not shown)

(a)

(b)

Fig. 5. Conservation law: (a) v with kv = sv − 1 ≥ 1, so that ν = ν 1 + . . . + ν sv , (b) v with sv = 2 and kv = 0. (The scale labels are not shown)

Notation 3.5. Given a tree θ , call 0 its root line and consider the internal lines 1 , . . . , p ∈ L(θ ) on scale −1 (if any) such that one has n ≥ 0 for all ∈ P( 0 , i ), i = 1, . . . , p; we shall say that 1 , . . . , p are the lines on scale −1 which are closest to the root of θ . For each such line i , call θi = θ i . Then we call pruned tree θ˘ the subgraph with set of nodes and set of lines N (θ˘ ) = N (θ )\

p

N (θi ),

i=1

L(θ˘ ) = L(θ )\

p

L(θi ),

i=1

respectively. By construction, θ˘ is a tree, except that, with respect to the constraints listed above, one has sv = 1 whenever kv = 0; moreover one has ν = σ e j (and hence n ≥ 0) for ˘ except possibly the root line. all internal lines ∈ L(θ) Constraint 3.6. The modes of the end nodes and the momenta of the lines are related as follows: if = v one has the conservation law νw − σw e jw = νw. ν = w∈E(θ) wv

w∈V0 (θ) wv

˘ w∈E(θ) wv

Note that by Constraint 3.6 one has ν = ν v if v ∈ E(θ ), and ν = ν 1 + · · · + ν sv if v ∈ V (θ ), kv ≥ 1, and 1 , . . . , sv are the lines entering v; see Fig. 5. Moreover for any line ∈ L(θ ) one has |ν | ≤ |E(θ˘ )|.

370


Remark 3.7. In the following we shall repeatedly consider the operation of changing the sign label of the nodes. Of course this change produces the change of other labels, consistently with the constraints mentioned above: for instance, if we change the label σv of an end node v into −σv , then also ν v is changed into −ν v ; if we change the sign labels of all the end nodes, then also the momenta of all the lines are changed, according to the conservation law (Constraint 3.6); and so on. Two unlabelled trees are called equivalent if they can be transformed into each other by continuously deforming the lines in such a way that they do not cross each other. We shall call equivalent two trees if the same happens in such a way that all labels match. Notation 3.8. We denote by Tkj,ν the set of inequivalent trees of order k with tree component j and tree momentum ν, that is, such that the component label and the momentum of the root line are j and ν, respectively. Finally for n ≥ −1 define Tkj,ν (n) the set of trees θ ∈ Tkj,ν such that n ≤ n for all ∈ L(θ ). − ˘ ˘ Remark 3.9. For θ ∈ Tkj,ν , by writing ν = (ν1 , . . . , νd ), one has νi = |E i+ (θ)|−|E i (θ )| ˘ = |E −σ (θ˘ )| + 1 ≥ 1, and for i = 1, . . . , d. In particular for ν = σ e j , one has |E σ (θ)| j

˘ = |E −σ ˘ |E σj (θ)| j (θ )| for all j = j.

j

Lemma 3.10. The number of unlabelled trees θ with N nodes is bounded by 4 N . If k(θ ) = k then |E(θ )| ≤ E 0 k and |V (θ )| ≤ V0 k, for suitable positive constants E 0 and V0 . Proof. The bound |V (θ )| ≤ |E(θ )| − 1 is easily proved by induction using that sv ≥ 2 for all v ∈ V (θ ). So it is enough to bound |E(θ )|. The definition of order and Remark 3.1 yield |E(θ )| = 1 + k(θ ) + |V0 (θ )|, and the bound |V0 (θ )| ≤ 2k(θ ) − 1 immediately follows by induction on the order of the tree, simply using that sv ≥ 2 for v ∈ V (θ ). Thus, the assertions are proved with E 0 = V0 = 3.

3.4. Tree expansion. Now we shall see how to associate with each tree θ ∈ Tkj,ν a (k) (k) contribution to the coefficients x j,ν and η j of the power series in (3.1). For all j = 1, . . . , d set c+j = c j and c−j = c∗j . We associate with each end node v ∈ E(θ ) a node factor Fv := cσjvv ,

(3.2)

and with each internal node v ∈ V (θ ) a node factor

Fv :=

⎧ sv,1 ! . . . sv,d ! ⎪ ⎪ f jv ,sv,1 ,...,sv,d , ⎪ ⎪ sv ! ⎨ ⎪ 1 ⎪ ⎪ ⎪ ⎩− σv , 2c jv

where the coefficients f j,s1 ,...,sd are defined in (1.5).

kv ≥ 1, (3.3) kv = 0,


371

Fig. 6. The functions ψ and n

Let ψ be a non-decreasing C ∞ function defined in R+ , such that (see Fig. 6) 1, for u ≥ 7γ /8, ψ(u) = (3.4) 0, for u ≤ 5γ /8, and set χ (u) := 1−ψ(u). For all n ∈ Z+ define χn (u) := χ (2n u) and ψn (u) := ψ(2n u), and set (see Fig. 6) n (u) = χn−1 (u) ψn (u),

(3.5)

where χ−1 (u) = 1. Note that χn−1 (u)χn (u) = χn (u), and hence {n (u)}n∈Z+ is a partition of unity. ] We associate with each line a propagator G := G [n j (ω · ν ), where ⎧ ⎪ ⎨ n (δ j (u)) , n ≥ 0, [n] u 2 − ω2j (3.6) G j (u) := ⎪ ⎩1, n = −1. Remark 3.11. The number of scale labels which can be associated with a line in such a way that G = 0 is at most 2. In particular, given a line with momentum ν = ν and scale n = n, such that n (δ j (ω · ν)) = 0, then (see Fig. 6) 2−(n+1) γ ≤

5 −n 7 2 γ ≤ δ j (ω · ν) ≤ 2−(n−1) γ ≤ 2−(n−1) γ , 8 8

(3.7)

and if n (δ j (ω · ν))n+1 (δ j (ω · ν)) = 0, then 5 −n 7 2 γ ≤ δ j (ω · ν) ≤ 2−n γ . 8 8 We define

⎛ V (θ ) := ⎝

∈L(θ)

⎞⎛ G ⎠ ⎝

(3.8)

⎞ Fv ⎠ ,

(3.9)

v∈N (θ)

and call V (θ ) the value of the tree θ . Remark 3.12. The number of trees θ ∈ Tkj,ν with V (θ ) = 0 is bounded proportionally to C k , for some positive constant C. This immediately follows from Lemma 3.10 and the observation that the number of trees obtained from a given unlabelled tree by assigning the labels to the nodes and the lines is also bounded by a constant to the power k (use Remark 3.11 to bound the number of allowed scale labels).

372


Remark 3.13. In any tree θ there is at least one end node with node factor cσj for each internal node v with kv = 0, σv = σ and jv = j (this is easily proved by induction on the order of the pruned tree): the node factors −1/2cσj do not introduce any singularity at cσj = 0. Therefore for any tree θ the corresponding value V (θ ) is well defined because both propagators and node factors are finite quantities. Remark 3.12 implies that also

V (θ )

θ∈Tkj,ν

is well defined for all k ∈ N, all j ∈ {1, . . . , d}, and all ν ∈ Zd . Lemma 3.14. For all k ∈ N, all j = 1, . . . , d, and any θ ∈ Tkj,σ e j , there exists θ ∈ σ Tkj,−σ e j such that c−σ j V (θ ) = c j V (θ ). The tree θ is obtained from θ by changing the sign labels of all the nodes v ∈ N0 (θ ).

Proof. The proof is by induction on the order of the tree. For any tree θ ∈ Tkj,e j consider

the tree θ ∈ Tkj,−e j obtained from θ by replacing all the labels σv of all nodes v ∈ N0 (θ ) with −σv , so that the mode labels ν v are replaced with −ν v and the momenta ν with −ν (see Remark 3.7). Call 1 , . . . , p the lines on scale −1 (if any) closest to the root of θ , and for i = 1, . . . , p denote by vi the node i enters and θi = θ i (recall (2) in Notation 3.2). As an effect of the change of the sign labels, each tree θi is replaced with a tree σ θi such that c−σ jvi V (θi ) = c jvi V (θi ), by the inductive hypothesis. Thus, for each node vi the quantity Fvi V (θi ) is not changed. Moreover, neither the propagators of the lines ˘ nor the node factors corresponding to the internal nodes v ∈ V (θ) ˘ with kv = 0 ∈ L(θ) σv v ˘ change, while the node factors c jv of the nodes v ∈ E(θ ) are changed into c−σ jv . On the other hand one has |E + (θ˘ )| = |E − (θ˘ )| for all i = j, whereas |E + (θ˘ )| = |E − (θ˘ )| + 1 i

i

j

j

−σ σ ˘ and |E +j (θ˘ )| + 1 = |E − j (θ )|. Therefore one obtains c j V (θ ) = c j V (θ ), and the assertion follows.

For k ∈ N, j ∈ {1, . . . , d}, and σ ∈ {±}, define (k)

η j,σ = −

1 cσj

V (θ ).

θ∈Tkj,σ e j (k)

(k)

Lemma 3.15. For all k ∈ N and all j = 1, . . . , d one has η j,+ = η j,− . Proof. Lemma 3.14 implies c−j

θ∈Tkj,e j

V (θ ) = c+j

V (θ )

θ∈Tkj,−e j

for all k ∈ N and all j = 1, . . . , d, so that the assertion follows from the definition (k) of η j,σ .


373

Lemma 3.16. Equations (1.10) formally hold, i.e., they hold to all perturbation orders, provided that for all k ∈ N and j = 1, . . . , d we set formally x j,ν =

∞

(k)

εk x j,ν ,

(k)

x j,ν =

V (θ ) ∀ν ∈ Zd \{±e j } ,

(k)

x j,±e j = 0 ,

(3.10)

θ∈Tkj,ν

k=1

ηj =

∞

(k)

(k)

εk η j ,

ηj = −

k=1

Proof. The proof is a direct check.

1 cj

V (θ ).

(3.11)

θ∈Tkj,e j

(k) Remark 3.17. In η j , defined as (3.11), there is no singularity in c j = 0 because V (θ˘ ) contains at least one factor c+j = c j by Remark 3.9.

In the light of Lemma 3.16 one can wonder why the definition of the propagators for ν = σ e j is so involved; as a matter of fact one could define G =

1 . (ω · ν )2 − ω2j

However, since n≥0 n (u) ≡ 1, the two definitions are equivalent. We use the definition (3.6) so that we can immediately identify the factors O(2n ) which could prevent the convergence of the power series (3.1). In what follows we shall make this idea more precise. 3.5. Clusters. A cluster T on scale n is a maximal set of nodes and lines connecting them such that all the lines have scales n ≤ n and there is at least one line with scale n; see Fig. 7. The lines entering the cluster T and the line coming out from it (unique if existing at all) are called the external lines of the cluster T . We call V (T ), E(T ), and L(T ) the set of internal nodes, of end nodes, and of lines of T , respectively; note that the external lines of T do not belong to L(T ). Define also E σj (T ) as the set of end nodes v ∈ E(T ) such that σv = σ and jv = j. By setting kv , k(T ) := v∈V (T )

we say that the cluster T has order k if k(T ) = k. 3.6. Self-energy clusters. We call self-energy cluster any cluster T such that (see Fig. 8) (1) T has only one entering line and one exiting line, (2) one has n ≤ min{n T , n T } − 2 for any ∈ L(T ), (3) one has |ν T − ν T | ≤ 2 and δ j T (ω · ν T ) = δ j (ω · ν T ). T

Notation 3.18. For any self-energy cluster T we denote by T and T the exiting and the entering line of T respectively. We call PT the path of lines ∈ L(T ) connecting T to T , i.e., PT = P( T , T ) (recall (1) in Notation 3.2), and set n T = min{n T , n T }.

374

(a)


(b)

Fig. 7. Example of tree and the corresponding clusters: once the scale labels have been assigned to the lines of the tree as in (a), one obtains the cluster structure depicted in (b)

Fig. 8. Example of self-energy cluster: consider the cluster T on scale 3 in Fig. 7, and suppose that the mode labels of the end nodes are such that |ν1 + ν2 + ν3 + ν4 + ν5 + ν6 | ≤ 2 and δ j (ω · ν T ) = δ j (ω · ν ). T

T

T

Then T is a self-energy cluster with external lines T (entering line) and T (exiting line). The path PT is such that PT = { }

Remark 3.19. Notice that, by Remark 2.3, for any self-energy cluster the label ν T is uniquely fixed by the labels j T , σ T , j T , σ T , ν T . In particular, for fixed ν and j such that δ j (ω · ν) ≤ γ , there are only 2d − 1 momenta ν = ν such that |ν − ν| ≤ 2 and δ j (ω · ν ) = δ j (ω · ν) for some j and σ , depending on ν . All the other ν with small divisor equal to δ j (ω · ν) are far away from ν, according to Lemma 2.1. We say that a line is a resonant line if it is both the exiting line of a self-energy cluster and the entering line of another self-energy cluster, that is, is resonant if there exist two self-energy clusters T1 and T2 such that = T1 = T2 ; see Fig. 9. Remark 3.20. The notion of self-energy cluster was first introduced by Eliasson, in the context of the KAM theorem, in [8], where it was called resonance. We prefer the term self-energy cluster to stress further the analogy with quantum field theory. The notion of equivalence given for trees can be extended in the obvious way to self-energy clusters.


375

Fig. 9. Example of resonant line: is resonant if both T1 and T2 are self-energy clusters

Fig. 10. A self-energy cluster in Ekj,σ, j,σ (ω · ν, n); T contains at least one line on scale ≤ n and n such that min{n , n } ≥ n + 2

Notation 3.21. We denote by Rkj,σ, j ,σ (ω · ν , n) the set of inequivalent self-energy clusters T on scale ≤ n of order k, such that ν T = ν , j T = j, σ T = σ , j T = j and σ T = σ . By definition of cluster for T ∈ Rkj,σ, j ,σ (ω · ν , n) one must have n ≤ n T − 2. For j = j and σ = σ define also Ekj,σ, j,σ (ω · ν , n) the set of selfenergy clusters T ∈ Rkj,σ, j,σ (ω · ν , n) such that (1) T enters the same node v which k

T exits and (2) kv = 0. We call vT such a special node and set R j,σ, j,σ (ω · ν , n) = Rkj,σ, j,σ (ω · ν , n)\Ekj,σ, j,σ (ω · ν , n); see Fig. 10. Notation 3.22. For any T ∈ Ekj,σ, j,σ (ω · ν , n) we call θT the tree which has as root line the line ∈ L(T ) entering vT (one can imagine to obtain θT from T by ‘removing’ the node vT ); see Fig. 11. Note that θT ∈ Tkj,σ e j (n). Notation 3.23. Consider a self-energy cluster T such that n = −1 for all lines ∈ PT . If T ∈ Ekj,σ, j,σ (ω · ν , n) for some k, j, σ, ν , n then we define the pruned self-energy cluster T˘ as the subgraph with N (T˘ ) = {vT } ∪ N (θ˘T ) and L(T˘ ) = L(θ˘T ). For all other self-energy clusters T , call 1 , . . . , p ∈ L(T ) the internal lines on scale −1 (if any) which are closest to the exiting line of T , that is, such that n ≥ 0 for all lines ∈ P( T , i ), i = 1, . . . , p. For each line i set θi = θ i . Then the pruned self-energy cluster T˘ is the subgraph with set of nodes and set of lines N (T˘ ) = N (T )\

p

i=1

respectively.

N (θi ),

L(T˘ ) = L(T )\

p

i=1

L(θi ),

376


Fig. 11. An example of self-energy cluster T ∈ Ekj,σ, j,σ (ω · ν, n) and the corresponding tree θT . (Only the mode labels of the end nodes are shown in T and θT .)

Remark 3.24. For T ∈ Rkj,σ, j ,σ (ω · ν , n) such that n ≥ 0 for all ∈ PT , one has σ ˘ ˘ |E i+ (T˘ )| = |E i− (T˘ )| for all i = j, j . If j = j then |E −σ j (T )| = |E j (T )| + 1 k ˘ and |E σj (T˘ )| = |E −σ j (T )| + 1; if j = j , σ = σ and T ∈ R j,σ, j,σ (ω · ν , n) then −σ ˘ σ ˘ ˘ |E σj (T˘ )| = |E −σ j (T )|, while if j = j and σ = −σ then |E j (T )| = |E j (T )| + 2. −σ Finally, for any T ∈ Ek (ω · ν , n) one has |E σ (T˘ )| = |E (T˘ )| + 1 ≥ 1. j,σ, j,σ

j

j

We shall define

⎛ V (T, ω · ν T ) := ⎝

⎞⎛

G ⎠ ⎝

∈L(T )

⎞ Fv ⎠ ,

(3.12)

v∈N (T )

where V (T, ω · ν T ) will be called the value of the self-energy cluster T . The value V (T, ω · ν T ) depends on ω · ν T through the propagators of the lines ∈ PT . Remark 3.25. The value of a self-energy cluster T ∈ Ekj,σ, j,σ (u, n) does not depend on u so that we shall write 1 V (T, u) = V (T ) = − σ V (θT ). 2c j We define also for future convenience M (k) j,σ, j ,σ (ω · ν , n) :=

V (T, ω · ν ).

(3.13)

T ∈Rkj,σ, j ,σ (ω·ν ,n) (k) (k) (n) + M (k) (k) Note that M j,σ, j,σ (ω · ν , n) = M j,σ, j,σ (ω · ν , n), where M j,σ, j,σ (n) j,σ, j,σ (k)

and M j,σ, j,σ (ω · ν , n) are defined as in (3.13) but for the sum restricted to the set k

Ekj,σ, j,σ (ω · ν , n) and R j,σ, j,σ (ω · ν , n) respectively. (k)

(k)

(k)

Remark 3.26. Both the quantities M j,σ, j ,σ (ω · ν , n) and the coefficients x j,ν and η j are well defined to all orders because the number of terms which one sums over is finite (by the same argument in Remark 3.12). At least formally, we can define

M j,σ, j ,σ (ω · ν ) =

∞ k=1

εk

n≥−1

(k)

M j,σ, j ,σ (ω · ν , n).


377

We define the depth D(T ) of a self-energy cluster T recursively as follows: we set D(T ) = 1 if there is no self-energy cluster containing T , and set D(T ) = D(T ) + 1 if T is contained inside a self-energy cluster T and no other self-energy clusters inside T (if any) contain T . We denote by S D (θ ) the set of self-energy clusters of depth D in θ , and by S D (θ, T ) the set of self-energy clusters of depth D in θ contained inside T . Notation 3.27. Call θ˚ = θ \S1 (θ ) the subgraph of θ formed by the set of nodes and lines of θ which are outside the set S1 (θ ) (the external lines of the self-energy clusters T ∈ ˚ and, analogously, for T ∈ S D (θ ) call T˚ = T \S D+1 (θ, T ) S1 (θ ) being included in θ), the subgraph of T formed by the set of nodes and lines of T which are outside the set S D+1 (θ, T ). We denote by V (T˚ ), E(T˚ ), and L(T˚ ) the set of internal nodes, of end nodes, and of lines of T˚ , and by k(T˚ ) the order of T˚ , that is, the sum of the labels kv of all the internal nodes v ∈ V (T˚ ). Lemma 3.28. Given a line ∈ L(θ ), if T is the self-energy cluster with largest depth containing (if any), ∈ PT and there is no line ∈ PT preceding with n = −1, one can write ν = ν 0 + ν T . Then one has |ν 0 | ≤ E 1 k(T˚ ), for a suitable positive constant E 1 , if k(T˚ ) ≥ 1, and |ν 0 | ≤ 2 if k(T˚ ) = 0.

Proof. We first prove that for any tree θ , if we denote by 0 its root line, one has E 1 k(θ˚ ) − 2, if 0 does not exit a self-energy cluster, (3.14) |ν 0 | ≤ if 0 exits a self-energy cluster, E 1 k(θ˚ ), for a suitable constant E 1 ≥ 4. The proof is by induction on the order of the tree θ . If k(θ ) = 1 (and hence θ˚ = θ ) then the only internal line of θ is 0 and |ν 0 | ≤ 2, so that the assertion trivially holds provided E 1 ≥ 4. If k(θ ) > 1 let v0 be the node which 0 exits. If v0 is not contained inside a self-energy cluster let 1 , . . . , m , m ≥ 0, be the internal lines entering v0 and θi = θ i for all i = 1, . . . , m. Finally let m+1 , . . . , m+m be the end-lines entering v0 . By definition we have k(θ˚ ) = kv0 + k(θ˚1 ) + · · · + k(θ˚m ). If kv0 > 0, we have ν 0 = ν 1 + · · · + ν m+m . This implies in turn |ν 0 | ≤ |ν 1 | + · · · + |ν m | + m ≤ E 1 k(θ˚1 ) + · · · + k(θ˚m ) + m ≤ E 1 (k(θ˚ ) − m − m + 1) + m . The assertion follows for E 1 ≥ 4 by the inductive hypothesis (the worst possible case is m = 0, m = 2). If kv0 = 0 then sv = 2 and m = 0. Moreover one of the lines, say 1 , is on scale n = −1 while for the other line one has ν 0 = ν 2 . Once more the bound follows from the inductive hypothesis since |ν 2 | ≤ E 1 k(θ˚2 ) ≤ E 1 (k(θ˚ ) − 1). Finally, if v0 is contained inside a self-energy cluster, then 0 exits a self-energy cluster T1 . There will be p self-energy clusters T1 , . . . , T p , p ≥ 1, such that the exiting line of Ti is the entering line of Ti−1 , for i = 2, . . . , p, while the entering line of T p does not exit any self-energy cluster. By Lemma 2.4, one has |ν 0 − ν | ≤ 2 and k(θ˚ ) = k(θ˚ ). Then, by the inductive hypothesis, one finds |ν 0 | ≤ 2 + E 1 k(θ˚ ) − 2 = E 1 k(θ˚ ). Now for and T as in the statement we prove, by induction on the order of the self-energy cluster, the bound E 1 k(T˚ ) − 2, if k(T˚ ) ≥ 1, 0 |ν | ≤ (3.15) 2 if k(T˚ ) = 0,

378


Fig. 12. The self-energy cluster T considered in the proof of Lemma 3.28, with m = 2, m = 3, and a chain of p self-energy clusters between and v (one has p ≥ 0, and = v if p = 0)

where T˚ is the set of nodes and lines of T˚ which precede . The bound is trivially satisfied when k(T˚ ) = 0. Otherwise let v be the node in V (T˚ ) between and T which is closest to . If kv = 0 the bound follows trivially by using the bound (3.14). If kv ≥ 1, call 1 , . . . , m , m ≥ 0, the internal lines entering v which are not along the path PT , and m+1 , . . . , m+m the end lines entering v; one has m + m ≥ 1. There is a further line 0 ∈ PT entering v such that ν 0 = ν 0 0 + ν T ; see Fig. 12. Using also Lemma 2.4 one has |ν 0 | ≤ 2 + |ν 0 0 | + |ν 1 | + · · · + |ν m | + m . As n 0 ≤ n T − 2 one has k(T˚ 0 ) ≥ 1 and hence, by (3.14) and the inductive hypothesis, one has |ν 0 | ≤ 2 + E 1 k(T˚ 0 ) − 2 + E 1 k(θ˚1 ) + · · · + k(θ˚m ) + m , where θi = θ i for all i = 1, . . . , m. Thus, since k(T˚ 0 ) + k(θ˚1 ) + · · · + k(θ˚m ) + (m + m ) = k(T˚ ) and m + m ≥ 1, one finds |ν 0 | ≤ E 1 k(T˚ ) − m − m + m ≤ E 2 k(T˚ ) − 2, provided E 1 ≥ 4. Therefore, the assertion follows with, say, E 1 = 4.

Notation 3.29. Given a tree θ and a line ∈ L(θ ), call = (θ ) the subgraph formed by the set of nodes and lines which do not precede ; see Fig. 13. Let us call

˚ the set of nodes and lines of which are outside any self-energy cluster contained inside . Lemma 3.30. Given a tree θ let 0 and be the root line and an arbitrary internal line preceding 0 . If k( ˚ ) ≥ 1 one has |ν 0 − ν | ≤ E 2 k( ˚ ), for a suitable positive constant E 2 .


379

Fig. 13. The set = (θ ) and the subtree θ determined by the line ∈ L(θ ). If is the root line then

= ∅

Proof. We prove by induction on the order of the bound E 2 k( ˚ ) − 2, if 0 does not exit a self-energy cluster, |ν 0 − ν | ≤ E 2 k( ˚ ), if 0 exits a self-energy cluster.

(3.16)

We mimic the proof of (3.14) in Lemma 3.28. The case k( ˚ ) = 1 is trivial provided E 2 ≥ 3, so let us consider k( ˚ ) > 1 and call v0 the node which 0 exits. If v0 is not contained inside a self-energy cluster and kv0 ≥ 1 then ν 0 = ν 1 + · · · + ν m+m , where 1 , . . . , m are the internal lines entering v0 , with (say) m ∈ P( 0 , ) ∪ { }, and m+1 , . . . , m+m are the end lines entering v0 . Hence k( ˚ ) = kv0 + k(θ˚1 ) + · · · + k(θ˚m−1 ) + k( ˚ m ), where θi = θ i and m = (θ m ) ( m = ∅ if m = ). Thus, the assertion follows by (3.14) and the inductive hypothesis. If v0 is not contained inside a self-energy cluster and kv0 = 0 then two lines 1 and 2 enter v0 , and one of them, say 1 , is such that |ν 1 | = 1. If = 2 the result is trivial. If 2 ∈ P( 0 , ) the bound follows once more from the inductive hypothesis. If = 1 one has |ν 0 − ν | ≤ |ν 0 | + 1 ≤ E 1 k(θ˚2 ) + 1 ≤ E 2 k( ˚ ) − 2, where θ2 = θ 2 , provided E 2 ≥ E 1 + 3, if E 1 is the constant defined in Lemma 3.28. If 1 ∈ P( 0 , ) denote by 1 the line on scale −1 along the path { 1 } ∪ P( 1 , ) which is closest to . Again call θ2 = θ 2 and J1 the subgraph formed by the set of nodes and lines preceding 1 (with 1 included) but not ; define also θ1 as the tree obtained from J1 by (1) reverting the arrows of all lines along { 1 , } ∪ P( 1 , ), (2) replacing 1 with an end line carrying the same sign and component labels as 1 , and (3) replacing all the labels σv , v ∈ N0 (J1 ) with −σv . One has, by using also (3.14), |ν 0 − ν | ≤ |ν 0 | + |ν | ≤ E 1 k(θ˚1 ) + E 1 k(θ˚2 ) ≤ E 2 k( ˚ ) − 2, provided E 2 ≥ E 1 + 2 so that the bound follows once more. Finally, if v0 is contained inside a self-energy cluster, then 0 exits a self-energy cluster T1 . There will be p selfenergy clusters T1 , . . . , T p , p ≥ 1, such that the exiting line of Ti is the entering line of Ti−1 , for i = 2, . . . , p, while the entering line of T p does not exit any self-energy cluster. By Lemma 2.4, one has |ν 0 −ν | ≤ 2 and k( ˚ ) = k( ˚ ), where = (θ ). Then, ˚ the inductive hypothesis yields |ν 0 − ν | ≤ 2 + |ν − ν | ≤ 2 + E 2 k( ˚ ) − 2 = E 2 k( ). Therefore the assertion follows with, say, E 2 = E 1 + 3 (and hence E 2 = 7 if E 1 = 4).

Remark 3.31. Lemma 3.28 will be used in Sect. 5 to control the change of the momenta as an effect of the regularisation procedure (to be defined). Furthermore, both Lemmas 3.28 and 3.30 will be used in Sect. 7 to show that the resonant lines which are not regularised cannot accumulate too much.

380


4. Dimensional Bounds In this section we discuss how to prove that the series (3.10) and (3.11) converge if the resonant lines are excluded. We shall see in the following sections how to take into account the presence of the resonant lines. Call Nn (θ ) the number of non-resonant lines ∈ L(θ ) such that n ≥ n, and Nn (T ) the number of non-resonant lines ∈ L(T ) such that n ≥ n. The analyticity assumption on f yields that one has |Fv | ≤ sv +kv

∀v ∈ V (θ )\V0 (θ ),

(4.1)

for a suitable positive constant . Lemma 4.1. Assume that 2−(n +2) γ ≤ δ j (ω · ν ) ≤ 2−(n −2) γ for all trees θ and all lines ∈ L(θ ). Then there exists a positive constant c such that for any tree θ one has Nn (θ ) ≤ c 2−n/τ k(θ ). Proof. We prove that Nn (θ ) ≤ max{0, c 2−n/τ k(θ ) − 2} by induction on the order of θ . 1. First of all note that for a tree θ to have a line on scale n ≥ n one needs k(θ ) ≥ kn = E 0−1 2(n−2)/τ , as it follows from the Diophantine condition (2.2a) and Lemma 3.10. Hence the bound is trivially true for k < kn . 2. For k(θ ) ≥ kn , let 0 be the root line of θ and set ν = ν 0 and j = j 0 . If n 0 < n the assertion follows from the inductive hypothesis. If n 0 ≥ n, call 1 , . . . , m the lines with scale ≥ n − 1 which are closest to 0 (that is, such that n ≤ n − 2 for all p = 1, . . . , m and all lines ∈ P( 0 , p )). The case m = 0 is trivial. If m ≥ 2 the bound follows once more from the inductive hypothesis. 3. If m = 1, then 1 is the only entering line of a cluster T . Set ν = ν 1 , j = j 1 and n = n 1 . By hypothesis one has δ j (ω · ν) ≤ 2−(n−2) γ and δ j (ω · ν ) ≤ 2−(n−3) γ , so that, by Lemma 2.2, either |ν − ν | > 2(n−5)/τ or |ν − ν | ≤ 2 and δ j (ω · ν) = δ j (ω · ν ). In the first case, since νw − σw e jw = νw, ν − ν = w∈E(T )

w∈V (T ) kw =0

w∈E(T˘ )

the same argument used to prove Lemma 3.10 yields |ν − ν | ≤ |E(T )| ≤ E 0 k(T ), and hence k(T ) ≥ E 0−1 2(n−5)/τ . Thus, if θ1 = θ 1 , one has k(θ ) = k(T ) + k(θ1 ), so that Nn (θ ) = 1 + Nn (θ1 ) ≤ c 2−n/τ k(θ1 ) − 1 ≤ c 2−n/τ k(θ ) − c 2−n/τ k(T ) − 1 ≤ c 2−n/τ k(θ ) − 2, provided c ≥ E 0 25/τ . 4. If instead |ν − ν | ≤ 2 and δ j (ω · ν) = δ j (ω · ν ), then the only way for T not to be a self-energy cluster is that n 1 = n 0 − 1 = n − 1 and there is at least a line ∈ T with n = n − 2. But then δ j (ω · ν) = δ j (ω · ν ) so that |ν − ν | > 2(n−6)/τ and we can reason as in the previous case provided c ≥ E 0 26/τ . Otherwise T is a self-energy cluster and 1 can be either resonant or not-resonant. Call 1 , . . . , m the lines with scale ≥ n − 1 which are closest to 1 . Once more the cases m = 0 and m ≥ 2 are trivial.


381

5. If m = 1, then 1 is the only entering line of a cluster T . If θ1 = θ 1 , then Nn (θ ) = 1 + Nn (θ1 ) if 1 is resonant and Nn (θ ) ≤ 2 + Nn (θ1 ) if 1 is non-resonant. Consider first the case of 1 being non-resonant. Set ν = ν 1 , j = j 1 and n = n 1 . By reasoning as before we find that one has either |ν − ν | > 2(n−5)/τ or |ν − ν | ≤ 2 and δ j (ω · ν ) = δ j (ω · ν ). If |ν − ν | > 2(n−5)/τ then k(T ) ≥ E 0−1 2(n−5)/τ ; thus, by using that k(θ ) = k(T ) + k(T ) + k(θ1 ), we obtain Nn (θ ) ≤ 2 + Nn (θ1 ) ≤ c 2−n/τ k(θ ) − c 2−n/τ k(T ) − c 2−n/τ k(T ) ≤ c 2−n/τ k(θ ) − c 2−n/τ k(T ) ≤ c 2−n/τ k(θ ) − 2, provided c ≥ 2E 0 25/τ . 6. Otherwise one has |ν−ν | ≤ 2, |ν −ν | ≤ 2, and δ j (ω·ν) = δ j (ω·ν ) = δ j (ω·ν ). Since we are assuming 1 to be non-resonant then, T is not a self-energy cluster. But then there is at least a line ∈ T with n = n − 2 and we can reason as in item 4. 7. So we are left with the case in which 1 is resonant and hence T is a self-energy cluster. Let 1 be the entering line of T . Once more 1 is either resonant or nonresonant. If it is non-resonant we repeat the same argument as done before for 1 . If it is resonant, we iterate the construction, and so on. Therefore we proceed until either we find a non-resonant line on scale ≥ n, for which we can reason as before, or we reach a tree θ of order so small that it cannot contain any line on scale ≥ n (i.e., k(θ ) < kn ). 8. Therefore the assertion follows with, say, c = 2E 0 26/τ .

Remark 4.2. One can wonder why in Lemma 4.1 did we assume 2−(n +2) γ ≤ δ j (ω · ν ) ≤ 2−(n −2) γ when Remark 3.11 assures the stronger condition 2−(n +1) γ ≤ δ j (ω · ν ) ≤ 2−(n −1) γ . The reason is that later on we shall need to slightly change the momenta of the lines, in such a way that the scales in general no longer satisfy the condition (3.7) noted in Remark 3.11. However the condition assumed for proving Lemma 4.1 will still be satisfied. For any tree θ we call L R (θ ) and L NR (θ ) the sets of resonant lines and of non-resonant lines respectively, in L(θ ). Then we can write ⎛ ⎛ ⎞ ⎞⎛ ⎞ V (θ ) = ⎝ G ⎠ V NR (θ ), V NR (θ ) := ⎝ G ⎠ ⎝ Fv ⎠ , (4.2) ∈L R (θ)

∈L NR (θ)

v∈N (θ)

where each propagator G can be bounded as C0 2n , for some constant C0 . Lemma 4.3. For all trees θ with k(θ ) = k one has | V NR (θ )| ≤ C k 3k (c), where

(c) := max{|c1 |, . . . , |cd |, 1} and C is a suitable positive constant. Proof. One has

⎛

|V NR (θ )| ≤ C0k 3k (c)k ⎝

⎞ 2n ⎠ ≤ C0k 3k (c)k

∈L N R (θ)

≤ C0k 3k (c)k exp c log 2 k

∞

2−n/τ n .

n=1

The last sum converges: this is enough to prove the lemma.

∞ n=0

2n Nn (θ)

382


Fig. 14. A chain of self-energy clusters

So far the only bound that we have on the propagators of the resonant lines is |G | ≤ 1/ω j δ j (ω · ν ) ≤ C0 2n . What we need is to obtain a gain factor proportional to 2−n for each resonant line with n ≥ 1. Lemma 4.4. Given θ such that V (θ ) = 0, let ∈ L(θ ) be a resonant line and let T be the self-energy cluster of largest depth containing (if any). Then there is at least one non-resonant line in T on scale ≥ n − 1. Proof. Set n = n . There are in general p ≥ 2 self-energy clusters T1 , . . . , T p , contained inside T , connected by resonant lines 1 , . . . , p−1 , and is one of such lines, while the entering line p of T p and the exiting line 0 of T1 are non-resonant. Moreover δ(ω · ν i ) = δ(ω · ν ) for all i = 0, . . . , p, so that all the lines 0 , . . . , p have scales either n, n − 1 or n, n + 1, by Remark 3.11. In any case the lines 0 , p must be in T by definition of the self-energy cluster.

5. Renormalisation Now we shall see how to deal with the resonant lines. In principle, one can have trees containing chains of arbitrarily many self-energy clusters (see Fig. 14), and this produces an accumulation of small divisors, and hence a bound proportional to k! to some positive power for the corresponding values. Let K 0 be such that E 1 K 0 = 2−8/τ . For T ∈ Rkj,σ, j ,σ (u, n), define the localisation operator L by setting ⎧ ⎪ k(T˚ ) ≤ K 0 2n T /τ , n ≥ 0 ∀ ∈ PT , ⎨V (T, σ ω j ), L V (T, u) := (5.1) ⎪ ⎩0, otherwise, which will be called the localised value of the self-energy cluster T . Define also R := 1 − L , by setting, for T ∈ Rkj,σ, j ,σ (u, n), R V (T, u) ⎧ 1 ⎪ ⎪u − σ ω dt ∂ V (T, σ ω + t (u −σ ω )), k(T˚ ) ≤ K 2n T /τ , n ≥ 0 ∀ ∈ P , ⎪ u 0 T ⎨ j j j 0 (5.2) = ⎪ ⎪ ⎪ ⎩ V (T, u), otherwise,

so that (k)

L M j,σ, j ,σ (u, n) =

L V (T, u),

(5.3a)

T ∈Rkj,σ, j ,σ (u,n)

R M (k) j,σ, j ,σ (u, n) =

R V (T, u).

(5.3b)

T ∈Rkj,σ, j ,σ (u,n)

We shall call R the regularisation operator and R V (T, u) the regularised value of T .


383

Remark 5.1. If T ∈ Ekj,σ, j,σ (u, n) the localisation operator acts as

L V (T ) =

⎧ n /τ ⎪ ⎨V (T ), k(T˚ ) ≤ K 0 2 T , ⎪ ⎩0,

k(T˚ ) > K 0 2n T /τ .

Remark 5.2. If in a self-energy cluster T there is a line ∈ PT such that ν = σ e j (and hence n = −1) then L V (T , u) = 0 for all self-energy clusters containing T such that ∈ PT . Recall the definition of the sets S D (θ ) and S D (θ, T ) after Remark 3.26. For any tree θ we can write its value as ⎞⎛ ⎛ ⎞⎛ ⎞ V (T, ω · ν T )⎠ ⎝ G ⎠ ⎝ Fv ⎠ , (5.4) V (θ ) = ⎝ T ∈S1 (θ)

∈L(θ\S1 (θ))

v∈N (θ\S1 (θ))

and, recursively, for any self-energy cluster T of depth D we have ⎛ V (T, ω · ν T ) = ⎝

⎞⎛

T ∈SD+1 (θ,T )

⎛

×⎝

V (T , ω · ν )⎠ ⎝

T

⎞ Fv ⎠ .

⎞ G ⎠

∈L(T \SD+1 (θ,T ))

(5.5)

v∈N (T \SD+1 (θ,T ))

Then we modify the diagrammatic rules given in Sect. 3 by assigning a further label OT ∈ {R, L }, which will be called the operator label, to each self-energy cluster T . Then, by writing V (θ ) according to (5.4) and (5.5), one replaces V (T, ω · ν T ) with L V (T, ω · ν T ) if OT = L and with R V (T, ω · ν T ) if OT = R. When considering the regularised value of a self-energy cluster T ∈ Rk (u, n) with k(T˚ ) ≤ K 0 2n T /τ j,σ, j ,σ

and n ≥ 0 for all ∈ PT , then we have also an interpolation parameter t to consider: we shall denote it by tT to keep trace of the self-energy cluster which it is associated with. We set tT = 1 for a regularised self-energy cluster T with either k(T˚ ) > K 0 2n T /τ or PT containing at least one line with n = −1. We call renormalised trees the trees θ carrying the further labels OT , associated with the self-energy clusters T of θ . As an effect of the localisation and regularisation operators the arguments of the propagators of some lines are changed. Remark 5.3. For any self-energy cluster T the localised value L V (T, u) does not depend on the operator labels of the self-energy clusters containing T .

Given a self-energy cluster T ∈ Rkj,σ, j ,σ (u, n) such that no line along PT is on scale −1, let be a line such that (1) ∈ PT , and (2) T is the self-energy cluster with largest depth containing . If one has OT = R, then the quantity ω · ν is changed according to the operator labels of all the self-energy clusters T such that (1) T contains T , (2) no line along PT has scale −1, and (3) ∈ PT . Call T p ⊂ T p−1 ⊂ · · · ⊂ T1 such

384


self-energy clusters, with T p = T . If OTi = R for all i = 1, . . . , p, then ω · ν is replaced with ω · ν (t ) = ω · ν 0 + σ p ω j p + t p ω · ν 0 p + σ p−1 ω j p−1 − σ p ω j p +

p−1

t p . . . ti ω · ν 0 i + σ i−1 ω ji−1 − σ i ω ji

i=2

+ t p . . . t1 ω · ν 1 − σ 1 ω j1 ,

(5.6)

where we have set t = (t1 , . . . , t p ), Ti = i and tTi = ti for simplicity. Otherwise let Tq be the self-energy cluster of highest depth, among T1 , . . . , T p−1 , with OTq = L (so that OTi = R for i ≥ q + 1). In that case, instead of (5.6), one has ω · ν (t ) = ω · ν 0 + σ p ω j p + t p ω · ν 0 p + σ p−1 ω j p−1 − σ p ω j p +

p−1

t p . . . ti ω · ν 0 i + σ i−1 ω ji−1 − σ i ω ji ,

(5.7)

i=q+1

with the same notations used in (5.6). If OT p = L , since ω · ν is replaced with ω · ν 0 + σ T ω j for ∈ PT , we can write T

ω · ν 0 + σ T ω j as in (5.6) by setting t p = 0. More generally, if we set tT = 0 whenever T

OT = L , we see that we can always claim that, under the action of the localisation and regularisation operators, the momentum ν of any line ∈ PT is changed to ν (t ), in such a way that ω · ν (t ) is given by (5.6). Lemma 5.4. Given θ such that V (θ ) = 0, for all ∈ L(θ ) one has 4 δ j (ω · ν ) ≤ 5 δ j (ω · ν (t )) ≤ 6 δ j (ω · ν ). Proof. The proof is by induction on the depth of the self-energy cluster. 1. Consider first the case that ∈ PT , with OT = L . Set n = n T , ν = ν T , σ = σ T , and j = j T . Then ω · ν is replaced with σ ω j , and, as a consequence, ω · ν is replaced with ω · ν (t ) = ω · ν 0 + σ ω j . Define n˜ such that 2−(n˜ +1) γ ≤ δ j (ω · ν 0 + σ ω j ) ≤ 2−(n˜ −1) γ ,

(5.8)

where δ j (ω · ν 0 + σ ω j ) = |ω · ν 0 + σ ω j − σ ω j | ≥ γ |ν 0 |−τ by the Diophantine condition (2.2b). Therefore 2n˜ −1 ≤ |ν 0 |τ ≤ (E 1 k(T˚ ))τ ≤ (E 1 K 0 )τ 2n = 2n−8 , and hence n˜ ≤ n − 7. Since |ω · ν − σ ω j | ≤ 2−n+2 γ by the inductive hypothesis, one has δ j (ω · ν ) = ω · ν 0 + ω · ν − σ ω j 15 ≥ ω · ν 0 + σ ω j − σ ω j − ω · ν − σ ω j ≥ δ j (ω · ν 0 + σ ω j ), 16 because δ j (ω · ν 0 + σ ω j ) ≥ 2−(n˜ +1) γ ≥ 2−n+6 γ ≥ 24 |ω · ν − σ ω j |. In the same way one can bound δ j (ω · ν ) ≤ |ω · ν 0 + σ ω j − σ ω j | + |ω · ν − σ ω j |, so that we conclude that


15 17 δ j (ω · ν 0 + σ ω j ) ≤ δ j (ω · ν ) ≤ δ j (ω · ν 0 + σ ω j ). 16 16

385

(5.9)

This yields the assertion. 2. Consider now the case that OT = R. In that case ω · ν (t ) is given by (5.6). Define n˜ as in (5.8), with σ = σ p and j = j p . We want to prove that 7 9 δ j (ω · ν 0 + σ ω j ) ≤ δ j (ω · ν (t )) ≤ δ j (ω · ν 0 + σ ω j ) 8 8

(5.10)

for all t = (t1 , . . . , t p ), with ti ∈ [0, 1] for i = 1, . . . , p. This immediately implies the assertion because, by using also (5.9), we obtain 7 14 δ j (ω · ν ) ≤ δ j (ω · ν 0 + σ ω j ) ≤ δ j (ω · ν (t )) 17 8 9 18 ≤ δ j (ω · ν 0 + σ ω j ) ≤ δ j (ω · ν ), 8 15 and hence 4δ j (ω · ν ) ≤ 5δ j (ω · ν (t )) ≤ 6δ j (ω · ν ). By the inductive hypothesis and the discussion of the case 1, in (5.8) we have i = 1, . . . , p, ω · ν 0 i + σ i−1 ω ji−1 − σ i ω ji ≤ 2−n i +2 γ , where n i = n i . Moreover one has n i ≥ n i+1 for i = 1, . . . , p − 1, so that we obtain δ j (ω · ν (t )) ≥ δ j (ω · ν 0 + σ ω j ) −

p

2−n i +2 γ ≥ δ j (ω · ν 0 + σ ω j ) − 2−n+3 γ .

i=1

Since δ j (ω · ν 0 + σ ω j ) ≥ 2−(n˜ +1) γ and n˜ ≤ n − 7, one finds δ j (ω · ν (t )) ≥ (1 − 2−3 )δ j (ω · ν 0 + σ ω j ). In the same way one has δ j (ω · ν (t )) ≤ (1 + 2−3 )δ j (ω ·

ν 0 + σ ω j ), so that (5.10) follows. Remark 5.5. Given a renormalised tree θ , with V (θ ) = 0, if a line ∈ L(θ ) has scale n then n (δ j (ω · ν )(t )) = 0, and hence, by Lemma 5.4, one has 2−(n +2) γ ≤ δ j (ω · ν ) ≤ 2−(n −2) γ . Therefore, Lemma 4.1 still holds for the renormalised trees without any changes in the proof (see also Remark 4.2). Remark 5.6. Another important consequence of Lemma 5.4 (and of Inequality (3.8) in Remark 3.11) is that the number of scale labels which can be associated with each line of a renormalised tree is still at most 2. 6. Symmetries and Identities Now we shall prove some symmetry properties on the localized value of the self-energy clusters. Lemma 6.1. If T ∈ Ekj,σ, j,σ (u, n) is such that T˘ does not contain any end node v with k

Fv = c−σ j then there exists T ∈ R j,σ, j,σ (u, n) such that −2L V (T ) = L V (T , u).

386


Fig. 15. The self-energy cluster T , the tree θT , and the self-energy cluster T in the proof of Lemma 6.1

Fig. 16. The sets F1 (T ) = {T1 , T2 } and F2 (T ) = {T3 } corresponding to the self-energy cluster T in Fig. 11

˘ Proof. If T ∈ Ekj,σ, j,σ (u, n) one has |E σj (T˘ )| = |E −σ j (T )|+1 (see Remark 3.24), so that −σ ˘ σ ˘ if |E (T )| = 0, then also |E (T )| = 1. This means that jv = j for all v ∈ E(T˘ )\{v0 }, j

j

k if E σj (T˘ ) = {v0 }. Consider the self-energy cluster T ∈ R j,σ, j,σ (u, n) obtained from θT by replacing the line exiting v0 with an entering line carrying a momentum ν such that ω · ν = u and n T = n T ; see Fig. 15. With the exception of v0 , the nodes of θT have the same node factors as T ; in particular they have the same combinatorial factors. If we compute the propagators G of ∈ L(T ), by setting u = σ ω j , then they are the same as the corresponding propagators of θT . Finally, as n T = n T , one has L V (T ) = 0 if and only if also L V (T , u) = 0. Thus, by recalling also Remark 3.25, one finds −2L V (T ) = L V (T , u).

For T ∈ Ekj,σ, j,σ (u, n) let us call F1 (T ) the set of all inequivalent self-energy clusters k

T ∈ R j,σ, j,σ (u, n) obtained from θT by replacing a line exiting an end node v ∈ E σj (θ˘T ) with an entering line carrying a momentum ν such that ω·ν = u and with n T = n T . Call also F2 (T ) the set of all inequivalent self-energy clusters T ∈ Rkj,σ, j,−σ (u , n), with u = u − 2σ ω j , obtained from θT by replacing a line exiting an end node v ∈ E −σ (θ˘T ) j

(if any) with an entering line carrying a momentum ν such that ω · ν = u and with n T = n T ; see Fig. 16.

Lemma 6.2. For all T ∈ Ekj,σ, j,σ (u, n) one has ⎛ ⎞ ⎝2cσj L V (T ) + cσj L V (T , u)⎠ = c−σ j T ∈F

1 (T )

T ∈F

L V (T , u ),

2 (T )

where u = u − 2σ ω j and the right hand side is meant as zero if F2 (T ) = ∅. Proof. The case k(T ) > K 0 2n T /τ is trivial so that we consider only the case k(T ) ≤ K 0 2n T /τ . By construction any T ∈ Ekj,σ, j,σ (u, n) is such that T˘ contains at least an end


387

˘ node v such that Fv = cσj , hence |E σj (T˘ )| ≥ 1. By Lemma 6.1 either |E −σ j (T )| ≥ 1 k

or there exists T ∈ R j,σ, j,σ (u, n) such that 2L V (T ) + L V (T , u) = 0. Hence the ˘ assertion is proved if E −σ j (T ) = ∅. ˘ So, let us consider the case |E −σ j (T )| ≥ 1. First of all note that there is a 1-to-1 correspondence between the lines of θT and the lines and external lines, respectively, of both T ∈ F1 (T ) and T ∈ F2 (T ); the same holds for the internal nodes. Moreover the propagators both of any T ∈ F1 (T ) and of any T ∈ F2 (T ) are equal to the corresponding propagators of T when setting u = σ ω j and u = −σ ω j , respectively. Also the node factors of the internal nodes of all self-energy clusters T ∈ F1 (T ) ∪ F2 (T ) are the same as those of T . For T ∈ F1 (T ) one has |E i+ (T˘ )| = |E i− (T˘ )| for all i = 1, . . . , d, whereas for T ∈ F2 (T ) one has |E i+ (T˘ )| = |E i− (T˘ )| for all i = j and ˘ |E σj (T˘ )| = |E −σ j (T )| + 2; thus, one has ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ σ σ σ ⎝ c v ⎠ = cσj ⎝ c v ⎠ = c−σ ⎝ c v⎠ v∈E(T˘ )

jv

v∈E(T˘ )

jv

j

v∈E(T˘ )

for all T ∈ F1 (T ) and all T ∈ F2 (T ). Therefore, if we write

⎛

⎞

−2cσj L V (T ) = V (θT ) = A (T ) ⎝

jv

v∈E(T˘ )

cσjvv ⎠ ,

(6.1)

where A (T ) depends only on T , then one finds ⎛ ⎞ 1 L V (T , u) = A (T ) σ ⎝ cσjvv ⎠ rv, j,σ , cj T ∈F1 (T )

v∈E(T˘ )

v∈V (T˘ )

with the same factor A (T ) as in (6.1). Analogously one has ⎛ ⎞ σ 1 L V (T , u ) = A (T ) −σ ⎝ c jvv ⎠ rv, j,−σ , cj T ∈F (T ) ˘ ˘ v∈E(T )

2

v∈V (T )

again with the same factor A (T ) as in (6.1), so one can write ⎞ ⎛ ⎝−2cσj V (T ) + cσj L V (T , u)⎠ − c−σ L V (T , u ) j ⎛

T ∈F1 (T )

= B(T ) ⎝−1 +

rv, j,σ

T ∈F2 (T )

⎞ − rv, j,−σ ⎠ ,

v∈V (T˘ )

where

⎛ B(T ) = A (T ) ⎝

v∈E(T˘ )

⎞ cσjvv ⎠ .

(6.2)

388


Fig. 17. A self-energy cluster T and the corresponding sets G1 (T ) = {T, T1 }, G2 (T ) = {T2 , T3 }, and G3 (T ) = {T4 , T5 }

On the other hand one has

rv, j,σ = |E σj (T˘ )|,

v∈V (T˘ )

˘ so that the term in the last parentheses of (6.2) gives −1 + |E σj (T˘ )| − |E −σ j (T )| = 0. Therefore the assertion is proved.

For T ∈ Rkj,σ, j ,σ (u, n) with j = j and n ≥ 0 for all ∈ PT , call G1 (T ) the set of self-energy clusters T ∈ Rkj,σ, j ,σ (u, n) obtained from T by exchanging the entering line with a line exiting an end node v ∈ E σ (T˘ ) (if any). Call also G2 (T ) the set of T

j

self-energy clusters T ∈ Rkj,σ, j ,−σ (u , n), with u = u −2σ ω j , obtained from T by (1) replacing the momentum of T with a momentum ν such that ω · ν = u , (2) changing ˘ the sign label of an end node v ∈ E −σ j (T ) into σ , and (3) exchanging the lines T and v . Finally call G3 (T ) the set of self-energy clusters T ∈ Rkj,−σ, j ,σ (u, n), obtained from T by (1) replacing the entering line T with a line exiting a new end node v0 with σv0 = σ and ν v0 = σ e j , (2) replacing all the labels σv of the nodes v ∈ N0 (T ) ∪ {v0 } with −σv and (3) replacing a line exiting an end node v ∈ E σj (T˘ ), with the entering line T ; see Fig. 17. Again we force n T = n T for all T ∈ G1 (T ) ∪ G2 (T ) ∪ G3 (T ). Lemma 6.3. For all T ∈ Rkj,σ, j ,σ (u, n), with j = j and n ≥ 0 for all ∈ PT , one has cσj L V (T , u) = c−σ L V (T , u ), j T ∈G1 (T )

σ L c−σ j c j T ∈G1 (T )

T ∈G2 (T )

V (T , u) =

cσj c−σ L j T ∈G3 (T )

V (T , u).


389

Proof. Again we consider only the case k(T˚ ) ≤ K 0 2n T /τ . For fixed T ∈ Rkj,σ, j ,σ (u, n), with j = j , let θ ∈ Tkj,σ e j (n) be the tree obtained from T by replacing the entering line T with a line exiting a new end node v0 with σv0 = σ and ν v0 = σ e j . Note that ˘ in particular one has |E σj (θ˘ )| = |E −σ j (θ )|. Any T ∈ G1 (T ) can be obtained from θ by replacing a line exiting an end node v ∈ E σ (θ˘ ) with an entering line , with the same j

labels as T , so that

cσj

T

˘ V (θ ). L V (T , u) = |E σj (θ)|

T ∈G1 (T )

On the other hand, any T ∈ G2 (T ) can be obtained from θ by replacing a line exiting an ˘ end node v ∈ E −σ j (θ) with an entering line T , with labels ν − 2σ e j , j , −σ , hence c−σ j

˘ L V (T , u) = |E −σ j (θ )| V (θ ),

T ∈G2 (T )

so that the first equality is proved. Now, let θ ∈ Tkj,−σ e j (n) be the tree obtained from θ by replacing all the labels σv of the nodes v ∈ N0 (θ ) with −σv . Any T ∈ G3 (T ) can be obtained from θ by replacing a line exiting an end node v ∈ E σj (θ˘ ) with an entering line T , carrying the same labels as T . Hence, by Lemma 3.14,

σ c−σ j c j

−σ ˘ σ ˘ σ L V (T , u) = c−σ j |E j (θ )| V (θ ) = c j |E j (θ )| V (θ )

T ∈G1 (T )

= cσj c−σ j

T ∈G

L V (T , u),

3 (T )

which yields the second identity, and hence completes the proof.

Lemma 6.4. For all k ∈ Z+ , all j, j = 1, . . . , d, and all σ, σ ∈ {±}, one has (i) η(k) = η(k) (|c1 |2 , . . . , |cd |2 ), i.e., η(k) depends on c only through the quantities |c1 |2 , . . . , |cd |2 ; (k) (k) σ (k) (ii) L M j,σ, j ,σ (u, n) = c−σ j c j M j, j (n), where M j, j (n) does not depend on the indices σ, σ . (k)

Proof. One works on the single trees contributing to L M j,σ, j ,σ (u, n). Then the proof follows from Lemma 3.14 and the results above.

Remark 6.5. Note that Lemma 6.4 could be reformulated as (k)

(k)

L M j,σ, j ,σ (u, n) = ∂cσ cσj L M j,σ, j,σ (n), j

(k)

with M j,σ, j,σ (n) defined after (3.13). We omit the proof of the identity, since it will not be used.

390


7. Cancellations and Bounds We have seen in Sect. 4 that, as far as resonant lines are not considered, no problems arise in obtaining ‘good bounds’, i.e., bounds on the tree values of order k proportional to some constant to the power k (see Lemma 4.3). For the same bound to hold for all tree values we need a gain factor proportional to 2−n for each resonant line on scale n ≥ 1. Let us consider a tree θ , and write its value as in (5.4). Let be a resonant line. Then exits a self-energy cluster T2 and enters a self-energy cluster T1 ; see Fig. 9. By construction T1 ∈ Rkj1,σ , j ,σ (ω · ν T , n 1 ) and T2 ∈ Rkj2,σ , j ,σ (ω · ν T , n 2 ), for suitable 1

1

1

2

1

1

2

2

2

2

values of the labels, with the constraint j1 = j2 = j and σ1 = σ2 = σ . If OT1 = OT2 = L , we consider also all trees obtained from θ by replacing T1 and T2 with other clusters T1 ∈ Rkj1,σ , j ,σ (ω · ν T , n 1 ) and T2 ∈ Rkj2,σ , j ,σ (ω · ν T , n 2 ), 1 1 1 1 2 2 2 2 1 2 respectively, with OT1 = OT2 = L . In this way ] L V (T1 , ω · ν T ) G [n j (ω · ν ) L V (T2 , ω · ν T2 ) 1

is replaced with (k )

(k2 ) (ω ,σ , j2 ,σ2

1 ] L M j1 ,σ (ω · ν T , n 1 ) G [n j (ω · ν ) L M j 1 , j ,σ 1

· ν T , n 2 ).

(7.1)

2

Then consider also all trees in which the factor (7.1) is replaced with (k )

(k2 ) (ω ,−σ , j2 ,σ2

1 ] L M j1 ,σ (ω · ν T , n 1 ) G [n j (ω · ν ) L M j 1 , j ,−σ 1

· ν T , n 2 ),

(7.2)

2

with ν such that ω · ν − σ ω j = ω · ν + σ ω j ; see Fig. 18. Because of Lemmas 6.2 and 6.3 the sum of the two contributions (7.1) and (7.2) gives (k1 ) (k2 ) [n ] [n ] , n1) G L M j1 ,σ (ω · ν (ω · ν ) + G (ω · ν ) L M j ,σ (ω · ν T , n 2 ), j j , j ,σ , j ,σ 1 T

1

2

2

2

where n (δ j (ω · ν )) 1 1 + (ω · ν − σ ω j ) ω · ν + σ ω j ω · ν − σ ω j 2n (δ j (ω · ν )) , (7.3) = (ω · ν + σ ω j )(ω · ν − σ ω j )

[n ] ] G [n j (ω · ν ) + G j (ω · ν ) =

[n ] −2 ] −n ) and hence |G [n j (ω · ν ) + G j (ω · ν )| ≤ 2ω j . This provides the gain factor O(2 we were looking for, with respect to the original bound C0 2n on the propagator G . ˚ If OT1 = R then if k(T˚1 ) > K 0 2n T1 /τ we can extract a factor C k(T1 ) from V (T1 , ω · ˚ ˚ ν T ) (C is the constant appearing in Lemma 4.3), and, after writing C k(T1 ) = C 2k(T1 ) 1

˚ C −k(T1 ) , use −n O(2 ).

that C −k(T1 ) ≤ C −K 0 2 ˚

n T /τ 1

≤ const.2−n T1 in order to obtain a gain factor

If k(T˚1 ) ≤ K 0 2n T1 /τ and n ≥ 0 for all ∈ PT , we obtain a gain factor proportional to 2−n because of the first line of (5.2). Of course whenever one has such a case,


391

Fig. 18. Graphical representation of the cancellation mechanism discussed in the text: ν = ν − 2σ e j . If we sum the two contributions we obtain a gain factor O(2−n )

then one has a derivative acting on V (T, u) – see (5.2). Therefore one needs to control derivatives like ⎛ ⎞⎛ ⎞ ∂u G ⎝ G ⎠ ⎝ Fv ⎠ , (7.4) ∂u V (T, u) = ∈PT

∈L(T )\{ }

v∈N (T )

where ∂u G =

n (δ(ω · ν )) ∂u n (δ(ω · ν )) − 2ω · ν . 2 2 (ω · ν ) − ω j ((ω · ν )2 − ω2j )2

(7.5)

The derived propagator (7.5) can be easily bounded by |∂u G | ≤ C1 22n ,

(7.6)

for some positive constant C1 . In principle, given a line , one could have one derivative of G for each self-energy cluster containing . This should be a problem, because in a tree of order k, a propagator G could be derived up to O(k) times, and no bound proportional to some constant to the power k can be expected to hold to order k. In fact, it happens that no propagator has to be derived more than once. This can be seen by reasoning as follows. Let T be a self-energy cluster of depth D(T ) = 1. If OT = R then a gain factor O(2−n T ) is obtained. When writing ∂u V (T, u) according to (7.4) one obtains |PT | terms, one for each line ∈ PT . Then we can bound the derivative of G according to −n (7.6). By collecting together the gain factor and the bound (7.6) we obtain 22n 2 T . We can interpret such a bound by saying that, at the cost of replacing the bound 2n of −n the propagator G with its square 22n , we have a gain factor 2 T for the self-energy cluster T . Suppose that is contained inside other self-energy clusters besides T , say T p ⊂ T p−1 ⊂ · · · ⊂ T1 (hence T p is that with largest depth, and D(T p ) = p + 1). Then, when taking the contribution to (7.4) with the derivative ∂u acting on the propagator G , we consider together the labels OTi = R and OTi = L for all i = 1, . . . , p (in other words we do not distinguish between localised and regularised values for such self-energy

392


clusters), because we do not want to produce further derivatives on the propagator G . Of course we have obtained no gain factor corresponding to the entering lines of the self-energy clusters T1 , . . . , T p , and all these lines can be resonant lines. So, eventually we shall have to keep track of this. Then we can iterate the procedure. If the self-energy cluster T does not contain any line whose propagator is derived, we split its value into the sum of the localised value plus the regularised value. On the contrary, if a line along the path PT of T is derived we do not separate the localised value of T from its regularised value. Note that, if T is contained inside a regularised self-energy cluster, then both ω · ν and ω · ν in (7.1) and (7.2) must be replaced with ω · ν (t ) and ω · ν (t ), respectively, but still ω · ν (t ) − σ ω j = ω · ν (t ) + σ ω j , so that the cancellation (7.3) still holds. Let us call a ghost line a resonant line such that (1) is along the path PT of a regularised self-energy cluster T and either (2a) enters or exits a self-energy cluster T ⊂ T containing a line whose propagator is derived or (2b) the propagator of is derived. Then, eventually one obtains a gain 2−n for all resonant lines , except for the ghost lines. In other words we can say that there is an overall factor proportional to ⎛ ⎞⎛ ⎞ −n n ⎝ 2 ⎠ ⎝ 2 ⎠ , (7.7) ∈L R (θ)

∈L G (θ)

where L G (θ ) is the set of ghost lines. Indeed, in case (2a) there is no gain corresponding to the line , so that we can insert a ‘good’ factor 2−n provided we allow also a compensating ‘bad’ factor 2n . In case (2b) one can reason as follows. Call (with some abuse of notation) T1 and T2 the self-energy clusters which enters and exits, respectively. If OT1 = OT2 = L , we consider ] L V (T1 , ω · ν T ) ∂u G [n j (ω · ν (t )) L V (T2 , ω · ν T2 ), 1

and, by summing over all possible self-energy clusters as done in (7.1), we obtain (k2 ) [n ] 1) L M (k j1 ,σ1 , j ,σ (ω · ν T , n 1 ) ∂u G j (ω · ν (t )) L M j ,σ , j ,σ (ω · ν T , n 2 );

1

2

2

2

then we sum this contribution with (k )

(k2 ) (ω ,−σ , j2 ,σ2

1 ] L M j1 ,σ (ω · ν T , n 1 ) ∂u G [n j (ω · ν (t )) L M j 1 , j ,−σ 1

· ν T , n 2 ), 2

where ν = ν − 2σ e j ; again we can use Lemmas 6.2 and 6.3 to obtain (k1 ) [n ] [n ] , n 1 ) ∂u G (ω · ν (ω · ν (t )) + ∂ G (ω · ν (t )) L M j1 ,σ u j j , j ,σ 1 T 1

×L

(k2 ) M j ,σ (ω , j2 ,σ2

· ν T , n 2 ), 2

where 2∂u n (δ(ω · ν ( t ))) (ω · ν ( t ) + σ ω j )(ω · ν ( t ) − σ ω j ) 4(ω · ν ( t ) − σ ω j )n (δ(ω · ν ( t ))) , − (ω · ν ( t ) + σ ω j )2 (ω · ν ( t ) − σ ω j )2

[n ] ] ∂u G [n j (ω · ν ( t )) + ∂u G j (ω · ν ( t )) =


393

so that we have not only the gain factor 2−n due to the cancellation, but also a factor 2n because of the term ∂u n (δ(ω · ν )). A trivial but important remark is that all the ghost lines contained inside the same self-energy cluster have different scales: in particular there is at most one ghost line on a given scale n. Therefore we can rely upon Lemma 4.4 and Lemma 5.4, to ensure that for each such line there is also at least one non-resonant line on scale ≥ n − 3 (inside the same self-energy cluster). Therefore we can bound the second product in (7.7) as ⎛ ⎞ ∞ ⎝ 2n ⎠ ≤ 2n Nn−3 (θ) , ∈L G (θ)

n=1

which in turn is bounded as a constant to the power k = k(θ ), as argued in the proof of Lemma 4.3. Finally if k(T˚1 ) ≤ K 0 2n T1 /τ and T1 contains at least one line ∈ PT1 with n = −1, in general there are p ≥ 1 self-energy clusters T p ⊂ T p−1 ⊂ · · · ⊂ T1 = T1 such that ∈ PTi for i = 1, . . . , p, and T p is the one with largest depth containing . For i = 1, . . . , p call i the exiting line of the self-energy cluster Ti and θi = θ i . Denote also, for i = 1, . . . , p − 1, by i = i+1 (θi ) (recall Notation 3.29). By Lemma 3.30 one has |ν i − ν i+1 | ≤ E 2 k( ˚ i ) for i = 1, . . . , p − 1. Moreover one has |ν 1 − σ e j | ≤ E 2 (k( ˚ 1 ) + · · · + k( ˚ p−1 )). On the other hand one has γ −n +2 ≤ δ ji (ω · ν i ) + δ ji+1 (ω · ν i+1 ) ≤ 2 Ti+1 γ , |ν i − ν i+1 |τ γ −n ≤ δ j1 (ω · ν 1 ) ≤ 2 T1 γ , τ |ν 1 − σ e j | so that one can write C

k( ˚ 1 )+···+k( ˚ p−1 ))

≤C

3k( ˚ 1 )+···+k( ˚ p−1 )) −n T1

2

p

2

−n T i

,

(7.8)

i=2

which assures the gain factors for all self-energy clusters T1 , . . . , T p . To conclude the analysis, if OT1 = L but OT2 = R, one can reason in the same way by noting that |n T − n | ≤ 1. 2

Lemma 7.1. Set (c) = max{|c1 |, . . . , |cd |, 1}. There exists a positive constant C such that for k ∈ N, j ∈ {1, . . . , d} and ν ∈ Zd one has | θ∈Tk V (θ )| ≤ C k 3k (c). j,ν

Proof. Each time one has a resonant line , when summing together the values of all self-energy clusters, a gain B1 2−n is obtained (either by the cancellation mechanism described at the beginning of this section or as an effect of the regularisation operator R). The number of trees of order k is bounded by B2k for some constant B2 ; see Remark 3.12. The derived propagators can be bounded by (7.6). By taking into account also the bound of Lemma 4.3, setting B3 = C0 , and bounding by B4k , with ∞ −n/τ B4 = exp 3c log 2 2 n , n=0

394


the product of the propagators (both derived and non-derived) of the non-resonant lines times the derived propagators of the resonant lines, we obtain the assertion with C = B1 B2 B3 B4 .

Lemma 7.2. The function (1.7), with x j,ν as in (3.10), and the counterterms η j defined in (3.11) are analytic in ε and c, for |ε| 3 (c) ≤ η0 with η0 small enough and (c) = max{|c1 |, . . . , |cd |, 1}. Therefore the solution x(t, ε, c) is analytic in t, ε, c for |ε| 3 (c)e3|ω| |Im t| ≤ η0 , with η0 small enough. Proof. Just collect together all the results above, in order to obtain the convergence of the series for η0 small enough and |ε| ξ (c) ≤ η0 , for some constant ξ . Moreover (k) x j,ν = 0 for |ν| > ξ k, for the same constant ξ . Lemma 3.10 gives ξ = 3.

A. Momentum-Depending Perturbation Here we discuss the Hamiltonian case in which the perturbation depends also on the coordinates y1 , . . . , yd , as in (1.13). As we shall see, differently from the y-independent case, here the Hamiltonian structure of the system is fundamental. It is more convenient to work in complex variables z, w = z ∗ , with z j = (y j + iω j x j )/ 2ω j , where the Hamilton equations are of the form −i˙z j = ω j z j + ε∂w j F(z, w, ε) + η j z j , (A.1) iw˙ j = ω j w j + ε∂z j F(z, w, ε) + η j w j , with F(z, w, ε) =

∞

εp

s+

s+

s−

s−

as + ,...,s + ,s − ,...,s − z 11 . . . z dd w11 . . . wdd . (A.2) 1

p=0 s + ,...,s + ,s − ,...,s − ≥0 1 d d 1 s1+ +···+sd+ +s1− +···+sd− = p+3

d

1

d

Note that, since the Hamiltonian (1.11) is real, one has a s+ ,s− = a s∗− ,s+ , s± = (s1± , . . . , sd± ) ∈ Zd+ .

(A.3)

Let us write f j+ (z, w, ε) = ε∂w j F(z, w, ε),

f j− (z, w, ε) = ε∂z j F(z, w, ε)

so that f jσ (z, w, ε) =

∞

εp

s+

s+

s−

s−

f j,σ s+ ,s− z 11 . . . z dd w11 . . . wdd ,

σ = ±,

s+ , s− ∈Zd+ s1+ +···+sd+ +s1− +···+sd− = p+1

p=1

− + with f j,+ s+ ,s− = (s − j + 1)a s + , s− +e j and f j, s+ , s− = (s j + 1)a s + +e j , s− , and hence ∗ f j,−s+ ,s− = f j,+ s− ,s+ , j = 1, . . . , d, s+ , s− ∈ Zd , (A.4a)

KAM Theory in Configuration Space and Cancellations in the Lindstedt Series − (s +j2 + 1) f j+1 ,s+ +e j ,s− = (s − j1 + 1) f j2 , s+ , s− +e j1 , 2

(s − j2

+ 1) f j+1 ,s+ ,s− +e j 2

(s +j2

+ 1) f j−,s+ +e ,s− 1 j2

395

j1 , j2 = 1, . . . , d, s+ , s− ∈ Zd ,

=

(s − j1

+ 1) f j+2 ,s+ ,s− +e j , 1

(A.4b) j1 , j2 = 1, . . . , d, s , s ∈ Zd ,

=

(s +j1

+ 1) f j−,s+ +e ,s− , 2 j1

(A.4c) j1 , j2 = 1, . . . , d, s , s ∈ Zd .

+

+

−

−

(A.4d) Expanding the solution (z(t), w(t)) in Fourier series with frequency vector ω, (A.1) gives (ω · ν − ω j )z j,ν = η j z j,ν + f j,+ν (z, w, ε), (A.5) (−ω · ν − ω j )w j,ν = η j w j,ν + f j,−ν (z, w, ε). We write the unperturbed solutions as (0)

z j (t) = c+j eiω j t ,

(0)

w j (t) = c−j e−iω j t ,

j = 1, . . . , d,

with c j = c+j ∈ C and c−j = c∗j . As in Sect. 1.2 we can split (A.5) into f j,+ e j (z, w, ε) + η j z j,e j = 0,

j = 1, . . . , d,

− f j,− e j (z, w, ε) + η j w j,−e j = 0, (ω · ν) − ω j z j,ν = f j,+ν (z, w, ε) + η j z j,ν , −(ω · ν) − ω j w j,ν = f j,−ν (z, w, ε) + η j w j,ν ,

j = 1, . . . , d,

(A.6a) (A.6b)

j = 1, . . . , d, ν = e j , (A.6c) j = 1, . . . , d, ν = −e j , (A.6d)

so that first of all one has to show that the same choice of η j makes both (A.6a) and (A.6b) hold simultaneously, and that such η j is real. We consider a tree expansion very close to the one performed in Sect. 3: we simply drop (3) in Constraint 3.4. We denote by Tkj,ν ,σ the set of inequivalent trees of order k, tree component j, tree momentum ν and tree sign σ that is, the sign label of the root line is σ . We introduce θ˘ and θ˚ as in Notation 3.5 and 3.27 respectively, and we define the value of a tree as follows. The node factors are defined as in (3.2) for the end nodes, while for the internal nodes v ∈ V (θ ) we define ⎧ + + !s − ! . . . s − ! sv,1 ! . . . sv,d σ ⎪ v,1 v,d ⎪ f j ,vs+ ,s− , kv ≥ 1, ⎪ ⎪ v v v ⎨ sv ! Fv = (A.7) ⎪ ⎪ 1 ⎪ ⎪ kv = 0. ⎩− σv , 2c jv The propagators are defined as G = 1 if ν = σ e j and ] G = G [n j (σ ω · ν ),

otherwise, and we define V (θ ) as in (3.9).

G [n] j (u) =

n (|u − ω j |) , u − ωj

(A.8)

396


Finally we set z j,e j = w ∗j,−e j = c j , and formally define z j,ν =

∞

(k)

(k)

εk z j,ν ,

z j,ν =

w j,ν =

V (θ ),

ν = e j ,

θ∈Tkj,ν,+

k=1 ∞

ε

k

w (k) j,ν ,

w (k) j,ν

=

(A.9) V (θ ),

ν = −e j ,

θ∈Tkj,ν,−

k=1

and η j,σ =

∞

εk η(k) j,σ ,

η(k) j,σ = −

k=1

1 cσj

V (θ ).

(A.10)

θ∈Tkj,σ e ,σ j

Note that Remarks 3.9, 3.13 and 3.17 still hold. Lemma A.1. With the notations introduced above, one has η∗j,+ = η j,− and z ∗j,ν = w j,−ν . Proof. By definition we only have to prove that for any θ ∈ Tkj,ν ,+ there exists θ ∈ Tkj,−ν ,− such that V (θ )∗ = V (θ ). The proof is by induction on the order of the tree. Given θ ∈ Tkj,ν ,+ , let us consider the tree θ obtained from θ by replacing the labels σv of all the nodes v ∈ N0 (θ ) with −σv and the labels σ of all the lines ∈ L(θ ) with −σ . Call 1 , . . . , p the lines on scale −1 (if any) closest to the root of θ , and denote by vi the node i enters and by θi the tree with root line i . Each tree θi is then replaced with a tree θi such that V (θi )∗ = V (θi ) by the inductive hypothesis. Moreover, as for any internal line in θ the momentum becomes −ν , the propagators do not change. Finally, for any v ∈ V (θ˘ ) the node factor is changed into ⎧ − − + ! · · · s+ ! sv,1 ! · · · sv,d !sv,1 ⎪ −σ v,d ⎪ ⎪ f j ,s −v ,s+ , kv ≥ 1, ⎪ ⎪ v v v ⎨ sv ! Fv = (A.11) ⎪ ⎪ 1 ⎪ ⎪ kv = 0. ⎪ ⎩− 2c−σv , jv

Hence by (A.4a) one has V (θ )∗ = V (θ ).

Lemma A.2. With the notations introduced above, one has η j,+ ∈ R. Proof. We only have to prove that for any θ ∈ Tkj,e j ,+ there exists θ ∈ Tkj,e j ,+ such that c+j V (θ )∗ = c−j V (θ ). Let v0 ∈ E +j (θ˘ ) (existing by Remark 3.9) and let us consider the tree θ obtained from θ by (1) exchanging the root line 0 with v0 , (2) replacing all the labels σv of all the nodes v ∈ N0 (θ )\{v0 } with −σv , and (3) replacing all the labels σ of all the internal lines with −σ , except for those in P( v0 , 0 ) which remain the same. The propagators do not change; this is trivial for the lines outside P( v0 , 0 ), while for ∈ P( v0 , 0 ) one


397

can reason as follows. The line divides E(θ˘ )\{v0 } into two disjoint sets of end nodes E(θ˘ , p) and E(θ˘ , s) such that if = w one has E(θ˘ , p) = {v ∈ E(θ˘ )\{v0 } : v ≺ w} and E(θ˘ , s) = (E(θ˘ )\{v0 })\E(θ˘ , p). If ν ( p) =

νv ,

˘ p) v∈E(θ,

ν (s) =

νv ,

˘ v∈E(θ,s)

one has ν ( p) + ν (s) = 0. When considering as a line in θ one has ν = ν ( p) + e j while in θ one has ν = −ν (s) + e j . Hence, as we have not changed the sign label σ , also G does not change. The node factors of the internal nodes are changed into their complex conjugates; this can be obtained as in Lemma A.1 for the internal nodes w such that w ∈ / P( v0 , 0 ) while for the other nodes one can reason as follows. First of all if v is such that v ∈ P( v0 , 0 )∪{ v0 }, there is a line v ∈ P( v0 , 0 )∪{ 0 } entering v. We shall denote j v = j1 , σ v = σ , j v = j2 , and σ v = σ . Moreover we call siσ the number of lines outside P( v0 , 0 ) ∪ { 0 } with component label i and sign label σ entering v. Let us consider first the case σ = σ = +. When considering v as node of θ one has + ∗ s1 ! · · · sd+ !s1− ! · · · sd− !(s +j2 + 1) + ∗ Fv = f j1 ,s+ +e j ,s− 2 sv ! =

s1+ ! · · · sd+ !s1− ! · · · sd− !(s +j2 + 1) sv !

f j−,s− ,s+ +e . 1

j2

+ When considering v as node of θ one has s+v = s− + e j1 and s− v = s , so that

Fv =

s1+ ! · · · sd+ !s1− ! · · · sd− !(s − j1 + 1) sv !

f j+2 ,s− +e j ,s+ , 1

and hence by (A.4b) Fv∗ = Fv . Reasoning analogously one obtains Fv∗ = Fv also in the cases σ = σ = − and σ = σ , using again (A.4b) when σ = σ = −, and (A.4c) and (A.4d) for σ = −, σ = + and σ = +, σ = − respectively. Hence the assertion is proved.

We define the self-energy clusters as in Sect. 3.6, but replacing the constraint (3) with (3 ) one has |ν T − ν T | ≤ 2 and |σ T ω · ν T − ω j T | = |σ T ω · ν T − ω j |. We T

introduce T˘ and T˚ as in Notation 3.23 and 3.27 respectively, and we can define V (T ) as in (3.12) and the localisation and the regularisation operators as in Sect. 5. Note that the main difference with the y-independent case is in the role of the sign label σ . In fact, here the sign label of a line does not depend on its momentum and component labels, and the small divisor is given by δ j,σ (ω · ν) = |σ ω · ν − ω j |. Hence the dimensional bounds of Sect. 4 and the symmetries discussed in Sect. 6 and summarised in Lemma 6.1 can be proved word by word as in the y-independent case, except for the second equality in Lemma 6.3 where one has to take into account a change of signs. More precisely for T ∈ Rkj,σ, j ,σ (u, n), with j = j and n ≥ 0 for all ∈ PT , we define G1 (T ) as in Sect. 6 and G3 (T ) as in Sect. 6 but replacing also the sign labels σ of the lines ∈ L(T ) with −σ .

398


Lemma A.3. For all T ∈ Rkj,σ, j ,σ (u, n), with j = j and n ≥ 0 for all ∈ PT , one has σ c−σ L V (T , u) = cσj c−σ L V (T , u). (A.12) j c j j T ∈G1 (T )

T ∈G3 (T )

Proof. We consider only the case k(T˚ ) ≤ K 0 2n T /τ . For fixed T ∈ Rkj,σ, j ,σ (u, n), with j = j , let θ ∈ Tkj,σ e j ,σ (n) be the tree obtained from T by replacing the entering line T with a line exiting a new end node v0 with σv0 = σ and ν v0 = σ e j . As in the proof of Lemma 6.3 one has cσj L V (T , u) = |E σj (θ˘ )| V (θ ). T ∈G1 (T )

Now, let θ ∈ Tkj,−σ e j ,−σ (n) be the tree obtained from θ by replacing all the labels σv of the nodes v ∈ N0 (θ ) with −σv , and the labels σ of all the lines ∈ L(θ ) with −σ . Any T ∈ G3 (T ) can be obtained from θ by replacing a line exiting an end node v ∈ E σj (θ˘ ) with entering line T , carrying the same labels as T . Hence, by Lemma A.1, −σ σ σ ˘ σ ˘ ∗ c−σ L V (T , u) = c−σ j c j j |E j (θ)| V (θ ) = c j |E j (θ )| V (θ ) T ∈G1 (T ) −σ = c−σ j c j

(L V (T , u))∗ .

T ∈G3 (T )

On the other hand, exactly as in Lemma A.2 one can prove that for any T ∈ G3 (T ) there exists T ∈ G3 (T ) such that ∗ σ c−σ j (L V (T , u)) = c j L V (T , u),

and hence the assertion follows.

The cancellation mechanism and the bounds proved in Sect. 7 follow by the same reasoning (in fact it is even simpler); see the next appendix for details. B. Matrix Representation of the Cancellations As we have discussed in Sect. 5 the only obstacle to convergence of the formal power series of the solution is given by the accumulation of resonant lines; see Fig. 14. The cancellation mechanism described in Sect. 7 can be expressed in matrix notation. This is particularly helpful in the y-dependent case. For this reason, and for the fact that the formalism introduced in Appendix A includes the y-independent case, we prefer to work here with the variables (z, w). We first develop a convenient notation. Given ν such that σ (ν, 1) = + and δ1,+ (ω·ν) < γ let us group together, in an ordered set S(ν), all the ν such that ν = ν ( j, σ ) := ν − e1 + σ e j , σ = ±1 and j = 1, . . . , d, see Remark 3.19. By definition one has δ1,+ (ω · ν) = δ j,σ (ω · ν ( j, σ )) for all j = 1, . . . , d and σ = ±. Then we construct a 2d × 2d localised self-energy matrix L M (k) (ω · ν, n) with entries L M (k) j,σ, j ,σ (ω · ν ( j , σ ), n). We also define the 2d × 2d diagonal propagator matrix G [n] (ω · ν) with


399

[n] [n] [n] entries G j,σ, j ,σ (ω · ν) = δ j, j δσ,σ G j (ω · ν ( j, σ )), with G j (u) defined according to (A.8), and δa,b is the Kronecker delta. As in Sect. 7 let us consider a chain of two self-energy clusters; see Fig. 9. By definition its value is ] L V (T1 , ω · ν 1 ) G [n j (ω · ν ) L V (T2 , ω · ν 2 ),

with ν 1 = ν T and ν 2 = ν T2 . 1 Notice that, if one sets also for the sake of simplicity, σ1 = σ T , j1 = j T , σ2 = σ T2 , 1

1

and j2 = j T2 , by the constraint (3 ) in the definition of self-energy clusters given in Appendix A, one has ν 1 − ν = σ1 e j1 − σ e j and ν − ν 2 = σ e j − σ2 e j2 ; moreover ν 1 , ν , ν 2 all belong to a single set S(ν) for some ν. As done in Sect. 7 let us sum together the values of all the possible self-energy clusters T1 and T2 with fixed labels associated with the external lines, and of fixed orders k1 and k2 , respectively. We obtain (k )

(k )

1 2 (ω · ν ( j , σ ), n T1 ) G j[n ,σ ] , j ,σ (ω · ν) L M j ,σ (ω · ν ( j2 , σ2 ), n T2 ). L M j1 ,σ 1 , j ,σ , j2 ,σ2

If we also sum over all possible values of the labels j , σ we get d σ =± j =1

[n ]

(k )

(k )

2 L M j 1,σ , j ,σ (ω · ν ( j , σ ), n T1 ) G j ,σ (ω · ν)L M j ,σ (ω · ν ( j2 , σ2 ), n T2 ) 1 1 , j ,σ , j2 ,σ2

= L M (k1 ) (ω · ν, n T1 ) G [n ] (ω · ν) L M (k2 ) (ω · ν, n T2 )

j1 ,σ1 , j2 ,σ2

,

(i.e. the entry j1 , σ1 , j2 , σ2 of the matrix in square brackets). By the definition (A.8) of the propagators and by the symmetries of Lemma 6.1, G [n] (ω · ν) and L M (k) (ω · ν, n) have the form ⎛

1 0 ⎜ 0 −1 ⎜ ⎜ (|ω · ν − ω |) 0 n 1 ⎜ [n] ⎜ G (ω · ν) = ⎜ .. ω · ν − ω1 ⎜ . ⎜ ⎝ 0

⎞ ···

0 .. ..

. .

···

0

⎟ ⎟ ⎟ .. ⎟ . ⎟, ⎟ .. . 0 ⎟ ⎟ 1 0 ⎠ 0 0 −1 .. .

(B.1)

and c1∗ c1 (k) ⎜ M1,1 (n) ⎜ c1 c1 ⎜ ⎜ .. ⎜ . L M (k) (ω · ν , n) = ⎜ ⎜ ⎜ ⎜ c ∗ c1 ⎝ (k) Md,1 (n) d cd c1 ⎛

c1∗ cd (k) (n) · · · M 1,d ∗ c1 c1 c1 cd ∗ .. c j c j c∗j c∗j (k) M j, j (n) . ∗ c j c j c j c j cd∗ cd cd∗ c1∗ (k) (n) · · · M d,d cd c1∗ cd cd

c1∗ c1∗

c1∗ cd∗

⎞

⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ cd∗ cd∗ ⎟ ⎠ cd cd∗

c1 cd∗

400


respectively. A direct computation gives L M (k1 ) (ω · ν, n T1 ) G [n ] (ω · ν) L M (k2 ) (ω · ν, n T2 ) =

n (|ω · ν − ω1 |) −σ1 σ2 c j1 c j2 ω · ν − ω1

d

j1 ,σ1 , j2 ,σ2

M j1 , j (n T1 ) M j, j2 (n T2 ) |c j |2

j=1

(−1)1+σ 1 = 0,

σ =±

(B.2) for all choices of the scales n , n T1 , n T2 and of the orders k1 , k2 . This proves the necessary cancellation. Note that this is an exact cancellation in terms of the variables (z, w): all chains of localised self-energy clusters of length p ≥ 2 can be ignored as their values sum up to zero. In the y-independent case, and in terms of the variables x, the cancellation is only partial, and one only finds L M (k1 ) G [n] L M (k2 ) = O(2−n ), as discussed in Sect. 7. C. Resummation of the Perturbation Series The fact that the series obtained by systematically eliminating the self-energy clusters converges, as seen in Sect. 4, suggests that one may follow another approach, alternative to what we have described so far, and leading to the same result. Indeed, one can consider a resummed expansion, where one really gets rid of the self-energy clusters at the price of changing the propagators into new dressed propagators – again terminology is borrowed from quantum field theory. This is a standard procedure, already exploited in the case of KAM tori [10], lower-dimensional tori [10,12], skew-product systems [11], etc. The convergence of the perturbation series reflects the fact that the dressed propagators can be bounded proportionally to (a power of) the original ones for all values of the perturbation parameter ε. In our case, the latter property can be seen as a consequence of the cancellation mechanism just described. In a few words – and oversimplifying the strategy – the dressed propagators are obtained starting from a tree expansion where no self-energy clusters are allowed, and then ‘inserting arbitrary chains of self-energy clusters’: this means that each propagator G [n] = G [n] (ω · ν) is replaced by a dressed propagator

[n] = G [n] + G [n] MG [n] + G [n] MG [n] MG [n] + · · · ,

(C.1)

where M = M(ω · ν) denotes the insertion of all possible self-energy clusters compatible with the labels of the propagators of the external lines (M is the matrix with entries M j,σ, j σ (ω · ν ( j , σ )) formally defined in Remark 3.26). Then, formally, one can sum together all possible contributions in (C.1), so as to obtain −1 −1

[n] = G [n] 1 − MG [n] = A−1 − B ,

A := G [n] ,

B := M. (C.2)

For sake of simplicity, let us also identify the self-energy values with their localised parts, so as to replace in (C.1), and hence in (C.2), M with L M, if L is the localisation operator. Then, in the notations we are using, the cancellation (B.2) reads B AB = 0, which implies

[n] = A + AB A.


401

Therefore one finds [n] ≤ A + A2 B = O(22n ). So the values of the trees appearing in the resummed expansion can be bounded as done in Sect. 4, with the only difference that now, instead of the propagators G bounded proportionally to 2n , one has the dressed propagators [n ] bounded proportionally to 22n . Of course, the argument above should be made more precise. First of all one should have to take into account also the regularised values of the self-energy clusters. Moreover, the dressed propagators should be defined recursively, by starting from the lower scales: indeed, the dressed propagator of a line on scale n is defined in terms of the values of the self-energy clusters on scales < n, as in (C.2), and the latter in turn are defined in terms of (dressed) propagators on scales < n, according to (3.13). As a consequence, the cancellation mechanism becomes more involved because the propagators are no longer of the form (B.1); in particular the symmetry properties of the self-energy values should be proved inductively on the scale label. In conclusion, really proceeding by following the strategy outlined above requires some work (essentially the same amount as performed in this paper). We do not push forward the analysis, which in principle could be worked out by reasoning as done in the papers quoted above.

References 1. Bartuccelli, M.V., Gentile, G.: Lindstedt series for perturbations of isochronous systems: a review of the general theory. Rev. Math. Phys. 14(2), 121–171 (2002) 2. Berretti, A., Gentile, G.: Bryuno function and the standard map. Commun. Math. Phys. 220(3), 623–656 (2001) 3. Bollobás, B.: Graph theory. An introductory course. Graduate Texts in Mathematics 63, New York-Berlin: Springer-Verlag, 1979 4. Bricmont, J., Gaw¸edzki, K., Kupiainen, A.: KAM theorem and quantum field theory. Commun. Math. Phys. 201(3), 699–727 (1999) 5. Bryuno, A.D.: Analytic form of differential equations. I, II. Trudy Moskov. Mat. Obšˇc. 25, 119–262 (1971); ibid. 26, 199–239 (1972). English translations: Trans. Moscow Math. Soc. 25, 131–288 (1971); ibid. 26, 199–239 (1972) 6. de la Llave, R., González, A., Jorba, À., Villanueva, J.: KAM theory without action-angle variables. Nonlinearity 18(2), 855–895 (2005) 7. De Simone, E., Kupiainen, A.: The KAM theorem and renormalization group. Erg. Th. Dynam. Syst. 29(2), 419–431 (2009) 8. Eliasson, L.H.: Absolutely convergent series expansions for quasi periodic motions. Math. Phys. Electron. J. 2, Paper 4, 33 pp. (electronic) (1996) 9. Gallavotti, G.: Twistless KAM tori. Commun. Math. Phys. 164(1), 145–156 (1994) 10. Gallavotti, G., Bonetto, F., Gentile, G.: Aspects of ergodic, qualitative and statistical theory of motion. Texts and Monographs in Physics, Berlin: Springer-Verlag, 2004 11. Gentile, G.: Resummation of perturbation series and reducibility for Bryuno skew-product flows. J. Stat. Phys. 125(2), 321–361 (2006) 12. Gentile, G.: Degenerate lower-dimensional tori under the Bryuno condition. Erg. Th. Dynam. Syst. 27(2), 427–457 (2007) 13. Gentile, G.: Diagrammatic methods in classical perturbation theory. Encyclopedia of Complexity and System Science, Vol. 2, Ed. R.A. Meyers, Berlin: Springer, 2009, pp. 1932–1948 14. Gentile, G.: Quasi-periodic motions in strongly dissipative forced systems. Erg. Th. Dynam. Syst. 30(5), 1457–1469 (2010) 15. Gentile G. (2010) Quasi-periodic motions in dynamical systems. Review of a renormalisation group approach. J. Math. Phys. 51, no. 1, 015207, 34 pp (2010) 16. Gentile, G., Bartuccelli, M., Deane, J.: Summation of divergent series and Borel summability for strongly dissipative equations with periodic or quasi-periodic forcing terms. J. Math. Phys. 46, no. 6, 062704, 21 pp (2005) 17. Gentile, G., Mastropietro, V.: Methods for the analysis of the Lindstedt series for KAM tori and renormalizability in classical mechanics. A Review with Some Applications. Rev. Math. Phys. 8(3), 393–444 (1996) 18. Harary, F.: Graph theory. Reading, MA-Menlo Park, CA-London: Addison-Wesley Publishing Co., 1969

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19. Levi, M., Moser, J.: A Lagrangian proof of the invariant curve theorem for twist mappings. In: Smooth ergodic theory and its applications (Seattle, WA, 1999), Proc. Sympos. Pure Math. 69, Providence, RI: Amer. Math. Soc., 2001, pp. 733–746 20. Moser, J.: Convergent series expansions for quasi–periodic motions. Math. Ann. 169, 136–176 (1967) 21. Poincaré, H.: Les méthodes nouvelles de la mécanique céleste. Vol. I–III, Paris: Gauthier-Villars, 1892– 1899 22. Salamon, D., Zehnder, E.: KAM theory in configuration space. Comment. Math. Helv. 64, 84–132 (1989) Communicated by G. Gallavotti


Communications in


On the C ∗ -Algebra of a Locally Injective Surjection and its KMS States Klaus Thomsen Institut for Matematiske Fag, Ny Munkegade, 8000 Aarhus C, Denmark. E-mail: [email protected] Received: 10 March 2010 / Accepted: 2 August 2010 Published online: 20 November 2010 – © Springer-Verlag 2010

Abstract: It is shown that a locally injective surjection on a compact metric space is a factor of a local homeomorphism in such a way that the associated C ∗ -algebras are isomorphic. This is subsequently used to obtain upper and lower bounds for the possible β-values of KMS-states for generalized gauge actions on the C ∗ -algebra. 1. Introduction In [Th] the construction of a C ∗ -algebra from an étale groupoid, as introduced by J.Renault in [Re1], was generalized to a larger class of locally compact groupoids called semi-étale groupoids, where the range and source maps are locally injective, but not necessarily open. The main purpose with the generalization was to make the powerful techniques for étale groupoids available to the study of dynamical systems via the groupoid constructed in increasing generality by Renault, Deaconu and AnantharamanDelaroche, [Re1,D,A], also when the underlying map is not open. In particular, as shown in [Th] this makes it possible to handle general (one-sided) subshifts. One of the intriguing connections between dynamical systems and C ∗ -algebras is the relation between the thermodynamical formalism of Ruelle, as described in [Ru], and quantum statistical mechanics, as described in [BR]. One relation between these formalisms is very concrete and direct and manifests itself in almost all of the C ∗ -algebraic settings of quantum statistical mechanics through a bijective correspondence between KMS states and measures fixed by a dual Ruelle operator. This relation is implicit in the work of J. Renault, [Re1 and Re2], and has been developed further by R. Exel, [E]. By using this correspondance Kumjian and Renault, [KR], were able to use Walters’ results, [W2], on the convergence of the Ruelle operator to extend most results on the existence and uniqueness of KMS states for the generalized gauge actions on Cuntz-Krieger algebras which has been one of the favourite models in quantum statistical mechanics. The main purpose of the present work is to show that there is a canonical way to pass from a locally injective continuous surjection to a local homeomorphism in such

404

K. Thomsen

a way that the C ∗ -algebras of the corresponding groupoids, one of them defined as in [Th], are isomorphic. The construction is a generalization of W. Krieger’s construction of a canonical extension for a sofic shift, [Kr1,Kr2], now known as the left Krieger cover. The canonical local homeomorphic extension of a general locally injective surjection which we construct is undoubtedly useful for other purposes, and it seems to deserve a more thorough investigation. Here we use it to investigate the KMS states of the generalized gauge actions. In fact, we restrict our considerations even further by focusing only on the possible values of the inverse temperature β for such KMS states. The results we obtain give bounds on the possible β-values and ensure the existence of KMS states under mild conditions on the potential function. We depart from the work of Exel in [E] and the main tool to prove existence of KMS states is a method developed by Matsumoto, Watatani and Yoshida in [MWY] and Pinzari, Watatani and Yonetani in [PWY]. Concerning bounds on the possible β-values of KMS states, the main novelty is the observation that it is not so much the entropy of the map which provides the bounds but rather the exponential growth rate of the number of pre-images. The relevant entity is thus an invariant h m which was introduced by Hurley in [Hu] and studied further in [FFN], among others. For forward expansive maps the invariant of Hurley is equal to the topological entropy, but in general it is smaller than the topological entropy. The invariant of Hurley controls the existence of KMS states completely when the potential function is strictly positive or strictly negative: For such potential functions there is a KMS-state if and only if h m is not zero. We refer to Sect. 6 for more details on our results on KMS states. 2. Recap about C r∗ ϕ Let X be a locally compact Hausdorff space and ϕ : X → X a continuous map. We assume that ϕ is locally injective, meaning that there is a basis for the topology of X consisting of sets on which ϕ is injective. Set ϕ = (x, k, y) ∈ X × Z × X : ∃a, b ∈ N, k = a − b, ϕ a (x) = ϕ b (y) . This is a groupoid with the set of composable pairs being ϕ(2) = (x, k, y), (x , k , y ) ∈ ϕ × ϕ : y = x . The multiplication and inversion are given by (x, k, y)(y, k , y ) = (x, k + k , y ) and (x, k, y)−1 = (y, −k, x). To turn ϕ into a locally compact topological groupoid, fix k ∈ Z. For each n ∈ N such that n + k ≥ 0, set ϕ (k, n) = (x, l, y) ∈ X × Z × X : l = k, ϕ k+i (x) = ϕ i (y), i ≥ n . This is a closed subset of the topological product X ×Z× X and hence a locally compact Hausdorff space in the relative topology. Since ϕ is locally injective ϕ (k, n) is an open subset of ϕ (k, n + 1), and hence the union ϕ (k) = ϕ (k, n) n≥−k

Local Homeomorphisms

405

is a locally compact Hausdorff space in the inductive limit topology. The disjoint union ϕ = ϕ (k) k∈Z

is then a locally compact Hausdorff space in the topology where each ϕ (k) is an open and closed set. In fact, as is easily verified, ϕ is a locally compact groupoid in the sense of [Re1]. Note that the unit space ϕ0 of ϕ equals X via the identification x → (x, 0, x). The local injectivity of ϕ ensures that the range map r (x, k, y) = x is locally injective, i.e. ϕ is semi étale.Wecan therefore define the corresponding C ∗ -algebra Cr∗ ϕ as in [Th]. Briefly Cr∗ ϕ is the completion of the ∗-algebra alg∗ ϕ generated by the continuous and compactly supported function on ϕ under the convolution product f g(x, k, y) = f (x, n, z)g(z, m, y), z,m+n=k

and the involution f ∗ (x, k, y) = f (y, −k, x). The elements of alg∗ ϕ are all bounded and of compact support, but not necessarily continuous. The elements of alg∗ ϕ whose supports are contained in the unit space, identified with X as it is, generate under the completion an abelian C ∗ -algebra Dϕ which contains C0 (X ) and consists of bounded functions vanishing at infinity. The restriction map extends to a conditional expectation Pϕ : Cr∗ ϕ → Dϕ . Let us now restrict the attention to the case where X is compact and metrizable. One ∗ R × N of the results from [Th] is that Cr∗ ϕ can then be realized as a crossed C ϕ r ϕ

∗ R ∗ -subalgebra of C ∗ is the C generated in the sense of Paschke, where C ϕ ϕ r r ϕ is the endomorphism of Cr∗ ϕ given by conjugation with the by Cc ϕ (0) and

isometry Vϕ , where ⎧ ⎨ m(x)− 21 when k = 1 and y = ϕ(x) Vϕ (x, k, y) = ⎩ 0 otherwise. The function m : X → N which enters here is also going to play an important role in the present paper and it is equal to m = N ◦ ϕ, with N (x) = #ϕ −1 (x). While this crossed product descriptionis useful for several purposes, including the cal culation of the K -theory groups of Cr∗ ϕ , it is going to be instrumental here to relate to a crossed product description in the sense of Exel, [E]. 3. C r∗ ϕ as a Crossed Product in the Sense of Exel Let f ∈ Dϕ . Then Pϕ Vϕ f Vϕ∗ (x) = m(x)−1 f (ϕ(x)). Since m ∈ Dϕ this shows that f ◦ ϕ ∈ Dϕ . We can therefore define a ∗-endomorphism αϕ of Dϕ such that αϕ ( f ) = f ◦ ϕ.

(3.1)

406

K. Thomsen

Note that αϕ is unital, and injective since ϕ is surjective. Let f ∈ Dϕ , and let 1ϕ (1,0) be the characteristic function of the open and compact subset ϕ (1, 0) of ϕ . Then 1∗ϕ (1,0) f 1ϕ (1,0) ∈ Dϕ and 1∗ϕ (1,0) f 1ϕ (1,0) (x) = Hence the function X x →

f (z).

(3.2)

z∈ϕ −1 (x)

f (z) is in Dϕ . In particular, the function 1 N (x) = #ϕ −1 (x) = z∈ϕ −1 (x)

z∈ϕ −1 (x)

is in Dϕ . This allows us to define Lϕ : Dϕ → Dϕ such that f (z). Lϕ ( f )(x) = N (x)−1 z∈ϕ −1 (x)

Lϕ is a unital positive linear map and Lϕ f αϕ (g) = Lϕ ( f )g for all f, g ∈ Dϕ . Hence Lϕ is a transfer operator in the sense of Exel, cf. [E and EV], so that the crossed product Dϕ αϕ ,Lϕ N is defined. Observe that Lϕ is faithful and that the Standing Hypotheses of [EV], Hypotheses 3.1, are all satisfied. The following result generalizes Theorem 9.2 in [EV], and to some extent also Theorem 4.18 of [Th]. Theorem 3.1. There is a ∗-isomorphism Dϕ αϕ ,Lϕ N → Cr∗ ϕ which is the identity on Dϕ and takes the isometry S of Exel (cf. [E]) to the isometry Vϕ ∈ Cr∗ ϕ . k Proof. Since ϕ is locally injective there is a partition of unity {bi }i=1 in C(X ) ⊆ Dϕ such that ϕ is injective on supp bi for each i. It is then straightforward to check that

f =

k

1 1 (bi m) 2 αϕ ◦ Lϕ (bi m) 2 f

i=1

1 k is a quasi-basis for the conditional expectation for all f ∈ Dϕ , so that (bi m) 2 i=1 αϕ ◦ Lϕ of Dϕ onto αϕ Dϕ in the sense of [EV]. It is also straightforward to check that Vϕ f = αϕ ( f )Vϕ and Vϕ∗ f Vϕ = Lϕ ( f ) for all f ∈ Dϕ . Furthermore,

1=

k

1

1

(bi m) 2 Vϕ Vϕ∗ (bi m) 2 .

i=1

It follows therefore from Corollary 7.2 of [EV] that there is a ∗-homomorphism ρ : Dϕ αϕ ,Lϕ N → Cr∗ ϕ which is the identity on Dϕ and takes the isometry S to the isometry Vϕ ∈ Cr∗ ϕ . To see that ρ is surjective we must show that Cr∗ ϕ n is generated by Dϕ and Vϕ . From the expresssion for Vϕn Vϕ∗ given in the proof of Theorem 4.8 of [Th], combined with Corollary 4.5 from [Th], it follows that the


407

C ∗ -algebra generated by Vϕ and Dϕ contains the characteristic function 1 R(ϕ n ) for each n. It follows then that it contains C(X ) 1 R(ϕ n ) C(X )

(3.3)

since C(X ) ⊆ Dϕ . Among the functions in (3.3) are the elements of C (R (ϕ n )) which are restrictions to R (ϕ n ) of product type functions, X × X (x, y) → f (x)g(y), with f, g ∈ C(X ). These functions generate C(X × X ) and their restriction generate C (R (ϕ n )) so it follows that the C ∗ -algebra generated by Vϕ and Dϕ contains C (R (ϕ n )) for each n. Since C (R (ϕ n )), Cr∗ Rϕ = n

we conclude from Theorem 4.6 of [Th] that it coincides with Cr∗ ϕ , proving that ρ is surjective. Finally, it follows from Theorem 4.2 of [EV] that ρ is injective since the gauge action on Cr∗ ϕ can serve as the required T-action.

4. A Canonical Local Homeomorphism Extending (X, ϕ) In this section we show that the continuous map ψ from the Gelfand spectrum of Dϕ to itself which corresponds to the endomorphism (3.1) of Dϕ is a local homeomorphism and that the corresponding dynamical system is a canonical extension of (X, ϕ). The proof is based on the well-known contravariant equivalence between compact Hausdorff spaces and unital abelian C ∗ -algebras.

be the Gelfand spectrum of Dϕ . Recall To simplify notation, set D = Dϕ and let D

that D consists of the unital ∗-homomorphisms c : D → C, also known as the char is closed in the weak*-topology of the unit ball in the dual space D ∗ acters of D. D of D and obtains in this way a compact topology. Since X is compact and metrizable

is metrizable. Finally, recall it follows that D is separable and it follows that also D

in the natural way; viz. that every element d ∈ D becomes a continuous function on D d(c) = c(d), and this recipe gives rise to an (isometric) ∗-isomorphism between D and

which we suppress in the notation by simply identifying D and C( D)

whenever C( D) it is convenient.

→ X arising from the fact that every character of C(X ) comes There is a map π : D

of D there is a unique point from evaluation at a point in X : Given a character c ∈ D π(c) ∈ X such that c( f ) = f (π(c)) for all f ∈ C(X ). Note that π is continuous. We

→ D

such that ψ(c)(g) = c (g ◦ ϕ) for all g ∈ D. It follows straightfordefine ψ : D

that ψ is continuous. Hence D,

ψ is wardly from the definition of the topology of D a dynamical system. Note that f ((ϕ ◦ π ( c)) = f ◦ ϕ (π(c)) = c ( f ◦ ϕ) = ψ(c)( f ) = f (π ◦ ψ(c))

ψ) is equivariant. Define ι : X → D

for all f ∈ C(X ), proving that π : (X, ϕ) → ( D,

by ι(x) = cx ∈ D, where cx is the character defined such that cx (g) = g(x) for all g ∈ D. Since g (ψ ◦ ι(x)) = cx (g ◦ ϕ) = g (ϕ(x)) = cϕ(x) (g) we see that also

ψ) is equivariant. Furthermore π ◦ ι(x) = x for all x ∈ X , proving ι : (X, ϕ) → ( D, that ι is injective and π surjective. Note, however, that ι is generally not continuous.

Since g ∈ D, cx (g) = 0 ∀x ∈ X ⇒ g = 0, the range ι(X ) of ι is dense in D.

408

K. Thomsen

ψ) is canonical in the following sense: If It is evident that the construction of ( D, ϕ : X → X is another locally injective surjection of a compact Hausdorff space X ,

ψ) to ( then a conjugacy from (X, ϕ) to (X , ϕ ) induces a conjugacy from ( D, D, ψ ) which extends the given conjugacy in the sense that the diagram

π

D

/ D

X

/ X

π

commutes. It remains now only to establish the following Proposition 4.1. ψ is a surjective local homeomorphism.

and set z = π(ψ(c)) = ϕ (π(c)). By Lemma Proof. ψ is locally injective: Let c ∈ D 3.6 of [Th] there is an open neighborhood U of z and open sets Vi , i = 1, 2, . . . , j, where j = #ϕ −1 (z), such that 1) ϕ −1 U ⊆ V1 ∪ V2 ∪ · · · ∪ V j , 2) Vi ∩ Vi = ∅ when i = i , and 3) ϕ is injective on Vi for each i. Without loss of generality we may assume that π(c) ∈ V1 . Let h, H ∈ C(X ) be such that 0 ≤ h ≤ 1, h (π(c)) = 1, ϕ (supp h) ⊆ U , H h = h and supp H ⊆ V1 . Set

: c (h) > 0 ; W = c ∈ D

To show that c ∈ W we choose a sequence {z k } in X such clearly an open subset of D. that limk ι (z k ) = c. Then π(c) = limk π ◦ ι (z k ) = limk z k so that c(h) = lim ι (z k ) (h) = lim h (z k ) = h (π(c)) = 1. k

k

W is therefore an open neighborhood c in D. To show that ψ is injective of on W , let c , c ∈ W and z in X such that lim and choose sequences z ι z k = c and k k k limk ι z k = c . Since lim h z k = lim ι z k (h) = c (h) > 0, k

k

it follows that h z k > 0 for all large k. Hence ϕ z k ∈ U , H z k = 1 and z k ∈ V1 for all large k. It follows that ⎛ ⎛ ⎞ ⎞ f H (v)⎠ = lim ι ◦ ϕ z k ⎝ f H (v)⎠ ψ(c ) ⎝ k

v∈ϕ −1 (·)

= lim k

for all f ∈ D. Similarly,

⎛ ψ(c ) ⎝

v∈ϕ −1 (·)

f H (v) = lim f (z k ) = c ( f )

v∈ϕ −1 (ϕ(z k ))

k

⎞ f H (v)⎠ = c ( f )

v∈ϕ −1 (·)

for all f ∈

D. It follows that ψ(c ) = ψ(c ) ⇒ c = c , proving that ψ

is injective on W .


409

ψ is open: Let f ∈ D be a non-negative function and set

: c( f ) > 0 . V = c∈D

so we consider an element c ∈ V , and set It suffices to show that ψ(V ) is open in D, ⎧ ⎫ ⎞ ⎛ ⎨ c( f ) ⎬

: c ⎝ W = c ∈ D f (v)⎠ > . ⎩ 2 ⎭ −1 v∈ϕ

(·)

Let {z k } be a sequence in X such that limk ι (z k ) = c and note that ⎛ ⎞ ⎛ ⎞ ψ(c) ⎝ f (v)⎠ = lim ι (ϕ(z k )) ⎝ f (v)⎠ v∈ϕ −1 (·)

k

= lim k

v∈ϕ −1 (·)

f (v) ≥ lim f (z k )

v∈ϕ −1 (ϕ(z k ))

k

c( f ) . k 2 It follows that W is an open neighborhood of ψ(c). It suffices therefore to show that W ⊆ ψ(V ). Let c ∈ W and choose a sequence {z k } in X such that limk→∞ ι(z k ) = c

For all large k, in D. ⎛ ⎞ c( f ) , f (v) = ι z k ⎝ f⎠ > 2 −1 −1 = lim ι(z k )( f ) = c( f ) >

v∈ϕ

(z k )

v∈ϕ

(·)

so for all large k there are elements vk ∈ ϕ −1 (z k ) such that f (vk ) ≥ c(2Mf ) , where

of the sequence {ι(vk )}. For M = max x∈X #ϕ −1 (x). Let c be point in D a condensation the corresponding subsequence vki we find that ψ(c ) = limi ϕ vki = limi z k i = c . Since c( f ) c ( f ) = lim f vki ≥ > 0, i 2M it follows that c ∈ V , proving that W ⊆ ψ(V ).

= D,

there is an element f ∈ D such that f = 0 and f ≥ 0, ψ is surjective: If ψ( D)

Since ψ(c)( f ) = c ( f ◦ ϕ) it follows that f ◦ ϕ = 0. while ψ(c)( f ) = 0 for all c ∈ D. This is impossible since f = 0 and ϕ is surjective.

The dynamical system ( D, ψ) will be called the canonical local homeomorphic

ψ) is the left Krieger cover of (X, ϕ) when extension of (X, ϕ). It can be shown that ( D, (X, ϕ) is a one-sided sofic shift. 5. Isomorphism of the C ∗-Algebras C r∗ ϕ and C r∗ ψ Since ψ is a local homeomorphism the C ∗ -algebras Cr∗ Rψ and Cr∗ ψ coincide with the one considered in [A]. In particular, the abelian C ∗ -algebra Dψ is equal to

= Dϕ . In this section we show that this identification, Dϕ = Dψ , is the C( D) restriction of an isomorphism between Cr∗ ϕ and Cr∗ ψ . As above we let N ∈ D be the function N (x) = #ϕ −1 (x), and set m = N ◦ ϕ.

410

K. Thomsen

Lemma 5.1. c(N ) = #ψ −1 (c) for all c ∈ D. Proof. For any f ∈ D, let I ( f ) denote the function I ( f )(x) = f (v). v∈ϕ −1 (x)

and let {z k } be a sequence in X such It follows from (3.2) that I ( f ) ∈ D. Let c ∈ D that limk ι(z k ) = c. Set z = π(c), and let be U, V1 , V2 , . . . , V j as in Lemma 3.6 of [Th], i.e. 1)-3) from the proof of Proposition 4.1 hold. Since limk N (z k ) = c(N ) we can assume that N (z k ) = c(N ) for all k, and since limk z k = limk π ◦ ι (z k ) = z in X we can assume that z k ∈ U for all k. Choose functions h i , Hi ∈ Cc (X ), i = 1, 2, . . . , j, such that 0 ≤ h i ≤ 1, h i (wi ) = 1, where wi = Vi ∩ ϕ −1 (z), ϕ (supp h i ) ⊆ U , Hi h i = h i and supp Hi ⊆ Vi for all i. Observe that c(N ) ≤ j and set gF = I (h i ) ∈ D i∈F

for every subset F ⊆ {1, 2, . . . , j} with c(N ) elements. For all sufficiently large k there is a subset F ⊆ {1, 2, . . . , j} with c(N ) elements such that g F (z k ) ≥ 21 . Indeed, since N (z k ) = c(N ) there is for each k a subset Fk ⊆ {1, 2, . . . , j} with c(N ) elements and elements vki ∈ Vi , i ∈ Fk , such that ϕ −1 (z k ) = vki : i ∈ Fk . When g Fk (z k ) < 21 there must be at least one i k ∈ Fk for which 1 c(N1 ) . h ik vkik < 2 Hence, if g Fk (z k )
0 . Wi = c ∈ D


411

To show that #ψ −1 (c) ≤ N (c) it suffices therefore to show that every element c of ψ −1 (c) is contained in Wi for some i ∈ F . To this end we pick a sequence {yk } in X

Set z = ϕ(yk ) and note that limk ι z = ψ(c ) = c such that limk ι (yk ) = c in D. k k while limk z k = limk π ◦ ψ ◦ ι (yk ) = limk π ◦ ψ c = z. In particular, N (z k ) = c(N )

(5.2)

z k ∈ U

(5.3)

and

for all sufficiently large k. Furthermore, by using (5.1) we find that lim g F (z k ) = c (g F ) = lim g F (z k ) ≥ k

k

1 . 2

(5.4)

By combining (5.2), (5.3) and (5.4) we find that 1 −1 −1 ϕ ,∞ zk ⊆ hi 4 i∈F

for all large k. Since yk ∈

z k , it follows that 1 −1 ,∞ hi yk ∈ 4

ϕ −1

i∈F

for all large k. Hence there is an i ∈ F such that yk ∈ h i−1 k which implies that c (h i ) = lim h i (yk ) ≥ k

Hence

c

∈ Wi .

1

4, ∞

for infinitely many

1 . 4

Corollary 5.2. #ψ −1 (ψ(c)) = c(m) for all c ∈ D. Proof. Using Lemma 5.1 for the first equality we find that #ψ −1 (ψ(c)) = ψ(c)(N ) = c(N ◦ ϕ) = c(m).

and all f ∈ D. Lemma 5.3. 1∗ψ (1,0) f 1ψ (1,0) (c) = c f (z) for all c ∈ D −1 z∈ϕ (·)

it suffices to establish Proof. Since both sides are continuous in c and ι(X ) is dense in D the identity when c = cx for some x ∈ X . It follows from Proposition 4.1 that we can apply (3.2) with ψ replacing ϕ to conclude that 1∗ψ (1,0) f 1ψ (1,0) (cx ) = c ( f ). c ∈ψ −1 (cx )

In comparison we have that ⎛ cx ⎝

z∈ϕ −1 (·)

⎞ f (z)⎠ =

z∈ϕ −1 (x)

f (z).

412

K. Thomsen

So it remains only to show that ψ −1 (cx ) = cz : z ∈ ϕ −1 (x) .

(5.5)

In fact, since the two sets have the same number of elements by Lemma 5.1, it suffices to check that ψ(cz ) = cx when z ∈ ϕ −1 (x). This is straightforward: ψ(cz )( f ) = cz ( f ◦ ϕ) = f (ϕ(z)) = f (x) = cx ( f ) for all f ∈ D.

Note that (5.5) means that ψ −1 (ι(X )) = ι(X ).

(5.6)

We can now adopt the proof of Theorem 3.1 to get the following: Theorem 5.4. There is a ∗-isomorphism Cr∗ ϕ → Cr∗ ψ which is the identity on Dϕ and takes the isometry Vϕ ∈ Cr∗ ϕ to Vψ ∈ Cr∗ ψ . Proof. We will appeal to Theorem 3.1 above and combine itwith Corollary 7.2 of [EV] for the existence of a ∗-homomorphism Cr∗ ϕ → Cr∗ ψ with the stated properties. We need therefore to check that 1) Vψ f = f ◦ ϕVψ,

and 2) c(Vψ∗ f Vψ ) = c N (·)−1 z∈ϕ −1 (·) f (z) , c ∈ D, k 1 1 3) 1 = i=1 (bi m) 2 Vψ Vψ∗ (bi m) 2 , where f ∈ D. To check 1) note first that cx , 1, c y : ϕ(x) = y

the openness of is dense in ψ (1, 0). This follows from the density of ι(X ) in D, ψ and (5.6). Since both sides of 1) are elements in Cc ψ (1, 0) it suffices therefore to check 1) on elements of the form (cx , 1, c y ) with ϕ(x) = y, where it is easy: 1 Vψ f cx , 1, c y = Vψ cx , 1, c y f c y = m(x)− 2 f (ϕ(x)) = f ◦ ϕVψ cx , 1, c y . The identity 2) is established in a similar way: Since both sides are continuous functions

it suffices to check it on elements from ι(X ): on D cx Vψ∗ f Vψ =

Vψ∗ cx , −1, c c ( f )Vψ c , 1, cx

c ∈ψ −1 (cx )

=

−1 #ψ −1 ψ(c ) c (f)

c ∈ψ −1 (cx )

=

N (x)−1 f (y)

y∈ϕ −1 (x)

⎛

= cx ⎝ N (·)−1

z∈ϕ −1 (·)

(by Corollary 5.2 and (5.6)) ⎞

f (z)⎠ .


413

k 1 1 To check 3) note that i=1 (bi m) 2 Vψ Vψ∗ (bi m) 2 ∈ Cc (R(ψ)). Since elements of the form cx , c y with (x, y) ∈ R(ϕ) are dense in R(ψ) it suffices to show that for (x, y) ∈ R(ϕ), k

1 1 (bi m) 2 Vψ Vψ∗ (bi m) 2 cx , c y =

i=1

0 1

when x = y when x = y.

So let (x, y) ∈ R(ϕ). Then ϕ(x) = ϕ(y) and we find that k

1 1 (bi m) 2 Vψ Vψ∗ (bi m) 2 cx , c y

i=1

=

k

1

1

1

1

1

1

bi (x) 2 m(x) 2 m(x)− 2 m(y)− 2 bi (y) 2 m(y) 2 (using Corollary 5.2)

i=1

=

0 1

when x = y when x = y

k since ϕ is injective on supp bi and i=1 bi = 1. This establishes the existence of a ∗-homomorphism μ : Cr∗ ϕ → Cr∗ ψ which is the identity on Dϕ and takes Vϕ to Vψ . The injectivity of μ follows from the faithfulness of the conditional expectation Pϕ : Cr∗ ϕ → Dϕ and the observation that Pψ ◦ μ = Pϕ . And, finally, the surjectivity of μ follows from the fact that Cr∗ ψ is generated by Vψ and Dψ = Dϕ .

By Theorem 5.4 we can identify Cr∗ ϕ with Cr∗ ψ and we will do that freely in the following. Remark 5.5. The isomorphism of Theorem 5.4 is clearly equivariant with respect to the gauge actions and it induces therefore an isomorphism between the correspond T T ing fixed point algebras, Cr∗ ϕ and Cr∗ ψ . Since ψ is a local homeomorphism T we have the equality Cr∗ ψ = Cr∗ Rψ . Since there are subshifts σ for which T Cr∗ (Rσ ) Cr∗ ( σ ) it follows that in general the isomorphism in Theorem 5.4 does not take Cr∗ Rϕ onto Cr∗ Rψ . 6. KMS States Let F : X → R be a real-valued function from D. Such a function defines a continuous action α F : R → Aut Cr∗ ϕ such that αtF (d) = d when d ∈ Dϕ and αtF Vϕ = ei Ft Vϕ , cf. [E]. The action α F can also be defined from the one-cocycle on ϕ defined by F as in the last line on p. 2072 in [KR], but the definition above allows us to combine Theorem 3.1 with the work of Exel in [E] to establish the connection between the KMS states

of α F and the Borel probablity measures on D fixed by the dual of a Ruelle-type operator. Let β ∈ R\{0}. A state ω on Cr∗ ψ is a KMS state with inverse temperature β for α F (or just a β-KMS state for short) when F (x) (6.1) ω(x y) = ω yαiβ for all α F -analytic elements x, y of Cr∗ ϕ .

414

K. Thomsen

Let τλ , λ ∈ T, bethe gauge action on Cr∗ ψ (so that τeit = αtF when F is constant 1) and let Pψ : Cr∗ ψ → D be the conditional expectation. Let S(D) denote the set of states on D. When χ ∈ S(D) the composition χ ◦ Pψ is a state on Cr∗ ψ . Note that χ ◦ Pψ is gauge-invariant since Pψ ◦ τλ = Pψ for all λ ∈ T. Let Q : Cr∗ ψ → Cr∗ Rψ be the conditional expectation ! τλ (x) dλ. Q(x) = T

Lemma 6.1. Let ω be a β-KMS state for α F . Then ω ◦ Q is a gauge-invariant β-KMS state for α F . Proof. Let x, y ∈ Cr∗ ϕ be analytic for α F . Since τ commutes with α F we find that ! ω ◦ Q (x y) = ω (τλ (x y)) dλ !T ! F = ω (τλ (x)τλ (y)) dλ = ω τλ (y)αiβ (τλ (x) ) dλ T !T F F = ω τλ (y)τλ (αiβ (x) dλ = ω ◦ Q yαiβ (x) . T

For any β ∈ R, define L −β F : D → D such that e−β F(y) g(y). L −β F (g)(x) = y∈ϕ −1 (x)

Theorem 6.2. Let β ∈ R\{0}. The map χ → χ ◦ Pψ is a bijection from the states χ ∈ S(D) which satisfy that χ ◦ L −β F = χ

(6.2)

onto the gauge-invariant β-KMS states for α F . Proof. Consider first the case β > 0. By Proposition 9.2 and Sect. 11 in [E] it suffices to show that any gauge-invariant β-KMS state ω of α F factorizes through Pψ , and this follows from Lemma 2.24 gauge-invariant we have of [Th] in the following way. Since ω Fis " " that ω = ω ◦ Q. Let d j be a partition of unity in D. Since αiβ d j = d j it follows " " from the KMS condition (6.2) that j ω d j x d j = ω(x) for all x ∈ Cr∗ ϕ . In " " d j Q(x) d j and hence ω Pψ (Q(x)) = ω(Q(x)) particular, ω(Q(x)) = jω by Lemma 2.24 of [Th] because Q(x) ∈ Cr∗ Rψ . Since Pψ ◦ Q = Pψ this shows that ω = ω ◦ Pψ as desired. The case β < 0 follows from the preceding case by observing that ω is a β-KMS state for α F if and only if ω is a (−β)-KMS state for α −F .

It follows from [E] that every β-KMS state is gauge invariant when F is strictly positive or strictly negative. This is not the case in general, but note that if there is a β-KMS state for α F then there is also one which is gauge invariant by Lemma 6.1. We have deliberately omitted β = 0 as an admissable β-value for KMS-states because they correspond to trace states and they exist only in rather exceptional cases, e.g. when ϕ has a fixed point x0 for which ϕ −1 (x0 ) = {x0 }.


415

6.1. Bounds on the possible β-values. Define Iβ F : D → D such that Iβ F (g)(x) =

eβ F(x) g ◦ ϕ(x). m(x)

Then L −β F ◦ Iβ F (g) = g for all g ∈ D, so if χ ∈ S(D) satisfies (6.2) we find that χ = χ ◦ Iβ F . (6.3) Thus 1 ∈ Spectrum L ∗−β F ∩ Spectrum Iβ∗F when there is a state χ ∈ S(D) for which (6.2) holds. Let ρ(T ) be the spectral radius of an operator T . Since Spectrum L ∗−β F ∩ Spectrum Iβ∗F = Spectrum L −β F ∩ Spectrum Iβ F , cf. [DS], we find that 1 ≤ ρ Iβ F

(6.4)

1 ≤ ρ L −β F

(6.5)

and

when (6.2) holds. To get the most out of these inequalities we consider a non-invertible invariant h m which has been introduced for general dynamical systems by M. Hurley in [Hu] and developed further in [FFN]. For a locally injective map like the map ϕ we consider here, the invariant h m (ϕ) is simply given by the formula 1 h m (ϕ) = lim log max #ϕ −n (x) , (6.6) n→∞ n x∈X cf. [FFN], or, alternatively, as h m (ϕ) = sup lim sup x∈X

n

1 log #ϕ −n (x), n

cf. Corollary 2.4 of [FFN]. For forward expansive maps, and hence in particular for one-sided subshifts, h m equals the topological entropy h, but in general we only have the inequality h m (ϕ) ≤ h(ϕ). It can easily happen that h m (ϕ) < h(ϕ) even when ϕ is a local homeomorphism. The next lemma shows that for a locally injective surjection, as the map ϕ we consider, the invariant h m agrees with that of its canonical local homeomorphic extension. Lemma 6.3. h m (ψ) = h m (ϕ). −k −k Proof. It follows from (5.6) that #ψ (ι(x)) = #ϕ (x) for all x ∈ X . Since −k

#ψ (c) = c ∈ψ −k (c) 1 depends continuously on c ∈ D and ι(X ) is dense in D, −k −k we conclude that maxc∈ D

#ψ (c) = max x∈X #ϕ (x). Hence h m (ψ) = h m (ϕ), cf. (6.6).

416

K. Thomsen

In the following we let M(X ) denote the set of Borel probability measures on X and Mϕ (X ) the subset of M(X ) consisting of the ϕ-invariant elements of M(X ). Similarly,

be the set of Borel probability measures on D

and Mψ ( D)

the set of we let M( D)

ψ-invariant elements in M( D). Lemma 6.4. Let β ∈ R and assume that there is a state χ ∈ S(D) such that (6.2) holds.

such that It follows that there are measures ν, ν ∈ Mψ ( D) ! F dν ≤ h m (ϕ) (6.7) β

D

and

!

D

log #ψ −1 (c) dν (c) ≤ β

!

D

F dν .

(6.8)

Proof. Let δ > 0. It follows from (6.5) that ρ L −β F ≥ 1 which implies that ⎛ ⎞ # # k−1 j 1 1 −β # # j=0 F ψ (c ) ⎠ −δ ≤ log #L k−β F (1)# = log ⎝ sup e ∞ k k

c∈ D −k c ∈ψ

(c)

such that for all large k. There is therefore, for each large k, a point ck ∈ D $ % j k−1 1 −β F ψ (c ) −k k −2δ ≤ log e j=0 sup #ψ (c) . k

c∈ D Let ν be a weak* condensation point of the sequence 1 δψ j (ck ) k k−1

j=0

Then ν ∈ Mψ ( D)

by Theorem 6.9 of [W1] and in M( D). 1 −β F ψ j (ck ) ≤ k k−1

!

D

j=0

−β F dν + δ

for infinitely many k. It follows that −2δ ≤

1 log sup #ψ −k (c) + k

c∈ D

!

D

−β F dν + δ

for infinitely many k, and we conclude therefore that 0 ≤ h m (ψ) + h m (ψ) = h m (ϕ) by Lemma 6.3 we get (6.7). Similarly it follows from (6.4) that ⎛

β

k−1

e 1 ≤ lim sup ⎝ 'k−1 k→∞ c∈ D

j=0

j=0 m

F ψ j (c)

ψ j (c)

⎞1 k

⎠ ,

&

−β Fdν. D

Since


417

which implies that % $ j j k−1 1 β F ψ (c) −log m ψ (c) −δ ≤ log sup e j=0 k

c∈ D for all large k. We can then work as before with −β & F replaced by β F − log m to pro such that −2δ ≤ β F − log m dν + δ. We omit the duce the measure ν ∈ Mψ ( D) & D & −1 repetition. Since ν is ψ-invariant we have that D

log m dν = D

log #ψ (c) dν (c). In this way we get (6.8).

ϕ

When H : X → R is a bounded real-valued function, set A H (k) = inf x∈X j ϕ ϕ ϕ ϕ (x) . Then A H (k + n) ≥ A H (k) + A H (n) for all n, k and we can set ϕ

ϕ

ϕ

ϕ

k−1 j=0

H

A (k) A (n) = sup H . = lim H k→∞ k n n j ϕ Similarly, we set B H (k) = supx∈X k−1 j=0 H ϕ (x) and ϕ AH

ϕ

B H = lim

k→∞

B H (k) B (n) = inf H . n k n

Proposition 6.5. When β > 0 is the inverse temperature of a KMS state for α F we have ϕ ϕ ϕ that Alog m ≤ β B F and β A F ≤ h m (ϕ). ϕ When β < 0 is the inverse temperature of a KMS state for α F we have that Alog m ≤ ϕ ϕ β A F and β B F ≤ h m (ϕ). Proof. Let ν and ν be the measures from Theorem 6.4. When β > 0 we find that !

1 h m (ϕ) ≥ β F dν = β n

D

! n−1

D k=0

ϕ

F ◦ ψ k dν ≥ β

A F (n) n

and !

ϕ

Alog m (n) n

≤

D

! n−1 n−1 ϕ B (n) 1 1 k log m ◦ ψ dν ≤ β F ◦ ψ k dν ≤ β F n n D n

k=0

k=0

for all n. The two first inequalities of Theorem 6.5 follow from this. The case β < 0 is handled similarly.

ϕ

Corollary 6.6. Assume that h m (ϕ) = 0. There are no KMS states for α F unless A F ≤ ϕ 0 ≤ BF . Lemma 6.7. Assume that there is a β-KMS state for α F . It follows that there is a measure

such that ν ∈ Mψ ( D) ! 1 F dν ≥ lim sup log inf #ψ −n (c). (6.9) β

n

n c∈ D D

418

K. Thomsen

Proof. Let χ ∈ S(D) be a state such that χ ◦ L −β F = χ . Then ⎛ ⎞ k−1 j χ⎝ e−β j=0 F◦ψ (c ) ⎠ = 1

(6.10)

c ∈ψ −k (·)

for all k and hence

⎛

inf #ψ −k (c)χ ⎝

c∈ D

1 #ψ −k (·)

e−β

k−1 j=0

⎞ F◦ψ j (c )

⎠≤1

(6.11)

c ∈ψ −k (·)

for all k ∈ N. Since log is concave we can apply Jensen’s inequality to the state μ on D defined by ⎛ ⎞ 1 μ(g) = χ ⎝ −k g(c )⎠ . #ψ (·) −k c ∈ψ

Then (6.11) gives the estimate

⎛

log inf #ψ −k (c) − βμ ⎝

c∈ D

k−1

(·)

⎞ F ◦ ψ j⎠ ≤ 0

(6.12)

j=0

of the sequence for all k. We can therefore choose a condensation point ν ∈ Mψ ( D) μk , k = 1, 2, . . ., where ⎛ ⎞ k−1 1 μk (g) = μ ⎝ g ◦ ψ j⎠ , k j=0

such that (6.9) holds.

Theorem 6.8. Assume that F is continuous and that there is a β-KMS state for α F . Set 1 −n m = lim log min #ϕ (x) n→∞ n x∈X and

1 log max #ϕ −n (x) . n→∞ n x∈X

M = lim

There is then a ϕ-invariant Borel probability measure μ ∈ Mϕ (X ) such that ! β F dμ ∈ [m, M]. X

such that Proof. By Proposition 6.5 and& Lemma 6.7 there are measures ν, ν ∈ Mψ ( D) & β D

F dν ≤ M and m ≤ β D

F dν . Since F is continuous on X by assumption we

It follows that with an appropriate convex have that F(c) = F(π(c)) for all c ∈ D. combination we have that m ≤ β

μ = sν ◦ π −1 + (1 − s)ν ◦ π −1

& X

F dμ ≤ M.


419

6.2. Existence of KMS states. While Proposition 6.5 and Theorem 6.8 give upper and lower bounds on the possible β-values of a KMS state for α F they say nothing about existence. This is where the work of Matsumoto, Watatani and Yoshida, [MWY], and Pinzari, Watatani and Yonetani, [PWY], comes in. Theorem 6.9 (cf. [PWY] and [MWY]). Let B be a unital commutative C ∗ -algebra and L : B → B a positive linear operator with spectral radius ρ(L). Then ρ(L) is in the spectrum of L and there is a state ω ∈ S(B) such that ω ◦ L = ρ(L)ω. Proof. We adopt arguments from [PWY] to show that ρ(L) is in the spectrum of L and then arguments from [MWY] to produce the state ω. Recall that Spectrum(L) = Spectrum(L ∗ ), cf. [DS]. By definition of ρ(L) there is an element z ∈ Spectrum (L ∗ ) with |z| = ρ(L). Let {z n } be a sequence of complex numbers such that |z n | > ρ(L) for all n and limn z n = z. It follows then from the principle of uniform boundedness that there is an element μ ∈ B ∗ such that lim R(z n )μ = ∞,

n→∞

where R(z) = (z − L ∗ )−1 is the resolvent. Since B ∗ is spanned by the states we may assume that μ ∈ S(B). Since |z n | > ρ (L ∗ ) the resolvent R(z n ) is given by the norm convergent Neumann series R(z n ) =

∞

z n−k−1 L ∗ k .

k=0

Since μ is a state and L a positive operator it follows that |R(z n )μ| ≤

∞

|z n |−k−1 L ∗ k μ = R (|z n |) μ

k=0

in B ∗ where |R(z n )μ| is the total variation measure of R(z n )μ. Hence R(z n )μ ≤ R (|z n |) μ , and we conclude that limn→∞ R (|z n |) μ = ∞, which implies that ρ(L) = limn→∞ |z n | is in Spectrum (L ∗ ) = Spectrum(L). Set μn =

R (|z n |) μ . R (|z n |) μ

A glance at the Neumann series shows that μn is a state since L is positive. As ρ(L) − L ∗ μn = (ρ(L) − |z n |) μn + R (|z n |) μ−1 μ converges to 0 in norm, any weak* condensation point ω of {μn } will be a state such that ω ◦ L = ρ(L)ω.

Corollary 6.10. Let β ∈ R\{0} satisfy that the spectral radius ρ L −β F of L −β F is 1. It follows that there is a gauge invariant β-KMS state for α F . Proof. Combine Theorem 6.9 with Theorem 6.2.

420

K. Thomsen

be a closed subset such that ψ −1 (A) ⊆ A. Assume that Lemma 6.11. Let A ⊆ D ψ

A F| A > 0. It follows that there are states ω, ν, ν ∈ S(D) and a β ∈ [0, ∞) such that ν ◦ ψ = ν, ν ◦ ψ = ν , ω(A) = ν(A) = ν (A) = 1, βν(F) ≤ limn→∞ n1 log maxc∈A #ψ −k (c) ≤ βν (F), and ω ◦ L −β F = ω. k ψ −1 Proof. Set δ = A F| A = limn inf c∈A n1 n−1 k=0 F ψ (c) . Since ψ (A) ⊆ A we can 1) 2) 3) 4)

A for any t ∈ R define a positive linear operator L −t F : C(A) → C(A) such that A e−t F(c ) g(c ). L −t F (g)(c) = c ∈ψ −1 (c)

Then A L −t F ◦ r A = r A ◦ L −t F ,

(6.13)

A where r A : D → C(A) is the restriction map. To estimate the spectral radius of L −t F we observe that when t ≥ 0 we get the estimate n n−1 k A (1)(c) = sup e−t k=0 F ψ (c ) sup L −t F c∈A

c∈A

c ∈ψ −n (c)

≤ sup c∈A

tδ

tδ

e−n 2 ≤ e−n 2 sup #ψ −n (c)

c ∈ψ −n (c)

c∈A

for infinitely many n. It follows that 1 n n A A sup lim ρ L −t = lim L lim (1)(c) = 0. F −t F

t→∞

t→∞ n→∞

c∈A

On the other hand ρ Since

L 0A

1 n −n = lim sup #ψ (c) ≥ 1. n→∞

c∈A

( ( ( ( ( ( ( A A ( (ρ L −t F − ρ L −t F ( ≤ t − t F∞

A for all t, t ∈ R, cf. Proposition 2.2 of [ABL], it follows that [0, ∞) t → ρ L −t F is continuous. Hence the intermediate value theorem of calculus implies the existence of a A β ∈ [0, ∞) such that ρ L −β F = 1. Then Theorem 6.9 implies the existence of a state A ω ∈ S (C(A)) such that ω ◦ L −β F = ω . Set ω = ω ◦ r A and note that (6.13) implies that ω ◦ L −β F = ω. Since ω( f ) = 0 for all f ∈ D with support in X \A it follows that ω(A) = 1.


421

To construct the ψ-invariant states ν and ν let > 0 and note that ⎛ ⎞ n−1 k 1 −β F ψ (c ) k=0 ⎠ = 0. lim log ⎝sup e n→∞ n c∈A −n c ∈ψ

(6.14)

(c)

For n ∈ N there are cn , cn ∈ ψ −n (A) such that n−1

F ψ k (cn ) =

k=0

≤

n−1

inf

c ∈ψ −n (A)

sup

k=0 n−1

c ∈ψ −n (A) k=0

F ψ k (c ) n−1 F ψ k (c ) = F ψ k (cn ) . k=0

Then −β

n−1 1 k 1 F ψ (cn ) + log sup #ψ −n (c) ≤ 0 n n c∈A k=0

≤ −β

n−1 1 1 k F ψ (cn ) + log sup #ψ −n (c) n n c∈A

(6.15)

k=0

asymptotically as n goes to infinity. Let ν and ν be states of D such that the correspond

are weak* condensation points of the sequences 1 n−1 δψ k (c ) and ing measures on D k=0 n n n−1 1 , = 1, 2, 3, . . . , respectively. Then 1) holds by Theorem 6.9 of [W1] δ k k=0 ψ (cn ) n and ν(A) = ν (A) = 1 since A is closed and ψ k (cn ) , ψ k cn ∈ A for all k, n. The estimates 3) follow from (6.15).

Theorem 6.12. Assume that h m (ϕ) > 0. ϕ

ϕ

ϕ

1) If A F > 0 there is a β-KMS state for α F such that β A F ≤ h m (ϕ) ≤ β B F . ϕ ϕ ϕ 2) If B F < 0 there is a β-KMS state for α F such that β B F ≤ h m (ϕ) ≤ β A F . 3) When F is continuous there is in both cases, 1) or 2), a ϕ-invariant Borel probability measure μ ∈ Mϕ (X ) such that ! F dμ = h m (ϕ).

β

(6.16)

X

and 2) follows by Proof. 1) follows directly from Lemma 6.11 applied with A = D applying 1) to −F. ϕ ϕ 3) Assume now that F is continuous. Since we either have that β A F ≤ h m (ϕ) ≤ β B F ϕ ϕ or β B F ≤ h m (ϕ) ≤ β A F there is a sequence n 1 < n 2 < · · · in N and points xi , yi ∈ X such that h m (ϕ) −

n i −1 1 1 ≤ β F ◦ ϕ j (xi ) i ni j=0

422

K. Thomsen

and n i −1 1 1 β F ◦ ϕ j (yi ) ≤ h m (ϕ) + ni i j=0

for all i. For each i we can then find a number si ∈ [0, 1] such that n i −1 ! 1 1 1 F ◦ ϕ j dνi ≤ h m (ϕ) + , h m (ϕ) − ≤ β i ni i X

(6.17)

j=0

where νi = si δxi + (1 − si )δ yi . Any weak* condensation point of the sequence n i −1 1 νi ◦ ϕ − j ni j=0

will be ϕ-invariant by Theorem 6.9 of [W1] and β

& X

F dμ = h m (ϕ) thanks to (6.17).

Corollary 6.13. Assume that F is continuous and either strictly positive or strictly negative. There is no KMS-state for α F if h m (ϕ) = 0. If h m (ϕ) > 0 there is a β-KMS-state for α F such that β=&

h m (ϕ) X F dμ

for some μ ∈ Mϕ (X ). Proof. The first assertion follows from Corollary 5.2 and the second from Theorem 6.12.

Example 6.14. Assume that ϕ : X → X is uniformly n-to-1, i.e. that #ϕ −1 (x) = n for all x ∈ X . Note that n ≥ 2 since we assume that ϕ is not injective. Then h m (ϕ) = log n and it follows from Theorem6.12 and Theorem 6.8 that there is exactly one β such that the gauge action on Cr∗ ϕ has a β-KMS state, namely β = log n. In many cases log n is also the topological entropy, h(ϕ). This is for example the case when ϕ is an affine map on Tk . To see that in general log n is smaller than the topological entropy, let f : Y → Y be an arbitrary homeomorphism of a compact metric space Y . Then ϕ × f : X × Y → X × Y is also locally injective and n-to-1. In particular h m (ϕ × f ) = log n, while the topological entropy is h(ϕ) + h( f ) which can be any number ≥ log n.

References [A] [ABL] [BKR]

Anantharaman-Delaroche, C.: Purely infinite C ∗ -algebras arising from dynamical systems. Bull. Soc. Math. France 125, 199–225 (1997) Antonevich, A.B., Bakhtin, V.I., Lebedev, A.V.: T-entropy and variational principle for the spectral radius of transfer and weighted shift operators. http://arXiv.org/abs/0809.3116v2 [math.DS], 2008 Boyd, S., Keswari, N., Raeburn, I.: Faithful Representations of Cross ed Products by Endomorphisms. Proc. Amer. Math. Soc. 118, 427–436 (1993)


[BR]

423

Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics II. New York-Heidelberg-Berlin: Springer Verlag, 1981 [D] Deaconu, V.: Groupoids associated with endomorphisms. Trans. Amer. Math. Soc. 347. 1779–1786 (1995) [DS] Dunford N., Schwartz, J.T.: Linear Operators, Part I : General Theory. New York: Interscience Publishers, 1966 [E] Exel, R.: Crossed products by finite index endomorphisms and KMS states. J. Func. Anal. 199, 153–158 (2003) [EV] Exel, R., Vershik, A.: C ∗ -algebras of Irreversible Dynamical Systems. Canad. J. Math. 58, 39–63, (2006) [FFN] Fiebig, D., Fiebig, U., Nitecki, Z.: Entropy and preimage sets. Erg. Th. & Dyn. Sys. 23, 1785–1806 (2003) [Hu] Hurley, M.: On topological entropy of maps. Erg. Th. Dyn. Sys. 15, 557–568. (1995) [Kr1] Krieger, W.: On sofic systems I. Israel J. Math. 48, 305–330 (1984) [Kr2] Krieger, W.: On sofic systems II. Israel J. Math. 60, 167–176 (1987) [KR] Kumjian, A., Renault, J.: KMS-states on C ∗ -algebras associated to expansive maps. Proc. Amer. Math. Soc. 134, 2067–2078 (2006) [MWY] Matsumoto, K., Watatani, Y., Yoshida, M.: KMS states for gauge actions on C ∗ -algebras associated with subshifts. Math. Z. 228, 489–509 (1998) [PWY] Pinzari, C., Watatani, Y., Yonetani, K.: KMS states, entropy and the variational principle in full C ∗ -dynamical systems. Commun. Math. Phys. 213, 331–379 (2000) [Re1] Renault, J.: A Groupoid Approach to C ∗ -algebras. LNM 793, Berlin-Heidelberg-New York: Springer Verlag, 1980 [Re2] Renault, J.: A F-equivalence relations and their co-cycles. Operator Algebras and Mathemaical Physics, Conference Proceedings, Constanza 2001, Bucharest: The Theta Foundation, 2003, pp. 365–377 [Ru] Ruelle, D.: Thermodynamic Formalism. Encyclopedia of Mathematics and its Applications 5, Reading, MA: Addison-Wesley, 1978 [Th] Thomsen, K.: Semi-étale groupoids and applications. Ann. l’Inst. Fourier 60(3), 759–800 (2010) [W1] Walters, P.: An Introduction to Ergodic Theory. New York-Heidelberg-Berlin: Springer Verlag, 1982 [W2] Walters, P.: Convergence of the Ruelle operator for a function satisfying Bowen’s condition. Trans. Amer. Math. Soc. 353, 327–347 (2000) Communicated by A. Connes


Communications in


Meixner Class of Non-commutative Generalized Stochastic Processes with Freely Independent Values II. The Generating Function Marek Bo˙zejko1 , Eugene Lytvynov2 1 Instytut Matematyczny, Uniwersytet Wrocławski, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland.


2 Department of Mathematics, Swansea University, Singleton Park, Swansea SA2 8PP, U.K.

E-mail: [email protected] Received: 15 March 2010 / Accepted: 6 June 2010 Published online: 15 September 2010 – © Springer-Verlag 2010

Abstract: Let T be an underlying space with a non-atomic measure σ on it. In [Comm. Math. Phys. 292, 99–129 (2009)] the Meixner class of non-commutative generalized stochastic processes with freely independent values, ω = (ω(t))t∈T , was characterized through the continuity of the corresponding orthogonal polynomials. In this paper, we derive a generating function for these orthogonal polynomials. The first question we have to answer is: What should serve as a generating function for a system of polynomials of infinitely many non-commuting variables? We construct a class of operator-valued functions Z = (Z (t))t∈T such that Z (t) commutes with ω(s) s, t ∈ T . Then a generating function can be understood as any(n) for G(Z , ω) = ∞ n=0 T n P (ω(t1 ), . . . , ω(tn ))Z (t1 ) · · · Z (tn ) σ (dt1 ) · · · σ (dtn ), where P (n) (ω(t1 ), . . . , ω(tn )) is (the kernel of the) n th orthogonal polynomial. We derive an explicit form of G(Z , ω), which has a resolvent form and resembles the generating function in the classical case, albeit it involves integrals of non-commuting operators. We finally discuss a related problem of the action of the annihilation operators ∂t , t ∈ T . In contrast to the classical case, we prove that the operators ∂t related to the free Gaussian and Poisson processes have a property of globality. This result is genuinely infinitedimensional, since in one dimension one loses the notion of globality.

1. Introduction and Preliminaries This paper serves as a continuation of our research started in [10]. We recall that the Meixner class of non-commutative generalized stochastic processes with freely independent values was characterized in [10] through the continuity of the corresponding orthogonal polynomials. The main aim of the present paper is to derive the generating function for these orthogonal polynomials. Let us first briefly recall some known results on the generating function of Meixner polynomials, in both the classical and free cases. Below, when speaking of orthogonal

426

M. Bo˙zejko, E. Lytvynov

polynomials on the real line, we will always assume that their measure of orthogonality has infinite support and is centered. According to e.g. [11] (see also the original paper [16]), the Meixner class of orthogonal polynomials on R consists of all monic orthogonal polynomials (P (n) )∞ n=0 whose (exponential) generating function has the form G(z, x) :=

∞ ∞ P (n) (x) n 1 z = exp(x(z) + (z)) = (x(z) + (z))k , n! k! n=0

k=0

where z is from a neighborhood of zero in C, and are analytic functions in a neighborhood of zero such that (0) = (0) = (0) = 0. This assumption automatically implies that (z) = −C((z)),

(1.1)

where C(z) is the cumulant generating function of the measure of orthogonality, μ: C(z) =

∞ n z C (n) , n! n=1

C (n) being the n th cumulant of μ. Recall that zs C(z) = log e μ(ds) . R

Each system of Meixner polynomials is characterized by three parameters k > 0, λ ∈ R, and η ≥ 0. The corresponding orthogonal polynomials satisfy the recursion relation x P (n) (x) = P (n+1) (x) + λn P (n) (x) + (kn + ηn(n − 1))P (n−1) (x), and the generating function takes the form G(z, x) = exp xλ,η (z) − kCλ,η (λ,η (z)) ,

(1.2)

(1.3)

where the functions λ,η (z) and Cλ,η (z) are determined by the parameters λ and η only. In particular, Cλ,η (z) is the cumulant generating function of the measure of orthogonality corresponding to the parameters k = 1, λ and η. We refer the reader to e.g. [16] for an explicit form of λ,η (z) and Cλ,η (z). We also note that these functions continuously depend on their parameters λ and η, see [18] for details. Let us now outline the infinite dimensional case, see [13,14,18] for further details. Let T be a complete, connected, oriented C ∞ Riemannian manifold and let B(T ) be the Borel σ -algebra on T . Let σ be a Radon, non-atomic, non-degenerate measure on (T, B(T )). (For simplicity, the reader may think of T as Rd and of σ as the Lebesgue measure). Let D denote the space of all real-valued infinitely differentiable functions on T with compact support. We endow D with the standard nuclear space topology. Let D denote the dual space of D with respect to the center space L 2 (T, σ ). Thus, D consists of generalized functions (distributions) on T . Let C denote the cylinder σ -algebra on D , i.e., the minimal σ -algebra on D with respect to which, for any ξ ∈ D, the mapping D ω → ω, ξ ∈ R is Borel-measurable. Here and below, ·, · denotes the pairing between elements of a given linear topological space and its dual space.

Meixner Class of Non-commutative Generalized Stochastic Processes II

427

Let μ be a probability measure on (D , C ) (a generalized stochastic process). The cumulant generating function of μ is given by C(ξ ) = log

D

e ω,ξ μ(dω) , ξ ∈ D.

The Meixner class of generalized stochastic processes with independent values may be identified as follows. We fix arbitrary smooth functions λ : T → R and η : T → [0, ∞), and define a probability measure μ on (D , C) whose cumulant generating function is Cλ(t),η(t) (ξ(t)) σ (dt), ξ ∈ D. C(ξ ) = T

Here, Cλ(t),η(t) (·) is as in (1.3). Consider the set of all continuous polynomials on D , i.e., functions on D which have the form F(ω) =

n

ω⊗i , f (i) , n ∈ N0 .

i=0

Here, for each i, f (i) belongs to the i th symmetric tensor power of D, i.e., f (i) ∈ D i , where denotes symmetric tensor product. Note that D i consists of all smooth symmetric functions on T i with compact support. For each f (n) ∈ D n , we denote by P( f (n) ) = P( f (n) , ω) the orthogonal projection of the monomial ω⊗n , f (n) onto the n th chaos, i.e., onto the orthogonal difference in L 2 (D , μ) of the closures of the polynomials of order ≤ n and of order ≤ n − 1, respectively. Then P( f (n) ) is a continuous polynomial. By construction, for any f (n) ∈ D n and g (m) ∈ D m with n = m, the polynomials P( f (n) ) and P(g (m) ) are orthogonal. Furthermore, for each ω ∈ D , one can recursively define P (n) (ω) ∈ D n , n ∈ N, so that P( f (n) , ω) = P (n) (ω), f (n) . The (exponential) generating function of these polynomials is defined by G(ξ, ω) :=

∞ 1 P (n) (ω), ξ ⊗n , n! n=0

where ξ is from a neighborhood of zero in D. We have:

G(ξ, ω) = exp ω(·), λ(·),η(·) (ξ(·)) −

T

Cλ(t),η(t) (λ(t),η(t) (ξ(t))) σ (dt) , (1.4)

compare with (1.3). Note that the measure σ now plays the role of the parameter k in (1.3). Below, in the free case, we will use, for many objects, the same notations as those used for their counterpart in the classical case. However, it should always be clear from the context which objects are meant.

428


Introduced by Anshelevich [1] and Saitoh, Yoshida [19], the free Meixner class of orthogonal polynomials on R consists of all monic orthogonal polynomials (P (n) )∞ n=0 on R whose (usual) generating function has the form G(z, x) :=

∞

P (n) (x)z n = (1 − x(z) − (z)))−1 =

n=0

∞ (x(z) + (z))k , k=0

where z is from a neighborhood of zero and and satisfy the same conditions as in the classical case. Then the function (z) automatically takes the form as in (1.1), but with C(·) being the free cumulant generating function of the measure of orthogonality, μ: C(z) :=

∞

z n C (n) ,

n=1

where C (n) is the n th free cumulant of μ, see [1,3]. A system of such polynomials is also characterized by three parameters k > 0, λ ∈ R, η ≥ 0 and the polynomials satisfy the recursion relation as in (1.2) but with the factors n and n − 1 being replaced by [n]0 and [n − 1]0 , respectively. Here, for q ∈ R and n = 0, 1, 2 . . ., we denote [n]q := (1 − q n )/(1 − q) and so [n]0 = 0 for n = 0 and = 1 for all n = 1, 2, . . . . Thus, P (0) (x) = 1, P (1) (x) = x, x P (1) (x) = P (2) (x) + λP (1) (x) + k P (0) (x), x P (n) (x) = P (n+1) (x) + λP (n) (x) + (k + η)P (n−1) (x), n ≥ 2. Furthermore, the generating function G(z, x) takes the form as in (1.3) but with the the resolvent function replacing the exponential function. In fact, we have [1] λ,η (z) =

z 2z 2 , C (z) = , λ,η 1 + λz + ηz 2 1 − λz + (1 − λz)2 − 4z 2 η Cλ,η (λ,η (z)) =

z2 , 1 + λz + ηz 2

(1.5) (1.6)

so that G(z, x) = 1 − x

z z2 +k 2 1 + λz + ηz 1 + λz + ηz 2

−1 .

(1.7)

We also note that the class of orthogonal polynomials which is now called the free Meixner class, was derived in the conditionally free central limit theorem and in the conditionally free Poisson limit theorem in [9], see also [6] for a characterization of these polynomials in terms of a regression problem. In [3] (see also [2]), Anshelevich introduced and studied multivariate orthogonal polynomials of non-commuting variables with a resolvent-type generating function. He, in particular, noticed that the generating function G(z, x) should be defined for noncommuting indeterminates (z 1 , . . . , z k ) = z (which form coefficients by the orthogonal


429

polynomials) and non-commuting indeterminates (x1 , . . . , xk ) = x (which are variables of the polynomials), and the z i -variables must commute with the x j -variables for all i, j = 1, . . . , k. The generating function is then supposed to have the form −1 k G(z, x) = 1 − xi i (z) − (z) . (1.8) i=1

We refer to [2,3] for an extension of formula (1.1) to the multivariate case, for a recursion relation satisfied by the corresponding orthogonal polynomials, for an operator model of these polynomials, and for further related results. In part 1 of this paper, [10], we identified the Meixner class of non-commutative generalized stochastic processes ω = (ω(t))t∈T as those a) which have free independent values; b) whose orthogonal polynomials are continuous in ω. The main aim of the present paper is to derive the generating function for a system of orthogonal polynomials as in b). However, when discussing a generating function for a system of polynomials of infinitely many non-commuting variables, the first question we have to answer is: What should serve as a generating function? Developing the idea of [3], we will proceed in this paper as follows. Think informally of each polynomial of ω as P (n) (ω), f (n) , where P (n) (ω) is an operator-valued distribution on T n and f (n) is a test function on T n . We will consider a class of test operator-valued functions on T , denoted by Z(T ). We assume that, for each Z ∈ Z(T ) and t ∈ T , the operator Z (t) commutes with each polynomial P (n) (ω), f (n) . (However, for s, t ∈ T , Z (s) and Z (t) do not need to commute.) In Sect. 2, we give a rigorous meaning to a ‘dual pairing’ P (n) (ω), Z n and define a generating function G(Z , ω) =

∞ P (n) (ω), Z n ,

Z ∈ Z(T ).

n=0

Here Z n (t1 , . . . , tn ) := Z (t1 ) · · · Z (tn ) for (t1 , . . . , tn ) ∈ T n . We also show that the generating function G(Z , ω) uniquely characterizes the corresponding system of polynomials. In Sect. 3, we prove that the generating function of the Meixner system is given by −1 G(Z , ω) = 1 − ω(·), λ(·),η(·) (Z (·)) + Cλ(t),η(t) (λ(t),η(t) (Z (t))) σ (dt) = 1 − ω,

Z + 1 + λZ + ηZ 2

T

T

Z (t)2 σ (dt) 1 + λ(t)Z (t) + η(t)Z (t)2

−1 , (1.9)

where ω is Meixner’s non-commutative generalized stochastic processes with freely independent values corresponding to functions λ and η. The reader is advised to compare formula (1.9) with the generating function in the classical infinite dimensional case, formula (1.4), and with the generating function in the finite-dimensional free case, formulas (1.7) and (1.8). In Sect. 4, we discuss a related problem of the action of the annihilation operator at point t ∈ T , denoted by ∂t in [10]. Recall that, in the classical infinite-dimensional

430


case, the annihilation operator ∂t can be represented as an analytic function of the Hida– −1 (Dt ). (Recall that Dt is the derivMalliavin derivative Dt , more precisely ∂t = λ(t),η(t) ative in the direction of the delta-function δt .) We discuss a free counterpart of this result in the free Gauss–Poisson case, i.e., when η ≡ 0. A striking difference from the classical case is that we represent ∂t not just as a function of the free derivative Dt in the direction δt (this being impossible), but rather as a function of an operator Dt G. More precisely, −1 (Dt G). Here G is a ‘global’ operator, which is independent we show that ∂t = λ(t),0 of t. In fact, G is a sum of certain integrals of Ds over the whole space T . It should be stressed that this result is genuinely infinite-dimensional, since in one dimension we lose the notion of ‘globality’. We expect that a similar result should also hold in the general case, not necessarily when η ≡ 0, and we hope to return to this problem in our future research. We finish the paper with a discussion of a free differential equation satisfied by the cumulant generating function for a free Meixner class. Such an equation in the multivariate case was first derived by Anshelevich [2]. We show how this equation may be properly interpreted in our infinite dimensional setting. 2. Generating Function: Construction and Uniqueness of Corresponding Polynomials Just as in [10], we will assume that T is a locally compact Polish space. We denote by B(T ) the Borel σ -algebra on T , and by B0 (T ) the collection of all relatively compact sets from B(T ). For any fixed A ∈ B0 (T ), we will denote by B(A) the trace σ -algebra of B(T ) on A, i.e., {B ∈ B(T ) | B ⊂ A}. 2.1. Construction of the integral of an operator-valued function with respect to an operator-valued measure. Let G be a real separable Hilbert space, and let L (G) denote the Banach space of all bounded linear operators in G. We will call a mapping Z : T → L (G) simple if it has a form Z (t) =

n

Z i χ i (t),

(2.1)

i=1

where Z 1 , . . . , Z n ∈ L (G), 1 , . . . , n ∈ B0 (T ), n ∈ N, and χ i (t) denotes the indicator function of the set i . We denote by Z(T ) the set of all mappings Z : T → L (G) such that there exists a set A ∈ B0 (T ) and a sequence of simple mappings {Z n }∞ n=1 which vanish outside A and satisfy sup Z (t) − Z n (t)L (G ) → 0 as n → ∞.

(2.2)

t∈T

Clearly, Z(T ) is a normed vector space equipped with the norm Z ∞ := sup Z (t)L (G ) . t∈T

By construction, the set of all simple mappings forms a dense subspace in Z(T ). Remark 2.1. It can be easily shown that any mapping Z : T → L (G) which is continuous and which vanishes outside a compact set in T , belongs to Z(T ).


431

Let H be another real, separable Hilbert space. We consider a mapping B0 (T ) → M( ) ∈ L (H). We assume: (A1) M(∅) = 0. (A2) M(·) admits a decomposition M( ) = U ( ) + V ( ), ∈ B0 (T ), with U ( ), V ( ) ∈ L (H) being such that, for any mutually disjoint sets 1 , 2 ∈ B0 (T ), we have Ran U ( 1 ) ⊥ Ran U ( 2 ), Ran V ( 1 )∗ ⊥ Ran V ( 2 )∗ , where Ran A denotes the range of a bounded linear operator A, and the symbol ⊥ refers to orthogonality in H. (A3) For any A ∈ B0 (T ), any sequence of mutually disjoint sets n ∈ B(A), n ∈ N, and any F ∈ H, ∞ ∞ ∗ ∞ ∞ U n F = U ( n )F, V n F = V ( n )∗ F, n=1

n=1

n=1

n=1

where the series converges in H. Remark 2.2. The reader will see below that Assumptions (A1)–(A3) are sufficient for our purposes. For each Z ∈ Z(T ), we will now identify an integral T Z ⊗ d M as a bounded linear operator in the Hilbert space G ⊗ H. We fix any A ∈ B0 (T ). Let Z be a simple mapping as in (2.1) such that i ⊂ A for all i = 1, . . . , n. Without loss of generality, we may assume that the sets 1 , . . . , n are mutually disjoint. We define Z ⊗ dU :=

n

T

Z i ⊗ U ( i ) ∈ L (G ⊗ H).

i=1

By (A2), Ran(Z i ⊗ U ( i )) ⊥ Ran(Z j ⊗ U ( j )), i = j, where ⊥ refers to orthogonality in G ⊗ H. Hence, for each F ∈ G ⊗ H, 2 n Z ⊗ dU F = Z i ⊗ U ( i )F2G ⊗H T

G ⊗H

i=1

≤

n

Z i 2L (G ) 1 ⊗ U ( i )F2G ⊗H

i=1

≤

max Z i 2L (G ) i=1,...,n

n i=1

1 ⊗ U ( i )F2G ⊗H

432


=

Z 2∞ 1 ⊗ U

n i=1

2 i F

G ⊗H

≤ Z 2∞ 1 ⊗ U (A)F2G ⊗H .

(2.3)

Note that the latter estimate follows from the inequality 1 ⊗ U (A1 )FG ⊗H ≤ 1 ⊗ U (A2 )FG ⊗H ,

A1 , A2 ∈ B0 (T ), A1 ⊂ A2 ,

which, in turn, is a consequence of (A2) and (A3). Hence, by (2.3), Z ⊗ dU ≤ Z ∞ U (A)L (H) .

(2.4)

L (G ⊗H)

T

Let now Z be an arbitrary element of Z(T ), and let {Z n }∞ n=1 be an approximating sequence of simple mappings as in the definition of Z(T ). By (2.4), for any m, n ∈ N, Z n ⊗ dU − Z m ⊗ dU = (Z n − Z m ) ⊗ dU T

L (G ⊗H)

T

L (G ⊗H)

T

≤ Z n − Z m ∞ U (A)L (H) .

∞ Hence, T Z n ⊗ dU n=1 is a Cauchy sequence in L (G ⊗ H), and so it has a limit, which we denote by T Z ⊗ dU . Clearly, the definition of T Z ⊗ dU does not depend on the choice of approximating sequence of simple mappings. Note that, if Z (·) belongs to Z(T ), then also Z (·)∗ belongs to Z(T ). We can therefore define, for each Z ∈ Z(T ), ∗ Z ⊗ d V := Z∗ ⊗ dV ∗ . (2.5) T

T

Finally, we set

Z ⊗ d M := T

By (2.4) and (2.5), Z ⊗ d M T

Z ⊗ dU + T

L (G ⊗H)

Z ⊗ d V. T

≤ Z ∞ U (A)L (H) + V (A)L (H) .

(2.6)

Thus, we have proved Proposition 2.1. Let M satisfy (A1)–(A3). Then, for each A ∈ B0 (T ), there exists a constant C1 (A) ≥ 0, such that, for each Z ∈ Z(T ) satisfying Z (t) = 0 for all t ∈ A, we have Z ⊗ d M ≤ C1 (A)Z ∞ . T

L (G ⊗H)

Remark 2.3. The reader is advised to compare our construction of constructions of operator-valued integrals available in [7,12,17].

T

Z ⊗ d M with


433

Let us consider the special case where G = R, and so L (G) = R. As easily seen, the set Z(T ) is now the space B0 (T ) of all bounded measurable functions f : T → R with compact support. Furthermore, for each f ∈ B0 (T ), the operator T f d M := f ⊗ d M ∈ L (H) is characterized by the formula T f d M F1 , F2 := f d M F1 ,F2 , F1 , F2 ∈ H. (2.7) H

T

T

Here, for any A ∈ B0 (T ) and any F1 , F2 ∈ H, the mapping B(A) → M F1 ,F2 ( ) := (M( )F1 , F2 )H ∈ R is a signed measure on (A, B(A)). By Proposition 2.1, the total variation of M F1 ,F2 on A satisfies |M F1 ,F2 |(A) ≤ C1 (A)F1 H F2 H .

(2.8)

Remark 2.4. Assume that T = R and M(·) is an orthogonal resolution of the identity in H, i.e., a projection-valued measure on (R, B(R)). Then M(·) clearly satisfies the above assumptions and R f d M is a usual spectral integral (see e.g. [5,20]). 2.2. Generating function uniquely identifies polynomials. We will now consider a n sequence (M (n) )∞ n=1 of operator-valued measures on B0 (T ), respectively. Our initial (n) assumptions on each M will be slightly weaker than those in Subsect. 2.1. We assume that, for each n ∈ N, we are given a function B0 (T n ) → M (n) ( ) ∈ L (H) which satisfies the following assumption: (B) For any F1 , F2 ∈ H and any A ∈ B0 (T ), the mapping (n)

B0 (An ) → M F1 ,F2 ( ) := (M (n) ( )F1 , F2 )H ∈ R is a signed measure on (An , B(An )) whose total variation on An satisfies (n)

|M F1 ,F2 |(An ) ≤ C2 (A)n F1 H F2 H ,

(2.9)

where the constant C2 (A) only depends on A, and is independent of F1 , F2 ∈ H and n ∈ N. to (2.7), we may then identify, for each f (n) ∈ B0 (T n ), the integral Analogously (n) (n) d M as an element of L (H). (This operator may be thought of as a polynoTn f mial of the n th order.) For any Z 1 , . . . , Z n ∈ Z(T ), we define (Z 1 Z 2 · · · Z n )(t1 , t2 , . . . , tn ) := Z 1 (t1 )Z 2 (t2 ) · · · Z n (tm ), where the right-hand side is understood in the sense of the usual product of operators. Note that, in the case where G = R, for any f 1 , f 2 . . . , f n ∈ Z(T ) = B0 (T ), we evidently have f1 ⊗ f2 ⊗ · · · ⊗ fn = f1 f2 · · · fn .

434


For each Z ∈ Z(T ), we would like to identify an integral T n Z n ⊗ d M (n) as an element of L (G ⊗ H). However, we cannot do this under the above assumptions, so we define a four-linear form Z n ⊗ d M (n) (G 1 , F1 , G 2 , F2 ) n T := (Z (t1 ) · · · Z (tn )G 1 , G 2 )G d M F(n) (t , . . . , tn ), G 1 , G 2 ∈ G, F1 , F2 ∈ H. 1 ,F2 1 Tn

(2.10) As easily follows from the definition of Z(T ) and (B), the function T n (t1 , . . . , tn ) → (Z (t1 ) · · · Z (tn )G 1 , G 2 )G ∈ R is indeed measurable, the integral in (2.10) is finite, and moreover,

Tn

Z n ⊗ d M (n) (G 1 , F1 , G 2 , F2 ) ≤ Z n∞ C2 (supp Z )n G 1 G G 2 G F1 H F2 H .

Here, supp Z denotes the support of Z . Hence, continuous) form.

(2.11)

Tn

Z n ⊗ d M (n) is a bounded (and so

Remark 2.5. If there exists an operator Q (n) ∈ L (G ⊗ H) such that (Q (n) G 1 ⊗ F1 , G 2 ⊗ F2 )G ⊗H = Z n ⊗ d M (n) (G 1 , F1 , G 2 , F2 ), G 1 , G 2 ∈ G, F1 , F2 ∈ H, Tn

then we can identify T n Z n ⊗ d M (n) with the operator Q (n) . However, the estimate (2.11) is not sufficient for this to hold. We define a generating function of (M (n) )∞ n=1 as follows. We set Dom(G) := {Z ∈ Z(T ) : Z ∞ C2 (supp Z ) < 1}. Note that for each Z ∈ Z(T ), one can find ε > 0 such that, for each a ∈ (−ε, ε), a Z belongs to Dom(G). By virtue of (2.11), for each Z ∈ Dom(G), ∞ Z n ⊗ d M (n) (2.12) G(Z ) := 1 + n n=1 T

defines a bounded four-linear form on G × H × G × H. Here, 1 denotes the form which corresponds to the identity operator in G ⊗ H. Remark 2.6. Just as in Remark 2.5, if there exists an operator Q ∈ L (G ⊗ H) such that (QG 1 ⊗ F1 , G 2 ⊗ F2 )G ⊗H = G(Z )(G 1 , F1 , G 2 , F2 ), G 1 , G 2 ∈ G, F1 , F2 ∈ H, then we can identify G(Z ) with the operator Q. The following proposition shows that the generating function uniquely identifies the sequence (M (n) )∞ n=1 .


435

˜ (n) ∞ Proposition 2.2. Let (M (n) )∞ n=1 and ( M )n=1 satisfy condition (B). Assume that ˜ ), Z ∈ Dom(G) ∩ Dom(G). ˜ G(Z ) = G(Z

(2.13)

˜ ) denotes the generating function of ( M˜ (n) )∞ .) Then, for each n ∈ N, (Here, G(Z n=1 (n) M = M˜ (n) . Proof. Let Z ∈ Z(T ). Fix ε > 0 such that, for each a ∈ (−ε, ε), a Z ∈ Dom(G) ∩ ˜ Then, by (2.13), for each G 1 , G 2 ∈ G, F1 , F2 ∈ H and each a ∈ (−ε, ε), Dom(G). ∞ n=1

a

n

Z

n

Tn

⊗ dM

(n)

∞ n (G 1 , F1 , G 2 , F2 ) = a n=1

Hence, for each n ∈ N, Z n ⊗ d M (n) = Tn

Tn

Tn

Z n ⊗ d M˜ (n) (G 1 , F1 , G 2 , F2 ).

Z n ⊗ d M˜ (n) ,

Z ∈ Z(T ).

Now, take as Hilbert space G the full Fock space over 2 : G = F(2 ). Fix n ∈ N and choose any mutually orthogonal vectors e1 , . . . , en in 2 with norm 1. Fix arbitrary 1 , . . . , n ∈ B0 (T ) and define Z ∈ Z(T ) by Z (t) :=

n

a + (ei )χ i (t),

i=1

a + (ei ) being the creation operator at ei . Set G 1 := — the vacuum, and G 2 := e1 ⊗ e2 ⊗ · · · ⊗ en . Then, for any F1 , F2 , ∈ H,

Z n ⊗ d M (n) (G 1 , F1 , G 2 , F2 ) ei1 ⊗ ei2 ⊗ · · · ⊗ ein , e1 ⊗ e2 ⊗ · · · ⊗ en F ( =

Tn

i 1 , i 2 ,...,i n =1,...,n (n)

×M F1 ,F2 ( i1 × i2 × · · · × in ) (n)

= M F1 ,F2 ( 1 × 2 × · · · × n ). Therefore, (n) (n) M F1 ,F2 ( 1 × 2 × · · · × n ) = M˜ F1 ,F2 ( 1 × 2 × · · · × n ).

Hence, by (B), for any ∈ B0 (T n ), (n) (n) M F1 ,F2 ( ) = M˜ F1 ,F2 ( ),

which implies the proposition.

2)

436


3. Generating Function for a Free Meixner Process We start with a brief recalling of the construction of a free Meixner process from [10]. Let T be as in Sect. 2, and we denote D := C0 (T ). Let σ be a Radon non-atomic measure on (T, B(T )) which satisfies σ (O) > 0 for each open, non-empty set O in T . Fix some functions λ, η ∈ C(T ), which play the role of parameters of the free Meixner process. Consider the extended Fock space F=R⊕

∞

L 2 (T n , γn ).

(3.1)

n=1

Here, for n ∈ N, the measure γn on (T n , B(T n )) satisfies

f (n) (t1 , . . . , tn ) γn (dt1 , . . . , dtn ) = f (n) (t1 , . . . , t1 , . . . , ti , . . . , ti )ηl1 −1 (t1 ) · · · ηli −1 (ti )σ (dt1 ) · · · σ (dti ) Ti

Tn

i∈N, l1 ,...,li ∈N, l1 +···+li =n

l1 times

li times

for any measurable function f (n) : T n → [0, ∞]. In particular, γn = σ ⊗n if and only if η ≡ 0. The free Meixner process is defined as the family (X ( f )) f ∈D of bounded linear operators in F given by X ( f ) = X + ( f ) + X 0 ( f ) + X − ( f ), where the creation operator X + ( f ), the neutral operator X 0 ( f ) and the (extended) annihilation operator X − ( f ) are defined by formulas (4.1)–(4.3) in [10]. We also have a representation of each X ( f ) as σ (dt) f (t)ω(t) = ω, f , X( f ) = T

where ω(t) = ∂t† + λ(t)∂t† ∂t + ∂t + η(t)∂t† ∂t ∂t

(3.2)

with ∂t† and ∂t being the creation and annihilation operator at point t, respectively (see [10, Cor. 4.2]). The corresponding system of orthogonal polynomials is denoted in this paper by P (n) (ω), f (n) ,

f (n) ∈ D(n) := C0 (T n ), n ∈ N0 .

These are the bounded linear operators in F which are recursively defined through P (0) (ω) = 1, P

(n)

P (1) (ω)(t) = ω(t),

(ω)(t1 , . . . , tn ) = ω(t1 )P (n−1) (ω)(t2 , . . . , tn ) − δ(t1 , t2 )λ(t1 )P (n−1) (ω)(t2 , . . . , tn ) − δ(t1 , t2 )P (n−2) (ω)(t3 , . . . , tn ) − [n − 2]0 δ(t1 , t2 , t3 )η(t1 )P (n−2) (ω)(t3 , . . . , tn ), n ≥ 2,


437

where δ(t1 , t2 ) and δ(t1 , t2 , t3 ) are the ‘delta-functions’ defined as in [10, Sect. 2]. In particular, for any f 1 , . . . , f n ∈ D, n ≥ 2, P (n) (ω), f 1 ⊗ · · · ⊗ f n = ω, f 1 P (n−1) (ω), f 2 ⊗ · · · ⊗ f n

− P (n−1) (ω), (λ f 1 f 2 ) ⊗ f 3 ⊗ · · · ⊗ f n

− f 1 (t) f 2 (t) σ (dt) P (n−2) (ω), f 3 ⊗ · · · ⊗ f n

T

−[n − 2]0 P (n−2) (ω), (η f 1 f 2 f 3 ) ⊗ f 4 ⊗ · · · ⊗ f n .

(3.3)

Recall also that we may extend the definition of X ( f ) and of P (n) (ω), f (n) to the case where f ∈ B0 (T ) and f (n) ∈ B0 (T n ), respectively. Our aim now is to derive the generating function for these orthogonal polynomials. So, let us fix a Hilbert space G. From now on, for simplicity of notation, we will sometimes identify operators X ∈ L (G) and Y ∈ L (F) with the operators X ⊗ 1 and 1 ⊗ Y in L (G ⊗ F). For each f ∈ D, we clearly have ω, f = T f d M, where for each ∈ B0 (T ), M( ) := X (χ ). Note that M satisfies conditions (A1)–(A3) with U ( ) = X + (χ ) + X 0 (χ ), V ( ) = X − (χ ). Indeed, (A1) is trivially satisfied. For any ∈ B0 (T ), we have, by (4.1) and (4.2) in [10], X + ( ) = χ ,

X 0 ( ) = 0,

(3.4)

where is the vacuum in F, and for any n ∈ N and any f (n) ∈ L 2 (T n , γn ), (X + ( ) f (n) )(t1 , . . . , tn+1 ) = χ (t1 ) f (n) (t2 , . . . , tn+1 ), (X ( ) f 0

(n)

)(t1 , . . . , tn ) = χ (t1 )λ(t1 ) f

(n)

(t1 , . . . , tn ).

(3.5) (3.6)

Furthermore, X − ( )∗ = X + ( ).

(3.7)

Now, (A2) and (A3) easily follow from (3.4)–(3.7). Therefore, by Subsect. 2.1, we define, for each Z ∈ Z(T ), ω, Z := Z ⊗ d M ∈ L (G ⊗ F). T

It easily follows from (2.6) and the definition of the space F that ω, Z L (G ⊗F) ≤ Z ∞ C3 (supp Z ), where

C3 (A) := 2 σ (A) + 2 sup η(t) + sup |λ(t)|, t∈A

Z ∈ Z,

(3.8)

A ∈ B0 (T ).

(3.9)

t∈A

For any n ∈ N and any Z 1 , . . . , Z n ∈ Z(T ), we recurrently define an operator P (n) (ω), Z 1 · · · Z n from L (G ⊗ F) as follows. By analogy with (3.3), we set P (0) (ω), Z 0 := 1, P (1) (ω), Z := ω, Z and for n ≥ 2,

438


P (n) (ω), Z 1 · · · Z n = ω, Z 1 P (n−1) (ω), Z 2 · · · Z n

− P (n−1) (ω), (λZ 1 Z 2 ) Z 3 · · · Z n

− Z 1 (t)Z 2 (t) σ (dt) P (n−2) (ω), Z 3 · · · Z n

T

−[n − 2]0 P (n−2) (ω), (ηZ 1 Z 2 Z 3 ) Z 4 · · · Z n . (3.10) Note that, for any Z 1 , Z 2 ∈ Z(T ), the point-wise (non-commutative) product Z 1 Z 2 belongs to Z(T ), and for each Z ∈ Z(T ), λZ and ηZ also belong to Z(T ). In formula (3.10) and below, for each Z ∈ Z(T ), the integral T Z (t) σ (dt) is understood in Bochner’s sense, see e.g. [5,20]. It then easily follows by induction from (3.8)–(3.10) and a standard estimate of the norm of a Bochner integral that, for any A ∈ B0 (T ), n ∈ N, and any Z 1 , . . . , Z n ∈ Z(T ) with support in A: P (n) (ω), Z 1 · · · Z n L (G ⊗F) ≤ C4 (A)n Z 1 ∞ · · · Z n ∞ ,

(3.11)

where C4 (A) := C3 (A) + σ (A) + sup |λ(t)| + sup η(t) t∈A

t∈A

= 2 σ (A) + σ (A) + 2 sup η(t) + 3 sup |λ(t)|. t∈A

(3.12)

t∈A

Hence, for each Z ∈ Z(T ) such that Z ∞ C4 (supp Z ) < 1, the sum G(Z ) = 1 +

∞

P (n) (ω), Z n

(3.13)

n=1

defines an operator from L (G ⊗ F). Next, we set, for each n ∈ N and ∈ B0 (T n ), M (n) ( ) := P (n) (ω), χ . Analogously to (3.11), we conclude that the sequence (M (n) )∞ n=1 satisfies condition (B), and so the function G defined by (3.13) is the generating function of the operator-valued measures (M (n) )∞ n=1 in the sense of Subsect. 2.2. Hence, by Proposition 2.2, the generating function G uniquely identifies (M (n) )∞ n=1 , and hence also polynomials P (n) (ω), f (n) , f (n) ∈ D(n) . To stress the dependence of the generating function G(Z ) on the free generalized stochastic process ω, we will write G(Z , ω). Theorem 3.1. Fix any A ∈ B0 (T ). Then there exists a constant C5 (A) > 0 such that, for any Z ∈ Z(T ) satisfying supp Z ⊂ A and Z ∞ < C5 (A), formula (1.9) holds. Furthermore, we have −1 G(Z , ω) = 1 − f (Z ) ω(·), λ(·),η(·) (Z (·)) f (Z ), (3.14) where

f (Z ) := 1 +

−1 Z (t)2 σ (dt) . (3.15) 2 T 1 + λ(t)Z (t) + η(t)Z (t) Remark 3.1. The right hand side of formula (1.9)should be understood in the follown ing sense: for any real-valued function f (x) = ∞ n=0 an x which is real-analytic on


439

(−r, r ), we write, for a bounded linear operator B whose norm is less than r : f (B) := ∞ Zl n n=0 an B . Under our assumption on Z ∈ Z(T ), we then have 1+λZ +ηZ 2 ∈ Z(T ), l = 1, 2. Proof. We divide the proof into several steps. Step 1. First, for a fixed A ∈ B0 (T ), let us explicitly specify a possible choice of a constant C5 (A) in the theorem. For each t ∈ T , define α(t), β(t) ∈ C so that α(t) + β(t) = λ(t), α(t)β(t) = η(t). Hence, for each x ∈ R, 1 + λ(t)x + η(t)x 2 = (1 − α(t)x)(1 − β(t)x). The right hand side of formula (1.9) now reads as 1 − ω, Z (1 − α Z )−1 (1 − β Z )−1

−1

Z (t) (1 − α(t)Z (t)) 2

+

−1

(1 − β(t)Z (t))

σ (dt)

−1 (3.16)

T

(we consider the above operator in the complexification of the real Hilbert space G ⊗ F, for which we keep the same notation). Set α A := sup |α(t)|, β A := sup |β(t)|. t∈A

t∈A

Choose C6 (A) > 0 so that ∞ ∞ k k l l α A C6 (A) β A C6 (A) C6 (A) C3 (A) + C6 (A)σ (A) < 1. (3.17) k=0

l=0

Then, by virtue of (3.8), we have that, for each Z ∈ Z(T ) such that supp Z ⊂ A and Z ∞ ≤ C6 (A), formula (3.16) defines a bounded linear operator in L (G ⊗ F). Recalling (3.11)–(3.13), we set C5 (A) := min{C4 (A)−1 , C6 (A)}.

(3.18)

Then, for each Z ∈ Z(T ) such that supp Z ⊂ A and Z ∞ < C5 (A), the left- and right-hand sides of formula (1.9) identify bounded linear operators in G ⊗ F. Let us denote the operators on the left- and right-hand sides of formula (1.9) by L(Z ) and R(Z ), respectively. Fix any , ϒ ∈ G ⊗ F. It follows that, for any Z ∈ Z(T ) such that supp Z ⊂ A, the functions f (L) (z) := (L(z Z ), ϒ)G ⊗F , f (R) (z) := (R(z Z ), ϒ)G ⊗F are analytic on z ∈ C : |z| < C5 (A)Z −1 ∞ . Step 2. Fix any A ∈ B0 (T ). Choose any set partition P = { 1 , . . . , J } of A, i.e., A=

J j=1

j , j ∈ B0 (T ), j = 1, . . . , J, J ∈ N

440


and the sets j are mutually disjoint. Set λ j := inf λ(t), η j := inf η(t), t∈ j

t∈ j

j = 1, . . . , J,

and define a function λP (t) :=

λ j , if t ∈ j , j = 1, . . . , J, 0, if t ∈ Ac ,

and analogously a function ηP (t). Now, we define a generalized operator-valued process ωP (t) and corresponding non-commutative polynomials P (n) (ωP ), f (n) , f (n) ∈ B0 (T n ), in the same way as ω(t) and P (n) (ω), f (n) were defined, but by using the functions λP and ηP instead of λ and η, respectively. We stress that these are also defined in the extended Fock space F constructed through the function η. Hence, generally speaking, the operators P (n) (ωP ), f (n) are not self-adjoint in F. This, however, does not lead to any problem when we define a generating function G P (Z ) of these polynomials. In particular, the corresponding operator-valued measure MP ( ) := ωP , χ , ∈ B0 (T ), 0 (χ ) and V ( ) = X − ( ), satisfies conditions (A1)–(A3) with U ( ) = X + (χ ) + X P P where 0 X + (χ ) := ∂t† σ (dt), X P (χ ) := λP (t)∂t† ∂t σ (dt), † − X P (χ ) := (∂t + ηP (t)∂t ∂t ∂t ) σ (dt),

− compare with (3.2). (We leave the evaluation of the adjoint operator of X P (χ ) in F to the interested reader.) Furthermore, analogously to (3.11), we get, for any A ∈ B0 (T ), n ∈ N and any Z 1 , . . . , Z n ∈ Z(T ) with support in A,

P (n) (ωP ), Z 1 · · · Z n L (G ⊗F) ≤ C4 (A)n Z 1 ∞ · · · Z n ∞ ,

(3.19)

with the same constant C4 (A) given by (3.12). (We, in particular, used that ηP (t) ≤ η(t) for all t ∈ A.) ⊗n ∞

n=1 Step 3. By definition, for each j = 1, . . . , J , the polynomials P (n) (ωP ), χ j satisfy the recursion relation (n−1) ⊗(n−1) ⊗n P

= ω , χ

− λ (ωP ), χ j

P (n) (ωP ), χ j P j j ⊗(n−2) −(σ ( j ) + [n − 2]0 η j ) P (n−2) (ωP ), χ

, n ≥ 2. j

Therefore, (n)

⊗n

= Pλ j ,η j ,σ ( j ) ( ωP , χ j ), P (n) (ωP ), χ j

(3.20)


441

(n)

where (Pλ j ,η j ,σ ( j ) )∞ n=0 is a system of polynomials on R recursively defined by (u) = 1, Pλ(0) j ,η j ,σ ( j )

Pλ(1) (u) = u, j ,η j ,σ ( j )

(n)

(n−1)

Pλ j ,η j ,σ ( j ) (u) = (u − λ j )Pλ j ,η j ,σ ( j ) (u) − (σ ( j ) (n−2)

+[n − 2]0 η j )Pλ j ,η j ,σ ( j ) (u), n ≥ 2.

(3.21)

(n)

By [1], the generating function of (Pλ j ,η j ,σ ( j ) )∞ n=0 is given by ∞

z

n

(n) Pλ j ,η j ,σ ( j ) (u)

= 1−u

n=0

σ ( j )z 2 z + 1 + λ j z + η j z2 1 + λ j z + η j z2

−1 . (3.22)

More precisely, for each r > 0, there exists εr,A > 0 such that formula (3.22) holds for each u ∈ R with |u| ≤ r and for each z ∈ C such that |z| < εr,A . Let Z j ∈ L (G) be such that Z j L (G ) < C5 (A), where C5 (A) is given by (3.18). Then, by (3.20) and (3.22), we get 1+

∞

⊗n Z nj P (n) (ωP ), χ

j

n=1

= 1 − ωP , χ j

Zj 1 + λ j Z j + η j Z 2j

+ σ ( j )

−1

Z 2j 1 + λ j Z j + η j Z 2j

,

j = 1, . . . , J.

(3.23) Denote U j := ωP , χ j

Z 2j Zj − σ ( ) , j 1 + λ j Z j + η j Z 2j 1 + λ j Z j + η j Z 2j

j = 1, . . . , J.

Then (3.23) is equivalent to ∞

P (n) (ωP ), (Z j χ j )n =

n=1

∞

U nj ,

j = 1, . . . , J.

(3.24)

n=1

Step 4. We claim that, for any n ∈ N and any j1 , j2 , . . . , jn ∈ {1, 2, . . . , J } such that j1 = j2 , j2 = j3 ,…, jn−1 = jn , and any k1 , k2 , . . . kn ∈ N, we have ⊗kn ⊗k1 ⊗k2 P (k1 +k2 +···+kn ) (ωP ), χ ⊗ χ ⊗ · · · ⊗ χ

j j j 1

= P

(k1 )

n

2

⊗kn ⊗k1 ⊗k2 (ωP ), χ

P (k2 ) (ωP ), χ

· · · P (kn ) (ωP ), χ

. j1 j2 jn

(3.25)

Indeed, first we can prove by induction in k1 ∈ N that, for any fixed k2 ∈ N, and any j1 , j2 ∈ {1, 2, . . . , J }, j1 = j2 , ⊗k1 ⊗k2 ⊗k1 ⊗k2 ⊗ χ

= P (k1 ) (ωP ), χ

P (k2 ) (ωP ), χ

. P (k1 +k2 ) (ωP ), χ j j j j 1

2

1

2

442


Then, we prove (3.25) by induction in n ∈ N. Step 5. Now, fix any Z 1 , . . . , Z J ∈ L (G) such that C5 (A) . J

max Z j L (G )
0, we define the quantity . (s , s ) = h

h

1 · |s | + |s |

τ τ

h

h −1

τ

τ

h

h −1

σ (s − ξ ) − σ (ξ ) dξ dξ ,

(4.10)

and extend such a definition to the case where one of the two waves s , s have neg ative size. Instead, for pairs of subwaves s h , s h of the interacting portions s [i] , s [i] of two k-waves s , s (see Definition 5), we define the quantity (s h , s h ) as in (4.10) with 1/(|s [i] | + |s [i] |) in place of 1/(|s | + |s |). Furthermore, for subwaves s h , s h [sp] r of the same k-wave s (or of the same composite portion of the split wave s as in Definition 6), according with Definition 4 we define (s h , s h ) as in (4.10) with 1/|s| (or 1/|s [sp]r |) in place of 1/(|s | + |s |). Remark 8. By Definitions 4-5 and Definition 8 it follows ! " that, ! for"every given pairs of h waves s , s of the same characteristic family, letting s , s h be the correspondh h ing partitions provided by Proposition 2, there holds h h , (4.11) s , s (s , s ) = h ,h

A Sharp Convergence Rate of the Glimm Scheme

e (s , s ) =

605

h h , s , s

(4.12)

h ,h

where the summation in (4.11) runs over all subwaves s h , s h of s and s , respec tively, while the summation in (4.12) runs over all subwaves s h , s h of s [i] and s [i] , respectively. The equality (4.11) remains valid in the case s = s = s (s being either a whole elementary wave or a composite portion s [sp]r split by an interaction). Notice that, in this case, the subwaves s h , s h that appear in (4.11) are all portions of the same wave s, and whenever s h , s h are subwaves of a same shock component of s, by Remark 4 h h one has (s , s ) = 0.

We next define the oscillation of the terms (skh (i, ), skh (i, )) for every pair of k-subwaves skh (i, ), skh (i, ). In view of Proposition 2 we define such oscillations q

q

for pairs of k-subwaves sk (i, (i , j ,k,h ,i) ), sk (i, (i , j ,k,h ,i) ), associated to pairs of points (i ε, j ε), (i ε, j ε) that they have previously crossed (or where they have been originated), setting Osc k,i ((i , j , h ), (i , j , h )) q q . = sk (i , j ,k,h ,i) (i, (i , j ,k,h ,i) ), sk (i , j ,k,h ,i) (i, (i , j ,k,h ,i) ) + − skh (i −1, (i , j ,k,h ,i -1) ), skh (i −1, (i , j ,k,h ,i -1) ) ,

(4.13)

(l, q being the maps in (4.1)). We also define the oscillation for a pair of k-subwaves skh (i − 1, (i , j ,k,h ,i -1) ), skh (i − 1, (i , j ,k,h ,i -1) ), issuing from two consecutive nodes of the layer t = (i −1)ε, that join together as subwaves of a shock component of a k-wave issuing from a node (iε, ), with = (i , j ,k,h ,i) = (i , j ,k,h ,i) , by setting Osc k,i ((i , j , h ), (i , j , h )) . = skh (i −1, (i , j ,k,h ,i -1) ), skh (i −1, (i , j ,k,h ,i -1) ) + − skh (i −1, (i , j ,k,h ,i -1) ), skh (i −1, (i , j ,k,h ,i -1) ) , (4.14) where skh , skh is defined as in (4.10) viewing skh , skh as subwaves of the interacting portions of waves sk[i] (i −1, (i , j ,k,h ,i -1) ), sk[i] (i −1, (i , j ,k,h ,i -1) ) (cfr. Definition 5). q

q

Instead, for the k-subwaves sk (i, (i , j ,k,h ,i) ), sk (i, (i , j ,k,h ,i) ), of a composed por[sp] tion of k-wave sk r issuing from a node of the layer t = iε, generated by the splitting of a (portion of a) shock component of a k-wave issuing from a node of the previous layer t = (i − 1)ε (cfr. Definition 6), we set Osc k,i ((i , j , h ), (i , j , h )) q q . = sk (i , j ,k,h ,i) (i, (i , j ,k,h ,i) ), sk (i , j ,k,h ,i) (i, (i , j ,k,h ,i) ) + q q − sk (i , j ,k,h ,i) (i, (i , j ,k,h ,i) ), sk (i , j ,k,h ,i) (i, (i , j ,k,h ,i) ) , (4.15) q q [sp] where sk () , sk () is defined as in (4.10) viewing skh , skh as subwaves of sk r . To

compare these definitions with the one given in (3.27), consider the k-waves sk , sk , k =

606

F. Ancona, A. Marson

1, . . . , N , issuing from two consecutive nodes ((i − 1)ε, ( j − 1)ε), ((i − 1)ε, jε), that take part of the solution s1 , . . . , s N of the Riemann problem at the node (iε, jε). Let {sk h }h , {sk h }h and {skh }h , be the corresponding partitions of sk , sk and sk . Denote with . I = {h ; ι(i -1, j -1,k,h ) ≥ i}, . I = {h ; ι(i -1, j,k,h ) ≥ i},

(4.16)

the index sets of k-subwaves of sk , sk that are not canceled by the interaction and, in the case sk · sk > 0, denote with ! " [i] . h [i] I = h ; sk , is a subwave of sk ! " (4.17) [i] . h [i] I = h ; sk is a subwave of sk , the index sets of k-subwaves that are effectively interacting (cfr. Remark 7), while set . I [i] = I [i] = ∅ whenever sk · sk ≤ 0, or sk [i] = sk[i] = 0. Moreover, setting . . qh = q(i -1, j -1,k,h ,i) , qh = q(i -1, j,k,h ,i) , let ! " q [sp] . [sp] I r = h ; sk h is a subwave of sk r , ! " (4.18) q [sp] . [sp] I r = h ; sk h is a subwave of sk r , denote the index set of k-subwaves of those portions sk [s]r of shock components of one [sp] of the two k-waves sk , sk , that are split in composed portions sk r of sk after the inter. . action at (iε, jε) (cfr. Definition 6 and Remark 7). Set I [sp] = ∅ (I [sp] = ∅) if no such splitting takes place after the interaction for a shock component of sk (sk ), and let ) . ) [sp]r [sp] [sp] [sp] I [sp] = I I r × I r . (4.19) × I r ∪ r

r

Then, defining * (h , h ) ∈ (I ∪ I )×(I ∪ I ); (h , h ) ∈ / I [i] ×I [i] . H= (h , h ) ∈ (I ∪ I )×(I ∪ I ) (h , h ) ∈ / I [sp] . = H (h , h ) ∈ (I ∪ I )×(I ∪ I ); (h , h ) ∈ /H ,

if sk ·sk > 0, if sk ·sk ≤ 0, (4.20) (4.21)

comparing definitions (3.21), (3.27), and Definition 6, with definitions (4.13)–(4.15), and relying on Remarks 7-8, there holds

si

k =

Osc k,i ((i − 1, j − 1, h ), (i − 1, j, h )) +

(h , h )∈H

+

Osc k,i ((i − 1, j − 1, h ), (i − 1, j, h )),

(4.22)

(h , h )∈H

where the terms of the second summand are defined as in (4.14) if (h , h ) ∈ I [i] ×I [i] , and as in (4.15) if (h , h ) ∈ I [sp] . Here the summations may run over a countable set


607

of indices, in which case their sums is well defined being the corresponding series abso lutely convergent since, by Lemma 1 and because of (3.14), one has h ,h Osc k,i + 2 h ,h Osc k,i = O(1) · (V (iε)) . 4.3. A functional measuring the oscillation of the interaction potential. In order to provide a definition of the total amount of oscillations of the terms (skh (i, ), skh (i, )) over a time interval [mε, nε], we now introduce two sets of pairs of indices F(m,k,i, , ) , S(m,k,i, , ) . They identify the pairs of points and of portions of wave from which are originated pairs of first and second generation k-subwaves, respectively, that: 1. eventually reach the nodes (iε, ε), (iε, ε); . 2. in the case = = , (a) don’t join together as subwaves of a k-shock issuing from (iε, ε), (b) don’t belong to a composed wave generated by a splitting at (iε, ε). (m,k,i,) , S(m,k,i,) , the sets of pairs of indices as in Instead, we shall denote by F F(m,k,i,,) , S(m,k,i,,) , corresponding to pairs of k-subwaves that, when reaching the node (iε, ε), either join together as portions of a shock, or are portions of a composed wave issuing from (iε, ε). To this end, for every m ≤ i < i ≤ n, 1 ≤ k ≤ N , ∈ Z, denote with . (4.23) I(i ,k,i,) = {( j, h); (i , j,k,h,i) = }, the index sets of k-subwaves that were present at time t = i ε, and eventually reach the node (iε, ε). Moreover, in the case where the k-waves sk (i − 1, − 1), sk (i − 1, ), have the same sign, let ! " q . [i] I (i ,k,i,) = ( j, h) ∈ I(i ,k,i,) ; sk (i , j,k,h,i−1) (i −1, −1) is a subwave of sk[i] (i − 1, − 1) , ! " q . [i] I (i ,k,i,) = ( j, h) ∈ I(i ,k,i,) ; sk (i , j,k,h,i−1) (i −1, ) is a subwave of sk[i] (i −1, ) ,

(4.24) denote the index sets of k-subwaves that are effectively interacting (cfr. Definition 5 and Remark 7) and eventually reach the node (iε, ε). Instead, whenever sk (i − 1, − 1) · sk (i − 1, ) < 0, or sk[i] (i − 1, − 1) = sk[i] (i − 1, ) = 0, as well as in the case where one of the two Riemann solutions at the nodes (i − 1, − 1), (i − 1, ), contains no k-wave taking part of the solution of the Riemann problem at (iε, ε), set . I (i[i] ,k,i,) = I (i[i] ,k,i,) = ∅. Furthermore, in the case where the k-wave sk (i, ) contains [sp]

composite waves sk r produced by the splitting of (portions of) shock components of k-waves issuing from the previous layer t = (i − 1)ε (cfr. Definition 6), let ! " q . [sp]r I(i ,k,i,) = ( j, h) ∈ I(i ,k,i,) ; sk (i , j,k,h,i) (i, ) is a subwave of sk[sp]r (i, ) , (4.25) denote the index sets of k-subwaves that when reaching the node (iε, ε) are portions [sp] of sk r , and set . ) [sp]r [sp] [sp]r I(i ,k,i,) × I(i ,k,i,) , (4.26) I(i ,i ,k,i,) = r

608


. [sp] while set I(i ,k,i,) = ∅, if no splitting takes place at (i, ) of a shock component of a k-wave present at t = (i − 1)ε. Then, recalling that {(i ε, j ε); (i , j ) ∈ Gm,k } is the set of all generation points of new waves within the strip ]mε, nε] × R, we define, for m < i ≤ n, 1 ≤ k ≤ N , , ∈ Z, the sets . . F(m,k,i, , ) = (( j , h ), ( j , h )) ∈ I(m,k,i, ) × I(m,k,i, ) ; if = = , [i]

[i]

(( j , h ), ( j , h )) ∈ / I (m,k,i,) ×I (m,k,i,) [sp] (( j , h ), ( j , h )) ∈ / I(m,m,k,i,) , there holds

(m,k,i,) F S(m,k,i, , )

(4.27) . = (( j , h ), ( j , h )) ∈ I(m,k,i,) × I(m,k,i,) ; (4.28) / F(m,k,i,,) , (( j , h ), ( j , h )) ∈ . = ((i , j , h ), (i , j , h )); (( j , h ), ( j , h )) ∈ I(i ,k,i, ) ×I(i ,k,i, ) , (i , j ), (i , j ) ∈ Gm,k ∪ {m} × Z, . max{i , i } > m, and, if = = , [i]

[i]

(( j , h ), ( j , h )) ∈ / I (i ,k,i,) ×I (i ,k,i,) [sp] (( j , h ), ( j , h )) ∈ / I(i ,i ,k,i,) ,

(4.29)

(i , j ), (i , j ) ∈ Gm,k ∪ {m} × Z, max{i , i } > m, ((i , j , h ), (i , j , h )) ∈ / S(m,k,i,,) .

(4.30)

there holds

. S(m,k,i,) = ((i , j , h ), (i , j , h )); (( j , h ), ( j , h )) ∈ I(i ,k,i,) ×I(i ,k,i,) ,

By construction, the maps

q q (( j , h ), ( j , h )) → sk (m, j ,k,h ,i) (i, ), sk (m, j ,k,h ,i) (i, ) q q ((i , j , h ), (i , j , h )) → sk (i , j ,k,h ,i) (i, ), sk (i , j ,k,h ,i) (i, )

(4.31)

(m,k,i, , ) , and S(m,k,i, , ) , associate to every pair of indices in F(m,k,i, , ) , F S(m,k,i, , ) , a pair of k-subwaves issuing from the nodes (iε, ε), (iε, ε). The sets (m,k,i, , ) identify the pairs of first generation subwaves, while F(m,k,i, , ) , F S(m,k,i, , ) , S(m,k,i, , ) identify the pairs of subwaves containing at least one second generation wave. Remark 9. By definition (4.24), and recalling Definition 5 and Remark 7, it follows that, q q whenever we consider two subwaves sk (i , j ,k,h ,i−1) (i − 1, − 1), sk (i , j ,k,h ,i−1) (i − 1, ), so that [i]

[i]

(( j , h ), ( j , h )) ∈ I (i ,k,i,) ×I (i ,k,i,) ,

(4.32) q

for some m ≤ i , i < i ≤ n, the corresponding k-subwaves sk (i , j ,k,h ,i) (i, ), q sk (i , j ,k,h ,i) (i, ) on the layer t = iε result in being portions of a same shock component of the k-wave sk (i, ) issuing from (iε, ε). Therefore, by Remark 8, we have q q sk (i , j ,k,h ,i) (i, ), sk (i , j ,k,h ,i) (i, ) = 0. (4.33)


609

Similarly, by definitions (4.25)–(4.26), and because of Definition 6 and Remark 7, it q q follows that, when we consider two subwaves sk (i , j ,k,h ,i) (i, ), sk (i , j ,k,h ,i) (i, ), so that [sp]

(( j , h ), ( j , h )) ∈ I(i ,i ,k,i,) , q

q

(4.34)

the corresponding k-subwaves sk (i , j ,k,h ,i−1) (i, ), sk (i , j ,k,h ,i−1) (i, ) on the layer t = (i − 1)ε will be portions of a same shock component of a k-wave issuing from one of the two nodes ((i − 1)ε, ( − 1)ε), ((i − 1)ε, ε). Hence, by Remark 8, there holds q q sk (i , j ,k,h ,i−1) (i, ), sk (i , j ,k,h ,i−1) (i, ) = 0. (4.35) Given a Glimm approximate solution, we now fix n > 0, and consider the corresponding wave partition provided by Proposition 2. With the above notations, and using the definitions of oscillation given in (4.13)–(4.15), we define a functional G n (t), t ∈ [0, nε[ , by setting for every 0 ≤ m < n: N n . = GF n,m

k=1 i=m+1 , ∈Z

Osc k,i ((m, j , h ), (m, j , h )),

(( j ,h ), ( j ,h ))∈F

(m,k,i, , )

(4.36) . GF n,m =

N

n

Osc k,i ((m, j , h ), (m, j , h )),

(4.37)

(m,k,i,) k=1 i=m+1 ∈Z (( j ,h ), ( j ,h ))∈F N n . GS = n,m

Osc k,i ((i , j , h ), (i , j , h )),

k=1 i=m+2 , ∈Z ((i , j ,h ), (i , j ,h ))∈S(m,k,i, , )

(4.38) . GS n,m =

N

n

Osc k,i ((i , j , h ), (i , j , h )),

(4.39)

(m,k,i,) k=1 i=m+2 ∈Z ((i , j ,h ), (i , j ,h ))∈S

S [i] [i] (the terms of G F n,m , G n,m , being defined as in (4.14) if (( j , h ), ( j , h )) ∈ I (·) ×I (·) , [sp]

and as in (4.15) if (( j , h ), ( j , h )) ∈ I(·) ), and letting . . F S S G n (t) = G n,m = G F n,m +G n,m +G n,m +G n,m

∀ t ∈ [mε, (m + 1)ε[,

(4.40)

for all 0 ≤ m < n. As observed for (4.22), the summations in (4.36)–(4.36) when running over countable sets of pairs of indices are well defined since the corresponding sums are absolutely convergent thanks to Lemma 1 and by (3.14). We wish to compare the variation of the functional G n (t) at a time t = mε with the oscillation of the terms (sα , sβ ) in Q(t) at t = mε. To this end, let si (mε, ε) denote a quantity defined as in (3.27)–(3.28), expressing the variation of the self-interacting terms of (1.23) related to the waves involved in the solution of the Riemann problem at the node (mε, ε), and denote with (mε, α ε, β ε) the variation of the interacting terms of (1.23) related to the solutions of the Riemann problems at different nodes (mε, α ε), (mε, β ε), α = β , , s , k = 1, . . . , N , be the k-waves issuing from the nodes defined as follows. Let sα,k α,k ((m − 1)ε, (α − 1)ε), ((m − 1)ε, α ε), that take part of the solution sα,1 , . . . , sα,N , of

610


the Riemann problem at the node (mε, α ε), and adopt similar notations for the k-waves , s , k = 1, . . . , N , issuing from ((m − 1)ε, ( − 1)ε), ((m − 1)ε, ε), and sβ,k β β β,k taking part of the solution sβ,k , k = 1, . . . N , of the Riemann problem at (mε, β ε). Then, set + , . , sβ,k ) − (sα,k , sβ,k ) + k (mε, α ε, β ε) = (sα,k , sβ,k ) − (sα,k + , − (sα,k , sβ,k ) − (sα,k , sβ,k ) , (4.41) and . (mε, α ε, β ε) = k (mε, α ε, β ε). N

(4.42)

k=1

Comparing definitions (4.13)–(4.15), (4.36)–(4.40) with (3.21), (3.27)–(3.28), Definition 6 and (4.41–(4.42, and relying on Remarks 7-8, we observe that the variation of the functional G n (t) at any time t = mε < nε, is precisely

G n,m−1 − G n,m =

si (mε, ε) +

∈Z

(mε, α ε, β ε).

(4.43)

α ,β ∈Z α =β

Hence, applying proposition 1, we deduce the estimates stated in the following proposition on the variation of the functionals G n (t) and Q(t) + G n (t) across grid-times t = mε < nε. Proposition 3. For every compact set K ⊂ , there exist positive constants χ2 , c2 , C, such that the following holds. Let u ε = u ε (t, x) be a Glimm approximate solution of (1.1), (1.6), and assume that lim x→−∞ u ε (mε−, x) ∈ K , V (mε−) < χ2 , for some . . . m > 0. Then, letting V − = V (mε−), Q − = Q(mε−), G − n = G n (mε−), and . . . + + + V = V (mε+), Q = Q(mε+), G n = G n (mε+), denote the values of V, Q, G n . (n > m), related to u ε (mε−, ·) and u ε (mε+, ·), respectively, and setting V (mε) = . . V + − V − , Q(mε) = Q + − Q − , G n (mε) = G +n − G − n , there hold ⎡

⎤

⎢ ⎢ G n (mε) = − ⎢ si (mε, ε) + ⎣ ∈Z

⎡

α ,β ∈Z α =β

⎥ ⎥ (mε, α ε, β ε)⎥, ⎦

(4.44) ⎤

⎢ ⎥ ⎢ ⎥ |sα sβ | + C(sα , sβ ) + e (sα , sβ )⎥, [V + C · (Q + G n )] (mε) ≤ −c2 · ⎢ ⎣ ⎦ kα xβ

kα =kβ

kα =kβ sα ·sβ >0

(4.45) where sα denotes a wave in u ε (mε−, ·) of the kα th family located in xα , C(sα , sβ ) denotes the amount of cancellation defined in (3.17), and e (sα , sβ ) is the interaction quantity defined in (3.21).


611

4.4. Uniform bound on the oscillations of the interaction potential of quadratic order in the total variation. We establish here an a-priori bound on the functional G n (t), t ∈ [0, nε[ , defined in (4.36)–(4.40), uniform with respect to n ∈ N. To this end, we fix n > 0 and, in connection with the wave partition provided by Proposition 2, we introduce a [sp] set of indices Gm,k , 0 ≤ m < n, that identify the points (i, j) in the strip ]mε, nε[×R, where a k-shock, incoming from the previous layer t = (i − 1)ε, is split in a composed wave because of the interaction occurring at (iε, jε) involving waves of the same k th family with opposite sign, or waves of other families. Namely, with the notations of Definition 6 and Proposition 2, we set ! " [sp] . [sp] Gm,k = (i, j) ∈ ]mε, nε] × R; I = ∅ for some m ≤ i , i < i . (4.46) (i ,i ,k,i, j) [sp]

Definition 9. Given any point (i, j) ∈ Gm,k , 0 ≤ m < n, we define the maximal (back[sp]

ward) tree of the kth family with vertex at (i, j) ∈ Gm,k , denoted by Tk,m (i, j), as the [sp]

collection of all k-subwaves originated at some point (i 0 , j0 ) ∈ Gm,k ∪ Gm,k , i 0 < i, or at time t = mε, that eventually reach a composed wave issuing from (i, j), generated by the splitting of a (portion of a) shock wave incoming from the previous layer t = (i −1)ε. More precisely, let . p r,+ Tm,k (i, j) = sk (i, j) l r −1 (i, j)< p≤l r (i, j) , k

(4.47)

k

be the collection of k-subwaves issuing from (i, j) that are portions of a same split [sp] component sk r (i, j) of sk (i, j) (cfr. Definition 6), and set . ) r,+ + (i, j) = Tm,k (i, j), Tm,k

(4.48)

r

where the (possibly countable) union is taken over all split components of sk (i, j). By Proposition 2, consider for each r the family of k-subwaves ! " h . r,− Tm,k (i, j) = sk p (i p , j p ) r −1 lk

(i, j)< p≤lkr (i, j)

(i p ≥ m)

(4.49)

r,+ that are in one-to-one correspondence with the subwaves of Tk,m (i, j) and enjoy, for r −1 r every lk (i, j) m ⇒ i, (i p , j p ,k,h p ,i) ∈ /

q(i p , j p ,k,h p ,i ) = p,

[sp] (i p , j p ) ∈ Gm,k ∪ Gm,k , [sp] ∀ ip < i Gm,k ∪ Gm,k

(4.50) (4.51)

< i,

(4.52)

612


(, q being the maps in (4.1)). Then, set ! q " (i , j ,k,h ,i) . Tm,k, p (i, j) = sk p p p (i, (i p , j p ,k,h p ,i) ); i p ≤ i ≤ i , r (i, Tm,k

lkr (i, j)

)

. j) =

Tm,k, p (i, j),

p=lkr −1 (i, j)+1

− Tm,k (i,

(4.53)

. ) r,− j) = Tm,k (i, j), r

. ) r Tm,k (i, j) = Tm,k (i, j). r p

p

r,+ Given any pair of subwaves sk (i, j), sk (i, j) ∈ Tm,k (i, j), we shall denote by [i] τ ( i, j, k, p , p ) the minimum index i ∈ max {i p , i p }, . . . , i with the property that, for every i < i ≤ i, ∈ Z, there holds

( j p , h p ), ( j p , h p ) ∈ / I (i[i] ,k,i,) ×I (i[i] ,k,i,) , p

(4.54)

p

[i] [i] (letting I (m,k,i,) , I (m,k,i,) be the sets defined in (4.23)–(4.24), and adopting the notations of Proposition 2). [sp]

Remark 10. A maximal backward tree Tm,k (i, j) with vertex at a point (i, j) ∈ Gm,k , is associated to a collection of polygonal lines passing through the grid points where the subwaves of Tm,k (i, j) have emanated. Hence, with a slight abuse of notations, we will equivalently speak of Tm,k (i, j) as a collection of waves or as a collection of lines (or of points connected by the lines). Such a tree has two key properties. (i) Every (backward) branch Tm,k, p (i, j), lkr −1 (i, j) < p ≤ lkr (i, j), starts at the point [sp] (i, j) ∈ Gm,k , and terminates at the point (i p , j p ), which is either a point of Gm,k where the subwaves of Tm,k, p (i, j) are generated, or a point of the line t = mε, or [sp] a point of Gm,k , where a k-wave is split in subwaves of Tm,k (i, j) and possibly in other subwaves travelling through points outside of Tm,k (i, j). (ii) Every two (backward) branches Tm,k, p (i, j), Tm,k, p (i, j), lkr −1 (i, j) < p , p ≤ lkr (i, j), coincide on a polygonal line starting at (i, j), and after splitting at the point (τ p p , p , p ), where . τ p , p = τ [i] ( i, j, k, p , p ), . p , p = (i p , j p ,k,h p ,τ p , p ) = (i p , j p ,k,h p ,τ p , p ) ,

(4.55)

they can again possibly join together only at their terminal points if (i p , j p ) = (i p , j p ). Moreover, τ p , p i p , one has

[i] ( j p , h p ), ( j p , h p ) ∈ I (i ,k,τ p

p , p , p , p )

[i]

×I (i

p ,k,τ p , p , p , p )

.

(4.56)


613

The properties (i)-(ii) will be useful to analyze the variation of the interaction terms q q q q r (i, j), and to estimate the total (sk , sk ) for pairs of waves sk , sk belonging to Tm,k r (i, j). amount of such a variation in terms of the quantity of wave interaction within Tm,k p

p

r,+ (i, j), with (i, j) ∈ Indeed, consider a pair of k-subwaves sk (i, j), sk (i, j) ∈ Tm,k [sp]

Gm,k , lkr −1 (i, j) < p , p ≤ lkr (i, j), and, to fix the ideas, assume that i p ≥ i p . Then, by properties (i)-(ii) and recalling definitions (4.23)–(4.24), for every ∈ Z there holds [i] [i] / I (i ,k,i,) ×I (i ,k,i,) ∀ i p ≤ i ≤ i, i = τ p , p , ( j p , h p ), ( j p , h p ) ∈ p p (4.57) [sp] ( j p , h p ), ( j p , h p ) ∈ / I(i ,i ,k,i,) ∀ i p < i < i. p

p

For i p < i ≤ i, letting , q be the maps in (4.1), set . i = (i p , j p ,k,h p ,i) , . qi = q(i p , j p ,k,h p ,i) ,

. i = (i p , j p ,k,h p ,i) , . qi = q(i p , j p ,k,h p ,i) .

(4.58) (4.59)

Because of (4.57), and by definitions (4.27)–(4.30), we deduce that the overall variation q

q

of the interaction term (sk i (i, i ), sk i (i, i )) for i p ≤ i ≤ i, is equal to the sum of all the terms Osc k,i , Osc k,i , i p < i ≤ i, of the functionals (4.36)–(4.39) that corre . . spond to pairs of waves in T p = Tm,k, p (i, j), T p = Tm,k, p (i, j), plus the variation q q of (sk i (i, i ), sk i (i, i )) between i = τ p , p − 1 and i = i. In fact, in accordance with (4.27)–(4.30), consider the sets of pairs of indices IiF

* ) . (m,k,i,) ; = (( j , h ), ( j , h )) ∈ F(m,k,i,) ∪ F ∈Z q(m, j ,k,h ,i) (i, (m, j ,k,h ,i) ) ∈ T p, sk

sk (m, j

∈Z q(i , j ,k,h ,i) sk (i, (i , j ,k,h ,i) ) ∈ T p,

sk (i

q

,k,h ,i)

p

(i, (i , j ,k,h ,i) ) ∈ T

p

* ) . S(m,k,i,) ∪ S(m,k,i,) ; IiS = ((i , j , h ), (i , j , h )) ∈ q

, j ,k,h ,i)

" ,

(i, (m, j ,k,h ,i) ) ∈ T

(4.60) " ,

F that are associated to the terms Osc k,i , Osc k,i of the functionals G F n,m , G n,m and

S GS n,m , G n,m , related to pairs of subwaves in T Observe that, for every i p m,

∅ {((i p , j p , h p ), (i p , j p , h p ))}

if i p = m, if i p > m.

(4.61) (4.62)

614


. Next, set τ − = τ p , p − 1, and

. k (i, j; p , p ) =

⎧ % & q − q − ⎪ p p τ τ − − ⎪ ⎪ ⎨ sk (i, j), sk (i, j) − sk (τ , τ − ), sk (τ , τ − ) ⎪ ⎪ p p ⎪ ⎩ sk (i, j), sk (i, j)

(4.63)

if τ p , p > i p , if τ p , p = i p ,

& % q − q − p p , sk , sk , are defined with the same conventions adopted for where sk τ , sk τ the corresponding terms in (4.14)–(4.15). Then, assuming to fix the ideas that i p = m, and recalling definitions (4.13)–(4.15), property (4.50), and Remark 9, we deduce that, with the notation (4.63), there holds h h p p sk (i, j), sk (i, j) − sk p (m, j p ), sk p (m, j p ) = k (i, j; p , p ) + + Osc k,i ((m, j p , h p ), (m, j p , h p )) + + Osc k,i ((m, j p , h p ), (m, j p , h p )) + i p i p , 0 otherwise.

(4.65)

[sp] (i, j; p , p ) , (i, j) ∈ Gm,k , in (4.63) is bounded The total amount of variations k by (V (mε))2 , as shown in the next Lemma 2. For every compact set K ⊂ , there exist constants χ3 , c3 > 0 such that the following holds. Let u ε = u ε (t, x) be a Glimm approximate solution of (1.1), (1.6), assume that V (t) < χ3 , lim x→−∞ u ε (t, x) ∈ K , and consider a wave parti[sp] tion as in Proposition 2 for some fixed n ∈ N. Then, letting Gm,k be the set in (4.46), r,+ (i, j) in (4.47), and lkr (i, j) the index associated to the collection of subwaves Tm,k [sp] k (i, j; p , p ), (i, j) ∈ Gm,k , be the quantity defined in (4.63) (with max{i p , i p } in place of i p ), for every 0 < m < n there holds N

k=1 (i, j)∈G [sp] m,k

r

lkr −1 (i, j)< p , p ≤lkr (i, j)

k (i, j; p , p ) ≤ c3 · (V (mε))2 , (4.66)

[sp]r

where the third summation runs over all split components sk sk (i, j) issuing from (i, j).

(i, j) of the k-wave


615

Proof. The proof is given in three steps. Step 1. Consider the potential interaction functional introduced in [7]: 1 sα sβ . sα sβ + σα (ξ ) − σβ (ξ ) dξ dξ Q(t) = 4 0 0 kα xβ (t)

(4.67)

kα =kβ

(where xα (t) denotes the position of the wave sα of the kα th characteristic family in the approximate solution u ε (t), and σα (·) is the wave-speed map associated to sα ). By the analysis in [7, Sect. 4] it follows that, if V (t) is sufficiently small, Q(t) is decreasing at every interaction, and that the total amount of wave interaction taking place within a time interval ]mε, nε] is bounded by O(1) · [Q(nε+) − Q(mε+)]. Namely, if we let s1 , . . . , s N , s1 , . . . , s N denote the incoming waves taking part of the solution of the Riemann problem at a node (i, j) (in the same setting of Lemma 1), and define the amount of interaction at (i, j) as . Int(i, j) = |sα sβ | + J (sα , sα ) (4.68) 1≤α,β≤N α>β

1≤α≤N

(J (sα , sβ ) being the amount of interaction between sα , sβ defined as in (2.19)–(2.22), then there holds Int(i, j) = O(1) · Q(mε+) ∀ n > 0, (4.69) m 0. (4.71) m k and (n, j,k,h) ≥ (n, j ,k ,h ) ), there holds ⎡ ⎤ h s k (m, j) + skh (m, j) skh (m, j )⎦ u βR (t) − u βL (t) = O(1) · ⎣ β

j,k,h

= O(1) · m,n ϒ ,

CW

where the summand on the left-hand side runs over all secondary fronts in ψ(t), while the second summand on the right-hand side runs over all pairs of crossing primary waves in u ε . 6. All secondary fronts travel with speed 2, strictly larger than all characteristic speeds.


629

Step 2. Using the same arguments of [12, Sect. 5], relying on (1.12), (1.16), (5.3), (5.4), one can prove that S(n−m)ε ψ(mε) − ψ(nε) 1 L # $ 1 + log(n − m) = O(1) · m,n ϒ + + ε (n − m)ε, n−m ε u (nε) − ψ(nε) 1 = O(1) · m,n ϒ · (n − m)ε, (6.1) L where S(n−m)ε ψ(mε) is the semigroup trajectory of (1.4), with initial datum ψ(mε) = u ε (mε), evaluated at time t = (n − m)ε. Step 3. As in [12, Sect. 6], let T = mε + ε , for some m ∈ N, 0 ≤ ε < ε, and fix a positive constant ρ > 2ε. Then, we inductively define integers 0 = m 0 < m 1 < · · · < m κ = m with the following procedure. Assuming m i given: 1. if ϒ(m i ε) − ϒ ((m i + 1)ε) ≤ ρ, let m i+1 be the largest integer less than or equal to m such that (m i+1 − m i )ε ≤ ρ and ϒ(m i ε) − ϒ(m i+1 ε) ≤ ρ; . 2. if ϒ(m i ε) − ϒ ((m i + 1)ε) > ρ, set m i+1 = m i + 1. On every interval [m i ε, m i+1 ε] where Case 1 holds, we construct a piecewise constant approximation of u ε according to Step 1. Then, using (6.1) we derive ε u (m i+1 ε) − S(m −m )ε u ε (m i ε) 1 i+1 i L # $ 1 + log(m i+1 − m i ) = O(1) · m i ,m i +1 ϒ + + ε (m i+1 − mi) ε. (6.2) m i+1 − m i On the other hand, on each interval [m i ε, m i+1 ε] where Case 2 is verified, by the Lipschitz continuity of u ε and applying (1.16) we find ε u (m i+1 ε) − S(m −m )ε u ε (m i ε) 1 = O(1) · ε. (6.3) i+1 i L Hence, observing that the cardinality of both classes of intervals Cases 1-2 is bounded by O(1) · ρ −1 , from (6.2)–(6.3) we finally deduce % &$ # ε u (T ) − ST u 1 = O(1) · ρ + ε log ρ + ε 1 + 1 , L ρ ε ρ . √ which yields (1.15) choosing ρ = ε · log | log ε|. Acknowledgements. The authors wish to thank Tong Yang and an anonymous referee for having pointed out two inconsistencies in the previous version of the interaction potential functional presented in this paper.

References 1. Ancona, F., Marson, A.: A note on the Riemann Problem for general n × n conservation laws. J. Math. Anal. Appl. 260, 279–293 (2001) 2. Ancona F., Marson A.: Well-posedness for general 2 × 2 systems of conservation laws. Mems. Amer. Math. Soc. 169(801) (2004) 3. Ancona, F., Marson, A.: A wave front tracking algorithm for N × N non genuinely nonlinear conservation laws. J. Diff. Eqs. 177, 454–493 (2001) 4. Ancona, F., Marson, A.: Existence theory by front tracking for general nonlinear hyperbolic systems. Arch. Rat. Mech. Anal. 185(2), 287–340 (2007)

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5. Ancona, F., Marson, A.: A locally quadratic Glimm functional and sharp convergence rate of the Glimm scheme for nonlinear hyperbolic systems. Arch. Rat. Mech. Anal. 196(2), 455–487 (2010) 6. Bianchini, S.: On the Riemann problem for non-conservative hyperbolic systems. Arch. Rat. Mech. Anal. 166, 1–26 (2003) 7. Bianchini, S.: Interaction estimates and Glimm functional for general hyperbolic systems. Dis. Cont. Dyn. Syst. 9, 133–166 (2003) 8. Bianchini, S., Bressan, A.: On a Lyapunov functional relating shortening curves and viscous conservation laws. Nonlin. Anal. TMA 51(4), 649–662 (2002) 9. Bianchini, S., Bressan, A.: Vanishing viscosity solutions to nonlinear hyperbolic systems. Ann. Math. 161, 223–342 (2005) 10. Bressan, A.: The unique limit of the Glimm scheme. Arch. Rat. Mech. Anal. 130, 205–230 (1995) 11. Bressan A. Hyperbolic Systems of Conservation Laws - The one-dimensional Cauchy problem. Oxford: Oxford Univ. Press, 2000 12. Bressan, A., Marson, A.: Error bounds for a deterministic version of the Glimm scheme. Arch. Rat. Mech. Anal. 142, 155–176 (1998) 13. Bressan, A., Yang, T.: On the convergence rate of vanishing viscosity approximations. Comm. Pure Appl. Math. 57, 1075–1109 (2004) 14. Colombo, R.M.: On a 2 × 2 hyperbolic traffic flow model. Math. Comput. Modelling 35, 683–688 (2002) 15. Dafermos C.M.: Hyperbolic conservation laws in continuum physics. Berlin: Springer-Verlag (2000) 16. DiPerna, R.: Uniqueness of solutions to hyperbolic conservation laws. Indiana Univ. Math. J. 28, 137–188 (1979) 17. Glass, O., LeFloch, P.G.: Nonlinear hyperbolic systems: nondegenerate flux, inner speed variation, and graph solutions. Arch. Rat. Mech. Anal. 185(3), 409–480 (2007) 18. Glimm, J.: Solutions in the large for nonlinear hyperbolic systems of equations. Comm. Pure Appl. Math. 18, 697–715 (1965) 19. Hua, J., Jiang, Z., Yang, T.: A new Glimm functional and convergence rate of Glimm scheme for general systems of hyperbolic conservation laws. Arch. Rat. Mech. Anal. 196(2), 433–454 (2010) 20. Hua, J., Yang, T.: An improved convergence rate of Glimm scheme for general systems of hyperbolic conservation laws. J. Diff. Eqs. 231, 92–107 (2006) 21. Iguchi, T., LeFloch, P.G.: Existence theory for hyperbolic systems of conservation laws with general flux-functions. Arch. Rat. Mech. Anal. 168, 165–244 (2003) 22. Lax, P.D.: Hyperbolic systems of conservation laws II. Comm. Pure Appl. Math. 10, 537–566 (1957) 23. Liu, T.P.: The determnistic version of the Glimm scheme. Commun. Math. Phys. 57, 135–148 (1975) 24. Liu, T.P.: The Riemann problem for general 2 × 2 conservation laws. Trans. Amer. Math. Soc. 199, 89–112 (1974) 25. Liu, T.P.: The Riemann problem for general systems of conservation laws. J. Diff. Eqs. 18, 218–234 (1975) 26. Liu T.P.: Admissible solutions of hyperbolic conservation laws. Mems. Amer. Math. Soc. 30(240) (1981) 27. Liu, T.P., Yang, T.: Weak solutions of general systems of hyperbolic conservation laws. Commun. Math. Phys 230, 289–327 (2002) 28. Muracchini, A., Ruggeri, T., Seccia, L.: Mixture of Euler’s fluids and second sound propagation in superfluid helium. Z. Angew. Math. Phys. 57, 567–585 (2006) 29. Ruggeri, T., Muracchini, A., Seccia, L.: Continuum approach to phonon gasand shape changes of second sound via shock wave theory. Nuovo Cimento D. 16, 15–44 (1996) 30. Ruggeri, T., Muracchini, A., Seccia, L.: Second sound and characteristic temperature in solids. Phys. Rev. B 54, 332–339 (1996) 31. Yang, T.: Convergence rate of Glimm scheme for general systems of hyperbolic conservation laws. Taiwanese J. Math. 7, 195–205 (2003) Communicated by P. Constantin


Communications in


A Uniqueness Theorem for Stationary Kaluza-Klein Black Holes Stefan Hollands1 , Stoytcho Yazadjiev2 1 School of Mathematics, Cardiff University, Cardiff CF24 4AG, UK. E-mail: [email protected] 2 Department of Theoretical Physics, Faculty of Physics, Sofia University, 5 J. Bourchier Blvd., Sofia 1164,

Bulgaria. E-mail: [email protected] Received: 5 May 2009 / Accepted: 9 June 2010 Published online: 28 January 2011 – © Springer-Verlag 2011

Abstract: We prove a uniqueness theorem for stationary D-dimensional Kaluza-Klein black holes with D − 2 Killing fields, generating the symmetry group R × U (1) D−3 . It is shown that the topology and metric of such black holes is uniquely determined by the angular momenta and certain other invariants consisting of a number of real moduli, as well as integer vectors subject to certain constraints. 1. Introduction The classic black hole uniqueness theorems state that four dimensional, stationary, asymptotically flat black hole spacetimes are uniquely determined by their mass and angular momentum in the vacuum case, and by their mass, angular momentum, and charge in the Einstein-Maxwell case. The solutions are in fact given by the Kerr metrics in the first case and the Kerr-Newman metrics in the second. This was proven in a series of papers [1,2,23,24,30,38,48]; for a coherent exposition clarifying many important details and providing a set of consistent technical assumptions see [7]. The black hole uniqueness theorem is not true as stated in general spacetime dimensions D ≥ 5. For example, in D = 5 dimensions, there exist asymptotically flat, stationary vacuum black holes with the same mass and angular momenta, but with nonisometric spacetime metrics, and in fact even different topology [4,11–13,15,41,45]. One would nevertheless hope that a similar uniqueness theorem still applies if additional invariants (“parameters”) are specified beyond the mass and angular momenta. Unfortunately, except in the static case [19,49,50], such a classification result is not known, nor is it known what could be the nature of the additional invariants. In this paper, we study the special case of stationary vacuum black hole spacetimes in dimension D ≥ 4 with a compact, non-degenerate, connected horizon, admitting D − 3 commuting additional Killing fields with closed orbits. The spacetimes that we consider asymptote to a flat Kaluza-Klein space with 1, 2, 3 or 4 large spatial dimensions and a corresponding number of toroidal extra dimensions. For simplicity, we do not consider

632

S. Hollands, S. Yazadjiev

spacetimes with net “monopole charge”, see also footnote 3. Examples of such metrics have been given by [34,47]. We will first show how to associate certain invariants to such a spacetime consisting of a collection of “moduli” {li ∈ R>0 } and certain generalized “winding numbers” {a i ∈ Z D−3 }. The moduli may be thought of as the length of various rotation “axis” within the spacetime, whereas the winding numbers characterize the nature of the action of the D − 3 rotational symmetries near a given axis. The collection of these winding numbers uniquely characterizes the topology and symmetry structure of the exterior of the black hole, and we refer to it as the “interval structure” of the manifold. This analysis also implies that the horizon must be topologically the cartesian product a torus of the appropriate dimension and either a 3-sphere, ring (S2 × S1 ), or Lens-space L( p, q). Our notion of interval structure may be viewed as a generalization of what has been called “weighted orbit space” in the mathematics literature on 4-manifolds with torus action [43,44], but the latter notion does not involve the moduli {li }. Also, in the context of stationary black holes, a similar notion called “rod structure” was first considered by [20,21]; see [14] for the static case. The main difference between this and our notion is that our winding numbers are found to obey an integrality condition as well as certain other constraints, whereas there were no such constraints in [20,21]. The latter are a necessary and sufficient condition for the spacetime to have the structure of a smooth manifold with torus action. These topological considerations are described in detail in Sect. 3. We will then prove a uniqueness theorem which states that there can be at most one black hole spacetime with the same angular momenta and interval structure.1 Our uniqueness theorem generalizes a theorem proved in a previous paper [28] on asymptotically flat vacuum black holes in D = 5 dimensions; see also [29] for the Einstein-Maxwell case. The proof of the theorem proceeds in two steps: First, one reduces the full Einstein equations onto the space of symmetry orbits. Because the spacetime is assumed to have a total number of D − 2 independent commuting Killing fields, the space of symmetry orbits is two-dimensional—in fact it is shown to be a manifold with boundaries and corners homeomorphic to a half-plane. The parameters {li } are essentially the lengths of the various boundary segments. The arguments in the first step are topological in nature, and the only role of Einstein’s equations is to provide additional information about the fundamental group of the manifold via the topological censorship theorem [9]. That information is needed to rule out the presence of conical singularities in the orbit space.2 Our results in this part may be thought of as a generalization of [43,44] to a higher dimensional situation. The second step is to cast the reduced Einstein equations on the orbit space into a suitable form. Here, we make use of a formulation due to [35] involving certain potentials. The form of the equations leads to a partial differential equation for a quantity representing the “difference” between any two black hole metrics of the type considered which has been called “Mazur identity” [38]. Using this identity, one can prove the uniqueness theorem. The vectors {a i } and parameters {li } are important to treat the boundary conditions of this differential equation. The arguments in the second step are geometrical/analytical, and involve the use of Einstein’s equations in an essential way. 1 It has been brought to our attention that a conjecture in this direction had been made at the talk [22], see also [21]. 2 Here our analysis also fills a gap in our previous paper [28], where the absence of such conical singularities had to be assumed by hand.

Uniqueness Theorem for Stationary Kaluza-Klein Black Holes

633

The simpler case of a 5-dimensional spherical black hole with trivial interval structure was previously treated by a similar method in [37]. While our uniqueness theorem in higher dimensions is in some ways similar to the corresponding theorem in four dimensions, there are some notable differences. The first, more minor, difference is that higher dimensional black holes are not only classified by the mass and angular momenta, but in addition depend on the interval structure. In D = 4 the interval structure of a single black hole spacetime is trivial. A more substantial difference is that in D = 4 dimensions, the additional axial Killing field is in fact guaranteed by the rigidity theorem [6,16,24,39,46]. While a generalized rigidity theorem can be established in D dimensions [26,27,40], this theorem now only guarantees at least one additional axial Killing field. For the arguments of the present paper to work, we need however D − 3 commuting axial Killing fields. It does not seem likely that our theorem covers all asymptotically Kaluza-Klein, stationary black hole spacetimes in D dimensions. A third difference is that we have not been able so far to establish for which given set of angular momenta and interval structure there actually exists a regular black hole solution. The situation in this regard is in fact unclear even in five asymptotically large dimensions with no small extra dimensions. Here, solutions corresponding to various simple interval structures have been constructed. These include solutions with horizon topology S3 , S2 × S1 , L( p, q), which are the possible topologies allowed by our uniqueness theorem. However, by contrast with the cases S3 , S2 × S1 [11,13,41,45], the black holes with lens space horizon topology found so far [4,15] are not regular, and are thus actually not covered by our theorem. The situation is very different in four dimensions. Here the interval structure for single black hole spacetimes only involves the specification of a single parameter (related to the area of the horizon), and a regular black hole solution is known to exist for any choice of this parameter and the angular momentum—the corresponding Kerr solution. The mass, surface gravity, angular velocity of the horizon etc. of the solution can all be expressed in terms of these parameters.

2. Description of the Problem, Assumptions, Notations Let (M, g) be a D-dimensional, stationary black hole spacetime satisfying the vacuum Einstein equations, where D ≥ 4. The asymptotically timelike Killing field is called t, so £t g = 0. We assume that M has s + 1 asymptotically flat large spacetime dimensions and D − s − 1 asymptotically small extra dimensions, where s > 0. More precisely, we assume that a subset of M is diffeomorphic to the cartesian product of Rs with a ball removed—corresponding to the asymptotic region of the large spatial dimensions—and R × T D−s−1 —corresponding to the time-direction and small dimensions.3 We will refer to this region as the asymptotic region and call it M∞ . The metric is required to behave in this region like g = −dτ 2 +

s i=1

dxi2 +

D−s−1

dϕi2 + O(R −s+2 ),

(1)

i=1

3 In particular, we thereby exclude situations such as D = 5, s = 3, where the extra dimension T1 is fibered non-trivially over the sphere S 2 at infinity. Solutions of this kind have been given in [47]. The Euler class

of the fibration corresponds to a net “monopole charge”. It would be interesting to generalize our analysis to include monopole charge.

634


where O(R −α ) stands for metric components that drop off faster than R −α in the radial coordinate R =

x12 + · · · + xs2 , with k th derivatives in the coordinates x1 , . . . , xs drop-

ping off at least as fast as R −α−k . These terms are also required to be independent of the coordinate τ , which together with xi forms the standard cartesian coordinates on Rs,1 . The remaining coordinates ϕi are 2π -periodic and parametrize the torus T D−s−1 . The timelike Killing field is assumed to be equal to ∂/∂τ in M∞ . We call spacetimes satisfying these properties “asymptotically Kaluza-Klein” spacetimes.4 The domain of outer communication is defined by M = I + (M∞ ) ∩ I − (M∞ ) ,

(2)

where I ± denote the chronological past/future of a set. The black hole region B is defined as the complement in M of the causal past of the asymptotic region, and its boundary ∂ B = H is called the (future) event horizon. In this paper, we also assume the existence of D −3 further linearly independent Killing fields, ψ1 , . . . , ψ D−3 , so that the total number of Killing fields is equal to the number of spacetime dimensions minus two. These are required to mutually commute, to commute with t, and to have periodic orbits. The Killing fields ψi are referred to as “axial” by analogy to the four-dimensional case, even though their zero-sets are generically higher dimensional surfaces rather than “axis” in D > 4. We also assume that, in the asymptotic region M∞ , the action of the axial symmetries is given by the standard rotations in the cartesian product of flat Minkowski spacetime Rs,1 times the standard flat torus T D−s−1 . In other words, ψi = ∂/∂ϕi for i > [s/2] and5 ψ j = x2 j−1 ∂x2 j − x2 j ∂x2 j−1 for j = 1, . . . , [s/2] in M∞ . The group of isometries is hence G = R × T , where R corresponds to the flow of τ , and where T = T D−3 corresponds to the commuting flows of the axial Killing fields. Looking at the action of G on the asymptotic region, it is evident that an asymptotically Kaluza-Klein spacetime can have at most [s/2] + D − s − 1 commuting axial Killing fields. If this number is more than or equal to D − 3 as we are assuming, then s can be either 1, 2, 3 or 4. A more general class of spacetimes admitting G as their isometry group would be ones that are asymptotically the direct product Rs,1 × Y , where Y is a compact manifold of dimension D − s − 1. By the classification of manifolds with torus action given in [43,44] and Sect. 3 of this paper, one would have the following possibilities: 1. When s = 4, then Y is a (D − 5)-dimensional compact manifold admitting an action of T D−5 , hence Y = T D−5 . 2. When s = 3, then Y is a (D − 4)-dimensional compact manifold admitting an action of T D−4 , hence Y = T D−4 3. When s = 2, then Y is a (D − 3)-dimensional compact manifold admitting an action of T D−4 . The possibilities are summarized in Thm. 2, i.e. Y ∼ = S3 × T D−6 , S2 × D−5 D−3 D−6 ∼ T ,T , or Y = L( p, q) × T , where L( p, q) is a Lens space. 4. When s = 1 then Y is a (D − 2)-dimensional compact manifold admitting an action of T D−3 . The possibilities are again as summarized in Thm. 2, i.e. Y ∼ = S3 × T D−5 , S2 × T D−4 , T D−2 , or Y ∼ = L( p, q) × T D−5 , where L( p, q) is a Lens space. In this paper, we will treat explicitly only the first case, i.e. when the asymptotics of the spacetime is R4,1 × T D−5 (and D ≥ 5), but we will occasionally comment on the other 4 For the axisymmetric spacetimes considered in this paper, we will derive below a stonger asymptotic expansion, see Eq. (83). 5 The notation [x] means the largest integer n such that n ≤ x.


635

cases. The second case is rather similar, and the statement and proof of our main result would apply with minor changes. The third and fourth cases are qualitatively somewhat different. We are going to analyze the uniqueness properties of the asymptotically Kaluza-Klein spacetimes just described. Unfortunately, in order to make our arguments in a consistent way, we will have to make certain further technical assumptions about the global nature of (M, g) and the action of the symmetries. Our assumptions are in parallel to those made by Chru´sciel and Costa in their study [7] of 4-dimensional stationary black holes. The requirements are (a) that M contains an acausal, spacelike, connected hypersurface S asymptotic to the τ = 0 surface in the asymptotic region M∞ , whose closure has as its boundary ∂ S = H, a cross section of the horizon. We assume H to be compact and (for simplicity) to be connected. (b) We assume that the orbits of t are complete. (c) We assume that the horizon is non-degenerate. (d) We assume that M is globally hyperbolic. We will also assume (e) that the spacetime, the metric, and the group action are analytic, rather than only smooth. This will serve us to transfer information gathered about the metric in M to all of M, and it is also convenient to exclude certain pathologies of the action of isometries. This condition could be relaxed to smoothness without major difficulties. For the spacetimes described, one of the following two statements is true: (i) t is tangent to the null generators of H . In the asymptotically flat case, the spacetime must be static by the results of [51]. In the asymptotically Kaluza-Klein case, no such general result is known to our knowledge, but it is plausible that this statement might still hold true. (ii) t is not tangent to the null generators of H . In this case, the rigidity theorem [26,40] implies6 that there exists a linear combination K = t + 1 ψ1 + · · · + D−3 ψ D−3 , i ∈ R

(3)

so that the Killing field K is tangent and normal to the null generators of the horizon H , and g(K , ψi ) = 0 on H .

(4) κ2

From K , one may define the surface gravity of the black hole by = −(1/2) lim H (∇a K b )∇ a K b , and it may be shown that κ is constant on H [52]. In the first case (i), one can prove that the spacetime is actually unique [30], and in fact isometric to the Schwarzschild spacetime when D = 4. For higher dimensions, the same has been proven in the asymptotically flat case by [19,49,50]. We also expect a statement of this type to be true in the asymptotically Kaluza-Klein case with the Schwarzschild spacetime replaced by an appropriate generalization, but this is presently still open. In this paper, we will be concerned exclusively with the second case (ii), and we will give a uniqueness theorem for such spacetimes. Of particular importance for us will be the orbit space Mˆ = M/G, so in the next section we will look in detail at this space. 3. Analysis of the Orbit Space 3.1. Manifolds with torus actions. To begin, we consider a somewhat simpler situation, namely an orientable, analytic, compact connected Riemannian manifold of 6 The rigidity theorem was proved in these references only in the asymptotically flat case. However, to prove the relation (3), only a local analysis of the geometry at the horizon is needed, and the asymptotic conditions do not play a role for this.

636


dimension s ≥ 3, with a smooth effective7 action of the N -dimensional torus T = T N . Thus, we assume that Diff( ) contains a copy of T . Such actions have been analyzed and classified in the case s = 4 in a classic work by Orlik and Raymond [43,44], and— repeating many of their arguments—in [28]. Some of our arguments for general s are in parallel with this case, others are not. Essentially all of our arguments do not require the analyticity of the manifold or group action, and would also hold e.g. if the quantities were only of class C 1 , or even just C 0 , However, since our application will be to analytic spacetimes, we may as well assume this here. We may equip with a Riemannian metric h, and by averaging h with the action of T if necessary, we may assume that T acts by isometries of h. Later, will be a spatial slice of our physical spacetime (so that s = D − 1) and N will be taken to be D − 3, but for the moment this is not relevant. It will be useful to view the N -torus as the quotient R N / N , where N = (2π Z) N is the standard 2π -periodic N -dimensional lattice. Elements k ∈ T will consequently be identified with equivalence classes of N -dimensional vectors, k = [τ1 , . . . , τ N ] ∈ R N / N . The standard basis of N will be denoted b1 , . . . , b N , i.e., bi = (0, . . . , 0, 2π, 0, . . . , 0) , where the non-zero entry is in the i th position. Various facts about lattices that we will use in this section may be found in the classic monograph [3]. We denote the commuting Killing fields generating the action of T by ψi , i = 1, . . . , N . The flows of these vector fields are denoted Fiτ , and we assume that they are normalized so that the flows are periodic with period 2π , so Fi2π (x) = x for any x ∈ , and any i. The action of a group element k = [τ1 , . . . , τ N ] on a point is abbreviated by k · x = F1τ1 ◦ · · · ◦ FNτ N (x).

(5)

We also abbreviate the action of k on a tensor field T = Ta1 ...aq b1 ...br on by k · T = [F1τ1 ◦ · · · ◦ FNτ N ]∗ T , where the ∗ denotes the push-forward/pull-back of the tensor field. Because the Killing fields commute, we have in particular k · ψi = ψi for any k ∈ T . If ψ1 , . . . , ψ N are Killing fields as above, then so are ψˆ 1 , . . . , ψˆ N , where ⎞ ⎛ A11 . . . A N 1 N ⎜ .. ⎟ ∈ S L(N , Z). ψˆ i = (6) Ai j ψ j , ±A = ± ⎝ ... . ⎠ i=1 A1 N . . . A N N Another way of saying this is that we may conjugate the action of T = T N by the inner automorphism8 α A ([τ ]) = [τ A T ] of T , and the modified Killing fields ψˆ i generate the conjugated action. The freedom of choosing different 2π -periodic Killing fields to generate the action of T = R N / N is closely related to the possibility of choosing different bases in the lattice N , because any such change of basis is implemented by an integer matrix A with det A = ±1. As is standard, we define the orbit and the isotropy subgroup associated with a point by, respectively Ox = {k · x | k ∈ T },

I x = {k ∈ T | k · x = x}.

(7)

7 This means that if k · x = x for all x ∈ , then k is necessarily the identity. Given an action of the above type, one may always pass to an effective action by taking a quotient of T if necessary. 8 The automorphism property is α (kk ) = α (k)α (k ) for all k, k ∈ T . A A A


637

I x is a closed (hence compact) subgroup of T , and Ox is a smooth manifold that can be identified with T /I x . Being compact and abelian, I x must be isomorphic to Tn × Z p j . A more precise description of the action I x in an open neighborhood of x ˆ = {Ox | x ∈ } is called the factor space will be given below. The set of all orbits

ˆ = /T . It is not a manifold for general group actions. and is also written as

It will be useful to define the non-negative, symmetric N × N Gram matrix of the Killing fields, f i j = h(ψi , ψ j ).

(8)

It will also be convenient to distinguish points in according to the dimension of their orbit. For this, we define Sr = {x ∈ | dim Ox = r } = {x ∈ | rank[ f (x)] = r } = {x ∈ | dim I x = n = N − r }.

(9)

Evidently, n = N − r is also equal to the number of independent linear combinations of the Killing fields ψ1 , . . . , ψ N that vanish at points of Sr . Clearly, we have

=

N

Sr .

(10)

r =0

Lemma 1. Let ( , h) be a Riemannian manifold of dimension s, with N mutually commuting Killing fields ψi , i = 1, . . . , N . Let f i j be the Gram matrix, and let x be a point such that rank[ f (x)] = r . Then it follows that N − r ≤ [(s − r )/2]. Proof. Let Vx ⊂ Tx be the span of the Killing fields ψi |x , i = 1, . . . , N at x, and let Wx be the orthogonal complement. The assumptions of the lemma mean that the dimension of Vx is r , and that there exist N − r linear combinations of ψi |x , i = 1, . . . , N that vanish. By forming suitable linear combinations of the Killing fields, we may hence assume that span{ψi |x , i = 1, . . . , r } = Vx , and that ψi |x = 0, i = r + 1, . . . , N . Let D be the derivative operator of h, and let ti = Dψi |x , where i = r + 1, . . . , N . Then each ti is a linear map ti : Tx → Tx . The Killing equation implies that ti is skew symmetric with respect to the bilinear form h : Tx × Tx → R, i.e. h(ti X, Y ) = −h(X, ti Y ). Evaluating the D-derivative of the commutator [ψi , ψ j ] = 0 at x for r < i, j ≤ N then implies that the corresponding commutator ti t j − t j ti = 0 vanishes, too. Evaluating the derivative of the commutator [ψi , ψ j ] = 0 at x for r < i ≤ N and 0 < j ≤ r then furthermore shows that ti Vx = Vx , and consequently ti Wx = Wx . Now let us choose an orthogonal basis {e1 , . . . , es−r } of Wx , and use that to identify ti , r < i ≤ N with a linear map Rs−r → Rs−r . These linear maps must hence be skew symmetric, i.e., commuting elements of the Lie-algebra o(s − r, R). They must also be linearly independent. Indeed, assume on the contrary that a non-trival linear combination λ1 tr +1 +· · ·+λ N −r t N vanishes. Then both the Killing field s = λ1 ψr +1 +· · ·+λ N −r ψ N , as well as its derivative Ds vanish at the point x. It is a well-known property of Killing fields (see e.g. [52]) that a Killing field vanishes identically on a connected Riemannian manifold if it vanishes at a point together with its derivative. Hence, the Killing fields ψi , r < i ≤ N would be linearly dependent, a contradiction. Thus, we conclude that the linear maps ti , r < i ≤ N may be viewed as forming a (N − r )-dimensional abelian subalgebra of o(s − r, R). Any maximal abelian subalgebra of o(s − r, R) has dimension [(s − r )/2], so N − r ≤ [(s − r )/2].

638


In the situation considered later in this section, we have N = s − 2 Killing fields. The lemma then implies that the sets Sr are non-empty only for r = s − 2, s − 3, s − 4, so we have = Ss−2 ∪ Ss−3 ∪ Ss−4 . Our task will now be to construct, for each orbit Ox , an open neighborhood of it and a coordinate system in which we can explicitly understand the action of the group T . We will then be able to locally take the quotient of this neighborhood by T and thereby get a local description of the orbit space. By patching the local regions together, we will be able to characterize the manifold structure of the orbit space. Let x be an arbitrary but fixed point in Sr . Then the dimension of Ox is r , and the dimension of the isotropy group I x is n = N − r . As we have just seen, n may only take on the values 0, 1, . . . , [(s − r )/2]. We first show that if x ∈ Sr , there exists a matrix ±A ∈ S L(N , Z) such that the vector fields ψˆ i , 0 < i ≤ N defined as in Eq. (6) satisfy ψˆ i |x = 0, r < i ≤ N and such that ψˆ i |x , 0 < i ≤ r span the tangent space Tx Ox . We start our discussion with a general lemma. Lemma 2. Let L ⊂ T = T N be an n-dimensional closed subgroup. Then there are matrices of integers (Ai j )i,N j=1 and (vi j )ri, j=1 , where r = N − n and det A = ±1, with the property that L = α A (L0 × L1 ). Here L0 = {0r } × R N −r / N −r ,

(11)

L1 = (v −1 r )/r × {0 N −r },

(12)

where r has been identified with the lattice generated by b1 , . . . , br with origin denoted 0r , and where N −r has been identified with the lattice generated by br +1 , . . . , b N , with origin denoted 0 N −r . We have also written v −1 r for the lattice of Rr generated by r −1 j=1 (v )i j b j , where i = 1, . . . , r . Hence L0 is connected, L1 is finite, α α L1 ∼ = Z pα1 × · · · × Z pα M , |L1 | = p1 1 . . . p MM = | det(vi j )ri, j=1 |, 1

(13)

M

with p j > 0 prime. Proof. Let us first assume that L is also connected. Then L is a compact, abelian, connected Lie-group and so must be isomorphic to Tn . Let β : Tn → L be the isomorphism. We identify T = T N with R N / N , where N is the standard lattice. Similarly, N we identify Tn with Rn /n , with n = spanZ (bi )i=r +1 . Let a i = β(bi ) ∈ N , where i = r + 1, . . . , N . If λi ∈ R are such that c = λ1 a r +1 + · · · + λn a N = β(λ1 br +1 + · · · + λn b N ) ∈ N ,

(14)

then it follows that λi ∈ Z. We conclude from [3, Cor. 3, I.2.2] that there are vectors a 1 , . . . , a r ∈ N such that a 1 , . . . , a N form a basis of N . We now let A be the N × N matrix of integers such that bi A T = a i for i = 1, . . . , N . Then det A = ±1 because the matrix relates two bases of the lattice N . Since L0 viewed as a subgroup of T N is generated precisely by br +1 , . . . , b N , this proves the lemma when L is connected. In the general case, L is isomorphic to the cartesian product of a torus

and cyclic groups of order given by a prime power, i.e. there is an isomorphism β : Tn × Z pα j → j

L. For j = 1, . . . , M, let c j be the image under β of the generator of the j th cyclic finite group in the decomposition, projected onto the (real) span of a 1 , . . . , a r . The


639

vectors c1 , . . . , c M together with a 1 , . . . , a r generate an r -dimensional lattice r . Let γ 1 , . . . , γ r be a basis of the lattice r . It follows from [3, Thm. 1,I.2.2] that there are integers vi j such that vii > 0, vii > v ji for j > i, and a 1 = v11 γ 1 a 2 = v21 γ 1 + v22 γ 2

(15)

.. .

.. . a r = vr 1 γ 1 + vr 2 γ 2 + · · · + vrr γ r .

It is evident that L is given by the image under α A of the cartesian product of the group given by the real multiples of a r +1 , . . . , a N mod N and the group of integer multiples of γ 1 , . . . , γ r mod N . The first group is the image under α A of L0 , while the second is the image of L1 . This proves that L = α A (L0 × L1 ). From the system (15) one sees that the order of L1 is given by |L1 | =

r

vii = det(vi j )ri, j=1 .

i=1 n On the other hand, α −1 A ◦ β is an isomorphism between T ×

Z pα j and L0 × L1 .

j αj

The number of connected components of the first group is given by p j = | Z pα j |, while it is given by |L1 | for the second. This finishes the proof of the lemma.

.

j

We apply this lemma to the isotropy group I x ⊂ T , and we formulate the intermediate result as another lemma for future reference: Lemma 3. Let x ∈ Sr . There are integer matrices (vi j )ri, j=1 and (Ai j )i,N j=1 (depending on x) with det A = ±1 such that I x = α A (L0 × L1 ), with L0 and L1 the groups given above in Eq. (11). Alternatively, we can say that I x is generated by the elements ⎡ ⎞⎤ ⎛ N r 1 ⎝ k(τ1 , . . . , τ N ) := α A ⎣ τi bi + (v −1 )i j τi b j ⎠⎦ , (16) 2π i=r +1

i, j=1

where τi ∈ R for r + 1 ≤ i ≤ N , and where τi ∈ 2π Z for 1 ≤ i ≤ r . If we define ψˆ i = Ai j ψ j , then Lemma 3 implies that ψˆ i |x = 0 for i = r + 1, . . . , N , and ψˆ j |x span Tx Ox for j = 1, . . . , r . We now continue our analysis by inspecting the action of I x on the tangent space Tx . Let k ∈ I x . Then, because k · x = x, this induces a linear map k : Tx → Tx satisfying h(k · X, k · Y ) = h(X, Y ) for all X, Y ∈ Tx . In fact, because k · ψi = ψi for any of our Killing fields, it follows that k leaves each vector in the tangent space Tx Ox invariant. But then it also leaves the orthogonal complement Wx invariant. Let {e1 , . . . , es−r } be an orthogonal basis of Wx . So for every k ∈ I x , we get a representing orthogonal matrix (ki j ), 0 < i, j ≤ s − r acting on the orthognonal basis by k · ei = ki j e j . Because

is assumed to be orientable, we have a distinguished non-vanishing rank s totally antisymmetric tensor field (determined up to sign by a1 ...as b1 ...bs h a1 b1 . . . h as bs = s!). This tensor is invariant under the isometries of , so in particular k · = at point x,

640


for any k ∈ I x . Because k · ψi for any of our Killing fields, this implies that the action of k on Wx preserves the orientation, so the matrix (ki j ) representing this action has determinant det (ki j ) = +1, and (ki j ) ∈ S O(s − r ). In particular, (ki j ) must have an even number of −1 eigenvalues. The matrices (ki j ) commute for different choices of k ∈ I x , and so we may put them simultaneously into Jordon normal form. By making a change of basis of the {e1 , . . . , es−r } with an orthogonal element g ∈ O(s − r ), we may achieve that k · (e2 j−1 + ie2 j ) = eiθ j (e2 j−1 + ie2 j ), 0 < j ≤ [(s − r )/2] if s − r even (17) together with k · es−r = es−r when s − r is odd.9 The phases θ j depend on k. For the elements of the isotropy group given by Lemma 3, we have in fact k(0, . . . , 2π, . . . 0) · (e2 j−1 + ie2 j ) r = exp 2πi (v −1 )lm wm j (e2 j−1 + ie2 j ), 0 < j ≤ [(s − r )/2]

(18)

m=1

if s − r is even together with k(0, . . . , 2π, . . . 0) · es−r = es−r when s − r is odd. Here, the 2π is in the l th slot, with l ≤ r . The wi j are integers, which follows from the fact that the group elements k( j vi j b j ) are the identity, by Lemma 3. The above formula becomes somewhat more transparent if we note that the elements γ i = rj=1 (v −1 )i j b j defined for i = 1, . . . , r generate a copy of the isotropy subgroup I x ∼ = (v −1 r )/r ∼ =

∼ α j = γ mod r , see Lemma 2. Thus, we may view the exponential expression Z j p i j

in the above formula as a homomorphism ϑ j : (v −1 r )/r → S1 = {z ∈ C | |z| = 1}, ϑ j (γ k ) = e2πi

r

m=1 (v

−1 )

km wm j

. (19)

We also have k(0, . . . , τl , . . . 0) · (e2 j−1 + ie2 j ) = exp(iτl wl j )(e2 j−1 + ie2 j ) , 0 < j ≤ [(s − r )/2] (20) together with k(0, . . . , τi , . . . 0) · es−r = es−r when s − r is odd. Here, the τl is in the l th slot, and r + 1 ≤ l ≤ N . The wi j are again integers. As yet, the basis {e1 , . . . , es−r } has only been defined in Wx , but we now wish to define it for any W y , with y ∈ Ox . Let x(τ1 , . . . , τr ) = k(τ1 , . . . , τr , 0, . . . , 0) · x, 0 ≤ τi < 2π,

(21)

where k(τ ) is as in Lemma 3. Note that x(τ ) is periodic in τ with period 2π in each component of τ , and that τ ∈ [0, 2π )r → x(τ ) ∈ Ox provide (periodic) coordinates in Ox . We define our basis elements in Wx(τ ) by transporting {e1 , . . . , es−r } to x(τ ) with the group element in Eq. (21). We call this basis {e1 (τ ), . . . , es−r (τ )}. We note that this is still an orthonormal system, because it was obtained by an isometry between 9 Here it has been used that (k ) has determinant +1. Otherwise (k ) could also act as a reflection on an ij ij odd number of basis vectors.


641

Wx → Wx(τ ) . Note that this basis is not periodic in τ , by Eq. (17). To obtain an orthonormal basis {e˜1 (τ ), . . . , e˜s−r (τ )} that is periodic in τ , we set ⎛ ⎞ r e˜2 j−1 (τ ) + i e˜2 j (τ ) = exp ⎝−i τl (v −1 )lm wm j ⎠ (e2 j−1 (τ ) + ie2 j (τ )) , (22) m,l=1

for 0 < j ≤ [(s − r )/2], together with e˜s−r (τ ) = es−r (τ ) when s − r is odd. In an open neighborhood of Ox , we now define coordinates as follows. First, on Ox , we use the coordinates (ys−r +1 , . . . , ys ) ∈ [0, 2π )r → x(ys−r +1 , . . . , ys ). In a neighborhood of Ox we use ⎞ ⎛ s−r (y1 , . . . , ys ) → Expx(ys−r +1 ,...,ys ) ⎝ y j e˜ j (ys−r +1 , . . . , ys )⎠ . (23) j=1

Here, “Exp” is the exponential map for our metric h, i.e., (y1 , . . . , ys−r ) are Riemannian normal coordinates transverse to Ox . They cover an open neighborhood of Ox . From the construction of the coordinates, the action of the isometry group T in these coordinates is described by the following lemma: Lemma 4. Let x ∈ Sr , let (vi j ) be the matrix and k(τ1 , . . . , τ N ) ∈ I x be as in Lemma 3. Then, in terms of the coordinates (23) covering a neighborhood of Ox , the action of T is given by k(σ1 , . . . , σr , 0, . . . , 0) · (y1 + i y2 , . . . , ys−r −1 + i ys−r , ys−r +1 , . . . , ys ) ⎛ ⎡ ⎤ ⎞ r [(s−r )/2] = ⎝(exp ⎣i σl (v −1 )lm wm j ⎦ (y2 j−1 + i y2 j )) j=1 , (ys−r +i + σi )ri=1 ⎠ l,m=1

(24) when s − r is even. When s − r is odd, ys−r remains unchanged. Furthermore, k(0, . . . , 0, σr +1 , . . . , σ N ) · (y1 + i y2 , . . . , ys−r −1 + i ys−r , ys−r +1 , . . . , ys ) N [(s−r )/2] = (exp i σl wl j (y2 j−1 + i y2 j )) j=1 , (ys−r +i )ri=1

(25)

l=r +1

when s − r is even. When s − r is odd, ys−r remains unchanged. Let A be the matrix in Lemma 4, and let ψˆ i = j Ai j ψ j . By Lemma 4, the Killing ˆ fields ψi are related to the coordinate vector fields ∂ yi as: ⎛

⎞ ⎛ ψˆ 1 v11 ⎜ . ⎟ ⎜ . ⎜ . ⎟ ⎜ . ⎜ . ⎟ ⎜ . ⎜ ⎟ ⎜ ψˆ r ⎟ ⎜ vr 1 ⎜ ⎟=⎜ ⎜ ψˆ ⎟ ⎜ 0 ⎜ r +1 ⎟ ⎜ ⎜ . ⎟ ⎜ . ⎜ . ⎟ ⎜ ⎝ . ⎠ ⎝ .. 0 ψˆ N

. . . v1r w1 1 . . . . . . . . . vrr wr 1 . . . 0 wr +1 1 . . . . . . ... 0 wN 1

⎞⎛ w1 [(s−r )/2] ∂ ys−r +1 ⎟⎜ . . . . ⎟⎜ . . ⎟⎜ ⎟⎜ . . . wr [(s−r )/2] ⎟ ⎜ ∂ ys ⎟⎜ . . . wr +1 [(s−r )/2] ⎟ ⎜ y1 ∂ y2 − y2 ∂ y1 ⎟⎜ ⎟⎜ . . .. . ⎝ ⎠ . ys−r −1 ∂ ys−r − ys−r ∂ ys−r −1 . . . w N [(s−r )/2] ...

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(26)

642


when s − r is even. When s − r is odd, there is an analogous expression. Let us denote the N × (r + [(s − r )/2]) matrix in this expression as C. When N − r = [(s − r )/2], C is a square N × N matrix. Furthermore, each of the commuting, locally defined Killing fields ∂/∂ yi and y2 j−1 ∂/∂ y2 j − y2 j ∂/∂ y2 j−1 on the right side of the above equation is periodic, with period precisely 2π . Hence, when N − r = [(s − r )/2], the matrix C must have determinant ±1. So we get the condition −r det (vi j )ri, j=1 · det (w(r +i) j )i,N j=1 = det C = ±1.

(27)

Because both determinants on the left are integers, we conclude that they must be ±1. −r In view of Lemma 2, this means p1 = · · · = pr = 1 and det (w(r +i) j )i,N j=1 = ±1. We summarize our findings in another lemma: Lemma 5. Let ψ1 , . . . , ψ N be Killing fields as above, x ∈ Sr , n = N −r = [(s −r )/2]. Then p1 = · · · = pr = 1 (see Lemma 2), and det (w(r +i) j )i,n j=1 = ±1. Furthermore, in that case I x is connected. With the help of the above lemmas, we are now ready to analyze the orbit space ˆ in the case when N = s − 2. We first cover by the coordinate systems defined

in Eq. (23). Within each such coordinate system, we can then separately perform the quotient by T . We need to distinguish the cases n = 0, 1, 2, where n = s − 2 − r , and where the coordinate system covers a point x ∈ Sr . Case 0. For n = 0 and hence r = s − 2, the isotropy group I x is discrete and is isomor −1 phic to the group generated by the elements γ i = s−2 j=1 (v )i j b j , see Lemmas 2, 3. It

is also isomorphic to j Z pα j . Furthermore, by combining Lemmas 3 and 4, the action j

of these isotropy groups in a neighborhood of Ox can be written as k(0, . . . , 2π, . . . , 0) · (y1 + i y2 , y3 , . . . , ys ) = ϑ(γ j )(y1 + i y2 ), y3 , . . . , ys ,

(28)

where we are using the notation introduced in Eq. (19) for the homomorphism ϑ :

1 th α j Z p j → S , and where the “2π ” is in the j slot. Consider now the kernel ker ϑ. If j

g is an element in its kernel, then it is evident from the above formula that the corresponding isometry of acts by the identity both in a full neighborhood of Ox . Consequently, g must be the identity element of the group, since we are assuming the action to be effective. In particular, ϑ is injective. Consider next the image ran ϑ. This is a finite subgroup of the circle group S1 . Hence it is given by ran ϑ = {e2πik/q | k = 0, . . . , q − 1} ∼ = Zq for some q. It follows from the fact that ϑ is injective that

αj |ran ϑ| = q = | Z pα j | = pj . (29) j

j

j

Furthermore, it follows that the inverse ϑ −1 is a well-defined map on Zq , which can be viewed as taking values in the isotropy group I x ⊂ T . It follows from the discussion that, within the neighborhood considered, the quotient

a is modeled upon R2 /Zq , where q = p j j = | det v | (see Lemma 2, 3), and where the cyclic group of q elements acts on the coordinates y1 + i y2 by complex phases e2πi/q . Thus, in a neighborhood of Ox , the quotient space is an orbifold R2 /Zq . In particular,


643

we see that the orbits having non-trivial discrete isotropy group must be isolated points ˆ These orbits are also called “exceptional orbits”. The other orbits in case (0) have in . no isotropy group and are called “principal orbits”. Case 1. For n = 1, Lemma 5 applies and pi = 1 for all i and w(s−3)1 = ±1. We first factor by the group elements [0, . . . , 0, σs−2 ], see Eq. (24), and afterwards by the group elements [σ1 , . . . , σs−3 , 0], see Eq. (25). Then it is quite clear that the resulting quotient space of our neighborhood of Ox is locally modeled upon R × R>0 . The first factor

corresponds to the variable y3 , while the second factor to the variable y12 + y22 . w(s−3)1 w(s−2)1 = ±1. Case 2. For n = 2, Lemma 5 applies and pi = 1 for all i and w(s−3)2 w(s−2)2 We first factor by the group elements [0, . . . , 0, σs−3 , σs−2 ], see Eq. (24), and afterwards by the group elements [σ1 , . . . , σs−4 , 0, 0], see Eq. (25). Then it is quite clear that the resulting quotient space of our neighborhood of O x is locally modeled upon R>0 × R>0 . The first factor corresponds to the variable factor to the variable y12 + y22 . Thus, we have proven the following theorem:

y32 + y42 , while the second

Theorem 1. Let be a compact orientable connected s-dimensional Riemannian manifold (without boundary) with s − 2 pairwise commuting Killing fields generatˆ = /T ing an action of the group T = Ts−2 by isometries. Then the quotient space

is an orbifold with conical singularities, boundary segments, and corners. Thus, each ˆ has a neighborhood modeled on a neighborhood of the tip of a cone R2 /Zq , point of

on a half-space R × R>0 , or on a corner, R>0 × R>0 . In the first case, the corresponding isotropy group is finite and q is given by the order of this group. Each point of the boundary segments, corners, or orbifold points in is associated with an isotropy group I x as in Lemma 3. It follows from our discussion in Case 1) that, as long as we stay within one boundary segment, the isotropy group does not change. Furthermore, by Lemmas 5 and 3, the isotropy group I x is connected for points x associated with boundaries and corners. For x associated with conical singularities, I x is discrete, again by Lemmas 5 and 3. It also follows from our discussion of Cases 1) and 2) that, for each boundary segment and each corner, the isotropy group is completely characterized by an integer matrix A of determinant ±1. Furthermore, it follows from our discussion in Case 0) that the isotropy group I x is characterized by an integer q and an injective homomophism ϑ −1 : Zq → Ts−2 , whose image is I x . There is one such matrix A for each boundary segment, one for each corner, and one such q, ϑ −1 for each conical singularity. The matrices A are actually not completely characterized by the corresponding isotropy subgroup I x . In fact, by Lemma 2 (with L = I x , x ∈ Sr ) the position of the isotropy subgroup within T is uniquely determined by the class (N = s − 2) [A] ∈

S L(N , Z) , U (N − r, r ; Z)

(30)

where U (N − r, r ; Z) is the group of block-upper triangular matrices with block sizes N − r, r with integer entries and determinant ±1. The quotient by such matrices U takes into account the fact that left-multiplying an A by such a matrix gives the same isotropy subgroup. When N − r = n = 1 (corresponding to Case 1, and a boundary segment),

644


the class of A is determined by the last row (a N 1 , . . . , a N N ) of the matrix A, and we have a N i ψi |x = 0 for each point x in M corresponding to the boundary segment under consideration. When N − r = n = 2 (corresponding to Case 2, and a corner), the class of A is determined by the last two rows (a(N −1)1 , . . . , a(N −1)N ), (a1N , . . . , a N N ) up to a S L(2, Z) transformation acting on each column of the N × 2 matrix formed from these. We have a(N −1)i ψi |x = 0 and a N i ψi |x = 0 for each point x in

corresponding to the corner under consideration. ˆ is the collection of boundary segments, and if Ii j = Ii ∩ I j are the If {I j } ⊂ ∂

corresponding corners, then for each Ii , we have a vector a(Ii ) ∈ Z N which is the last row of the matrix A corresponding to that boundary segment. The greatest common divisor (g.c.d.) of the entries of the vector may be assumed to be equal to 1, g.c.d.{ai (I j ) | i = 1, . . . , D − 3} = 1.

(31)

For each corner Ii j , the corresponding vectors a(Ii ) and a(I j ) must be such that the N × 2 matrix formed from these vectors can be supplemented by N − 2 rows of integers to an S L(N , Z)-matrix, and this introduces a constraint on the pair a(Ii ), a(I j ). In the case s = 4 (i.e., N = 2), the constraint at each corner Ii j is simply that det (a(Ii ), a(I j )) = ±1. In general, the constraint on the vectors adjacent to a corner Ii j can be restated as follows applying [3, Lemma 2, I.2.3]: ˆ be the boundary segments. With each boundary segment Proposition 1. Let {I j } ⊂ ∂

there is associated a vector a(I j ) ∈ Zs−2 and ai (I j )ψi = 0 at the corresponding points of . At a corner Ii j = Ii ∩ I j , the vectors are subject to the constraint g.c.d. {Q kl | 1 ≤ k < l ≤ D − 3} = 1. Here, the numbers Q kl ∈ Z are defined by a (I ) ak (I j ) |. Q kl = | det k i al (Ii ) al (I j )

(32)

(33)

ˆ be the conical singularities. With each one, there is associated a natural Let {xî } ⊂

number qi > 1, specifying the type R2 /Zqi of the conical singularity, and a homomorphism ϑi−1 : Zqi → T , whose image is the discrete isotropy subgroup at xi = any point ˆ The collections {Ii } and {xî } are finite. in in the class xî ∈ . Remarks. (1) The data consisting of (i) the vectors {a(I j )}, (ii) the pairs {q j , ϑ −1 j }, ˆ (iv) the topological type of

ˆ (genus) has been called (iii) the orientation of , the “weighted orbit space” by Orlik and Raymond [43,44] for the case s = 4. Our proposition hence may be viewed as a generalization of their analysis to higher dimensions. ˆ is empty, then, as explained in detail in [43, Sect. 1.3], there are (2) If the boundary ∂

additional invariants associated with the T -space . These may be characterized as obstructions to lift certain cross sections on the boundaries of tubular neighborhoods of the orbifold-type orbits xî to and may be thought of as a class in the space m ˆ H 2 , Di2 ; Zs−2 ∼ = Zs−2 , i=1

where each Di2 is a disk around xî . This class has to be added to the data.

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645

Proof. It only remains to be shown that the collections {Ii } and {xî } are finite. Suppose ˆ is a compact manifold with e.g. that there was an infinite set of points {xî }. Because

boundaries and corners, there would then have to be a convergent subsequence with limit ˆ We claim that yˆ cannot be on ∂ . ˆ Indeed, the boundary ∂

ˆ consists of orbits for yˆ ∈ . which we have already constructed open neighborhoods with an action of T described above in Cases 1) and 2) after Lemma 5. Clearly these open neighborhoods thus do not contain any points with discrete isotropy group, and hence none of the xi . This shows ˆ We next show that I y ⊂ T , the isotropy subgroup of y, must be discrete that yˆ ∈ / ∂ . and non-trivial. Let {xîk } be the subsequence converging to yˆ , and let Tk := I xik be the corresponding isotropy subgroups. Clearly, each Tk is a discrete, non-trivial subgroup of the compact abelian group T . It easily follows from these general facts that there is a sequence gk ∈ Tk with limit g ∈ T not equal to the identity element, and it also easily follows that g ∈ I y . Hence the isotropy group of the point y is not empty, and because yˆ ∈ / ∂ , must be discrete. However, we may now construct a chart around the orbit of y as in Case 0) below Lemma 5, and this shows that there are no other points x with non-trivial isotropy subgroup in an open neighborhood of the orbit of y besides the orbit itself. In particular, xî cannot converge to yˆ , a contradiction. Hence {xî } is finite. The argument that the collection {Ii } is finite proceeds in a similar way. By a similar analysis we can also prove the following theorem on cohomogeneity-1 torus actions: Theorem 2. Let (H, γ ) be a connected, orientable, compact Riemannian manifold of dimension s − 1 > 1 with an isometry group containing an (s − 2)-dimensional torus T = Ts−2 . Then the orbit space Hˆ = H/T is diffeomorphic to a closed interval as a manifold with boundary, or to a circle. In the first case, we have the following possibilities concerning the topology of H: ⎧ 2 s−3 ⎪ ⎨S × T 3 s−4 ∼ H= S ×T ⎪ ⎩ L( p, q) × Ts−4

(35)

Here L( p, q) is a 3-dimensional Lens space. In the second case, H ∼ = Ts−1 . Proof. Let ψi , i = 1, . . . , s −2 be the commuting Killing fields of period 2π generating the action of T on H. In the decomposition H = ∪Sr defined as in Eqs. (9), (10), only the sets with r = s − 1 and r = s − 2 may be non-zero, by Lemma 1. We consider these cases separately. Case 0). Let x ∈ Ss−1 , and let Tx H = Tx Ox ⊕ Wx be the orthogonal decomposition into vectors tangent to Ox and those orthogonal to Ox . By assumption, the dimension of Wx is one. If k ∈ I x is in the isotropy group, then it leaves Tx Ox invariant, as k · ψi = ψi for all i. So k acts as ±1 on Wx . But k also preserves the rank (s − 1) anti-symmetric tensor compatible with the metric, which exists since H is orientable. So k acts as +1 on Wx , and hence as the identity on Tx H. The action of k must hence leave invariant any piecewise smooth geodesic on (H, γ ) through x, and therefore k must act as the identity on all of H, since this is a connected manifold. Thus, the isotropy group I x is trivial

646


in Case 0). Consequently, near Ox , Hˆ = H/T has the structure of a 1-dimensional manifold, i.e., an open interval. Case 1). Let x ∈ Ss−2 . By exactly the same arguments as given above using Lemmas 4 and 5, the action of T is given near Ox in local coordinates (y1 , . . . , ys−1 ) by k(σ1 , . . . , σs−2 ) · (y1 + i y2 , y3 , . . . , ys−1 ) s−2 = exp i wl σl (y1 + i y2 ), y3 + σ1 , . . . , ys−1 + σs−3 .

(36)

l=1

Here, ±A is some S L(s − 2, Z) matrix, thenumbers wl are integers, and ws−2 = ±1 (see Lemma 5). It is evident from this that y 2 + y 2 furnishes a coordinate for Hˆ in a 1

2

neighborhood of Ox , thus identifying this neighborhood locally with a half-open interval. Because Hˆ can be covered by neighborhoods of the kind described in Cases 0) and 1), i.e., open and half open intervals, and because Hˆ is compact in a natural topology and connected, it follows that Hˆ must be a 1-dimensional connected compact manifold with or without boundaries. In the first case, Hˆ is diffeomorphic to a closed interval, in the second case to a circle. In the first case, the two boundary points of this closed interval correspond to orbits Ox respectively O y in H, where an integer linear combination ai,1 ψi respectively ai,2 ψi vanishes. We can redefine our action of T using Ai j ψ j for some integer matrix A with det A = ±1 instead the Killing fields ψˆ i = in such a way that on Ox we have ψˆ 1 = 0, while on O y we have p ψˆ 1 + q ψˆ 2 = 0. Consider now the subgroup L ⊂ T generated by ψˆ 3 , . . . , ψˆ s−2 . Clearly, L is isomorphic to Ts−4 . It follows from the discussion of the Cases 0) and 1) that there are no points in H which are fixed under a non-trivial element of L, so H ∼ = (H/L) × Ts−4 . Then, H/L is a three-dimensional manifold on which there acts the subgroup of isometries in T generated by ψˆ 1 , ψˆ 2 . It is not difficult to see, and argued carefully in [28], that H/L is isomorphic to S3 if ( p, q) = (0, 1), isomorphic to S2 × T1 if ( p, q) = (1, 0), and a Lens-space L( p, q) otherwise. In the second case, H must be diffeomorphic to the direct product of T and a circle, i.e. to Ts−1 . 3.2. The fundamental group of . In the previous section, we have analyzed oriented s-dimensional manifolds with an effective action of T = Ts−2 . We showed that ˆ = /T was an orientable 2-manifold with a finite number of the quotient space

conical singularities in the interior, and with boundaries and corners. With each of the ˆ there was associated an integer qi ∈ Z and an injective conical singularities xî ∈

−1 homomorphism ϑi : Zqi → T . These homomorphisms may be written as 2πi/q j ϑ −1 ) = (e2πi p1, j /q j , . . . , e2πi ps−2, j /q j ), j (e

(37)

where g.c.d.{q j , g.c.d.{ p1, j , . . . , ps−2, j }} = 1. Furthermore, with each of the boundary intervals Ii ⊂ ∂ , there was associated a vector a i = (a1,i , . . . , as−2,i ) ∈ Zs−2 . On a ˆ is corner, the vectors are subject to the constraint (32), (33). If is compact, then

a compact oriented 2-dimensional topological manifold, and hence topologically of the form


ˆ ∼ ˆ g\

=

d

647

D 2j ,

(38)

j=1

ˆ g is a closed Riemann surface of where each D 2j is a 2-dimensional disk, and where

genus g. One can show that the manifold with T -action is fixed up to equivariant isomorˆ {Ii }, {xî }, {qi , p }, {a i }; we will indicate how to prove phism by the data consisting of , i this in Subsect. 3.3. Therefore, any topological invariant of must be expressible in terms of these data. It is evident that the fundamental group π1 ( ) should provide a strong invariant for the topology of . It is given in the next theorem: Theorem 3. Let be a compact orientable manifold with an effective action of T = ˆ = ∅. Then the fundamental group can be presented as: Ts−2 such that ∂

π1 ( ) = k1 , . . . , ks−2 , d1 , . . . , dc , h 1 , . . . , h d , m 1 , . . . , m g , l1 , . . . , l g [m 1 , l1 ] · · · [m g , l g ] · d1 · · · dc · h 1 · · · h d ; [m i , k j ] ; [li , k j ] ; [di , k j ] ; [h i , k j ] ; [ki , k j ] ; a

a

a

a

s−2,1 s−2,b , . . . , k1 1,b · · · ks−2 ; k1 1,1 · · · ks−2

ps−2,1 ps−2,c q p q p . d1 1 k1 1,1 . . . ks−2 , . . . , dc c k1 1,c . . . ks−2

(39)

Here, we are using the usual notation for a finitely generated group in terms of its relations, and [x, y] = x yx −1 y −1 is the commutator of group elements. Above, g is the ˆ c is the number of conical singularities, b is the number of number of handles of , ˆ homeomorphic to intervals {Ii } , and d is the number of boundary components in ∂

circles, see Eq. (38). Proof. The proof is essentially an application of the Seifert-Van Kampen theorem, which is described e.g. in [36, Chap. 4]. Let x ∈ be any point with trivial isotropy group, and let ki , i = 1, . . . , s − 2 be the closed loops obtained by a applying the i th generator of π1 (T ) (=generator of the i th copy of T1 in Ts−2 ) to x. Let di , i = 1, . . . , c be lifts ˆ and let h i , i = 1, . . . , d be lifts of loops going around the i th conical singularity in ∂ , ˆ (=boundary component in ∂ ). ˆ We cut out a of loops going around the i th hole of

ˆ we cut out a small neighsmall disk Di2 around each of the conical singularities in , ˆ and we consider the corresponding subset of . This borhood of the boundary in , subset will have a homotopy group generated by k1 , . . . , ks−2 , d1 , . . . , dc , h 1 , . . . , h d , ˆ The relations are and generators m 1 , l1 , . . . , m g , l g corresponding to the g handles of . [m 1 , l1 ] · · · [m g , l g ] · d1 · · · dc · h 1 · · · h d ; [m i , k j ] ; [li , k j ] ; [di , k j ] ; [h i , k j ] ; [ki , k j ].

(40)

We now glue back in the neighborhood of the boundary. Since, near the i th boundary a as−2,i segment Ii , the generator k1 1,i · · · ks−2 shrinks to zero size, we receive the relations a

a

a

a

s−2,1 s−2,b k1 1,1 · · · ks−2 ; . . . ; k1 1,b · · · ks−2

(41)

via the Van Kampen theorem. We finally glue in the disks around the conical singularities, each of which corresponds to a tube D 2 ×Ts−2 . We must perform the gluing in such

648


a way that the standard action of T on D 2 × Ts−2 matches up with the action of T on

near the exceptional orbits. This action is characterized by the homomorphism (37) for the j th tube; we receive the relations q

p

p

q

p

p

s−2,1 s−2,c ; . . . ; dc c k1 1,c . . . ks−2 d1 1 k1 1,1 . . . ks−2

from this operation, again via the Van Kampen theorem.

(42)

The theorem has an interesting corollary in s = 4 if the action of T has a fixed point, i.e. when the orbit space has a corner. The vectors associated with the intervals Ii , Ii+1 adjacent to the corner, a i , a i+1 , must then satisfy det(a i , a i+1 ) = ±1 [see Eq. (32)]. This imposes the relation k1 = k2 = e in Eq. (39). Then, if π1 ( ) = 0, this will imply that g = d = 0, and q1 , . . . , qc = 0. In other words, if s = 4, if the action has fixed point, and if is simply connected, then there are no conical singularities, i.e., exceptional orbits. This generalizes a result first proved using methods from singular cohomology in [43]. The above theorem has another related corollary which will be relevant below in our ˆ be any disk in the interior of the application to the structure of black holes. Let D 2 ⊂

orbit manifold not intersecting any of the boundaries or conical singularities. Thus, the orbits are all (s − 2)-dimensional tori, with no fixed points. The inverse image of D 2 in

is homeomorphic to D 2 × Ts−2 , with T acting on the second factor. Let us denote the generators of π1 (D 2 × Ts−2 ) by k1 , . . . , ks−2 , which are the s − 2 generators of π1 (Ts−2 ) = Zs−2 . Without loss of generality, we may assume that k j are the image of the paths generated by the action of the j th copy on T = Ts−2 on a point x ∈ D 2 ×Ts−2 . From the inclusion f : D 2 × Ts−2 → , we get a corresponding homomorphism f ∗ : π1 (D 2 × Ts−2 ) → π1 ( ). The way we have set things up, we may assume that f ∗ (k j ) = k j , using the same notation and assumptions as in the above Theorem 3. Lemma 6. If f ∗ : π1 (D 2 × Ts−2 ) → π1 ( ) is surjective, then we have g = d = 0, ˆ is a topologically a disk, and there are no conical q1 = · · · = qc = 1. In other words,

singularities. Proof. Using Eq. (39) and the formula f ∗ (k j ) = k j , we see that f ∗ π1 (D 2 × Ts−2 ) is a normal subgroup of π1 ( ). By assumption, the factor group π1 ( )/ f ∗ π1 (D 2 × Ts−2 ) is trivial. From the quotient, the group π1 ( ) [see Eq. (39)] receives the additional relations k j = e for j = 1, . . . , s − 2. This means that the factor group is isomorphic to π1 ( )/ f ∗ π1 (D 2 × Ts−2 ) ∼ = d1 , . . . , dc , h 1 , . . . , h d , m 1 , . . . , m g , l 1 , . . . , l g

q q [m 1 , l1 ] · · · [m g , l g ] · d1 · · · dc · h 1 · · · h d ; d1 1 ; . . . ; dc c .

(43) Evidently, this group is non-trivial unless g = d = 0, q1 = · · · = qc = 1, from which the lemma follows. 3.3. Model spaces, examples. In the previous sections, we showed how a closed oriented manifold of dimension s with an action of T = Ts−2 gives rise to a number ˆ see Prop. 1, and Thm. 3. In this of invariants and decoration data on an orbit space , section we will outline to what extent the converse of these data determine the original manifold with T -action. In other words, given another such manifold , does there


649

exist a diffeomorphism h : → , and an automorphism α A : T → T such that h(k · x) = α A (k) · h(x) for all x ∈ , k ∈ T ? As shown in the case s = 4 in [44, Para. I], ˆ = 0, the decoration the answer to this question is in the affirmative. (In the case that ∂

data must include also the invariant mentioned in remark (2) after Prop. 1.) The proof of this theorem really extends straightforwardly to the case of with arbitrary dimension, so we will not describe it here in detail. ˆ and given decoration data as described in A related question is whether for a given

Prop. 1, we can find a corresponding manifold with T -action described by these data. The question is again in the affirmative, and we now outline how one can construct such ˆ which is an oria manifold. Thus, let us assume that we are given (i) an orbit space

ented two-dimensional manifold with boundaries, corners, and conical singularities, (ii) ˆ satisfying the constraints (32), (iii) a vectors {a(I j )}, one for each component I j ⊂ ∂ , ˆ as described in around (37). collection {qi , pi }, one for each conical singularity xî ∈ , We want to construct a corresponding manifold with T -action. ˆ is a half-plane R>0 × R, with finitely many coniFor simplicity, let us assume that

cal singularities in the interior, and with boundary divided into the segments I1 , . . . , Ib . We first consider the conical singularities in the interior. We may assume that they are all in a disk D 2 ⊂ R>0 × R. We cut out this disk, and we consider D 2 × T with standard action of T on the second factor. We cut out from this region c tubes of the form Di2 × T , with each Di2 a small disk containing the i th of the c conical singularities. Near the conical singularities, we would like the T -action to be described by the homomorphisms ϑi−1 : Zqi → T given in Eq. (37). A model space for this action is Di2 ×ϑ −1 T , i

Di2 = {z ∈ C | |z − z i | ≤ 1},

(44)

where g ∈ Zqi ⊂ S1 acts on the disk by multiplication with the complex phase. We glue in these model spaces along the boundaries where we cut out the c tubes Di2 × T with diffeomorphisms h i : ∂(Di2 ×ϑ −1 T ) → ∂(Di2 × T ) in such a way that the T -actions i

match up. We call the manifold with boundary obtained from D 2 × T in this way 0 . We now construct a second T -space 1 that incorporates the data {a(I j )}. These data were constructed above by giving, for each orbit, a neighborhood together with a set of coordinates in which the action of T was explicitly given. It is intuitively clear that we can turn this around and define 1 to be the collection of these coordinate charts with corresponding T -action, and we now briefly explain how this can be done. For simplicity and concreteness, we consider explicitly the case when s = dim = 4. The construction is well-known in topology and is sometimes called “linear plumbing”, see [25]. We present the construction in such a way that the generalization to general s should be fairly obvious, details will be given in [8]. The construction of 1 is as follows. Let b ≥ 2 be the number of boundary segments {I j }. On the boundary S 3 of the four-dimensional solid ball B 4 = {y12 + y22 + y32 + y42 < 1}, we consider the disjoint subsets 3 S+ := {(y1 , y2 , y3 , y4 ) ∈ S | y32 + y42 < 1/4}, (45) S− := {(y1 , y2 , y3 , y4 ) ∈ S3 | y12 + y22 < 1/4}. Both of these subsets are topologically solid tori. We consider the disjoint union of b − 1 copies of the solid ball B 4 , and on the i th copy we define an action of T = T2 generated

650


by the two 2π -periodic vector fields ψ1 , ψ2 given by y1 ∂ y2 − y2 ∂ y1 a1 (Ii ) a2 (Ii ) ψ1 = . a1 (Ii+1 ) a2 (Ii+1 ) ψ2 y3 ∂ y4 − y4 ∂ y3

(46)

The consistency condition on the i th corner (33), (32) guarantees that the determinant of the above matrix is ±1. We wish to glue the S+ -part of the boundary of the i th copy of the ball B 4 to the S− -part of the boundary of the (i + 1)th copy in such a way that the actions of T on these copies are compatible. It is not difficult to see that this is achieved if we identify these parts by the maps f i : S− → S+ defined by f i (y1 , y2 , y3 , y4 ) = (y3 , y4 , y1 sin(n i ϕ) + y2 cos(n i ϕ), y1 cos(n i ϕ) − y2 sin(n i ϕ)) , (47) where ϕ = arctan yy43 and n i = a1 (Ii )a2 (Ii+2 )−a2 (Ii )a1 (Ii+2 ), i.e. we have f i ∗ ψ1 = ψ1 and f i ∗ ψ2 = ψ2 . Thus, for b > 2 we define10

1 = (. . . ((B 4 ∪ f1 B 4 ) ∪ f2 B 4 ) · · · ∪ fb−3 B 4 ) ∪ fb−2 B 4 .

(48)

For b = 2 we define 1 = B 4 . The space 1 has a 3-dimensional boundary whose structure is determined by the first and last vector a(I1 ), and a(Ib ). It is either T1 × S2 , S3 , or a lens space L( p, q), see Thm. 2. We may cut out from 1 a tube D 2 × T , and glue the boundary obtained in this way onto ∂ 0 . The manifold obtained in this way is the desired T -space in the special case considered. The general case may be treated in a similar way, as we will discuss in a future paper [8]. We may call the manifold constructed from the decoration data ˆ {a(Ii )}, {qi , p }], where is an orientation, and

ˆ an oriof the orbit space X [, , i ented two-dimensional manifold with boundaries and corners. We give some examples (without conical singularities): ˆ = D 2 , ∂ D 2 = I1 ∪ I2 ∪ I3 , and consider the Example 1. (from [43]) Let s = 4,

data {(1, 0), (0, 1), (1, 1)}. Then the space X [D 2 , {(1, 0), (0, 1), (1, 1)}] is the complex projective space CP 2 = C3 / ∼, where the equivalence relation is (z 1 , z 2 , z 3 ) ∼ (λz 1 , λz 2 , λz 3 ) and the action of T = T2 is [τ1 , τ2 ]·(z 1 , z 2 , z 3 )∼ = (eiτ1 z 1 , eiτ2 z 2 , z 3 )∼ . The equivalence X [D 2 , {(1, 0), (0, 1), (1, 1)}] ∼ = CP 2 can be seen e.g. by noting that the axis in CP 2 corresponding to the vectors (1, 0), (0, 1), (1, 1) are given by the set of points (z 1 , z 2 , z 3 )∼ ∈ CP 2 such that, respectively, z 1 = 0, z 2 = 0, z 3 = 0. ˆ = D 2 and consider the data {(1, 0), (0, 1), (1, 0), (0, 1)} (four Example 2. Let s = 4,

intervals). Then the space X [D 2 , {(1, 0), (0, 1), (1, 0), (0, 1)}] is S2 × S2 , with the standard action of T . This is easily seen by considering the isotropy groups of the action. In fact, Examples 1 and 2 constitute in some sense the most general case in s = 4 because one can show that [43,44], topologically, is a connected sum of projective spaces on S2 × S2 ’s in the situation under consideration. ˆ = D 2 and consider the data {(1, 0, 0), (q1 , q2 , p), (0, 1, 0)}. Example 3. Let s = 5,

The constraints on the corners are fulfilled if we have g.c.d.( p, q1 ) = 1 = g.c.d.( p, q2 ). The corresponding space X [D 2 , {(1, 0, 0), (q1 , q2 , p), (0, 1, 0)}] is a generalized lens 10 If X, Y are sets and f is a map f : A ⊂ X → Y , then X ∪ Y is the set defined as the quotient of the f disjoint union X ∪ Y by the equivalence relation x ∼ y :⇔ (x, y) ∈ graph f .


651

space L( p; q1 , q2 ). The generalized lens space is defined as the quotient of S5 (realized as the unit sphere in C3 ) by the discrete subgroup of isometries of order p generated by an element λ acting as λ · (z 1 , z 2 , z 3 ) = (e2πi/ p z 1 , e2πiq1 / p z 2 , e2πiq2 / p z 3 ). The action of T = T3 on an equivalence class (z 1 , z 2 , z 3 )∼ ∈ L( p; q1 , q2 ) under this action is [τ1 , τ2 , τ3 ] · (z 1 , z 2 , z 3 )∼ = (eiτ3 / p z 1 , ei(τ1 +q1 τ3 / p) z 2 , ei(τ2 +q2 τ3 / p) z 3 )∼ .

(49)

The axis corresponding to the vectors (1, 0, 0), (q1 , q2 , p), (0, 1, 0) are, respectively, z 2 = 0, z 2 = z 3 = 0, z 3 = 0. Note that π1 (L( p; q1 , q2 )) ∼ = Z p , so for p = 1 this space is not simply connected. ˆ be as in the previous example, but let the data now be Example 4. Let s,

{(1, 0, 0), (q1 , q2 , p), (0, 1, 0), (1, 1, 0)}. The constraints on the corners are fulfilled if we have g.c.d.( p, q1 ) = 1 = g.c.d.( p, q2 ). The manifold in question is now topologically (combining Examples 1 and 3) X [D 2 , {(1, 0, 0), (q1 , q2 , p), (0, 1, 0), (1, 1, 0)}] ∼ = L( p; q1 , q2 )#(CP 2 × S1 ). (50) 3.4. The orbit space of the domain of outer communication. We next want to determine the orbit space of a D-dimensional asymptotically Kaluza-Klein stationary black hole spacetime (M, g) with D − 3 axial Killing fields ψi , i = 1, . . . , D − 3 generating an (effective) action of T = T D−3 . Thus, the total group isometries is G = T × R, with R the additive group generated by the asymptotic timelike Killing field t. The asymptotic behavior of the spacetime is assumed, as always, to be given by R4,1 × T D−5 . We have the following theorem: Theorem 4. Let (M, g) be a stationary, asymptotically Kaluza-Klein, D-dimensional vacuum black hole spacetime with isometry group G = R × T , satisfying the technical assumptions stated in Sect. 2. Then the orbit space Mˆ = M/G of the domain of outer communication is a 2-dimensional manifold with boundaries and corners homeomorphic ˆ The possibilities to a half-plane. In particular, there are no conical singularities in M. for the horizon topology are Eqs. (35), with s = D − 1. One of the boundary segments I j ⊂ ∂ Mˆ is the quotient of the horizon Hˆ = H/G, while the remaining I j correspond to the various “axis”, where ai (I j )ψi = 0. The vectors a(I j ) ∈ Z D−3 are subject to the constraint (32) on each corner Ii ∩ I j . Remarks. 1) In the statement concerning the horizon topology, Eq. (35), we do not mean that the torus factors (such as in H ∼ = S2 × T D−4 ) correspond to the rotations in the extra dimensions near infinity. 2) If the asymptotic behavior of the spacetime is instead R3,1 × T D−4 , then the statement and proof of the theorem remains more or less unchanged. However, if it is R2,1 ×T D−3 , then the orbit space is no longer homeomorphic to an upper half-plane, but instead is a plane minus a disk. We will not discuss this further here. Proof. The “structure theorem” 4.3 of [7] states that M contains a smooth, spacelike, acausal slice whose boundary is a cross section H of the horizon, which is asymptotic to a τ = const. slice in the exterior under the identification of the exterior with (part of) Rs,1 ×T D−s−1 , see Eq. (83). Furthermore, is invariant under the action of T = T D−3 ,

652


and it is transversal to the orbits of t represented by the factor R in G. Also, if F τ is the flow of t, then M = ∪τ F τ ( ). This result is going to allow us to reduce the proof of Thm. 4 to Thm. 1, and to use Lemma 6. We first factor M by R. Then, because M = ∪τ F τ ( ), we can identify the resulting space with ( , h), with h the Riemannian metric induced from g. Evidently, T acts as a group of isometries on ( , h). Asymptotically, h approaches the standard flat metric on R4 × T D−5 , and the T -action is, by assumption, the standard action on this space in the asymptotic region. Namely, the action of T = T2 × T D−5 is the product of the action of T2 on R4 by rotations in the 12- and 34-plane, and of the action of T D−5 on itself. It also leaves the horizon H = −∂ invariant, because this is the boundary. We would like to apply the classification results derived in the previous sections to the manifold with T -action . However, there we assumed that is closed, and this is now evidently not the case. In fact, has an inner boundary H, and a (conformal) boundary at infinity. But we can reduce this case to the one which we have discussed by passing to the closed manifold

= ∪ X ∪ ({pt} × T ),

(51)

which is obtained by gluing in a suitable manifold with boundary ∂ X = H with T -action along the horizon, and another one at the end at infinity. These manifolds are glued across the boundaries in such a way that the resulting compactified space is orientable and carries a smooth action of T . It is quite obvious how this should be done at the end at infinity, because action of T is then conjugate in the exterior region to the standard action on R4 × T D−5 there, and we therefore will not describe it in detail. On the other hand, the choice of X requires some comment. First, since H is itself a compact manifold of dimension D −2 with an action of T , we can apply the classification (35). In the first two cases, we simply choose X = B 4 ×T D−5 respectively X = B 3 × T D−4 with the standard actions, and it is then quite obvious from the proof of Thm. 2 that these actions will match up with that from across the boundary H. In the last case, it is not so obvious how to choose X , and we now explain this following mainly [44]. First, as we have seen in the proof of Thm. 2, there are two degenerate T -orbits in H corresponding to places where a linear combination ai,1 ψi = 0 or ai,2 ψi vanish. By Lemma 2, we can find a matrix B ∈ S L(D − 3, Z) such that a 1 B T = (1, 0, . . . , 0), and a 2 B T = (q, p, 0, . . . , 0). Thus, redefining the axial Killing −1 if necessary, we can assume without loss of fields as ψi → j Ai j ψ j and A = B generality that a 1 = (1, 0, . . . , 0) and a 2 = (q, p, 0, . . . , 0). Now, if pq = 0 mod 2 it is possible to see that we can find a continued fraction representation of qp , q = n1 − p

1 n2 −

(52)

1 n3 −

1 ...

1 nk

in such a way that there are (u 1 , v1 ), . . . , (u k , vk ) ∈ Z2 with 0 u2 u u u u n k−1 = det 1 3 , n k−2 = det 2 4 , n k = det 1 v2 v1 v3 v2 v4 p u . . . n 1 = det k−1 . vk−1 q

(53)


653

Then our choice for X is 1 X = B 4 ∪ f1 (B 4 ∪ f2 (· · · ∪ fk B 4 )) × T D−5 = X [ D 2 , {(u i , vi , 0, . . . , 0)}], (54) 2 where the gluing maps f i are the same as in Eq. (47), and the action of T = T2 × T D−5 on X is the T2 -action described there on the first Cartesian factor, and it is the standard T D−5 action on itself on the second Cartesian factor. This action matches precisely that of across the joint boundary ∂ X = H = −∂ . The orbit space Xˆ = 21 D 2 is half of a 2-disk, with the cut corresponding to the orbit space Hˆ = H/T of the horizon, and with the semi-circle corresponding to the various “axis” of the T -action. When pq is odd, a similar construction can be made. We are now in a position to apply Thm. 1 to , which tells us that its orbit space is ˆ g with g handles, with orbifold points (conical singularities) given by a handle body

and with removed disks, ˆg\

/T ∼ =

d

D 2j .

(55)

j=1

To rule out the presence of handles, removed disks, and points with conical singularities, we now use the topological censorship theorem for asymptotically Kaluza-Klein spaces [9], see also [17,18]. This theorem implies that any curve γ with endpoints in the asymptotic region ∞ of can be continuously deformed to a curve entirely within

∞ , and it follows from our gluing construction that the same is still true in the compactified space . Furthermore, any closed loop in ∞ is homotopic to a closed loop in {pt} × T . These facts together imply that if f : {pt} × T → is the embedding map, then f ∗ : π1 ({pt} × T ) → π1 ( ) is surjective. We can now apply Lemma 6, and thereby conclude that there can be no handles, removed disks, nor conical singularities, and that /T ∼ = D 2 . We now recall that was the union (51), the pieces of which are separately invariant, and whose orbit spaces are, respectively X/T ∼ = 21 D 2 ˆ = /T = 1 D 2 \{pt} is a half-disk 1 D 2 , with a and ({pt} × T )/T ∼ = {pt}. Thus,

2 2 ˆ itself point {pt} removed somewhere on the arc-shaped boundary. In other words,

is homeomorphic to a half-plane, and there are no conical singularities. The boundary component of 21 D 2 \{pt} along the straight cut is the horizon interval I H , i.e. it is coming from the quotient H by T . The point {pt} corresponds to the point at infinity in the upper half plane picture. ˆ this proves the theorem. Since Mˆ = M/G ∼ = , The proof of the theorem also implies the following corollary: Corollary 1. Topologically, the domain of outer communication M is given by M ∼ = R × ( \{pt}\X ),

(56)

where “pt” represents the point at infinity, where is a compact, connected manifold without boundary with T -action, and where X (the ‘black hole’) is a compact connected manifold with T -action and boundary ∂ X = H. Furthermore, the action of T has no points with discrete isotropy group.

654


Remark. In D = 5, the corollary implies together with results of [43,44], that 2

∼ = # k · CP 2 # k · CP # l · (S2 × S2 ).

(57)

Note that a generic compact 4-manifold could also contain K 3 and K 3’s. If one additionally assumes that M is spin, then the complex projective spaces are absent in the decomposition. All known black hole solutions in fact have k = k = l = 0, and this may well be the only possibilities. In D = 7, a similar decomposition applies if the second Stiefel-Whitney class of the spacetime vanishes. Then we have a decomposition of the type [42]:

∼ = # k · (S2 × S4 ) # (k − 1) · (S3 × S3 ).

(58)

This pattern presumably persists in all dimensions, but we have not been able to show this. 4. Stationary Vacuum Black Holes in D Dimensions 4.1. Canoical coordinates. In the previous section, we looked at the topology of the domain of outer communication M and the structure of the orbits of the symmetries. In this section, we investigate the spacetime metric, i.e. the implications of the Einstein equations Rab = 0. These equations imply a set of coupled differential equations for the metric on the ˆ described above in Thm. 4. To understand these equatwo-dimensional factor space M, tions in a geometrical way, we note that the projection π : M → M/G = Mˆ (with G = T D−3 × R the isometry group) defines a G-principal fibre bundle over the interior ˆ because we argued in the previous section that such points correspond to points in of M, the domain of outer communication with trivial isotropy group. At each point x ∈ M ˆ we may uniquely decompose the tangent space in a fibre over π(x) in the interior of M, at x into a subspace of vectors tangent to the fibres, and a space Wx of vectors orthogonal to the fibres. Evidently, the distribution of vector spaces Wx is invariant under the group G of symmetries, and hence forms a “horizontal bundle” in the terminology of principal fibre bundles [33]. According to standard results in the theory of principal fibre bundles [33], a horizontal bundle is equivalent to the specification of a G-gauge connection I d x α ∧ d x β , with Dˆ on the factor space, whose curvature we denote by Fˆ = TI Fˆαβ TI , I = 0, . . . , D − 3 the generators of the abelian group G. Roman indices α, β, . . . take the values 1, 2. The horizontal bundle gives an isomorphism Wx → Tπ(x) Mˆ for any x, and this isomorphism may be used to uniquely construct a smooth covariant tensor field tˆαβ...γ on the interior of Mˆ from any smooth G-invariant covariant tensor field tab...c on M. For example, the metric gab on M thereby gives rise to a symmetric tensor gˆ αβ on ˆ M. One can show with a significant amount of labor [8] (see also [7]) that the D − 2 dimensional subspaces spanned by the Killing fields at points of M corresponding to interior points of Mˆ always contain a timelike vector. Hence the bilinear form induced from gab on Wx has signature (++), so gˆ αβ is in fact a Riemannian metric. We let Dˆ act on ordinary tensors tˆαβ...γ as the Levi–Civita connection of gˆ αβ , with Ricci tensor denoted Rˆ αβ .


655

By performing the well-known Kaluza-Klein reduction of the metric gab along the orbits of G, we can locally write the Einstein equations as a system of equations I of on the interior of the factor space Mˆ in terms of the metric gˆ αβ , the components Fˆαβ the curvature, and the Gram matrix field G I J , t if I = 0, G I J = g(X I , X J ), X I = (59) ψi if I = i = 1, . . . , D − 3. The resulting equations are similar in nature to the “Einstein-equations” on Mˆ for gˆ αβ , I and the “scalar fields” G , see [5,32]. We will not coupled to the “Maxwell fields” Fˆαβ IJ write these equations down here, as we will not need them in this most general form. In our case, the equations simplify considerably because one can show (see e.g. [7]) that the distribution of horizontal subspaces Wx is locally integrable, i.e., locally tangent to a family of two-dimensional submanifolds. In that case, the connection is flat, I = 0, and the dimensionally reduced equations may be written as Fˆαβ Dˆ α (r G −1 Dˆ α G) = 0 ,

(60)

1 Rˆ αβ = Dˆ α Dˆ β log r − Tr Dˆ α G −1 Dˆ β G . 4

(61)

together with

Greek indices have been raised with gˆ αβ . The equations are well-defined a priori only at points in the interior of Mˆ where the Gram determinant r 2 = −det G

(62)

does not vanish. Chru´sciel has shown [8] (based on previous work of Carter [2] and also ˆ The reduced Einstein equations are of [7]) that r 2 > 0 away from the boundary of M. hence well-defined there. On the other hand, r vanishes on any boundary component I j of Mˆ corresponding to an axis, i.e. where a linear combination ai (I j )ψi = 0 vanishes, because the Gram matrix then has a non-trivial kernel. It also vanishes on the segment of ∂ Mˆ corresponding to the horizon H , because the span of X I , I = 0, . . . , D − 3 is tangent to H and hence a null space, with the signature of G consequently being (0 + + · · · +) there. Taking the trace of the first reduced Einstein equation (60), one finds that r is a ˆ harmonic function on the interior of M, Dˆ α Dˆ α r = 0.

(63)

Since Mˆ is an (orientable) simply connected 2-dimensional analytic manifold with connected boundary and corners by Thm. 1, we may map it analytically to the upper complex half plane {ζ ∈ C | Im ζ > 0} by the Riemann mapping theorem. Furthermore, since r is harmonic, we can introduce a harmonic scalar field z conjugate to r Dˆ α z = ˆ αβ Dˆ β r,

(64)

where ˆαβ is the anti-symmetric tensor on Mˆ satisfying ˆ αβ ˆαβ = 2. Thus both r, z are ˆ g), ˆ Combining this with the fact Mˆ is harmonic functions on ( M, ˆ and r = 0 on ∂ M.

656


homeomorphic to a half-plane, one can argue (see e.g. [7, 6.3] or [53]) that r and z are globally defined coordinates, and identify Mˆ with {z + ir ∈ C | r > 0}. In these coordinates, the metric gˆ globally takes the form gˆ = e2ν(r,z) (dr 2 + dz 2 ).

(65)

Since Eq. (60) is invariant under conformal rescalings of gˆ αβ , and since a 2-dimensional metric is conformally flat, it decouples from Eq. (61). In fact, writing the Ricci tensor Rˆ αβ of (65) in terms of ν, one sees that Eq. (61) may be used to determine ν by a simple integration, see e.g. [21] for details. The boundary r = 0 of Mˆ consists of several segments according to our Classification Theorem 4. In the description of Mˆ as the upper complex half plane Mˆ = {z + ir ∈ C | r > 0}, these are represented by a collection of intervals {I j } of the z-axis. The length of the j th interval as measured by the coordinate z is called l(I j ). Because the coordinates (r, z) were canonically defined, the numbers l(I j ) ≥ 0 are invariantly defined, i.e. are the same for isometric spacetimes. Each segment is either an axis for which there is a vector a(I j ) ∈ Z D−3 such that i ai (I j )ψi = 0, or it corresponds to the horizon. In that case, we put the corresponding vector to zero, a H = 0, because no non-trivial linear combination of the axial Killing fields vanishes in the interior of the corresponding interval I H , see Thm. 4. Concerning the length l H of the horizon segment, we have the following lemma. Lemma 7. The length of the horizon interval satisfies (2π ) D−3l H = κ A H ,

(66)

where A H is the area of the horizon cross section H, and where κ > 0 is the surface gravity. The proof of Lemma 7 is given in Appendix A. We call the collection of real positive numbers {l(I j )} and integer vectors {a(I j )} associated with the intervals the “interval structure” of the spacetime. As we explained in the previous section, the collection {a(I j )} determines the manifold structure of M and the action of G on this space up to diffeomorphism. In particular, the vector fields X I are determined up to diffeomorphism. Furthermore, if we are given G I J and gˆ (i.e., ν) as functions of r, z, then we can reconstruct the metric g of the spacetime in the domain of outer communication. In a local coordinate system consisting of r, z and ξ I , I = 0, . . . , D − 3, such that the Killing fields are given by X I = ∂/∂ξ I , the metric locally takes the form g = e2ν(r,z) (dr 2 + dz 2 ) + G I J (r, z) dξ I dξ J .

(67)

For M = R4,1 × T D−5 with the standard flat metric g0 , the axial symmetries are the rotations in the 12-plane of R4,1 generated by the Killing field ψ1 , the rotations in the 34-plane of R4,1 generated by the Killing field ψ2 and the rotations of the D − 5 compact extra dimensions generated by Killing fields ψ3 , . . . , ψ D−3 . The coordinates r, z

as constructed above are given by r = R1 R2 and z = 21 (R12 − R22 ), with R1 = x12 + x22 and R2 = x32 + x42 , and with xi the standard spatial Cartesian coordinates of R4,1 . The


657

√ conformal factor is given by e2ν = 1/2 r 2 + z 2 , and the Gram matrix of g0 is given by ⎞ ⎛ −1 0 0 0 0 0 ⎟ ⎜ 0 ρ(1 − cos θ ) , (68) G0 = ⎝ 0 0 ρ(1 + cos θ ) 0 ⎠ 0 0 0 δi j where i, j = 3, . . . , D − 3. Here, we have introduced the coordinates ρ, θ which are related to r, z by r = ρ sin θ, z = ρ cos θ, or

(69)

2 (x12 + x22 )(x32 + x42 ) 1 2 ρ = (x1 + x22 + x32 + x42 ), θ = arctan 2 x12 − x32 + x22 − x42

(70)

in terms of the spatial cartesian coordinates xi of R4,1 . The metric g0 is hence given explicitly by g0 = −dτ 2 + +

D−3

1 (dρ 2 + ρ 2 dθ 2 ) + ρ(1 − cos θ )dϕ12 + ρ(1 + cos θ )dϕ22 2ρ

dϕi2 ,

(71)

i=3

where the last line is the metric of the small dimensions T D−5 . 4.2. Asymptotic behavior. A general D-dimensional asymptotically Kaluza-Klein spacetime (M, g) with asymptotically flat 5-dimensional part will differ in the asymp

totic region by terms of order O(R −2 ), with R = x12 + x22 + x32 + x42 , by our general assumptions about the asymptotic behavior of the spacetime metric, see Eq. (1). We would like to know exactly what this means in terms of our canonical coordinates (r, z) or alternatively (ρ, θ ), as we will need this type of information later in the proof of our classification theorem. In the spacetime M = R4,1 × T D−5 with standard metric g0 , we had ρ = 21 (R12 + R22 ) and tan θ = R1 R2 , where R1 =

x12 + x22 and R2 =

x32 + x42 ,

but in a spacetime whose metric is only asymptotic to g0 up to terms of order O(R −2 ), this will no longer be the case exactly. In order to analyze this issue, we recall that, in the asymptotic region of (M, g), we assumed the coordinates to be chosen in such a way that the Killing fields are given exactly by ψ1 = x1 ∂x2 − x2 ∂x1 , ψ2 = x3 ∂x4 − x4 ∂x3 , and t = ∂τ , as well as ψi = ∂ϕi for i = 3, . . . , D − 3. It is then evident from Eq. (1) that g(t, t) = −1 + O(R −2 ) , g(t, ψ1 ) = R1 O(R −2 ), g(t, ψ2 ) = R2 O(R −2 ), g(ψ1 , ψ1 ) = R12 (1 + O(R −2 )), g(ψ2 , ψ2 ) = R22 (1 + O(R −2 )) , g(ψ1 , ψ2 ) = R1 R2 O(R −2 ), g(ψi , ψ j ) = δi j + O(R g(ψi , ψ1 ) = R1 O(R

−2

−2

) , g(ψi , t) = O(R

(72) −2

) , g(ψi , ψ2 ) = R2 O(R

),

−2

),

658


where i, j = 3, . . . , D − 3. These equations determine the Gram-matrix G, whose determinant (62) gives us the coordinate r . We find r = R1 R2 (1 + O(R −2 )).

(73)

The second canonical coordinate z was defined to be the dual harmonic coordinate to r , see Eq. (64). For this, we need gˆ = d R12 + d R22 + O(R −2 ) , ˆ = (1 + O(R −2 )) d R1 ∧ d R2 ,

(74)

which follows from the definitions of the orbit space metric and the asymptotic conditions. This leads to dz = (1 + O(R −2 )) (R1 d R2 − R2 d R1 ) + O(R −2 ).

(75)

When we integrate this we find z=

1 2 (R − R22 ) + O(1) , 2 1

(76)

as we could have anticipated. As above, let us define r = ρ sin θ, z = ρ cos θ . Inverting the relations for (r, z) just given in terms of (R1 , R2 ) gives, after some straightforward analysis: 2 = ρ(1 ± cos θ )(1 + O(ρ −1 )) + O(sin2 θ ) , R1/2

(77)

and in particular R 2 = 2ρ(1 + O(ρ −1 )). Inserting this into Eq. (1) and taking into account Eqs. (72) then delivers the following asymptotic form of the metric g: g = g0 + O(1)(dρ 2 + ρ 2 dθ 2 ) +O(ρ −1 )dτ 2 + O(1)(1 + cos θ )dϕ12 + O(1)(1 − cos θ )dϕ22 +O(1) sin θ dϕ1 dϕ2 +

D−3

O(ρ −1 )dϕi dϕ j +

i, j=3

+O(ρ +

D−3 i=3

−1/2

)(1 + cos θ )

1/2

dϕ1 dτ + O(ρ

D−3

O(ρ −1 )dϕi dτ

i=3 −1/2

O(ρ −1/2 )(1 + cos θ )1/2 dϕ1 dϕi +

)(1 − cos θ )1/2 dϕ2 dτ

D−3

O(ρ −1/2 )(1 − cos θ )1/2 dϕ2 dϕi ,

i=3

(78) where g0 is the flat background metric given above in Eq. (71). We emphasize that these estimates hold uniformly in ρ, θ , including the axes θ = 0, π . This will be needed later. Similar expansions can be carried out when the number of asymptotically large spatial dimensions is 3, but when this number is ≤ 2, the analysis would be substantially different. Using Einstein’s equations, one would expect that it is possible to determine the asymptotics of the metric g in considerably more detail than (78), and we now outline how this can be done. However, we emphasize that for the purposes of this paper, already Eq. (78) will be sufficient. First, one writes G = G 0 F, with G 0 the diagonal Gram-matrix for R4,1 × T D−5 given above. The matrix function F represents the corrections and it satisfies the second order non-linear elliptic equation Dˆ α (F −1 Dˆ α F) +


659

ˆ ( Dˆ α F −1 )G −1 0 ( Dα G 0 )F = 0, and F → I for ρ → ∞. By doubling the half space ˆ M across its boundary, one would then expect to be able to show that F satisfies an asymptotic expansion of the general form Fn,m (sin θ, cos θ )ρ −n logm ρ , (79) F∼I+ n,m≥1

for large ρ. If we assume that such an asymptotic expansion indeed holds for F, then it is straightforward to determine explicitly its first terms. We do not give the details of the straightforward but somewhat lengthy calculation but only quote the solution. It can be stated as saying that ⎛

⎞ −2M B1 B2 bi −1 −1 −1 M − A1 Cρ ci ρ ⎟ ⎜B ρ F = ρ −1 ⎝ 1 −1 ⎠ + ··· . B2 ρ Cρ −1 M − A2 di ρ −1 bi ci di hi j

(80)

Here, dots represent higher terms in the asymptotic expansion, and the quantities M, A1,2 , B1,2 , C, h i j , bi , ci , di are undetermined real constants and i, j range through 3, . . . , D − 3 in this block-matrix. Because we must have −det G = r 2 , they are subject to the constraint det F = 1, from which it follows that A1 + A2 =

D−3

h ii .

i=3

According to Eq. (64), we are still free to change the coordinate z by adding a constant. This will result in adding a constant ±η to A1/2 , and we may thus fix the remaining ambiguity in z in order to set A1 = A2 = A. We will do this in the following. The asymptotic form of the conformal factor e2ν can similarly be determined by the second reduced Einstein equation, Eq. (61), together with the asymptotic form of the Gram matrix G = G 0 F. Again, we omit the straightforward but somewhat lengthy calculation and give only the result, which is e2ν =

M−A 1 + + ··· , 2ρ 4ρ 2

(81)

where the dots represent terms that go to zero faster as ρ → ∞. Thus, in a coordinate system (τ, ρ, θ, ϕ1 , . . . , ϕ D−3 ) such that t = ∂/∂τ, ψi = ∂/∂ϕi , i = 1, . . . , D − 3,

(82)

we obtain the following asymptotic form of the metric Eq. (67) for large ρ: Asymptotic form of the metric. for stationary black hole spacetime with D − 3 axial Killing fields, behaving as R4,1 × T D−5 near infinity: M − A 2 2M 2 1 dτ + 1+ (dρ + ρ 2 dθ 2 ) g = − 1− ρ 2ρ 2ρ M − A 2 M − A 2 dϕ1 + ρ(1 + cos θ ) 1 + dϕ2 +ρ(1 − cos θ ) 1 + ρ ρ

660


+

D−3

δi j +

i, j=3

+

hi j 2C sin2 θ dϕi dϕ j + dϕ1 dϕ2 ρ ρ

D−3 2B1 (1 − cos θ ) 2B2 (1 + cos θ ) 2 dϕ1 dτ + dϕ2 dτ + bi dϕi dτ ρ ρ ρ i=3

+

2(1 + cos θ ) ρ

D−3

di dϕ2 dϕi +

i=3

2(1 − cos θ ) ρ

D−3

ci dϕ1 dϕi + · · · ,

(83)

i=3

D−3 where the dots represent terms that are higher order in 1/ρ, and where A = 21 i=3 h ii . The constants bi are proportional to the angular momenta of the solutions in the asymptotically small dimensions, and the constants B1/2 are proportional to the two independent angular momenta in the asymptotically large dimensions. They can be defined e.g. by the Komar expressions Ji = ∗dψi , (84) S3 ×T D−5

where the integration is over a surface at infinity, and where dψi denote the 2-forms obtained by taking the exterior differential of ψi after lowering the index. The constant M is related to the ADM-mass of the solution, see e.g. [31, Sec. 3], and the constant A to the “tension” of the small extra dimensions. We finally remark that the analysis given above would be different for different numbers of asymptotically large dimensions. For example, for M = R3,1 × T D−4 , the axial symmetries may be taken as the rotations in the 12-plane of R3,1 and rotations of the D − 4 compact extra dimensions. The functions r, z are then given by r =

x12 + x22

and z = x3 , with xi the standard spatial Cartesian coordinates on R3,1 . The conformal factor is just e2ν = 1. For a general D dimensional asymptotically Kaluza-Klein spacetime with asymptotically flat 4-dimensional part, we may again derive an expression for the asymptotic form of the metric which is similar but not identical to that given above. In the case M = R2,1 × Y , we have again r =

x12 + x22 but z can now be e.g. a

periodic coordinate, depending on Y . For example if Y = T D−3 , then z parametrizes the orbit space T = T D−3 /T D−4 . The analysis of a general spacetime with this asymptotic behavior would also be quite different. In the case M = R1,1 × Y , the function r is simply a constant, and the definition of the (r, z) coordinates is not possible any longer. 5. Uniqueness Theorem for Stationary Black Holes with ( D − 3) Axial Symmetries In the previous two sections, we have analyzed stationary black hole spacetimes that are asymptotically R4,1 × T D−5 , and that have an isometry group G = R × T D−3 . We have derived a number of “invariants” associated with such solutions: • We showed that the orbit space of the domain of outer communication by G is a half plane Mˆ = {z + ir | r > 0}. The boundary of the half-plane is divided into a finite collection of intervals {I j }. With each interval, there is associated its length11 11 For a half infinite interval, this would be ∞.


661

l(I j ) ∈ R>0 , and a vector a(I j ) ∈ Z D−3 subject to the normalization (31). One of the intervals corresponds to the orbit space Hˆ of the horizon and is associated with the zero vector, while the others correspond to an “axis” in spacetime, i.e. points where the linear combination i ai (I j )ψi = 0 vanishes. For adjacent intervals I j and I j+1 (not including the horizon), there is a compatibility condition stating that the collection of minors Q kl ∈ Z, 1 ≤ k < l ≤ D − 3 given by Q kl = | det

ak (I j+1 ) ak (I j ) | al (I j+1 ) al (I j )

(85)

have greatest common divisor g.c.d.{Q kl } = 1, see the discussion around (32). The data {l(I j )} together with {a(I j )} were called the “interval structure”. • Because the spacetime is asymptotically Kaluza-Klein, we can define its mass, and the angular momenta {Ji } corresponding to the axial Killing fields, i = 1, . . . , D −3. Some of the angular momenta correspond to the large, and some to the small (extra) dimensions. • The asymptotic form of the metric (83) contains additional real parameters {h i j }, {ci }, {di }, . . . which are related to the asymptotic metric on the tori generated by the axial Killing fields ψi , i = 1, . . . , D − 3 in the region of spacetime near infinity. These numbers are invariantly defined. • The collection of angular velocities {i }, the surface gravity κ, and horizon area. It is natural to ask the following questions: Is the spacetime (M, g) under consideration uniquely determined by the above data? To what extent can the data be specified independently? The following theorem provides an answer to the first question and a partial answer to the second question. Theorem 5. There can be at most one stationary, asymptotically Kaluza-Klein spacetime (M, g) with D − 3 axial Killing fields and 5 asymptotically large dimensions, satisfying the technical assumptions stated in Sect. 2, for a given interval structure {a(I j ), l(I j )} and a given set of angular momenta {Ji }, i = 1, . . . , D − 3. This uniqueness theorem is the main result of this paper. The same result is true if the number of asymptotically large dimensions is only 4. The only difference in the proof would be the analysis of the asymptotic behavior. For 3 or less asymptotically large dimensions, we still expect a result of this type to be true, and the proof to be similar. However, in that case, the nature of the orbit space would also be different, so the differences in the proof would presumably be greater. A consequence of the theorem is that the interval structure and angular momenta uniquely determine the other invariants mentioned above, such as e.g. the mass of the spacetime. In D = 4 with no extra dimensions, the only non-trivial interval structure for a single black hole spacetime is given by the intervals (−∞, −z 0 ], [−z 0 , z 0 ], [z 0 , ∞). The middle interval corresponds to the horizon, while the half-infinite ones correspond to the axis of the rotational Killing field. The interval vectors a(I j ) are 1-dimensional integer vectors in this case and hence trivial. For each z 0 > 0 and for each angular momentum J , there exist a solution given by the appropriate member of the Kerr-family of metrics. Thus, the Kerr metrics exhaust all possible stationary, axially symmetric single black hole spacetimes (satisfying the technical assumptions stated in Sec. 2). This is of course just the classical uniqueness theorem for the Kerr-solution [1,2,24,38,48], see [7] for a rigorous account. The mass m of the non-extremal Kerr solution charac terized by z 0 , J is related to these parameters by z 0 = m 2 − J 2 /m 2 > 0. Hence the

662


uniqueness theorem may be stated equivalently in terms of m and J , which is more commonly done. Note that the length of the horizon interval, l H = 2z 0 tends to zero in the extremal limit, in accordance with Lemma 7. In higher dimensions, one may similarly derive relations between the interval structure and angular momenta on the one side, and the other invariants on the other side for any given solution. Such formulae are provided for the Myers-Perry or black-ring solutions e.g. in [21], but they would not be expected to be universal. Of course, for most interval structures it is not known whether there actually exists a solution, so in this sense much less is known in higher dimensions than in D = 4. Proof of Thm. 5. We will show that the domains of outer communication of any two spacetimes as in the theorem must be isometric. It then follows from analyticity that they are globally isomorphic, including the interior of the black hole. Without assuming analyticity, one could obtain information about the interior if one can generalize the argument given in [16] based on the characteristic initial value formulation of the Einstein equations from D = 4 to higher dimensions. We will not do this here. The key step is to define from the reduced Einstein equations (60) a set of equations which describe the difference between two solutions as described in the theorem. This formulation is due to [35,38], see also [37], and it involves certain potentials which we define first. We first consider the twist 1-forms, ωi = ∗(ψ1 ∧ · · · ∧ ψ D−3 ∧ dψi ) i = 1, . . . , D − 3 ,

(86)

where the Killing fields have been identified with 1-forms via the metric. Using the vacuum field equations and standard identities for Killing fields [52], one shows that these 1-forms are closed, dωi = 0. Since the Killing fields commute, the twist forms are invariant under G, and so we may define corresponding 1-forms ωˆ i on the interior of the factor space Mˆ = {z + ir ∈ C | r > 0}. These 1-forms are again closed. Thus, the “twist potentials”

xˆ

χi =

ωˆ i

(87)

0

are globally defined on Mˆ and independent of the path connecting 0 and the point ˆ and dχi = ωˆ i . The twist potentials and the Gram matrix of the axial Killing xˆ ∈ M, fields f i j = g(ψi , ψ j ), satisfy a system of coupled differential equations on Mˆ which follow from the reduced Einstein equation (60). They are 0 = Dˆ α r (det f )−1 χ i Dˆ α χi + r Dˆ α log det f , (88) (89) 0 = Dˆ α r (det f )−1 f i j Dˆ α χ j , 0 = Dˆ α r f jk Dˆ α f ki + r (det f )−1 f jk χi Dˆ α χk , (90) 0 = Dˆ α −r Dˆ α χi + r χi Dˆ α log det f + r ( f jk Dˆ α f i j )χk (91) + r (det f )−1 χ j ( Dˆ α χ j )χi . Here we are using the summation convention and f i j denotes the components of the inverse of the matrix f i j , which is used to raise indices on χi . To verify these equations,


663

it is necessary to use the relations Dˆ α α i = r (det f )−1 ˆ αβ f i j Dˆ β χ j ,

(92)

β = f i j αi α j − (det f )−1r 2

(93)

as well as

for the scalar products αi = g(t, ψi ) and β = g(t, t). Again, α i means f i j α j . The above equations can be written in a compact matrix form. For this, one introduces the (D − 2) × (D − 2) matrix field which is written in an obvious block-matrix notation as (det f )−1 −(det f )−1 χi . (94) = −(det f )−1 χi f i j + (det f )−1 χi χ j The matrix satisfies T = , det = 1, and is positive semi-definite, being the sum of two positive semi-definite matrices. Hence it may be written in the form = S T S for some matrix S of determinant ±1, i.e. up to sign S ∈ S L(D − 2, R). Because S is defined only up to S → RS, where R is a rotation, we can think of it as an element of the right coset S L(D − 2, R)/S O(D − 2, R), and this coset therefore parametrizes the possible . Equations (88) can be stated in terms of as Dˆ α (r −1 Dˆ α ) = 0.

(95)

These can be viewed as the equations for a sigma-model on the target space S L(D − 2, R)/S O(D −2, R). We will not use this viewpoint, but it could be used to give an alter˜ g) native proof of our theorem. Consider now two black hole solutions (M, g) and ( M, ˜ as in the statement of the theorem. We denote the corresponding matrices defined as ˜ and we use the same “tilde” notation to distinguish any other quanabove by and , tities associated with the two solutions. M as a manifold with a G-action is uniquely determined by the interval structure modulo diffeomorphisms preserving the action of G and similarly for the tilde spacetime. Therefore, since the interval structures are assumed ˜ are isomorphic as manifolds with to be the same for both spacetimes, M and M a G action, and we may hence assume that t˜ = t, ψ˜ i = ψi for i = 1, . . . , D − 3, and we may also assume that r˜ = r and z˜ = z. Consequently, it is possible to combine the divergence identities (95) for the two solutions into a single identity on the upper complex half plane, called “Mazur identity”. It is given by Dˆ α (r Dˆ α σ ) = r gˆ αβ Tr Nˆ αT Nˆ β , (96) and it can be proven in almost exactly the same way as the identity given in [38]. Here, we have written ˜ −1 − I ), σ = Tr(

˜ − −1 Dˆ α )S, ˜ −1 Dˆ α Nˆ α = S˜ −1 (

(97)

˜ = S˜ T S˜ hold. The key where in turn S and S˜ are matrices such that = S T S and point about the Mazur identity (96) is that on the left side we have a total divergence, while the term on the right hand side is non-negative. This structure can be exploited in various ways. In this paper, we follow a strategy invented by Weinstein [53,54], which differs from that originally devised by Mazur.

664


The basic idea is to view r, z as cylindrical coordinates in an auxiliary space R3 consisting of the points x = (r cos γ , r sin γ , z), and to view σ as a rotationally symmetric function defined on this R3 , minus the z-axis. The Mazur identity then gives x σ ≥ 0 on R3 \{z − axis},

(98)

where x is the ordinary Laplacian on R3 . As we will show, σ is globally bounded on R3 , including at infinity and the z-axis. Furthermore, we claim that σ ≥ 0 at any ˜ −1 , we have σ = Tr (F T F) − (D − 2). point away from the axis: Writing F = SS T T ˜ det −1 = 1, so we may bring F T F into the Now, F F ≥ 0, and det F F = det u u −u −···−u 1 1 D−3 D−3 form diag(e , . . . , e ,e ) by a similarity transformation. Thus, σ will be non-negative if and only if 1 (eu 1 + · · · + eu D−3 + e−u 1 −···−u D−3 ) ≥ 1, D−2

(99)

which in turn follows directly because the exponential function is convex. Thus, we are in a position to apply the maximum principle arguments in [54], which imply that σ = 0 everywhere. As we now see, this implies that the metrics g and g˜ are isometric on the domain of outer communication, thus proving the theorem. ˜ = everywhere First, σ = 0 implies that u 1 = · · · = u D−3 = 0, and hence that ˆ in M. Therefore, the twist potentials and the Gram matrices of the axial Killing fields are identical for the two solutions, f˜i j = f i j and χ˜ i = χi . To see that the other scalar products between the Killing fields coincide for the two solutions, let αi = g(t, ψi ), β = g(t, t) as above, and define similarly the scalar products α˜ i , β˜ for the other spacetime. The right side of Eq. (92) does not depend upon the conformal factor ν, so since χ˜ i = χi and f˜i j = f i j , it also follows that α˜ i = αi up to a constant. That constant has to vanish, since it vanishes at infinity. Furthermore, from Eq. (93) we have β˜ = β. Thus, all scalar products of the Killing fields are equal for the two solutions, G˜ I J = G I J on the entire upper half plane. Viewing now the second reduced Einstein equation (61) as an equation for ν respectively ν˜ , and bearing in mind that ν = ν˜ at infinity, one concludes that ν˜ = ν. Thus, summarizing, if we could show that the field σ of Eq. (97) is globally bounded on R3 , then we would know that G˜ I J = G I J , r˜ = r , z˜ = z and ν˜ = ν. Since t˜ = t, ψ˜ i = ψi it would follow from Eqs. (67) and (65) that g˜ = g in the domain of outer communication. Thus, it remains to be shown that σ is globally bounded, including near the z-axis ˆ and near infinity. It is here that the assumptions of the theorem (corresponding to ∂ M) about the interval structures and angular momenta are needed. Away from the z axis in R3 (or equivalently, the boundary of Mˆ = {z + ir | r > 0}, all fields f i j , χi , ν, f˜i j , χ˜ i , ν˜ and the matrix inverses are smooth (in fact analytic) functions, and this consequently also applies to σ , which is just an algebraic function of these. However, σ could diverge ˆ To see that this is not the case, we cover Mˆ (including the at infinity, or the boundary ∂ M. boundary) with different open neighborhoods (1)—(5). These neighborhoods are chosen as follows: (1) In an open neighborhood of (z j + δ, z j − δ), where I j = (z j , z j+1 ) ⊂ ∂ Mˆ corresponds to a rotation axis. The quantity δ > 0 has been introduced so that the neighborhood does not include any intersection points of two axis, or an axis with the horizon. (2) is a similar neighborhood, where the interval I H in question now represents the horizon, (3) covers the outside of a large ball r 2 + z 2 ≤ R 2 , i.e. infinity. We also need to consider separately (4) open neighborhoods of the intersection points z j of adjacent


665

intervals I j−1 = (z j−1 , z j ) and I j = (z j , z j+1 ) which do not represent the horizon, and (5) open neighborhoods of the intersection points of the horizon interval and the adjacent intervals representing axis. The cases (4) and (5) represent “corners” of the orbit space ˆ which were “straightened out” by introducing the coordinates (r, z). M, (1) Axis. On each segment z ∈ I j = (z j , z j+1 ), r = 0 of the boundary ∂ Mˆ representing an axis, we know that the null spaces of the Gram matrices f i j and f˜i j coincide, because we are assuming that the interval structures of both solutions are identical. Furthermore, from Eq. (87), and from the fact that ωˆ i vanishes on any axis by definition, the twist potentials χi are constant on the z-axis outside of the segment (z h , z h+1 ) representing the horizon. The difference between the constant value of χi on the z-axis left and right to the horizon segment can be calculated as follows: z h+1 χi (r = 0, z h ) − χi (r = 0, z h+1 ) = ωˆ i zh 1 = ∗(dψi ) (2π ) D−3 H 1 1 = ∗(dψi ) = Ji . D−3 (2π ) (2π ) D−3 S3 ×T D−5 The first equality follows from the definition of the twist potentials, the second from the defining formula for the twist 1-forms and the fact that these are invariant under the action of the D − 3 independent rotation isometries each with period 2π (with H a horizon cross section), the third equation follows from Gauss’ theorem and the fact that d(∗dψi ) = 0 because ψi is a Killing vector on a Ricci-flat manifold, and the last equality follows from the Komar expression for the angular momentum. ˜ g). The analogous expressions hold in the spacetime ( M, ˜ Because we assume that ˜ Ji = Ji , we can add constants to χi , if necessary, so that χi = χ˜ i on the axis. From the definition (87), it then follows that in fact χi − χ˜ i = O(r 2 ) near any axis, or more precisely, near any compact subset of the open interval I j = (z j , z j+1 ) of the ˆ boundary ∂ M. In order to analyze the behavior of σ near (z j + δ, z j+1 − δ), we now calculate σ = −1 +

f i j (χi − χ˜ i )(χ j − χ˜ j ) det f + + f i j ( f˜i j − f i j ). ˜ ˜ det f det f

(100)

We wish to show that each term on the right side is uniformly bounded near (z j + δ, z j+1 − δ). Let a(I j ) ∈ Z D−3 be the vector generating the kernel of the matrix f in on our interval I j . By Lemma 2, we can find a matrix B ∈ S L(D − 3, Z) such that a(I j )B T = (1, 0, . . . , 0). Thus, redefining the axial Killing fields as ψi → j Ai j ψ j and A = B −1 if necessary, we can assume without loss of generality that a(I j ) = (1, 0, . . . , 0), so that the axis under consideration corresponds to zeros of ψ1 . Let x ∈ M be a point on this axis. By Lemma 4, we can introduce coordinates (τ, y1 , . . . , y D−1 ) in a neighborhood of x such that the action of the rotational part of the isometry group takes the form described in Lemma 4, and such that the time-part of the isometry group acts by simply shifting τ by a constant. In other words, the isometry group acts locally by shifting τ, y4 , . . . , y D−1 , and it acts by

666


rotating y1 + i y2 by a phase. Thus, if we define R =

y12 + y22 , Y = y3 , then (Y, R) ∈ R × R>0 are a coordinate system that displays the nature of Mˆ as a manifold with boundary near (z j + δ, z j+1 − δ). It also follows that any analytic function on M that is defined near x and is invariant under the action of G has an absolutely convergent expansion in Y and even powers of R locally near R = 0. In particular this applies to the function r 2 = − det G. Since r = 0 when R = 0, and since g = d R 2 + R 2 dϕ12 + . . . near x, we see from Eq. (67) that r 2 = R 2 (a + O(R 2 )) near our point on the axis for some a > 0. Thus, we may exchange O(R 2 ) with O(r 2 ), and similarly, we may exchange O( R˜ 2 ) (defined with respect to corresponding ˜ for O(r 2 ). coordinates near x˜ ∈ M) Actually, by arguments parallel to those in the proof of Lemma 7 and [21, Sect. 3], the matrix f takes the following form near (z j + δ, z j+1 − δ): f =

O(r 2 ) r 2 e2ν + O(r 4 ) , O(r 2 ) di j + O(r 2 )

(101)

where di j is an invertible (D − 4)-dimensional square matrix, and similarly for f˜i j of the second solution (here e2 ν is replaced by e2ν˜ and di j by d˜i j ). It follows from this expression and Eq. (100) that σ is finite near (z j + δ, z j+1 − δ), i.e. near any axis away from places where they intersect each other or the horizon, and away from infinity. (2) Horizon. On any compact sub-interval of the interval I H associated with the horizon, the matrices f i j , f˜i j are invertible, so σ is finite there. (3) Infinity. Near infinity, we use the asymptotic form of the metric (78), where we recall that r = ρ sin θ, z = ρ cos θ . This asymptotic form holds of course for both g and g, ˜ and gives us the asymptotic behavior of f i j and f˜i j since we can simply read off the scalar products of the axial Killing fields. We thereby immediately find that f − f˜ ⎛

⎞ O(1)(1 + cos θ ) O(1) sin θ O(ρ −1/2 )(1+cos θ )1/2 =⎝ O(1) sin θ O(1)(1 − cos θ ) O(ρ −1/2 )(1−cos θ )1/2 ⎠. −1/2 1/2 −1/2 1/2 O(ρ )(1 + cos θ ) O(ρ )(1−cos θ ) O(ρ −1 ) (102)

It is also easy to estimate the size of the matrix elements of the inverse; we find f −1 = ρ −2 sin−2 θ (1 + O(ρ −1 )) ⎞ ⎛ ρ(1 − cos θ ) O(1) sin θ O(ρ 1/2 )(1 − cos θ)1/2 ×⎝ O(1) sin θ ρ(1 + cos θ ) O(ρ 1/2 )(1 + cos θ)1/2 ⎠ , O(ρ 1/2 )(1 − cos θ )1/2 O(ρ 1/2 )(1 + cos θ )1/2 ρ 2 sin2 θ(δi j + O(ρ −1 ))

(103) with the same expression for f˜−1 , and we also find det f = ρ 2 sin2 θ (1 + O(ρ −1 ))

(104)


667

again with the same expression for det f˜. It is important to emphasize that these expressions hold for large ρ, and uniformly in θ including the axis θ = 0, π . From these expressions we immediately conclude that −1+

det f = O(ρ −1 ) , det f˜

f i j ( f˜i j − f i j ) = O(ρ −1 ) ,

(105)

again uniformly in θ , including the axis. These expressions show that the first two, and last term in Eq. (100) go to zero at infinity (including at the axis). For the middle term in Eq. (100), we also need to analyze the twist potentials χi and χ˜ i . To evaluate the twist potentials at a generic coordinate (ρ, θ ), we may take a path in Eq. (87) that moves outwards to infinity along the axis θ = 0, and then follows a half circle of constant ρ in the asymptotic region, or alternatively one that moves outwards along the axis θ = π . Either paths will give the same result, because the twist 1-forms are closed. On the axis and away from the horizon, the twist 1-forms vanish, whereas in the asymptotic region, we can use the asymptotic form of the metric, Eq. (78) derived above, and the corresponding asymptotic expansion of the twist 1-forms ωi . In the coordinates (ρ, θ ), the projected twist 1-forms ωˆ i on Mˆ are found to be given by ωˆ i = O(ρ 1/2 ) sin θ dθ + O(sin θ )O(ρ −1/2 ) dρ.

(106)

The same expression holds for the twist 1-forms of the metric g. ˜ 12 We now form the difference between the twist 1-forms for both metrics g and g, ˜ and we integrate this difference along the paths just described. The twist potentials χi respectively χ˜ i are already known to be identical on the axis θ = 0, π for sufficiently large ρ, since we have already argued that they have to be proportional to Ji and J˜i respectively, and we are assuming that J˜i = Ji . Therefore, the integration gives χi − χ˜ i = O(sin2 θ )O(ρ 1/2 ).

(108)

Together with the estimates for f i j and det f˜ given above this yields f i j (χi − χ˜ i )(χ j − χ˜ j ) = O(ρ −1 ) , det f˜

(109)

uniformly in θ , including the axis θ = π, 0. Thus, we have shown that all three terms in σ in Eq. (100) are of order O(ρ −1 ) uniformly in θ including the axis. Thus, σ tends to zero near infinity, as we wanted to show. 12 A more accurate analysis using the asymptotics (83) would give the better estimate

ωˆ i = ×

1 (2π )3−D Ji 2 (1 − (−1)i cos θ ) sin θ (1 + O(ρ −1 )) dθ + O(sin2 θ )dρ for i = 1, 2, sin θ (1 + O(ρ −1 )) dθ + O(sin2 θ )dρ

and corresponding expressions for the potentials.

for i = 3, . . . , D − 3,

(107)

668


(4) An axis meets the horizon. Consider two adjacent intervals I H , I H +1 , where I H represents the horizon. On open interval I H , the matrices f and f˜ are non-singular, because no linear combination of the axial Killing fields ψi can vanish at any point of the horizon away from the axis. On I H +1 a linear combination a(I H +1 )ψi = 0 vanishes. By Lemma 2, we can find a matrix B ∈ S L(D − 3, Z) such that T a(I . , 0). Thus, redefining the axial Killing fields as ψi → H +1 )B = (1, 0, . .−1 A ψ and A = B if necessary, we can assume without loss of generality j ij j that a(I H +1 ) = (1, 0, . . . , 0), so that ψ1 = 0 at points represented by I H +1 . Shifting the z-coordinate by a constant if necessary, we may also assume that I H = (z 1 , 0), Ih+1 = (0, z 2 ), so that the intersection point of interest is (r, z) = (0, 0). Let x ∈ M be a point on the horizon that is also on the axis where ψ1 = 0, so ˆ By Lemma 4, that the orbit Ox corresponds to (0, 0) in the upper half plane M. we can introduce coordinates (τ, y1 , . . . , y D−1 ) in a neighborhood of x such that the action of the rotational part of the isometry group takes the form described in Lemma 4, and such that the time-part of the isometry group acts by simply shifting τ by a constant. In other words, the isometry group acts locally by shifting τ, y4 , . . . , y D−1 , and it acts by rotating y1 +i y2 by a phase. It is then straightforward to see that ∂/∂ y3 must be orthogonal to the horizon and outward pointing in the ˆ tangent space at x. Thus, we can parametrize the orbit space M by the coordinates R = y12 + y22 and Y = y3 . Outside the horizon, the range of these coordinates is {(R, Y ) | R > 0, Y > Y (R)}, where Y (R) is an analytic function which has an expansion in even powers of R, and for which Y (0) = 0. Actually, as shown in [7], by modifying the coordinates by a polynomial transformation, one can furthermore achieve that as many derivatives of Y (R) at R = 0 are made to vanish, and we are assuming that this has been done. The choice of coordinates (R, Y ) displays the nature of the orbit space as a manifold with a corner at the place where the horizon meets the axis. The canonical orbit space coordinates (r, z) in effect straighten out this corner. Any function F on M that is analytic in an open neighborhood of x and is invariant under G has a convergent power series expansion in even powers of R, F(x) = Fm,n Y m R n , (110) m∈Z,n∈2Z

as is in particular the case for any component of f i j or χi . In fact, since R = 0 corresponds to the places near x where ψ1 = 0, we have f =

R 2 (1 + O(R 2 )) R 2 O(1) , 2 2 2 R O(1) di j + O(R + Y )

(111)

where di j is an invertible (D − 4)-dimensional matrix, and that χi = Ji + O(R 2 ). It follows that the matrix inverse of f i j is of the form f −1 =

R −2 O(1) O(1) . O(1) O(1)

(112)

˜ Here, we have The exact same analysis can obviously also be carried out for M. ˜ Y˜ ), and a corresponding expansion for f˜i j , χ˜ i . coordinates ( R, In order to show that σ is bounded in a neighborhood of the horizon where an axis meets the horizon, we use Eq. (100), and the estimates for f i j , χi resp. f˜i j , χ˜ i


669

that we have just derived. To make this work, however, we must understand first ˜ Y˜ ) → (r, z) near (0, 0). A detailed the relation (map) (R, Y ) → (r, z) and ( R, analysis of these maps was carried out in Sect. 6.5 of [7], and the relevant results of this analysis for our purpose may be summarized as follows: In the sector r < z, we have a R 2 = r (1 + O(z)), Y 2 = a(z + r 2 + z 2 ) + O(z 5/2 ) 4 (113) a ˜ 2 2 5/2 2 2 ˜ ˜ ˜ + r + z ) + O(z ). R = r (1 + O(z)), Y = a(z 4 In the sector r ≥ z, we have instead (Y + i R)2 = a(z + ir ) + O(r 2 + z 2 ) , ˜ 2 = a(z (Y˜ + i R) ˜ + ir ) + O(r 2 + z 2 ).

(114)

These estimates can be used, in either case, to prove | f i j ( f i j − f˜i j )| ≤ O(1),

| f i j (χi − χ˜ i )(χ j − χ˜ j )| |det f | ≤ O(1), ≤ O(1), ˜ |det f | |det f˜| (115)

in either sector. In view of Eq. (100), it follows that σ is uniformly bounded in a neighborhood of any intersection point of an axis and a horizon. (5) Two axis meet. Consider two adjacent intervals I j−1 , I j , each representing an axis where the linear combinations ai (I j−1 )ψi respectively ai (I j )ψi vanish. The first step is again to change the action of the rotational symmetries generated by the Killing fields ψi to a convenient form. By Lemma 2, we can find a matrix B ∈ S L(D − 3, Z) such that a(I j )B T = (1, 0, . . . , 0) and such that a(I j−1 )B T = (0, 1, . . . , 0). Thus, redefining the axial Killing fields as ψi → j Ai j ψ j and A = B −1 if necessary, we can assume without loss of generality that a(I j ) = (1, 0, . . . , 0) and a(I j−1 ) = (0, 1, . . . , 0). Furthermore, we redefine the coordinate z by an additive constant so that the intersection point z j of the two intervals is 0, as this is going to simplify some of our formulas. Now let x ∈ M be a point covering our intersection point (0, 0) = Ox on the boundary of our orbit space Mˆ = {z + ir | r ≥ 0}. We have just argued that, at x, we may assume that ψ1 = 0 = ψ2 , and that the remaining ψi are non-vanishing there. Hence, we can introduce canonical coordinates (τ, y1 , . . . , y D−1 ) in a neighborhood of x such that the action of the rotational part of the isometry group takes the form described in 4, and such that the time-part of the isometry group acts by simply shifting τ by a constant. In other words, the isometry group acts locally by shifting τ, y5 , . . . , y D−1 , and it acts by rotating y1 + i y2 resp.y3 + i y4

by independent phases. Thus, the quantities R1 = y12 + y22 and R2 = y32 + y42 are alternative coordinates of M/G near Ox displaying clearly the character of the intersection point as a corner. We need to understand the relationship of the coordinates (R1 , R2 ) to (r, z) near r = 0 = z, and we need to understand the behavior of the fields f i j , χi in terms of either coordinate system. In a similar way, let x˜ ∈ M˜ be a point covering our intersection point (0, 0) = Ox , and introduce as above coordinates R˜ 1 , R˜ 2 . We also need to understand the relationship of the coordinates

670


( R˜ 1 , R˜ 2 ) to (r, z) near r = 0 = z, and we need to understand the behavior of the fields f˜i j , χ˜ i in terms of either coordinate system. We first note that, if F is any analytic function on M that is defined near x and that is invariant under the action of G, it will locally have a convergent expansion of the form F(x) = Fm,n R1m R2n , (116) m,n∈2Z

i.e. in even powers of R1 , R2 . This applies e.g. to any component of f i j , χi . It follows from this and the explicit form of that action of G in the canonical coordinates introduced above that ⎛ ⎞ R12 (1 + O(R12 )) R12 R22 O(1) R12 O(1) ⎜ ⎟ f = ⎝ R12 R22 O(1) (117) R22 (1 + O(R22 )) R22 O(1) ⎠, R22 O(1) di j + O(R12 + R22 ) R12 O(1) where di j is an invertible (D − 5)-dimensional matrix, and that χi = R12 R22 O(1). As a consequence, the matrix inverse of f i j is of the form ⎛

f −1

⎞ O(1) O(1) R1−2 O(1) ⎜ ⎟ = ⎝ O(1) R2−2 O(1) O(1) ⎠ . O(1) O(1) O(1)

(118)

Similarly, any corresponding function F˜ on M˜ that is defined near x˜ will have an even expansion in R˜ 1 , R˜ 2 . This applies to any component of f˜i j , χ˜ i , and in fact these fields will again have the form (117) and (118), with R1 , R2 now replaced by R˜ 1 , R˜ 2 . We would now like to exploit these estimates above in Eq. (100) to estimate σ near, but to do this we must first relate (R1 , R2 ) and ( R˜ 1 , R˜ 2 ) to the coordinates (r, z). This was done in a very similar context in Sect. 6.5 of [7]. A relevant result can be stated saying that

r 2 + z 2 + O(r 2 + z 2 ) , = a˜ 1/2 ±z + r 2 + z 2 + O(r 2 + z 2 ),

R1/2 = a R˜ 1/2

1/2

±z +

(119)

in an open neighborhood of (r, z) = (0, 0), where a, a˜ are positive constants. We divide this open neighborhood into three sectors: The sector where r > |z|, the sector where 0 ≤ r ≤ z, and the sector where 0 ≤ r ≤ −z. In the first sector where r > |z|, we have from Eq. (119), −2 R1/2 ≤ a −1 r −1 + O(r 1/2 ) ,

−2 R˜ 1/2 ≤ a˜ −1r −1 + O(r 1/2 ) ,

(120)

as well as |χi | ≤ cr 2 + O(ρ 7/2 ) and |χ˜ i | ≤ cr ˜ 2 + O(r 7/2 ). It follows immediately from these estimates together with Eqs. (117) and (118) that the estimates (115) hold uniformly in the sector r > |z|. Hence, σ is uniformly bounded in this sector


671

by Eq. (100). Next, consider the sector 0 ≤ r ≤ z. We claim that in this sector, the estimate (119) can be improved to a R22 = r (1 + O(z)) , R12 = a(z + r 2 + z 2 ) + O(z 5/2 ), 4 (121) a ˜ 2 ˜ + r 2 + z 2 ) + O(z 5/2 ) , R˜ 2 = r (1 + O(z)) , R˜ 12 = a(z 4 uniformly throughout the sector. With these estimates, it again follows that σ ≤ O(1) in uniformly in the sector using Eqs. (117), (118) and (100). The remaining sector 0 ≤ r ≤ −z is treated in exactly the same way, except that the roles of R1 and R2 resp. R˜ 1 and R˜ 2 are now reversed. Thus, what remains is to show Eq. (121). Consider the mappings w, w˜ which assign to each (r, z) ∈ Mˆ in an open neighborhood of (0, 0) in the upper complex half-plane the complex numbers w = (R1 + i R2 )2 resp. w˜ = ( R˜ 1 + i R2 )2 , where R1/2 resp. R˜ 1/2 are the coordinates of the orbit labeled by (r, z) that we introduced above. If the spacetime manifolds M and M˜ were R4,1 × T D−5 with the standard action of U (1) D−3 , then the mappings z + ir → w and z + ir → w˜ would be the identity map on the complex upper half plane. In the present case, we are of course not in this situation, but in Riemannian normal coordinates (y1 , . . . , y D−1 ) resp. ( y˜1 , . . . , y˜ D−1 ) on M resp. M˜ that we introduced above in Lemma 4, the action of U (1) D−3 is identical, and furthermore, the metrics g and g˜ do not differ much in a neighborhood of the origin in these Riemannian normal coordinates. Hence, it is plausible that the maps w resp. w˜ do not differ much from the identity map. By going through the precise definition of r, z carefully, it was shown13 in Sect. 6.5 of [7] that this is indeed the case in the sense that both w and w˜ can be extended to smooth maps in an open neighborhood of the origin in the complex plane, satisfying w = a(z + O(z 2 ) + ir (1 + O(z))) , w˜ = a(z ˜ + O(z 2 ) + ir (1 + O(z))), (122) in our sector. Solving for R1/2 resp. R˜ 1/2 in terms of w resp. w˜ then gives the desired result (121). Thus, we have altogether shown that σ remains bounded in the neighborhood of any point where two intervals representing an axis meet. Thus we have argued that σ remains bounded, including the axis, horizon segment, and tends to zero near infinity. As we have explained, this concludes the proof of the theorem. 6. Conclusions and Outlook In this paper, we have proved a uniqueness theorem for D-dimensional stationary, asymptotically Kaluza-Klein black hole spacetimes satisfying the vacuum Einstein equations, allowing a group of isometries G = R×T D−3 . We showed that the solutions are uniquely determined by certain combinatorial data specifying the group action, certain moduli, and the angular momenta. This combinatorial data in particular determines the topology of the spacetime outside the black hole, and the topology of the horizon. 13 In that reference, the authors in fact considered a neighborhood of a point where an axis hits the horizon. However, the relevant calculations leading to the relevant conclusions about w and w˜ remain exactly the same.

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To be able to prove our uniqueness theorem, we also had to make a number of technical assumptions. They mainly concern the analyticity of the metric and the causal structure of the spacetime. One feels that it ought to be possible to remove these assumptions, but it is not clear to us how this could be done in practice. The more unsatisfactory aspect of our analysis is that we have not been able to prove or disprove the existence of smooth black hole solutions associated with more elaborate topological structure/combinatorial data, such as “black lenses” etc. Some partial results have been obtained in the literature on this (see e.g. [4]), but the general situation is still unclear. Acknowledgements S.H. would like to thank Iskander Aliev for discussions about lattices, and Piotr Chru´sciel for extensive discussions on manifolds with torus actions. S.Y. gratefully acknowledges support by the Alexander von Humboldt Foundation and the Sofia University Research Fund under grant No 111.

A. Proof of Lemma 7 Lemma 7. The length of the horizon interval satisfies (2π ) D−3l H = κ A H ,

(123)

where A H is the area of the horizon cross section H, and where κ > 0 is the surface gravity. Proof. We take the horizon to correspond to the interval z ∈ (z 1 , z 2 ), r = 0 on the boundˆ Let v = (1, 1 , . . . , D−3 ). Then by definition G I J v I v J = ary of the orbit space M. g(K , K ), where K is the Killing vector (3), which is tangent to the null generators of the horizon H , so G I J v I v J = 0 on H . It then follows e.g. from the min-max principle that G I J v J = 0 on the horizon, so limr →0 G I J v J = 0 in the orbit space for z ∈ (z 1 , z 2 ). As was shown in [21, Sec. 3], one can furthermore use the first reduced Einstein equation (60) to show that limr →0 G I J v J /r = 0 for z ∈ (z 1 , z 2 ). Let us now choose coordinates (u, r, ϕ1 , . . . , ϕ D−3 ) near H such that K = ∂/∂u, ψi = ∂/∂ϕi . Let us define Xˆ I as X I above in Eq. (59), with t replaced by ˆ and K , and let Gˆ I J = g( Xˆ I , Xˆ J ). Then the reduced Einstein equations also hold for G, furthermore, near r = 0 and z ∈ (z 1 , z 2 ), we have 2 −r det f −1 O(r 2 ) , Gˆ ∼ O(r 2 ) fi j

(124)

up to terms of higher order in r . Here, z ∈ (z 1 , z 2 ), and f i j (z) is the limit as r → 0 of g(ψi , ψ j ). Following [21, Sect. 3], the second reduced Einstein equation (61) furthermore gives 1 ∂r ν → 0, ∂z ν → − ∂z log det f, as r → 0, z ∈ (z 1 , z 2 ). 2

(125)

We conclude from the last relation that e−2ν → c2 det f for some constant c > 0 as r → 0, z ∈ (z 1 , z 2 ). From the form of the metric given in Eq. (67) (with G replaced by ˆ it follows that, near H , we have G),


g ∼ e2ν (dz 2 + dr 2 − c2 r 2 du 2 ) +

D−3

673

f i j (z) dϕi dϕ j + 2r 2

i, j=1

= e2ν (dz 2 + dU dV ) +

D−3 i, j=1

f i j (z) dϕi dϕ j +

D−3

O(1) dudϕi

i=1 D−3 1 O(1) (V dU − U dV )dϕi , c i=1

(126) The minus sign in front of the du 2 -term follows from the fact that K is timelike in a neighborhood outside H , which in turn follows directly from ∇a (K b K b ) = −2κ K a . In the last line we switched to Kruskal-like coordinates U, V defined by U V = r 2 , U/V = e2cu . It is apparent in these coordinates that H corresponds to V = 0. The restriction of K = ∂/∂u to H is found to be cU ∂/∂U , from which one concludes in view of the equation K a ∇a K b = κ K b on H that c = κ. The lemma may now be proven by calculating the horizon area in the coordinates z, ϕi using the above form of the metric and e−2ν = κ 2 det f . It is z 2 2π 1 AH = dz dϕi e2ν det f = (2π ) D−3 (z 2 − z 1 ), (127) κ z1 0 i

from which the lemma follows immediately in view of l H = z 2 − z 1 .

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21. Harmark, T.: Stationary and axisymmetric solutions of higher-dimensional general relativity. Phys. Rev. D 70, 124002 (2004) 22. Harmark, T.: Talk available at http://online.itp.ucsb.edu/online/highdgr06/harmark1/pdf/Harmark_KITP. pdf, 2006 23. Hawking, S.W.: Black holes in general relativity. Commun. Math. Phys. 25, 152–166 (1972) 24. Hawking, S.W., Ellis, G.F.R.: The large scale structure of space-time. Cambridge: Cambridge University Press, 1973 25. Hirzebruch, F.: Differentiable manifolds and quadratic forms. Lect. Notes. Univ. of California, Berkely (1962) 26. Hollands, S., Ishibashi, A., Wald, R.M.: A higher dimensional stationary rotating black hole must be axisymmetric. Commun. Math. Phys. 271, 699 (2007) 27. Hollands, S., Ishibashi, A.: On the ‘Stationary Implies Axisymmetric’ Theorem for Extremal Black Holes in Higher Dimensions. Commun. Math. Phys 291, 403–441 (2009) 28. Hollands, S., Yazadjiev, S.: Uniqueness theorem for 5-dimensional black holes with two axial Killing fields. Commun. Math. Phys. 283, 749 (2008) 29. Hollands, S., Yazadjiev, S.: A Uniqueness theorem for 5-dimensional Einstein-Maxwell black holes. Class. Quant. Grav. 25, 095010 (2008) 30. Israel, W.: Event horizons in static vacuum space-times. Phys. Rev 164, 1776–1779 (1967) 31. Kastor, D., Ray, S., Traschen, J.: The First Law for Boosted Kaluza-Klein Black Holes. JHEP 0706, 026 (2007) 32. Kerner, R.: Generalization of Kaluza-Klein theory for an arbitrary non-abelian gauge group. Ann. Inst. H. Poincarè 9, 143 (1968) 33. Kobayshi, S., Nomizu, K.: Foundations of Differential Geometry I. New york: Wiley, 1969 34. Larsen, F.: Rotating Kaluza-Klein black holes. Nucl. Phys. B 575, 211 (2000) 35. Maison, D.: Ehlers-Harrison-type Transformations for Jordan’s extended theory of graviation. Gen. Rel. Grav. 10, 717 (1979) 36. Massey, W. S.: Algebraic Topology: An Introduction. Berlin-Heidelberg-New york: Springer, 1977 37. Morisawa, Y., Ida, D.: A boundary value problem for five-dimensional stationary black holes. Phys. Rev. D 69, 124005 (2004) 38. Mazur, P.O.: Proof of uniqueness of the Kerr-Newman black hole solution. J. Phys. A 15, 3173–3180 (1982) 39. Moncrief, V., Isenberg, J.: Symmetries of cosmological Cauchy horizons. Commun. Math. Phys. 89, 387–413 (1983) 40. Moncrief, V., Isenberg, J.: Symmetries of Higher Dimensional Black Holes. Class. Quant. Grav. 25, 195015 (2008) 41. Myers, R.C., Perry, M.J.: Black holes in higher dimensional space-times. Annals Phys 172, 304 (1986) 42. Oh, H.S.: Topology and Its Applications 13, 137–154 (1982) 43. Orlik, P., Raymond, F.: Actions of the torus on 4-manifolds I. Transactions of the AMS 152(2), 531–559 (1972) 44. Orlik, P., Raymond, F.: Actions of the torus on 4-manifolds II. Topology 13, 89–112 (1974) 45. Pomeransky, A.A., Sen’kov, R.A.: Black ring with two angular momenta. http://arXiv.orglabs/hep-th/ 0612005v1, 2006 46. Racz, I.: On further generalization of the rigidity theorem for spacetimes with a stationary event horizon or a compact Cauchy horizon. Class. Quant. Grav 17, 153 (2000) 47. Rasheed, D.: The Rotating dyonic black holes of Kaluza-Klein theory. Nucl. Phys. B 454, 379 (1995) 48. Robinson, D.C.: Uniqueness of the Kerr black hole. Phys. Rev. Lett. 34, 905–906 (1975) 49. Rogatko, M.: Uniqueness theorem of static degenerate and non-degenerate charged black holes in higher dimensions. Phys. Rev. D 67, 084025 (2003) 50. Rogatko, M.: Classification of static charged black holes in higher dimensions. Phys. Rev. D 73, 124027 (2006) 51. Sudarsky, D., Wald, R.M.: Extrema of mass, stationarity, and staticity, and solutions to the Einstein Yang-Mills equations. Phys. Rev. D 46, 1453–1474 (1992) 52. Wald, R.M.: General Relativity. Chicago: University of Chicago Press, 1984 53. Weinstein, G.: On rotating black holes in equilibrium in general relativity. Commun. Pure Appl. Math. 43, 903 (1990) 54. Weinstein, G.: On the Dirichlet problem for harmonic maps with prescribed singularities. Duke Math. J. 77(1), 135–165 (1995) (See Lemma 8) Communicated by P.T. Chru´sciel

Commun. Math. Phys. 302, 675–696 (2011) Digital Object Identifier (DOI) 10.1007/s00220-010-1181-x

Communications in


On Semi-Classical States of Quantum Gravity and Noncommutative Geometry Johannes Aastrup1 , Jesper Møller Grimstrup2 , Mario Paschke1 , Ryszard Nest3 1 Mathematical Institute, University of Münster, Einsteinstrasse 62, D-48149 Münster, Germany.


2 The Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17,

DK-2100 Copenhagen, Denmark. E-mail: [email protected]; [email protected]

3 Mathematical Institute, University of Copenhagen, Universitetsparken 5,

DK-2100 Copenhagen, Denmark. E-mail: [email protected] Received: 14 August 2009 / Accepted: 26 August 2010 Published online: 11 February 2011 – © Springer-Verlag 2011

Abstract: We construct normalizable, semi-classical states for the previously proposed model of quantum gravity which is formulated as a spectral triple over holonomy loops. The semi-classical limit of the spectral triple gives the Dirac Hamiltonian in 3+1 dimensions. Also, time-independent lapse and shift fields emerge from the semi-classical states. Our analysis shows that the model might contain fermionic matter degrees of freedom. The semi-classical analysis presented in this paper does away with most of the ambiguities found in the initial semi-finite spectral triple construction. The cubic lattices play the role of a coordinate system and a divergent sequence of free parameters found in the Dirac type operator is identified as a certain inverse infinitesimal volume element. Contents 1. 2. 3. 4.

5. 6. 7.

Introduction . . . . . . . . . . . . . . . Noncommutative Geometry . . . . . . . Ashtekar Variables and Holonomy Loops Spectral Triples of Holonomy Loops . . 4.1 Holonomy loops . . . . . . . . . . . 4.2 Generalized connections . . . . . . 4.3 A spectral triple over A . . . . . . 4.4 The limiting spectral triple . . . . . The Space of Connections . . . . . . . . The Quantization of the Poisson Bracket Semiclassical Analysis . . . . . . . . . . 7.1 Coherent states on a Lie group . . . 7.2 Product states . . . . . . . . . . . . 7.3 Semi-classical states: one copy of G 7.4 Determining the sequence {an } . . . 7.5 Three copies of G . . . . . . . . . .

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J. Aastrup, J. M. Grimstrup, M. Paschke, R. Nest

7.6 Semiclassical states on A . . . . . . . . . . . . . . . . . . . . . . . . . 691 7.7 The Dirac Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . 692 Discussion & Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694

1. Introduction A critical test of any quantum model is the existence of a semi-classical limit. This limit - its existence once established - should make contact to known physics, explain qualitative and quantitative results, and thereby render credibility to the model. Most importantly, the semi-classical limit serves to confirm the operational interpretation of the observables of the model. Furthermore, as there exist infinitely many in-equivalent quantizations of classical field theories, the semi-classical limit often provides an important tool to distinguish physical relevant models. The semi-finite spectral triple over a configuration space of connections constructed in [1–6] constitute a non-perturbative quantum model. The spectral triple emerges from a fusion between noncommutative geometry [7,8] and canonical quantum gravity [9–11]. It involves an algebra of holonomy loops and a Dirac type operator that resembles a global functional derivation operator. Its existence - as a mathematical entity - was established in [4,5]. Its interpretation in terms of a non-perturbative quantum field theory is immediate since the interaction between the algebra and the Dirac type operator reproduces the Poisson bracket of general relativity, formulated in terms of Ashtekar variables, and of Yang-Mills theory. What remained unresolved, in the papers [1–6], was the exact physical interpretation of the spectral triple construction. It was not clear whether the model should be understood in terms of gravity or Yang-Mills theory, or something else. In particular, no substantial results concerning a semi-classical limit were obtained. In this paper we make the first steps towards a semi-classical analysis. Drawing on results by Hall [12,13] concerning coherent states on compact Lie-groups, we construct semi-classical states over the configuration space of connections. This analysis enlightens us on two fronts: First, at a conceptual level, the semi-classical analysis entails a clearer physical interpretation of the semi-finite spectral triple. In particular, we find that the Dirac type operator descends, in this limit, to a Dirac Hamiltonian on a 3+1 dimensional ultra-static space-time. Through a careful analysis of the Poisson structure of general relativity we first obtain an interpretation of the constituents of the Dirac type operator as quantized triad field operators. In short, the Dirac type operator appears as an infinite sum of quantized triad field operators. In the semi-classical limit, these triad operators entail classical triad fields which appear in the classical Dirac operator. The special class of semi-classical states constructed in this paper suggest an interpretation as one-fermion states for a spinor field on the ultra-static space-time. This interpretation has, however, a problem since the scalar product induced on this space depends on the chosen coordinates. Nevertheless, we believe that our analysis indicates that the semi-finite spectral triple should be understood in terms of quantum gravity coupled to quantized matter fields. Indeed, if the time-scale is chosen appropriately, then the scalar product becomes coordinate independent. Second, at a more technical level, the semi-classical analysis resolves several questions and ambiguities concerning the construction of the semi-finite spectral triple. For instance, the triple is built over a countable system of nested graphs. In [5] it was clear that the construction would work for a large class of such systems of graphs and no

On Semi-Classical States of Quantum Gravity and Noncommutative Geometry

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mechanism was found to single out one system of graphs from another. Furthermore, it was also clear that two spectral triples, based on different systems of graphs, would constitute entirely different models. This ambiguity is resolved through the semi-classical analysis: we find that a system of cubic lattices is singled out as “natural” with an interpretation as a choice of a coordinate system. This coordinate system is made to coincide with the coordinate system used to write down the Ashtekar variables and their Poisson bracket. Moreover, the construction of the Dirac type operator involves an infinite series of free parameters which is required to diverge in order for the operator to have a compact resolvent. In the papers [1–6] no clear physical interpretation of these parameters were found. Again, the semi-classical analysis resolves this ambiguity: it identifies the series of free parameters as the inverse infinitesimal, Euclidean volume element, the divergence arising through a continuum limit where the volume elements approach zero. Clearly, the introduction of finite graphs breaks diffeomorphism invariance. In loop quantum gravity [9–11], which is also based on an inductive system of graphs [14–16], the philosophy is to include all1 possible graphs and thereby restore the symmetries in the inductive limit of graphs and Hilbert spaces. This renders the limiting Hilbert space non-separable, something which probably obstructs the construction of a spectral triple [1]. In this paper we find that the constructed semi-classical limit does not depend on finite parts of the inductive system of lattices. Thus, in this limit the lattices seemingly disappear and the symmetries, broken by the initial choice of graphs, are restored. This means that the expressions for the classical Dirac operator and the Dirac Hamiltonian, found in the semi-classical limit, are coordinate covariant. The finding that cubic lattices are singled out by the semi-classical analysis plays well with recent results by Flori and Thiemann which state that, in loop quantum gravity, only lattices with cubic topology give the right semi-classical limit [17]. This paper is organized as follows: In Sect. 2 we briefly review noncommutative geometry and Connes work on the standard model. In Sect. 3 we introduce Ashtekar variables together with their dual variables, the loop and flux variables. In Sect. 4 we then review the construction of the semi-finite spectral triple. First, a spectral triple is constructed on a fixed graph, and subsequently a continuum limit of spectral triples is taken over an infinite system of ordered graphs. In Sect. 5 we comment on the underlying space of generalized connections and Sect. 6 is concerned with a careful analysis of the relationship between the spectral triple construction and the Poisson bracket between flux and loop variables. Finally, Sect. 7 is concerned with the semi-classical states. In Sect. 8 we give a conclusion. 2. Noncommutative Geometry It is a central observation in noncommutative geometry, due to Connes, that the metric of a compact manifold can be recovered from the Dirac operator together with its interaction with the smooth functions on the manifold [7]. In other words the metric is completely determined by the triple (C ∞ (M), L 2 (M, S), D). This observation leads to a noncommutative generalization of Riemmanian geometries. Here the central objects are spectral triples (A, H, D), where A is a not necessarily commutative algebra; H a Hilbert space and D an unbounded self-adjoint operator called 1 To be precise, all piece-wise analytic graphs.

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the Dirac operator. The triple is required to satisfy some interplay relations between A, H, D mimicking those of (C ∞ (M), L 2 (M, S), D). The choice of the Dirac operator D is strongly restricted by these requirements. In physics, a key example of a noncommutative geometry comes from particle physics. Again, it was Connes who realized that the entire data of the standard model coupled to general relativity can be understood as a single, gravitational model formulated in terms of a spectral triple [8,18–23]. Here, the algebra is an almost commutative algebra A = C ∞ (M) ⊗ A F , where A F is the algebra C⊕H⊕ M3 (C). The corresponding Dirac operator then consists of two parts, D = DM + DF , one of which is the standard Dirac operator D M on M. The other part, D F , is given by a matrix-valued function on the manifold M, that encodes the metrical aspects of the states over the algebra A F . It is a highly nontrivial and very remarkable fact that the above mentioned requirements for Dirac operators force D F to contain the non-abelian gauge fields of the standard model and the Higgs-field together with their couplings to the elementary fermion fields. In particular the Higgs-field thus obtains a geometrical interpretation as being a part of the gravitational field on a noncommutative space. Even more so, the classical action of the standard model coupled to the Einstein-Hilbert action, in the Euclidean signature, emerges from the spectral triple through the so-called spectral action principle [18], which states that physics only depends on the spectrum of the Dirac operator. In view of the widely held opinion that quantum effects of the gravitational field will necessarily lead to a noncommutativity of space-time this observation indicates that the gauge interactions and the appearance of the Higgs field may be interpreted as quantum effects of the gravitational interactions. In other words they are the first shadows of the noncommutativity of space-time, visible at the length scale corresponding to the Z -mass. It should be mentioned in this respect that the spectral action does not directly reproduce the correct coupling constants of the standard model. In fact it only allows for lesser free parameters than the standard model. In order to obtain the measured coupling constants for the electromagnetic and strong interactions to a fairly good approximation, Connes and Chamseddine applied renormalization group methods in [19] and subsequent publications. This analysis ultimately leads to a prediction of the Higgs mass [21]. The predicted value, which was based on the assumption of “the big desert”, was recently excluded by Tevatron data. Nevertheless it is very remarkable that the use of quantum field theoretical concepts is absolutely essential here to obtain a physically reasonable classical action. To our point of view this strongly indicates that the spectral triple used by Connes and Chamseddine should be viewed as the semi-classical low energy limit of some genuine quantum theory. One may then also hope that other quantum corrections present in the full theory provide a more realistic value for the Higgs mass. Since the noncommutative description of the standard model is entirely gravitational this full theory should, presumably, be a theory of the quantized gravitational field. Thus, if there were already a theory of quantum gravity one should certainly investigate whether it admits some semi-classical states that resemble this almost commutative spectral triple. It was these considerations which motivated the construction of the semi-finite spectral triple over a configuration space of connections [1–6]. The idea is to seek a general


679

framework which combines the machinery and ideas of noncommutative geometry with elements of quantum gravity. The final goal, then, is to make contact to Connes work on the standard model through the formulation of a semi-classical analysis. 3. Ashtekar Variables and Holonomy Loops We start with some notation. Let M be a 4-dimensional globally hyperbolic manifold with a vierbein E μA and a space-time metric G μν = E μA E νB η AB , where η AB = diag(−1, 1, 1, 1) is the corresponding tangent space metric. Here the letters μ, ν, . . . and A, B, . . . denote curved and flat space-time indices respectively. Next, take a foliaa e be tion of M according to M = R × , where is a spatial manifold. Let gmn = em na a the corresponding spatial metric and em the spatial dreibein. Here the letters m, n, . . . and a, b, . . . denote curved and flat spatial indices. The Ashtekar variables [24,25] consist first of a complex SU (2) connection Aam (x) on . The Ashtekar connection is a certain complex linear combination of the spatial spin connection and the extrinsic curvature of in M. The canonically conjugate variable to Aam (x) is the inverse densitized dreibein E¯ am = eeam , a ). This set of variables satisfy the Poisson bracket where e = det(em n (3) δ (x, y), {Aam (x), E¯ bn (y)} = κδba δm

where κ is the gravitational constant. The formulation of canonical gravity in terms of connection variables permits a shift to loop variables which are taken as the holonomy transform h l (A) = Pexp Am d x m , l

along a loop l in . To define a conjugate variable to h l (A) let d Fa be the flux of the triad field E¯ am corresponding to an infinitesimal area element of the spatial manifold , which can be written d Fa = mnp E¯ am d x n ∧ d x p . Given a 2 dimensional surface S in we write the total flux of E¯ am through S, FSa = d F a . S

Next, consider a surface S and let l = l1 · l2 be a line segment in which intersect S at the point l1 ∩ l2 . The Poisson bracket between the flux and holonomy variables read [9] {h l , FSa } = ι(S, l)κh l1 τa h l2 , where τ denote the generators of the Lie algebra of G. Here, ι is given by ι(S, l) = ±1, 0, depending on the intersection between S and l.

(1)

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Fig. 1. A plaquet in the lattice

4. Spectral Triples of Holonomy Loops In this section we outline the construction of the semi-finite spectral triple first presented in [3,4] and further developed in [5]. This spectral triple combines ideas and techniques of canonical gravity and noncommutative geometry. We first construct a spectral triple at the level of a finite graph. Next we take the limit of such spectral triples, over an infinite system of ordered graphs, to obtain a limiting spectral triple. 4.1. Holonomy loops. Let be a 3-dimensional, finite, cubic lattice. Let {vi } and {l j } denote vertices and edges in , respectively. The edges in are oriented according to the three main directions in , the x 1 - x 2 - and x 3 -directions, see Fig. 1. Thus, an edge l is a map l : {0, 1} → {vi }, where l(0) and l(1), the start and endpoints of l, are adjacent vertices in . A sequences of edges {li1 , li2 , . . . , lin }, where li j (1) = li j+1 (0) is a based loop if li1 (0) = lin (1) = v0 , where v0 ∈ {vi } is a preferred vertex in called the basepoint. An edge has a natural involution given by reversing its orientation. Thus, li∗ (t) = li (1 − t), and the involution of a loop L = {li1 , li2 , . . . , lin } is given by L ∗ = {li∗n , . . . , li∗2 , li∗1 }. In the following we shall discard trivial backtracking which means that we introduce the equivalence relation {. . . , li j−1 , li j , li∗j , . . .} ∼ {. . . , li j−1 , . . .}, and let a loop L be an equivalence class with respect hereto. The product between two loops L 1 = {li j } and L 2 = {lil } is simply given by gluing the loops to form a new sequence of edges: L 1 · L 2 = {li1 , . . . , lin , lk1 , . . . , lkm }. One easily checks that the involution equals an inverse which gives the set of loops in the structure of a group, called the hoop group.


681

Finally, we consider finite series of loops a=

ai L i , ai ∈ C,

(2)

i

with the involution a∗ =

a¯ i L i∗ ,

i

and the product between a and a second element b = a·b =

j

bj L j,

(ai b j )L i · L j .

i, j

The set of elements of the form (2) is a -algebra. We denote this algebra by B . 4.2. Generalized connections. Next, let G be a compact, connected Lie-group. For the aim of this paper it is natural to choose G = SU (2). We shall, however, develop the formalism for general groups. Let ∇ be a map ∇ : {li } → G, which satisfies ∇(li ) = ∇(li∗ )−1 , and denote by A the set of all such maps. Clearly, A G n , where the total number of vertices in is written n . Given a loop L = {li1 , li2 , . . . , lin }, let ∇(L) = ∇(li1 ) · ∇(li2 ) · . . . · ∇(lin ). This turns ∇ into a homomorphism from the hoop group into G and provides a norm on B ,

a = sup ∇∈A

ai ∇(L i ) G , a ∈ B ,

i

where the norm on the rhs is the matrix norm given by a chosen representation of G. The closure of the -algebra of loops with respect to this norm is a C -algebra.2 We denote this loop algebra by B . 2 Note that the natural map from B to B is not necessarily injective.

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Fig. 2. Subdivision of a cubic lattice cell into 8 new cells

4.3. A spectral triple over A . First, let H be the Hilbert space L 2 (G n , Cl(T ∗ G n ) ⊗ Ml (C)), where L 2 is with respect to the Haar measure and where l is the size of the matrix representation of G. Here, Cl(T ∗ G n ) is the Clifford bundle of the cotangent bundle over G n with respect to a chosen left and right invariant metric. There is a natural representation of the loop algebra on H given by f L · (∇) = (1 ⊗ ∇(L)) (∇), ∈ H , where the first factor acts on the Clifford bundle and the second factor acts on the matrix factor in H . Next, denote by D a Dirac operator on A . The precise expression for D will be determined below through the process of taking the continuum limit of the construction. D acts on the factor of H which involves the Clifford bundle. In total, the triple (B , H , D ) is a geometrical construction over A . 4.4. The limiting spectral triple. The goal is to obtain a spectral triple over the space A. To do this we take the limit of spectral triples over the intermediate spaces A . Let {i }, i ∈ N, be an infinite sequence of 3-dimensional, finite, cubic lattices where i+1 is the lattice obtained from i by subdividing each elementary cell in i into 8 new cells. This process involves the subdivision of each edge l j in i into two new edges in i+1 together with the addition of new vertices and edges, see Fig. 2. We denote the initial lattice by 0 . Corresponding to this sequence of cubic lattices there is a projective system {Ai } of spaces obtained from the graphs {i }, together with natural projections between these spaces Pi,i+1 : Ai+1 → Ai .

(3)

Consider now a system of triples (Bi , Hi , Di ), with the restriction that these triples are compatible with the projections (3). This requirement is easily satisfied for the algebras and the Hilbert spaces, see [4]. For the Dirac type operators, however, some care must be taken. The problem reduces to the simple case where an edge in i subdivided into two edges in i+1 , see Fig. 3.1, which corresponds to the projection P : G 2 → G, (g1 , g2 ) → g1 · g2 ,

(4)


683

Fig. 3. A subdivision of an edge into two and the new parameterization of the edge

and a corresponding map between Hilbert spaces P ∗ : L 2 (G, Cl(T ∗ G) ⊗ Ml ) → L ∗ (G 2 , Cl(T ∗ G 2 ) ⊗ Ml ). The compatibility condition for the Dirac type operator reads P ∗ (D1 v)(g1 , g2 ) = D2 (P ∗ v)(g1 , g2 ), v ∈ L 2 (G, Cl(T ∗ G) ⊗ Ml ). Here D1 is the Dirac operator on G, and D2 is the corresponding Dirac operator on G 2 . Consider the following change of variables:

: G 2 → G 2 ; (g1 , g2 ) → (g1 · g2 , g1 ) =: (g1 , g2 ),

(5)

for which projection (4) obtains the simple form P(g1 , g2 ) = g1 .

(6)

This change of variables corresponds to a new parameterization of the edge, see Fig. 3.2. It is now straightforward to write down a Dirac operator on G 2 which is compatible with the projection (6). Basically, we can pick any Dirac operator of the form D2 = D1 + a D2 , a ∈ R, where D2 is a Dirac operator on the copy of G in G 2 whose coordinates are eliminated by the projection (6). At this point the choice of the operator D2 is essentially unrestricted with a being an arbitrary real parameter. However, for reasons explained in [5] it turns out that D1 and D2 should of the form j ei · L e j , (7) Di = j

i

j

where the product is Clifford multiplication. In Eq. (7) {ei } denotes a left-translated orthonormal basis of T ∗ G, where G is the i th copy in G n . L e j denotes the correspondi ing differential. For later reference we denote by Re j the right translated vector fields. i This line of analysis is straightforwardly generalized to repeated subdivisions. At the level of the n th subdivision of the edge the change of variables which generalizes (5) reads

: Gn → Gn ; (g1 , g2 , . . . , gn ) → (g1 · g2 · . . . · gn , g2 · . . . · gn , . . . , gn ) := (g1 , g2 , . . . , gn ),

(8)

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Fig. 4. Two different types of partition which will lead to different Dirac type operators. The second partition is the one which we will later argue is “natural”

which corresponds to the structure maps ). Pn,n/2 : G n → G n/2 ; (g1 , g2 , g3 , . . . , gn ) → (g1 , g3 , . . . , gn−1

Again, it is straightforward to construct a Dirac type operator compatible with these structure maps. This construction gives rise to a series of free parameters {ai }, one for each subdivision. Thus, by solving the G 2 → G problem repeatedly, and by piecing together the different edges, we end up with a Dirac type operator on the level of n of the form ai Di , (9) Dn = i

where Di is a Dirac type operator corresponding to the i th level of subdivision in An . The change of variables in (8) is the key step to construct Dn . However, there will be many different partitions of the line segment which simplify the structure maps and lead to different Dirac type operators, see Fig. 4. This ambiguity was also commented on in [5]. In subsequent sections we will argue that a single type of subdivision stand out as “natural” due to the classical interpretation of the corresponding Dirac type operator. We are now ready to take the limit of the triples (Bi , Hi , Di ). First, the Hilbert space H is the inductive limit of the intermediate Hilbert spaces Hi . That is, it is constructed by adding all the intermediate Hilbert spaces H = ⊕∈{i } L 2 (G n() , Cl(T ∗ G n() ) ⊗ Ml (C))/N , where N is the subspace generated by elements of the form (. . . , v, . . . , −Pi∗j (v), . . .), where Pi∗j are the induced maps between Hilbert spaces. The Hilbert space H is then the completion of H . The inner product on H is the inductive limit inner product. This Hilbert space is manifestly separable. Next, the algebra B := lim B

−→

contains loops defined on a cubic lattice n in {n }. Note that the algebra B differs from the algebra used in loop quantum gravity on two points: first, we only consider loops running in cubic lattices, whereas the algebra in loop quantum gravity is generated


685

by piece-wise analytic loops. Second, we consider loops which correspond to untraced holonomy loops. Thus, the algebra B is noncommutative, in contrast to the algebra of traced holonomy loops in loop quantum gravity, which is commutative. Finally, the Dirac-like operator Dn descends to a densely defined operator on the limit Hilbert space H, D = lim Dn .

−→

We factorize H in lim L 2 (G n() , Ml ) ⊗ lim Cl(Tid∗ (G n() )). On lim Cl(Tid∗ (G n() )) there is an action of the algebra lim Cl(Tid∗ (G n() )). The completion of this algebra with respect to this action is the CAR algebra and admits a normalized trace, i.e. tr (1) = 1. Let T r be the ordinary operator trace on the operators on lim L 2 (G n() , Ml ) and define τ = T r × tr . In [4] we prove that for a compact Lie-group G the triple (B, H, D) is a semi-finite spectral triple with respect to τ when the sequence {an } converges to infinity. This means that: 1. (1+ D 2 )−1 is τ -compact, i.e. can be approximated in norm with finite trace operators, and 2. the commutator [D, a] is bounded. 5. The Space of Connections Let us now turn to the spaces Ai and their projective limit. Denote by A := lim A .

←−

Further, given a trivial principal G-bundle denote by A the space of all smooth connections herein. In [4] we prove that A is densely embedded in A: A → A. This fact justifies the terminology generalized connections for the completion A and shows that the semi-finite spectral triple (B, H, D) is indeed a geometrical construction over the space A of smooth connections. 6. The Quantization of the Poisson Bracket To determine the relation between the construction of the spectral triple (B, H, D) and the formulation of canonical gravity in terms of loop and flux variables satisfying the Poisson bracket (1), we calculate the commutator between the Dirac type operator D and an element in the loop algebra B. Consider first a single line element li and the corresponding group element ∇(li ) ∈ G. We assume that the copy of G in An assigned to li corresponds to the m th subdivision of the initial cubic lattice. We then find eik · ∇(li )σ k , [D, ∇(li )] = am

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Fig. 5. The surface Si

where σ k are generators in the Lie-algebra g. Also, consider a loop L = {li1 , li2 , . . . , lin } and the commutator [D, f L ] = [D, ∇(l1 )] · ∇(l2 ) . . . ∇(ln ) + ∇(l1 )[D, ∇(l2 )] . . . ∇(ln ) + · · · . These formula show that a commutator between D and an element of the algebra B inserts Lie-algebra generators at vertices in the graphs {i }. This general structure is similar to the structure of the Poisson bracket (1) and suggest that the interaction between the Dirac type operator D and the loop algebra B is related to a representation of the Poisson bracket (1). Consider again a single edge li which we now for simplicity assume to belong to the initial lattice. Let li (0) = v j and li (1) = v j+1 , where v j and v j+1 are vertices in 0 . Let us also assume that li runs in the x 1 -direction in 0 . Also, let ∇(li ) belong to the i th copy of G in A0 . The commutator between the left-invariant vector field eia and the group element ∇(li ) gives [L eia , ∇(li )] = ∇(li )σ a . This shows that L eia corresponds to a quantization of a flux variable FSa , where the surface S intersects li at v j+1 . Actually, the surface S is of no significance here except for its intersection point with the vertex v j+1 . Let Si be a surface which intersects the vertex v j+1 and is perpendicular to li , see Fig. 5. The size of Si corresponds to the initial lattice 0 in the sense that it spans an area corresponding to a side in a single cell. The operator L eia should then, due to the Poisson bracket (1), be interpreted as a quantization a , of the flux variable FS i quantization

a −→ l 2P L eia , iFS i

where l P is the Planck length. It is important to realize that the inverse, densitized triad a is located at the endpoint of l . Thus, F a involves the quantity field involved in FS i Si i E¯ am (v j+1 ) through a d x 2 ∧ d x 3 E¯ a1 (v j+1 ). FSi = Si

. Consider next the first subdivision of li into two edges, which we denote li and li+1 Thus, ∇(li ) = ∇(li ) · ∇(li+1 ).

Also, denote the new vertex which subdivides li by v j+1/2 . Now, the new copy of G is associated to the first half of the line segment li , which means to li . For notational


687

Fig. 6. An alternative partition of an edge into two

simplicity, let us assume that this new copy of G is the (i +1)th copy of G in A1 whereas the full line segment li corresponds to the i th copy of G. At first hand, it seems that the corresponding left-invariant vector fields L ek should be interpreted according to i+1

quantization

a a iFS −→ l 2P L ei+1 i+1

(first guess).

(10)

However, this cannot be correct since L ek commutes with ∇(li ) which belongs to the i+1

i th copy of G. If Eq. (10) should be correct then the commutator between L ek and i+1 ∇(li ) should split up ∇(li ) and insert a Lie-algebra generator at the new vertex v j+1/2 , since the edge li intersects the surface Si+1 at v j+1/2 . Instead, we find that relation (10) obtains an additional term: quantization

a iFS −→ l 2P L ek + l 2P Rgi+1 ek g−1 . i+1 i+1

i i+1

is located at the new vertex v j+1/2 . If Notice here that the triad field involved in we had chosen a different partition of the line segment, see Fig. 6, then the left-invariant vector field corresponding to the new copy of G would have an interpretation in terms of a flux variable and triad field located at v j+1 . Thus, the classical interpretation of D distinguishes between the different modes of subdividing the line segment. Notice also that the surfaces Si must shrink with each subdivision, in order to have one intersection point between the lattice and each surface. Thus, if we set the area of the initial surface equal to one, then the size of the surfaces decrease with subdivisions like k FS i+1

|Si | = 2−2n .

(11)

Consider the next subdivision of li into four edges. The notation is as indicated in a a Fig. 7. We find that the two new flux operators FS and FS have the following i+2 i+3 correspondences: quantization

a 2 2 a +l R iFS −→ l 2P L ei+2 P gi+2 ea g −1 + l P R gi+2 ea i+2 i

−1 i+1 gi+2

i+2

,

and quantization

a 2 a +l R iFS −→ l 2P L ei+3 P gi+1 gi+3 ea g −1 g −1 . i+3 i

i+3 i+1

Once more, the particular subdivision of li is singled out by this interpretation. If we had chosen the alternative subdivision of the edge into two, as pictured in Fig. 7, then this interpretation would not have been possible. There exist, however, at this level the possibility to choose the subdivision in Fig. 4.1. At this point of the analysis, there is no particular reason to choose between the two modes

688


Fig. 7. Partition of an edge into four

of subdivision pictured in Fig. 4, except perhaps that the subdivision in Fig. 4.2 is more symmetrical since new copies of G are all assigned to edges of the same length. In general, at the n th level of subdivision of li we obtain the correspondence quantization

k k FS −→ l 2P L ek + l 2P i+s−1 , i+s i+s

(12)

k where i+s−1 is a combination of twisted, right-invariant vector fields acting on the copies of G assigned to edges which are situation “higher” in the inductive system of k lattices. Put differently, i+s−1 probes information which is more coarse grained relative to the line segment to which the (i + s)th copy of G is assigned. k In the following we shall ignore the correction terms i+s−1 when we apply relation (12) to translate quantized quantities involving the Dirac type operator D to their classical counterparts. The reason for this will become clear in the next section where we construct semi-classical states. These states have the property that any dependency on finite parts of the inductive system of lattices vanishes in the semi-classical limit. In the limit of repeated subdivision of lattices we find that the semi-finite spectral triple (B, H, D) encodes information tantamount to a representation of the Poisson bracket of general relativity. Thus, the triple carries information of the kinematical sector of quantum gravity. Clearly, the triple is based on a different set of variables than loop quantum gravity and hence the “representation” it encodes is different to the representation used there.

7. Semiclassical Analysis In this section we construct semi-classical states in H and evaluate their expectation value of D.

7.1. Coherent states on a Lie group. We will first recall the results for coherent states on compact connected Lie groups that we are going to use. For simplicity we will only consider the case of most interest, namely SU (2). Let {ea } be a basis for su(2). Given g0 in SU (2) and given three momenta (real numbers) p 1 , p 2 , p 3 there exist families φt ∈ L 2 (SU (2)) such that lim φ t , t L ea φ t = i pa ,

t→0


689

and lim φ t ⊗ v, gφ t ⊗ v = (v, g0 v),

t→0

where v ∈ M2 (C), and (, ) denotes the inner product hereon. Corresponding statements hold for operators of the type f (g)P(t L e1 , t L e2 , t L e3 ), where P is a polynomial in three variables, and f is a smooth function on SU (2), i.e. lim φ t , f (g)P(t L e1 , t L e2 , t L e3 )φ t = f (g0 )P(i p 1 , i p 2 , i p 3 ).

t→0

This statement also carries over to symbols, i.e. functions on T ∗ SU (2) with certain properties. The construction of these states follows from work of Hall, see [12,13], and are more explicitly described in [26] Sect. 3.1. The states have further important physical properties, which we are however not going to use at the present stage of the analysis. Also, the precise construction of these states, in particular the choice of complexifier, is irrelevant for the results presented in this paper. 7.2. Product states. Let us consider the n th level in a subdivision of lattices. We split the edges into {li }, and {li }, where {li } denotes the edges appearing in the n th subdivision but not in the n − 1th subdivision, and {li } the rest. Define φlti to be the coherent state on SU (2) such that lim φlti ⊗ v, gφlti ⊗ v = (v, h li (A)v),

t→0

and lim φlti , t L eia , φlti = 2−2n iE am (v j+1 ),

t→0

where v ∈ M2 (C); v j+1 denotes the right endpoint of li , and the m in the E am refers to the direction of li . The factor 2−2n comes from the scaling (11). Furthermore define the states φli by lim φlt ⊗ v, gφlt ⊗ v = (v, h li (A)v),

t→0

i

i

and lim φlt , t L eaj φlt = 0.

t→0

i

i

Finally define φnt to be the product of all these states as a state in L 2 (An ). These states are essentially identical to the states constructed in [26] except that they are based on cubic lattices and a particular mode of subdivision. In the limit n → ∞ these states produce the right expectation value on all loop operators in the infinite lattice.

690


Fig. 8. A single edge

7.3. Semi-classical states: one copy of G. We now proceed to construct semi-classical states in H. From here on we set t = l 2P and rescale the left-invariant vector fields in the Dirac type operator accordingly L eia → t L eia . The first step is to consider again a single edge. Let ψ(x) be a field on . A priori, ψ(x) can either be a two-spinor or a two-by-two matrix valued field. For reasons which shall become clear later, we choose the second option. Consider again an edge li with endpoints v j and v j+1 , see Fig. 8. The states in L 2 (G, Cl(T ∗ G) ⊗ M2 (C)) which we are interested in have the form3 t (li ) = (gi ψ(v j+1 ) + ieia σ a ψ(v j ))φlti , where the spinor field is evaluated at the endpoints of the edge li . A straightforward computation gives the expectation value of D on this state ¯ j )σ a E am (ψ(v j+1 ) − ψ(v j )) ¯ t |D|t = 2−2n an − ψ(v lim t→0

¯ j+1 ) − ψ(v ¯ j ))σ a E am ψ(v j ) + (ψ(v ¯ j ){ Am , σ a E am }ψ(v j ) , + ψ(v

(13)

where we applied the expansion g = 1 + Am + O( 2 ), with = 2−n . Also, the index m denotes the direction of the edge li . 7.4. Determining the sequence {an }. Formula (13) indicates that the sequence {an } of free parameters in D plays a specific role in the semiclassical analysis. In particular, note the term (ψ(v j+1 ) − ψ(v j )). If we consider the limit where the edge li lies increasingly deep in the inductive system of graphs, then this term approaches (ψ(v j+1 ) − ψ(v j )) → ∂m ψ(v j )d x m , (no sum over m), where d x m is the infinitesimal line segment, which goes as 2−n . Here n denotes the level of subdivisions of graphs. Thus, if we choose the sequence an = 23n , then the expression (13) converges, when one considers edges of increasing depth in the inductive system of lattices, towards the quantity ¯ t |D|t = ψ(v ¯ j )σ a E am ∇m ψ(v j ) − ∇m ψ(v ¯ j )σ a E am ψ(v j ), lim lim

n→∞ t→0

3 Here we assume that ψ(x) is matrix valued. If ψ(x) was a two-spinor field then we would instead consider the Hilbert space L 2 (G, Cl(T ∗ G) ⊗ C2 ) and states therein.


691

Fig. 9. Three edges, connected in one vertex

(again, no sum over m) with ∇m = ∂m + Am . This is the expectation value (in a point) of the self-adjoint operator σ a E am ∇m + ∇m σ a E am

(no sum over m).

Here, we applied what amounts to a partial integration (this will be justified shortly where an integral over emerges).

7.5. Three copies of G. Next, we consider instead three edges, denoted for simplicity by l1 , l2 , l3 , all leading out of the same vertex, with three copies of G associated to them, correspondingly. First, consider the state t (g1 , g2 , g3 ) = e2a e3a g1 ψ(v1 ) − e1a e3a g2 ψ(v2 ) + e1a e2a g3 ψ(v3 ) i + e1a e2b e3c δ ab σ c + δ ac σ b + δ bc σ a ψ(v0 ) φlt1 φlt2 φlt3 , 5

(14)

where the enumeration of the vertices are shown Fig. 9. We find that the expectation value of D on this state leads to the operator σ a E am ∇m + ∇m σ a E am

(15)

in the limit where the edges li lie increasingly deep in the inductive system of lattices. In Eq. (15) we now sum over m.

7.6. Semiclassical states on A. To obtain semiclassical states on the full space A we need to prescribe a procedure to sum up the results for the individual copies of G, or rather, for vertices. First, at the n th level in the inductive system of lattices, where we have n n copies of G, we write down the state ⎛ ⎞ tn (An ) = n ⎝ v j ⎠ φnt , (16) vj

692


Fig. 10. Three more edges

where n equals 2−3(n−1)/2 . This will, in the limit taken below, converge to the Lebesque measure. Also, we define v j = eaj2 eaj3 g j1 ψ(v j1 ) − eaj1 eaj3 g j2 ψ(v j2 ) + eaj1 eaj2 g j3 ψ(v j3 ) i + eaj1 ebj2 ecj3 δ ab σ c + δ ac σ b + δ bc σ a ψ(v j ), 10

(17)

see Fig. 10. The sum in (16) runs over a certain subclass of vertices in n . At the n th level, these vertices are the midpoints of the minimal cubes present at the (n − 1)th level. This discrimination between vertices admittedly appears to be somewhat arbitrary and it might be possible to take into account all edges. This, however, complicates matters. We shall return to this point in a later publication. With (16) we have a sequence {tn } of states in H and we can calculate the limit of the expectation value of D on these states. We call this limit the continuum limit. We find 1 √ a m √ t t ¯ ¯ lim lim n |D|n = d 3 x ψ(x)( gσ ea ∇m + ∇m gσ a eam )ψ(x). (18) n→∞ t→0 2 Thus, the sequence of states {tn } defines a semi-classical limit where D, to lowest order, is a spatial Dirac operator on . Notice that the integral in (18) is the invariant integral √ μ over . The factor g, where g is the determinant of the spatial metric, comes from E¯ a . Here, however, it should be stressed that the emerging normalization of spinors ψ(x) is not coordinate invariant. We shall comment on this below. Note that the emergence of the integral in (18) crucially depends on the way the CAR algebra appears in the expression (17). Interestingly, the elements of the CAR algebra play the role of localizers in the construction.

7.7. The Dirac Hamiltonian. In Eqs. (14) and (17) we ignored certain degrees of freedom. To take these into account we modify the expression in Eq. (17) to ˜ v j = eaj eaj g j1 ψ(v j1 ) − eaj eaj g j2 ψ(v j2 ) + eaj eaj g j3 ψ(v j3 ) 2 3 1 3 1 2 i a b c ab c + e j1 e j2 e j3 δ σ + δ ac σ b + δ bc σ a , Mv j ψ(v j ), 20


693

where Mv j is an arbitrary self-adjoint two-by-two matrix. Write Mv j = N (v j )1 + iN a (v j )σ a , where N and N a are real fields on , scalar and vectorial respectively. Finally, we let N m = N a eam ; we define

⎛ ˜ tn (An ) = n ⎝

⎞ ˜ v j ⎠ φnt ,

vj

and repeat the calculations leading to (18). We obtain ¯˜ t |D| ˜ tn lim lim n 1 √ √ m √ a m 3 ¯ a m ( g N σ ea ∇m + N ∇m gσ ea ) + i g N ∂m ψ(x) = d x ψ(x) 2 1 1 √ a m √ m √ m 3 ¯ + d x ψ(x) g N Am + (∂m g N ) + (∂m N ) gσ ea ψ(x). 2 2

n→∞ t→0

(19)

Here, the first line is the principal part of the Dirac Hamiltonian in 3+1 dimensions. The second line contains additional zero-order terms. The fields N and N m are seen to play the role of the lapse and shift fields respectively. The additional zero-order terms appearing in (19) are not identical to the zero-order terms in the Dirac Hamiltonian. This, however, is not to be expected since the Dirac Hamiltonian is not self-adjoint while the Dirac type operator is. For the reconstruction of the 4-metric only the principal part is used. We shall return to a discussion of the zeroth order part later. We believe that the correct treatment of the zeroth order terms can only be performed once the Wheeler-DeWitt constraint is formulated and implemented and thereby the freedom in choosing the foliation, i.e. the lapse and the shift fields, is eliminated. This might also be a possible solution to another problem arising at this point. The ˜ tn now depends on the lapse and shift fields, norm of the semi-classical states ¯˜ t | t ˜ ¯ lim lim = d 3 x ψ(x)ψ(x)(N , N m ), n n n→∞ t→0

where the function (N , semi-classical states as constituting the one-fermion states problematic as the induced scalar product is obviously not appropriate. Interestingly, however, the lapse and shift √ fields may be chosen such that (N , N m ) = g. Thus, an appropriate choice of the time-coordinate restores the invariance of the norm under spatial diffeomorphisms. However, we are not aware of a compelling physical reason for such a choice of lapse and shift fields. Nevertheless, it might be conceivable that there is such a reason, as in quantum field theory, the one-particle space is not invariant under general coordinate changes. Thus, our restriction to one-particle states may well imply a restriction of the choice of coordinates. The solution to the above problem might also lie in the construction of the states, i.e. it might be possible to modify the construction of the semi-classical states such that the N m ) is readily computed. This renders the interpretation of the

694


norm of the semi-classical spinors is automatically coordinate independent. We shall investigate this problem in future work. Finally, disregarding lapse and shift fields, √ we should note that it would also be possible to remedy the deficiency of the missing g in the inner product by assigning the zero-order expectation value of the Halls coherent states to the non-densitized triad field and then adding, appropriately, the density in the semi-classical state. With this alteration the inner product of semi-classical states renders the correct inner product of spinors. However, this choice would spoil the interpretation of the left invariant vector fields as flux operators. Note that ψ(x) takes values in M2 (C). In view of the action of the σ a ’s this can consistently be interpreted as a Dirac 4-spinor. The space spanned by these fields ψ(x) can thus be interpreted as the space of solutions of the Dirac equation for the static 4-metric described by the 3-metric, the lapse and the shift fields (see [27]). 8. Discussion & Outlook In this paper we have shown that to certain states for the previously constructed spectral triple over holonomy loops, one can associate gravitational and fermionic matter fields. This clearly indicates that one should interpret this model as describing quantized gravitational fields coupled to quantized matter fields. To this concern we have constructed a small class of semi-classical states. Disregarding for the moment the open problem of identifying the correct scalar product, these semi-classical states can be interpreted as one-fermion states in a given foliation and given gravitational background field. We have identified the expectation value of the Dirac type operator of the spectral triple, in these semi-classical states, as the expectation value of the energy of the corresponding matter fields. This raises the question whether one can generally interpret the Dirac type operator as the energy operator for the matter fields present in the model. Thus, future work must clarify, first, whether there are many-particle fermionic states present in the model, and, of course, whether additional matter fields, for example photons, can be found. A consistent interpretation of the Dirac type operator then requires that it can also be interpreted as the energy of these states. At the present state of the project the investigation of these issues is certainly within reach. A further strong indication that the model should be interpreted in terms of quantum gravity is the fact that it encodes information tantamount to a representation of the Poisson bracket of general relativity. This has been carefully analyzed for the first time in this paper and should therefore be seen as one of its central results. All this being said, we should stress that our Hilbert space can only be viewed as the kinematical Hilbert space of quantum gravity. The Wheeler-DeWitt constraint has not been constructed nor implemented. In the construction above, this fact is nicely reflected by the appearance of the lapse and shift fields. Yet, as the Wheeler-DeWitt equation should in principle eliminate these unphysical degrees of freedom, the concreteness of their appearance raises the hope that our analysis may lead to a novel approach to the construction and implementation of the Hamiltonian constraint in quantum gravity. Apart from the physical interpretation of the model, the semi-classical analysis has also proven beneficial at a more technical level: it turned out that the system of nested, cubic lattices, on which the semi-finite spectral triple is based, simply plays the role of a coordinate system. In particular, the lattices form the coordinate system already used to


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write down the Ashtekar variables and their Poisson bracket. This choice of background structure does, however, not imply lack of background invariance: there is no choice of background metric and the semi-classical limit is coordinate independent. This shows that it is possible to recover the spatial symmetries with a countable system of lattices. Yet, it is an issue for future work to establish the full covariance of the model under change of the chosen coordinate system. These observations are all based on the fact that any dependency on finite parts of the lattices vanishes in the limits (18) and (19). That is, only the continuum limit contributes to the integrals in (18) and (19). It is as if the lattices, which we have used to construct the spectral triple, disappear in this semi-classical limit. Furthermore, the free parameters {an }, which appear in the Dirac type operator, play an important role in the semi-classical limit. A priori, this sequence is only required to diverge in order for the resolvent of the Dirac type operator to be compact. In the semiclassical limit, however, the sequence is identified as the inverse, infinitesimal volume element. This fixes the sequence. We should stress that we only found states living on static 4-manifolds. This had to be expected since we interpret these states as one-particle states and it is well known in quantum field theory that such states would not exist on non-static space-times, e.g. in accelerating frames (which would be described by time-dependent lapse and shift fields). In the future it is certainly an interesting question whether one can find and describe semi-classical states which correspond to states of a quantized fermion field on a non-static space-time. The application of the CAR algebra as a tool to form the local Riemann integral in Eqs. (18) and (19) is highly intriguing. It would certainly be very interesting and important to investigate the role played by the CAR algebra more thoroughly. Moreover, the analysis in this paper is based on a real SU (2) connection whereas the Ashtekar connection is complex. A real SU (2) connection corresponds either to a Euclidean metric or to a more involved Hamiltonian. We believe it is desirable to work with the original Ashtekar connection. One may speculate whether the complexity of the connection only appears in the semi-classical limit. If so, then one might exploit the techniques presented in this paper to obtain a complex connection via a doubling of the Hilbert space. Immediate tasks to be addressed are: to compute quantum corrections for the semiclassical states in higher order of the Planck length; to investigate the operational interpretation of the loop algebra in the semi-classical states; to construct many particle states. Hopefully this will provide further evidence that the spectral triple over holonomy loops is a viable candidate for quantum gravity coupled to matter fields. Acknowledgements. J.A. and M.P. were supported by the SFB 478 grant “Geometrische Strukturen in der Mathematik” of the Deutsche Forschungsgemeinschaft.

References 1. Aastrup, J., Grimstrup, J. M.: Spectral triples of holonomy loops. Commun. Math. Phys. 264, 657 (2006) 2. Aastrup, J., Grimstrup, J. M.: Intersecting Connes noncommutative geometry with quantum gravity. Int. J. Mod. Phys. A 22, 1589 (2007) 3. Aastrup, J., Grimstrup, J.M., Nest, R.: On Spectral Triples in Quantum Gravity I. Class. Quant. Grav. 26, 065011 (2009) 4. Aastrup, J., Grimstrup, J.M., Nest, R.: On Spectral Triples in Quantum Gravity II. J. Noncommut. Geom. 3, 47 (2009)

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5. Aastrup, J., Grimstrup, J.M., Nest, R.: A new spectral triple over a space of connections. Commun. Math. Phys. 290, 389 (2009) 6. Aastrup, J., Grimstrup, J.M., Nest, R.: Holonomy Loops, Spectral Triples & Quantum Gravity. to appear in Class. Quant. Grav. 26, 6500 (2009) 7. Connes, A.: Noncommutative Geometry. London-New York: Academic Press, 1994 8. Connes, A.: Gravity coupled with matter and the foundation of non-commutative geometry. Commun. Math. Phys. 182, 155 (1966) 9. Thiemann, T.: Introduction to modern canonical quantum general relativity. http://arxiv.org/abs/gr-qc/ 0110034vL, 2001 10. Rovelli, C.: Quantum gravity. Cambridge, UK: Cambridge Univ. Pr, 2004 11. Ashtekar, A., Lewandowski, J.: Background independent Quantum Gravity: A status report. Class. Quant. Grav. 21, R53 (2004) 12. Hall, B.C.: The Segal-Bargmann “coherent state” transform for compact Lie groups. J. Funct. Anal. 122(1), 103–151 (1994) 13. Hall, B. C.: Phase space bounds for quantum mechanics on a compact Lie group. Commun. Math. Phys. 184(1), 233–250 (1997) 14. Ashtekar, A., Lewandowski, J.: Representation theory of analytic holonomy C* algebras. http://arxiv. org/abs/gr-qc/9311010v2, 1993 to appear in J. Baez (ed.): Knotz and Quantum Gravity, Oxford: Oxford Univ. Press, 1994 15. Ashtekar, A., Lewandowski, J.: Differential geometry on the space of connections via graphs and projective limits. J. Geom. Phys. 17, 191 (1995) 16. Ashtekar, A., Lewandowski, J.: Quantum theory of geometry. I: Area operators. Class. Quant. Grav. 14, A55 (1997) 17. Flori, C., Thiemann, T.: Semiclassical analysis of the Loop Quantum Gravity volume operator: I. Flux Coherent States. http://arxiv.org/abs/0812.1537v1, 2008 18. Chamseddine, A.H., Connes, A.: Universal formula for noncommutative geometry actions: Unification of gravity and the standard model. Phys. Rev. Lett. 77, 4868 (1996) 19. Chamseddine, A.H., Connes, A.: A universal action formula. Phys. Rev. Lett. 77, 4868 (1996) 20. Chamseddine, A.H., Connes, A.: The spectral action principle. Commun. Math. Phys. 186, 731 (1997) 21. Chamseddine, A.H., Connes, A., Marcolli, M.: Gravity and the standard model with neutrino mixing. Adv. Theor. Math. Phys. 11, 991–1089 (2007) 22. Chamseddine, A.H., Connes, A.: Why the Standard Model. J. Geom. Phys. 58, 38–47 (2008) 23. Chamseddine, A.H., Connes, A.: A Dress for SM the Beggar. with different title, 2007; appeared in Phys. Rev. Lett. 99, 9160 (2007) http://arxiv.org/abs/0706.3690v1 [hep-th] 24. Ashtekar, A.: New Variables for Classical and Quantum Gravity. Phys. Rev. Lett. 57, 2244 (1986) 25. Ashtekar, A.: New Hamiltonian Formulation of general relativity. Phys. Rev. D 36, 1587 (1987) 26. Thiemann, T., Winkler, O.: Gauge field theory coherent states (GCS). IV: Infinite tensor product and thermodynamical limit. Class. Quant. Grav. 18, 4997 (2001) 27. Paschke, M., Kopf, T.: A spectral quadruple for de Sitter space. J. Math. Phys. 43, 818 (2002) Communicated by Y. Kawahigashi

Commun. Math. Phys. 302, 697–736 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1187-z

Communications in


A Model Problem for Conformal Parameterizations of the Einstein Constraint Equations David Maxwell Department of Mathematics, University of Alaska Fairbanks, P. O. Box 757500, Fairbanks, AK 99775, USA. E-mail: [email protected] Received: 30 September 2009 / Accepted: 19 September 2010 Published online: 8 February 2011 – © Springer-Verlag 2011

Abstract: We study the conformal and conformal thin sandwich (CTS) methods as candidates for parameterizing the set vacuum initial data for the Cauchy problem of general relativity. To this end we consider a small family of symmetric conformal data. Within this family we obtain an existence result so long as the mean curvature has constant sign. When the mean curvature changes sign we find that solutions either do not exist, or they are not unique. In some cases solutions are shown to be non-unique. Moreover, the theory for mean curvatures with changing sign is shown to be extremely sensitive with respect to the value of a coupling constant in the Einstein constraint equations. 1. Introduction Initial data for the Cauchy problem of general relativity consist of a Riemannian manifold and a second fundamental form that satisfy a system of nonlinear PDEs known as the Einstein constraint equations. A longstanding goal has been to find a constructive description of the full set of solutions of these equations on a given manifold, and hence a method of producing all possible initial data. Although this problem remains open in general, the conformal method of Lichnerowicz and Choquet-Bruhat and York provides an elegant and complete solution to the problem of constructing constant-mean curvature (CMC) solutions. For example, on compact manifolds the solutions of the Einstein constraint equations are effectively parameterized by selection of conformal data consisting of a conformal class for the metric, a so-called transverse-traceless tensor, and a (constant) mean curvature. The conformal method can also be used to construct non-CMC solutions of the constraint equations, but much less is known in this case. Ideally one would like to show that selection of generic conformal data leads to a unique corresponding solution of the constraint equations. Until recently, virtually all results for the conformal method only applied to nearCMC initial data. The first construction using the conformal method of a family of initial data with arbitrarily specified mean curvature was given by Holst, Nagy, and Tsogtgrel

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in [9]. Although this result represents a breakthrough for the conformal method, it has a number of important limitations: – The near-CMC hypothesis is replaced by a smallness assumption on the transversetraceless tensor (i.e. a small-TT hypothesis). – It is not known if small-TT conformal data determine a unique solution. – The construction only works on Yamabe-positive compact manifolds. – The construction requires non-vanishing matter fields. It was subsequently shown in [18] that the construction could be extended to vacuum initial data, but the other restrictions remain. These results are compatible with the possibility that a large set of conformal data lead to no solutions or multiple solutions; from the point of view of parameterizing the full set of solutions one would like to show that this does not occur. In this paper we investigate the conformal method and its variation, the conformal thin sandwich (CTS) method, by studying a model problem obtained from a quotient of certain symmetric conformal data. Despite the simplicity of the model problem, it captures the core issues of the conformal method, including the nonlinear coupling and difficulties regarding conformal Killing fields. Moreover, the model problem is easily studied numerically, and thus gives an important tool for suggesting theorems which might be proved in the future. We consider a three-parameter family of model conformal data that allow for simultaneous violations of both the near-CMC and small-TT conditions on a Yamabe-null manifold. The mean curvatures in this family are written as the sum of an average mean curvature, t, and a fixed zero-mean function describing departure from the mean. If t is chosen so that the mean curvature does not change sign, we find that there exists a solution of the constraint equations so long as the transverse-traceless tensor in the family is not identically zero. When the mean curvature changes sign, the situation is more delicate. We observe in this regime non-existence for certain large transverse-traceless tensors, non-uniqueness for certain small transverse traceless tensors, and a critical value of t (depending on the choice of lapse function in the CTS method and the choice of conformal class representative in the standard conformal method) for which there is an infinite family of solutions when the transverse-traceless tensor vanishes identically. Previous non-uniqueness results for the conformal method have been obtained by adding separate, poorly behaved terms to the equations, either in the form of non-scaling matter sources [4,20] or from coupling with a separate PDE in the extended conformal thin sandwich method [19,20]. We prove here the first nontrivial non-uniqueness result for the standard, vacuum conformal method. It arises from the nonlinear coupling of the equations, and indicates that the standard conformal and CTS methods already contain poorly behaved terms. Non-existence results for the conformal method are available in the CMC case for vacuum and scaling matter [16], as well as for scalar field matter sources [10]. In vacuum, non-existence only occurs for certain non-generic, well-understood conformal data and therefore does not pose a difficulty from the point of view of parameterization. The non-existence result proved here for the model equations translates to either non-existence or non-uniqueness for the full vacuum conformal method, with the final outcome not known. This can be compared to a similar dichotomy shown by Rendall for certain Yamabe-positive data [14]. Unlike Rendall’s example, where non-existence can be thought of as an extension of a CMC non-existence result, neither of the possible outcomes shown here are favorable for using the conformal method as a parameterization scheme.

A Model Problem for Conformal Parameterizations of the Einstein Constraint Equations

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Intriguingly, we find that for mean curvatures in the three-parameter family with changing sign, the existence theory depends sensitively on the values of the constants involved in the nonlinear coupling of the conformal method. We show that these constants are balanced in such a way that any arbitrarily small adjustment to their values lead to one of two different existence theories. All previous results for the conformal method depend only on the signs of the constants in these equations. This sensitivity suggests why it has been so difficult to obtain general large-data results for the conformal method. The conformal data used in this study has one potential drawback: the mean curvature is not continuous, but has jump discontinuities. This level of regularity is lower than has previously been considered for the fully coupled conformal method. We note, however, that the CMC theory of the conformal method readily constructs solutions of the constraint equations with certain kinds of discontinuous second fundamental forms ([5,17,9]), and we use the CMC results of [5] to cope with the discontinuities in the mean curvature. From this perspective the singularities in the mean curvature are comparatively mild. It would be interesting to know if low regularity techniques introduced in [17] and extended in [9] could be generalized to non-CMC conformal data of the regularity we consider here. 1.1. Conformal parameterizations. Let (M n , h) be a Riemannian manifold and let K be a second fundamental form on M n , i.e. a symmetric (0, 2)-tensor. The vacuum Einstein constraint equations for (h, K ) are Rh − |K |2h + tr h K 2 = 0 divh K − d tr h K = 0

[Hamiltonian constraint], [momentum constraint],

(1a) (1b)

where Rh is the scalar curvature of h. For simplicity, we restrict our attention to compact manifolds. Problem 1 (Conformal Parameterization Problem). Let (M n , g) be a compact Riemannian manifold. Find a constructive parameterization of the set of solutions (h, K ) of Eq. (1) such that h belongs to the conformal class of g. If (h, K ) is a solution of Eq. (1) with h in the conformal class of g, we may write h = φ q−2 g for some positive function φ, where q=

2n . n−2

(2)

Without loss of generality we can write K = φ −2 S + Tn g , where S is a traceless (0, 2)-tensor and T is a scalar field. The constraint equations (1) for (h, K ) can then be written in terms of (φ, S, T ) as − 2κq g φ + Rg φ − |S|2g φ −q−1 + κ T 2 φ −q−1 = 0, divg S − κφ q d φ −q T = 0,

(3a) (3b)

where κ=

n−1 . n

(4)

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The conformal parameterization problem then amounts to parameterizing the solutions (φ, S, T ) of (3). The conformal method [6] and its variation, the conformal thin sandwich (CTS) method [21], provide possible approaches for solving Problem 1. An overview of these methods can be found in [3]. We summarize the techniques here to establish notation and to state known results that impact our analysis of the model problem. With the conformal method, one specifies a mean curvature τ and a transversetraceless tensor σ (i.e. a symmetric, trace-free, divergence-free (0, 2)-tensor). We write T = φ q τ and S = σ + L W , where W is an unknown vector field and L is the conformal Killing operator defined by (L V )i j = ∇i V j + ∇ j Vi −

2 k ∇ Vk gi j . n

(5)

Equations (3) then become − 2κq g φ + Rg φ − |σ + L W |2g φ −q−1 + κτ 2 φ q+1 = 0

[conformal Hamiltonian constraint],

(6a) divg L W − κφ q dτ = 0.

[conformal momentum constraint],

(6b) These are coupled nonlinear elliptic equations to solve for unknowns (φ, W ). For the CTS approach one specifies σ and τ along with an additional positive scalar function N which represents a lapse.1 The CTS method is then obtained by replacing L W 1 with 2N L W wherever it appears in the discussion for the conformal method. Although operationally similar to the conformal method, the CTS method has the advantage of being conformally covariant. Specifically, if θ is a positive function, then conformal data (θ q−2 g, θ −2 σ, θ q N , τ ) yields the solution (h, K ) if and only if (g, σ, N , τ ) does. From the perspective of working with a fixed background metric g, the standard conformal method simply corresponds to the CTS method with the choice of N = 1/2. We can think of the CTS approach as providing many different parameterizations, one for each choice of N . It is not known if certain choices of N are superior for the purposes of finding a parameterization. From the conformal covariance we observe that the choice of N in the conformal-thin sandwich method is equivalent to the choice of background metric for the conformal method: the solution theory for the standard conformal method with the background metric gˆ = θ q−2 g is equivalent to the solution theory for the conformal thin sandwich method with lapse function N = 21 θ −q . A conformal thin sandwich solution exists for (g, σ, N , τ ) if and only if a standard conformal method solution exists for (g, ˆ θ −2 σ, τ ), and the resulting solutions of the Einstein constraint equations are the same. In the event that τ is constant, it is easy to see that the existence theory for system (6) reduces to the study of the Lichnerowicz equation − 2κq g φ + Rg φ − |σ |2g φ −q−1 + κτ 2 φ q+1 = 0.

(7)

The obstruction to the existence of solutions of (7) is stated in terms of the metric’s Yamabe invariant 1 Although the CTS method is not usually presented as specifying σ (compare [21]) it is straightforward to show that the presentation here is equivalent to the usual one.


Yg =

inf ∞

M

2κq |∇ f |2g + Rg f 2 d Vg

f ∈C (M) f ≡0

|| f ||2L q

,

701

(8)

and we have the following theorem from [16]. Theorem 1. Let (M, g) be a smooth compact Riemannian manifold, let σ be a transverse-traceless tensor, and let τ be a constant. Then there exists a positive solution of (7) (and hence a solution of the conformally parameterized constraint equations (6)) if and only if one of the following hold: 1. 2. 3. 4.

Yg Yg Yg Yg

> 0, σ ≡ 0, = 0, τ = 0, σ ≡ 0, < 0, τ = 0, = 0, τ = 0, σ ≡ 0.

When a solution exists it is unique, except in case 4) in which case any two solutions are related by a positive scalar multiple. Hence the set of CMC solutions of (1) having a metric conformally related to g is essentially parameterized by choosing pairs (σ, τ ). The following non-CMC variation of Theorem 1 appeared in [18]. Theorem 2. Let (M, g) be a smooth compact Riemannian 3-manifold with no conformal Killing fields. Suppose σ and τ are a transverse-traceless tensor and a mean curvature such that one of the following hold: 1. Yg > 0, σ ≡ 0, 2. Yg = 0, σ ≡ 0, τ ≡ 0, 3. Yg < 0 and there exists gˆ in the conformal class of g such that Rgˆ = −τ 2 . If there exists a global upper barrier for (g, σ, τ ), then there exists at least one solution of the conformally parameterized constraint equations (6). The reader is referred to [18] for the definition of a global upper barrier (where it is called a global supersolution2 ); see also Appendix B. Cases 1-3 of Theorem 2 reduce to those of Theorem 1 if τ is constant. Moreover, the condition on τ in Case 3 is necessary if Yg < 0[17]. Until now, all results for the conformal method are consistent with the possibility that (aside from the exceptional Case 4 of Theorem 1), the conditions of Cases 1–3 of Theorem 2 are necessary and sufficient for the unique solvability of Eq. (6). We show in this paper that this is not the case. In particular we find certain data satisfying the conditions of Case 2 for which there are nontrivially related multiple solutions. We also find other symmetric data satisfying the conditions of Case 2 for which there are no symmetric solutions (and hence there are either no solutions or there are multiple solutions). Global upper barriers can be found if the conformal data is CMC, satisfies a near-CMC condition such as max |∇τ | is sufficiently small, (9) min |τ | or if Yg > 0 and σ is small-TT, i.e. max |σ |g is sufficiently small, with smallness depending on τ.

(10)

2 The terminology global supersolution is perhaps misleading since it is not clear that all solutions of (6) have associated global supersolutions.

702

D. Maxwell

This last upper barrier was first presented in [9] and led to the far-from CMC results of [9] and [18]. Uniqueness theorems are available for a general class of near-CMC data under additional hypotheses on the size of |∇τ | ([13,11]), but nothing is known concerning uniqueness in the small-TT case. Results of O’Murchadha and Isenberg [14] show that the condition σ ≡ 0 in Hypotheses 1 and 2 of Theorem 3 is necessary for certain non-CMC data. In particular, their “no-go” theorem proves that if Rg ≥ 0 (or if Yg ≥ 0 and we are using the CTS method), then there does not exist a solution of (6) if τ is near-CMC and σ ≡ 0. Rendall has also shown, as presented in [14], that there exists a class of Yamabe-positive far-from CMC conformal data with σ ≡ 0 such that if a solution to Eq. (6) exists, it is not unique. It is not known which of existence or uniqueness fails for Rendall’s data. Symmetries pose a difficulty for the conformal method, and this hampers the development of concrete examples. Essentially all non-CMC existence results require that (M n , g) has no conformal Killing fields.3 Analytically this condition arises to guarantee that the operator div L is surjective, but the need for this condition is more fundamental. If (M n , g) admits a nontrivial conformal Killing field X , then selection of a mean curvature poses an a-priori restriction on the solution φ of (6) even before σ is selected. If (h, K ) is a solution of the constraint equations, then the mean curvature τ = tr h K must satisfy M X (τ ) d Vh = 0; this identity is obtained by multiplying the momentum constraint (1b) by X and integrating by parts.4 Writing this equation in terms of g we find φ q X (τ ) d Vg = 0. (11) M

If τ is constant then Eq. (11) holds trivially. If it is possible to find a solution (φ, W ) for general data (σ, τ ), then W has to arise in such a way that φ, which solves a Lichnerowicz equation depending on W , also satisfies (11). The mechanism which might cause this for arbitrary conformal data is not understood, and the issue is sidestepped in the literature by assuming that there are no conformal Killing fields. 2. Conformally Flat Symmetric Data on the Torus Let Sr1 denote the circle of radius r and let M n = Sr11 × · · · × Sr1n with the product metric g. We can pick coordinates x k along each factor such that gi j = δi j and consider the following variation of Problem 1. Problem 2 (Reduced Parameterization Problem). Find all solutions (h, K ) of the Einstein constraint equations on M n such that h is conformally related to g and such that the Lie derivatives L∂k g and L∂k K vanish for 1 ≤ k ≤ n − 1. In practice we are seeking solutions such that h and K are periodic functions of x n alone; by an obvious scaling argument we may reduce to the case rn = 1 and x ≡ x n ∈ [−π, π ] 3 [12] contains an exception, but it requires the conformal data be constant along the integral curves of any conformal Killing fields. For the toroidal initial data we consider in Sect. 3 this amounts to assuming that τ is constant. 4 This condition should be compared with the Bourguignon-Ezin condition M X (R g )d Vh = 0 for the prescribed scalar curvature problem [2].


703

The maximal globally hyperbolic spacetime obtained from such data will be a Gowdy spacetime with a conformally flat Cauchy surface. Our focus is not so much to generate initial data for Gowdy spacetimes (the formulation of the constraint equations found in [7] is more convenient for that purpose), but to use the conformally flat torus as a test case for conformal parameterizations in general. We remark that the CMC version of Problem 2 (including more general toroidal background metrics) was effectively treated in [15]. For the moment we work in three dimensions and use the variables (φ, S, T ) introduced in the previous section. In coordinates we can write ⎡ ⎤ c d 1 ⎣ −a − b c −a + b e ⎦ . S= (12) 3 d e 2a Assuming that S and T are functions of x = x 3 alone, we have div S = 13 (d , e , 2a ), and hence the momentum constraint (3b) reads 1 2 (d , e , 2a ) = φ 6 (0, 0, (φ −6 T ) ). (13) 3 3 Here primes denote derivatives with respect to x. Note that S is transverse-traceless if and only if a, d, and e are constant, and that (φ, S, T ) satisfies the momentum constraint if and only if d and e are constant and a = φ 6 (φ −6 T ) .

(14)

Letting η2 = (b2 + c2 + d 2 + e2 )/9, and noting that (M n , g) is scalar flat, the Hamiltonian constraint (3a) reads

2 2 T − a 2 φ −7 = 0. − 8φ − 2η2 φ −7 + (15) 3 A similar derivation works in higher dimensions, and we obtain the reduced equations

−2κq φ − 2η2 φ −q−1 + κ T 2 − a 2 φ −q−1 = 0, (16) a − φ q (φ −q T ) = 0. Solving Problem 2 amounts to parameterizing the solutions (φ, η, a, T ) of (16). The conformal method can be described in this framework as follows. First we write T = φ q τ,

(17)

where τ is a prescribed mean curvature function and the conformal factor φ is unknown. Additionally, we decompose a = μ + w ,

(18)

where μ is a prescribed constant and w is an unknown function. The function w is related to the vector field W of the conformal method via 2W = w∂n . The constant μ is part of the transverse-traceless tensor; to specify the remainder we select an arbitrary function η. Equations (16) become −2κq φ − 2η2 φ −q−1 − κ(μ + w )2 φ −q−1 + κτ 2 φ q−1 = 0, w − φ q τ = 0.

(19)

704

D. Maxwell

For the CTS approach we additionally choose a positive function N and write a = μ + 1/(2N )w . The CTS equations are then −2κq φ − 2η2 φ −q−1 − κ(μ + (2N )−1 w )2 φ −q−1 + κτ 2 φ q−1 = 0, ((2N )−1 w ) − φ q τ = 0.

(20)

Equations (20) provide a model for the full CTS equations on a Yamabe-null manifold. The nonlinear coupling for this system is the same as for the original equations. Moreover, the background metric on S 1 has a nontrivial conformal Killing field (∂x ). Hence the central difficulties of the conformal method are present in the model. Appendix B outlines how standard techniques for the conformal method can be adapted to equations (20) if the data satisfy an additional evenness hypothesis. Our primary focus, however, is on examining a family of conformal data for which we obtain stronger results than are possible with the techniques of Appendix B.

3. A Family of Low Regularity Conformal Data The prescribed data for system (20) are a constant μ and a function η together with a mean curvature function τ . We will assume that η is constant and work with a one-parameter family of mean curvatures τt = t + λ,

(21)

where t is constant and λ(x) =

−1 −π < x < 0 1 0 < x < π.

(22)

This three-parameter family is suitable for exploring simultaneous violations of the near-CMC and small-TT hypotheses. The parameters η and μ control the size of the relevant pieces of the transverse-traceless tensor. On the other hand, t controls the departure from CMC in the sense that for large values of t the mean curvature has small relative deviation from its mean, and is hence near-CMC (see also Proposition 22 and the subsequent discussion in Appendix B). Data of this kind fall outside the current theory of the conformal method for two reasons. First, the manifold possesses a non-trivial conformal Killing field (∂x ) and the non-CMC data is not constant along it. Second, the discontinuities in τt make the data more singular than is treated in the current best low-regularity results of [9] for the full coupled system (6). We avoid both difficulties by showing that the reduced system (20) for this data can be decoupled, and the analysis will reduce to the study of Lichnerowicz-type equations. Just as for the CMC-conformal method, the decoupling removes potential obstructions posed by conformal Killing fields. Moreover, the data we consider are only modestly irregular for the Lichnerowicz equation alone. In particular, the results of [5] are applicable.


705

Fig. 1. Ranges of t considered by the theorems of Sect. 3.1

3.1. Summary of results. We wish to solve −2κq φ − 2η2 φ −q−1 − κ(μ + (2N )−1 w )2 φ −q−1 + κτt2 φ q−1 = 0, ((2N )−1 w ) − φ q τt = 0

(23)

on S 1 . Here N is a given smooth lapse function, η and μ are constants, and τt is defined 2, p by (21) and (22). We seek solutions (φ, w) ∈ W+ (S 1 ) × W 1, p (S 1 ) where p > 1; the subscript + denotes the subset of positive functions. An easy bootstrap argument shows that if such a solution exists it belongs to W+2,∞ (S 1 ) × W 1,∞ (S 1 ). If (φ, w) is a solution, so is (φ, w + c) for any constant c, and it determines the same solution of the constraint equations. We will say that (φ, w) is the unique solution of (23) if any other solution is of the form (φ, w + c). The existence theory turns out to depend on the choice of lapse function N in the conformal thin sandwich case (or equivalently, on the choice of conformal representative of the background metric in the standard conformal method). We define 1 λN γ N = − S . (24) S1 N It is easy to see that −1 < γ N < 1 and that if N is constant (as in the conformal method with the flat background metric), then γ N = 0. Our results depend on the value of t and its relationship with γ N (Fig. 1). The near-CMC regime is expressed in terms of the distance between t and γ N . Theorem 3 (Near-CMC results). If |t − γ N | > 2 there exists a solution (φ, w) of (23) if and only if η = 0 or μ = 0. Solutions are unique if μ = 0. Note that the condition η = 0 or μ = 0 is exactly the condition that the transverse traceless tensor is not identically zero. Hence Theorem 3 extends the near-CMC existence/uniqueness theorem of [11] and the “no-go” theorem of [14] to this family of data. We have not determined if uniqueness holds for μ = 0. The value t = γ N is special, and we have the following result that is a partial analogue of exceptional Case 4 of Theorem 1. Theorem 4 (Exceptional case: t = γ N ). If t = γ N and if μ = η = 0, then there exists a one-parameter family of solutions of (23). If μ = 0 and η = 0, there does not exist a solution.

706

D. Maxwell

It is not known if the non-existence result of Theorem 4 can be extended to include the case μ = 0. Given the non-existence result of Theorem 4, we can only expect a small-TT existence theorem if t = γ N . We have shown that if γ N = 0, then this is essentially the only condition needed to obtain small-TT solutions, and have obtained a partial result for γ N = 0. Theorem 5 (Small-TT results). Suppose |t| > |γ N | and |t| = 1. If μ = 0 or η = 0, and if μ and η are sufficiently small, then there exists at least one solution of (23). It is not known if existence holds if γ N = 0 and either t = −γ N or |t| < |γ N |. The case |t| = 1 remains open as well. The mean curvature changes sign if and only if |t| < 1. We have the following existence theorem that applies when |t| > 1. Note that since |γ N | < 1, the near-CMC condition |t − γ N | ≥ 2 is strictly stronger than the condition |t| > 1. Theorem 6 (Non-vanishing mean curvature). Suppose |t| > 1 and either μ = 0 or η = 0. Then there exists at least one solution of (23). We have not determined if solutions are unique in this case, nor do we have an extension of the “no-go” theorem to this regime. The existence theory for |t| < 1 is quite different than that for the near-CMC regime. If μ = 0, we can show that when solutions exist, there are usually at least two, and that if μ = 0 and η is sufficiently large, then there are no solutions. Hence a small-TT hypothesis is necessary if μ = 0. Theorem 7 (Nonexistence/non-uniqueness). Suppose |t| < 1 and μ = 0. There exists a critical value η0 ≥ 0 such that if |η| < η0 there exist at least two solutions of (23), and if |η| ≥ η0 there are no solutions. If in addition |t| > |γ N |, then η0 > 0. The preceding theorems omit the case t = ±1. These values of t are interesting as they correspond to mean curvatures τt that are equal to zero on a large set. The techniques for working with such mean curvatures are somewhat specialized, and for simplicity we do not consider these values. We conjecture, however, that Theorem 5 can be extended to include t = ±1. The following theorem collects the results of Theorems 3 through 7 specialized to the case μ = 0 and γ N = 0 where they are most complete. Theorem 8. Suppose μ = 0 and γ N = 0. 1. If |t| > 2, there exists a solution of (23) if and only if η = 0. If a solution exists it is unique. 2. If |t| > 1 and η = 0, there exists at least one solution. 3. If 0 < |t| < 1, there is a critical value η0 > 0. If 0 < |η| < η0 , there are at least two solutions. If |η| > η0 there are no solutions. 4. If t = 0 there exists a solution if and only if η = 0, in which case there is a oneparameter family of solutions. Figure 2 illustrates Theorem 8. We have a fairly complete picture of the existence/uniqueness theory when μ = 0; we are missing a non-existence result for 0 < t < 2 if η = μ = 0, a uniqueness result for 1 < |t| < 2, and results for |t| = 1. A little care is required in translating the results for the model problem to the full conformal method. Because we are seeking solutions within a symmetry class, the number of


707

Fig. 2. Multiplicity of solutions for t ≥ 0 and η ≥ 0 when μ = 0. Dashed lines correspond to curves where the multiplicity is unknown. The shape of the curve separating the existence and non-existence regions for t < 1 is conjectural

solutions we find is a lower bound for the total number of solutions. Non-uniqueness for the model problem implies non-uniqueness for the full conformal method, but uniqueness only implies that there is a single solution with symmetry. Solutions without symmetry (of which there must be more than one if there are any) may be present. Similarly, nonexistence for the model problem implies either non-existence or non-uniqueness for the full conformal method. 3.2. Reduction to root finding. In this section we show how for the specific choice of mean curvatures τt in Eq. (22), the existence theory of system (23) can be reduced to the question of finding roots of a certain real valued function. We first show that the solution of the momentum constraint can be determined exactly, up to knowledge of the value of φ(0). Proposition 1. Suppose (φ, w) ∈ W+2,∞ (S 1 ) × W 1,∞ (S 1 ) is a solution of (23). Let 1 λN γ N = − S . (25) S1 N Then 1 w = φ(0)q λ + γ N . 2N

(26)

Proof. Notice that τt = 2 [δ0 − δπ ], where δx denotes the Dirac delta distribution with singularity at x. If (φ, w) is a solution of (23), then ((2N )−1 w ) = 2φ q [δ0 − δπ ] = 2φ(0)q δ0 − 2φ(π )q δπ . (27) Since ((2N )−1 w ) , 1 = 0 (where ·, · denotes the pairing of distributions on test functions) we have 0 = φ q (δ0 − δπ ), 1 = φ(0)q − φ(π )q . (28) Hence φ(0) = φ(π ).

708

D. Maxwell

The momentum constraint then reads 1 w = φ(0)q λ . 2N

(29)

Hence

for some constant C. Since

1 w = φ(0)q [λ + C] 2N

S1

(30)

w = 0 the value of C is determined by 2N [λ + C] = 0.

(31)

S1

This occurs precisely when C = γ N . Substituting Eq. (26) into the Hamiltonian constraint of system (23) we obtain a nonlocal equation for φ. Proposition 2. Suppose (φ, w) ∈ W+2,∞ (S 1 ) × W 1,∞ (S 1 ) solves (23). Then φ satisfies − 2κq φ − 2η2 φ −q−1 − κ[μ + φ(0)q (γ N + λ)]2 φ −q−1 + κ(t + λ)2 φ q−1 = 0.

(32)

W+2,∞ (S 1 ) is a solution of (32). Then there exists a solution

Conversely, suppose φ ∈ w ∈ W 1,∞ (S 1 ) (uniquely determined up to a constant) of (26) and (φ, w) is a solution of (23). Proof. If (φ, w) is a solution of (23) then Proposition 1 implies w solves (26). Substituting this solution into the Lichnerowicz equation, we obtain Eq. (32). Conversely, suppose φ solves (32). By the choice of γ N , Eq. (26) is integrable and the solution w ∈ W 1,∞ (S 1 ) is determined up to a constant. Let w be such a solution. By construction, w solves the momentum constraint for φ, and φ solves the Hamiltonian constraint for w. That is, (φ, w) is a solution of (23). To study the nonlocal Eq. (32) we introduce a family of Lichnerowicz equations depending on a positive parameter d: −q−1

− 2κq φd − 2η2 φd

−q−1

− κ[μ + (γ N + λ)d q ]2 φd

q−1

+ κ(t + λ)2 φd

= 0. (33)

Clearly the solutions of (32) are in one-to-one correspondence with the solutions φd of (33) satisfying φd (0) = d. The functions φd tend to grow as d increases, and it will be more convenient to work with a rescaled function that is bounded as d → ∞. The following result follows easily from Proposition 2 after defining ψd = d −1 φd . We omit the proof. Proposition 3. The solutions of (23) are in one-to-one correspondence with the functions 2, p ψd ∈ W+ (S 1 ) satisfying 2q

−q−1

− 2κqd − n ψd − 2η2 d −2q ψd

−q−1

− κ(μd −q + γ N + λ)2 ψd

q−1

+ κ(t + λ)2 ψd

=0

(34) and ψd (0) = 1

(35)

for some d > 0. Given a solution ψd solving (34) and satisfying ψd (0) = 1, the corresponding solution φ of (32) is dψd .


709

Equation (34) can be written as a Lichnerowicz equation of the form − u − α 2 u −q−1 + β 2 u q−1 = 0,

(36)

where α ≡ 0 and β ≡ 0. We have the following facts for this equation, which are proved in Appendix A. Proposition 4. Suppose α and β in Eq. (36) belong to L ∞ (S 1 ) and that α ≡ 0 and β ≡ 0. Let p > 1. 2, p

1. There exists a unique solution u ∈ W+ (S 1 ), and moreover u ∈ W 2,∞ (S 1 ). 2. If w ∈ W+2,∞ (S 1 ) is a subsolution of (36), (i.e. −w − α 2 w −q−1 + β 2 wq−1 ≤ 0) then w ≤ u. 3. If v ∈ W+2,∞ (S 1 ) is a supersolution of (36), (i.e. −v − α 2 v −q−1 + β 2 v q−1 ≥ 0) then v ≥ u. 2, p 4. The solution u ∈ W+ depends continuously on (α, β) ∈ L ∞ × L ∞ . We can now define the real valued function F that will be the focus of our study. Definition 1. Let t be a constant and let τt be defined by Eqs. (21) and (22). Let N be a smooth lapse function and let γ N be defined by Eq. (24). Finally, let η and μ be constants. For d > 0, Proposition 4 Part 1 implies that there exists a corresponding solution ψd ∈ W+2,∞ (S 1 ) of Eq. (34). We define F : R>0 → R>0 by F(d) = ψd (0).

(37)

We define F0 to be the analogous function corresponding to the same mean curvature but vanishing transverse-traceless tensor (i.e. for μ = η = 0). From Proposition 3 it is clear that the existence theory of the CTS method for this family of data reduces to the study of the (algebraic) solutions of F(d) = 1. Proposition 5. The solutions (φ, w) ∈ W+2,∞ (S 1 ) × W 1,∞ (S 1 ) of system (23) are in one to one correspondence with the positive solutions of F(d) = 1. 3.3. Solutions of F(d) = 1. Theorems 3 through 7 follow from Proposition 5 and facts about F and F0 proved in this section. Figure 3 shows representative graphs of F and F0 obtained by numerical computation for certain values of t, η and μ. Key features are the singular behaviour of F at d = 0, the limit of F0 at d = 0, and the rapid convergence of F and F0 to a common limit at ∞. We note that for the illustrated choice of t, η and μ it appears there is exactly one solution of F(d) = 1 and none of F0 (d) = 1. 3.3.1. Elementary estimates for F. In this section we establish: 1. If μ = 0 or η = 0 (i.e. if the transverse-traceless tensor is not identically zero) then F(d) is O(d −1 ) for d sufficiently small. 2. If μ = η = 0 then F is uniformly bounded on (0, ∞). 3. For all values of μ and η, F(d) is bounded above for values of d sufficiently large. 4. If μ = 0 or η = 0, then a solution of F(d) = 1 exists if and only if F(d) ≤ 1 for some d > 0. These facts are all demonstrated by examining constant sub- and supersolutions.

710

D. Maxwell

Fig. 3. Functions F and F0 for t = 3/2, μ = 0, and η = 3

Lemma 1. Suppose |t| = 1. We define the constants m d = min(Md,+ , Md,− ), Md = max(Md,+ , Md,− ),

(38)

where

Md,±

2η2 d −2q + κ(μd −q + γ N ± 1)2 = κ(t ± 1)2

2q1

.

(39)

Then m d ≤ ψd ≤ Md for all d > 0 and in particular m d ≤ F(d) ≤ Md .

(40)

Proof. A constant M is a supersolution of (34) so long as 2 − 2η2 d −2q M −q−1 − κ μd −q + γ N + λ M −q−1 + κ(t + λ)2 M q−1 ≥ 0.

(41)

Since λ = ±1 on S 1 , this is ensured if 2 κ(t ± 1)2 M 2q ≥ 2η2 d −2q + κ μd −q + γ N ± 1 .

(42)

In particular, Md is a supersolution. Proposition 4, Part 3 now implies that ψd ≤ Md on S 1 . A similar proof shows that m d is a subsolution if |t| = 1, and hence Proposition 4, Part 2 implies ψd > m d on S 1 . From the limiting behaviour of m d and Md as d → ∞ we have estimates for ψd (and hence F(d)) for large values of d. Lemma 2. Suppose |t| = 1. Let M∞

1 1 − γN q = max 1−t

1 1 + γN q , 1+t

(43)


and m∞

1 1 − γN q = min 1−t

1 1 + γN q . , 1+t

711

(44)

Given > 0, m ∞ − ≤ ψd ≤ M∞ +

(45)

holds for d sufficiently large. If μ = η = 0 then m ∞ ≤ ψd ≤ M∞

(46)

lim Md = M∞

(47)

lim m d = m ∞ .

(48)

for all d > 0. Proof. We note that d→∞

and d→∞

Hence the bounds m ∞ − ≤ ψd ≤ M∞ + hold for d sufficiently large. If μ = η = 0, then m d = m ∞ and Md = M∞ for all d > 0, so m ∞ ≤ ψd ≤ M∞ for all d > 0. The singular or bounded behaviour of F near zero follows from the analogous behaviour of the associated sub- and supersolutions. Lemma 3. Suppose |t| = 1. If η = μ = 0 then F(d) ≤ M∞

(49)

for all d > 0. Otherwise there is a positive constant c such that F(d) ≥ cd −1

(50)

for d sufficiently small. Proof. We note that if η = 0 or μ = 0, then Md,+ and Md,− are both O(d −1 ) at d = 0 and hence so is m d . The uniform upper bound (49) when μ = η = 0 was proved in Lemma 2. The singularity of F at d = 0 gives a simple test for determining if there is at least one solution of F(d) = 1. Lemma 4. Suppose η = 0 or μ = 0. There exists a solution of F(d) = 1 if and only if for some d > 0, F(d) ≤ 1. Proof. By Lemma 3, F(d) > 1 for d sufficiently small. Fixing p > 1, from Proposition 4, Part 4 it follows that the map d → ψd from (0, ∞) to W 2, p (S 1 ) is continuous. From the continuous imbedding of W 2, p (S 1 ) → C(S 1 ) it follows that F is continuous and the result now follows from the Intermediate Value Theorem.

712

D. Maxwell

3.3.2. Proof of Theorem 3 (near-CMC results). In this section we show that in the nearCMC regime (|t − γ N | > 2) the following hold: 1. lim supd→∞ F(d) < 1. 2. F is differentiable and F (d) < 0 if F(d) = 1 (and μ = 0). 3. F(d) < 1 for all d if μ = η = 0. The existence of a solution of F(d) = 1 if μ = 0 or η = 0 follows from Fact 1 and Lemma 4. The uniqueness of solutions of F(d) = 1 if μ = 0 follows from Fact 2. And the non-existence of solutions of F(d) = 1 if μ = η = 0 follows from Fact 3. The upper bounds of Facts 1 and 3 follow from the constant supersolutions of Lemma 1. In effect, F(d) < 1 because ψd < 1 everywhere. Lemma 5. Suppose |t − γ N | > 2. Then M∞ < 1.

(51)

Proof. Note that since |γ N | < 1, if |t − γ N | > 2, then |t| > 1 and in particular |T a| = 1. Suppose first that t > 1. Then 1 − γN 1 + γN 1 − γN 1 + γN q = max , M∞ = max . (52) , t −1 t +1 t −1 t +1 So M∞ < 1 if 1 − γ N < t − 1 and 1 + γ N < t + 1. The first equality holds since 2 < |t − γ N | = t − γ N . The second holds since γ N < 1 < t. The case where t < −1 is proved similarly. Corollary 1. Suppose η = 0 or μ = 0. If |t − γ N | > 2,

(53)

then there exists a solution of F(d) = 1. Proof. From Lemma 2, lim sup F(d) ≤ M∞ .

(54)

d→∞

From Lemma 5 M∞ < 1. Existence of a solution now follows from Lemma 4. If η = 0 and μ = 0 (i.e. for vanishing transverse-traceless tensors) we have a corresponding non-existence result which generalizes the “no-go” theorem of [14] to this family of data. Recall that F0 corresponds to F with η = μ = 0. Corollary 2. If |t − γ N | > 2,

(55)

then F0 (d) < 1 for all d > 0. In particular, there are no solutions of F0 (d) = 1. Proof. If η = 0 and μ = 0 then Md = M∞ for all d > 0. By Lemma 5, M∞ < 1. Hence F(d) < 1 for all d > 0. To show solutions of F(d) = 1 are unique we show that F is decreasing at any solution of F(d) = 1. We start by showing that F is differentiable.


713

Lemma 6. The function F is differentiable. Moreover, F (d) = h(0),

(56)

− 2κq h + d q−2 V h = −R,

(57)

where h ∈ W 2, p (S 1 ) solves

and where

−q−2 q−2 V = (q + 1) 2η2 d −2q + κ(μd −q + γ N + λ)2 ψd + (q − 1)κ(t + λ)2 ψd

(58)

and −q−1

−q−1

+ 2qκμd −3 (μd −q + γ N + λ)ψd R = (q + 2)2η2 d −q−3 ψd

q−1 −q−1 × κ(t + λ)2 ψd − κ(μd −q + γ N + λ)2 ψd .

+ (q − 2) (59)

2, p

Proof. Consider the function M : R>0 × W+ (S 1 ) → L p (S 1 ) defined by M(d, ψ) = −2κq ψ − 2η2 d −q−2 ψ −q−1 −κ(μd −q + γ N + λ)2 d q−2 ψ −q−2 s + κ(t + λ)2 ψ q−1 .

(60)

Using the fact that 2q/n = q − 2 it follows that M(d, ψd ) = 0 for all d > 0. It is tedious but routine to show that M is Fréchet differentiable and M [d, ψ](δ, h) = −2κq h + V h + Rδ.

(61)

From the continuous embedding W 2, p (S 1 ) → C(S 1 ) it follows that the operators V and R are continuous as functions of ψ and d; see, for example, Lemma 11 below that can be used to show that they are locally Lipschitz. So the map (d, ψ) → M [d, ψ] is continuous. The operator from W 2, p (S 1 ) → L p (S 1 ), h → −2κq h + V h

(62)

has a continuous inverse as V ∈ L ∞ ≥ 0 and V ≡ 0 (see, e.g. [5] Theorem 7.7). The Implicit Function Theorem ([1] Cor. 4.2) then implies that given a solution of M(d0 , ψ0 ) = 0 there is a unique function G defined near d0 such that M(d, G(d)) = 0, and G is continuously differentiable. But M(d, ψd ) = 0 for all d, so by the uniqueness of G we have G(d) = ψd . Let h = G (d). Then by the chain rule, 0=

∂ M(d, G(d)) = −2κ h + V h + R. ∂d

(63)

Now F(d) = ψd (0). Since the evaluation map ψ → ψ(0) is linear and continuous on W 2, p (S 1 ), it follows that F is continuously differentiable and F (d) = G (d)(0). That is, F (d) = h(0) where h solves (63). Proposition 6. Suppose |t − γ N | > 2. If μ = 0 there exists at most one solution of F(d) = 1.

714

D. Maxwell

Proof. Suppose F(d) = 1. We will show that F (d) < 0, and hence there can be at most one solution. Consider the functions of a real variable z, g± (z) = −(γ N ± 1)2 z −q−1 + (t ± 1)2 z q−1

(64)

f ± (z) = −2η2 d −2q z −q−1 + g± (z).

(65)

and

Note that g± and f ± are increasing in z for z > 0 and f + (M+ ) = f − (M− ) = 0 (where M± is defined in Lemma 1). Let I− = (−π, 0) and I+ = (0, π ). Then 2q

− 2κqd − n ψd + f ± (ψd ) = 0

(66)

on I± . Since the coefficients of the differential equation (66) are constant on I± , the function ψd is smooth on these intervals. Suppose without loss of generality that M− ≥ M+ . By Lemma 1, M+ ≤ ψd ≤ M− on S 1 . Since g+ (M+ ) ≥ f + (M+ ) = 0, we have g+ (ψd ) ≥ 0 on I+ . To show that g− (ψd ) ≥ 0 on I− we use the near-CMC assumption. Since ψd ≤ M− and f − (M− ) = 0, it follows from Eq. (66) that ψd ≤ 0 on I− . Since ψd (−π ) = ψd (0) = 1, it follows that ψd ≥ 1 on I− . Since g− is increasing, we conclude that (γ N − 1)2 2 2 2 . (67) g− (ψd ) ≥ g− (1) = −(γ N − 1) + (t − 1) = (t − 1) 1 − (t − 1)2 Now (γ N − 1)2 2q ≤ M∞ < 1 (t − 1)2

(68)

by the definition of M∞ and Lemma 5. Hence g− (ψd ) > 0 on I− . By Lemma 6, F (d) = h(0) where − 2κq h + V h = −R,

(69)

and where V and R are defined in Eqs. (58) and (59). Since μ = 0, −q−1

R = (q + 2)2η2 d −2q d −q−3 ψd

+ (q − 2)d q−3 κg± (ψd )

(70)

on I± . Since g± (ψd ) ≥ 0 and g− (ψd ) > 0 we conclude that R ≥ 0,

R ≡ 0.

(71)

Since V ≥ 0 and V ≡ 0, the strong maximum principle ([8] Th. 9.6) then implies that h < 0 on S 1 . In particular, F (d) = h(0) < 0. Corollaries 1 and 2, together with Propositions 6 and 5 imply Theorem 3 – in the near-CMC regime |t − γ N | > 2 there exists a solution of (23) if and only if the TT-tensor is not identically zero. If μ = 0 the solution is unique. Although we have not determined uniqueness if μ = 0, we note that Proposition 6 is the first uniqueness result for the conformal method that does not make use of a bound for |∇τ |.


715

3.3.3. Proof of Theorem 4 (exceptional case: t = γ N ). The value t = γ N is special. We have a partial result that is parallel to the exceptional Case 4 of Theorem 1. Lemma 7. Suppose t = γ N . If μ = η = 0, then F(d) = 1 for all d > 0, and hence there is a one-parameter family of solutions. On the other hand, if μ = 0 but η = 0, then there are no solutions. Proof. If t = γ N and μ = η = 0, then the unique solution of (34) is clearly ψd = 1 (for any d). Hence φ solves (32) if and only if φ is a positive constant. On the other hand, suppose that μ = 0 and η = 0. Then the constant 1 is evidently a subsolution of (34), as is 1 + for sufficiently small. Hence F(d) > 1 for all d > 0. Theorem 4 follows from Lemma 7 and Proposition 5. 3.3.4. Proof of Theorem 5 (small-TT results). In this section we wish to show that solutions of F(d) = 1 exist for small, nonzero, transverse-traceless tensors (i.e. if μ and η are small but not both zero). From Lemma 7 we know that if t = γ N , then there are no solutions of F(d) = 1 when μ = 0 and η = 0. So we cannot expect to find small-TT solutions if t = γ N . We show here that if γ N = 0, then this is the only obstacle. We also obtain a partial result for γ N = 0, showing small-TT solutions exist if |t| > |γ N |. Recall that F0 (d) = ψ0,d (0), where ψ0,d is defined analogously to ψd , but using μ = η = 0. We will establish the following facts: 1. If |γ N | < |t|, then limd→0+ F0 (d) < 1. 2. For any fixed d > 0, F(d) approaches F0 (d) as μ and η approach zero. So if |γ N | < |t|, and if μ and η are sufficiently small, there is a d such that F(d) < 1. If in addition μ = 0 or η = 0, then Lemma 4 implies that there is at least one solution of F(d) = 1. If μ = 0 or η = 0, Lemma 3 shows that F(d) → ∞ as d → 0, but that this 2q singularity is not present if η = μ = 0. In this case, the term −d − n ψ0,d dominates Eq. (34) as d → 0 and we expect the solutions to be nearly constant. The following lemma computes the value of this constant, which is less than one if |γ N | < |t|. Lemma 8. Let N ,t =

1 + γ N2 1 + t2

1

2q

.

(72)

If |t| = 1, then ψ0,d −−−→ N ,t

(73)

d→0

in W 2, p (S 1 ), and hence uniformly on S 1 . In particular lim F0 (d) = N ,t .

(74)

d→0+

Proof. Recall that ψ0,d is the solution of nq

−q−1

− κ(γ N + λ)2 ψ0,d − 2d − 2 κq ψ0,d

q−1

+ κ(t + λ)2 ψ0,d = 0.

(75)

716

D. Maxwell

From Lemma 2 (since μ = η = 0), 0 < m ∞ ≤ ψ0,d ≤ M∞ for all d, and consequently there exists a positive constant C such that 1 − (γ N + λ)2 ψ −q−1 + 1 (t + λ)2 ψ q−1 ≤ C (76) 0,d 0,d 2q 2q for all d > 0. Since ψ0,d satisfies (75) it follows that ||ψ0,d || L p ≤ 2πCd q−2 p

(77)

for all d > 0. The Poincaré inequality implies that there is a constant c p such that if u ∈ W 2, p (S 1 ), (78) ||u||W 2, p (S 1 ) ≤ c p ||u || L p + u . Let Ad =

1 2π

S1

||d ||W 2, p (S 1 )

ψ0,d , and let d = ψ0,d − Ad . Then ≤ c p ||d || L p + d = c p ||ψ0,d || L p (S 1 ) ≤ (2πc p C d 2q/n )1/ p → 0 1 S1

S

(79) as d → 0. We will now show that Ad → N ,t as d → 0. Since ψ0,d = Ad + d , it then follows that ψ0,d → N ,t in W 2, p (S 1 ) as d → 0. Let (dk ) be any positive sequence converging to zero. Since m ≤ Adk ≤ M,

(80)

some subsequence {Adkl } converges to a constant A ∈ [m, M]. Moreover, ψ0,dkl → A uniformly. = 0, and hence Then ψ0,dkl = Adkl + dk → A in W 2, p (S 1 ). For all d, S 1 ψ0,d −q−1 q−1 κ(γ N + λ)2 ψ0,d − κ(t + λ)2 ψ0,d = 0. (81) S1

Using the uniform convergence of ψ0,dkl to A and the fact that 0 < m ≤ ψ0,d ≤ M for all d, we conclude that κ(γ N + λ)2 − Aq−1 κ(t + λ)2 = 0. (82) A−q−1 S1

Now

S1

Similarly,

S1

(γ N + λ)2 = π (γ N + 1)2 + (γ N − 1)2 = 2π 1 + γ N2 . S1

(t + λ)2 = 2π 1 + t 2 .

Hence

A=

1 + γ N2 1 + t2

(83)

(84)

1

2q

= N ,t .

The uniqueness of the limit A now implies that Ad → N ,t as d → 0.

(85)


717

Proposition 7. Suppose |t| > |γ N | and |t| = 1. Then there exists at least one solution of F(d) = 1 if 1. η = 0 or μ = 0 and 2. |η| and |μ| are sufficiently small. Proof. Since |t| > |γ N | it follows that the constant N ,t from Lemma 8 is less than 1. In particular, F0 (d) < 1 for d sufficiently small. Fix a particular value of d such that this holds. By Proposition 4, Part 4 it follows that F(d) → F0 (d) as (η, μ) → (0, 0). In particular, F0 (d) < 1 if μ and η are sufficiently small. The existence result now follows from Lemma 4. 3.3.5. Proof of Theorem 6 (non-vanishing mean curvature). From the definition of the mean curvatures τt , we see that τt has constant sign if |t| > 1, but changes sign if |t| ≤ 1. In this section we wish to show that there are solutions of F(d) = 1 so long as τt has constant sign. Recall that our near-CMC existence result Corollary 1 was obtained by showing that ψd (x) < 1 for all x ∈ S 1 if d is sufficiently large and |t − γ N | > 2. We only need to show, however, that F(d) = ψd (0) < 1 if d is sufficiently large. Section 4 contains an asymptotic analysis that allows us to compute the exact value of limd→∞ F(d) (as well as the speed of the convergence). Assuming the results of Sect. 4 for now, we show in this section that: 1. limd→∞ F0 (d) < 1 if |t| > 1. 2. limd→∞ F0 (d) = 1 if |t| < 1. 3. limd→∞ F(d) = limd→∞ F0 (d) if |t| = 1. In particular, if |t| > 1, then F(d) < 1 for some d > 0. If μ = 0 or η = 0, then Lemma 4 then implies that there is a solution of F(d) = 1. Definition 2. We say that f (x) → L rapidly at infinity if lim | f (x) − L| x n = 0

x→∞

(86)

for all n ∈ N. We say that f (x) → L rapidly at 0 if lim | f (x) − L| x −n = 0

x→0

(87)

for all n ∈ N. Recall that F0 (d) = ψ0,d (0), where ψ0,d is defined analogously to ψd , but with η = μ = 0. Proposition 8. Assume that |t| = 1. Then lim ψ0,d (0) =

d→∞

and this convergence is rapid.

1 − q1

|t|

|t| < 1 |t| > 1,

(88)

718

D. Maxwell

Proof. Assuming the results of Sect. 4, it follows from Theorem 9 applied to Eq. (75) q √ (taking = 2κqd − n ) that

κ |γ N + 1| + κ |γ N − 1| lim ψd (0) = d→∞ κ |t + 1| + κ |t − 1|

1 q

(89)

and this convergence is rapid. Note that since |γ N | < 1, |1 + γ N | + |1 − γ N | = 2. If |t| < 1, then |1 + t| + |1 − t| = 2, otherwise |1 + t| + |1 − t| = 2 |t|. The result now follows. We would like to establish a corresponding limit without the hypothesis η = μ = 0. For large values of d the contribution of the terms involving η and μ in Eq. (34) are small. So we expect that ψ0,d should be a good approximation for ψd , and we expect to obtain the same limit. To make this idea precise, we will show that small perturbations of ψ0,d are sub- and supersolutions of the equation for ψd . Recall from Lemma 2 that 0 < m ∞ ≤ ψ0,d ≤ M∞ for all d > 0. We define Gd : [−m ∞ /2, M∞ ] → L ∞ by Gd (K ) = Nd (ψ0,d + K ),

(90)

2, p

where Nd : W+ (S 1 ) → L p (S 1 ) is the nonlinear Lichnerowicz operator 2q

Nd (w) = −2κqd − n w − 2η2 d −2q w −q−1 − κ[μd −q + λ + γ N ]2 wq−1 .

(91)

So ψ0,d + K is a sub- or supersolution of (34) if and only if Gd (K ) ≤ 0 or ≥ 0 almost everywhere. Using the fact that ψ0,d solves Eq. (75) we can write Gd (K ) = D(K ) + E(K ), where

(92)

q+1 D(K ) = (t + λ)2 (ψ0,d + K )q+1 − ψ0,d , −q−1

E(K ) = (γ N + λ)2 [ψ0,d − (ψ0,d + K )−q−1 ]

2 − 2η2 d −2q + (μd −q + γ N + λ)2 ) (ψ0,d + K )−q−1 .

(93)

Lemma 9. There exist positive constants D− , D+ , E − and E + such that D− K ≤ D(K ) ≤ D+ K D+ K ≤ D(K ) ≤ D− K and

K ≥ 0, K ≤ 0,

−q−1 E − K ≤ (γ N + λ)2 ψ0,d − (ψ0,d + K )−q−1 ≤ E + K

−q−1 E + K ≤ (γ N + λ)2 ψ0,d − (ψ0,d + K )−q−1 ≤ E − K

for all d > 1 and all K ∈ [−m ∞ /2, M∞ ].

(94)

K ≥ 0, K ≤0

(95)


719

Proof. First consider the expression f A (h) = A−q−1 −(A +h)−q−1 for A ∈ [m ∞ , M∞ ] and h ∈ [−m ∞ /2, M∞ ]. Then 1 (q + 1)(A + th)−q−2 dt h. (96) f A (h) = 0

If h ≥ 0 then (q + 1)(2M∞ )−q−2 h ≤ f A (h) ≤ (q + 1)(m ∞ /2)−q−2 h.

(97)

(q + 1)(m ∞ /2)−q−2 h ≤ f A (h) ≤ (q + 1)(2M∞ )−q−2 h.

(98)

If h ≤ 0 then Inequalities (95) now follow letting E + = max[(γ N −1)2 , (γ N +1)2 ](q +1)(m ∞ /2)−q−2 and E − = min[(γ N − 1)2 , (γ N + 1)2 ](q + 1)(2M∞ )−q−2 . The argument for inequality (94) is similar. Proposition 9. There exists a constant c > 0 such that ||ψ0,d − ψd ||∞ < cd −q

(99)

for all d sufficiently large. In particular, lim F(0) = lim ψd (0) = lim ψ0,d (0).

d→∞

d→∞

d→∞

(100)

Proof. For each d sufficiently large, we will find constants K − (d) and K + (d) that are O(d −q ) and that satisfy Gd (K − (d)) < 0 and Gd (K + (d)) > 0. Assuming this for the moment, we see that ψ0,d + K − (d) and ψ0,d + K + (d) are sub- and supersolutions of (34) and hence ψ0,d + K − (d) ≤ ψd ≤ ψ0,d + K + (d) for d sufficiently large. The asymptotics of K ± (d) then imply inequality (99). Notice that D(K ) has the same sign as K . So Gd (K ) > 0 if K > 0 and E(K ) > 0. Now if 0 < K ≤ M∞ then Lemma 9 implies −q−1

E(K ) = (γ N + λ)2 [ψ0,d − (ψ0,d + K )−q−1 ]

2 − 2η2 d −2q + (μd −q + λ + γ N )2 ) (ψ0,d + K )−q−1

≥ E − K − (2η2 + μ2 )d −2q + 4|μ|d −q (m ∞ /2)−q−1 . Let

2 (2η + μ2 )d −q + 4|μ| (m ∞ /2)−q−1 −q K + (d) = d . E−

(101)

(102)

Then 0 < K + (d) ≤ M∞ if d is sufficiently large, and we have E(K + (d)) ≥ 0 and Gd (K + (d)) ≥ 0 also. On the other hand, if −m ∞ /2 ≤ K < 0, then Lemma 9 implies E(K ) ≤ E − K − (2η2 + μ2 )d −2q (2M∞ )−q−1 + 4|μ|d −q (m ∞ /2)−q−1 .

(103)

Let K − (d) = −

4|μ|(m ∞ /2)−q−1 −q d , E−

(104)

720

D. Maxwell

so −m ∞ /2 ≤ K − (d) < 0 if d is sufficiently large. We then have E(K − (d)) ≤ 0 and Gd (K − (d)) ≤ 0 also. Since K − (d) and K + (d) are both O(d −q ), we have proved the desired result. We now summarize the argument that, along with Proposition 5, proves Theorem 6. Proposition 10. Suppose |t| > 1. If η = 0 or μ = 0, there exists at least one solution of F(d) = 1. Proof. By Propositions 8 and 9, if |t| > 1 then 1 1 q lim F(d) = < 1. d→∞ |t|

(105)

So F(d) < 1 for d sufficiently large. Since η = 0 or μ = 0, Lemma 4 now implies there exists a solution of F(d) = 1. 3.3.6. Proof of Theorem 7 (nonexistence/non-uniqueness). In this section we restrict our attention to the case μ = 0, so that η alone controls the size of the transverse-traceless tensor. We show that if |t| < 1, (i.e. when τt changes sign), then there is a critical threshold η0 ≥ 0 for the size of η. If η > η0 , then there are no solutions of F(d) = 1, whereas if η < η0 there are at least two. In some cases we can show that η0 > 0 and hence there are multiple solutions for small values of η. The choice of η plays a critical role in this section, so we use the notation F[η] to distinguish different functions F corresponding to different values of η. Since F[η] (d) only depends on η2 , we can assume that η ≥ 0. We will show the following facts (assuming μ = 0 and |t| < 1): 1. limd→∞ F[η] (d) = 1, and this limit is approached from above. 2. For any fixed d > 0, F[η] (d) is strictly increasing in η. 3. On any finite interval (0, d0 ] we can find η sufficiently large so that F[η] (d) > 1 on (0, d0 ]. The idea of the proof proceeds as follows. Picking an arbitrary η > 0, Fact 1 implies F[η] (d) > 1 for d larger than some d0 . Using Fact 3 we then increase η to ensure that F[η] (d) > 1 on (0, d0 ]. Fact 2 ensures that after having increased η, we still have the condition F[η] (d) > 1 for d > d0 . So F[η] > 1 for all d > 0 and there are no solutions of F[η] (d) = 1. The existence of a critical value of η follows from Fact 2: if no solutions exist for some η, then F[η] (d) > 1 for all d and raising the value of η maintains this inequality. On the other hand, since F[η] (d) > 1 for d large (by Fact 1) and for d near zero (since F[η] is singular there), if F[η] (d) < 1 for some d, then there will be at least two solutions. Proposition 11. For fixed d, the value of F[η] (d) is strictly increasing in η. Moreover,

2η2 F[η] (d) ≥ κ(1 + |t|)2

2q1

d −1 .

(106)

Proof. Fix d > 0 and suppose 0 ≤ η1 ≤ η2 . Let ψd,1 and ψd,2 be the corresponding solutions of (34). Then substituting ψ1 into the equation for ψ2 we have 2q

−q−1

−2κqd − n ψd,1 − 2η22 d −2q ψd,1 − κ[μd −q + λ + γ N ]2 ψd

= 2(η12 − η22 )d −2q ψd,1 < 0.

q−1

+ κ(t + λ)2 ψd

(107)


721

So ψd,1 is a subsolution of the equation for ψd,2 and ψd,1 ≤ ψd,2 . A similar computation shows that ψd,1 + is also a subsolution for > 0 sufficiently small and hence ψd,1 < ψd,2 everywhere. In particular, F[η1 ] (d) < F[η2 ] (d). To obtain the estimate (106) we note that a constant k is a subsolution of (34) if − 2η2 d −2q k −q−1 + κ(t + λ)2 k q−1 ≤ 0.

(108)

This holds in particular if − 2η2 d −2q k −q−1 + κ(1 + |t|)2 k q−1 ≤ 0,

(109)

and therefore if k 2q =

2η2 d −2q . κ(1 + |t|)2

(110)

Since F[η] (d) ≥ k if k is a subsolution of (34), we have established inequality (106). Proposition 12. Suppose μ = 0 and η = 0. Then there exists a constant c > 0 such that ψd ≥ ψ0,d + cd −2q

(111)

for all d sufficiently large. Proof. We use the function Gd : [−m ∞ /2, M∞ ] → L ∞ defined in Sect. 3.3.5. Recall that ψ0,d + K is a subsolution of the equation for ψd if Gd (K ) ≤ 0 almost everywhere. Recall also that Gd can be written Gd (K ) = D(K ) + E(K ),

(112)

where D and E are defined in Eq. (93). If 0 < K ≤ M∞ , then by Lemma 9, D(K ) ≤ D+ K

(113)

for a certain constant D+ > 0. Also, −q−1

E(K ) = (γ N + λ)2 [ψ0,d

− (ψ0,d + K )−q−1 ] − 2η2 d −2q (ψ0,d + K )−q−1

≤ E + K − 2η2 (2M∞ )−q−1 d −2q

(114)

for a certain constant E + > 0. Let K− =

2η2 (2M∞ )−q−1 −2q d . D+ + E +

(115)

If d is sufficiently large, then 0 < K − ≤ M∞ and we then have Gd (K − ) = D(K − ) + E(K − )

≤ D+ K − + E + K − − 2η2 (2M∞ )−q−1 d −2q = 0.

(116)

So ψ0,d + K − is a subsolution, and we have obtained inequality (111) with c = 2η2 (2M∞ )−q−1 /(D+ + E + ).

722

D. Maxwell

The following proposition formalizes the arguments made at the start of this section and, along with Proposition 5, completes the proof of Theorem 7. Proposition 13. Suppose |t| < 1 and μ = 0. There exists η0 ≥ 0 such that if 0 < |η| < η0 , there exists at least two solutions of F(d) = 1, while if |η| > η0 , there are no solutions. If |t| > γ N then η0 > 0. Proof. We first show that ψd (0) > 1 for d sufficiently large. From Proposition 9 we know that limd→∞ ψ0,d (0) = 1 and that this convergence is rapid. On the other hand, from Proposition 12 there is a positive constant c such that ψd (0) > ψ0,d (0) + cd −2q . Hence

ψd (0) − 1 ≥ (ψ0,d (0) − 1) + cd −2q = (ψ0,d (0) − 1)d 2q + c d −2q . (117) From the rapid convergence we have (ψ0,d (0) − 1)d 2q → 0 as d → ∞ and hence ψd (0) > 1 for d large enough. To show that there are no solutions for η sufficiently large, fix a given η1 and pick d0 so that if d > d0 then F[η1 ] (d) > 1. From inequality (106) we can find η2 so that F[η2 ] (d) > 1 for all d ∈ (0, d0 ]. Letting η = max(η1 , η2 ), it follows from Proposition 11 that F[η] (d) > 1 for all d > 1. Let A = inf{η ≥ 0 : F[η] (d) > 1 for alld > 0}; we have just shown that A is nonempty. Suppose η ∈ A and η ≥ η. Proposition 11 implies that for any d > 0, F[η ] (d) ≥ F[η] (d) > 1, and hence η ∈ A. Let η0 = inf A. If η > η0 then η ∈ A and there are no solutions of F[η] (d) = 1. Suppose 0 < η < η0 , and pick η so η < η < η0 . Then η ∈ A and for some d0 , F[η ] (d0 ) ≤ 1. By Proposition 11, F[η] (d0 ) < F[η ] (d0 ) ≤ 1. From Lemma 3 we know that F[η] (d) > 1 for d sufficiently small, and we have already shown that F[η] (d) > 1 for d sufficiently large. From the continuity of F it follows that there are at least two solutions of F[η] (d) = 1, one for d < d0 and one for d > d0 . Proposition 7 implies that η0 > 0 if |t| > |γ N |; if η0 = 0 then there can only be solutions of (34) if η = 0. We have now proved all the results of Sect. 3.1, up to the asymptotic analysis cited in the proof of Proposition 8. 3.4. Sensitivity with respect to a coupling coefficient. The results of the previous sections depend in a sensitive way on coupling constants in Eq. (20). Consider the following variation of the Einstein constraint equations: Rh − |K |2h + tr h K 2 = 0, divh K − (1 + ) d tr h K = 0.

(118)

The case = 0 corresponds with the standard constraint equations. Repeating the analysis above for these perturbed constraint equations the analogue of Eq. (34) is −q−1

2q

−2κqd − n ψd − 2η2 d −2q ψd +d

−q

μ]

2

−q−1 ψd

+ κ(t + λ)

2

− κ[(γ N + λ)(1 + )

q−1 ψd

= 0.

(119)

One readily shows that estimate (50) of Lemma 3 holds for this equation, as does Lemma 4, so long as > −1. Hence there exists a solution of the constraints for this data if and only if F(d) ≤ 1 for some d > 0.


723

Recall that for the standard conformal method (i.e. when = 0), limd→∞ F(d) = 1 if |t| < 1. Since we are seeking solutions of F(d) = 1, it is as if there is a solution of F(d) = 1 at d = ∞. Adjusting affects the value of this limit. We will show that when < 0, limd→∞ F(d) < 1, and the solution at d = ∞ becomes a true solution. On the other hand, for > 0, limd→∞ F(d) > 1, and this allows for there to be no solutions at all of F(d) = 1 for sufficiently small transverse-traceless tensors. We first show that when < 0, we have existence under rather general conditions, and lose the non-existence results of Theorems 4 and 7. Proposition 14. Suppose −1 < < 0 and t = 1. If either μ = 0 or η = 0 then there exists at least one solution of Eq. (119). Proof. Following the the arguments leading to Proposition 8 we see that lim ψ0,d (0) =

d→∞

1

|1 + | q

1

|1 + | q |t|

− q1

|t| < 1 |t| > 1.

(120)

Since |1 + | < 1, we see that for any choice of t = 1, ψ0,d (0) < 1 for d sufficiently large. The arguments of Sect. 3.3.6 can then be repeated to show that limd→∞ ψd (0) = limd→∞ ψ0,d (0), and hence ψd (0) < 1 for d sufficiently large. Hence there exists at least one solution. Raising the value of the coupling coefficient, i.e. when > 0, we lose the small-TT result 5. Proposition 15. Suppose > 0. If t is sufficiently close to γ N , and if μ = 0, then there does not exist a solution of (119). Proof. We will show that φ = 1 + δ is a subsolution of (119) for any d > 0 if δ > 0 is sufficiently small and t is sufficiently close to γ N . Having shown this we conclude that F(d) ≥ 1 + δ for all d > 0, and hence there are no solutions. Note that φ = 1 + δ is a subsolution (for μ = 0) if − 2η2 d −2q (1 + δ)−q−1 − (1 + )2 (γ N + λ)2 (1 + δ)−q−1 + (t + λ)2 (1 + δ)q−1 ≤ 0. (121) First, consider the case δ = 0. We then wish to show that − 2η2 d −2q − (1 + )2 (γ N + λ)2 + (t + λ)2 ≤ 0.

(122)

− (1 + )2 (γ N + λ)2 + (t + λ)2

(123)

Since > 0,

is strictly negative if t = γ N . Hence the left-hand side of (121) is negative if δ = 0, and it is easy to see that it remains negative if δ > 0 is sufficiently small. For any such δ, we observe that this condition also holds for t sufficiently close to γ N .

724

D. Maxwell

4. A Singularly Perturbed Lichnerowicz Equation The most interesting results of Sect. 3 concerning non-existence/non-uniqueness depend on the asymptotic analysis of this section. We consider the singularly perturbed Lichnerowicz equation − 2 u − α 2 u −q−1 + β 2 u q−1 =0

(124)

on S 1 , which we take to be [−π, π ] with endpoints identified. We assume that the functions α and β are constant on the intervals I− = (−π, 0) and I+ = (0, π ) taking on the values α± and β± . Proposition 4 implies that there exists a (unique) solution u ∈ W+2,∞ (S 1 ) of (124) so long as one of α± = 0 and one of β± = 0. By uniqueness of the solution we note that it is even about x = π/2. As → 0, Eq. (124) becomes an algebraic equation for u and we expect that, away from the points of discontinuity of α and β, u converges to the algebraic solution u 0 = |α± /β± |1/q on I± ; see Fig. 4. We are concerned with the behaviour of u at the point of discontinuity, i.e. lim→0+ u (0). The principal result of this section is the following. Theorem 9. Suppose that β− = 0 and β+ = 0. Then 1 |α+ | + |α− | q , lim u (0) = →0 |β+ | + |β− |

(125)

and this convergence is rapid (as defined in Definition 2). To obtain the limit at zero, we use a blow-up argument, guessing an asymptotic form of the solution. We start with a boundary value problem on [0, ∞). Proposition 16. Let u 0 > 0. There exists a solution on [0, ∞) of − U = U −q−1 − U q−1

(126)

satisfying U (0) = u 0 and lim x→∞ U (x) = 1 (with U converging rapidly to its limit at ∞). Moreover, U satisfies the first order equation

2 −q/2 U (127) U = − U q/2 q and U (x) → 0 rapidly as x → ∞.

Fig. 4. Functions u and their limit as → 0


725

Proof. We construct a solution by means of the method of reduction of order. Suppose 0 0. Hence X has an increasing inverse function U : [0, ∞) → [u 0 , 1) satisfying U (0) = u 0 and lim x→∞ U (x) = 1. Moreover,

2 1 U (x)−q/2 − U (x)q/2 . U (x) = = X (U (x)) q

(129)

(130)

An easy computation involving the chain rule and Eq. (127) now shows that U satisfies the ODE (126) and hence U is the function we seek. If u 0 > 1 one shows similarly that the inverse function of 2 u 0 v q/2 dv (131) X (u) = q u vq − 1 defined on (1, u 0 ] is the desired function. When u 0 = 1, then U (x) ≡ 1 is the solution. To show the rapid convergence at infinity we focus on the case 0 0 and lim x→∞ W (x) = 0. Now

2 (1 − W )q/2 − (1 − W )−q/2 = H (W )W, W = (132) q where H is a continuous function near 0 and

d 2 q/2 −q/2 H (0) = (1 − W ) = − 2q. − (1 − W ) dW W =0 q Since W (x) → 0 as x → ∞, there exists x0 so that if x ≥ x0 , √ H (W (x)) < − q.

(133)

(134)

Hence √ W ≤ − q W

(135)

for x ≥ x0 and by Gronwall’s inequality

√ W (x) ≤ W (x0 ) exp(− q x).

(136)

Since W ≥ 0 also, we conclude that W converges rapidly to 0 and U converges rapidly to 1. The rapid convergence when u 0 > 1 is proved similarly, while the result is trivial if u 0 = 1. Finally, we note that the rapid convergence of U to 0 at infinity follows from the rapid convergence of U to 1 at infinity and Eq. (127).

726

D. Maxwell

We now turn to a boundary value problem on R with piecewise constant coefficients. Consider − v − α 2 v −q−1 + β 2 v q−1 = 0

(137)

on R where α and β are equal to the constants α± and β± on the intervals (0, ∞) and (−∞, 0). Proposition 17. Suppose β± = 0. Let L ± = |α± /β± |1/q . There exists a solution v ∈ 2,∞ Wloc (R) of (137) satisfying lim v(x) = L ± .

(138)

x→±∞

Moreover, v converges rapidly to its limits at ±∞, v converges rapidly to 0 at ±∞, and 1 |α+ | + |α− | q . (139) v(0) = |β+ | + |β− |

1 q+1 q−1 2q Proof. Let ω± = α± β± . Given any c > 0 we define x >0 L + U+ (ω+ x) vc = , L − U− (−ω− x) x < 0

(140)

where U± is the solution of (126) provided by Proposition 16 satisfying U± (0) = c/L ± and lim x→∞ U± (x) = 1. Then vc is continuous, satisfies the differential equation (137) on (0, ∞) and (−∞, 0), and has the correct limiting behaviour at ±∞. If for some c, vc is differentiable at 0, then vc will be a weak solution on R and by elliptic regularity the desired strong solution. From Proposition 16 we have q/2 c −q/2 2 c vc (0+) = L + ω+ U+ (0) = L + ω+ − , (141) q L+ L+ and similarly

c −q/2 2 c q/2 vc (0−) = −L − ω− − . q L− L−

Setting these quantities equal we obtain

−q/2 −q/2 q/2 q/2 L + ω+ L + cq = L + ω+ L + + L − ω− L − . + L − ω− L −

(142)

(143)

From the definitions of L ± and ω± we have the identities −q

q

2 2 L 2± ω± = α+ ±2 L ± = β± L ±,

(144)

and hence cq =

|α+ | + |α− | . |β+ | + |β− |

With this choice of c we obtain a solution of (137) satisfying Eq. (139).

(145)


727

Using the function found in Proposition 17 we can construct approximate solutions of the differential equation (124). Our strategy for proving Theorem 9 will be to show that these approximate solutions improve as → 0 and can be corrected using Newton’s method to obtain solutions satisfying the limit (125). We form the approximate solutions first on [−π/2, π/2], defining w (x) = v(x/) + h (x),

(146)

where h will be a small correction term. We will pick h so that w (±π/2) = 0, and hence can we can extend w to be defined on S 1 by declaring it to be even about x = π/2. To define the correction term, we first let 1 2 x 0 ≤ x ≤ π/2 ζ (x) = π , (147) 0 −π/2 < x ≤ 0 and note that ζ (π/2) = 1. Let h (x) = −d,+ ζ (x) − d,− ζ (−x),

(148)

where d,± =

1 v (±π/(2)).

(149)

With this choice of h , w (±π/2) = 0. For p > 1 we define the nonlinear Lichnerowicz operator N : W 2, p (S 1 ) → L p (S 1 ) by N (w) = − 2 w − α 2 w −q−1 + β 2 wq−1 .

(150)

The error E = N (w (x)) is even about x = π/2 and one readily computes that on [−π/2, π/2],

E = −α 2 (v(x/) + h (x))−q−1 − v(x/)−q−1

2 2 d,+ χ+ + d,− χ− , (151) + β 2 (v(x/) + h (x))q−1 − v(x/)q−1 + π where χ± are the characteristic functions of (0, π ) and (−π, 0) respectively. Lemma 10. ||E || L ∞ (S 1 ) → 0

(152)

rapidly as → 0. Proof. From Proposition 17 we know that v (x) → 0 rapidly as x → ∞. Consequently the constants d,± converge rapidly to zero as → 0. Moreover, d,+ χ + and d,− χ− converge rapidly to 0 in L ∞ (S 1 ). 1 Consider F(v) = v −q−1 . Then F(v + h) − F(v) = (−q − 1) 0 (v + th)−q−2 dt h and therefore |F(v + h) − F(v)| ≤ (q + 1) max (v + th)−q−2 |h|. t∈[0,1]

(153)

728

D. Maxwell

Now v (x) ≥ min(L + , L − ) > 0 and h converges rapidly to 0 in L ∞ ([−π/2, π/2]). So there is an m such that v + th ≥ m > 0

(154)

for all t ∈ [0, 1] and all sufficiently small. It follows that ||(v + h )−q−1 − v−q−1 || L ∞ ([−π/2,π/2]) ≤ (q + 1)m −q−2 ||h || L ∞ ([−π/2,π/2]) (155) for sufficiently small. From the rapid convergence of h to zero we conclude that

α 2 (v + h )−q−1 − v−q−1 → 0 (156) rapidly in L ∞ ([−π/2, π/2]) as → 0. A similar argument establishes

β 2 (v + h )q−1 − vq−1 → 0

(157)

rapidly as → 0. We have considered all terms of E and conclude that ||E || L ∞ ([−π/2,π/2]) → 0

(158)

rapidly as → 0. Since E is even about x = π/2, we have the same convergence in L ∞ (S 1 ). p

For constants 0 < m < M and p > 1 we define the slab Sm,M = {u ∈ W 2, p (S 1 ) : m ≤ u ≤ M}. 2, p

Lemma 11. For u ∈ W+ (S 1 ), let Fr (u) = u r . There exists a constant K (m, M, r ) such that ||Fr (u) − Fr (v)|| L p (S 1 ) ≤ K (m, M, r )||u − v|| L p (S 1 )

(159)

p

for all u, v ∈ Sm,M . Let L u,r : W 2, p → L p be the linear function L u,r v = Fr (u)v.

(160)

p

The map u → L u,r is Lipschitz continuous on Sm,M . Proof. Note that if m ≤ x, y ≤ M then 1 r ((1 − t)x + t y)r −1 dt (x − y), x r − yr =

(161)

0

and hence |x r − y r | ≤ r (m r −1 + M r −1 ) |x − y| .

(162)

||Fr (u) − Fr (v)|| L p (S 1 ) ≤ r (m r −1 + M r −1 )||u − v|| L p (S 1 ) .

(163)

Consequently

Inequality (159) now follows setting K = r (m r −1 + M r −1 ).


729

p

If u 1 , u 2 ∈ Sm,M and v ∈ W 2, p , then ||L u 1 ,r v − L u 2 ,r v|| L p (S 1 ) = ||(Fr (u 1 ) − Fr (u 2 ))v|| L p (S 1 ) ≤ ||Fr (u 1 ) − Fr (u 2 )|| L p (S 1 ) ||v|| L ∞ (S 1 ) ≤ K (m, M, r )||u 1 − u 2 || L p (S 1 ) ||v||W 2, p (S 1 ) ≤ K (m, M, r )||u 1 − u 2 ||W 2, p (S 1 ) ||v||W 2, p (S 1 ) . (164) Hence ||L u 1 ,r − L u 2 ,r || ≤ K (m, M, r )||u 1 −u 2 ||W 2, p (S 1 ) which establishes the Lipschitz continuity. One readily shows that the linearization of N at w is the operator N [w] defined by N [w]h = − 2 h + [(q + 1)α 2 w −q−2 + (q − 1)β 2 wq−2 ]h.

(165)

As an immediate consequence of Lemma 11 we see that N is Lipschitz continuous. Corollary 3. Suppose 0 < m < M. There exists a constant C(m, M) such that for all p v, w ∈ Sm,M , ||N [v] − N [w]|| L(W 2, p (S 1 ),L p (S 1 )) < C(m, M)||v − w||W 2, p (S 1 ) .

(166)

Our application of Newton’s method requires an estimate of the size of N −1 as → 0, which we obtain next. Proposition 18. Let V ∈ L ∞ (S 1 ) and consider the operator L = − 2 +V

(167)

as a map from W 2, p (S 1 ) to L p (S 1 ), where p > 1. Suppose there is a constant m such that V ≥ m > 0. Then L is continuously invertible. Moreover, there is a constant C such that if is sufficiently small, −4 ||L−1 || ≤ C .

(168)

Proof. The fact that L is continuously invertible follows from standard elliptic theory and the positivity of V . We turn our attention to obtaining the estimate (168). Let Sr1 denote the circle of radius r , and let ir : Sr1 → S 1 be the natural diffeomorphism. For a function u defined on S 1 let u r = u ◦ ir . Suppose − 2 u + V u = f

(169)

on S 1 . Letting r = 1/ we then have − u r + Vr u r = fr .

(170)

Let I be an interval of length 1 in Sr1 and let I be the interval of length 1/2 at the center of I . From interior L p estimates ([8] Th. 9.11) we have ||u r ||W 2, p (I ) ≤ C1 || fr || L p (I ) + ||u r || L p (I ) , (171) where C1 depends on ||V ||∞ but does not depend on I or r . Averaging these interior estimates over all intervals I in Sr1 we obtain

(172) ||u r ||W 2, p (Sr1 ) ≤ C2 || fr || L p (Sr1 ) + ||u r || L p (Sr1 ) ,

730

D. Maxwell

where C2 (and all subsequent constants Ck ) is independent of r (and ). One readily verifies that for any function w on S 1 , ||∇ k w|| L p (S 1 ) = r

k− 1p

||∇ k wr || L p (Sr1 ) .

(173)

Assuming that r > 1 (i.e. < 1) it then follows that ||u||W 2, p (S 1 ) ≤ C3r

2− 1p

||u r ||W 2, p (Sr1 ) ,

(174)

and therefore

C2 C3 || fr || L p (Sr1 ) + ||u r || L p (Sr1 ) = −2 C2 C3 || f || L p (S 1 ) + ||u|| L p (S 1 ) .

||u||W 2, p (S 1 ) ≤ r

2− 1p

(175)

By Sobolev embedding in S 1 we have for some constant C4 , ||u|| L p (S 1 ) ≤ C4 ||u||W 1,2 (S 1 ) . Suppose
0. Let N : Br (x) → Y be a differentiable map with Lipschitz continuous derivative, i.e. there exists k > 0 such that ||N [x1 ] − N [x2 ]|| L(X,Y ) ≤ k||x1 − x2 || X

(180)


731

for all x1 , x2 ∈ Br (x). Suppose x is a point where N [x] has a continuous inverse. Let c1 = ||N (x)|| and let c2 = ||N [x]−1 ||. If 2kc12 c2 < 1 and 2c1 c2 < r , then there exists a solution of N (u) = 0 satisfying ||u − x|| X ≤ 2c1 c2 . We apply this method to the operators N . Let m = inf v and M = sup v, where v is the asymptotic solution found in Proposition 17. Taking sufficiently small we can assure that m/2 < w < 2M. By the imbedding of W 2, p (S 1 ) into C 0 (S 1 ) we can find an r such that if m/2 < w < 2M and u ∈ Br (w), then m/3 < u < 3M. Let k be p the Lipschitz constant for N on Sm/3,3M obtained in Corollary 3. So for sufficiently small, N is Lipschitz continuous with constant k on Br (w ). Let c1 () = ||N (w )|| L p and let c2 () = ||N −1 ||. By Lemma 10 c1 () converges rapidly to zero, while by Proposition 18, c2 () is O( −4 ). Hence 2kc1 c22 and 2c1 c2 converge rapidly to zero, and for sufficiently small we obtain a solution of N (u ) = 0 with ||u −w ||W 2, p (S 1 ) < 2c1 c2 . By the continuous imbedding of W 2, p (S 1 ) into C 0 (S 1 ) we have in particular that u (0) converges rapidly to w (0) = v(0) as → 0. Since u is the unique solution of (124), we have proved the result. 5. Conclusion By working with a concrete model problem, we have observed a number of new phenomena for the vacuum conformal and CTS methods. For certain conformal data violating both a small-TT and a near-CMC condition we have shown that there cannot be a unique solution: there will either be no solutions or more than one. For other small-TT data violating a near-CMC we have shown that there are multiple solutions. We have also found existence of certain solutions under a very weak near-CMC hypotheses (τ has constant sign), dependence of the solution theory on the lapse function or conformal class representative, and extreme sensitivity of the solution theory with respect to a coupling constant in the Einstein constraint equations. This work was motivated by the following questions that arise from the Yamabepositive small-TT existence theorems of [9] and [18]: 1. Is the small-TT hypothesis required to ensure existence for arbitrary mean curvatures? 2. Are small-TT solutions necessarily unique? 3. Can the Yamabe-positive restriction be relaxed? Our examples were obtained using a Yamabe-null background metric, and therefore do not directly address questions 1) and 2). The answers to these questions in the Yamabe-null case, however, are that the small-TT hypothesis is necessary (at least for the existence of symmetric solutions for symmetric data), and that small-TT solutions need not be unique. Moreover, our coefficient sensitivity results also suggest that if it is possible to extend the existence results of [9] and [18] to Yamabe-null manifolds, the proof will be difficult. These negative results suggest that the conformal and CTS methods do not lead to a good parameterization scheme for solutions of the Einstein constraint equations. Since the conformal method, in its CMC formulation, is so successful, one is lead to wonder if there is some other generalization of it that does lead to a parameterization. This remains to be seen, and the model problem developed here could provide a useful test case for investigating possible alternatives. Acknowledgements. I would like to thank Daniel Pollack and Jim Isenberg for useful comments and discussions.

732

D. Maxwell

A. The Lichnerowicz Equation We give the proof here of Proposition 4 concerning solutions of the differential equation − u − α 2 u −q−1 + β 2 u q−1 = 0

(181)

on S 1 . Proposition 19. Suppose α and β in Eq. (181) belong to L ∞ (S 1 ) and that α ≡ 0 and β ≡ 0. Let p > 1. 2, p

1. There exists a unique solution u ∈ W+ (S 1 ), and u ∈ W 2,∞ (S 1 ). 2. If w ∈ W+2,∞ (S 1 ) is a subsolution of (36), (i.e. −w − α 2 w −q−1 + β 2 wq−1 ≤ 0) then w ≤ u. 3. If v ∈ W+2,∞ (S 1 ) is a supersolution of (36), (i.e. −v − α 2 v −q−1 + β 2 v q−1 ≥ 0) then v ≥ u. 2, p 4. The solution u ∈ W+ depends continuously on (α, β) ∈ L ∞ × L ∞ . Proof. We consider the differential equation (181) to hold on T n = (S 1 )n rather than S 1 so as to be able to cite existing work (recall that n is related to q by q = 2n/(n − 2)). That is, we consider − u − α 2 u −q−1 + β 2 u q−1 = 0

(182)

on (T n , g), where α and β depend only on x n . Since α 2 ≡ 0 and β 2 ≡ 0, [5] Theorem 4.10 and Corollary 4.11 imply that there exists a positive solution in W 2, p for p > n/2. Uniqueness of this solution follows from [5] Theorem 4.9. From uniqueness we know that u is a function of x n alone (otherwise translation along x k with 1 ≤ k ≤ n − 1− would yield a different solution). But then Eq. (182) reduces to Eq. (181). This establishes Part 1 for p > n/2. On the other hand, 2, p if a solution exists for some p > n/2, it also belongs to W+ for any p ∈ (1, n/2]. 2, p Moreover, if u is a solution of (181) in W+ for some p ∈ (1, n/2], then an easy boot2, p strap shows that u ∈ W+ for all p > n/2 and must therefore agree with the unique solution previously found. Thus Part 1 also holds for 1 < p ≤ n/2. Suppose u − ∈ W+2,∞ is a subsolution of (181). Then it is also a subsolution of (182). Let u be the positive solution of (181). Arguing as in [18] Lemma 2 it follows that Mu is a supersolution for any M > 1. Pick M so that u − ≤ Mu. Proposition 8.2 of [5] implies there is a solution v of (182) such that u − ≤ v ≤ Mu. By uniqueness of the solution it follows that v = u. Hence u − ≤ u and we have proved Part 2. Part 3 is proved similarly. To show continuity, we use the Implicit Function Theorem. Consider the map N : 2, p W+ × (L ∞ × L ∞ ) → L p taking (u, α, β) → −u − α 2 u −q−1 + β 2 u q−1 .

(183)

This map is evidently continuous (since W 2, p is an algebra). One readily shows that its Fréchet derivative at (u, α, β) with respect to u in the direction h is N [u, α, β]h = −h + [(q + 1)α 2 u −q−2 + (q − 1)β 2 u q−2 ]h

(184)

The continuity of the map (u, α, β) → N [u, α, β] follows from the fact that W 2, p (S 1 ) is an algebra continuously embedded in C 0 (S 1 ) along with Lemma 11. Since α ≡ 0 and β ≡ 0 the potential V = [(q +1)α 2 u −q−2 +(q −1)β 2 u q−2 ] is not identically zero. By [5]


733

Theorem 7.7, − +V : W 2, p → L p is an isomorphism. The Implicit Function Theorem (see, e.g. [1] Theorem 4.1) then implies that if u 0 is a solution for data (α0 , β0 ), there is a continuous map defined near (α0 , β0 ) taking (α, β) to the corresponding solution of (181). This establishes Part 4. We remark that the hypothesis u ± ∈ W 2,∞ in Parts 2 and 3 can be weakened; we make it only for convenience so as to be able to apply Proposition 8.2 of [5] in a straightforward way. In our applications in Sect. 3, the sub- and supersolutions are either constants or the sum of a constant and an element of W 2,∞ . B. Theory for Even Conformal Data In this section we sketch how, despite the presence of a conformal Killing field, existing techniques for the conformal method can be adapted to the model problem (20) if the conformal data satisfy an evenness hypothesis. For simplicity, we assume all data in this section are smooth, and we focus on the standard conformal method (i.e. N = 1/2). The coupled system to solve is −2κq φ − 2η2 φ −q−1 − κ(μ + w )2 φ −q−1 + κτ 2 φ q−1 = 0, w = φ q τ .

(185)

From Theorem 1 and dimensional reduction we have the following result for the Lichnerowicz equation: − φ − α 2 φ −q−1 + β 2 φ q−1 = 0

(186)

on S 1 . Proposition 20. Suppose α and β belong to C ∞ (S 1 ). There exists a smooth positive solution φ of (186) if and only if 1. α ≡ 0 and β ≡ 0 or 2. α ≡ 0 and β ≡ 0. The solution in Case 1 is unique. In Case 2 the solutions are the positive constants. For the momentum constraint we consider w = f

(187)

on S 1 . The following result is trivial to prove. Proposition 21. Suppose f ∈ C ∞ (S 1 ). There exists a solution w ∈ C ∞ (S 1 ) of (187) if and only if f = 0. (188) S1

Any two solutions of (187) differ by an additive constant. Recall that we are working with functions on S 1 with domain of definition [−π, π ]. We say that a function f on S 1 is even or odd if f (−x) = f (x) or f (−x) = − f (x) for all x ∈ [−π, π ]. Subscripts e and o denote subspaces of even and odd functions. Using the uniqueness results of Propositions 20 and 21 we have the following easy corollaries.

734

D. Maxwell

Corollary 4. Suppose α and β are in Ce∞ (S 1 ). If Condition 1 or 2 of Proposition 20 holds, then the solution φ of (186) belongs to Ce∞ (S 1 ). Corollary 5. Suppose f ∈ Co∞ (S 1 ) and N ∈ Ce∞ (S 1 ). Then there exists a unique solution w ∈ Co∞ (S 1 ) of (187) satisfying w(0) = 0. Any other solution of (187) can be written as a + w where a is constant. Assume η, τ ∈ Ce∞ (S 1 ) and μ is constant. We define a map N : Ce∞ (S 1 ) → Ce∞ (S 1 ) as follows. Let φ ∈ Ce∞ (S 1 ). Then φ q τ is odd and hence there exists a unique function w ∈ Co∞ (S 1 ) solving w = φ q τ .

(189)

1 1 2 [2η2 + κ(μ + 1/(2N )w )2 ] and β = 2q τ , so α and β belong to Ce∞ (S 1 ). Let α = 2κq Finally, define N (φ) to be the solution of (186) for this choice of α and β. The existence of a smooth solution of (19) is equivalent to the existence of a fixed point of N . By assuming that η and τ are even, we have ensured that N is well defined and thus avoiding the trouble with conformal Killing fields. The existence theory of [18] for the standard conformal method now proceeds without change and we have the following generalization of Theorem 2.

Theorem 10. Suppose N ∈ Ce∞ (S 1 ), η, τ ∈ Ce∞ (S 1 ) and μ ∈ R. Suppose further that τ ≡ 0 and that either η ≡ 0 or μ = 0. If there exists a global upper barrier for (η, μ, τ ), then there exists a solution (φ, w) ∈ Ce∞ × Co∞ of (6). Recall that a global upper barrier is defined as follows. Given a smooth even positive function φ, let wφ be an odd solution of wφ = φ q τ .

(190)

Then wφ is uniquely defined. We say that a smooth positive even function is a global upper barrier if for all smooth even functions φ satisfying 0 < φ ≤ , then − 2κq − 2η2 −q−1 − κ(μ + wφ )2 −q−1 + κτ 2 q−1 ≥ 0.

(191)

Following [14] and [11] one readily shows that there is a constant global upper barrier if max |∇τ | is sufficiently small. min |τ |

(192)

To conclude this section, we show how we can use such near-CMC data to construct data that violate both the small-TT condition and the near-CMC condition (192) arbitrarily. To see this we ‘double’ the frequency of the mean curvature: if f is a periodic function with period 2π , let f [k] (x) = f (2k x). Proposition 22. Suppose τ satisfies the near-CMC condition (192), and η and μ are constant. Then for any k ∈ N there exists a solution of (6) (η, μ, τ [k] ) so long as one of η or μ is non-zero. Proof. Let k ∈ N. Since τ is near-CMC, there exists a solution (φ, w) of (6) for confornk

mal data (2−nk η, 2−nk μ, τ ). One verifies then that (2 q φ [k] , 2(n−1)k w [k] ) is a solution for conformal data (η, μ, τ [k] ).


735

[k]

|∇τ | By taking k sufficiently large, we can make the ratio max as large as we please. For min|τ [k] | each of these mean curvatures, we can solve (6) for certain arbitrarily large TT-tensors. This result seems to suggest that large relative gradients of τ are not, by themselves, a source of trouble. The kind of near-CMC violation described above introduces large gradients without affecting the deviation of τ from its mean. On the other hand, we can write a given mean curvature τ as

τ = t + λ,

(193)

1 where t is constant and S 1 λ = 0. If |t| is large relative to, say, 2π S 1 |λ|, then the ratio (192) will be small (and τ will be near-CMC). This weaker notion of being near-CMC is similar to one used in [14]. It is not violated by the mean curvatures of Proposition 22, and extends to the rough mean curvatures considered in Sect. 3.

References 1. Akerkar, R.: Nonlinear functional analysis. New Dehli: Narosa Publishing House, 1999 2. Bourguignon, J.P., Ezin, J.P.: Scalar curvature functions in a conformal class of metrics and conformal transformations. Trans. Amer. Math. Soc. 2, 723–736 (1987) 3. Bartnik, R., Isenberg, J.: The constraint equations. In: Chrusciel, P.T., Friedrich, H. (eds.) The Einstein Equations and the Large Scale Behavior of Gravitational Fields: 50 Years of the Cauchy Problem in General Relativity. Basel: Birkhäuser, 2004 4. Baumgarte, T.W., Ó Murchadha, N., Pfeiffer, H.P.: Einstein constraints: uniqueness and nonuniqueness in the conformal thin sandwich approach. Phys. Rev. D 75(4), 044009 (2007) 5. Choquet-Bruhat, Y.: Einstein constraints on compact n-dimensional manifolds. Class. Quant. Grav. 21(3), S127–S151 (2004) 6. Choquet-Bruhat, Y., York, J.W. Jr.: The Cauchy problem. In: Held, A. (ed.) General Relativity and Gravitation. New York: Plenum, 1980 7. Chrusciel, P.T.: On space-times with U (1) × U (1) symmetric compact Cauchy surfaces. Ann. Phys. 202(1), 100–150 (1990) 8. Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Berlin-HeidelbergNew York: Springer-Verlag, 1999 9. Holst, M., Nagy, G., Tsogtgerel, G.: Rough solutions of the Einstein constraint equations on closed manifolds without near-CMC conditions. Commun. Math. Phys. 288(2), 547–613 (2008) 10. Hebey, E., Pacard, F., Pollack, D.: A variational analysis of Einstein-scalar field Lichnerowicz equations on compact Riemannian manifolds. Commun. Math. Phys. 278, 117–132 (2008) 11. Isenberg, J., Clausen, A., Allen, P.T.: Near-constant mean curvature solutions of the Einstein constraint equations with non-negative Yamabe metrics. Class. Quant. Grav. 25(7), 075009 (2008) 12. Isenberg, J., Choquet-Bruhat, Y., Moncrief, V.: Solutions of constraints for Einsteins equations. C. R. Acad. Sci., Ser. I: Math. 315, 349–355 (1992) 13. Isenberg, J., Moncrief, V.: A set of nonconstant mean curvature solutions of the Einstein constraint equations on closed manifolds. Class. Quant. Grav. 13(7), 1819–1847 (1996) 14. Isenberg, J., Ó Murchadha, N.: Non-CMC conformal data sets which do not produce solutions of the Einstein constraint equations. Class. Quant. Grav. 21(3), S233–S241 (2004) 15. Isenberg, J.: The construction of spacetimes from initial data, Ph.D. dissertation, University of Maryland, 1979 16. Isenberg, J.: Constant mean curvature solutions of the Einstein constraint equations on closed manifolds. Class. Quant. Grav. 12(9), 2249–2274 (1995) 17. Maxwell, D.: Rough solutions of the constraint equations on compact manifolds. J. Hyp. Diff. Eqs. 2(2), 521–546 (2005) 18. Maxwell, D.: A class of solutions of the vacuum Einstein constraint equations with freely specified mean curvature. Math. Res. Lett. 16(4), 627–645 (2009) 19. Pfeiffer, H.P., York, J.W. Jr.: Uniqueness and nonuniqueness in the Einstein constraints. Phys. Rev. Lett. 95(9), 091101 (2005)

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20. Walsh, D.M.: Non-uniqueness in conformal formulations of the Einstein constraints. Class. Quant. Grav. 24(8), 1911–1925 (2007) 21. York, J.W. Jr.: Conformal “thin-sandwich” data for the initial-value problem of general relativity. Phys. Rev. Lett. 82(7), 1350–1353 (1999) Communicated by P.T. Chru´sciel


Communications in


Ergodic Solenoidal Homology: Realization Theorem Vicente Muñoz1 , Ricardo Pérez-Marco2, 1 Facultad de Matemáticas, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040 Madrid, Spain.


2 CNRS, LAGA UMR 7539, Université Paris XIII, 99, Avenue J.-B. Clément, 93430 Villetaneuse, France.

E-mail: [email protected] Received: 15 October 2009 / Accepted: 13 September 2010 Published online: 11 February 2011 – © Springer-Verlag 2011

Abstract: We define generalized currents associated with immersions of abstract oriented solenoids with a transversal measure. We realize geometrically the full real homology of a compact manifold with these generalized currents, and more precisely with immersions of minimal uniquely ergodic solenoids. This makes precise and geometric De Rham’s realization of the real homology by only using a restricted geometric subclass of currents. 1. Introduction We consider a smooth compact connected oriented manifold M of dimension n ≥ 1. Any closed oriented submanifold N ⊂ M of dimension 0 ≤ k ≤ n determines a homology class in Hk (M, Z). This homology class in Hk (M, R), as dual of De Rham cohomology, is explicitly given by integration of the restriction to N of differential k-forms on M. Also, any immersion f : N → M defines an integer homology class in a similar way by integration of pull-backs of k-forms. Unfortunately, because of topological reasons dating back to Thom [13,14], not all integer homology classes in Hk (M, Z) can be realized in such a way. Geometrically, we can realize any class in Hk (M, Z) by topological k-chains. The real homology Hk (M, R) classes are only realized by formal combinations with real coefficients of k-cells. This is not satisfactory for various reasons. In particular, for diverse purposes it is important to have an explicit realization, as geometric as possible, of real homology classes. The first contribution in this direction came in 1957 from the work of S. Schwartzman [9]. Schwartzman showed how, by a limiting procedure, one-dimensional curves embedded in M can define a real homology class in H1 (M, R). More precisely, he proved that this happens for almost all curves solutions to a differential equation admitting an Partially supported through Spanish MEC grant MTM2007-63582.

Second author supported by CNRS (UMR 7539).

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V. Muñoz, R. Pérez-Marco

invariant ergodic probability measure. Schwartzman’s idea is very natural. It consists of integrating 1-forms over large pieces of the parametrized curve and normalizing this integral by the length of the parametrization. Under suitable conditions, the limit exists and defines an element of the dual of H 1 (M, R), i.e. an element of H1 (M, R). This procedure is equivalent to the more geometric one of closing large pieces of the curve by relatively short closing paths. The closed curve obtained defines an integer homology class. The normalization by the length of the parameter range provides a class in Hk (M, R). Under suitable hypothesis, there exists a unique limit in real homology when the pieces exhaust the parametrized curve, and this limit is independent of the closing procedure. In the article [5], we study the different aspects of the Schwartzman procedure, that we extend to higher dimension. Later in 1975, D. Ruelle and D. Sullivan [8] defined, for arbitrary dimension 0 ≤ k ≤ n, geometric currents by using oriented k-laminations embedded in M and endowed with a transversal measure. They applied their results to stable and unstable laminations of Axiom A diffeomorphisms. In a later article Sullivan [11] extended further these results and their applications. The point of view of Ruelle and Sullivan is also based on duality. The observation is that k-forms can be integrated on each leaf of the lamination and then all over the lamination using the transversal measure. This makes sense locally in each flow-box, and then it can be extended globally by using a partition of unity. The result only depends on the cohomology class of the k-form. In [4] we review and extend Ruelle-Sullivan theory. It is natural to ask whether it is possible to realize every real homology class using a topologically minimal (i.e. all leaves are dense) Ruelle-Sullivan current. In order to achieve this goal we must enlarge the class of Ruelle-Sullivan currents by considering immersions of abstract oriented solenoids. We define a k-solenoid to be a Hausdorff compact space foliated by k-dimensional leaves with finite dimensional transversal structure (see the precise definition in Sect. 2). For these oriented solenoids we can consider k-forms that we can integrate provided that we are given a transversal measure invariant by the holonomy group. We define an immersion of a solenoid S into M to be a regular map f : S → M that is an immersion in each leaf. If the solenoid S is endowed with a transversal measure μ, then any smooth k-form in M can be pulled back to S by f and integrated. The resulting numerical value only depends on the cohomology class of the k-form. Therefore we have defined a closed current that we denote by ( f, Sμ ) and that we call a generalized current. This gives a homology class [ f, Sμ ] ∈ Hk (M, R). Our main result is: Theorem 1.1 (Realization Theorem). Every real homology class in Hk (M, R) can be realized by a generalized current ( f, Sμ ), where Sμ is an oriented, minimal, uniquely ergodic solenoid. Minimal and uniquely ergodic solenoids are defined later on. This result strengthens De Rham’s realization theorem of homology classes by abstract currents, i.e. forms whose coefficients are distributions. It is a geometric De Rham’s Theorem where the abstract currents are replaced by generalized currents that are geometric objects. We can ask why we do need to enlarge the class of Ruelle-Sullivan currents. The result does not hold for minimal Ruelle-Sullivan currents due to the following result from [4] (compare with [3]). Theorem 1.2 [4, Cor. 10.2]. Homology classes with non-zero self-intersection cannot be represented by Ruelle-Sullivan currents with no compact leaves.

Ergodic Solenoidal Homology: Realization Theorem

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Therefore it is not possible to represent a real homology class in Hk (M, R) with nonzero self-intersection by a minimal Ruelle-Sullivan current that it is not a submanifold. Note that this obstruction only exists when n − k is even. This may be the historical reason behind the lack of results on the representation of an arbitrary homology class by minimal Ruelle-Sullivan currents. The space of solenoids is large, and we would like to realize the real homology classes by a minimal class of solenoids enjoying good properties. We are first naturally led to topological minimality. As we prove in [4], the spaces of k-solenoids is inductive and therefore there are always minimal k-solenoids. However, the transversal structure and the holonomy group of minimal solenoids can have a rich structure. In particular, such a solenoid may have many distinct transversal measures, each one yielding a different generalized current for the same immersion f . Also when we push Schwartzman ideas beyond 1-homology for some nice classes of solenoids, we see that in general, even when the immersion is an embedding, the generalized current does not necessarily coincide with the Schwartzman homology class of the immersion of each leaf (actually not even this Schwartzman class needs to be well defined). Indeed the classical literature lacks information about the precise relation between Ruelle-Sullivan and Schwartzman currents. One would naturally expect that there is some relation between the generalized currents and the Schwartzman current (if defined) of the leaves of the lamination. We study this problem in [5]. The main result is that there is such a relation for the class of minimal, ergodic solenoids with a trapping region. A solenoid with a trapping region (see the definition in Sect. 2) has holonomy group generated by a single map. Then the bridge between generalized currents and Schwartzman currents of the leaves is provided by Birkhoff’s ergodic theorem. The main result of [5] is the following. Theorem 1.3 [5, Theorems 1.1 & 1.2]. Let Sμ be a minimal solenoid endowed with an ergodic transversal measure μ and possessing a trapping region W . Let f : Sμ → M be an immersion of Sμ into M such that f (W ) is contained in a ball of M. Then for μ-almost all leaves l ⊂ Sμ , the Schwartzman homology class of f (l) ⊂ M is well defined and coincides with the homology class [ f, Sμ ]. If moreover S is uniquely ergodic, then this happens for all leaves. (We recall the definition of Schwartzman homology class and trapping region in Sect. 2.) The solenoids constructed for the proof of the Realization Theorem do satisfy the hypothesis of this theorem and the transversal measure is unique, that is, the solenoids are uniquely ergodic. Solenoidal Hodge Conjecture. The Hodge Conjecture is an statement about the geometric realization of an integral class of pure type ( p, p) in a complex (projective) manifold. If we drop the condition of the class being integral, then Theorem 1.1 suggests a natural conjecture for real homology classes of pure type as follows. For a compact Kähler manifold M of complex dimension n, a complex immersed solenoid f : Sμ → M (that is, a solenoid where the images f (l) of the leaves l ⊂ Sμ are complex immersed submanifolds), of dimension k = 2(n − p), defines a class in Hn− p,n− p (M) = H p, p (M)∗ ⊂ Hk (M, R), as proved in Proposition 9.3 of [4]. It is natural to formulate the following conjecture: Conjecture 1.4 (Solenoidal Hodge Conjecture). Let M be a compact Kähler manifold. Then any class in H p, p (M) is represented by a complex immersed solenoid of dimension k = 2(n − p).

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Note that the standard Hodge Conjecture is stated for projective complex manifolds, since it fails for Kähler manifolds [16]. The counterexamples of [16] are non-algebraic complex tori. It is easy to see that Conjecture 1.4 holds for complex tori (using nonminimal complex solenoids).

2. Solenoids and Generalized Currents Let us review the main concepts introduced in [4]. Definition 2.1. A k-solenoid, where k ≥ 0, of class C r,s , is a compact Hausdorff space endowed with an atlas of flow-boxes A = {(Ui , ϕi )}, ϕi : Ui → D k × K (Ui ), where D k is the k-dimensional open ball, and K (Ui ) ⊂ Rl is the transversal set of the flow-box. The changes of charts ϕi j = ϕi ◦ ϕ −1 j are of the form ϕi j (x, y) = (X (x, y), Y (y)),

(1)

where X (x, y) is of class C r,s and Y (y) is of class C s . Let S be a k-solenoid, and U ∼ = D k × K (U ) be a flow-box for S. The sets L y = × {y} are called the (local) leaves of the flow-box. A leaf l ⊂ S of the solenoid is a connected k-dimensional manifold whose intersection with any flow-box is a collection of local leaves. The solenoid is oriented if the leaves are oriented (in a transversally continuous way). A transversal for S is a subset T which is a finite union of transversals of flow-boxes. Given two local transversals T1 and T2 and a path contained in a leaf from a point of T1 to a point of T2 , there is a well-defined holonomy map h : T1 → T2 . The holonomy maps form a pseudo-group. A k-solenoid S is minimal if it does not contain a proper sub-solenoid. By [4, Sect. 2], minimal sub-solenoids do exist in any solenoid. If S is minimal, then any transversal is a global transversal, i.e., it intersects all leaves. In the special case of an oriented minimal 1-solenoid, the holonomy return map associated to a local transversal, Dk

RT : T → T is known as the Poincaré return map (see [4, Sect. 4]). Definition 2.2. Let S be a k-solenoid. A transversal measure μ = (μT ) for S associates to any local transversal T a locally finite measure μT supported on T , which are invariant by the holonomy pseudogroup, i.e. if h : T1 → T2 is a holonomy map, then h ∗ μT1 = μT2 . We denote by Sμ a k-solenoid S endowed with a transversal measure μ = (μT ). We refer to Sμ as a measured solenoid. Observe that for any transversal measure μ = (μT ) the scalar multiple c μ = (c μT ), where c > 0, is also a transversal measure. Notice that there is no natural scalar normalization of transversal measures.


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Definition 2.3 (Transverse ergodicity). A transversal measure μ = (μT ) on a solenoid S is ergodic if for any Borel set A ⊂ T invariant by the pseudo-group of holonomy maps on T , we have μT (A) = 0 or μT (A) = μT (T ). We say that Sμ is an ergodic solenoid. Definition 2.4. Let S be a k-solenoid. The solenoid S is uniquely ergodic if it has a unique (up to scalars) transversal measure μ and its support is the whole of S. Now let M be a smooth manifold of dimension n. An immersion of a k-solenoid S into M, with k < n, is a smooth map f : S → M such that the differential restricted to the tangent spaces of leaves has rank k at every point of S. The solenoid f : S → M is transversally immersed if for any flow-box U ⊂ S and chart V ⊂ M, the map f : U = D k × K (U ) → V ⊂ Rn is an embedding, and the images of the leaves intersect transversally in M. If moreover f is injective, then we say that the solenoid is embedded. Note that under a transversal immersion, resp. an embedding, f : S → M, the images of the leaves are immersed, resp. injectively immersed, submanifolds. Definition 2.5 (Generalized currents). Let S be an oriented k-solenoid of class C r,s , r ≥ 1, endowed with a transversal measure μ = (μT ). An immersion f :S→M defines a current ( f, Sμ ) ∈ Ck (M), called generalized Ruelle-Sullivan current (or just generalized current), as follows. Let ω be a k-differentialform in M. The pull-back f ∗ ω defines a k-differential form on the leaves of S. Let S = i Si be a measurable partition such that each Si is contained in a flow-box Ui . We define f ∗ ω dμ K (Ui ) (y),

( f, Sμ ), ω = K (Ui )

i

L y ∩Si

where L y denotes the horizontal disk of the flow-box. The current ( f, Sμ ) is closed, hence it defines a real homology class [ f, Sμ ] ∈ Hk (M, R), called Ruelle-Sullivan homology class. Note that this definition does not depend on the measurable partition (given two partitions consider the common refinement). If the support of f ∗ ω is contained in a flow-box U then

( f, Sμ ), ω =

K (U )

f ∗ω

Ly

dμ K (U ) (y).

In general, take a partition of unity {ρi } subordinated to the covering {Ui }, then ∗

( f, Sμ ), ω = ρi f ω dμ K (Ui ) (y). i

K (Ui )

Ly

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Let us see that ( f, Sμ ) is closed. For any exact differential ω = dα we have ∗ ρi f dα dμ K (Ui ) (y)

( f, Sμ ), dα = =

i

K (Ui )

i

K (Ui )

−

i

K (Ui )

Ly

d(ρi f ∗ α)

dμ K (Ui ) (y)

Ly

∗

dρi ∧ f α Ly

dμ K (Ui ) (y) = 0.

The first term vanishes using Stokes in each leaf (the form ρi f ∗ α is compactly supported on Ui ), and the second term vanishes because i dρi ≡ 0. Therefore [ f, Sμ ] is a well defined homology class of degree k. In their original article [8], Ruelle and Sullivan defined this notion for the restricted class of solenoids embedded in M. When M is a compact and oriented n-manifold, the Ruelle-Sullivan homology class [ f, Sμ ] ∈ Hk (M, R) gives an element [ f, Sμ ]∗ ∈ H n−k (M, R), under the Poincaré duality isomorphism Hk (M, R) ∼ = H n−k (M, R). We have the following result (Theorem 10.1 in [4]) which proves Theorem 1.2. Theorem 2.6 (Self-intersection of embedded solenoids). Let M be a compact, oriented, smooth manifold. Let f : Sμ → M be an embedded oriented measured solenoid, such that the transversal measures (μT ) have no atoms. Then we have [ f, Sμ ]∗ ∪ [ f, Sμ ]∗ = 0 in H 2(n−k) (M, R). This indicates that we cannot use only embedded solenoids to represent real homology classes in general. Now let us recall the notions of Schwartzman theory that we are going to need, and that are extensively studied in [5]. Let M be a compact smooth Riemannian manifold. Given a Riemannian immersion c : N → M from an oriented complete smooth manifold N of dimension k ≥ 1, we consider exhaustions (Un ) of N with Un ⊂ N being k-dimensional compact submanifolds with boundary ∂Un . We close Un with a k-dimensional oriented manifold n with boundary ∂n = −∂Un (that is, ∂Un with opposite orientation, so that Nn = Un ∪ n is a k-dimensional compact oriented manifold without boundary), in such a way that c|Un extends to a piecewise smooth map cn : Nn → M. We may consider the associated homology class [cn (Nn )] ∈ Hk (M, Z). Suppose that Volk (cn (n )) → 0. Volk (cn (Nn )) If the following limit exists: lim

n→+∞

1 [cn (Nn )] ∈ Hk (M, R), Volk (cn (Nn ))

we call it a Schwartzman asymptotic k-cycle.

(2)


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Definition 2.7. The immersed manifold c : N → M represents a homology class a ∈ Hk (M, R) if for all exhaustions (Un ), the class (2) exists and equals a. We denote [c, N ] = a, and call it the Schwartzman homology class of (c, N ). For immersed solenoids f : S → M, we may consider the Schwartzman homology classes associated to its leaves. Definition 2.8 (Schwartzman representation of homology classes). Let f : Sμ → M be an immersion in M of an oriented measured k-solenoid S, and give S the induced Riemannian structure. The immersed solenoid f : Sμ → M fully represents a homology class a ∈ H1 (M, R) if for all leaves l ⊂ S, we have that ( f, l) is a Schwartzman asymptotic k-cycle with [ f, l] = a. A class of solenoids with good properties are those which have a trapping region, since for them the holonomy is represented by a single map. The definition is cumbersome but very natural [5, Definition 7.9]. Definition 2.9 (Trapping region). An open subset W ⊂ S of a solenoid S is a trapping region if there exists a continuous map π : S → T = R/Z such that (1) (2) (3) (4) (5)

For some 0 < 0 < 1/2, W = π −1 ((− 0 , 0 )). There is a global transversal T ⊂ π −1 ({0}). Each connected component of π −1 ({0}) intersects T in exactly one point. 0 is a regular value for π . For each connected component L of π −1 (T − {0}) we have L ∩ T = {x, y}, where {x} = L ∩ T ∩ π −1 ((− 0 , 0]) and {y} = L ∩ T ∩ π −1 ([0, 0 )). The main result of [5] is the following theorem.

Theorem 2.10 [5, Theorem 1.2]. Let S be a minimal oriented k-solenoid endowed with a transversal uniquely ergodic measure μ ∈ ML (S) and with a trapping region W ⊂ S. Consider an immersion f : S → M such that f (W ) is contained in a contractible ball in M. Then f : Sμ → M fully represents its Ruelle-Sullivan homology class [ f, Sμ ]. 3. Realization of H1 (M, R) Let M be a C ∞ smooth compact Riemannian manifold. Given a real 1-homology class a ∈ H1 (M, R), we want to construct an immersion f : S → M in M of a uniquely ergodic solenoid Sμ with generalized current [ f, Sμ ] = a. In some situations (depending on the dimension) we will achieve an embedding. Actually the abstract 1-solenoid S that we will construct is independent of a and of M, and moreover it has a 1-dimensional transversal structure. Let h : T → T be a diffeomorphism of the circle with an irrational rotation number (and therefore uniquely ergodic), which is a Denjoy counter-example, i.e. has the unique invariant probability measure supported on the minimal Cantor set K ⊂ T. Let μ K denote the invariant probability measure. For the original construction of Denjoy counter-examples see [1]. Actually for any given > 0, h can be taken to be of class C 2− (see [2]). The suspension of h, Sh = ([0, 1] × T)/(0,x)∼(1,h(x))

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Fig. 1. The 1-solenoid S

is C 2− -diffeomorphic to the 2-torus T 2 . More explicitly, the diffeomorphism is as follows: take c > 0 small, let h t , t ∈ [0, c], be a (smooth) isotopy from id to h, then we define the diffeomorphism H : T 2 → Sh by (t, h −1 (h t (x))), for t ∈ [0, c], H (t, x) = (t, x), for t ∈ [c, 1]. Note that Sh is foliated by the horizontal leaves, so T 2 is foliated accordingly. It can be considered also as a 1-solenoid of class C ω,2− . The sub-solenoid S = ([0, 1] × K )/∼ ⊂ Sh is an oriented 1-solenoid of class C ω,2− , with transversal T = ({0}×T)∩ S = {0}× K . The holonomy is given by the map h, which is uniquely ergodic. Moreover, the associated transversal measure is μ K on the transversal K ∼ = {0} × K . So S is an oriented and uniquely ergodic 1-solenoid. Using the diffeomorphism H , we may see the solenoid S inside the 2-torus, S ⊂ Sh ∼ = T 2 , consisting of the paths (t, x), x ∈ K , t ∈ [c, 1], together with the paths (t, h t (x)), x ∈ K , t ∈ [0, c]. The embedding S → T 2 is of class C ω,2− , so we shall think of S as an oriented 1-solenoid of regularity C ω,2− (Fig. 1). Theorem 3.1 Let M be a compact smooth manifold, and let a ∈ H1 (M, R) be a non-zero 1-homology class. If dim M ≥ 3 then (a positive multiple of) a can be fully represented by an embedding (of class C ∞,2− ) of the (oriented, uniquely ergodic) 1-solenoid S into M. If dim M = 2 then (a positive multiple of) a can be fully represented by a transversal immersion of S into M. Proof. Let C1 , . . . , Cb1 be (integral) 1-cycles which form a basis of the (real) 1-homology of M. Switch orientations and reorder the cycles if necessary so that there are real numbers λ1 , . . . , λr > 0 such that a = λ1 C1 + · · · + λr Cr . By dividing by λi if necessary, we can assume that λi = 1. Consider the solenoid S constructed above and partition the cantor set K into r disjoint compact subsets K 1 , . . . , K r in cyclic order, each of which with

μ K (K i ) = λi .


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Fig. 2. The open manifold U

Consider the transversal T = {0} × T in Sh . We consider angles θ1 , θ2 , . . . , θn ∈ T in the same cyclic order as the K i , such that K i is contained in the open subset Ui ⊂ T with boundary points θi and θi+1 (denoting θn+1 = θ1 ). We may assume that θ1 = 0. Remove the segments [c, 1] × {θi } from Sh to get the open 2-manifold (Fig. 2) U = Sh − ∪i ([c, 1] × {θi }). By construction, our solenoid S is included as a subset of U, S ⊂ U . Suppose that dim M ≥ 3. Then we can C ∞ -smoothly embed F : U → M as follows: suppose that all cycles Ci share a common base-point p0 ∈ M (and are otherwise disjoint to each other). Then embed the central part (0, c) × T ⊂ U in a small ball B around p0 and embed each of the [c, 1] × Ui in M − B in such a way that if we contract B to p0 then the images of [c, 1] × {t}, t ∈ Ui , represent cycles homologous to Ci . F

The embedding f of S into M is defined as the composition S → U → M. By Theorem 2.10, as S is uniquely ergodic, to prove that f : S → M fully represents a, it is enough to see that [ f, Sμ ] = a. Let α be any closed 1-form on M. Since H 1 (M) = H 1 (M, B), we may assume that α vanishes on B. We cover the solenoid S by the flow-boxes ((0, c) × T) ∩ S and [c, 1] × K i , i = 1, . . . , r . As f ∗ α vanishes in the first flow-box, we have

r r ∗

[ f, Sμ ], [α] = f α dμ K i (y) =

Ci , [α]dμ K i (y) =

i=1 K i r

[c,1]

Ci , [α]μ(K i ) =

i=1

i=1 r

Ki

λi Ci , [α] = a, [α],

i=1

proving that [ f, Sμ ] = a. Now suppose that dim M = 2. Let us do the appropriate modifications to the previous construction. Choose cycles Ci sharing a common base-point p0 ∈ M, and such that their intersections (and self-intersections) away from p0 are transversal. Changing Ci by 2Ci if necessary, we suppose that going around Ci does not change the orientation (that is, the normal bundle to Ci is oriented, hence trivial). From the manifold U in Fig. 2, remove [0, c] × {θ1 } to get the open 2-manifold (Fig. 3) V = ((0, c) × (0, 1)) ∪i ([c, 1] × Ui ) . The manifold V can be immersed into the surface M, F : V → M, in such a way that (0, c) × (0, 1) is sent to a ball B around p0 , [c, 1] ×Ui are sent to M − B, the images

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Fig. 3. The open manifold V

of [c, 1] × {t}, t ∈ Ui , represent cycles homologous to Ci if we contract B to a point, and the intersections and self-intersections of horizontal leaves are always transverse. Note that the solenoid S is not contained in V , since we have removed [0, c] × {θ1 } from U . So we cannot define an immersion f : S → M by restricting that of F. To define f in S ∩((0, c)×T), we need to write explicitly our isotopy h t . Consider h : T → T and ˜ lift it to h˜ : R → R with r := h(0) ∈ (0, 1). Consider a smooth function ρ : R → [0, 1], with ρ(t) = 1 for t ≤ 0, ρ(t) = 0 for t ≥ c, and ρ (t) < 0 for t ∈ (0, c). Then we can define ˜ h˜ −1 (x)ρ(t) + x(1 − ρ(t))) mod Z. h t (x) = h( Define the immersion f : S → M as follows: f equals F for (t, x) ∈ [c, 1] × K ⊂ V . For (t, h −1 (h t (x))) ∈ S ∩ ([0, c] × T), we set F(t, (h˜ −1 (x) + 1)ρ(t) + x(1 − ρ(t))), x ∈ K ∩ (0, r ), f (t, h −1 (h t (x))) = −1 x ∈ K ∩ (r, 1). F(t, h˜ (x)ρ(t) + x(1 − ρ(t))), It is easily checked that f sends S ∩ ([0, c] × T) into the ball B and the intersections of the leaves in this portion of the solenoid are transverse. The proof that the Ruelle-Sullivan homology class of f : S → M is [ f, Sμ ] = a goes as before. Remark 3.2. We do not need M to be compact for the above construction to work. If M is non-compact, take integer 1-cycles C1 , C2 , . . . (possibly infinitely many) which form a basis of H1 (M, R). Then for any a ∈ H1 (M, R) there exist an integer r ≥ 1 and λ1 , . . . , λr ∈ R with a = λi Ci . The construction of Theorem 3.1 works. The solenoid S is oriented, regardless of M being oriented or not. 4. Realization of Hk (M, R) Let M be a smooth compact oriented Riemannian C ∞ manifold and let a ∈ Hk (M, R) be a non-zero real k-homology class, 1 ≤ k ≤ n − 1. We are going to construct a uniquely ergodic k-solenoid f : S → M with a 1-dimensional transversal structure, immersed in M and fully representing a.


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To start with, fix a collection of compact k-dimensional smooth oriented manifolds S1 , . . . , Sr and positive numbers λ1 , . . . , λr > 0 such that λi = 1. For any fixed

> 0, let h : T → T be a diffeomorphism of the circle which is a Denjoy counterexample with an irrational rotation number and of class C 2− . Hence h is uniquely ergodic. Let μ K be the unique invariant probability measure, which is supported on the minimal Cantor set K ⊂ T. Partition the Cantor set K into r disjoint compact subsets K 1 , . . . , K r in cyclic order, each of which with μ K (K i ) = λi . We fix two points on each manifold Si , and remove two small balls, Di+ and Di− , around them. Denote Si = Si − (Di+ ∪ Di− ), so that Si is a manifold with oriented boundary ∂ Si = ∂ Di+ ∂ Di− . Fix two diffeomorphisms: ∂ Di+ ∼ = S k−1 , being orientation preserving, and ∂ Di− ∼ = S k−1 , being orientation reversing. There are inclusions A± :=

i± (∂ Di± × K i ) → S k−1 × T,

whose image is S k−1 × K ⊂ S k−1 × T. Define S= (Si × K i )/ ∼, (x, y) ∼ i +−1 ◦ (id ×h) ◦ i − (x, y), (x, y) ∈ A− . This is an oriented k-solenoid of class C ∞,2− , with 1-dimensional transversal dimension. As S k−1 × K ⊂ S in an obvious way, fixing a point p ∈ S k−1 we have a global transversal T = { p} × K ⊂ S k−1 × K ⊂ S. Identifying T ∼ = K , the holonomy pseudogroup is generated by h : K → K . Hence S is uniquely ergodic. Let μ denote the transversal measure corresponding to μ K . We want to give an alternative description of S. Fix an isotopy h t , t ∈ [0, 1], from id to h. Define the set (Fig. 4) W := {(t, x, h −1 (h t (y))) ; t ∈ [0, 1], x ∈ S k−1 , y ∈ K } ⊂ [0, 1] × S k−1 × T. Then we have that S=

(Si × K i ) W / ∼,

(x, y) ∼ (0, i − (x, y)), (x, y) ∈ ∂ Di− × K i , (x, y) ∼ (1, i + (x, y)), (x, y) ∈ ∂ Di+ × K i .

Strictly speaking, we should say that they are diffeomorphic, but we shall fix an identification. We define a map π : S → T = R/Z by (t, x, h −1 (h t (y))) ∈ W , π(t, x, h −1 (h t (y))) = t − 21 , π( p) = 21 ,

p ∈ S − W .

Then W = Int(W ) = π −1 (− 21 , 21 ) is a trapping region according to Definition 2.9. Consider angles τ1 , τ2 , . . . , τn ∈ T in the same cyclic order as the K i , such that K i is contained in the open subset Ui ⊂ T with boundary points τi and τi+1 (denoting

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Fig. 4. The trapping region W

Fig. 5. The manifold X

τn+1 = τ1 ). We may assume that τ1 = 0. Then the solenoid S sits inside the (k + 1)dimensional open manifold X =

(Si × Ui ) ([0, 1] × S k−1 × T)/ ∼,

(where (x, y) ∼ (0, i − (x, y)), (x, y) ∈ ∂ Di− × Ui , (x, y) ∼ (1, i + (x, y)), (x, y) ∈ ∂ Di+ × Ui ),

as the collection of points (x, y), x ∈ Si , y ∈ K i , together with the points (t, x, h −1 (h t (y))), x ∈ S k−1 , y ∈ K , t ∈ [0, 1] (Fig. 5). Remark 4.1. The 1-solenoid constructed in Sect. 3 corresponds to the case Si = S 1 , i = 1, . . . , r . Theorem 4.2 Let M be a compact oriented smooth Riemannian manifold of dimension n, and let a ∈ Hk (M, R) be a non-zero real k-homology class, 1 ≤ k ≤ n − 1. Then (a positive multiple of ) a can be fully represented by a transversal immersion f : S → M of a uniquely ergodic oriented k-solenoid. If moreover, n ≥ 2k + 1 then we can suppose that f is an embedding. Proof. By Proposition A.3, we may take a collection C1 , . . . , Cbk ∈ Hk (M, Z) which are a basis of Hk (M, Q) and such that Ci are represented by immersed submanifolds Si ⊂ M with trivial normal bundle and self-transverse intersections, and such that Si intersects S j transversally. Moreover, if n ≥ 2k +1, we may assume that there are neither intersections nor self-intersections. After switching the orientations of Ci if necessary, reordering the cycles and multiplying a by a suitable positive real number, we may suppose that a = λ1 C1 + · · · + λr Cr ,


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for some r ≥ 1, λi > 0, 1 ≤ i ≤ r , and λi = 1. We construct the solenoid S with the procedure above starting with the manifolds Si and coefficients λi . This is a uniquely ergodic k-solenoid with a 1-dimensional transversal structure, and a trapping region W ⊂ S. Now we want to define an immersion f : S → M, and to prove that it fully represents a. We have the following cases: (1)

n ≥ 2k + 1. The general position property on the Si implies that all Si are disjoint submanifolds of M. As the normal bundle to Si is trivial and Ui is an interval, we can embedded Si × Ui in a small neighbourhood of Si . Fix a base point p0 ∈ M off all Si . Take a small box B ⊂ M around p0 of the form B = [0, 1] × D n−1 , where D n−1 is the open (n − 1)-dimensional ball. Consider a circle T ⊂ D k+1 ⊂ D n−1 and let D k × T ⊂ D k+1 ⊂ D n−1 be a tubular neighbourhood of it, with boundary S k−1 × T. For each i = 1, . . . , r , fix yi ∈ Ui , and consider two paths in M − Int(B), say γi± , where γi− goes from the point (0, yi ) ∈ {0}×Ui ⊂ {0}×T ⊂ {0}× D n−1 ⊂ B to the point ( pi− , yi ) ∈ Si × Ui , and γi+ goes from (1, yi ) ∈ {1} × Ui ⊂ {1} × T ⊂ {1} × D n−1 ⊂ B to ( pi+ , yi ) ∈ Si × Ui . We arrange that γi± are transverse to Si × Ui at ( pi± , yi ) and are disjoint from all S j otherwise. We thicken γi± to immersions γi± × D k × Ui into M − Int(B) such that one extreme goes to Di± × Ui and the other goes to either D k × Ui × {0} ⊂ D k × T × {0} ⊂ D n−1 ×{0} ⊂ B for γi− , or D k ×Ui ×{1} ⊂ D k ×T×{1} ⊂ D n−1 ×{1} ⊂ B for γi− . It is possible to do this in such a way that the Ui directions match, since n ≥ k + 2. Recall that Si = Si − (Di+ ∪ Di− ), and set Si = Si ∪ (γi+ × S k−1 ) ∪ (γi− × S k−1 ), which is diffeomorphic to Si (to be rigorous, we should smooth out corners). Then we can define the set U := (Si × Ui ) ∪ (γi+ × S k−1 × Ui )∪ ∪(γi− × S k−1 × Ui ) ∪ ([0, 1] × S k−1 × T), which is a (k + 1)-dimensional open manifold embedded in M. The manifold U is foliated as follows: Si ×Ui is foliated by Si ×{y}, for y ∈ Ui , and [0, 1]× S k−1 ×T is foliated by L y = {(t, x, h −1 (h t (y))) ; t ∈ [0, 1], x ∈ S k−1 },

(2)

for y ∈ T. Clearly the solenoid S is a sub-solenoid of U, S ⊂ U . Restricting the embedding F : U → M to S we get an embedding f : S → M. By construction f (W ) ⊂ Int(B), i.e. the image of the trapping region is contained in a contractible ball. 1 < n − k ≤ k. The same construction as in (1) works now, with the modification that we have to allow intersections of different leaves, but we may take them to be always transversal. So we get a transversal immersion f : S → M.

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(3)

n − k = 1. The submanifolds Si have trivial normal bundle and they intersect each other transversally. We cannot avoid that the paths γi± intersect other S j , but we arrange these intersections to be transverse. This produces a transversal immersion f of the region S − W of the solenoid into M − Int(B). We have to modify the previous construction of the immersion of W into B, as codimension one does not leave enough room for it to work. Consider the box B = [0, 1] × D n−1 and remove the axis A = [0, 1] × {0}. Use polar coordinates to identify B − A = [0, 1] × S k−1 × (0, 1), where the third coordinate corresponds to the radius. By construction, W ⊂ S embeds into C = [0, 1] × S k−1 × T, as the set of points (t, x, h −1 (h t (y))), t ∈ [0, 1], x ∈ S k−1 and y ∈ K . We remove D = [0, 1] × S k−1 × τ1 from C, so that C − D = [0, 1] × S k−1 × (0, 1). Then W immerses into C − D, by using the process at the end of the proof of Theorem 3.1 (now there is an extra factor S k−1 which plays no role). This is a transversal immersion. There is one extra detail that we should be careful about. When connecting pi± with the two faces of B, the orientations of the Ui should match. This happens because the normal bundle to Si is trivial, and in this case Si ×Ui is (diffeomorphic to) the normal bundle to Si .

We prove now that f : S → M fully represents a; we use Theorem 2.10. The solenoid S has a trapping region W , and f (W ) ⊂ Int(B), a contractible ball in M. So we only need to see that [ f, Sμ ] = a. Recall that the associated transversal measure is μ K on the transversal K . Let α be any closed 1-form on M. Since H 1 (M) = H 1 (M, B), we may assume that α vanishes on B. We cover the solenoid S by the flow-boxes Si × K i , i = 1, . . . , r , and W (where the form α vanishes). Thus r r ∗

[ f, Sμ ], [α] = f α dμ K i (y) =

Ci , [α] dμ K i (y) =

i=1 K i r

Si

Ci , [α]μ(K i ) =

i=1

i=1 r

Ki

λi Ci , [α] = a, [α],

i=1

proving that [ f, Sμ ] = a. Remark 4.3. A similar comment to that of Remark 3.2 applies to the present situation, that is, the compactness of M is not necessary. Remark 4.4. The orientability of M is not necessary as well. If M is non-orientable, we may consider its oriented double cover π : M˜ → M. Then for a ∈ Hk (M, R), there ˜ R) with π∗ (a) ˜ = a. exists a˜ ∈ Hk ( M, We can consider immersed submanifolds f i : Si → M˜ with transversal selfintersections, and intersecting transversally each other. Then it is easy to perturb f i so that f˜i = π ◦ f i : Si → M are immersed oriented submanifolds with transversal self-intersections, and intersecting transversally each other. This will allow to construct a uniquely-ergodic oriented k-solenoid f : S → M transversally immersed in M˜ fully representing (a multiple of) a˜ such that π ◦ f : S → M is transversally immersed in M and fully represents (a multiple of) a. If n ≥ 2k + 1, then we can assume that f is an embedding (since transversal intersections in this dimension do not happen).


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Remark 4.5. Theorem 4.2 also holds (obviously) for k = 0, n. Remark 4.6. In the article [6], we prove that the currents that we have constructed are general enough in order to fill a dense subset of the space of currents. Therefore, the generalized Ruelle-Sullivan currents associated to immersed measured oriented uniquelyergodic solenoids are dense in the space of closed currents. This question was prompted to the authors by Dennis Sullivan. Acknowledgements. The authors are grateful to Alberto Candel, Etienne Ghys, Nessim Sibony, Dennis Sullivan and Jaume Amorós for their comments and interest on this work. In particular, Etienne Ghys early pointed out on the impossibility of realization in general of integer homology classes by embedded manifolds. The first author wishes to acknowledge Universidad Complutense de Madrid and Institute for Advanced Study at Princeton for their hospitality and for providing excellent working conditions. The second author thanks Jean Bourgain and the IAS at Princeton for their hospitality and facilitating the collaboration of both authors.

Appendix. Homology Classes Represented by Submanifolds By a theorem of Thom (see [13] and [14]), if a ∈ Hk (M, Z) then there exists N >> 1 such that N · a is represented by a smooth submanifold of M. This submanifold C ⊂ M is oriented because it represents a non-zero homology class (the top homology of a compact connected non-orientable manifold is zero). Moreover, if n ≥ 2k + 1 or n − k is odd then it can be arranged that the normal bundle of C is trivial [13,14]. If n − k is even then it can be arranged that the normal bundle is trivial if and only if a ∪ a = 0. Also according to Sullivan [12], using Thom’s method and the thesis of Wells [15] one can always represent N · a by an immersed submanifold f : C → M with trivial normal bundle. (Note that the normal bundle is defined for any immersed manifold.) Moreover, with a small perturbation, we may assume that f has only transversal self-intersections. For completeness, we give here a proof of these results by elementary methods. We start first with the case of odd codimension. Lemma A.1. Let M be a compact and oriented manifold of dimension n. Let 1 ≤ k ≤ n − 1 with n − k odd and a ∈ Hk (M, Z). There exists N >> 1 (dependent only on n and k) and a smooth map f : M → S n−k such that for a generic point p ∈ S n−k , C = f −1 ( p) ⊂ M is a smooth submanifold with trivial normal bundle such that [C] = N · a. Proof. Let aˆ ∈ H n−k (M, Z) be the Poincaré dual of a. We aim to construct a map f : M → S n−k such that f ∗ ([S n−k ]) is a multiple of a. ˆ For this, consider a CW decomposition of M. Let (C ∗ (M, Z), ∂) be the complex of CW-cochains, and let a¯ ∈ C n−k (M, Z) such that ∂ a¯ = 0 and [a] ¯ = a. ˆ We start by considering a map f from the (n − k − 1)-skeleton of M to a base point p ∈ S n−k . To define f in the (n − k)-skeleton, write n i Ci∗ , aˆ = i

with {Ci } being the (n − k)-cells of M, and {Ci∗ } the dual basis. Then define f |Ci in such a way that the induced map f |Ci : Ci /∂Ci → S n−k has degree n i .

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To extend f to the higher skeleta, we work as follows: let T be an (n − k + 1)-cell of M. Since a(∂ ˆ T ) = ∂ a(T ˆ ) = 0, we have that f |∂ T : ∂ T → S n−k has degree 0. Therefore, we can extend f to a map T → S n−k . Now by induction on l = 1, 2, . . . we assume that the map f has been extended to the (n − k + l − 1)-skeleton of M and we wish to extend it to the (n − k + l)skeleton. Let T be a (n − k + l)-cell. The map f |∂ T : ∂ T → S n−k gives, recalling that ∂T ∼ = S n−k+l−1 , an element [ f |∂ T ] ∈ πn−k+l−1 (S n−k ). By a result of Serre [10], this group is torsion (since n − k is odd). So there is a non-zero integer m l such that m l ·[∂ T ] = 0. Multiplying a by m l , the map f (in the (n −k +l −1)skeleton) corresponding to a = m l ·a is the composition of f with a map S n−k → S n−k of degree m l . Therefore [ f |∂ T ] = m l · [ f |∂ T ] = 0, and there is no obstruction to extend f to the cell T , and hence to the (n − k + l)-skeleton. In this way, we get an extension to the n-skeleton, i.e. to M. This gives a continuous map f : M → S n−k and it is trivial to verify that f ∗ ([S n−k ]) = N · a, ˆ for some large integer N (actually, N = m 2 m 3 · · · m k ). Now, we homotop f to a smooth function, which we call f again. Taking a regular value p ∈ S n−k , we have a smooth submanifold C = f −1 ( p) of dimension k, and with trivial normal bundle. Clearly, [C] = P D[N · a] ˆ = N · a. Lemma A.2. Let M be a compact and oriented manifold of dimension n. Let 1 ≤ k ≤ n − 1 with n − k even and a ∈ Hk (M, Z). There exists N >> 1 (only dependent of n and k), and an immersion i : C → M of an oriented compact manifold C with i ∗ [C] = N · a and whose normal bundle νC/M → C is trivial. Proof. We consider M × R, which is an (n + 1)-manifold. It is open, but the proof of Lemma A.1 works for it and for the homology class a ∈ Hk (M × R, Z) ∼ = Hk (M, Z). Note that (n + 1) − k is odd, so Lemma A.1 guarantees the existence of a smooth k-dimensional submanifold C ⊂ M × R with trivial normal bundle, and such that [C] = N · a, for some N ≥ 1. Denote by j : C → M × R the inclusion, and let π : M × R → M be the projection into the first factor. Denote by t the coordinate of the R direction, and by ∂t∂ the vertical vector field. Fixing a non-zero normal vector field X to C ⊂ M × R, the compression theorem in [7] allows to isotop the pair ( j, X ) to ( j , ∂t∂ ), where j : C → M × R is an embedding and ∂t∂ becomes a normal vector field to j (C). Therefore the composition i = π ◦ j : C → M is an immersion. Clearly, i ∗ [C] = π∗ j∗ [C] = π∗ [C] = π∗ (N · a) = N · a ∈ Hk (M, R) and the normal bundle to C in M is trivial. The precise result that we use in Sect. 4 is the following: Proposition A.3. Let M be a compact manifold of dimension n, and let bk = dim Hk (M, R). Then we may take a collection C1 , . . . , Cbk ∈ Hk (M, Z) which forms a basis of Hk (M, Q) and such that Ci are represented by immersed submanifolds Si ⊂ M with trivial normal bundle and self-transverse intersections, and such that Si intersects S j transversally. Moreover, if n ≥ 2k + 1, we may assume that there are neither intersections nor self-intersections.


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Proof. Using Lemma A.1 or Lemma A.2 (according to the parity or n − k), we may find a collection of immersed oriented compact submanifolds Si with trivial normal bundle representing a basis for the rational homology Hk (M, Q). Now a small perturbation of each Si makes all intersections of Si with S j , i = j, and all self-intersections of Si , transverse. If n ≥ 2k +1, the transversality of the intersections implies that there are no intersections at all. So the result follows. References 1. Denjoy, A.: Sur les courbes définies par les équations différentielles à la surface du tore. J. Math. Pures Et Appliquées 11(9. série), 333–375 (1932) 2. Herman, M.R.: Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations. Inst. Hautes Études Sci. Publ. Math. 49, 5–233 (1979) 3. Hurder, S., Mitsumatsu, Y.: The intersection product of transverse invariant measures. Indiana Univ. Math. J 40(4), 1169–1183 (1991) 4. Muñoz, V., Pérez-Marco, R.: Ergodic solenoids and generalized currents. Revista Matematica Complutense. In press, doi:10.1007/s13163-010-0050-7, 2010 5. Muñoz, V., Pérez-Marco, R.: Schwartzman cycles and ergodic solenoids. In: Essays in Mathematics and its Applications. Dedicated to Stephen Smale, eds. P. Pardalos, Th.M. Rassias. Berlin-Heidelberg-Newyork: Springer. In press 6. Muñoz, V., Pérez-Marco, R.: Ergodic solenoidal homology: Density of ergodic solenoids. Australian J. Math. Anal. Appl. 6(1), Article 11, 1–8 (2009) 7. Rourke, C., Sanderson, B.: The compression theorem. Geometry & Topology 5, 399–429 (2001) 8. Ruelle, D., Sullivan, D.: Currents, flows and diffeomorphisms. Topology 14(4), 319–327 (1975) 9. Schwartzman, S.: Asymptotic cycles. Ann. Math. 66(2), 270–284 (1957) 10. Serre, J.-P.: Groupes d’homotopie et classes de groupes abéliens.. Ann. Math. 58(2), 258–294 (1943) 11. Sullivan, D.: Cycles for the dynamical study of foliated manifolds and complex manifolds. Invent. Math. 36, 225–255 (1976) 12. Sullivan, D.: René Thom’s work on geometric homology class and bordism. Bull. AMS 41(3), 341–350 (2004) 13. Thom, R.: Sous-variétés et classes d’homologie des variétés différentiables. I et II. C. R. Acad. Sci. Paris 236, 453–454 and 573–575 (1953) 14. Thom, R.: Quelques propriétés globales des variétés différentiables. Commentarii Mathematici Halvetici 236, 17–86 (1954) 15. Wells, R.: Cobordisms groups of immersions. Topology 5, 281–294 (1966) 16. Zucker, S.: The Hodge conjecture for cubic fourfolds. Compositio. Math. 34, 199–209 (1977) Communicated by A. Connes

Commun. Math. Phys. 302, 755–788 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1188-y

Communications in


Sugawara-Type Constraints in Hyperbolic Coset Models Thibault Damour1 , Axel Kleinschmidt2 , Hermann Nicolai3 1 Institut des Hautes Etudes Scientifiques, 35, Route de Chartres, FR-91440 Bures-sur-Yvette, France 2 Physique Théorique et Mathématique, Université Libre de Bruxelles & International Solvay Institutes,

ULB-Campus Plaine C.P. 231, BE-1050 Bruxelles, Belgium. E-mail: [email protected]

3 Max-Planck-Insitut für Gravitationsphysik, Albert-Einstein-Institut, Am Mühlenberg 1,

DE-14476 Potsdam, Germany Received: 21 December 2009 / Accepted: 7 September 2010 Published online: 8 February 2011 – © Springer-Verlag 2011

Abstract: In the conjectured correspondence between supergravity and geodesic models on infinite-dimensional hyperbolic coset spaces, and E 10 /K (E 10 ) in particular, the constraints play a central role. We present a Sugawara-type construction in terms of the E 10 Noether charges that extends these constraints infinitely into the hyperbolic algebra, in contrast to the truncated expressions obtained in Damour et al. (Class. Quant. Grav. 24:6097, 2007) that involved only finitely many generators. Our extended constraints are associated to an infinite set of roots which are all imaginary, and in fact fill the closed past light-cone of the Lorentzian root lattice. The construction makes crucial use of the E 10 Weyl group and of the fact that the E 10 model contains both D = 11 supergravity and D = 10 IIB supergravity. Our extended constraints appear to unite in a remarkable manner the different canonical constraints of these two theories. This construction may also shed new light on the issue of ‘open constraint algebras’ in traditional canonical approaches to gravity. 1. Introduction In canonical formulations of gravity, the constraints are the essential ingredients for, and main obstacles to, carrying out a canonical quantization of gravity [1] (for an overview and bibliography see [2]). This applies in particular to the Hamiltonian (scalar) constraint determining evolution in ‘time’, and therefore the dynamics. The problem of properly setting up and defining the quantum constraints has been tackled in a variety of approaches but, arguably, the problem remains as open as in Bryce DeWitt’s seminal 1967 paper [1]. A further cause of difficulties, shared by all approaches so far, can be traced to the fact that the constraints form an open algebra, that is, the structure ‘constants’ are not constants, but field dependent. At the level of classical maximal supergravity, progress has been made in the last years towards establishing a correspondence between the equations of D = 11 supergravity on the one hand and a geodesic coset model based on the hyperbolic Kac–Moody structure E 10 [3] on the other (similar correspondences exist for other supergravity models). The

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T. Damour, A. Kleinschmidt, H. Nicolai

supergravity equations are treated canonically and therefore comprise dynamical (evolution) equations and constraint equations. There is a precise correspondence between a truncation of the dynamical equations and a truncation of the geodesic equation on the coset E 10 /K (E 10 ) [3]. The D = 11 supergravity constraint equations can similarly be mapped to constraints that can be imposed consistently on the geodesic motion [4]. For instance, imposition of the Hamiltonian constraint implies that the geodesic is null. According to [4] the weakly conserved constraints of D = 11 supergravity can be translated into weakly conserved coset model constraints, which in turn allow for a reformulation as bilinear expressions in terms of conserved charges, that is, as strongly conserved constraints.1 As noted there, this construction is very reminiscent of the wellknown Sugawara construction [5] for affine Lie algebras [6,7]. It is the purpose of the present paper to follow up on this observation, making it more precise and giving the beginning of a generalized Sugawara construction for hyperbolic Kac–Moody algebras which makes the analogy with the affine construction much more compelling. Understanding and reformulating supergravity in these algebraic terms could prove very useful for the transition to the quantum theory (see [8] for first steps towards the quantization of the E 10 /K (E 10 ) model and [9] for pure gravity). An analogy to be kept in mind in this discussion is that of (bosonic) string theory. There, the dynamical equation for the embedding (target space) coordinates can be written as a free wave equation if one adopts a conformal gauge. This free wave equation admits an infinite set μ of conserved charges αn . The price to pay for the simple dynamical equation is that one has to impose the (Fubini-Veneziano-)Virasoro constraints, L ∼ α α, on the solutions. In the quantum version, the Virasoro constraints and the existence of a proper Hilbert space imply the critical dimension [10]. Assuming the validity of the Kac–Moody/supergravity correspondence, the dynamical equations of supergravity also become simple, yielding geodesics on a symmetric space as their solutions. This system is fully integrable. It admits an infinite set of conserved charges, J , that do not (Poisson) commute among themselves, and one can formally write down the general solution in terms of J and some initial data. The complications and interesting structures are then again to be found in the constraints and their algebra. The fact that all constraints found so far admit a Sugawara-like structure, i.e., L ∼ J J , is tantalizing in this analogy, and may turn out to be crucial for the quantisation of the theory. The gauge symmetries encoded in the coset constraints are directly linked to the space-time and gauge symmetries that are known from the geometrical formulation of supergravity. The replacement of the supergravity constraints by coset model constraints with an underlying algebraic structure may also shed new light on the old problem of open constraint algebras alluded to above, circumventing some of the seemingly insurmountable difficulties of the usual canonical formulation. The main new feature here is that the ‘structure constants’, while still dependent on the dynamical degrees of freedom (fields), become constants of motion in the present formulation. More explicitly, suppose the classical constraints C A (φ) satisfy the first-class canonical (Poisson) algebra C A (φ), C B (φ) = f AB C (φ)C C (φ), (1.1) where φ denotes the canonical variables. In the standard formulation of canonical gravity and supergravity, the φ-dependent structure ‘constants’ f AB C (φ) do not (Poisson) 1 As usual, the term ‘weakly conserved constraints’ here refers to a set of constraints C satisfying (modulo the coset equations of motion) dC/dt = f (C) ≈ 0, where f (C) is a function vanishing on the constraint surface defined by C = 0, while ‘strongly conserved’ constraints satisfy dC/dt = 0 (upon use of the equations of motion).

Sugawara-Type Constraints in Hyperbolic Coset Models

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commute with the Hamiltonian and thus vary in time. By contrast, the structure constants obtained with the Sugawara-like form of the constraints do commute with the Hamiltonian constraint, and are thus preserved in time, even though they still depend on the canonical variables φ. Because the correspondence between the space-time based field theory and the one-dimensional E 10 /K (E 10 ) model is only very incompletely understood, it is, however, not clear how to translate the coset model constraints back into more conventional field theory language. At the very least, one can say that the relation between the field variables of the geometric theory and the E 10 variables must be extremely non-local. Obtaining a universal algebraic description of the constraints and their algebra is also desirable from an M-theory point of view. In the same way that the unique dynamical geodesic equation on E 10 /K (E 10 ) allows for maps to different maximal supergravity theories, depending on the level decomposition chosen to describe the infinite-dimensional Lie algebra [11–14], the constraints should also exhibit this ‘versatility’. Our construction below has this property, albeit in a novel way. More precisely, we will define a ‘universal scaffold’ of hyperbolic Sugawara constraints by using null root vectors α of the hyperbolic algebra, decomposed into a sum of two real roots β1 + β2 = α, and the hyperbolic Weyl group. This will define an infinite number of constraints Lα associated with a ‘skeleton’ of roots α on the light-cone in terms of current bilinears. (The notions of skeleton and scaffold are depicted in Figs. 2 and 3 below.) Extending (away from the real βi case) the set of current-bilinear contributions Lα ∼ Jβ1 Jβ2 to a given null-root constraint (α 2 = 0), or extending the skeleton of supporting roots α constraints into the light-cone (α 2 < 0), however, seems to require the choice of a subalgebra of the hyperbolic algebra that is kept manifest. In analogy with affine algebras, this procedure is very suggestive of a choice of ‘spectral parameters’ for the hyperbolic algebra, even though we do not know whether such a realization of the hyperbolic algebra exists. However, the picture that emerges from the present work is that if such realizations exist, they do so only in combination with suitable constraints. Furthermore, such realizations cannot be unique, giving the algebra a ‘chameleon-like’ aspect. This feature would be in line with the conjectured emergence of a space-time structure from the Lie algebra, where the dimension of the emergent space would depend on the decomposition and the chosen form of the constraints, such that the ‘spectral parameters’ would become associated to spatial coordinates.2 These points will be elaborated on and explained below by means of the constraints of D = 11 supergravity and of type IIB supergravity, respectively, but similar results are expected to hold for other decompositions, such as massive IIA theory, as well as for maximal supergravities in lower dimensions. Importantly, though the set of roots ‘supporting’ the constraints is clearly related to the weight diagram of particular highest-weight representations of E 10 , the constraints themselves do not form (under Poisson commutation) a highest or lowest weight representation of the hyperbolic E 10 , as already observed in [4], and explained in much more detail here. Rather, they indicate the existence of new unexplored algebraic structures inside the hyperbolic algebra and its enveloping algebra. We emphasize that our approach is canonical and crucially relies on a split of space and time, as well as certain gauge choices required for matching the supergravity and coset model degrees of freedom. An earlier and conceptually different M-theory proposal 2 However, this association is likely to be more subtle than just a simple equality, as can already be seen for the affine spectral parameter in D = 2 supergravities, cf. Eq. (2.1) of [15] with ρ = t (time) and ρ˜ = x 1 (space).

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based on the indefinite, but non-hyperbolic, ‘very extended’ Kac–Moody algebra E 11 has been developed by Peter West and collaborators [16,17]. In contradistinction to the present work, their approach is ‘covariant’ in the sense that neither a split of space-time nor gauge choices for the supergravity fields are required, and the issue of writing down canonical constraints thus does not arise in the same way. Instead, one needs to introduce extra gauge invariances encompassing the gauge transformations of supergravity, and the problem becomes one of ‘fitting’ such gauge symmetries into the E 11 framework [18]. However, despite many similarities at the kinematical level, especially with regard to embedding the bosonic sectors of maximal supergravities [19–22], it appears doubtful whether a gauge-fixed version of that approach matches with the structures presented here. From the mathematical point of view, it would also be desirable to associate a Sugawara-type construction to a hyperbolic algebra. In the affine case, the existence of this construction is directly linked to the realization of affine algebras as loop algebras via the so-called spectral parameter. A similar description and understanding is lacking for hyperbolic algebras; the only known description is in terms of generators and relations in the Chevalley–Serre basis. Any construction hinting at an alternative description could shed light on the deeper and to date elusive structure of hyperbolic Kac–Moody algebras. After all, even not knowing about the current algebra realization of affine algebras, the existence of a preferred set of bilinear Virasoro operators in the enveloping algebra would almost inevitably lead to this realization. Here, we are searching for a similarly distinguished structure in the enveloping algebra of the hyperbolic algebra. The remainder of the paper is structured as follows. In Sect. 2 we first review the affine Sugawara construction and rephrase it in a slightly unconventional form. We use this form to propose a (partly schematic) trial expression for Sugawara generators for hyperbolic algebras. In Sect. 3 we then explore this trial expression in more detail in the case of E 10 and show that our trial expression does not only serve to reproduce the D = 11 constraints but also those of type IIB supergravity. This also allows for a more precise definition of the Sugawara constraints and an exploration of their structure in terms of a skeleton of constraints associated with null roots and terms induced by covariantization. In appendices, we collect some known results on level decomposition in order to render the presentation self-contained, as well as some more detailed computations.

2. Sugawara Construction Before proceeding to the discussion of the hyperbolic Sugawara construction we first review briefly the definition of Sugawara operators for affine Lie algebras, see [7] (as well as [5,6] for earlier work and [23,24] for generalizations of Sugawara’s construction)

2.1. Affine Sugawara construction. A non-twisted affine Lie algebra can be defined for any finite-dimensional Lie algebra. Let the finite-dimensional Lie algebra gbe simple and generated by T A (A = 1, . . . , dim g) with commutation relations T A , T B = f AB C T C and non-degenerate invariant form T A |T B = κ AB . Then the corresponding affine Lie algebra gˆ has generators TmA (for m ∈ Z), c and d with non-trivial commutation relations

C TmA , TnB = f AB C Tm+n + κ AB mδm,−n c, d, TmA = −mTmA .

(2.1)


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The generator c commutes with all Lie algebra generators and is called the central element,3 while the generator d is called the derivation.4 In any irreducible highest weight representation, the central element c acts as a scalar; its eigenvalue k on that representation is called the level of the representation. For such a level k representation, the Sugawara generators are defined (within the enveloping algebra of the TmA ’s) by [7] (for n ∈ Z) Ln =

1 A : Tn−m TmB : κ AB , ∨ 2(k + h )

(2.2)

m∈Z

where the colons denote normal ordering as appropriate for the highest weight representation and κ AB is the inverse of κ AB ; h ∨ is the dual Coxeter number defined by f AC D f B D C = 2h ∨ κ AB . We note that there are two separate contributions to the normalization of the Sugawara generators (2.2): The first one is k, related to the central extension, the second one h ∨ comes from normal ordering. Both contributions are quantum effects. Below, we will treat these two contributions differently. In the hyperbolic extension, the central generator ceases to be central and is on par with all the other Lie algebra generators. Normal ordering, on the other hand, will be mostly ignored, as our discussion deals with the classical constraints only. Normal ordering ensures that the generators L m are well defined on any element of the representation. The operators (2.2) obey a Virasoro algebra [L m , L n ] = (m − n)L m+n +

k dim g m(m 2 − 1)δm,−n . 12(k + h ∨ )

Their commutators with the affine generators are A L m , TnA = −nTm+n .

(2.3)

(2.4)

Here, we would like to take a more formal point of view and rewrite (2.2) as a quadratic expression in the generators without resorting to an integrable representation. The reason is that the normalization in (2.2) involves the inverse of the (shifted) eigenvalue of the central generator c. However, in the full hyperbolic algebra the element c is no longer central (in fact, the hyperbolic algebra does not possess any central elements), and a direct generalization of (2.2) would thus necessarily involve the inverse of an operator, which furthermore is no longer singled out in the full algebra. For this reason, we formally multiply (2.2) by the central element and drop the normalization constant. We also recall that affine Lie algebras have two different kinds of roots: real roots and null roots. In particular, there is a primitive null root δ which can be used to describe all roots of the affine algebra via an affine ladder diagram: Let fin ≡ (g) be the set of roots of the finite-dimensional algebra g (where we include α = 0 for simplicity), then the root system of the affine extension gˆ is aff ≡ (ˆg) = α + nδ : α ∈ fin and n ∈ Z , (2.5) that is, there are Z copies of the finite root system. The roots nδ are null roots and the associated root space gˆ nδ has dimension given by the rank: mult(nδ) = dim gˆ nδ = rank(g) 3 The central element of the affine Lie algebra, here denoted c, is often denoted K ; it should not be confused with the central element of the Virasoro algebra associated to the affine algebra. 4 This terminology follows from the presentation of affine algebras as loop algebras where d is the derivative with respect to the spectral parameter [7].

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for n = 0. For n = 0 the dimension is equal to that of the Cartan subalgebra and takes the value rank(g) + 2 (the two extra elements are c and d). All other roots are real and the corresponding root spaces are one-dimensional. Using the structure of the affine root system we can rewrite the commutation relations (2.1) as Tα1 , Tα2 = f α1 α2 α1 +α2 Tα1 +α2 + κα1 ,α2 c, (2.6) where we have suppressed the multiplicity index for null roots. The values of f α1 α2 α1 +α2 and κα1 ,α2 can be obtained by comparison with (2.1). We furthermore define quadratic generators in the enveloping algebra U (ˆg) by L nδ := Tnδ−β Tβ , (2.7) β∈aff

where Tβ is a canonically normalized element in the root space gˆ β . If the β root space is degenerate, we choose an orthonormal basis and contract with the canonically conjugate basis. Then the definition (2.7) is unambiguous except when the root spaces of nδ − β and β have different dimensions. This happens only when one of nδ − β or β is equal to zero, i.e., when one of the generators belongs to the Cartan subalgebra. In that case the generators are to be contracted according to the definition (2.2), i.e., we omit any terms involving a contraction with c or d, but contract only with elements of the Cartan subalgebra of the horizontal g. Except for this point and the lack of normal ordering, the expression (2.7) is a reformulation of (2.2). Note that although we could have defined quadratic generators of the form (2.7) for any point on the root lattice, we do this only for null roots. To get a Virasoro algebra it is furthermore essential that the space of null roots has an additive structure since all null roots lie on a Z-graded line. The affine Weyl group is the semi-direct product of the finite Weyl group with a translation group [25]. After the standard embedding of the affine algebra into a hyperbolic algebra of over-extended type [26], the affine Weyl group can also be described as the subgroup of the hyperbolic Weyl group stabilizing an affine null root [27]; the so-called affine translations are then realized as Lorentz boosts along this null direction.5 Since null roots nδ are stabilized by the affine Weyl group W aff , the l.h.s. of the definition (2.7) is invariant under the action of the Weyl group. One can check that the r.h.s. is also invariant. Besides the convention for null root spaces, the definition (2.7) differs from the standard one (2.2) by its lack of normal ordering. However, as is well known, this affects only the generator L 0 for affine algebras. In addition, normal ordering is only required for the quantum theory, whereas we are here mainly concerned with the structure of the classical constraints. In the classical theory, one associates to each symmetry generator Tα a corresponding conserved charge, say Jα . Accordingly, we will below consider expressions such as (2.7) (with the replacement Tα → Jα ) as functions on phase space and leave open the quantum definition of the constraints. We also remark that the generator L 0 as defined in (2.2) differs from the Hamiltonian (quadratic Casimir) by a term proportional to cd. Omission of this term is admissible in the affine case, but not in the hyperbolic algebra. [In other words, our hyperbolic-algebra generalization of (2.2) will contain terms of the type cd, which do not enter the affine version of (2.2).] Correlatively, 5 We also note that the Weyl orbit of the ‘cusp’ δ is dense on the boundary of the hyperbolic space obtained by projecting the interior of the forward lightcone onto the unit hyperboloid. Equivalently, the rays through all the hyperbolic null roots cover the boundary of the lightcone densely.


761

while the affine Hamiltonian is bounded below, i.e., L 0 ≥ 0, the full Hamiltonian is not because the Cartan-Killing metric on the Cartan subalgebra is indefinite for hyperbolic algebras (with c|c = d|d = 0 and c|d = 1). We now proceed to compute the algebra of the constraints as defined by (2.7). In the course of the following computations we manipulate infinite sums formally, well aware that they are not well-defined and normally would require a normal ordered evaluation on a representation space. With this in mind one computes in the universal enveloping algebra, [L mδ , Tα ] = −2κα,−α c Tmδ+α ,

(2.8)

which is the same as (2.4), but now expressed in terms of affine roots. The important point we wish to emphasize here is that the r.h.s is bilinear in affine generators since we multiplied the Sugawara generators by the central element. Continuing now to the commutator of two Sugawara generators (2.7) leads to [L mδ , L nδ ] = 2(m − n) c L (m+n)δ ,

(2.9)

so that in this formulation the algebra closes with a pre-factor (= c) that is itself an algebra generator. Due to the lack of normal ordering one does not obtain the central term as in (2.3). Neither is the shift by the dual Coxeter number visible in this formal computation in the enveloping algebra. 2.2. Hyperbolic Sugawara construction. The expression (2.7) can be formally generalized to hyperbolic Lie algebras of the over-extended type [26].6 In the hyperbolic case the root system hyp is much more complicated than (2.5): Besides the real and null roots there are now time-like (purely imaginary) roots α with α 2 < 0. The multiplicities of these roots grows exponentially and no closed formula for their multiplicities is known although these can be computed algorithmically, for example via the Peterson recursion formula. For each α root space gα ⊂ g , we choose a basis Tα(s)

for s = 1, . . . , mult(α),

(2.10)

which is ‘null orthonormal’ (when using the standard bilinear form) with respect to the corresponding dual basis in the g−α root space: (s )

Tα(s) |Tβ = δs,s δα+β,0 . The commutation relations are then +α2 (s12 ) Tα(s1 1 ) , Tα(s2 2 ) = f (sα11)(sα22 ) α(s112 ) Tα1 +α2 .

(2.11)

(2.12)

Our hyperbolic generalization of the affine Sugawara construction (2.7) then consists of two elements: (i) the choice of a special set of ‘constraint’ generators, labelled by a subset, say C, of the set of pairs (α, s¯ ) labelling the roots (including their degeneracy); and (ii) a general expression for the hyperbolic Sugawara generator Lα,¯s (or ‘generalized Virasoro constraint’) associated to a particular pair7 (α, s¯ ) ∈ C of the form 6 By ‘over-extension’ we mean the canonical extension via the non-twisted affine extension, whereby two nodes are added to the Dynkin diagram; adding a third node would yield ‘very-extended’ algebras [28]. 7 Note that while α runs over a subset of , s¯ correspondingly runs over a subset of the full degeneracy of the root α ∈ .

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Lα,¯s =

β1 ,β2 ∈hyp s1 ,s2 β1 +β2 =α

(s )

(s )

Ms1 ,s2 (β1 , β2 )Tβ1 1 Tβ2 2 .

(2.13)

Here Ms1 ,s2 (β1 , β2 ) denote some numerical coefficients that we expect to be simply ±1 (s) or 0 (or possibly other rational numbers) for an appropriate choice of the dual bases T±α in the ±α root spaces. We do not have yet a full understanding of the precise set C of ‘constraint’ generators,8 nor of the numerical coefficients Ms1 ,s2 (β1 , β2 ) entering the definition of our generalized Virasoro constraints Lα,¯s . We will argue that a distinguished role is played by the ‘null subset’ of C, i.e., by the case where α is a null root. In that case, the corresponding constraint degeneracy index takes only one value (while the degeneracy of a null root within the hyperbolic algebra is equal to the rank). Moreover, still in the case where α is a null root, we will be able to verify that the coefficients Ms1 ,s2 (β1 , β2 ) in (2.13) are indeed simply equal to ±1 when both β1 and β2 (such that α = β1 + β2 ) are real roots. In the following, we shall refer to the better understood ‘null’ subset of C as being the skeleton of C; and we shall refer to the better understood set of special configurations (α, β1 , β2 ), with α null, β1 , and β2 real, and α = β1 + β2 , as being the universal scaffold at the basis of our construction. As the name ‘skeleton’ suggests, there are more constraints than those associated to null roots. Below, we shall give explicit examples of (‘fleshy’) constraints associated with strictly imaginary roots α 2 < 0. However, constraints associated to null roots play a distinguished role in our construction. The special role of light-like α is already suggested by the affine Sugawara construction (2.7) where constraints were only defined for null roots. In addition, the special configurations where both β1 and β2 are real introduce a significant simplification in our construction. Indeed, in that case the root spaces associated to β1 and β2 are one-dimensional, so that there exists a unique (up to sign) contraction between the associated step operators. By contrast, when not both β1 and β2 are real, the root spaces that are paired are multidimensional, and moreover not necessarily of equal dimension. This leaves open many possibilities for ‘contracting’ (s ) (s ) Tβ1 1 with Tβ2 2 in forming Lα,¯s . The information on how to contract the elements of different root spaces is then encoded in the choice of the coefficients Ms1 ,s2 (β1 , β2 ). Let us note, however, that, given a certain pair (α, s¯ ) ∈ C, i.e., given a certain Lie algebra (¯s ) generator Tα , there exists (when α = β1 + β2 ) a distinguished way of contracting (a part of) the β1 root space gβ1 with the β2 one gβ2 . Indeed, if we denote β1 = α − β, (¯s )

(¯s )

so that β2 = +β, the adjoint action of Tα , ad T (¯s ) x ≡ Tα , x maps g−β onto (a part α of) gβ1 = gα−β . We can then use the natural ‘dual’ pairing between g−β and g+β (i.e., between g−β2 and g+β2 ) to write putative constraints of the form9 (s) (s) Lα,¯s = N (α, β) Tα(¯s ) , T−β Tβ . (2.14) β∈hyp s

Here the coefficients N (α, β) no longer depend on the degeneracy index s within the dual spaces g±β , and the sum over s is easily seen to be independent of the choice of 8 The letter C is used here to evoke both the word ‘constraint’, and the fact that the set C appears to have the structure of a convex cone. 9 To see that expression (2.14) is indeed well-defined, one can invoke the invariance of the bilinear form, see Lemma 2.4 in [25].


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(s)

(dual) bases T±β (as long as the orthonormalization condition (2.11) is satisfied). We leave to future work further study of the usefulness of the special construction (2.14). One advantage of expressing the constraints as in (2.13) is that, contrary to the expressions derived in [4] (which were formulated in terms of the G L(10) level decomposition of E 10 ), such a definition a priori appears not to be tied to any particular level decomposition of the hyperbolic algebra. Therefore, this opens up the possibility of writing a ‘universal’ set of coset constraints, whose further (particular) level decompositions could give rise to the apparently different canonical constraints arising in different maximal supergravities (mIIA, IIB, . . .). However, we shall give evidence below that this hope of a universal constraint construction is not fulfilled in this simple way. Rather, we will encounter a more refined construction, where only the scaffold is universal. The reason appears to lie in the existence of various ways of contracting (multi-dimensional) root spaces, i.e., in the possibility of various consistent choices for the coefficients Ms1 ,s2 (β1 , β2 ). Each particular level decomposition might be tied to a particular corresponding choice for these coefficients. Even if this turns out to be the case, it seems that our construction still involves a universal part, namely the part of (2.13) involving the skeleton of ‘null’ constraints, and its associated scaffold of special configurations where a null root α is decomposed into two real roots β1 and β2 . As we shall emphasize below, this universal part is invariant under the Weyl group of the hyperbolic algebra and already yields an infinite number of constraints (associated to the intersection of the light-cone with the root lattice). This ‘universal part’ is, however, not invariant under the hyperbolic algebra itself. As we shall see below, one can associate to each choice of a finite-dimensional subalgebra (used as a way of ‘slicing’ the hyperbolic algebra by means of a corresponding level decomposition) a way of generating additional constraints by covariantizing under that subalgebra. Each such covariantization procedure allows one to ‘flesh out’ the skeleton by adding new constraints inside the light cone and also terms with β1 and β2 not both real. The prescription will be made more precise in Sect. 3 when we discuss the example of E 10 . A further general issue regarding (2.13) is the operator ordering. Below we will work with similar expressions involving functions on classical phase space which are commuting. [Note that they commute as functions, but do not ‘Poisson commute’.] For those the issue of ordering becomes relevant only after the transition to the quantum theory, which we will not consider here. Finally, as written, (2.13) is meant to define only one constraint per root even though null roots have multiplicity greater than one. The structure of null roots in hyperbolic over-extended algebras is known to be given by Weyl orbits through null = W · (n δ), (2.15) n∈Z\{0}

where W is the hyperbolic Weyl group and δ the primitive null root of the affine algebra embedded in the hyperbolic extension. Restricting the construction (3.5) to affine generators reduces all the Weyl orbits to points since δ is invariant under the affine Weyl group. Hence the construction gives constraints only for the roots α = n δ in agreement with the affine Sugawara construction (2.7). At this point, we stress a possible qualitative difference between the usual affine Sugawara construction (2.7) and the corresponding hyperbolic construction (2.13) at the present stage of our understanding of the construction. The affine Virasoro constraints L n δ form a two-sided tower, where n runs over the set of integers Z, while it seems consistent that the hyperbolic constraints Lα,¯s run over a set C which is a one-sided

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convex cone, contained within the past light-cone of the Lorentzian root lattice. This one-sided structure of the constraints was clearly apparent in [4], where only constraints Lα corresponding to negative imaginary α were found, as will be shown in Sect. 3 below.10 This asymmetry between the two-sidedness of the usual affine (Virasoro) constraints, and the one-sidedness of the hyperbolic ones, seems to be deeply rooted in the different physics (and mathematics) associated to the origin of these constraints. In the usual affine case, the origin of the constraints is a gauge invariance under reparametrizations of (two) periodic (world-sheet light-cone) variables σ± = τ ± σ . The periodic nature of these variables, and the real (or hermitian) character of the worldsheet embedding functions, e.g. ∂± X μ (τ, σ ), implies the existence of two-sided Fourier expansions involving, for each choice of sign in σ± the two complex-conjugated basis functions exp(+inσ± ) and exp(−inσ± ). By contrast, the hyperbolic coset models should describe the gravitational physics taking place near a spacelike singularity, i.e., in a time-asymmetric situation of the type t → 0+ , say. Moreover, the hyperbolic coset model is itself parametrized asymmetrically in terms of positive roots only. The analysis of the dynamics of supergravity in [3] found evidence for relating the supergravity fields to one-sided towers of coset variables. This tower consists of the so-called ‘gradient generators’ that are conjectured to correspond to multiple spatial gradients, roughly in terms of a spatial Taylor expansion. It is then natural to conjecture that the usual space-dependent supergravity constraints will also give rise to one-sided-only towers of ‘gradient cousins’ of the (already one-sided) low-level constraints discussed in [4]. Another (related) argument for expecting that the tower of coset constraints be onesided only, is the idea proposed in [4] that the set of constraints be just large enough to reduce the exponentially infinite number of variables entering the hyperbolic coset models to a much smaller number of degrees of freedom involving only a rather small vicinity of the future light-cone in root space (i.e., essentially the gradient generators, plus a relatively manageable set of extra M-theoretic degrees of freedom). To achieve such a strong reduction in the number of degrees of freedom, without killing them all, it is natural to have a set of constraints C which fills, like the coset variables, a onesided cone and whose degeneracies do not grow faster than the ones of the roots. Note, however, that our intuitive argument cannot exclude the possibility that the constraints fill a double-sided cone, if the degeneracies of the constraints are such that the sum of the positive-sided and negative-sided ones does not grow faster than the positive-root degeneracies. Whatever is the ultimate definition of the physically correct set of coset constraints, Lα,¯s , one would expect it to satisfy some commutation relations (of the general type [L, L] = O(L)) reflecting some aspects of the (currently unknown) underlying gauge symmetry of the hyperbolic models, in the same way that the Virasoro algebra (2.9) is a gauge-fixed remnant of the worldsheet diffeomorphism symmetry of the underlying (Nambu-Goto-type) string action. Given the trial expression (2.13) one can wonder what algebra these expressions satisfy, i.e., whether there is a generalization of the Virasoro algebra (2.9) associated with our construction. While a conclusive answer to this question would require a knowledge of the E 10 algebra which is presently not available, we can at least formulate the following expectation. Under the Poisson (or Dirac) bracket

10 There was a further one-sidedness in [4] related to the fact that we were working in a truncated coset whence only a Borel subalgebra of the hyperbolic algebra played a role. This effect is an artefact of the truncation and irrelevant to the present construction.


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the grading of the algebra implies that the simplest type of commutation relation one might have is of the form

Lα , Lβ = Jα+β−γ Lγ . (2.16) γ

As we shall discuss in the next section below, relations of the type (2.16) do hold if we consider only the (truncated, low-level) constraints of [4]. However, the vast generalization of the definition of the constraints introduced in the present paper makes the validity of a result of the type (2.16) highly non-trivial and dependent upon delicate structures that we do not currently understand in detail. Indeed, there are two non-trivial assertions contained in the expected result (2.16). The first one is that the trilinear11 expression in current components on the r.h.s. organizes itself into products between constraints and certain current components, much in the same way as for the affine Virasoro algebra (cf. (2.9) where the r.h.s. is a product of a constraint L nδ by a (conserved) algebra generator c). The second claim relates to the roots γ contributing on the r.h.s. and the question whether these only cover constraints that had been defined previously. Both points are important for ascertaining the closure of the constraint algebra. The fact that only strongly conserved coefficients appear in the algebra of constraints is important for the discussion of open algebras, as mentioned in the Introduction. We note one point concerning (2.16) in comparison to the affine Virasoro algebra (2.9). There it was important that an additive structure existed on the set of all roots for which generators L mδ were defined. Here, we expect that this additive structure will be replaced by a certain convexity-related structure of the cone C, akin to the structure of integrable highestweight representations [25]. Though we do not yet fully comprehend this structure, we shall see below that our proposed ‘fleshing out’ of the skeleton ensures (when starting from a past-light-cone-only skeleton) the convex structure of a solid cone, i.e., all α’s generated by our construction lie on or inside the light-cone. 3. Universality and Relation to Supergravity In this section we specialize to the case of E 10 whose Dynkin diagram is given in Fig. 1. The relation to supergravity will help to make the construction of the preceding section more concrete. An important role will be seen to be played by the relation between D = 11 supergravity (or type IIA in D = 10), and type IIB in D = 10. 3.1. Consistency with supergravity constraints: D = 11. The Sugawara constraints (2.13) can be interpreted as constraints to be imposed on geodesics on the infinitedimensional coset space E 10 /K (E 10 ) as follows [4]. The global E 10 symmetry gives rise to conserved Noether charges J ∈ Lie(E 10 ) that can be expanded in the orthonormal (s) basis {Tα | α ∈ hyp , s = 1, . . . , mult α} as J =

mult α

Jα(s) Tα(s) .

(3.1)

α∈hyp s=1 11 The hyperbolic Lie algebra structure {J, J } = J guarantees that the commutator of two J -bilinear constraints L is only trilinear in the J ’s.

766


Fig. 1. Dynkin diagram of E 10 with numbering of nodes

The pairing between charges and generators is as in [4]: J = ··· + +

(0) 1 (−1)m 1 m 2 m 3 Fm 1 m 2 m 3 + J m n K n m J 3!

1 (1) J m1m2m3 E m1m2m3 + · · · , 3!

(3.2)

where we have for definiteness chosen the gl(10) level decomposition of E 10 that is reviewed in Appendix A.1. An important point to note here is that tensor generators and coefficients transform contragrediently. For instance, for the Chevalley-Serre generators this translates into the following identification: Tα1 = K 1 2 [∼ e1 ] ,

(3.3)

Jα1 = J 2 1 [∼ f 1 = −ω(e1 )] ,

and so on, where ω is the Chevalley involution on E 10 . With this identification of algebra generators and current components we can work either in the universal enveloping algebra, generated by the Tα , or in the Poisson algebra, generated by the current components Jα . Namely, when considered as elements of a Poisson algebra on phase space, the (s) components Jα close into the same hyperbolic algebra under Poisson commutation, as follows directly from the Hamiltonian formulation of the coset space dynamics. That is, we have the canonical brackets (s )(s ) α +α (s ) Jα(s11 ) , Jα(s22 ) = f α11 α22 (s112 )2 Jα112 (3.4) +α2 , identical (including the sign) to the commutation relations of the hyperbolic algebra (2.11). The classically conserved charges of the E 10 /K (E 10 ) model are commuting functions on phase space in terms of which we write the classical constraints as Lα =

β∈hyp s,s

(s) Ms,s (α, β)Jα−β Jβ(s ) ,

(3.5)

without specifying the summation over the ‘internal’ degrees of freedom at this point (that is, the matrix Ms,s (α, β)). The Hamiltonian (scalar) constraint entering the coset model of [3] can be represented as the special member of the hierarchy of constraints (3.5) corresponding to α = 0, L0 ≡ H =

β mult β≥0 s=1

(s) (s)

J−β Jβ .

(3.6)


767 (s)

In this way one confirms that all Noether charges Jα are indeed classically conserved because they Poisson commute with H:

H, Jα(s) = 0.

(3.7)

This is a direct consequence of the fact that H is just the quadratic Casimir operator for the hyperbolic algebra (see Chap. 2 of [25] for a proof and the explicit computation). We note also that for the Hamiltonian constraint (3.6) the issues of contracting generators from root spaces of different dimensions are absent since the root spaces of α and −α always have the same dimension. Since all components of J are conserved, any expression of the type (3.5) is strictly conserved for any geodesic. We can therefore consistently constrain the geodesic motion on the coset space by demanding that the initial conditions satisfy Lα = 0. In [4] we have shown (with the same truncation of higher order spatial gradients as in [3]) that the canonical constraints of D = 11 supergravity can be successively rewritten in two different (but related) forms. Our analysis used an A9 = sl(10) level decomposition of the E 10 algebra, corresponding to the removal of node 10 in Fig. 1. The results of this level decomposition of [3,29] are reproduced in Appendix A.1. The explicit computation involved the determination of various numerical coefficients in the E 10 expressions that were originally fixed by requiring weak conservation of the constraint surface under the coset model equations of motion. Comparison with the canonical D = 11 supergravity constraints and use of the dictionary then showed precise agreement of these numerical coefficients, thus extending the correspondence between the E 10 /K (E 10 ) coset model and the (truncated) D = 11 supergravity equations of motion to the full canonical formulation. In Sect. 3.1.2, we shall show that, remarkably, these specific numerical coefficients found for the supergravity constraints in [4] coincide with our proposed sum over canonically normalized current components (3.5) when both β and α − β are real and for unit coefficients Ms,s (α − β, β). In addition to this unearthing of a hidden simplicity in the definition of the constraints, another advantage of writing the constraints in the form (3.5) is that this will allow us to evaluate them also for other level decompositions, and in this way to verify agreement with the canonical constraints of massive IIA and IIB supergravity as well. The agreement between the dynamical (evolution) equations of these theories with the coset model equations in appropriate truncations had already been established in [11,13,14]. Moreover, the form (3.5) is directly amenable to an affine reduction, and brings out more clearly the analogy with the affine Sugawara construction. 3.1.1. On the roots associated to the supergravity constraints. Let us first turn to the detailed consideration of the set of roots, including their multiplicities, that are associated to supergravity constraints. In the case of D = 11 supergravity, these constraints are, respectively, the diffeomorphism and Gauss constraints, and the Bianchi identities for the 4-form field strength and the Riemann tensor.12 The analysis of [4] was based on a gl(10) level decomposition truncated at level = 3, such that, when expressed in terms of the conserved E 10 Noether current in this decomposition, the constraints take 12 In a more conventional canonical analysis, one would not interpret the Bianchi identities as proper constraints, as they are not directly associated to gauge transformations, unlike the diffeomorphism and Gauss constraints. In the present setting, however, they would correspond to generators of gauge transformations on the dual fields, i.e., on the 7-form field and the ‘dual graviton’.

768


the form (−3)

L

(−4)

L

n 1 ...n 9

m 1 ...m 10 ||n 1 n 2

(−1)

(−2)

(−3)

(0)

= 28 J [n 1 n 2 n 3 J n 4 ...n 9 ] + 3 J p|[n 1 ...n 8 J n 9 ] p , (3.8a) (−2) (−2) (−3) (−1) 21 3 = J n 1 [m 1 ...m 5 J m 6 ...m 10 ]n 2 + J n 2 |[m 1 ...m 8 J m 9 m 10 ]n 1 10 2 −(n 1 ↔ n 2 ), (3.8b)

for the diffeomorphism and Gauss constraints and (−5)

L

(−6)

L

m 1 ...m 10 ||n 1 ...n 5

m 1 ...m 10 ||n 0 |n 1 ...n 7

(−2) (−3) m 1 m 2 [n 1 ...n 4 n 5 ]|m 3 ...m 10

=3 J

J

(−3) (−3) n 0 |m 1 ...m 8 m 9 |m 10 n 1 ...n 7

=9 J

J

,

(3.8c)

,

(3.8d)

for the Bianchi identities. Here, we have changed the normalization of the charge J (−3) compared to [4,12] so that all highest weight states are uniformly normalized to unity (the usefulness of this re-definition was already pointed out in footnote 19 of [4]). Explicitly, the normalizations of the E 10 generators, in their A9 decomposition are (0)

(0)

(−1) (1) a1 a2 a3 | J b1 b2 b3

J a b | J c d = δda δbc − δba δdc , J

(−2) (2) a1 ...a6 | J b1 ...b6

J

= 3! δba11ba22ba33 ,

...a6 = 6! δba11...b . 6

(3.9)

By contrast, for the mixed symmetry field on level || = 3 we shall take here a normalization that differs from the one given in Eq. (2.30) of [12] by a factor 1/9, viz.

(−3) (3) 8 · 8! a0 a1 ...a8 a0 a1 ...a7 a8 J a0 |a1 ...a8 | J b0 |b1 ...b8 = δb0 δb1 ...b8 − δ[b . (3.10) δ 1 b2 ...b8 ]b0 9 This normalization is chosen so that operators associated to real roots (two indices identical) have unit norm, like the highest weight (−3) (3) 10|3 4 5 6 7 8 9 10 | J 10|3 4 5 6 7 8 9 10

J

= 1,

(3.11)

whereas for operators associated to null roots (all indices different) (−3) (3) 2|3 4 5 6 7 8 9 10 | J 2|3 4 5 6 7 8 9 10

J

=

8 . 9

(3.12)

In addition to these normalizations, we have used in (3.8) the same implicit antisymmetrization conventions as in [4]. For instance, the expression in (3.8c), corresponding to a Bianchi constraint on the four-form field strength, is understood to be antisymmetrized (with weight one) over m 1 . . . m 10 ; furthermore the last relation (3.8d) is to be projected onto a (7, 1) hook for the indices n 1 . . . n 7 and n 0 . We note that for the constraints listed in (3.8) there are no ordering ambiguities in a possible transition to operator expressions in a quantum theory, except for L(−6) in (3.8d), since all commutator terms vanish by Jacobi or Serre relations; for instance (−1) (−2) (−3) (3.13) J [m 1 m 2 m 3 , J m 4 ...m 9 ] ∝ J [m 1 |m 2 ...m 9 ] = 0. Let us now exhibit the roots underlying the diffeomorphism constraint (3.8a). For this, we first consider its highest component, corresponding to the indices 2 3 4 5 6 7 8 9 10.


769

To identify the root α to which it belongs we must find the eigenvalues under the ten Cartan generators of E 10 . (Indeed, the ‘covariant’ components, αi ≡ α(h i ) of a root precisely encode the eigenvalues in [h i , eα ] = α(h i )eα .) Since we are working with the current components J we display the Cartan elements in this description. In the gl(10) basis the Cartan elements are h i = J i i − J i+1 i+1 (i = 1, . . . , 9),

2

1 1 h 10 = − J 1 + · · · + J 77 + J 8 8 + J 9 9 + J 10 10 . 3 3

(3.14)

Alternatively, one can do the calculation with Lie algebra elements, using the more familiar expressions of the Cartan generators h i in terms of Lie algebra generators recalled in Appendix A.1. (In that case, one notes that the constraint L(−3) 2 3 4 5 6 7 8 9 10 is associated with the contragredient Lie-algebra basis element F2 3 4 5 6 7 8 9 10 .) An easy calculation shows that the only non-zero eigenvalue corresponds to h 1 (first node in Fig. 1), and is equal to +1. Hence, the list of ‘covariant’ components αi ≡ α(h i ), also known as ‘Dynkin labels’, is [+1, 0, 0, 0, 0, 0, 0, 0, 0, 0]. This is equivalent to saying that the root associated to the highest component of the diffeomorphism constraint is equal to the fundamental weight 1 associated to the simple root α1 .13 To explicitly write the root α = 1 associated to the highest diffeomorphism constraint in terms of the simple roots, we must convert its Dynkin labels to root labels, i.e., pass from covariant indices to contravariant ones by using the inverse of the Cartan matrix Ai j = h i |h j . This leads to the corresponding root α = −(α2 + 2α3 + 3α4 + 4α5 + 5α6 + 6α7 + 4α8 + 2α9 + 3α10 ) ≡ −δ, where the (positive) root δ denotes the primitive null root of E 9 ⊂ E 10 . In particular, this shows that the root α = 1 associated to the highest component of the diffeomorphism constraint is a negative null root.14 We can therefore write for this particular component (−3)

L

2 3 4 5 6 7 8 9 10

≡ Lα

with α = 1 = −δ ≡ −δ (3) .

(3.15)

Let us now consider the roots associated to the other components of the diffeomorphism constraint (3.8a). They are obtained by the action of the permutation group S10 on the indices. Since the permutation group is the Weyl group of sl(10), we conclude that all components of the diffeomorphism constraint are associated with (negative) null roots, forming a single orbit of the Weyl group W (sl(10)). These null roots can be obtained by acting with the corresponding Weyl transformation on δ, such that w (Lα ) = Lw(α) ,

(3.16)

where w on the left-hand side acts on the indices of the constraint L by permuting them. 13 The fundamental weights are defined as dual to the simple roots α w.r.t. the Cartan inner product: i j i |α j = +δi j . The fact that 1 , and the integrable highest-weight representation L(1 ) built from it, is related to the tower of constraints was already discussed at some length in [4]. This relation does not mean, however, that L(−3) 2 3 4 5 6 7 8 9 10 is a highest weight vector for the action of all the E 10 generators. Actually, as was already shown in [4], and will be further discussed in Sect. 3.3, it fails to be one. 14 We note that the association of the ‘null’ ( or ‘cusp’) fundamental weight to the diffeomorphism 1 constraint is valid not only for maximal supergravity and E 10 , but also for other (super)gravity theories. For instance, for pure gravity in any spatial dimension d the basic (diffeomorphism) constraint is always associated to roots of the form −μa , where μa (with a = 1, . . . , d) denotes the null roots that are contained within the G L(d) multiplet of the ‘gravity root’. The notation μa = −β a + c β c is the notation used in [30]. Note that the null root −μ1 is indeed the fundamental weight associated with the ‘hyperbolic’ node of AE d (as explicitly dispayed in Eq. (3.14) of [31]).

770


Let us now proceed to considering the roots associated to the higher-level (or rather ‘lower-level’, as the levels are negative) constraints. To find the roots for the level = −4 and = −5 constraints in (3.8b) and (3.8c), we consider their highest weight components. These are L(−4) 1 2 3 4 5 6 7 8 9 10||9 10 and L(−5) 1 2 3 4 5 6 7 8 9 10||6 7 8 9 10 , respectively. A straightforward calculation gives the eigenvalues [0, 0, 0, 0, 0, 0, 0, 1, 0, −1] and [0, 0, 0, 0, 1, 0, 0, 0, 0, −1], respectively. The corresponding roots are again found to be null and negative. In view of the fact, recalled in (2.15), that all null roots are Weyl images of the basic one-dimensional string of affine null roots n δ, we can look for the specific affine root n δ from which they descend. We find that it is −δ, i.e., n = −1. In other words, in addition to being null, the roots associated to the level = −4 and = −5 constraints can be obtained from the ‘basic’ = −3 ‘diffeomorphism-constraint’ root α = 1 = −δ ≡ −δ (3) by applying some E 10 Weyl reflection: wα (β) = β − (α · β)α (here simplified by taking into account the fact that α · α = 2 for the roots of a simply laced algebra). More explicitly, we have: δ (4) = wθ (δ (3) ), θ := α1 + α2 + α3 + α4 + α5 + α6 + α7 + α10

(3.17)

δ (5) = wθ (δ (4) ), θ := α6 + 2α7 + 2α8 + α9 + α10 ,

(3.18)

and

where we have given the explicit Weyl reflections in W(E 10 ) that move between the different levels. Note that θ is the highest root of the embedded A8 algebra associated with the IIB theory, and θ is the highest weight of an embedded D5 algebra. Finally, similarly to the case of the roots associated to L(−3) , the fact that the Young tableaux describing the G L(10) index structure of L(−4) and L(−5) are totally antisymmetric guarantees that all the roots associated to the other components of these constraints are obtained from the basic ones (3.17) and (3.18) by G L(10) permutations, i.e., by further Weyl reflections. In particular, all of them are null. So far all the roots associated to the first three levels of constraints have been found to be light-like (and negative). The constraint L(−6) differs from the lower level ones in that it is the first in the hierarchy of constraints to involve a non-trivial Young tableau. As a consequence, we are going to see that it contains a mixture of null (α 2 = 0) and time-like (α 2 = −2) roots. More precisely, the highest weight component L(−6) 1 2 3 4 5 6 7 8 9 10||10|4 5 6 7 8 9 10 is easily checked to be associated to a null root, which can be obtained from −δ (5) by the following Weyl transformation: δ (6) = wθ

(δ (5) ), θ

= α4 + 2α5 + 2α6 + 2α7 + α8 + α10 .

(3.19)

Here, θ

is the highest root of an embedded D6 algebra. Covariantizing this component under the action of the sl(10) = A9 subalgebra gives a representation of (7, 1) hook type which is not a pure antisymmetric tensor unlike the constraints on levels −3, −4 and −5. From the point of view of the permutation group S10 = W(sl(10)) this means that there are two separate orbits under W(sl(10)). The ‘outer’ orbit consists of permutations of the lowest weight indices and corresponds to null roots of E 10 . The inner orbit corresponds to imaginary E 10 roots with α 2 = −2. In terms of the supergravity constraint (3.8d) these two orbits correspond to cases when there are two identical indices on the (7, 1) hook part or when they are all different, respectively. The ‘skeleton’ of null roots −δ (3) , −δ (4) , −δ (5) , . . . , together with their multiples (discussed below) and their time-like descendants, is sketched in Fig. 2.


771

Fig. 2. Sketch of the set C of roots (and notably its ‘skeleton’ of null roots on the past light-cone) labelling the extended set of constraints constructed in this paper

Let us finally note that all null roots α appearing in these constraints appear with multiplicity one, although the same roots, considered as E 10 roots have the non-trivial root multiplicity eight. That the null roots appear with multiplicity one in the Sugawara construction should be so by consistency with the affine case. By contrast, the purely imaginary roots belonging to the inner orbit of L(−6) have multiplicity seven as constraints compared to multiplicity 44 as roots of E 10 . 3.1.2. Supergravity constraints and canonical normalization. So far we have analyzed the roots α labelling the l.h.s. of our basic Sugawara-like expression (2.13). Next we analyze the roots β1 , β2 contributing to the right hand side of (2.13). Our principal aim here will be to see what are the values of the numerical coefficients Ms1 ,s2 (β1 , β2 ) that enter the Sugawara-like sum. We start here from the explicit G L(10)-decomposed form (3.8). To this aim let us consider the components of the currents J on the r.h.s. where the indices are distributed in a specific way. For example, we can pick out two representative terms where only operators for real roots appear and obtain (−3)

L

2 3 4 5 6 7 8 9 10

3! · 6! (−1)2 3 4 (−2)5 6 7 8 9 10 8! (−3)2|2 3 4 5 6 7 8 9 (0)10 28 · +3· J J J J 2 9! 9! 1 (−1)2 3 4 (−2)5 6 7 8 9 10 (−3)2|2 3 4 5 6 7 8 9 (0)10 (3.20) = + J J J J 2 . 3

Hence, we find the remarkable fact that the combinatorial factors appearing in (3.8a) are precisely such as to imply, in the root basis, a relative normalization equal to unity. As the overall prefactor 1/3 (as well as the corresponding 1/60 in the formulas below) is merely chosen to agree with the normalisations in [4], it might eventually be traded for a more convenient one. Thus, all terms in the bracket belong to real roots and are canonically normalized, justifying in retrospect the relative factor in (3.8a) by (3.5).

772


Fig. 3. Sketch of one of the basic elements of the infinite ‘scaffold’ of special Sugawara configurations α = β1 + β2 with α null and β1 , β2 real. The real roots β1 , β2 lie within the hyperplane tangent to the light-cone along the null root (here chosen to be α = −δ). One must imagine completing the structure shown here by all its Weyl images

For the Gauss constraint (3.8b) one similarly finds (−4)

L

1 2 3 4 5 6 7 8 9 10||9 10

21 2 · 5! · 5! (−2)9 1 2 3 10 4 (−2)5 6 7 8 9 10 · J J 10 10! 3 1 2 · 8! (−3)9|9 10 1 2 3 4 5 6 (−1)7 8 10 + · · J J 2 2 10! 1 (−2)9 1 2 3 10 4 (−2)5 6 7 8 9 10 (−3)9|9 10 1 2 3 4 5 6 (−1)7 8 10 . = + J J J J 60 (3.21)

Again, the terms appear with the same relative coefficient and confirm the expression (3.5) for real roots. For the constraints (3.8c) and (3.8d) on levels = −5 and = −6 there is nothing to check since there is only one type of term. The basic ‘scaffold’ of Sugawara constraints exhibiting a decomposition α = β1 + β2 with α null and β1 , β2 real is illustrated in Fig. 3. Note that the relations α 2 = 0 and β12 = β22 = 2 imply that α · β1 = 0 = α · β2 , i.e., that β1 and β2 are orthogonal to α, so that they belong to the hyperplane tangent to the light-cone along the considered null root (see Fig. 3, where one has chosen α = −δ). One has to imagine the infinite ‘scaffold’ made by the tangent hyperplanes associated to the infinite skeleton of Weyl images of −δ. 3.1.3. General structure of constraints. We note that there are also terms contributing to (3.5) where not both Jα−β and Jβ are real. For example, (3.8a) contains a term (−3)

L

2 3 4 5 6 7 8 9 10

1 (−3)1|2 3 4 5 6 7 8 9 (0)10 J J 1, 3

(3.22)


773

where an imaginary level three root is contracted with a real level zero root (albeit positive). Similar contractions appear also for the other constraints. Note that, though, after removing the same prefactor (1/3) as above, we have again a simple coefficient unity, the time-like-root generator associated to the level −3 root is such that its normalization involves the fraction 8/9, see (3.12). At this stage, we start seeing several patterns appearing within the structure of the constraints, and notably in the set C labelling the roots (together with their multiplicity) associated to the constraints. A first pattern is that, so far, all the constraints can be labelled by the members of the integrable highest-weight representation descending from the fundamental weight 1 , which is dual to the first (‘hyperbolic’) node of the Dynkin diagram, Fig. 1. A second, closely related, pattern is that the pattern of roots comprise many null roots, and that the null constraint-roots studied so far all belong to the Weyl orbit of 1 = −δ. A third pattern is the simple (unit) relative normalization of the contributions (α, β1 , β2 ) to the Sugawara expression (2.13) involving the decomposition of a null root α into two real roots (β1 , β2 ). A fourth pattern is that the null roots associated to non purely antisymmetric Young tableaux give rise, upon covariantization under G L(10), to a set of roots which ‘penetrate’ within the past light-cone, i.e., which are time-like (and past-directed) rather than light-like. It is tantalizing to generalize these patterns to the infinite tower of coset constraints that we are trying to construct. We can first assume that the set C of ‘constraint roots’ contains the full weight diagram, say P(1 ), of the fundamental representation L(1 ) based on 1 = −δ. By Proposition 10.1 of [25] we know that P(1 ) (including its multiplicities) is invariant under the full E 10 Weyl group, W(E 10 ). In particular, this would imply, in view of (2.15), that there is an infinite sequence of null constraints related to the orbit of minus the primitive null element −δ. Upon covariantization of the resulting highest weight vectors under sl(10) we obtain a series of constraints related to δ as indicated in the first row of the following table: −δ −2δ . . .

= −3 (−3) L(−δ)

= −4 (−4) L(−δ)

= −5 (−5) L(−δ)

= −6 (−6) L(−δ) (−6)

L(−2δ)

= −7 (−7) L(−δ)

= −8 (−8) L(−δ)

… …

L(−2δ)

…

(−8)

Here, we added a subscript (−δ) to all the constraints in the W(E 10 ) orbit of −δ and suppressed the labels for the W(sl(10)) suborbits in the columns. Let us also recall the existence of the Hamiltonian constraint, L0 , which could be thought of as being associated to the 0th multiple of δ. In addition to the ‘skeleton’ of null roots constituting the Weyl orbit of 1 = −δ, the weight diagram P(1 ) of L(1 ) contains all (past-directed) time-like roots. This follows from Proposition 11.2a of [25]. To apply this proposition, we need, for each putative 10 pi i , with pi ≥ 0), to conweight μ within the Weyl chamber (μ ∈ P+ ; i.e. μ = i=1 trol the ‘support’ of the root 1 − μ, i.e. the non-zero coefficients m j in its simple-root decomposition: 1 − μ = 10 j=1 m j α j . Using 1 |α j = δ1 j , and αi |α j = Ai j , the root-basis integers m j are easily seen to be related to the weight-basis integers pi via the knowledge of the inverse of the E 10 Cartan matrix Ai j . Now, by explicit inspection of this inverse Cartan matrix (see, e.g, [32]), one finds that the only place in it where there is a zero in the first column is in the first row. This shows that any element of the Weyl 10 chamber μ = i=1 pi i such that pi = 0 for at least one i among 2, . . . , 10, the vector 1 − μ has non-vanishing ‘support’ m 1 on the first node and hence is ‘non-degenerate

774


w.r.t 1 ’ (in the sense defined in Sect. 11.2 of [25]). Hence, by Kac’s Proposition 11.2a such μ’s are indeed weights (together with their Weyl images). The only exceptional case is when p j = 0 for j = 2, . . . , 10, which corresponds to μ = p1 1 . In other words, we have found that all the negative time-like weights belong to P(1 ), but that the multiples of 1 = −δ are not part of the weight diagram P(1 ).15 Though the set P(1 ) is already quite large, it only corresponds to the G L(10) covariantization of the first row in the table above. In view of the structure of the usual affine Virasoro-Sugawara constraints L n δ recalled above, together with the known structure of E 10 null roots (2.15), it is now quite natural to conjecture that the ‘null skeleton’ of C contains, in addition to the orbit of −δ (first row in the table) the Weyl orbits of (negative) multiples of δ: −nδ. This amounts to conjecturing that, besides the weight diagram P(1 ) of the fundamental representation L(1 ), we must add the weight diagrams P(n1 ) (with n = 2, 3, . . .) corresponding to the multiple tensor product of L(1 ) with itself: L(1 ) ⊗ L(1 ), L(1 ) ⊗ L(1 ) ⊗ L(1 ), etc. Besides this mathematical argument for conjecturing an extension of the set of constraints beyond the ones related to the Weyl orbit of −δ (and its covariantization), there is a physical argument suggesting the necessity of this extension. Indeed, all the constraints discussed so far correspond, in view of the ‘dictionary’ of [3], to the values at one spatial point, of some space-dependent supergravity constraints. For instance, L(−3) n 1 ...n 9 is the spatial n 1 ...n 10 dual of the diffeomorphism constraint Hm (x0 ), taken at the specific spatial point x0 around which one analyzes the asymptotic behaviour of the supergravity fields as t → 0. However, the full supergravity diffeomorphism constraint consists of imposing the vanishing of Hm (x) at all spatial points. When expanding the diffeomorphism constraint Hm (x) in a (ten-dimensional) spatial Taylor expansion around the base point x0 , we see that we should replace the unique constraint Hm (x0 ) ∼ L(−3) n 1 ...n 9 by an infinite gradient tower of spatial derivatives of the form ∂m 1 ...m k Hm (x0 ). For instance, at the first spatial-gradient level m = 1, we should be considering the two irreducible G L(10) tensors contained in ∂m Hn (x0 ), i.e., its symmetric and antisymmetric parts. Dualizing back these first-gradient constraints by means of n 1 ...n 10 , we are led to expecting that the ‘first-gradient descendants’ of L(−3) n 1 ...n 9 will comprise two G L(10) tensors bearing 18 contravariant indices, and belonging to two different Young tableaux: one with [9,9] boxes (corresponding to the symmetric combination) and one with [10,8] boxes (corresponding to the antisymmetric combination). The former corresponds to the null root −2δ = 21 , whereas the latter corresponds to the imaginary 2 and so lies inside the past light-cone. The extension of this gradient construction to the other supergravity constraints (Gauss, etc.) then naturally leads us to conjecture the existence of the second row of the table. Then, when considering higher spatial gradients we are led to conjecturing the existence of further rows ‘stemming’ from −3 δ, −4 δ, etc. One finds that the putative constraints associated with the Weyl orbit of −nδ start on level = −3n and are spaced by n. Finally, it seems that the full table is describing all possible weights on or inside the (past) light-cone. The notation in the table is condensed and does not display the sl(10) representation structure of the various constraints. For example, the set of constraints labelled by 15 Another way of seeing this is by using Proposition 11.3 of [25] where P( ) is described as the convex 1 hull of the Weyl orbit of 1 . The infinitely many Weyl images of 1 all lie on the light-cone (and densely approximate any null direction) and one might think that the convex hull covers all points on the light-cone. This is not true since one is constructing the convex hull as an infinite union of closed sets but this is not necessarily closed. In the present case it is open and misses exactly the multiples of 1 and their Weyl images but the convex hull covers all points inside the light-cone.

Sugawara-Type Constraints in Hyperbolic Coset Models (−6)

775

(−6)

L(−δ) and L(−2δ) transform in different sl(10) representations. The former one is in the hook representation of (3.8d), whereas the latter has two sets of antisymmetric 9-tuples. Explicitly, one has the following two index structures: (−6) m ...m ||n |n ...n 1 10 0 1 7 L (−δ)

and

(−6) m ...m |n ...n 1 9 1 9 . L (−2δ)

(3.23)

In the affine truncation to E 9 only one member in each infinite sequence (row) for a given −nδ is non-trivial because of the presence of 10-tuples of antisymmetrized indices in the higher components. In the example (3.23) above, the first tensor vanishes in the affine truncation, whereas the second one is non-zero. In addition, all the surviving constraints (−3n) from the beginning of each sequence reduce to singlets under sl(9). These are the L(−nδ) . This (one-sided) sequence of constraints naturally correspond to the generators L −nδ (for n > 0) of the affine Sugawara construction that we had introduced in (2.7). (We will return below to specific issues concerning the contractions of the null roots and Cartan subalgebra generators). We do not present an explicit expression for the second rung of constraints, like the second term in (3.23), but note that on the contractions of real root spaces it is given by the same general formula (3.5) as the other constraints we have considered so far. Among (−n) the other constraints in L(−nδ) , some have an index structure similar to the elementary (−)

(−8)

L(−δ) , but with all tuples replicated n times. For example, the index structure of L(−2δ) contains a tensor with two 10-tuples and two 2-tuples (−8) m ...m || p ... p ||n n |q q 1 10 1 10 1 2 1 2 . L (−2δ)

(3.24)

To complete this discussion, let us point out the following ‘experimental’ relation between the constraints and the level decomposition of the adjoint of E 10 under A9 [29]. ‘Admissible’ A9 representations in the level decomposition rarely appear with outer multiplicity zero. Here, ‘admissible’ refers to solving necessary diophantine conditions on the lowest weight vectors of a possible A9 representation occurring in the adjoint representation of E 10 , see Eqs. (6) and (7) in [3]. The only cases up to ≤ 28 for which the outer multiplicity of an admissible representation is zero are those when the associated lowest root in the representation is null.16 More precisely, the only entries with vanishing outer multiplicities in the tables of [29] occur at17 Level = 3n = 4n = 5n

E 10 root n(0, 1, 2, 3, 4, 5, 6, 4, 2, 3) n(1, 2, 3, 4, 5, 6, 7, 4, 2, 4) n(1, 2, 3, 4, 5, 7, 9, 6, 3, 5)

A9 weight [n, 0, 0, 0, 0, 0, 0, 0, 0] [0, 0, 0, 0, 0, 0, 0, n, 0] [0, 0, 0, 0, n, 0, 0, 0, 0]

The first line corresponds to the root nδ and both the second and the third line can be obtained from the first line by the Weyl transformations given explicitly in Eqs. (3.17) and (3.18). These entries have vanishing outer multiplicities since the corresponding E 10 generators are already contained in the gradient representations on the relevant 16 This is no longer necessarily true when considering Kac–Moody algebras different from E 10 [21] or decompositions other than that under A9 . 17 We use two different notations for describing elements α of the (self-dual) E root lattice, namely in terms 10 of either the basis of simple roots αi or of the basis of fundamental weights i : α = i m i αi = i pi i . In the former we write the ten-tuple of coefficients with round parentheses (m 1 , . . . , m 10 ) and in the latter with square brackets [ p1 , . . . , p10 ]. The pi are often referred to as Dynkin labels. The A9 weight is obtained from [ p1 , . . . , p10 ] by dropping the last entry p10 since this corresponds to the node that is deleted in the A9 level decomposition.

776


level. One potentially important implication of the vanishing outer multiplicities is that there are no ordering ambiguities because the relevant commutators always vanish, as in (3.13), whereas ordering ambiguities will occur in general for higher level constraints (−6) like L(−δ) . 3.1.4. Algebra of constraints. Let us now return to the question of the constraint algebra (2.16) raised at the end of Sect. 2. We discuss this issue by using the explicit expressions for the constraints (3.8). As discussed above, one would like the constraint algebra to close with structure constants given by current components. From the results of [4] it follows that one can generate higher level constraints from lower level constraints by the action of the negative level current operator, J (−1) , i.e., that schematically (−(+1)) (−1) (−) (3.25) L = J , L , is valid for = −3, −4, −5. This property is equivalent to the result of [4] that the + (i.e., that they form a repconstraints are ‘covariant’ under the upper Borel group E 10 + resentation of E 10 ; even if they do not form a representation of the full group E 10 ). In addition, the level-three truncated constraints (3.8) have the property that their Sugawara expression contains only negative level currents, i.e., schematically (−)

L =

(− p) (−q)

J · J .

(3.26)

p+q=

It is now easy to see that the two properties (3.25) and (3.26) imply that the Poisson bracket of two constraints closes in the desired manner of (2.16). This is certainly an encouraging result, which suggests that the structure of the constraints incorporates special features allowing for the existence of a closed algebra of the type of a generalized Virasoro algebra (2.16). However, it is not clear whether the two special properties (3.25) and (3.26) continue to hold for the generalized infinite tower of E 10 constraints whose construction was sketched above. We shall see that the property (3.26) is likely to be violated when implementing a certain ‘see-saw’ construction defined below. As for the property (3.25) + ), one reason for believing (which says that the constraints form a representation of E 10 that it might not be universally valid comes from the example of the affine Sugawara construction. There the constraints do not transform in a representation of the affine algebra: L(−−n) = J (−n) , L(−) . Rather one finds that it is the algebra which trans

forms under the constraints, i.e., J (−−n) = J (−n) , L(−) . We leave to future work further discussion of this important issue. 3.2. Universality: D = 11, IIB and massive IIA. The full E 10 Lie algebra can be obtained from the closure (via commutators) of two of its finite-dimensional sub-algebras: (i) its A9 subalgebra (relevant for D = 11 supergravity), and (ii) its A8 ⊕ A1 subalgebra (relevant for type IIB supergravity). The A9 subalgebra corresponds to nodes 1, 2, 3, 4, 5, 6, 7, 8, 9 of the Dynkin diagram in Fig. 1; the A8 ⊕ A1 algebra corresponds to the nodes 1, 2, 3, 4, 5, 6, 7, 10 and 9 in Fig. 1. The two subalgebras A9 and A8 ⊕ A1 together cover all ten nodes of the E 10 diagram, and therefore their closure is all of E 10 . For the A8 ⊕ A1 decomposition, the term ‘level’ refers to node 8. For low levels,


777

the decomposition under this A8 ⊕ A1 subalgebra, originally performed in [13,21], is reproduced in Appendix A.2. The two decompositions under A9 and A8 ⊕ A1 provide two different bases for the same Lie algebra E 10 . In order to distinguish them we use the letter J for the current components in the A9 decomposition, as already done for example in (3.8), and the letter I for current components in the A8 ⊕ A1 decomposition. Since the real root spaces are one dimensional it is usually straightforward to explicitly work out the ‘change of basis’ between the current components expressed in the J basis or the I basis. For example, the root space of the real root α = −α10 contains the current component J 8 9 10 in the A9 decomposition. In the A8 ⊕ A1 decomposition this root space is part of the A8 ‘gravity line’ and therefore one obtains the following relation between the vectors of the two bases in the α = −α10 root space (−1) 8 9 10

J

(0) 8

=I

9

corresponding to E −α10 ≡ f 10 .

(3.27)

That the two generators are not on the same level with regard to the two decompositions of the E 10 algebra will be of crucial importance for the construction we shall discuss next. In Appendix A, we also recall the association of the level decompositions with low-lying generators in an explicit tensor basis for the two decompositions. The fact that A9 and A8 ⊕ A1 together generate the whole E 10 algebra allows in principle to extend the lowest level supergravity constraints to arbitrarily high levels by the following mechanism (which for obvious reasons we will refer to as a ‘see-saw mechanism’). Among the root components contributing to a given known constraint in one level decomposition, there are some that correspond to ‘unknown’ levels in a different decomposition. Covariantizing the resulting expression with regard to the gl(n, R) subalgebra relevant for that new decomposition we generate new components, which in turn can be analyzed in terms of the first decomposition. Covariantizing again, but now with respect to the first decomposition, we again generate new components. It is easy to see that this procedure never stops, and so continues ad infinitum. To see how this construction works in a concrete example consider the following terms in the D = 11 diffeomorphism constraint (3.8a), see also (3.20), (−3) 1 (−1)2 3 4 (−2)5 6 7 8 9 10 (−3)9|2 3 4 5 6 7 9 10 (0)8 2 3 4 5 6 7 8 9 10 + J J J J 9 L 3 (−3) (−3) (0) (0) (3.28) + J 9|2 3 4 5 6 7 8 9 J 10 9 − J 8|2 3 4 5 6 8 9 10 J 7 8 . All the terms in the bracket correspond to canonically normalized real root components of the current. In analogy with (3.27) one can now convert these terms into the alternative basis provided by the A8 ⊕ A1 decomposition. In this way we obtain (see Appendix A for the notation) (−1) 234

J

(−3) 9|2 3 4 5 6 7 9 10

J

(−3) 9|2 3 4 5 6 7 8 9

J

(−3) 8|2 3 4 5 6 8 9 10

J

(−2) 2349

= I

,

(−2) 5 6 7 8 9 10

(−3) 2 3 4 5 6 7,1˙

= I

J

(0) 8

,

J

(−4) 2 3 4 5 6 7 8 9,1˙ 1˙

= I

(−4) 8|2 3 4 5 6 8 9

= I

,

(−1) 8 9,2˙

= I

9

(0) 10

,

J

(0) 7

J

(−2) 5678

= I

8

9

(0) ˙ 2

=I

(0) 7

=I

8,

, 1˙ ,

, (3.29)

778


where dotted indices refer to the sl(2, R) algebra associated with node 9. Putting this back into (3.28) one can see that this is part of a G L(9, R) × S L(2, R) covariant expression of the form18 (−4)

C

n 1 ...n 8

=

35 (−2)[n 1 ...n 4 (−2)n 5 ...n 8 ] 28 (−1)[n 1 n 2 ,α (−3)n 3 ...n 8 ],β − αβ I I I I 3 3 1 (−4)n 1 ...n 8 ,αγ (0)β 8 (−4) p|[n 1 ...n 7 (0)n 8 ] − I I γ αβ − I I p + ..., 3 3

(3.30)

where for clarity of notation we use the symbol C to denote the IIB constraints. Remarkably, this expression is exactly the diffeomorphism constraint of IIB supergravity when the correspondence with E 10 of [13] is used. This is explained in more detail in Appendix B.19 Indeed, using the expressions (A.5) and (A.6) for the Cartan generators expressed in IIB variables, one finds that the component 2 3 4 5 6 7 8 9 of the IIB diffeomorphism constraint is associated with the root space of −δ, just as is the component 2 3 4 5 6 7 8 9 10 of the D = 11 diffeomorphism constraint, see Appendix B. This suggests that, possibly the two expressions agree completely. Inspecting all the different root components and E 10 generators one verifies (−3)

L

(−4)

2 3 4 5 6 7 8 9 10

= C

2 3 4 5 6 7 8 9

real roots

= real roots

1 L−δ , 3 real roots

(3.31)

i.e., the expressions agree on the bilinear expressions involving two real root generators — as was, in fact, guaranteed by our use of the Weyl group in the covariantization procedure. We find it remarkable that there is such an agreement between the constraints of two different physical theories expressed in the simple algebraic fashion (3.5). However, considering the bilinear terms contributing to the two expressions, one finds that there are terms that differ, an explicit example can be found in Appendix C. One way to interpret this difference is the following: The full set of constraints can be divided in two parts: (i) a universal part, based on the ‘skeleton’ of null roots, and comprising the ‘scaffold’ of special configurations Lα =

Jβ1 Jβ2

for α null

(3.32)

β1 +β2 =α β1 ,β2 real

and, (ii) a non-universal part (the ‘flesh’) that depends on the choice of subgroup under which one covariantizes the ‘scaffold’ part (3.32). The universal part of the construction (3.32) has the property of being preserved by the action of the discrete Weyl group W (E 10 ) and its subgroups W (A9 ) and W (A8 ⊕ A1 ). By contrast, the covariantization of the skeleton under the corresponding continuous groups G L(10, R) (for D = 11) and G L(9, R) × S L(2, R) (for type IIB) leads to different results on the additional new terms inside the light-cone that are generated by the covariantization. That different new terms are possible is due to the fact that in those terms one has to specify the coefficients Ms1 ,s2 (β1 , β2 ) for the contraction of root 18 The 8-index tensor on the l.h.s. is fully antisymmetric. We use the convention that 1˙ 2˙ = +1 = − . 1˙ 2˙ 19 We take this opportunity to point out a typo in the Einstein equation (67) in [13]: The terms involving the

(self-dual) five-form field strength should be multiplied by 1/2. This does not affect the dictionary derived in that paper.


779

spaces of different dimensions.20 These are fixed by covariance under a chosen level decomposition subgroup.21 We believe there is some evidence that hyperbolic algebras may admit a realisation akin to the realisation of affine algebras in terms of a spectral parameter22 , but our results here strongly suggest that, if there is such a realisation, it will not be unique. Thinking of Sugawara constructions as being associated with spectral parameters, this can be interpreted by saying that the E 10 algebra may not possess a single or unique set of spectral parameters. Rather, one can and has to choose a set of spectral parameters by covariantizing under a subalgebra of one’s choice. If the spectral parameters are related to space variables (as is the case for the affine algebras appearing in D = 2 supergravities), then this would be in good agreement with the anticipation that one can make space-times of different dimensions emergent from E 10 , depending on the choice of level decomposition [34]. From this point of view the hyperbolic Sugawara construction considered here is less unique than in the affine case since it depends on the choice of level decomposition. At the same time it nicely incorporates the expected possibility of having different spaces emerging from an U-duality (Weyl group) invariant scaffold. On the other hand, restricting only to real root generators, we can now use the agreement between the two expressions to construct new terms involving higher level generators, showing the full power of the approach. The crucial point is that the IIB diffeomorphism constraint (3.30) also contains other components that are not contained in the previous expressions (3.8) corresponding to the A9 level decomposition with the level truncation appropriate to D = 11 supergravity, for example the real root combination (−4)

C

23456789

1 (−4)8|2 3 4 5 6 7 8 (0)9 I I 8. 3

(3.33)

Translating again between the two different bases of E 10 using (0) 9

I

(+1)

8

= J 8 9 10 ,

(−4) 8|2 3 4 5 6 7 8

I

(−4) 8 9 10|2 3 4 5 6 7 8 9 10

= J

,

(3.34)

we infer that this is part of an extended sl(10) covariant expression, namely (−3)

L

m 1 ...m 9

→ (3.8a) +

(−4) 1 (+1) J p1 p2 p3 J p1 p2 p3 |m 1 ...m 9 . 3 · 3!

(3.35)

The normalization is fixed by the term in the IIB expansion. This is also the only possible contraction between A9 level +1 and −4 contributing to the diffeomorphism constraint in D = 11. (The mass deformation generator on = 4 does not contribute to the diffeomorphism constraint [14].) We note that the generator appearing in this new piece of the D = 11 diffeomorphism constraint is a gradient generator in the language of [3]. The new term in the D = 11 constraint now has components on IIB level = −5 and = −6 that can be covariantized now under sl(9) ⊕ sl(2) generating new terms. We have carried out this procedure one step farther and found the following expressions for 20 A similar difference was already noted for the E contraction in [4]. 9 21 See, however, the suggestion above Eq. (2.14) that one might use the Lie algebra generator associated to

the considered constraint-root α to define a universal way of pairing the two different root spaces gβ1 , gβ2 . 22 Some evidence from the structure of the compact subgroup K (E ) was given in [33]. 10

780


the ‘diffeomorphism constraints’ in A9 decomposition (−3)

L

m 1 ...m 9

(−1)

(−2)

(−3)

(0)

= 28 J m 1 m 2 m 3 J m 4 ...m 6 + 3 J p|m 1 ...m 8 J m 9 p 1 (−4) p1 p2 p3 |m 1 ...m 9 (+1) 1 (−5) p1 ... p6 |m 1 ...m 9 (+2) + J J p1 p2 p3 + J J p1 ... p6 3 · 3! 3 · 6! +··· . (3.36)

with implicit antisymmetrization over [m 1 . . . m 9 ], and a corresponding expression in A8 ⊕ A1 decomposition (−4)

C

m 1 ...m 8

=

35 (−2)m 1 ...m 4 (−2)m 5 ...m 8 28 (−1)m 1 m 2 ,α (−3)m 3 ...m 8 ,β − αβ I I I I 3 3 1 (−4)m 1 ...m 8 ,αγ (0)β 8 (−4) p|m 1 ...m 7 (0)m 8 − I I γ αβ − I I p 3 3 1 (−5) p1 p2 |m 1 ...m 8 ,α (+1) 1 (−6) p1 ... p4 |m 1 ...m 8 (+2) + I I p1 p2 ,α + I I p1 ... p4 3 · 2! 3 · 4! +··· (3.37)

with implicit antisymmetrization over [m 1 . . . m 8 ]. Note that the index range of the world indices is different in the two decompositions: In the D = 11 case, corresponding to A9 , the index range is m = 1, . . . , 10 and in the type IIB case, corresponding to A8 ⊕ A1 , the index range is m = 1, . . . , 9. By construction, these two expressions have the property that they agree on the real roots. In this way one produces an expression for a Sugawara constraint L−δ which extends to arbitrarily positive and negative step operators. We also note the appearance of gradient generators precisely in accord with (2.2), as these generators are the ones that reduce to the higher level affine generators in the truncation of E 10 to E 9 [33]. The gradient generators are those generators related to real roots of the affine E 9 [3,29]. It is straightforward to see that the infinite prolongation of our procedure will give rise to all the terms needed to match with the full sum in (2.2), for negative values of n. Our see-saw mechanism not only demands the extension of the constraints Lα (for a given α) to infinite strings of bilinears of Noether charges in agreement with the affine Sugarawara construction, but also allows to switch between constraints that are distinct as supergravity constraints. For instance, certain components of the IIB diffeomorphism constraint metamorphose into components of the D = 11 Gauss constraint when viewed in a different level decomposition! To see this more explicitly, consider the following component of the IIB diffeomorphism constraint (3.37) (−4)

C

12345678

1 (−2)1 2 3 4 (−2)5 6 7 8 + ··· , I I 3

(3.38)

where we only picked out one real root combination for simplicity. Translating this to the A9 basis via (−2) 1234

I

(−2) 1 2 3 4 9 10

= J

,

(−2) 5678

I

(−2) 5 6 7 8 9 10

= J

,

(3.39)

− (n 1 ↔ n 2 ),

(3.40)

we find that it is part of a covariant expression (−4)

L

m 1 ...m 10 ||n 1 n 2

(−2) (−2) n 1 [m 1 ...m 5 m 6 ...m 10 ]n 2

= 42 J

J


781

where the overall normalization differs by a factor of 20 from (3.8b), see also (3.20) in comparison to (3.21). This is exactly the combination that appears in the Gauss constraint of D = 11 supergravity, a result not too surprising from the point of view of U-duality. Evidently, this process could now be continued ad libitum. One can similarly generate new terms for the Gauss constraint in D = 11, given up to = 3 in (3.8b). Starting from the following components of the IIB diffeomorphism constraint: (−4) (−4) (0) 1 (−4)1 2 3 4 5 6 7 8,1˙ 1˙ (0)2˙ 12345678 8|2 3 4 5 6 7 8 1 I I 1˙ + I I 8 + · · · . (3.41) C 3 They can be mapped to A9 quantities using the two distinct A9 level = 4 representations (−4) 1 2 3 4 5 6 7 8,1˙ 1˙

I

(−4) 9|9|1 2 3 4 5 6 7 8 9 10

= J

,

(−4) 8|2 3 4 5 6 7 8

I

(−4) 8 9 10|2 3 4 5 6 7 8 9 10

= J

(3.42) to give sl(10) covariant additions to (3.8b) via (−4)

L

m 1 ...m 10 ||n 1 n 2

→ (3.8b) +

2 (−4)m 1 ...m 10 | p[n 1 (0)n 2 ] J J p 3

10 (−4)m 1 ...m 9 |n 1 n 2 p (0)m 10 (3.43) J J p + ··· . 3 We note that here both the gradient and non-gradient generator on A9 level = 4 contribute. Since the mass deformation parameter of massive type IIA is contained in the non-gradient generator, this is in agreement with the fact that the Gauss constraint of massive IIA gets modified by the Romans mass [14,35]. +

3.3. General remarks on the construction. Let us summarize the construction and comment on some open questions concerning this procedure. Starting from a single constraint L−δ , associated to the primitive null root, we construct the ‘scaffold’ as in (3.32), based on the decomposition α = β1 + β2 for α null and β1 , β2 real.23 Since all root spaces involved are real, they are one-dimensional and there is no ambiguity in the contraction. There are also no ordering ambiguities at this level. We can then act on the expression (3.32) with W (E 10 ) to generate similar expressions for all null roots. This constitutes the full scaffold of the hyperbolic Sugawara constraints which is invariant (only) under W (E 10 ). The constraints of the type in (3.32) are both infinite in number and each consists of an infinite number of bilinears in the current components. In order to construct constraints for the full E 10 one then needs to choose a level decomposition under a regular, finite-dimensional subalgebra. Covariance under this subalgebra induces additional terms on top of those already contained in the skeleton. The precise form of these additional terms depends on the subalgebra one chose in a systematic way, as is apparent from the explicit expressions in Appendix C. From that point of view it is clear that our construction is not covariant with respect to the full E 10 Lie algebra, but involves only the Weyl group W(E 10 ) in a canonical way. Everything beyond that depends on the chosen subalgebra for the level decomposition.24 23 We note that decompositions of imaginary roots into real roots have been considered in a different context in [36]. 24 In some sense this is also true for the affine E Sugawara construction which uses as choice of subalgebra 9 for the level decomposition E 8 .

782


We can also bring out the lack of ‘E 10 covariance’ by relating our construction to the question of an E 10 representation structure in the bilinear expression in E 10 generators. As already pointed out in footnote 13 one might have liked to identify the constraint L−δ with a highest weight vector of an integrable E 10 representation with highest weight 1 = −δ. If this were the case the constraint should be annihilated by all raising operators. Here, we recall that we express the step operators in terms of current components (s) (rather than in terms of the ‘contragredient’ E 10 Lie algebra generators Tα ), so that for 2 example e1 = J 1 . Using the explicit expression for L−δ in A9 decomposition we find that (−3) (−3) (0) 2 3 4 5 6 7 8 9 10 i+1 2 3 4 5 6 7 8 9 10 ei , L = J i, L = 0 for i = 1, . . . , 9 (3.44) where all commutators should be read as Poisson (or Dirac) brackets in the canonical setting. However, for the e generator corresponding to the omitted node we get (−3) (−3) (1) e10 , L 2 3 4 5 6 7 8 9 10 = J 8 9 10 , L 2 3 4 5 6 7 8 9 10 = 0, (3.45) showing that this component of the constraint generator is only a highest weight state with respect to the A9 subalgebra, but not the full E 10 algebra. Since the A9 expression agrees with the A8 ⊕ A1 expression on the real roots, we can repeat the calculation in IIB variables to find (−4) 23456789 ei , C = 0 for i = 1, . . . , 7, 9, 10 (3.46) and

(−4)

e8 , C

23456789

= 0.

(3.47)

Being a highest weight vector now with respect to A8 ⊕ A1 , this is different from the result for the A9 decomposition, but again illustrates the lack of full E 10 covariance. Similar conclusions hold for the D9 ≡ S O(9, 9) decomposition of [11]. Without going into the details of the calculation, the lowest order constraint for the D9 decomposition is (−2)

L

I

=

(−1) 1 (0)K L (−2)I K L 1 (−1)A + J J J (C I ) AB J B + · · · , 2 2

(3.48)

where I, K , L = 1, . . . , 18 and A, B = 1, . . . , 256 are S O(9, 9) vector and spinor indices, respectively, and we use again the symbol J to denote the components of the conserved E 10 current, but now in the D9 decomposition. The 18 = 9 + 9 constraints in (3.48) correspond to the diffeomorphism constraint and the Gauss constraint for the Neveu-Schwarz 2-form field of IIA theory; alternatively, they might be interpreted as a doubled set of diffeomorphism constraints w.r.t. the nine spatial target space coordinates X i and their (world-sheet) ‘duals’ X˜ i [11]. As before it is the omitted node (i.e., node 9 for the massive IIA theory) which causes failure of the construction: by S O(9, 9) covariance, the dilaton field associated with this node cannot appear in (3.48). Accordingly, it is now the generator e9 which does not annihilate the relevant component of L(−2) . To summarize: The failure of the constraint to be a highest weight vector w.r.t. the full E 10 algebra is invariably associated with the node that has been deleted for the given


783

level decomposition. In Appendix C we show that a related statement applies to the dependence of the constraints on the Cartan subalgebra generators. One further interesting aspect of our construction is that, to start with, it associates a constraint with every Weyl image of the fundamental null root −δ. In the same way one can associate constraints to the Weyl images of −nδ and in this way obtain a constraint for every E 10 root on the (past) light-cone. After choosing a level decomposition subalgebra one generates additional constraints inside the light-cone by covariantization under this subalgebra. It is possible that, as indicated in Subsect. 3.1.1, the set of roots C ‘supporting’ the full set of constraints be universally given by all the weights inside the (past) light-cone. This set can also be described as the union of the weight diagrams of the representations L(1 ), L(1 ) ⊗ L(1 ), L(1 ) ⊗ L(1 ) ⊗ L(1 ), etc.. On the other hand, the precise Sugawara-like expression defining the constraint Lα associated to some α ∈ C seems to depend on the choice of a level decomposition. Finally, note that since we are defining an infinity of constraints associated with all null roots of the hyperbolic algebra E 10 , one might worry whether there are any solutions that satisfy the geodesic equation and all the Sugawara constraints. It is reassuring to note that there are such solutions, namely for example the Kasner cosmologies. These correspond to only non-vanishing Cartan subalgebra components of the current, and hence all constraints except the Hamiltonian constraint (3.6) are trivially satisfied. Other solutions correspond to specific cases of Bianchi cosmologies. The exact count of the remaining number of degrees of freedom is quite involved and beyond the scope of this paper. Acknowledgements. We would like to thank Ofer Gabber and Victor Kac for informative discussions. AK is a Research Associate of the Fonds de la Recherche–FNRS, Belgium, and would like to thank IHES and AEI for hospitality. This work has been supported in part by IISN-Belgium (conventions 4.4511.06, 4.4505.86 and 4.4514.08) and by the Belgian Federal Science Policy Office through the Interuniversity Attraction Pole P6/11.

A. Level Decompositions For the reader’s convenience, we collect in this appendix some results on the level decompositions of E 10 appropriate for D = 11 supergravity and for type IIB in D = 10. These appeared originally in [3,29] and [13,21], respectively. A.1. Level decomposition under A9 . The A9 ∼ = sl(10) subalgebra relevant for D = 11 supergravity is obtained by removing node 10 from the Dynkin diagram of Fig. 1. 0 0 1 2 3 3 4 4 4

A9 Dynkin labels [1, 0, 0, 0, 0, 0, 0, 0, 1] [0, 0, 0, 0, 0, 0, 0, 0, 0] [0, 0, 0, 0, 0, 0, 1, 0, 0] [0, 0, 0, 1, 0, 0, 0, 0, 0] [0, 1, 0, 0, 0, 0, 0, 0, 1] [1, 0, 0, 0, 0, 0, 0, 0, 0] [0, 0, 0, 0, 0, 0, 0, 0, 2] [0, 0, 0, 0, 0, 0, 0, 1, 0] [1, 0, 0, 0, 0, 0, 1, 0, 0]

E 10 root for lowest weight (−1, −1, −1, −1, −1, −1, −1, −1, −1, 0) (0, 0, 0, 0, 0, 0, 0, 0, 0, 0) (0, 0, 0, 0, 0, 0, 0, 0, 0, 1) (0, 0, 0, 0, 1, 2, 3, 2, 1, 2) (0, 0, 1, 2, 3, 4, 5, 3, 1, 3) (0, 1, 2, 3, 4, 5, 6, 4, 2, 3) (1, 2, 3, 4, 5, 6, 7, 4, 1, 4) (1, 2, 3, 4, 5, 6, 7, 4, 2, 4) (0, 1, 2, 3, 4, 5, 6, 4, 2, 4)

μ 1 1 1 1 1 0 1 0 1

α2 2 0 2 2 2 0 2 0 2

784


5 5 5 6 6 6 6 6 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 8 8

A9 Dynkin labels [0, 0, 0, 0, 0, 1, 0, 0, 1] [0, 0, 0, 0, 1, 0, 0, 0, 0] [1, 0, 0, 1, 0, 0, 0, 0, 0] [0, 0, 0, 1, 0, 0, 0, 1, 0] [0, 0, 1, 0, 0, 0, 0, 0, 1] [0, 1, 0, 0, 0, 0, 0, 0, 0] [1, 1, 0, 0, 0, 0, 0, 0, 1] [2, 0, 0, 0, 0, 0, 0, 0, 0] [0, 0, 0, 0, 0, 0, 0, 0, 1] [0, 0, 1, 0, 0, 1, 0, 0, 0] [0, 1, 0, 0, 0, 0, 0, 1, 1] [0, 1, 0, 0, 0, 0, 1, 0, 0] [1, 0, 0, 0, 0, 0, 0, 0, 2] [1, 0, 0, 0, 0, 0, 0, 1, 0] [2, 0, 0, 0, 0, 0, 1, 0, 0] [0, 0, 0, 0, 0, 0, 0, 1, 2] [0, 0, 0, 0, 0, 0, 0, 2, 0] [0, 0, 0, 0, 0, 0, 1, 0, 1] [0, 0, 0, 0, 0, 1, 0, 0, 0] [0, 0, 2, 0, 0, 0, 0, 0, 0] [0, 1, 0, 0, 1, 0, 0, 0, 1] [0, 1, 0, 1, 0, 0, 0, 0, 0] [1, 0, 0, 0, 0, 0, 1, 1, 0] [1, 0, 0, 0, 0, 1, 0, 0, 1] [1, 0, 0, 0, 1, 0, 0, 0, 0] [2, 0, 0, 1, 0, 0, 0, 0, 0]

E 10 root for lowest weight (1, 2, 3, 4, 5, 6, 8, 5, 2, 5) (1, 2, 3, 4, 5, 7, 9, 6, 3, 5) (0, 1, 2, 3, 5, 7, 9, 6, 3, 5) (1, 2, 3, 4, 6, 8, 10, 6, 3, 6) (1, 2, 3, 5, 7, 9, 11, 7, 3, 6) (1, 2, 4, 6, 8, 10, 12, 8, 4, 6) (0, 1, 3, 5, 7, 9, 11, 7, 3, 6) (0, 2, 4, 6, 8, 10, 12, 8, 4, 6) (2, 4, 6, 8, 10, 12, 14, 9, 4, 7) (1, 2, 3, 5, 7, 9, 12, 8, 4, 7) (1, 2, 4, 6, 8, 10, 12, 7, 3, 7) (1, 2, 4, 6, 8, 10, 12, 8, 4, 7) (1, 3, 5, 7, 9, 11, 13, 8, 3, 7) (1, 3, 5, 7, 9, 11, 13, 8, 4, 7) (0, 2, 4, 6, 8, 10, 12, 8, 4, 7) (2, 4, 6, 8, 10, 12, 14, 8, 3, 8) (2, 4, 6, 8, 10, 12, 14, 8, 4, 8) (2, 4, 6, 8, 10, 12, 14, 9, 4, 8) (2, 4, 6, 8, 10, 12, 15, 10, 5, 8) (1, 2, 3, 6, 9, 12, 15, 10, 5, 8) (1, 2, 4, 6, 8, 11, 14, 9, 4, 8) (1, 2, 4, 6, 9, 12, 15, 10, 5, 8) (1, 3, 5, 7, 9, 11, 13, 8, 4, 8) (1, 3, 5, 7, 9, 11, 14, 9, 4, 8) (1, 3, 5, 7, 9, 12, 15, 10, 5, 8) (0, 2, 4, 6, 9, 12, 15, 10, 5, 8)

μ 1 0 1 1 1 1 1 0 1 1 1 1 1 2 1 1 0 2 2 1 1 1 1 2 2 1

α2 2 0 2 2 0 −2 2 0 −4 2 2 0 0 −2 2 2 0 −2 −4 2 2 0 2 0 −2 2

The low-lying generators are denoted by (a, b, . . . = 1, . . . , 10) = 0 : K a b, = 1 : E abc = E [abc] , = 2 : E a1 ...a6 = E [a1 ...a6 ] , =3:E

a0 |a1 ...a8

=4:E

a1 a2 a3 |b1 ...b9

=E

a0 |[a1 ...a8 ]

=E

(A.1) ,

[a1 a2 a3 ]|[b1 ...b9 ]

and E a|b|c1 ...c10 = E (a|b)|[c1 ...c10 ]

(with the usual irreducibility conditions E [a0 |a1 ...a8 ] = 0, etc.). They are related to the Chevalley–Serre generators by ei = K i i+1 ,

f i = K i+1 i , h i = K i i − K i+1 i+1 (i = 1, . . . , 9)

(A.2)

e10 = E 8 9 10 , f 10 = F8 9 10 , 1 h 10 = − K + K 8 8 + K 9 9 + K 10 10 , 3

(A.3)

and

10 where K = a=1 K a a . Commutation relations for these generators can be found in [12,14] but note that we have rescaled all generators such that their lowest weight elements (e.g. E 10|3 4 5 6 7 8 9 10 ) have norm 1.


785

A.2. Level decomposition under A8 ⊕ A1 . The A8 ⊕ A1 ∼ = sl(9) ⊕ sl(2) subalgebra relevant for type IIB supergravity is obtained by removing node 8 from the Dynkin diagram of Fig. 1. 0 0 0 1 2 3 4 4 4 5 5 6 6 6 6 7 7 7 8 8 8 8 8 8 8 8

A8 ⊕ A1 Dynkin labels [1, 0, 0, 0, 0, 0, 0, 1][0] [0, 0, 0, 0, 0, 0, 0, 0][0] [0, 0, 0, 0, 0, 0, 0, 0][2] [0, 0, 0, 0, 0, 0, 1, 0][1] [0, 0, 0, 0, 1, 0, 0, 0][0] [0, 0, 1, 0, 0, 0, 0, 0][1] [0, 1, 0, 0, 0, 0, 0, 1][0] [1, 0, 0, 0, 0, 0, 0, 0][0] [1, 0, 0, 0, 0, 0, 0, 0][2] [0, 0, 0, 0, 0, 0, 0, 1][1] [1, 0, 0, 0, 0, 0, 1, 0][1] [0, 0, 0, 0, 0, 0, 1, 1][0] [0, 0, 0, 0, 0, 1, 0, 0][0] [0, 0, 0, 0, 0, 1, 0, 0][2] [1, 0, 0, 0, 1, 0, 0, 0][0] [0, 0, 0, 0, 1, 0, 0, 1][1] [0, 0, 0, 1, 0, 0, 0, 0][1] [1, 0, 1, 0, 0, 0, 0, 0][1] [0, 0, 0, 1, 0, 0, 1, 0][0] [0, 0, 1, 0, 0, 0, 0, 1][0] [0, 0, 1, 0, 0, 0, 0, 1][2] [0, 1, 0, 0, 0, 0, 0, 0][0] [0, 1, 0, 0, 0, 0, 0, 0][2] [1, 1, 0, 0, 0, 0, 0, 1][0] [2, 0, 0, 0, 0, 0, 0, 0][0] [2, 0, 0, 0, 0, 0, 0, 0][2]

E 10 root for lowest weight (−1, −1, −1, −1, −1, −1, −1, 0, 0, −1) (0, 0, 0, 0, 0, 0, 0, 0, 0, 0) (0, 0, 0, 0, 0, 0, 0, 0, -1, 0) (0, 0, 0, 0, 0, 0, 0, 1, 0, 0) (0, 0, 0, 0, 0, 1, 2, 2, 1, 1) (0, 0, 0, 1, 2, 3, 4, 3, 1, 2) (0, 0, 1, 2, 3, 4, 5, 4, 2, 2) (0, 1, 2, 3, 4, 5, 6, 4, 2, 3) (0, 1, 2, 3, 4, 5, 6, 4, 1, 3) (1, 2, 3, 4, 5, 6, 7, 5, 2, 3) (0, 1, 2, 3, 4, 5, 6, 5, 2, 3) (1, 2, 3, 4, 5, 6, 7, 6, 3, 3) (1, 2, 3, 4, 5, 6, 8, 6, 3, 4) (1, 2, 3, 4, 5, 6, 8, 6, 2, 4) (0, 1, 2, 3, 4, 6, 8, 6, 3, 4) (1, 2, 3, 4, 5, 7, 9, 7, 3, 4) (1, 2, 3, 4, 6, 8, 10, 7, 3, 5) (0, 1, 2, 4, 6, 8, 10, 7, 3, 5) (1, 2, 3, 4, 6, 8, 10, 8, 4, 5) (1, 2, 3, 5, 7, 9, 11, 8, 4, 5) (1, 2, 3, 5, 7, 9, 11, 8, 3, 5) (1, 2, 4, 6, 8, 10, 12, 8, 4, 6) (1, 2, 4, 6, 8, 10, 12, 8, 3, 6) (0, 1, 3, 5, 7, 9, 11, 8, 4, 5) (0, 2, 4, 6, 8, 10, 12, 8, 4, 6) (0, 2, 4, 6, 8, 10, 12, 8, 3, 6)

μ 1 1 1 1 1 1 1 0 1 1 1 1 0 1 1 1 1 1 1 1 1 2 1 1 0 1

α2 2 0 2 2 2 2 2 0 2 0 2 2 0 2 2 2 0 2 2 0 2 −2 0 2 0 2

The low-lying generators are (now a, b, . . . = 1, . . . , 9 are sl(9) vector indices and ˙ 2˙ are sl(2) vector indices) α, β = 1, = 0 : K a b and K α β (with δαβ K α β = 0), = 1 : E ab,α = E [ab],α ,

= 2 : E a1 a2 a3 a4 = E [a1 a2 a3 a4 ] , =3:E

a1 ...a6 ,α

=4:E

a0 |a1 ...a7

=E

[a1 ...a6 ],α

=E

(A.4)

,

a0 |[a1 ...a7 ]

and E a1 ...a8 ,αβ = E [a1 ...a8 ],(αβ) ,

The relation to the Chevalley–Serre generators is now given by ei = K i i+1 , f i = K i+1 i , h i = K i i − K i+1 i+1 (i = 1, . . . , 7), e10 = K 8 9 , f 10 = K 9 8 , h 10 = K 8 8 − K 9 9 , e9 = K

1˙

2˙ ,

f9 = K

2˙

1˙ ,

h9 = K

1˙

1˙

−K

2˙

(A.5)

2˙ .

The explicit dots on the indices indicate numerical values for sl(2) vector indices. For the deleted node 8 one has

786

T. Damour, A. Kleinschmidt, H. Nicolai ˙

e8 = E 8 9,2 , f 8 = F8 9,2˙ ,

1 1 1˙ ˙ K 1˙ − K 2 2˙ , h8 = − K + K 88 + K 99 − 4 2

(A.6)

9 where now K = a=1 K a a is the trace in gl(9). Commutation relations for these generators can be found in [13], where we used an so(1, 2) spinor and vector notation instead of sl(2) tensors as above. B. Constraints of Type IIB Supergravity and Universality The Einstein equation of motion of IIB supergravity can be written as 1 1 R AB = − S αA S B,α + FA C1 ...C4 FBC1 ...C4 4 96 1 1 + H A C1 C2 ,α H BC1 C2 ,α − η AB H C1 ...C3 ,α HC1 C2 C3 α 4 48

(B.1)

in flat indices, where we corrected a factor of two compared to [13]. The diffeomorphism constraint is obtained as the 0a component of this equation. Using self-duality of F and the dictionary of [13] one finds, up to overall normalization, the expression (−4)

C

m 1 ...m 8

=

35 (−2)m 1 ...m 4 (−2)m 5 ...m 8 28 (−1)m 1 m 2 ,α (−3)m 3 ...m 8 ,β + αβ I I I I 3 3 1 (−4)m 1 ...m 8 ,αγ (0)β 8 (−4) p|m 1 ...m 7 (0)m 8 + (B.2) I I γ αβ + I I p 3 3

in terms of the E 10 current components in A8 ⊕ A1 decomposition. C. Explicit Expressions Involving Cartan Generators In this appendix, we give explicit expressions for the contractions between the Cartan subalgebra and the δ root space to show that the A9 and A8 ⊕ A1 covariant expressions (3.36) and (3.37) differ, thereby illustrating that (3.31) is indeed only valid on contractions of real root spaces. Consider the contributions from the Cartan subalgebra to the highest component of (3.36). They come exclusively from the J (−3) J (0) contraction and are (−3) (0) (0) (−3) (−3) 3 L 2 3 4 5 6 7 8 9 10 J 2|3 4 5 6 7 8 9 10 J 2 2 + J 3|4 5 6 7 8 9 10 2 J 3 3 (0) (−3) + · · · + J 10|2 3 4 5 6 7 8 9 J 10 10 (−3) (−3) = J 2|3 4 5 6 7 8 9 10 (h 2 + h 3 + · · · + h 8 + h 9 ) + · · · + J 9|10 2 3 4 5 6 7 8 h 9 ,

(C.1) where the hook symmetry (−3) [2|3 4 5 6 7 8 9 10]

J

=0

(C.2)


787

of the level three element was used and we identified for simplicity the current component with the corresponding Cartan generators using (A.2) and (A.3). We see that only the Cartan generators of the A9 ‘gravity line’ appear in this contraction. The only ‘missing’ ones are the one from the deleted node 10 and the hyperbolic node 1. The latter is related to our choice of (highest) component. Repeating the same calculation for the A8 ⊕ A1 decomposition and (3.37) one finds similarly (−4)

3 C

23456789

(−4) (0) 2|3 4 5 6 7 8 9 2

I

I

2

(−4) (0) ˙ 2 3 4 5 6 7 8 9,1˙ 2˙ 1

(I

+ I

(−4) 2|3 4 5 6 7 8 9

= I

1˙ −

I

(0) ˙ 2

I

I

9

2˙ ) (−4) 8|9 2 3 4 5 6 7

(h 2 + · · · + h 7 + h 10 )+ I

(−4) 2 3 4 5 6 7 8 9,1˙ 2˙

+ I

(−4) (0) 9|2 3 4 5 6 7 8 9

+ ···+

h9.

h 10 (C.3)

The Cartan generators that appear in this expression are those from the A8 ⊕ A1 gravity line, so the ‘missing’ generators are that of the deleted node 8 and of the hyperbolic node 1. The latter is again related to our choice of component of the diffeomorphism constraint so that the real discrepancy between the two expressions can be traced again to the different deleted nodes. This is related to the failure of this constraint to be a highest weight vector, see the expressions (3.45) and (3.47). Finally, it is clear that the Cartan generator missing in (3.48) is h 9 , as the diagonal generators among the S O(9, 9) generators J (0) K L are identified with h 1 , . . . , h 8 , h 10 , while h 9 is associated with the dilaton, again confirming our general conclusion. References 1. DeWitt, B.S.: Quantum Theory of Gravity. 1. The Canonical Theory. Phys. Rev. 160, 1113 (1967) 2. Kiefer, C.: Quantum gravity. Int. Ser. Monogr. Phys. 124, Oxford: Oxford University Press, 2004 3. Damour, T., Henneaux, M., Nicolai, H.: E 10 and a ‘small tension expansion’ of M Theory. Phys. Rev. Lett. 89, 221601 (2002) 4. Damour, T., Kleinschmidt, A., Nicolai, H.: Constraints and the E 10 Coset Model. Class. Quant. Grav. 24, 6097 (2007) 5. Sugawara, H.: A Field theory of currents. Phys. Rev. 170, 1659 (1968) 6. Bardakçi, K., Halpern, M. B.: New dual quark models. Phys. Rev. D 3, 2493 (1971) 7. Goddard, P., Olive, D.I.: Kac-Moody And Virasoro Algebras In Relation To Quantum Physics. Int. J. Mod. Phys. A 1, 303 (1986) 8. Kleinschmidt, A., Koehn, M., Nicolai, H.: Supersymmetric quantum cosmological billiards. Phys. Rev. D 80, 061701 (2009) 9. Forte, L.A.: Arithmetical Chaos and Quantum Cosmology. Class. Quant. Grav. 26, 045001 (2009) 10. Goddard, P., Thorn, C.B.: Compatibility of the Dual Pomeron with Unitarity and the Absence of Ghosts in the Dual Resonance Model. Phys. Lett. B 40, 235 (1972) 11. Kleinschmidt, A., Nicolai, H.: E 10 and S O(9, 9) invariant supergravity. JHEP 0407, 041 (2004) 12. Damour, T., Nicolai, H.: Eleven dimensional supergravity and the E 10 /K (E 10 ) sigma-model at low A9 levels. In: Pogoyan, G.S., Vicent, L.E., Wolf, K.B. (eds.) Group Theoretical Methods in Physics. IOP conference series no. 185, pp. 93–111. IOP Publishing (2005) 13. Kleinschmidt, A., Nicolai, H.: IIB supergravity and E 10 . Phys. Lett. B 606, 391 (2005) 14. Henneaux, M., Jamsin, E., Kleinschmidt, A., Persson, D.: On the E 10 /Massive Type IIA Supergravity Correspondence. Phys. Rev. D 79, 045008 (2009) 15. Nicolai, H., Samtleben, H.A.J.: On K (E 9 ). Q.J. Pure Appl. Math. 1, 180 (2005) 16. West, P.C.: E 11 and M theory. Class. Quant. Grav. 18, 4443 (2001) 17. West, P.C.: E(11), S L(32) and central charges. Phys. Lett. B 575, 333 (2003) 18. Riccioni, F., West, P.: Local E 11 . JHEP 0904, 051 (2009) 19. Schnakenburg, I., West, P.C.: Kac-Moody symmetries of IIB supergravity. Phys. Lett. B 517, 421 (2001)

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20. Schnakenburg, I., West, P.C.: Massive IIA supergravity as a non-linear realisation. Phys. Lett. B 540, 137 (2002) 21. Kleinschmidt, A., Schnakenburg, I., West, P.C.: Very-extended Kac-Moody algebras and their interpretation at low levels. Class. Quant. Grav. 21, 2493 (2004) 22. West, P.C.: The IIA, IIB and eleven-dimensional theories and their common E(11) origin. Nucl. Phys. B 693, 76 (2004) 23. Morozov, A.Y., Perelomov, A.M., Roslyi, A.A., Shifman, M.A., Turbiner, A.V.: Quasiexactly Solvable Quantal Problems: One-Dimensional Analog of Rational Conformal Field Theories. Int. J. Mod. Phys. A 5, 803 (1990) 24. Halpern, M.B., Kiritsis, E.: General Virasoro Construction on Affine G. Mod. Phys. Lett. A 4, 1373 (1989) 25. Kac, V.G.: Infinite dimensional Lie algebras. Cambridge: Cambridge University Press, 1995 26. Damour, T., de Buyl, S., Henneaux, M., Schomblond, C.: Einstein billiards and overextensions of finitedimensional simple Lie algebras. JHEP 0208, 030 (2002) 27. Gebert, R.W., Nicolai, H.: An affine string vertex operator construction at arbitrary level. J. Math. Phys. 38, 4435 (1997) 28. Gaberdiel, M.R., Olive, D.I., West, P.C.: A class of Lorentzian Kac-Moody algebras. Nucl. Phys. B 645, 403 (2002) 29. Nicolai, H., Fischbacher, T.: Low level representations for E 10 and E 11 . Cont. Math. 343, Providence, RI: Amer. Math. Soc., 2004, p. 191 30. Damour, T., Henneaux, M., Nicolai, H.: Cosmological billiards. Class. Quant. Grav. 20, R145 (2003) 31. Damour, T., Henneaux, M., Julia, B., Nicolai, H.: Hyperbolic Kac-Moody algebras and chaos in KaluzaKlein models. Phys. Lett. B 509, 323 (2001) 32. Kac, V., Moody, R.V., Wakimoto, M.: On E 10 . In: Bleuler, K., Werner, M., (eds.) “Differential geometrical methods in theoretical physics”, pp. 109–128. Dordrecht, Kluwer (1988) 33. Kleinschmidt, A., Nicolai, H., Palmkvist, J.: K (E 9 ) from K (E 10 ). JHEP 0706, 051 (2007) 34. Damour, T., Nicolai, H.: Symmetries, singularities and the de-emergence of space. Int. J. Mod. Phys. D 17, 525 (2008) 35. Romans, L.J.: Massive N=2a supergravity in ten-dimensions. Phys. Lett. B 169, 374 (1986) 36. Brown, J., Ganor, O.J., Helfgott, C.: M-theory and E 10 : Billiards, branes, and imaginary roots. JHEP 0408, 063 (2004) Communicated by N.A. Nekrasov

Commun. Math. Phys. 302, 789–813 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1189-x

Communications in


Dorey’s Rule and the q-Characters of Simply-Laced Quantum Affine Algebras C. A. S. Young1 , R. Zegers2 1 Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan.


2 Laboratoire de Physique Théorique, Université Paris-Sud 11/CNRS, 91405 Orsay Cedex, France.

E-mail: [email protected] Received: 3 January 2010 / Accepted: 3 October 2010 Published online: 8 February 2011 – © Springer-Verlag 2011

Abstract: Let Uq ( g) be the quantum affine algebra associated to a simply-laced simple Lie algebra g. We examine the relationship between Dorey’s rule, which is a geometrical statement about Coxeter orbits of g-weights, and the structure of q-characters of fundamental representations Vi,a of Uq ( g). In particular, we prove, without recourse to the ADE classification, that the rule provides a necessary and sufficient condition for the monomial 1 to appear in the q-character of a three-fold tensor product Vi,a ⊗ V j,b ⊗ Vk,c . 1. Introduction 1.1. Background. This paper concerns the relationship between the representation theory of simply-laced quantum affine algebras on the one hand, and, on the other, the particle fusing rule originally given by Dorey in the context of affine Toda field theories. Recall that Affine Toda Field Theories (ATFTs) are integrable quantum field theories in 1+1 dimensions [Cor94]. Let g be any simply-laced simple Lie algebra, and I the set of nodes of the Dynkin diagram of g. The (real coupling, purely elastic) ATFT associated to the untwisted affine algebra g has rank g species of particles, labelled by the nodes i ∈ I . The root system data of g determine not only the masses of these particles, but also the allowed fusings: if particles of species j ∈ I and i ∈ I can interact to form a ¯ and this process can occur particle of species k¯ ∈ I one says there is a fusing j, i → k, only if the rapidities θi , θ j of the incoming particles are related by θi − θ j =

√ −1 θ kji ,

(1.1)

where θ kji is a real angle, called the fusing angle. If there is a fusing j, i → k¯ then there are also fusings i, k → j¯ and k, j → ı¯, and the fusing angles obey j

θ kji + θik + θki j = 2π.

(1.2)

790

C. A. S. Young, R. Zegers

The problem of determining the masses, fusings and fusing angles for the ATFTs associated to all simple Lie algebras (simply-laced or not) was solved in [BCDS90]. It was observed in that paper that the allowed fusings form a strict subset of the non-zero Clebsch-Gordon coefficients for g, in the sense that if i, j → k¯ is a fusing then Homg Vi ⊗ V j , Vk¯ ∼ = Homg Vi ⊗ V j ⊗ Vk , C = 0,

(1.3)

where Vi is the i th fundamental representation of g. It is a strict subset because the converse statement does not hold: the first counterexample is D5 ,

where there is a non-trivial homomorphism V2 ⊗ V2 → V2 of d5 modules but no fusing 2, 2 → 2 in the ATFT. Soon after, this same “hole” in the allowed interactions was also found in a different (and non-diagonal) scattering theory [Mac91], giving an indication of a more general underlying structure. Subsequently, Dorey gave a rule which encodes both the pattern of allowed fusings, and the fusing angles, in an elegant geometrical fashion for all the simply-laced cases [Dor91,Dor92b,Dor92a,FLO91,FO92].1 To state the rule, we introduce some standard notation: let (αi )i∈I be a set of simple roots of g, (λi )i∈I the corresponding fundamental weights, and ai j the Cartan matrix: αi · α j = ai j ,

αi · λ j = δi j .

(1.4)

Let W denote the Weyl group of g, generated by the reflections (si )i∈I in the simple roots. It is always possible to write I as a disjoint union I = I• I◦

(1.5)

in such a way that (I• , I◦ ) is a two-colouring of the Dynkin diagram (as, for example, in the case D5 above). Let then w ∈ W be the choice of Coxeter element given by2 w = w◦ w• ,

w◦ =

i∈I◦

si ,

w• =

si ,

(1.6)

i∈I•

and write = w for the cyclic subgroup of W generated by w, whose order h is the Coxeter number of g. 1 A generalization of the rule to non-simply laced cases was mentioned in [Dor93] and used in [CP96]; see also [Oot97,FKS00]. In the present work we shall focus exclusively on the simply laced cases but it would be very interesting to try to prove analogous results for any simple Lie algebra. 2 This choice will be convenient in what follows, but the rule itself is independent of the choice of Coxeter element.

Dorey’s Rule and the q-Characters of Simply-Laced Quantum Affine Algebras

791

Then the rule states that there is a fusing i, j → k¯ if and only if 0 ∈ λi + λ j + λk ;

(1.7)

that is, if and only if there are integers p, q, r such that 0 = w p λi + wq λ j + wr λk .

(1.8)

j

Moreover, the fusing angles, θikj θ ijk and θki are given by projecting this latter equation onto the exp (±2πi/ h) eigenplane of w, as discussed in [Dor91,Dor92b] and recalled in Sec. 3 below. The original statement of the rule involved Coxeter orbits of roots, but it was observed in [Bra92] that the statement above in terms of weights is equivalent, essentially because (one can show that) φi := (1 − w −1 )λi are a linearly independent collection of roots. Writing the rule in terms of weights is suggestive, because of the following Theorem 1.1 (PRV [PRRV67,Kum88,Mat89]). A necessary and sufficient condition for Homg Vi ⊗ V j ⊗ Vk , C = 0 (1.9) is that 0 ∈ W λi + W λ j + W λk .

(1.10)

Now clearly (1.7) implies (1.10), but not vice versa. So in light of this result, which connects the Weyl-orbits of weights to invariants of g-representations, it is very natural to suppose that the fusing rule (1.7) plays a similar role for representations of some larger (and hence more restrictive) algebraic structure. In [Mac92], MacKay conjectured that this is indeed the case and that the relevant algebra is the Yangian Y (g). Recall that the universal envelope U ( g) of the untwisted affine algebra g has a canonical Drinfel’d-Jimbo deformation Uq ( g), called a quantum affine algebra, and that the Yangian Y (g) is the rational degeneration of Uq ( g) [Dri85,Dri88]. Y (g) and Uq ( g) share essentially the same representation theory [Var00]. There is a notion of the fundamental representations Vi,a of Uq ( g), where i ∈ I , and a ∈ C=0 is an additional label which we will call the rapidity; see e.g. [CP].3 The Vi,a are finite-dimensional and Vi ⊂ Vi,a . In the classical cases, the following theorem was proved by Chari and Pressley, confirming the conjecture above. (In fact, [CP96] considered all the classical cases ABC D, but we quote here only the result for the classical simply-laced cases AD.) Theorem 1.2 ([CP96]). A necessary and sufficient condition for HomUq (g) Vi,a ⊗ V j,b ⊗ Vk,b , C = 0 ,

(1.11)

for some rapidities a, b, c ∈ C=0 , is that 0 ∈ λi + λ j + λk .

(1.12)

1.2. Motivations and outline. Despite the positive result above, it is fair to say that a satisfactory understanding of the link between the fusing rule and the representation theory of simply-laced quantum affine algebras is still missing. Most apparently, the 3 We are of course using the word “rapidity” in two, a priori different, senses: for the kinematical label of particles in ATFT and for the spectral parameter of representations of Uq ( g). The role of Uq ( g)-symmetry in real- and imaginary-coupling affine Toda field theory is indeed rather subtle. See [TW99,SWK00] and references therein.

792


proof in [CP96] was case-by-case and did not include the exceptional cases E 6 , E 7 and E 8 . More importantly, part of what makes the rule (1.7) elegant is that it encodes not only the triples (i, j, k) for which fusing can occur, but also the fusing angles, via the projection map mentioned above. This aspect played no role in [CP96], where the required rapidities were determined without reference to this projection map. One would like to understand why the rapidities emerge as they do from the geometry of Coxeter orbits of roots and weights. In the present paper we take a step in this direction, by relating the geometry of Coxeter orbits to the q-characters of fundamental representations of Uq ( g). The notion of q-characters, due to Frenkel and Reshetikhin [FR98], following [Kni95], is an important development in the representation theory of quantum affine algebras. Here they will allow us to give, in particular, a general proofthat Dorey’s rule is a necessary condition for the existence of invariant maps, HomUq (g) Vi,a ⊗ V j,b ⊗ Vk,c , C = 0, and singlets, HomUq (g) (C, Vk,c ⊗ V j,b ⊗ Vi,a ) = 0. The structure of this paper is as follows: in Sect. 2 we recall the definition of Uq ( g), and the necessary details of the theory of q-characters. Then in Sect. 3 we go on to prove our main result (Theorem 3.1), which states that Dorey’s rule provides a necessary and sufficient condition for the monomial 1 to occur in the q-character of a three-fold tensor product of fundamental representations. We prove this by first showing (Lemma 3.2) that the latter statement can be rephrased as a statement about the occurrence of quadratic monomials in the q-character of a single fundamental representation. We then prove that such quadratic monomials are in a certain precise correspondence with solutions to Dorey’s rule. Indeed, it will emerge that in fact every monomial in the q-character can very naturally be seen as specifying some identity among the Coxeter orbits of the fundamental weights of g (Proposition 3.3). The reverse direction however (going from identities to monomials) is more subtle, and one must work harder to show (Propositions 3.4 and 3.45) that it always holds for identities of the form (1.7) above. We conclude in Sect. 4 by commenting on the relationship of our result to Theorem 1.2 above, and noting some open questions. We assume, throughout this paper, that q ∈ C=0 is not a root of unity. 2. Quantum Affine Algebras and q-Characters g) is an associative algebra over C generated by The quantum affine algebra Uq ( ± )i∈I,n∈Z , (xi,n

(ki±1 )i∈I ,

(h i,n )i∈I,n∈Z=0 ,

(2.1)

and central elements c±1/2 . In this paper we study finite dimensional representations of Uq ( g) when g is simply laced. As we recall below, for this purpose it actually suffices to work with the quantum loop algebra Uq (Lg) = Uq ( g)/(c±1/2 − 1). Following [Dri88], let us arrange the generators into formal series xi± (u) :=

± −n xi,n u ,

(2.2)

n∈Z

φi± (u)

=

∞ n=0

± φi,±n u ±n

:=

ki±1 exp

±(q − q

−1

)

∞ m=1

h i,±m u

±m

,

(2.3)


and set δ(u) :=

793

un .

(2.4)

n∈Z

The defining relations of Uq (Lg) are then

± ∓ φi± (u), φ ± j (v) = φi (u), φ j (v) = 0,

(2.5)

1 − q −ai j uv + x (v) φi± (u), 1 − q ai j uv j ai j −ai j 1 − q uv x − (v) φi± (u), φi± (u) x − j (v) = q −a 1 − q i j uv j

δi j xi+ (u), x − δ(v/u)φi+ (1/v) − δ(u/v)φi− (1/u) , j (v) = −1 q −q ± ±a ±ai j i j u − v x ± (v) x ± (u), v xi (u) x ± u−q j (v) = q j i φi± (u) x +j (v) = q ai j

± ± −1 ± xi± (u)xi± (v)x ± j (w) − (q + q )x i (u)x j (v)x i (w) ± ± +x ± j (v)x i (u)x i (w) + (u ↔ v) = 0 if ai j =

−1,

(2.6) (2.7) (2.8) (2.9)

(2.10)

where ai j is the Cartan matrix of g. As we shall see, this presentation, which is a slightly modified version of Drinfel’d’s current presentation [Dri88], is convenient because the φi± (u) and xi± (u) behave analogously to the usual Cartan generators and raising/lowering operators in the representation theory of finite-dimensional simple Lie algebras. From its origin as a standard Drinfel’d-Jimbo deformation of U ( g), Uq ( g) admits a standard Hopf algebra structure Uq ( g)std – see e.g. [CP]. No closed form is known for the standard coproduct in the current presentation above. As we note in the conclusion, there does exist another (twist-equivalent [EKP07]) Hopf algebra structure for Uq ( g) better suited to the current presentation; for details see [Her05,Her07a,Gro01]. 2.1. Representations and characters. A representation V of Uq ( g) is of type 1 if c±1/2 acts as the identity on V and V is the direct sum of its Uq (g)-weight spaces, V = ⊕λ Vλ where Vλ = {v ∈ V : ki v = q αi ,λ v}

(2.11)

and λ in the weight lattice of g. We recall (see e.g. [CP] Chap. 12.2B) that any finitedimensional irreducible representation of Uq ( g) can be obtained by twisting, by an automorphism of Uq ( g), a finite-dimensional type 1 representation. Thus it suffices for our purposes to consider type 1 representations, and to regard them as representations of Uq (Lg). Any type 1 representation V of Uq ( g) also furnishes a representation of Uq (g) (the ± )i∈I , (ki± )i∈I ). Recall that the character latter being the subalgebra generated by (xi,0 χ (V ) of V regarded as a Uq (g)-module is defined as χ (V ) = dim (Vλ ) eλ . (2.12) λ

If Rep(Uq (g)) is the category whose objects are finite-dimensional representations of Uq (g) and whose morphisms are homomorphisms of Uq (g)-modules, then the Grothendieck ring Rep(Uq (g)) is the ring generated by the isomorphism classes of objects

794


in Rep(Uq (g)) subject to the relations [X ][Y ] = [X ⊗ Y ] and, for each exact sequence 0 → U → W → V → 0 of Uq (g)-modules, [W ] = [U ] + [V ]. The character map χ is a homomorphism of rings

(2.13) χ : Rep(Uq (g)) −→ Z yi±1 i∈I

yi±1

= e±λi . to the ring of polynomials in variables Let us pause to recall that Rep(Uq (g)), like Rep(U (g)), is a semisimple category: exact sequences 0 → U → W → V → 0 exist precisely when W = U ⊕ V as Uq (g)modules; and thus the defining relations of Rep(Uq (g)) are in fact just [U ][V ] = [U ⊗V ] and [U ] + [V ] = [U ⊕ V ]. In contrast, representations of Uq ( g) can be reducible but not fully-reducible. That is, it can happen that there is a short exact sequence 0 → U → W → V → 0 of Uq ( g)-modules, so that U is a submodule of W , but that W is not the direct sum U ⊕ V as a Uq ( g)-module. One says that W is indecomposable. Now for any type 1 representation V of Uq ( g), the decomposition above into Uq (g)weight spaces may be further refined by decomposing V into Jordan subspaces of the ± mutually commuting φi,±r defined in (2.3), [FR98]: ± ± V = ⊕γ Vγ , γ = (γi,±r )i∈I,r ∈N , γi,±r ∈ C,

where Vγ = {v ∈ V : ∃N ∈ N, ∀i ∈ I,

± N φi (u) − γi± (u) v = 0}.

If dim(Vγ ) > 0, we shall refer to the corresponding formal series ± ∀i ∈ I , γi± (u) := γi,±r u ±r

(2.14)

(2.15)

(2.16)

r ∈N

as an l-weight of V . It is known [FR98] that for every finite-dimensional type 1 repreg), these l-weights are of the form sentation of Uq ( γi± (u) = q deg Q i −deg Ri

Q i (uq −1 )Ri (uq) , Q i (uq)Ri (uq −1 )

(2.17)

where the right hand side is to be treated as a formal series in positive (negative) integer powers of u for γi+ (u) (respectively γi− (u)), and Q i and Ri are polynomials with constant term 1. These latter may be written as Q i (u) = Ri (u) = (2.18) (1 − ua)qi,a , (1 − ua)ri,a , a∈C=0

a∈C=0

and this allows one to assign to γ a monomial q −r mγ = Yi,ai,a i,a

(2.19)

i∈I,a∈C=0

in variables (Yi,a )i∈I ;a∈C=0 . The q-character map χq [FR98] is the injective homomorphism of rings

±1 g)) −→ Z Yi,a (2.20) χq : Rep(Uq ( i∈I,a∈C=0


795

defined by4 χq (V ) =

γ

dim Vγ mγ .

(2.21)

±1 The Yi,a are to be thought of as the quantum-affine analogues of the usual variables ±1 ±1 with y j = e±λ j appearing in character polynomials. In particular, one associates Yi,a the classical weight ±λi . An algorithm for computing q-characters of Uq ( g)-modules directly from the root-system data of g was proposed in [FR98,FM01]. It has been proven to work for all fundamental representations [FM01], which is all that we shall require in the present paper, although it is known not to work in general [HL09,NN08]. In [Nak04], Nakajima deduced an algorithm for computing the q-character of any irreducible representation, and formulas for the q-characters of fundamental representations were given in [Nak03a,Nak06]; see also [CM06]. We now turn to summarizing the properties of q-characters that we shall need. A

±1 monomial in Z Yi,a is said to be i-dominant if and only if it contains no

i∈I ; a∈C=0 −1 Yi,a ’s. It is said to be dominant if and only if it is i-dominant for all i monomials are similarly defined to be those not containing Yi,a ’s.

∈ I . Antidominant

2.2. Uq ( sl2 ) characters. We first summarize the situation for Uq ( sl2 ) characters. In ±1 this case the Dynkin diagram has one node, I = {1}, and we write Y1,a = Ya±1 . The sl2 ) have dimension two and are labelled by the fundamental representations Va of Uq ( rapidity a ∈ C=0 . Their q-characters are −1 −1 = Ya + Yaq χq (Va ) = Ya 1 + Aaq 2,

(2.22)

Aa = Yaq Yaq −1 .

(2.23)

where one defines

The tensor product Vb ⊗ Vc of two fundamental representations is irreducible whenever b/c ∈ / {q −2 , q +2 }. When b = aq and c = aq −1 for some a ∈ C=0 , there is an exact sequence of Uq ( sl2 )-modules ([CP91], and with their choice of coproduct) 0 → Wa(2) → Vaq ⊗ Vaq −1 → C → 0,

(2.24)

(2) where Wa(2) is a 3-dimensional irreducible submodule and C ∼ = Vaq ⊗ Vaq −1 Wa is the 1-dimensional module. If instead b = aq −1 and c = aq, one has the same exact sequence but with arrows reversed: 4 Note that the original definition of χ [FR98] was in terms of the universal R-matrix of U ( q q g), which makes its close relationship to the transfer matrix of physics more evident. But the above definition, cf. e.g. [CH], is more directly suited for our purposes.

796


0 → C → Vaq −1 ⊗ Vaq → Wa(2) → 0.

(2.25)

In either case, there is more than one dominant monomial in the q-character: χq (Vaq −1 ⊗ Vaq ) = χq (Vaq ⊗ Vaq −1 ) = χq (Vaq −1 )χq (Vaq ) −1 −1 Yaq + Yaq = Yaq −1 + Yaq 3 −1 −1 −1 = 1 + Yaq −1 Yaq + Yaq −1 Yaq + Y Y 3 aq aq 3 . (2.26) (2)

In the final line the quantity in brackets is χq (Wa ). More generally, for each r ∈ Z≥1 and a ∈ C=0 there is an irreducible submodule Wa(r ) ⊂ Vaq r −1 ⊗ Vaq r −3 ⊗ · · · ⊗ Vaq −r +1

(2.27)

sl2 ). It has dimension r + 1 and called the r -th Kirillov-Reshetikhin module of Uq ( 5 q-character r −1 −1 −1 −1 (r ) χq (Wa ) = Yaq −r +1 Yaq −r +3 . . . Yaq r −1 1 + Aaq r Aaq r −2 . . . Aaq r −2t t=0

=

Yaq −r +1 Yaq −r +3 . . . Yaq r −3 Yaq r −1 −1 + Yaq −r +1 Yaq −r +3 . . . Yaq r −3 Yaq r +1 ... −1 −1 −1 +Yaq −r +1 Yaq −r +5 . . . Yaq r −1 Yaq r +1

+

−1 −1 −1 −1 Yaq −r +3 Yaq −r +5 . . . Yaq r −1 Yaq r +1 .

(2.28)

Wa(r ) is completely characterised by the set of rapidities Sr (a) = aq −r +1 , aq −r +3 , . . . , aq r −1 appearing in its dominant monomial, which we shall refer to as a segment of length r centred on a. Two such segments are said to be in special position if their union is itself a segment and neither of them contains the other. We say aq −r +1 is the leftmost element of Sr (a), aq r −1 the rightmost. More generally we say that aq k is to the right (left) of aq l iff k > l (resp. k < l). Presented with any dominant monomial m + = s Yas one can reconstruct the unique irreducible Uq ( sl2 )-module V (m + ) such that m + is the highest weight monomial in χq (V (m + )). First split the factors Yas into a product of segments no two of which are in special position: say Yat q −rt +1 Yat q −rt +3 . . . Yat q rt −1 , (2.29) m+ = t∈T

for some index set T ; then V (m + ) ∼ =

Wa(rt t ) ,

(2.30)

t∈T (r )

5 W a is the pull-back of the usual spin r/2 representation of Uq (sl2 ) by the evaluation homomorphism 2 ) → Uq (sl2 ). See e.g [CP91]. eva : Uq (sl


797

which can be shown to be irreducible and, up to isomorphism, independent of the ordering of the tensor factors. Finally, there is an important caveat: reducible modules certainly have more than one dominant monomial, as in e.g. (2.26), but irreducible modules can also have multiple dominant monomials. This happens precisely when they fail to be regular, in the termi2 nology of [FR98]. Consider m + = Yaq −1 Yaq to see the problem. Note that the resulting

−1 . Thus, q-character contains the (dominant) monomial Yaq −1 but not the monomial Yaq in computing q-characters, one cannot treat all dominant monomials as though they were highest monomials. For that reason we shall need the following

Proposition 2.1. Let V be a simple finite dimensional Uq ( sl2 )-module of type 1. Suppose that for some a ∈ C=0 and n > 0, χq (V ) includes a dominant monomial m such that Yan is a factor6 of m and Yaq 2 is not. Then, either −p

i) χq (V ) includes the monomials m Aaq , 1 ≤ p ≤ n; or ii) there exists a k > 0 such that χq (V ) includes the monomial m Aaq k . Proof. Let T be an index set such that V can be written as in (2.30) above, with the Srt (at ) in pairwise general position and rt > 0 for all t ∈ T . By hypothesis there exist (r ) (m t )t∈T such that m t is a monomial of χq (Wat t ) for each t ∈ T and m=

mt .

(2.31)

t∈T −1 Let T = {t ∈ T : a ∈ Srt (at )}. Note that Yaq 2 is a factor of m t only if m t ∈ T . T is the disjoint union of the following three subsets:

T1 = {t ∈ T : m t is dominant and has both Ya and Yaq 2 as factors}, T2 = {t ∈ T : m t is dominant and has rightmost factor Ya }, T3 = {t ∈ T : m t is not dominant}. If there is a t ∈ T3 such that Ya−1 is not a factor of m t then the leftmost factor Y −1 in m t (rt ) −1 is Yaq 2 for some > 0. In that case m t Aaq 2 −1 appears in χq (Wat ), cf. (2.28), and ii) holds. It remains to consider the case that Ya−1 is a factor of every m t , t ∈ T3 . By defini−1 tion of T , Yaq 2 is then also a factor of every m t , t ∈ T3 . Suppose for a contradiction that there existed a t ∈ T such that m t is dominant with leftmost factor Yaq 2 . Since by assumption the total power of Yaq 2 in m is zero, that would require |T1 | < |T3 |; but also, by definition of general position, that |T2 | = 0 and hence |T1 | − |T3 | ≥ n > 0, a contradiction. Therefore there is no such t ∈ T and so in fact, by counting powers of Yaq 2 in m, |T1 | = |T3 |. Consequently the power of Ya in t∈T T m t is zero. It follows 1 3 that |T2 | ≥ n and hence that i) holds. uc 6 For every b ∈ C , k ∈ Z , we say that Y k is a factor of m = =0 =0 c∈C=0 Yc iff either u b ≥ k > 0 or b

u b ≤ k < 0.

798


2.3. Uq ( g) characters. Returning to the general case, we let Vi,a , i ∈ I, a ∈ C=0 denote the i th fundamental representation of Uq ( g) at rapidity a. (See e.g. [CP].) It may be shown [FR98,FM01] that χq (Vi,a ) contains the highest weight monomial Yi,a and that, if we define −1 Ai,a = Yi,aq −1 Yi,aq Y j,a , (2.32)

j,i

where the product j,i is over the nodes j of the Dynkin diagram that neighbour i,7 then every monomial in χq (Vi,a ) is of the form −1 Yi,a A−1 j1 ,a1 . . . A jn ,an

(2.34)

for some finite collection of n ≥ 0 pairs ( jk , ak ) ∈ I × C=0 . For each j ∈ I , let ( j) ± Uq ( sl2 ( j) ) ⊂ Uq ( g) be the subalgebra generated by x ± be the j (u), φ j (u). Let χq sl2 ( j) ) and q-character map of Uq (

±1 ±1 → Z Y j,a (2.35) β j : Z Yi,a i∈I ;a∈C=0

a∈C=0

±1 the ring homomorphism which sets to one all the Yk,a with k = j. Then every Uq ( g)( j) ( j) module V is also a Uq (sl2 )-module, and χq (V ) = β j ◦ χq (V ). In fact, more is true: there exists [FM01] an injective ring homomorphism

±1 ±1 ±1 τ j : Z Yi,a → Z Y j,a ⊗ Z Z k,b (2.36) i∈I ;a∈C=0

a∈C=0

k= j;b∈C=0

±1 refining β j , where Z i,a are certain new formal variables, and ( j) χq (V p ) ⊗ N p , τ j (χq (Vi,a )) =

(2.37)

p ±1 where the V p are Uq ( sl2 ( j) )-modules and the N p are monomials in (Z k,b )k= j,b∈C=0 . Furthermore, in the diagram

(2.38)

let the right vertical arrow be multiplication by β j (A−1 j,c )⊗1; then the diagram commutes

if and only if the left vertical arrow is multiplication by A−1 j,c . 7 That is

j,i = j:I ji =1 , where Ii j = 2δi j − ai j = is the incidence matrix.

1 0

if i, j are neighbouring nodes on the Dynkin diagram otherwise

(2.33)


799

Consequently, if one has found a term m + ⊗ N p in the r.h.s of (2.37), and one knows ( j) that m + is the highest weight monomial of χq (V p ), then one can construct all the ( j) remaining monomials in χq (V p ) ⊗ N p (as discussed in the previous subsection) and hence their (unique) preimages in χq (Vi,a ). Frenkel and Mukhin gave an algorithm for computing the q-character with a given highest monomial [FM01], by repeatedly completing Uq ( sl2 )-characters in this way. They proved that it works for any q-character with a unique dominant monomial (and so in particular for the q-characters of fundamental representations). The specific instance of this sort of reasoning which we will require, in Proposition 3.5, is the following, which follows immediately from the existence and property (2.38) of τ j together with Proposition 2.1 above. Proposition 2.2. Let j ∈ I, a ∈ C=0 and n > 0. Suppose m is a j-dominant monomial in χq (Vi,a ) such that Yan is a factor of β j (m) and Yaq 2 is not. Then, either −p

i) χq (Vi,a ) includes the monomials m A j,aq , 1 ≤ p ≤ n; or ii) there exists a k > 0 such that χq (Vi,a ) includes the monomial m A j,aq k . Also, in Proposition 3.3 below, we will need the following consequence of the Frenkel-Mukhin algorithm. Theorem 2.3 ([FM01]). Every monomial m = Yi,a in χq (Vi,a ) is of the form m A−1 j,aq r +1 for some j ∈ I and some r ∈ Z, where m is a monomial in χq (Vi,a ) having Y j,aq r as a factor. Equivalently but more intuitively, every monomial apart from the highest one is obtained from some (at least one) other monomial by a “lowering step” consisting of a replacement of the form −1 Y j,aq r → Y j,aq r A−1 = Y Yk,aq r +1 . (2.39) r +1 r +2 j,aq j,aq

k, j

The q-characters of any fundamental representation Vi,a thus has the structure of a connected directed graph, whose nodes are the monomials and whose edges are labelled by −1 . (An example is shown in Fig. 2.) factors Ai,a Finally, in the proof of Lemma 3.2, we will need the following results from [FM01]. We shall say that a monomial m has compact support of length n and base d if m ∈ ±1 Z[Yl,dq r ]l∈I,0≤r ≤n . Combining Lemma 6.1 and 6.13 of [FM01], we have Lemma 2.4. All the monomials in χq (Vl,d ), where l ∈ I and d ∈ C=0 , have compact support of length h and base d. Moreover, a monomial m=

p

l,r Yl,dq r ,

pl,r ∈ Z

(2.40)

(l,r )∈I ×N0

having compact support of length n and base d is said to be right negative (resp. left positive) if, in addition, there exists a (k, s) ∈ I × N0 such that pk,s < 0 (resp. pk,s > 0) and for each (l, r ) ∈ I × N0 such that pl,r > 0 (resp. pl,r < 0), r < s (resp. r > s). Lemma 2.5. For all i ∈ I, a ∈ C=0 , in the q-character χq (Vi,a ) i) every monomial except for the highest weight monomial, Yi,a , is right negative, and ii) every monomial except for the lowest weight monomial, Yı¯−1 , is left positive. ,aq h

800


Proof. Part 2.5) is Lemma 6.5 in [FM01]. Proposition 6.18 in [FM01] states (in the ±1 simply-laced case) that χq (Vı¯,aq −h ) and χq (Vi,a ) are related by exchanging Y j,aq n ↔

∓1 Y j,aq −n , for all j ∈ I, n ∈ {0, 1, . . . , h}. This map sends right-negative monomials to left-positive monomials (and vice versa). So part 2.5) for χq (Vı¯,aq −h ) implies part ii) for χq (Vi,a ).

Corollary 2.6. Let i ∈ I and a ∈ C=0 . The monomial 1 does not occur in χq (Vi,a ). Proof. The monomial 1 is not right negative and 1 = Yi,a . Thus, by Lemma 2.5, it cannot appear in χq (Vi,a ). 3. Coxeter Orbits and q-Characters In this section we relate the geometry of the Coxeter orbits of g-weights to the structure of q-characters of fundamental representations. Recalling our notations for the roots, weights and Coxeter element of g from the Introduction, let us begin by noting the following identities. Write λi = λi• (λi◦ ) when i ∈ I• (respectively I◦ ). Then λ◦j , w• λi◦ = λi◦ (3.1) w• λi• = λi• − αi = −λi• +

j,i

and likewise with ◦ ↔ •. Thus

◦ • 1 + w ±1 λi• = λ◦j .

(3.2)

j,i

We also define 2 P= h

n∈Z/ h Z

2π n cos h

wn ,

(3.3)

which is the orthogonal (with respect to the Killing form ·, ·) projector from the weight lattice of g to the exp (±2πi/ h)-eigenplane of w.8 Let θ be the map which returns the signed angle between the projections of two given vectors in weight space into this plane, i.e. the map defined by

Pμ, Pρ cos θ (μ, ρ) = √ ;

Pμ, Pμ Pρ, Pρ

im(θ ) = (−π, π ]

(3.4)

and, to fix the orientation, θ (μ, wμ) = +2π/ h. To fix a direction in the plane, let λ be any vector in weight space such that Pλ = 0. Our main result is then Theorem 3.1. Let i 1 , i 2 , i 3 ∈ I and a1 , a2 , a3 ∈ C=0 . The following are equivalent: i) The q-character χq Vi1 ,a1 ⊗ Vi2 ,a2 ⊗ Vi3 ,a3

(3.5)

includes the monomial 1. 8 Recall that the exponents of g are by definition those integers s ∈ Z/ hZ for which exp (2πis/ h) is an

eigenvalue of w, and that s = ±1 are always exponents.


801

ii) There exist n 1 , n 2 , n 3 ∈ Z and a ∈ C=0 such that w n 1 λi1 + w n 2 λi2 + w n 3 λi3 = 0

(3.6)

and h

ak = aq π θ

λ,wn k λi k

, k = 1, 2, 3.

(3.7)

Let us illustrate this with an example in the case of E 6 , for which the Coxeter number is h = 12. We label the nodes of the Dynkin diagram as in [BCDS90]:

(This labelling is related to the masses of the corresponding particles, H/heavy or L/light), in the Toda theory.) Among the solutions to the fusing rule (tabulated in [BCDS90]) is w −2 λ¯l + λL + w 5 λh = 0,

(3.8)

whose P-projection may be pictured as follows:

(3.9)

So the theorem asserts, in particular, that 1 occurs in the q-character χq V¯l,aq −5 ⊗ VL,a ⊗ Vh,aq 10 .

(3.10)

Proof of Theorem 3.1. We first express ii) in a less symmetric but more convenient form. The reference vector λ serves purely to make manifest the symmetry under permutations of {1, 2, 3}. It follows from (3.6) that, by using this symmetry if necessary, we can assume − π < θ (λi1 , w n 2 λi2 ) ≤ 0 < θ (λi1 , w n 3 λi3 ) ≤ π.

(3.11)

Then, by the freedom in the choice of a, we can assume that λ = λi1 and n 1 = 0. Let us also pick the two-colouring I = I• I◦ such that i 1 ∈ I• . Given that −λı¯2 is in the

802


Fig. 1. Picture of the e±2πi/ h -eigenplane of w, for h = 5 (left) and h = 6 (right) showing the directions (though not the lengths) of the projected Coxeter orbits of fundamental weights. Here λ• (λ◦ ) denotes any λi such that i ∈ I• (respectively I◦ )

Coxeter orbit of λi2 ,9 we can introduce an n ∈ Z such that w n 2 λi2 = −w n λı¯2 . Let us also write m := n 3 . Then (3.11) becomes 0 < θ (λi1 , w n λı¯2 ) ≤ π, 0 < θ (λi1 , w m λi3 ) ≤ π.

(3.12)

Thus the solution ii) has been brought to the form λi1 − w n λı¯2 + w m λi3 = 0

(3.13)

where, on examining Fig. 1, one sees that (3.12) is equivalent to the following conditions on n, m (modulo h): h2 h2 ı¯2 ∈ I• i 3 ∈ I• 0 0, vanish in the expression crj Y j,r − grj A j,r ? (3.39) E (c, g) := ( j,r )∈ Iˆ•

( j,r )∈ Iˆ◦

Or, equivalently, when is the factor Yi,aq n absent from the monomial ⎛ ⎞⎛ ⎞ cr −gr j⎠ ? m(c, g) := ⎝ Y j,aq r j ⎠ ⎝ A j,aq r ( j,r )∈ Iˆ•

(3.40)

( j,r )∈ Iˆ◦

It is clear that the answer is: if and only if gin−1 + gin+1 −

g nj = cin .

(3.41)

j,i

Let us regard c as a fixed source term. Then it is possible to satisfy (3.41) at every black node (i, n) with n > 0 by choosing an appropriate g. Assume that sufficiently far down the graph the source vanishes, i.e. that there is an N such that cin = 0 for all n > N . Then, furthermore, the solution is unique if we specify also that gin = 0 for all n > N , because Eq. (3.41) at each row n fixes uniquely the gin−1 in the row above. Proposition 3.4. Choose the two-colouring of I such that i 1 ∈ I• . Suppose that we have a solution to the fusing equation (3.6), written, as in (3.13), in the form λi1 − w n λı¯2 + w m λi3 = 0,

(3.42)

with n, m ∈ Z subject to (3.14). Then there exists a unique g : Iˆ◦ → Z such that −gt j , t Yı¯2 ,aq r Yi−1 A (3.43) s = Yi 1 ,a j,aq ,aq 3 ( j,t)∈ Iˆ◦

where r, s ∈ Z are as in (3.16), and such that, for some N ∈ N, gin = 0 for all n > N . Proof. Let c be the source function that vanishes everywhere except ci01 = +1,

cır¯2 = −1,

cis3 = +1.

(3.44)

Note that then (3.42) is E (c, 0) = 0. Consider solving (3.41) for g in the manner given above. The resulting expression E (c, g) has by construction no terms Yi,n with n > 0. So it can only be a linear combination of the Yi,0 = λi• and 10 Yi,−1 = λi◦ . But of course E (c, g) = 0 identically, since all we have done is to add various re-writings of zero (the A’s) to an expression (3.42) which was zero to begin with. Therefore, since the λi are linearly independent, the identity E (c, g) = 0 must be trivial, in the sense that the expression on the right-hand side of (3.39) consists entirely of cancelling pairs of terms and vanishes without appealing to properties of the Coxeter element. Consequently, we have also that m(c, g) = 1, which, on rearranging, is (3.43) as required. 10 For this proof only, we consider working on I × {−1, 0, 1 . . . }.

808


Fig. 4. Two copies of Iˆ in the case g = d5 , showing the solutions to the problem (3.41) for the source functions c associated, as in Proposition 3.4, to the identities shown. ⊕ denotes a node at which g = +1; elsewhere g=0

The right-hand side of (3.43) is of the right form to be a monomial in χq (Vi1 ,a ), cf. (2.34), but we are by no means done. A priori, it is perhaps not even clear from the procedure above that the gin need all be non-negative: indeed, although we stated the proposition for identities involving three terms, (cf. Fig. 4), the obvious generalization to arbitrary identities of the form (3.24) is valid, but the resulting gin are not all non-negative in general. Nonetheless, Proposition 3.5. Under the assumptions of the preceding proposition, the monomial Yı¯2 ,aq r Yi−1 s 3 ,aq

(3.45)

of (3.43) occurs in χq (Vi1 ,a ). Proof. First consider the following iterative procedure which generates a finite sequence m 0 , m 1 , . . . , m h−1 of monomials in χq (Vi1 ,a ). We set m 0 = Yi1 ,a . Roughly speaking, the idea is to lower fully in all black directions to obtain m 1 , then lower fully in all white directions to obtain m 2 , and so on. More precisely, suppose that for some even p ≥ 0 we have found an m p in χq (Vi1 ,a ) of the form ⎛ m p = ⎝

⎞⎛ bi ⎠⎝ Yi,aq p

i∈I•

⎞−1 bi ⎠ Yi,aq p+1

(3.46)

i∈I◦

for some non-negative integers bi , i ∈ I . Certainly (cf. 2.34) −gt j A j,aq t m p = Yi1 ,a ( j,t)∈ Iˆ◦

(3.47)


809

for some g tj ≥ 0 with, in view of (3.46), g tj = 0 ∀ t > p. Thus for all k > 0 and i ∈ I, m p Ai,aq p+k is not of the form (2.34) and so cannot be in χq (Vi1 ,a ). Proposition

−bi 2.2 thus guarantees that m p Ai,aq p+1 is in χq (Vi 1 ,a ), for i ∈ I• . By similar reasoning for each black direction in turn, we have that χq (Vi1 ,a ) contains −b m p+1 = m p Ai,aqi p+1 . (3.48) i∈I•

It too is of the form (3.46), but with p odd and the roles of black and white exchanged. With the obvious colour swaps, we then iterate. As stated, the iteration proceeds until we arrive at the lowest monomial m h−1 = −1 Yı¯ ,aq h of χq (Vi1 ,a ).11 The key observation is that, for all p ≤ h − 1, the g tj of (3.47) 1 solve the problem (3.41) in rows 1, 2 . . . , p − 1, for the source function c defined to be zero everywhere except for c0i1 = +1, and the initial conditions g 0i = 0 ∀i ∈ I . Note that for all p ≤ h − 1 the g tj of (3.47) are non-negative in rows 1, 2, . . . , p − 1; this is clear from their character-theoretic construction, and is a fact about the solution to (3.41) for the source c and initial conditions g 0i = 0 ∀i ∈ I that is not otherwise manifest. Now let g and c be the functions of the proof of Proposition 3.4. Since ∀n < r, cin = cni ,

(3.49)

and because each gin is determined by the values of c and g in rows above (when we think of solving from row 0 downwards), we have ∀n ≤ r, gin = g ni .

(3.50)

In particular, the gin are non-negative for all n ≤ r . On the other hand, by imagining turning the diagram upside-down and applying the same argument starting from the +1 source in row s, we conclude also that the gin are non-negative for all n ≥ r . Therefore all the gin are non-negative. (Note that this trick would not work if c were non-zero at more than three nodes.) Furthermore, again thinking of solving from row 0 downwards, crı¯2 = cır¯2 + 1

⇒

g rı¯2+1 = gı¯r2+1 + 1 > 0.

(3.51)

This relation is crucial, because if we are to obtain the desired quadratic monomial (3.43), we must modify the procedure on reaching row r : we set m 1 = m 1 , . . . , m r = m r , but then rather than lowering m r completely in the direction ı¯2 , we want to preserve one factor of Yı¯2 ,aq r – and the above inequality guarantees that there is at least one such factor. That is, if ⎛ ⎞⎛ ⎞−1 b b i i ⎠⎝ ⎠ , mr = ⎝ Yi,aq Yi,aq (3.52) r r +1 i∈I•

i∈I◦

11 This sequence of “lowering steps” is of the general type mentioned in [Her07b], Remark 2.16. Note that this particular sequence picks out a route through the graph of χq (Vi 1 ,a ), from the highest to the lowest 2 ) Kirillov-Reshetikhin modules, in the sense that at each lowering monomial, that avoids non-trivial Uq (sl 2 )-character is that of an (irreducible) tensor product of fundamental representations at step the relevant Uq (sl coincident rapidity. It is also interesting to note that the monomials m 0 , m 1 , . . . , m h−1 have the property that the sequence of their classical weights is a permutation of the Coxeter orbit of the highest weight λi 1 .

810


supposing in what follows, for the sake of definiteness, that ı¯2 ∈ I• , then we are guaranteed that bı¯2 ≥ 1. Setting n = bı¯2 , p = bı¯2 − 1 in Proposition 2.2 we deduce that −gt −bı¯ +1 −bi j m r +1 := m r Aı¯ ,aq2 r +1 A j,aq t Ai,aq (3.53) r +1 = Yi 1 ,a 2

i∈I• \{¯ı 2 }

( j,v)∈ Iˆ◦ :t≤r +1

is a monomial in χq (Vi1 ,a ). We would then like to continue to apply the above alternating black/white lowering procedure in subsequent rows, preserving the prefactor Yı¯2 ,aq r at each step. Once more we shall argue that this is possible by a finite recursion. Consider a white lowering step: suppose that for some odd p with s > p ≥ r + 1 we have shown that ⎛ ⎞⎛ ⎞−1 b b −gt i i j = Y ⎠⎝ ⎠ (3.54) m p := Yi1 ,a A j,aq t Yi,aq Yi,aq p ı¯2 ,aq r ⎝ p+1 ( j,t)∈ Iˆ◦ :t≤ p

i∈I◦

i∈I•

is a monomial in χq (Vi1 ,a ), for certain bi ∈ Z, i ∈ I . To begin the recursion, this is cerp+1 tainly true for p = r + 1, as in (3.53). Now observe that in fact, for all i ∈ I◦ , bi = gi (this is clear when thinking of solving for g row-by-row from row 0 downwards) and that these are non-negative as noted above. Thus we can lower in all white directions as before and find that −g p+1 j m p+1 := m p A j,aq p+1 (3.55) ( j, p+1)∈ Iˆ◦

is also a monomial in χq (Vi1 ,a ). This completes the white inductive step. For the black step, lowering in the directions I• \ {¯ı 2 } works in exactly the same way. It remains only to check that the lowering step in the direction ı¯2 is also valid: but this is clear because p+2 m p+1 is an ı¯2 -dominant monomial and βı¯2 (m p+1 ) = Yı¯2 ,aq r Yı¯n ,aq p+1 with n = gı¯2 ≥ 0, 2 which is still of the correct form to apply Proposition 2.2. Iterating, we have that every monomial in the sequence −gt j Yi1 ,a A j,aq t for p = 1, 2, . . . , s (3.56) ( j,v)∈ Iˆ◦ :t≤ p

is in χq (Vi1 ,a ). Finally then, at row s, we indeed arrive at −gt −1 j = Y A j,aq t Yi1 ,a ı¯2 ,aq r Yi 3 ,aq s ,

(3.57)

( j,t)∈ Iˆ◦

which is the required monomial.

4. Outlook It is an immediate corollary of our main result, Theorem 3.1, that Dorey’s rule provides a necessary condition for HomUq (g) Vi,a ⊗ V j,b ⊗ Vk,c , C = 0. We have not, however, given a general proof here of sufficiency; and it may be that such a proof would require more knowledge about the structure of fundamental Uq ( g)-modules than their q-characters alone provide. The correct statement should be the following. Under the


811

conditions of Theorem 3.1, the ordered triple of vectors (w n 1 λi1 , w n 2 λi2 , w n 3 λi3 ) can be said to be either cyclic or acyclic according to the order in which their projections occur in the oriented s = 1 eigenplane of w, cf. (3.4). In the example following the theorem, (w −2 λ¯l , λL , w 5 λh ) is cyclic, for instance. It should be that, in the cyclic case, HomUq (g) (C, Vi1 ,a1 ⊗Vi2 ,a2 ⊗Vi3 ,a3 ) = 0 and HomUq (g) (Vi3 ,a3 ⊗Vi2 ,a2 ⊗Vi1 ,a1 , C) = 0. (For the a- and d-series, one may verify that this statement indeed unpacks to give Theorems 6.1 and 7.1 of [CP96]. There the proof proceeds by induction on the rank, and relies on specific properties of these root systems.) Now, as mentioned in Sect. 2, there is a “current” Hopf algebra structure on Uq ( g), originally due to Drinfel’d. It restricts, over the quantum loop algebra, to the following relations: (φi± (u)) = φi± (u) ⊗ φi± (u),

(4.1)

(xi+ (u)) (xi− (u)) S(φi± (u)) S(xi+ (u)) (φi± (u))

(4.2)

= = = = =

1 ⊗ xi+ (u) + xi+ (u) ⊗ φi− (1/u), xi− (u) ⊗ 1 + φi+ (1/u) ⊗ xi− (u), φi± (u)−1 , −xi+ (u)φi− (1/u), S(xi− (u)) = 1, (xi± (u)) = 0,

(4.3) (4.4) −φi+ (1/u)xi− (u),

(4.5) (4.6)

where is the coproduct, S the antipode and the counit. This Hopf algebra structure is twist-equivalent to the standard one in a sense given in [EKP07]; note that the infinite sums on the right of the coproducts above require careful interpretation [Her05,Her07a, Gro01]. With respect to this “current” Hopf algebra structure,it is clear that the singlet −1 state in Vi1 ,a1 ⊗ Vi2 ,a2 ⊗ Vi3 ,a3 must be of the form |Yı¯−1 h ⊗ Yı¯1 ,a1 q h Yi 3 ,a3 ⊗ Yi 3 ,a3 ,a q 1 1 – where the first and last tensor factors are the lowest and highest weight vectors of the respective representations, and the middle factor is an eigenvector of φi± (u) with l-weight corresponding to the monomial shown. Finally, let us remark that it would be interesting to investigate whether generalizations of our results exist for representations other than the fundamental ones (as was suggested in [EKMY05] based on the structure of local charges in certain integra(k) ble sigma models). The natural candidates are the Kirillov-Reshetikhin modules Wi,a , which can be thought of as the “minimal affinizations” [CP95] of the highest weight representations Vkλi of g and for which the Frenkel-Mukhin algorithm is known to work [Nak03b,Her06]. The form of our arguments suggests that such generalizations may be possible, perhaps using the braid group actions of [Bec94,Cha02] to lift the periodicity of the Coxeter element. Acknowledgemets. We are grateful to Patrick Dorey and Niall MacKay for valuable discussions and suggestions. During much of the preparation of this work, C.A.S.Y. was funded by the Leverhulme Trust and R.Z. by an EPSRC postdoctoral fellowship. C.A.S.Y. is funded by a fellowship from the Japan Society for the Promotion of Science.

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Communications in


Breather Solutions in Periodic Media Carsten Blank, Martina Chirilus-Bruckner, Vincent Lescarret, Guido Schneider Institut für Analysis, Dynamik und Modellierung, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany. E-mail: [email protected] Received: 25 February 2010 / Accepted: 16 September 2010 Published online: 1 February 2011 – © Springer-Verlag 2011

Abstract: For nonlinear wave equations existence proofs for breathers are very rare. In the spatially homogeneous case up to rescaling the sine-Gordon equation ∂t2 u = ∂x2 u − sin(u) is the only nonlinear wave equation which is known to possess breather solutions. For nonlinear wave equations in periodic media no examples of breather solutions have been known so far. Using spatial dynamics, center manifold theory and bifurcation theory for periodic systems we construct for the first time such time periodic solutions of finite energy for a nonlinear wave equation s(x)∂t2 u(x, t) = ∂x2 u(x, t) − q(x)u(x, t) + r (x)u(x, t)3 , with spatially periodic coefficients s, q, and r on the real axis. Such breather solutions play an important role in theoretical scenarios where photonic crystals are used as optical storage. Contents 1.

2. 3. 4. 5.

6.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 The construction . . . . . . . . . . . . . . . . . . . . . . 1.2 On the spectral assumption (Spec) . . . . . . . . . . . . 1.3 Some remarks . . . . . . . . . . . . . . . . . . . . . . . Symmetries of the Spatial Dynamics Formulation . . . . . . Application of Floquet’s Theory . . . . . . . . . . . . . . . . An Example for a Suitable Choice of s = s(x) . . . . . . . . . The Reversibility . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Preparations . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The reversible change of variables . . . . . . . . . . . . 5.3 Conjugation of the old and the new reversibility operator The Center Manifold Reduction . . . . . . . . . . . . . . . .

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816 817 818 820 823 824 825 828 828 829 831 832

816

C. Blank, M. Chirilus-Bruckner, V. Lescarret, G. Schneider

7. Construction of a Homoclinic Solution . . . . . . . . . . . . . . . . . . . . 835 8. Persistence of the Homoclinic Solution . . . . . . . . . . . . . . . . . . . . 837 A. The Physical Motivation: Photonic Crystals as Optical Storage . . . . . . . . 838 1. Introduction We consider a nonlinear wave equation of the form s(x)∂t2 u(x, t) = ∂x2 u(x, t) − q(x)u(x, t) + r (x)u(x, t)3 ,

(1.1)

with x ∈ R, t ∈ R, u(x, t) ∈ R and a-periodic real-valued coefficient functions s, q and r , i.e., s(x) = s(x + a), q(x) = q(x + a), and r (x) = r (x + a), where w.l.o.g. in the following a = 1. It is the purpose of this paper to give an example of coefficient functions s, q, and r such that (1.1) possesses breather solutions u, i.e., for this choice of coefficient functions we prove the existence of spatially localized, 2π/ω-time periodic solutions of finite energy. With χ M being the characteristic function of the set M our result is as follows. Theorem 1.1. Let s(x) = χ[0,6/13] + 16χ(6/13,7/13) + χ[7/13,1] (x mod 1), q(x) = (q0 − ε2 )s(x) with q0 ∈ R defined explicitly in (4.5), and r (x) = 1. Then there exist an ε0 > 0 and a C > 0 such that for all ε ∈ (0, ε0 ), Eq. (1.1) possesses breather solutions with minimal period 2π/ω∗ , where ω∗ = 13π/16, i.e., there are solutions u : R × R → R of (1.1) which satisfy for a β > 0 that lim u(x, t)eβ|x| = 0, ∀t ∈ R,

|x|→∞

u(x, t) = u(x, t + 2π/ω∗ ), ∀x, t ∈ R, and

(1.2)

sup u(x, t) − u app (x, t) ≤ Cε2 ,

(1.3)

u app (x, t) = εη1 sech(εη2 x)w1 (π, x)eiπ x eiω∗ t + c.c.,

(1.4)

x,t∈R

where

with constants η1 , η2 and a 1-periodic function w1 (π, ·) which are all defined subsequently in Remark 1.5. The solution is C σ w.r.t. t for every fixed σ , but only piecewise smooth w.r.t. x and C 1 w.r.t. x at the jumps of s. A sketch of such a solution can be found in Fig. 1. The novelty of this result is as follows. For spatially homogeneous nonlinear wave equations up to rescaling the sine-Gordon equation ∂t2 u = ∂x2 u − sin(u) is the only nonlinear wave equation which is known to possess breather solutions. For nonlinear wave equations in periodic media the situation is expected to be different. However, no examples of breather solutions have been known so far. Using spatial dynamics, center

Breather Solutions in Periodic Media

817

Fig. 1. A breather solution in periodic media. The wavelength of the carrier wave and of the medium are of a comparable order

manifold theory and bifurcation theory for periodic systems, we construct for the first time such time periodic solutions of finite energy for a nonlinear wave equation with spatially periodic coefficients. Note that our method of construction heavily relies on the subsequent condition (Spec). The special choice for s in Theorem 1.1 is a carefully tuned example which fulfills this condition. It is not clear at this point if more such examples do exist which are not a trivial adaption of the present one, in other words, the genericity of breathers in periodic media is still an open question. Before we explain the relevance of Theorem 1.1 in applications, as for the use of photonic crystals as optical storage, and generalizations of the result, we explain our method to construct such solutions and the major mathematical difficulty associated with the problem. 1.1. The construction. The subsequent explanations are made for general 1-periodic coefficient functions s = s(x), q = q(x), and r = r (x) in order to put the problem in a general framework which allows us to discuss subsequently possible generalizations of Theorem 1.1. For the construction of the breather solutions we will use spatial dynamics, center manifold theory, and bifurcation theory. Spatial dynamics means that we write (1.1) as an evolutionary system w.r.t. x ∈ R in the phase space of 2π/ω-time periodic functions, i.e., we consider ∂x u(x, t) = v(x, t), ∂x v(x, t) = s(x)∂t2 u(x, t) + q(x)u(x, t) − r (x)u(x, t)3 .

(1.5)

Due to the periodic dependence of s, q, and r on x the system is non-autonomous. Using the symmetries of the system we can restrict ourselves to solutions which are odd w.r.t. t in order to reduce the dimensionality of the existence problem by a factor 2. If the spectral assumption (Spec): The linearization of the periodic spatial dynamics system (1.5) possesses two Floquet exponents with real part zero and the rest of the Floquet spectrum is uniformly bounded away from the imaginary axis, cf. the left panel of Fig. 2, holds, by using invariant manifold theory for periodic systems the infinite-dimensional spatial dynamics system (1.5) can be reduced to a two-dimensional system on the center

818


Fig. 2. The spectral picture of the linearized spatial dynamics formulation (1.5) for ε = 0 and ε > 0. All Floquet exponents possess imaginary part iπ

manifold which is associated with the two Floquet exponents with real part zero. As will be explained in Sect. 1.2 for constant coefficients s, q and r the spectral condition (Spec) is not satisfied and as a consequence breather solutions for (1.1) cannot be constructed with the spatial dynamics method in the constant coefficient case if the minimal period w.r.t. t is non-zero, cf. Remark 1.4. Thus, for a given minimal period 2π/ω∗ w.r.t. t the coefficients s, q and r have to be suitably chosen. By moving the two central Floquet exponents from the imaginary axis, cf. Fig. 2, bifurcating homoclinic solutions can be found in the lowest order approximation of the reduced system, cf. Fig. 10 in Sect. 8. Using reversibility arguments for the reduced system finally gives the persistence of the homoclinic solution w.r.t. higher order perturbations. These homoclinic solutions of the spatial dynamics formulation (1.5) in the phase space of time-periodic solutions correspond to breather solutions in the original formulation (1.1). In order to have the reversibility of the spatial dynamics formulation (1.5), i.e. the invariance of (1.5) under (x, u) → (−x, u) the coefficient functions s = s(x), q = q(x) and r = r (x) have to be even w.r.t. x, i.e., s(x) = s(−x), q(x) = q(−x), and r (x) = r (−x). Hence the strategy to establish the existence of such solutions is very clear. However, it is not clear at all how to choose s, q, and r such that the assumption (Spec) can become true. 1.2. On the spectral assumption (Spec). It is well known that the solutions of the linearization s(x)∂t2 u = ∂x2 u − q(x)u of (1.1) at the origin are given by oscillations of Bloch modes, namely eilx wn (l, x)eiωn (l)t ,

(1.6)

with wn (l, x) = wn (l, x + 1) and curves of eigenvalues l → ωn (l), where ωn (l) ∈ R for l ∈ (−π, π ] and n ∈ Z/{0}. They are ordered such that ωn (l) ≥ ωn−1 (l). Spectral


819

Fig. 3. If the dotted line l → mω∗ falls into a spectral gap between the curves of eigenvalues plotted over the Bloch wave numbers l of the linearized time evolutionary system (left upper panel), then in the spectral picture of the linearized space evolutionary system two Floquet exponents off the imaginary axis occur (right upper panel). In the other case they are on the imaginary axis (lower panels)

gaps can occur, i.e., the set { ωn (l) | l ∈ (−π, π ], n ∈ Z/{0} } ⊂ C is in general not connected for periodic s and q, cf. [7]. In case of Schrödinger operators with periodic potential a detailed discussion about the occurence of spectral gaps can be found for instance in [28, Sect. XIII.16], especially have a look at [28, Theorem XIII.91]. There is a one-to-one correspondence between the spectral pictures of the linearizations of the time evolutionary and of the space evolutionary system (1.1) and (1.5), respectively. Using Fourier series u(x, t) = m∈Z uˆ m (x)eimω∗ t with respect to time the spectral problem of the space evolutionary system (1.5) will split into infinitely many decoupled problems (3.1) which are indexed with m ∈ Z and which each modulo 2π i will create two Floquet exponents. When the integer multiple mω∗ of the basic temporal wavenumber ω∗ falls into a spectral gap of the time evolutionary system (1.1) then modulo 2π i there are two Floquet exponents off the imaginary axis in the m th space evolutionary system (1.5). In the other case the Floquet exponents are on the imaginary axis, see Fig. 3. In order to satisfy the spectral assumption (Spec) except of two (see Sect. 2) all integer multiples mω∗ of the basic temporal wavenumber ω∗ have to fall into a spectral gap of the time evolutionary system (1.1). For smooth s the spectral gaps become smaller and smaller for larger n and, therefore, integer multiples mω∗ of ω∗ in general do not fall into spectral gaps. In [7, Theorem 4.5.3] and [23] there are estimates on the asymptotic size ln =

inf

l∈(−π,π ]

ωn+1 (l) −

sup

l∈(−π,π ]

ωn (l)

820


Fig. 4. Graph of the function x → s(x)

of the n th spectral gap for n → ∞. For the sake of clarity we restate the findings of [7, Theorem 4.5.1]: i) ii) iii)

ln = O(n) if s(x) is piecewise smooth, ln = O(1) if s (x) exists and is piecewise smooth, ln = O(n −r −1 ) if s (r +2) (x) and q (r ) all exist and are piecewise smooth.

To avoid that gaps become smaller we have to ensure that s is at most once differentiable. A particularly handy choice for s seems therefore to be a step function, since, on one hand, it will ensure that the open gaps are O(1) (due to the lack of regularity) and on the other hand the resulting band structure can be computed explicitly which is important for tuning it to fulfill the crucial condition (Spec). An example of coefficient functions s and q leading to O(1)-spectral gaps around each value (2n + 1)ω∗ with ω∗ = 13π/16 and n ∈ N are s(x) = χ[0,6/13] + 16χ(6/13,7/13) + χ[7/13,1] (x mod 1)

and

q(x) = 0,

where χ M is the characteristic function of the set M (Fig. 4). Since q according to [7, Theorem 4.5.3 ii)] does not affect the asymptotics of the spectral gaps, we can choose q to adjust two Floquet exponents on the imaginary axis without destroying the overall spectral picture. It turns out that a choice q(x) = μs(x) with μ ∈ R is sufficient in our case. See Fig. 2. Since we have chosen a cubic nonlinearity like in nonlinear optics, cf. Remark 1.2, it is sufficient that only every second gap opens in the required way. See Sect. 4 for the details. 1.3. Some remarks. The plan of the paper is as follows. After exploiting the symmetries of the spatial dynamics formulation (1.5) in Sect. 2 we recall Floquet theory for periodic systems and analyze the spectral situation for the above choice of s and q in Sects. 3 and 4. The center manifold reduction is made in Sect. 6 and the reduced system is analyzed in Sects. 7 and 8. In the Appendix we explain how (1.1) can be derived as a model for the evolution of the electric field in photonic crystals. Before we start with this plan we close the Introduction with a number of remarks mainly explaining related results in mathematics, the relevance of the result w.r.t. applications, and possible generalizations of the result. Remark 1.2. The paper is motivated by theoretical scenarios where photonic crystals are used as optical storage [5]. Photonic crystals consist of a dielectric material, for instance


821

glass, with a periodic structure with a wavelength comparable to the wavelength of light. They are suitable tools for the construction of all optical devices in photonics which is loosely speaking electronics with photons instead of electrons. Due to the vanishing group velocities, which are implied by the horizontal tangencies in the left panel of Fig. 3, in principle, photonic crystals can be used as optical storage, where the breather solution of Theorem 1.1 stands for a one in the digital encoding of information. In the Appendix we explain more details and how Eq. (1.1) can be derived as a model for the dynamics of light in photonic crystals. Remark 1.3. The construction of breather solutions is a very active field of research. Breather solutions are known to exist in various systems, and so we refrain from giving a complete overview. In discrete systems such as Hamiltonian networks of weakly coupled oscillators or nonlinear Schrödinger lattices breather solutions have been constructed for instance in [20,25]. In S 1 -symmetric continuous systems such solutions are widely known to exist. In the spatially periodic case they have been constructed for instance in [24]. In various limits connecting the discrete and the continuous case, breather solutions have been constructed, very recently in [1,2,27]. For a recent overview of existence results for breathers in lattice equations see [19]. The result which is closest to the presented result is the construction of breathers in a diatomic FPU-model in [17]. Remark 1.4. For nonlinear wave equations existence proofs for breathers are very rare. In the homogeneous case there are no spectral gaps and so according with Fig. 3 all eigenvalues of the spatial dynamics formulation lie on the imaginary axis and hence up to rescaling, the sine-Gordon equation ∂t2 u = ∂x2 u − sin(u) is the only nonlinear wave equation in homogeneous medium which is known to possess breather solutions in NLS-form [3,6]. In general only solutions with small tails at infinity have been proven to exist [10–13]. However, for periodic media the situation is different. By the coefficient functions s, q, and r we have infinitely many parameters which can be adjusted to have an intersection of the one-dimensional weakly unstable and one-dimensional weakly stable manifold which are associated to the two Floquet exponents close to the imaginary axis. Remark 1.5. Solutions of (1.1) can be approximated via the ansatz u(x, t) = ε A(ε(x − cg t), ε2 t)wn (l0 , x)eil0 x eiωn (l0 )t + c.c. with A(X, T ) ∈ C, group velocity cg = ∂T A = −i

ωn (l0 )

∈ R and 0 < ε 1 by a NLS-equation

ωn (l0 ) 2 ∂ X A + iγn (l0 )A|A|2 2

with coefficient

1

γn (l0 ) = 3

(1.7)

r (x)|wn (l0 , x)|4 dx,

(1.8)

(1.9)

0

where cg = 0 for l0 = 0, ±π . The NLS-equation possesses pulse solutions A(X, T ) = ˜ if ω (l )γ (l ) < 0 of the form displayed in (1.4). In [4] an approximation ˜ )eiωT A(X n 0 n 0 result has been established that guarantees that solutions of (1.1) can be approximated on an O(ε−2 ) time scale via even more general solutions of this NLS-equation. The approximation (1.4) is obtained from (1.7) by choosing l0 = π . By the opening of the gaps which is induced by our choice of s and q we have to choose l0 = π in order to satisfy (Spec). See Sect. 4.

822


Fig. 5. In the right panel there is an example for which ω∗ touches a band edge at l0 = ±π . This is the situation of Theorem 1.1. In the left panel there is an example for which ω∗ touches a band edge at l0 = 0. This situation cannot be handled with the coefficient function s = s(x) chosen in this paper

Remark 1.6. Theorem 1.1 is formulated for the first spectral gap at ω∗ = 13π/16 and l0 = π as can be seen in the right panel of Fig. 5. It can also be formulated for all band edges at l0 = π with the same function s = s(x). For band edges at l0 = 0 as can be seen in the left panel of Fig. 5 the function s = s(x) cannot be used, as will be explained below. Moreover, r = 1 can be replaced by arbitrary γ r (x) as long as the non-degeneracy condition γn (l0 ) = 0 is satisfied. Breather solutions exist then either for γ = −1 or γ = 1. By rescaling time, other values of ω∗ can be reached. Remark 1.7. An alternative approach to spatial dynamics and center manifold theory would be a Lyapunov-Schmidt reduction as used in [26]. The subsequent infinite set of ODEs (2.3) is considered as an elliptic problem. In the subspace of reflection symmetric solutions the linearization around the approximate pulse solution possesses no zero eigenvalue such that the implicit function theorem can be applied to prove the persistence of the homoclinic solutions under higher order perturbations. Remark 1.8. The subsequent Lemma 4.1 explains how to construct step functions s and q in order to make (1.1) to possess breather solutions. By a simple perturbation argument it is also clear that the situation is structurally stable if small smooth perturbations are added to the step functions s and q. Remark 1.9. So far we were unable to find s and q for the construction of breather solutions with l0 = 0, cf. Remark 1.5, and breather solutions for general nonlinearities, in particular for those containing quadratic terms. Remark 1.10. The assumption that the functions s, q, and r have to be even is necessary to establish Theorem 1.1 with our method. However, we strongly expect that there are breather solutions in the non-even case, too. The functions s, q, and r give infinitely many parameters which we expect to allow us to bring the one-dimensional stable and the one-dimensional unstable manifold in the two-dimensional center manifold to an intersection. However, this is a different story. Remark 1.11. We refrain from speculating about the stability of the breather solutions constructed in Theorem 1.1. We only remark that since the solutions are not very smooth w.r.t. x we expect that some nonlinear stability result will be hard to obtain, even if some linear stability result can be established.


823

Notation. Constants which can be chosen independently of the small parameter 0 < ε 1 are often denoted with the same symbol C. A solution of a non-autonomous differential equation ddux = f (u, x) with the initial condition u|x=x0 = u 0 is denoted with u = u(x, x0 , u 0 ). 2. Symmetries of the Spatial Dynamics Formulation Since we are interested in time-periodic solutions of Eq. (1.1), i.e., u(x, t + 2π/ω) = u(x, t) for all x ∈ R we use Fourier series u(x, t) = m∈Z uˆ m (x)eimω∗ t with respect to time leading to the system of countably many ODEs, ∂x2 uˆ m (x) = −s(x)m 2 ω∗2 uˆ m (x) + q(x)uˆ m (x) − r (x)gˆ m (x),

m ∈ Z,

(2.1)

where gˆ m (x) =

uˆ n 1 (x)uˆ n 2 (x)uˆ n 3 (x),

m ∈ Z.

(2.2)

n 1 ,n 2 ,n 3 ∈Z,n 1 +n 2 +n 3 =m

There are a number of linear subspaces invariant under the evolution of (2.1). These are as follows. The invariant subspace corresponding to real solutions of (2.1) is given by UR = {(uˆ n )n∈Z | uˆ n = uˆ −n }. Since the associated first order system (1.5) is invariant under the transform S : (t, u, v) → (−t, −u, −v) also Uodd = {(uˆ n )n∈Z | uˆ n = −uˆ −n } is some invariant subspace. According to the fact that we have a cubic nonlinearity also UO = {(uˆ n )n∈Z | uˆ 2n = 0}, the space of solutions whose even coefficients vanish, is an invariant subspace. Therefore, the intersection of all these subspaces UR ∩ Uodd ∩ UO = {(uˆ m )m∈Z | Re uˆ m = 0, uˆ 2m = 0, uˆ m = −uˆ −m , m ∈ Z} = Xˆ is also invariant. In the following we restrict our analysis to those solutions of (2.1) which are in Xˆ for fixed x, i.e., in particular we can restrict ourselves to m ∈ Nodd = {1, 3, 5, . . .}. Since u m ∈ R introduced by uˆ m = iu m satisfies (2.1), except for the opposite sign in front of the nonlinearity, the subsequent systems have the properties of real-valued systems, i.e., we consider in the following ∂x2 u m (x) = −s(x)m 2 ω∗2 u m (x) + q(x)u m (x) + r (x)gm (x),

m ∈ Z,

(2.3)

where gm (x) =

n 1 ,n 2 ,n 3 ∈Z,n 1 +n 2 +n 3 =m

u n 1 (x)u n 2 (x)u n 3 (x),

m ∈ Z.

(2.4)

824


3. Application of Floquet’s Theory In order to analyze the linear part of the spatial dynamics system (2.3) which decouples into infinitely many linear second order ODEs with periodic coefficients, namely ∂x2 u m (x) = −s(x)m 2 ω∗2 u m (x) + q(x)u m (x),

(3.1)

we use tools from Floquet theory. We shortly recapitulate these for the system ∂x u m (x) = vm (x), ∂x vm (x) = −s(x)ω2 u m (x) + q(x)u m (x),

(3.2)

which is simply (3.1) written as a first order system and with the replacement ω2 = m 2 ω∗2 . In order to adjust the spectral picture we will consider ω ∈ R and, thus, first ignore the sampling via m ∈ Z. The fundamental matrix of (3.2) is denoted with = (x), where (0) = I . Floquet’s theorem [7] shows that x M˜ ˜ (x) = P(x)e

(3.3)

˜ ˜ + 1) and a x-independent matrix M. ˜ Note that M˜ is not unique with P(x) = P(x 2π in ˜ = 1 for n ∈ Z. The eigenvalues of M are called Floquet exponents. according to e ˜ The eigenvalues of the so called monodromy matrix C = e M are called Floquet multipliers. For our special system we have two Floquet multipliers ρ− and ρ+ satisfying ρ+ ρ− = 1. They can be computed via the characteristic polynomial, cf. [7], ρ 2 − D(ω2 )ρ + 1 = 0 and are given by ρ± (ω2 ) =

1 1 D(ω2 ) ± (D(ω2 ))2 − 4, 2 2

where the trace of the monodromy matrix, D(ω2 ) = trace Cω2 , is called the discriminant. We find that S1a) if |D(ω2 )| > 2 then the Floquet multipliers ρ± (ω2 ) are real. As a consequence the solutions have exponential growth or decay w.r.t. x. S1b) if |D(ω2 )| < 2 then the Floquet multipliers ρ± (ω2 ) are on the complex unit circle away from ±1. As a consequence the solutions are uniformly bounded w.r.t. x. S2) if |D(ω2 )| = 2 then the Floquet multipliers ρ± (ω2 ) are 1 or −1. In this case we have at most polynomial growth. In case S1) the Floquet multipliers are simple. In case S2) we have algebraic multiplicity 2, but geometric multiplicity 1, i.e., in M˜ a nontrivial Jordan-block occurs. In order to deduce from the discriminant the spectral relation (l, ω) belonging to (1.6) one uses 1 1 e±il = D(ω2 ) ± (D(ω2 ))2 − 4 (3.4) 2 2 for values where −2 ≤ D(ω2 ) ≤ 2, i.e., where the Floquet exponents are given by ±il. Why the spectral relation (l, ω) really consists of infinitely many curves l → ωn (l) becomes evident from inspecting the properties of D(ω2 ) and Eq. (3.4), in particular, by taking into account that solving for ω involves an inversion of D(ω2 ).


825

4. An Example for a Suitable Choice of s = s(x) In this section we show that the choice s(x) = χ[0,6/13] + 16χ(6/13,7/13) + χ[7/13,1] (x mod 1)

and

q(x) = μs(x) (4.1)

allows us to satisfy the spectral assumption (Spec) in case of ω∗ = 13π/16 and μ = q0 with q0 ∈ R a fixed number defined explicitly in (4.5). Lemma 4.1. For s and q defined in Eq. (4.1) we find the discriminant 16 8 25 9 ω2 − μ − cos ω2 − μ . Dμ (ω2 ) = cos 8 13 8 13

(4.2)

Proof. We have to solve the ODE u (x) + (ω2 − μ)s(x)u(x) = 0, i.e. u (x) + (ω2 − μ)u(x) = 0, for x ∈ [0, 6/13] ∪ [7/13, 1]

(4.3)

u (x) + 16(ω2 − μ)u(x) = 0, for x ∈ (6/13, 7/13),

(4.4)

and

˜ = (x) ˜ for continuous u and u . We set λ = ω2 − μ. The fundamental matrix of ˜ (4.3) with (0) = I is given by

√ √ √1 sin( λx) cos( λx) λ ˜ √ √ √ (x) = cos( λx) − λ sin( λx) and of (4.4) by ˜˜ (x) =

√ cos(4 λx) √ √ −4 λ sin(4 λx)

√ sin(4 λx) √ . cos(4 λx)

1 √ 4 λ

Hence, we find 6 6 ˜ 1 ˜ ˜ ˜ (1) = 13 13 13

√ 6 √ √1 sin( λ 6 ) cos( λ 13 ) 13 λ √ √ 6 √ = 6 ) cos( λ 13 ) − λ sin( λ 13 √ 1 √ 1

1 √ cos(4 λ 13 ) sin(4 λ 13 ) 4 λ √ 1 √ 1 √ × cos(4 λ 13 ) −4 λ sin(4 λ 13 )

√ 6 √ √1 sin( λ 6 ) cos( λ 13 ) 13 λ √ 6 √ 6 √ × . ) cos( λ 13 ) − λ sin( λ 13

826

C. Blank, M. Chirilus-Bruckner, V. Lescarret, G. Schneider 15

15

10

10

5

5

0

0

−5

−5

−10

−10

−15 −5

0

5

10

15

−15 −5

0

5

Fig. 6. The figure shows the discriminant and the associated dispersion relation for μ = 0. S1a) If |D0 (ω2 )| > 2 then the Floquet exponents are real. As a consequence the solutions have exponential growth or decay w.r.t. x and so a spectral gap occurs. S1b) If |D0 (ω2 )| < 2 then the Floquet exponents are purely imaginary. As a consequence the solutions are uniformly bounded w.r.t. x and curves of eigenvalues occur in the spectral picture of the time evolutionary system. S2) If |D0 (ω2 )| = 2 then the Floquet exponents are 0 or iπ . In this case we have a horizontal tangency in the spectral picture of the time evolutionary system. The right panel show the associated functions l → ωn (l). For the understanding of the relation between the left and right panel compare (1.6) with (3.3)

For the discriminant which is the trace of this matrix we find √ 6 2 √ 4 √ 6 2 √ 4 − 2 sin D(λ) = 2 cos λ cos λ λ cos λ 13 13 13 13 √ 6 √ 6 √ 4 17 cos sin . λ λ λ − sin 2 13 13 13

Using the representation of sin and cos as exponentials allows us to simplify this expression easily into (4.2). Note that in the simplification there is a cancellation since no terms 4 with argument 13 ω2 − μ occur. Both q and s are even as necessary for the reversibility of (1.5). The graph ω → D0 (ω2 ) of the discriminant for μ = 0 and the associated dispersion relation can be found in Fig. 6. For ω∗ = 13π/16 and ω2 = m 2 ω∗2 with m ∈ N we find for q = μ = 0 that ⎧ 34 ⎪ ⎨ 8, 2 2 D0 (m ω∗ ) = 2, ⎪ ⎩ 25 −8,

m ∈ 2 + 4Z, m ∈ 4Z, m ∈ 1 + 2Z,


827

Im

Im

Re

Re

Fig. 7. The Floquet multipliers for the above choice of s, q, and ω. Left: for m ∈ Z. Right: for m ∈ Nodd

with associated Floquet multipliers ⎧ ⎪ 34 1 34 2 ⎪ ± − 4 ∈ {−1, 1}, ⎪ 8 ⎨ 16 2 2 2 1, ρ± (m ω∗ ) = ⎪ 2 ⎪ ⎪ 25 1 25 ⎩ − 16 ± 2 − 4 ∈ {−1, 1}, 8

m ∈ 2 + 4Z, m ∈ 4Z, m ∈ 1 + 2Z.

The associated Floquet diagram is plotted on the left hand side of Fig. 7. For μ = 0 these five points remain as accumulation points of the Floquet multipliers. The fact that there are infinitely many Floquet multipliers on the unit circle prevents at a first view the application of center manifold theory. However, since we only need to look for m ∈ Nodd the Floquet multipliers on the unit circle have not to be taken into account. As a consequence we have only negative Floquet multipliers being off the unit circle as can be seen in the right hand side of Fig. 7. Since the asymptotics of the Floquet multipliers is not affected by q, cf. [7, Theorem 4.5.3 ii)] we can use numerics to check for which parameter value μ of a given family of functions q = q(x, μ) two of the multipliers collide in −1. For our choice, q = q(x, μ) = μs(x), there is a critical value μ = q0 ≈ 3.7703 for which we have exactly two Floquet multipliers at −1. This value can be computed explicitly by solving 2 Dμ (( 13π 16 ) ) = −2. This condition is equivalent to the solution of the following set of equations: 13π , p = 8 ω∗2 − q0 /13, ω∗ = 16 √ w.r.t q0 . Using cos(2 p) = 2 cos2 ( p) − 1 we find p = arccos((9 + 1881)/100) and finally 25 cos(2 p) − 9 cos( p) = −16,

q0 =

ω∗2

−

13 p 8

2

=

13π 16

2 −

2 √ 13 arccos((9 + 1881)/100) ≈ 3.7703. 8 (4.5)

828


Since

m 2 ω∗2 − μ = mω∗ (1 −

μ + . . .) 2(mω∗ )2

the role of μ in Dμ (m 2 ω∗2 ) becomes smaller and smaller for m → ∞. Hence the spectral gaps are located asymptotically around mω∗ and the spectral gaps are O(1) for m → ∞. Since the discriminant becomes asymptotically a periodic function the associated eigenvalues in the spatial dynamics formulation are uniformly bounded away from the unit circle and on the real axis. See Fig. 9. 5. The Reversibility In the spatial dynamics formulation (1.5) the breather solution is a homoclinic solution. We find this bifurcating homoclinic solution approximately in the two-dimensional center manifold associated with the spectral picture drawn in Fig. 2. The persistence of the approximate homoclinic solution w.r.t. higher order perturbations heavily relies on the reversibility of the spatial dynamics formulation. 5.1. Preparations. We introduce a number of abbreviations, namely Um = (u m , vm ), vm (x) , (m Um )(x) = −s(x)m 2 ω∗2 u m (x) + q0 s(x)u m (x) 0 , Nm (U ) = ε2 s(x)u m (x) + r (x)gm (x) and rewrite (2.3) as ∂x U = F(x, U ) = U + N (U ),

(5.1)

where U = (Um )m∈Nodd , = (m )m∈Nodd , and N = (Nm )m∈Nodd . Definition 5.1. The non-autonomous system (5.1) is called reversible if there is an operator R such that RF(x, U ) = −F(−x, RU ). R is called the reversibility operator. For system (5.1) we define a reversibility operator R by R = ⊕m∈Nodd R with R(u m , vm ) = (u m , −vm ). System (5.1) is reversible, i.e., invariant under (x, u m , vm ) → (−x, u m , −vm ), due to the assumption that s, q and r are even functions. Lemma 5.2. With x → U (x) solving (5.1), also x → V (x) = RU (−x) is a solution. Proof. We have V˙ (x) = −RU˙ (−x) = −RF(−x, U (−x)) = F(x, RU (−x)) = F(x, V (x)). Lemma 5.2 implies that with x → U (x) = (u m , vm )m∈Nodd (x) being a solution of (5.1), also x → RU (−x) = (u m , −vm )m∈Nodd (−x) is a solution of (5.1). In the following arguments the fixed space of reversibility plays a major role. It is given by R f i x = {U = RU } = {(u m , 0)m∈Nodd }.


829

15

15

———————10

10

———————5

5

0

0

−5

−5

−10

−10

−15 −5

0

5

10

——————————————-

−15 −5

15

0

5

Fig. 8. The figure shows the discriminant and the associated dispersion relation for μ = q0 = 3.7703. The large value of Dμ (ω2 ) at ω = 0 comes from the fact that cos(iu) = cosh(u) Im

Im

Re

Im

Re

Im

Re

Re

Fig. 9. The Floquet multipliers for different choices of μ. Upper left: μ = 0. Upper right: μ < q0 . Lower left: μ = q0 . Lower right: μ > q0

5.2. The reversible change of variables. Due to the above theorem of Floquet (see ˜ Sect. 3) the solutions of ∂x Um = m Um are given by Um (x) = P˜m (x)e x Mm Um (0) 2×2 with P˜m (x) = P˜m (x + 1) and M˜ m ∈ C . Since all Floquet multipliers in Fig. 9 have negative real part and vanishing imaginary part the associated Floquet exponents, i.e., the eigenvalues of the M˜ m , are of the form α ± iπ with α ∈ R. In order to have real Floquet exponents we apply Floquet’s theorem for 2-periodic functions, i.e., the solutions of ∂x Um = m Um are given by Um (x) = Pm (x)e x Mm Um (0) with Pm (x) = Pm (x + 2), Pm (0) = I and Mm ∈ R2×2 . In order to make the linear part of the system autonomous we could make a change of variables Um (x) = Pm (x)Vm (x). However, this choice would destroy the reversibility. Moreover, the linear part will not be in Jordan normal form. Hence we proceed as follows. We write Um (x) = Pm (x)e x Mm Um (0) = Pm (x)Sm−1 e x Jm Sm Um (0) = Q m (x)e x Jm Vm (0)

830


such that Vm (x) defined by Um (x) = Q m (x)Vm (x)

(5.2)

satisfies ∂x Vm = Jm Vm , where Jm is the Jordan normal form of Mm and Sm the associated transformation. Case S1). Assume first that real-valued the Floquet exponents λ j for fixed m satisfy λ1 = λ2 . The solutions of ∂x Um = m Um can be written as U (x) = c1 ψ1 (x) + c2 ψ2 (x) = c1 eλ1 x φ1 (x) + c2 eλ2 x φ2 (x) with constants c j and 2-periodic φ j here and in the following. Since the systems are reversible with x → eλ1 x φ1 (x) also x → e−λ1 x Rφ1 (−x) is a solution. Hence we define the second fundamental solution eλ2 x φ2 (x) = e−λ1 x Rφ1 (−x), which implies λ2 = −λ1 and φ2 (x) = Rφ1 (−x). We introduce the new variable V (x) = (v1 , v2 )(x) by v (x) , U (x) = v1 (x)φ1 (x) + v2 (x)φ2 (x) = (φ1 (x), φ2 (x)) 1 v2 (x) where by construction ∂x V (x) = BV (x) with B = diag(λ1 , λ2 ). Hence, the above change of variables (5.2) and the last change of variables coincide, i.e. B = Jm , and the linear system is now reversible w.r.t. the transformed reversibility operator R˜ m defined through v1 v2 ˜ = . Rm v2 v1 Case S2). Next assume that we have a Jordan-block for the Floquet exponent λ = 0. Then U (x) = c1 ψ1 (x) + c2 ψ2 (x) = c1 φ1 (x) + c2 (xφ1 (x) + φ2 (x)). Due to the reversibility φ1 (x) = Rφ1 (−x), and φ2 can be chosen such that φ2 (x) = −Rφ2 (−x). We introduce the new variable V (x) = (v1 , v2 )(x) by v1 (x) , U (x) = v1 (x)φ1 (x) + v2 (x)φ2 (x) = (φ1 (x), φ2 (x)) v2 (x) 0 1 where by construction ∂x V (x) = BV (x), with B = . In this case the repre0 0 sentation of the reversibility operator is preserved, i.e. v1 v1 ˜ Rm = . v2 −v2 With Um (x) = Q m (x)Vm (x) we find ∂x Vm (x) = Bm Vm (x) + N˜ m (x, V (x)),

(5.3)


831

where N˜ (x, V (x)) = Q −1 (x)N (x, Q(x)V (x)) = ((Q m (x))−1 Nm (x, (Q j (x)V j (x)) j∈Nodd ))m∈Nodd and Q(x) = ⊕n j ∈Nodd Q m (x). We have by construction that B1 has some Jordan block of size 2 with associated eigenvalue 0. All other Bm with m ≥ 3 possess one positive and one negative eigenvalue which are uniformly bounded away from the imaginary axis w.r.t. m, i.e., (5.3) has the spectral picture plotted in the right panel of Fig. 2. The change of variables is bounded in the following sense. q11,m (x) q12,m (x) . Then there exists a C > 0 Lemma 5.3. Let Q m (x) = q21,m (x) q22,m (x) such that for all m ∈ Nodd we have supx∈[0,2] (|q11,m (x)| + |q12,m (x)|) < C and supx∈[0,2] |(Q m (x))−1 | < C. Proof. The proof follows from explicit calculations using the representation of the fundamental matrix (x) given in Lemma 4.1 for ω2 = m 2 ω∗2 and m ∈ 1 + 2Z. We refrain from writing down all the formulas, especially from writing down (1) for these values of m. We restrict ourselves to some explanations why the result is true. From the form of (1) it follows that we can choose the eigenvectors to be of the form a a and ψ2 (1) = , ψ1 (1) = b −b with a and b non-vanishing numbers satisfying b/a = O(n). Since ψ j (1) = ρ j ψ j (0), where ρ j are the associated Floquet multipliers we can compute ψ1 (x) = λ (x)ψ1 (0). We find that ψ2 (x) = (x)ψ2 (0) satisfies ψ2 (x) = Rψ1 (−x). From ψ j (x) we can = φ j (x + 1). Note that φ j (x) and ψ j (x) have compute explicitly φ j (x) = ψ j (x)ρ −x j the same asymptotics w.r.t. m since all Floquet multipliers are uniformly bounded independently of m. The matrix Q m (x) which possesses φ1 (x) and φ2 (x) as columns is O(1) O(1) , where for the same reason its m-depentherefore again of the form O(m) O(m) dent determinants are by O(m) constants. Hence, we have that bounded from below O(1) O(1/m) Q −1 m is of the form O(1) O(1/m) . 5.3. Conjugation of the old and the new reversibility operator. The old reversibility ˜ = ⊕m∈N R˜ m are conjugated via the operator R and the new reversibility operator R odd transform U (x) = Q(x)V (x). We find ˜ = RQ(x), Q(−x)R ˜ −1 (x). By the analysis of the last subsection we already which implies Q −1 (−x)R = RQ know that the transformed linear operator is reversible w.r.t. to the new reversibility oper˜ ator R. ˜ i.e., Lemma 5.4. System (5.3) is reversible w.r.t. to the new reversibility operator R, especially we have ˜ N˜ (x, V ) = − N˜ (−x, RV ˜ ). R

832


Proof. This holds according to ˜ N˜ (x, V ) = RQ ˜ −1 (x)N (x, Q(x)V ) = Q −1 (−x)RN (x, Q(x)V ) R ˜ ) = −Q −1 (−x)N (−x, RQ(x)V ) = −Q −1 (−x)N (−x, Q(−x)RV ˜ ). = − N˜ (−x, RV 6. The Center Manifold Reduction It is the purpose of this section to construct the center manifold for (5.3) associated to the two Floquet multipliers on the unit circle. System (5.3) is extended with the equation ∂x ε = 0 which allows us to handle all terms with an ε as nonlinear. Before we do so we make a number of remarks. Remark 6.1. The combination of spatial dynamics and center manifold theory goes back to [21]. For a number of different formulations in the continuous case see [29]. There are a number formulations for discrete dynamical systems, too. The estimates below show that for instance, the abstract center manifold theorem [18, Theorem 6.2] applies to the time-one-map in our case. We do not use this discrete version since for the discussion of the reduced system in the following we would like to have an ordinary differential equation rather than a discrete system. Remark 6.2. Invariant manifolds for periodic systems are invariant in the following sense. We denote with V = V (x, x0 , V0 ) a solution V at x with initial condition V0 given at x0 . We introduce a nonlinear evolution operator Sx,x0 defined by Sx,x0 V0 = V (x, x0 , V0 ). Due to the 1-periodicity we have V (x, x0 , V0 ) = V (x + 1, x0 + 1, V0 ). Hence the time-1-maps x0 which are defined through x0 (V0 ) = V (x0 + 1, x0 , V0 ) play a crucial role. Time-1-maps for different x0 are conjugated to each other, i.e. x ◦ Sx,0 = Sx,0 ◦ 0 which is a direct consequence of V (x + 1, x, V (x, 0, V0 )) = V (x + 1, 1, V (1, 0, V0 )) = V (x, 0, V (1, 0, V0 )). The center manifold of the origin W c (x0 ) = {V0 | (x0 )n (V0 ) ≤ Ceη|n| for |n| → ∞} = {V0 |V (x, x0 , V0 ) ≤ Ceη|x−x0 | for |x| → ∞} for a C > 0 and a fixed small η > 0 satisfies W c (x0 ) = W c (x0 + 1) by construction. We have the invariance x0 W c (x0 ) ⊂ W c (x0 ) and the transport by the flow W c (x) = Sx,x0 W c (x0 ).

(6.1)

Similar statements are true for the unstable and stable manifold. Since our systems are only reversible, cf. Definition 5.1, for x0 = 0, we set x0 = 0 in the following.


833

In the following we fix σ ≥ 0. For m ≥ 3 we define projections Ps,m and Pu,m on the stable and unstable eigenspaces which are uniformly bounded in R2×2 w.r.t. m due to the diagonal form of the Bm . Introducing Vm,s (x) = Pm,s Vm (x) and Vm,u (x) = Pm,u Vm (x) for m ≥ 3 allows us to construct a center manifold as usual by applying a fixed point argument to the map F = (F1 , (Fm,s , Fm,u )m∈Nodd ) : Yη → Yη for a small but fixed η > 0, where x F1 (x) = e B1 x V1 (0) + e B1 (x−ξ ) Nˇ 1 (ξ, V (ξ )) dξ, 0 x Fm,s (x) = e Bm (x−ξ ) Pm,s Nˇ m (ξ, V (ξ )) dξ, −∞ ∞ Fm,u (x) = − e Bm (x−ξ ) Pm,u Nˇ m (ξ, V (ξ )) dξ, x

and Yη = {V ∈ C 0 (R, 1 (σ )) | sup |eη|x| V (x)1 (σ ) < ∞} x∈R

with V = (V1 , V3 , V5 , . . .) and V 1 (σ ) =

m∈Nodd

|Vm |m σ . Moreover, we let

Nˇ m (x, V ) = N˜ m (x, V )χ (V 1 (σ ) /δ) for a small, but fixed δ > 0, where χ is a C0∞ -function with values in [0, 1] satisfying χ (r ) = 1 for r ≤ 1, χ (r ) = 0 for r ≥ 2. Lemma 6.3. Nˇ (V ) = ( Nˇ m (V ))m∈Nodd is Lipschitz continuous in 1 (σ ) with Lipschitz constant proportional to δ 2 for δ → 0. Moreover, Nˇ (V ) is reversible w.r.t. the revers˜ i.e., we have ibility operator R, ˜ Nˇ (x, V ) = − Nˇ (−x, RV ˜ ). R

(6.2)

Proof. Since in Nm only the first coordinate of U occurs, after the transforms Um = Q m Vm only q11,m and q12,m occur in the transformed nonlinearity. Since both are uniformly bounded, since the same is true for Q −1 m according to Lemma 5.3, and since 1 (σ ) is closed under convolutions the Lipschitz continuity follows. The magnitude of the Lipschitz constant follows from the cut-off function and the fact that N does not contain quadratic terms. The reversibility of N˜ which is known from Lemma 5.4 is not destroyed by the cut-off function and so (6.2) follows. Moreover, due to the asymptotics of the discriminant which results in Fig. 2 it follows Lemma 6.4. There exist β such that for all η with β > η > 0 we have a C > 0 such that e B1 x R2 →R2 ≤ Ceη|x|/2 , sup e m

Bm x

Pm,s R2 →R2 ≤ Ce−βx ,

sup e Bm x Pm,u R2 →R2 ≤ Ceβx , m

∀x ∈ R, ∀x ≥ 0, ∀x ≤ 0.

Proof. See Fig. 9 and note that the Bm are diagonal for m ≥ 3.

834


Hence from Lemma 6.3 and 6.4 the existence of a fixed point V ∗ = V ∗ (x, V1∗ , ε) of F follows with the usual estimates, cf. [29]. Since the spectral gap, the cut-off and all estimates are O(1) w.r.t. ε, the size of the center manifold will be O(1) for ε → 0. We define the graph of the center manifold by a mapping h from the central subspace to the hyperbolic subspace by h(0, V1∗ , ε) = Ph V ∗ (0, V1∗ , ε), where Ph = (Pm,u + Pm,s )m∈Nodd is the projection to the hyperbolic subspace. After the sketch of the major steps of the proof of the center manifold theorem the result is summarized in Theorem 6.5. For all n ∈ N there exist ε0 > 0 and ϑ0 > 0 such that the spatial dynamics formulation (5.3) extended with ∂x ε = 0 possesses a three-dimensional invariant manifold Wc = {(V ∗ , ε) ∈ 1 (σ ) × R | (0, V3 , V5 , . . .) = h(0, V1∗ , ε)} tangential to the space E c = {(V1∗ , 0, 0, . . .) × R | V1∗ ∈ R2 } with h(0, ·, ·) ∈ C n ({V1 ∈ R2 | V1 1 (σ ) ≤ ϑ0 } × [0, ε0 ], 1 (σ )). The center manifold Wc = Wc (0) has been constructed for starting time x0 = 0. At the beginning of this subsection we explained that the invariant manifolds are transported by the flow, i.e. the center manifold Wc (x0 ) for starting time x0 and Wc (0) are related via Wc (x0 ) = Sx0 ,0 Wc (0), where Sx0 ,0 is the evolution operator of the spatial dynamics system (5.1) extended with ∂x ε = 0. Hence we define the reduction function h(x0 , ·, ·) for Wc (x0 ) by V ∗ (x) ⊕ ε = Sx,0 (V ∗ (0) ⊕ ε) = Sx,0 (V1∗ (0) ⊕ h(0, V1∗ (0)) ⊕ ε) = V1∗ (x) ⊕ h(x, V1∗ (x)) ⊕ ε. Since the dynamics in ε is trivial we suppress in the following the variable ε in our notation. Since with x → V (x) = V1 (x) ⊕ h(x, V1 (x)) being a solution on the center ˜ r h(−x, V1 (−x)) is a solution on the ˜ (−x) = RV ˜ 1 (−x) ⊕ R manifold also x → RV ˜ ˜ ˜ r h(−x, V1 ) = h(x, RV ˜ ˜ 1) center manifold, where R = R ⊕ Rr we can conclude that R ˜ r h(x, V1 ) = h(−x, RV ˜ 1 ). From this we find or equivalently R ˜ r h(x, V1 )) ˜ 1⊕R R˜ N1 (x, V1 ⊕ h(x, V1 )) = −N1 (−x, RV ˜ 1 ⊕ h(−x, RV ˜ 1 )) = −N1 (−x, RV such that the vector field of the reduced system ∂x V1 (x) = B1 V1 (x) + Nˇ 1 (x, V1 (x), h 3 (x, V1 (x), ε), . . .)) (6.3) is reversible w.r.t. the transformed reversibility operator R˜ which coincides for V1 with the original one reversibility operator R. As a consequence of the center manifold reduction all small bounded solutions of (5.3) can be found on the center manifold and it is sufficient to discuss the reduced system on the center manifold. Finally, we mention that the right-hand side of (6.3) is smooth w.r.t. (V, ε), but only piecewise smooth w.r.t. x with discontinuities at the jumps of the coefficient function s = s(x).


835

7. Construction of a Homoclinic Solution By the center manifold reduction of the last section the infinite dimensional spatial dynamics formulation (5.1) has been reduced to the 2-periodic two-dimensional ordinary differential equation (6.3) which is reversible w.r.t. the reversibility operator R. The reduced system (6.3) is analyzed with the help of bifurcation theory. The small bifurcation parameter has been introduced by q(x) = (q0 −ε2 )s(x). Hence, in (6.3) only powers of ε2 occur. Since V1 = 0 is a solution for all values of ε, terms depending on ε alone cannot occur. Moreover, the reduced system must have the same symmetries as the original one which reduces the number of possible terms. Especially the translation invariance w.r.t. t reduces the number of possible terms drastically. By our choice of coordinates B1 is a Jordan block of size two. Setting V1 = (a, b) this allows us to rewrite (6.3) into ∂x a = b + O(|ε2 a|, |ε2 b|, |a 3 |, . . . , |b3 |), ∂x b = O(|ε2 a|, |ε2 b|, |a 3 |, . . . , |b3 |) for small (a, b). Introducing a˜ and b˜ by ˜ a(x) = εa(x) ˜ and b(x) = ε2 b(x) yields ∂x a˜ = εb˜ + O(ε2 ),

∂x b˜ = εs1 (x)a˜ + εs3 (x)a˜ 3 + O(ε2 ),

(7.1)

with s1 = s1 (x) and s3 = s3 (x) being 2-periodic functions. In order to find the homoclinic solutions for (7.1) which is of the form z˙ = ε f (z, x, ε)

(7.2)

˜ we compare it with the averaged system with z = (a, ˜ b) y˙ = ε f (y) =

ε 2

2

f (y, x, 0)d x.

(7.3)

0

In order to analyze the averaged system we rescale time X = εx, and introduce A(X ) = a(x), ˜

˜ B(X ) = b(x).

(7.4)

In this scaling the averaged system is given ∂ X A = B,

∂ X B = s¯1 A + s¯3 A3 ,

(7.5)

where s¯1 =

2

s1 (x) dx/2

and

0

Lemma 7.1. We have s¯1 > 0 and s¯3 < 0.

2

s¯3 = 0

s3 (x) dx/2.

(7.6)

836


Proof. Since ε has been defined in such a way that ω∗ falls into a spectral gap in the right panel of Fig. 8 for ε > 0 or equivalently that in Fig. 2 a real positive and a real negative eigenvalue of order O(ε) occur, it follows for the ε independent coefficient s¯1 that s¯1 > 0. 1 (π ) The coefficient s¯3 is a positive multiple of 2γ as can be seen by comparing the ω1 (π ) formal derivation of (7.5) and the derivation of the associated NLS-equation, cf. Remark 1.5. According to (1.9) the coefficient γ1 (π ) is positive due to r (x) = 1 and according to the right panel of Fig. 8 the coefficient ω1 (π ) is negative such that s¯3 has a negative sign. Remark 7.2. Due to the fact that we have a cubic nonlinearity the coefficient function s3 = s3 (x) is independent of the reduction function h and so it can be computed from Q −1 1 (x)N1 (x, Q 1 (x)V1 ) εA alone. According to our scalings V1 = we have ε2 B εA εq11 A q11 (x) q12 (x) + O(ε2 ). Q 1 (x)V1 = = q21 (x) q22 (x) εq12 A ε2 B Since

0 N1 (x, V1 ) = and ε2 s(x)V1 + 3r (x)V13 q22 (x) −q12 (x) −1 −1 Q 1 (x) = (det Q 1 (x)) −q12 (x) q11 (x)

we find

0 −1 + O(ε4 ) Q −1 (x)N (x, Q (x)V ) = Q (x) 1 1 1 1 1 ε3 s(x)q11 (x)A + 3ε3 r (x)q11 (x)3 A3 O(ε3 ) + O(ε4 ) = ε3 (det Q 1 (x))−1 (s(x)q11 (x)2 A + 3r (x)q11 (x)4 A3

and so s1 (x) = (det Q 1 (x))−1 s(x)q11 (x)2 and s3 (x) = 3(det Q 1 (x))−1r (x)q11 (x)4 . (7.7) In Q 1 (x), cf. Sect. 5.2 S2), only φ1 (x) plays a role and so q11 is nothing else than a multiple of w1 (x, π )eiπ x Comparing (7.6) and (7.7) with the formula (1.9) for the coefficient γ1 (π ) in front of the cubic terms in the associated NLS equation shows that det Q 1 (x) is a multiple ω1 (π ). The magnitude of the multiple depends on the scaling of w1 (x, π )eiπ x . Since s¯3 has a negative sign system (7.5) possesses a pair of homoclinic solutions qhom = (Ahom , Bhom ) which is given by √ 2s1 Ahom (X ) = ± sech s1 X , ∂ X Ahom = Bhom . −s 3 Undoing the scaling (7.4) shows that system (7.3) possesses a pair of homoclinic solutions, too.


837

8. Persistence of the Homoclinic Solution Remark 8.1. The homoclinic orbit qhom lies in the intersection of the stable manifold and the unstable manifold of System (7.3). In general systems if higher order terms are added the intersection will break up and the perturbed stable manifold and the unstable manifold will no longer intersect. In reversible systems the situation is different. The persistence of the homoclinic solution is established by proving a transversal intersection of the stable manifold with the fixed space of reversibility. This gives the homoclinic orbit for x ∈ [0, ∞). Applying the reversibility operator R to this part of the solution gives the homoclinic orbit also for x ∈ (−∞, 0]. The persistence proof consists of three steps: i)

Beyond other things in [14, Theorem 4.1.1] the following is shown: Lemma 8.2. There exists a C r -w.r.t. y-change of coordinates z = y + εw(y, x, ε) under which (7.2) becomes y˙ = ε f (y) + ε2 f 1 (y, x, ε),

(8.1)

where f 1 is of period 2 w.r.t. x.

ii)

Hence in an O(1)-neighborhood the stable manifold W s of the averaged system (7.3) and the stable manifold Ws of the full system (7.2), resp. (8.1), are O(ε)-close together. In addition to the statement in Lemma 8.2 in [14, Theorem 4.1.1] it is shown Lemma 8.3. If z(x) and y(x) are solutions of (7.2) and (7.3) with |z(−1/ε) − y(−1/ε)| = O(ε), then supx∈[−1/ε,0] |z(x) − y(x)| = O(ε).

iii)

Applying the approximation result from Lemma 8.3 shows that the stable manifold W s of the averaged system (7.3) and the stable manifold Ws of the full system (7.2), resp. (8.1), are O(ε)-close together on a scale O(1/ε). Hence O(ε)-close to the intersection point of the averaged system (7.3) with the fixed space of reversibility there is an intersection point of the full system (7.2), resp. (8.1). See Fig. 11 As a consequence we have a solution V1 (x) of (7.2) for x ∈ [0, ∞) which satisfies lim x→∞ V1 (x) = 0 and V1 (0) ∈ {B = 0}. Finally, we use the reversibility of the reduced system (6.3), resp. (7.2). It allows us to extend V1 (x) for x ∈ [0, ∞) by V1 (−x) = RV1 (x) to x ∈ R. Hence we constructed a homoclinic solution to the origin for (6.3) and so as a consequence of the exact center manifold reduction finally for the original system (2.3).

Remark 8.4. We cannot use [14, Theorem 4.1.1 iii)] directly which is a statement about the closedness of the stable manifolds, since there the closedness for x = 0 is assumed, which was exactly the goal of the above steps i) and ii). We used i) to control x ∈ (−∞, −1/ε) and ii) to control x ∈ (−1/ε, 0]. Remark 8.5. Finally we remark that in the statement of [14, Theorem 4.1.1] also smoothness of the vector field w.r.t. x is assumed. However, looking at the proof shows that piecewise continuity w.r.t. x is sufficient. Inserting z = y + εw(y, x, ε) into z˙ = ε f (z, x, ε) shows that y˙ = (1 + ε∂1 w(y, x, ε))−1 (ε∂2 w(y, x, ε) + ε f (y + εw(y, x, ε), x, ε) = ε∂2 w(y, x, ε) + ε f (y, x, 0) + O(ε2 ).

838


0

0

Fig. 10. The homoclinic orbits for the reduced system

Fig. 11. The combination of local estimate for the difference from (i) with the approximation result from ii). The dotted/full line is the stable manifold of the averaged system (7.3)/full system (7.2)

As a consequence the x-dependent parts of f (y, x, ε) = m∈Z f m (y, ε)eiπ mx can be eliminated by choosing w(y, x, ε) = m∈Z\{0} (iπ m)−1 f m (y, ε)eiπ mx . Hence w is at least continuous w.r.t. x and so the transformed system has the same regularity w.r.t. x as the original one. Lemma 8.3 follows from a simple application of Gronwall’s inequality for which the given regularity w.r.t. x is sufficient. Acknowledgements. The authors are grateful for discussions with K. Busch, D. Pelinovski, L. Tkeshelashvili, and H. Uecker. The research is partially supported by the Graduiertenkolleg 1294: Analysis, Simulation, and Design of Nanotechnological Processes sponsored by the Deutsche Forschungsgemeinschaft (DFG) and the Land Baden-Württemberg. Finally, we would like to thank the referees for a number of useful suggestions, especially for pointing out the possibility of writing down the explicit formula (4.5) for q0 .

A. The Physical Motivation: Photonic Crystals as Optical Storage This research is motivated by theoretical scenarios where photonic crystals are used as optical storage [5]. Photonic crystals consist of a dielectric material, for instance glass, with a periodic structure with a wavelength comparable to the wavelength of light. They are suitable tools for the construction of all optical devices in photonics which is loosely


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speaking electronics with photons instead of electrons. The light pulses in the photonic crystal are described by Maxwell’s equations (cf. [9]) in media which are given by ∇ · (E + P) = ∇×E = ∇×B= ∇ · (B + M) =

ρ, ∂t (B + M), −∂t (E + P) + j, 0,

(A.1) (A.2) (A.3) (A.4)

where E is the electric field, B the magnetic field, P the polarization, M the magnetization, j the electric current density, and ρ the electric charge density, where by rescaling all coefficients have been set to one. In photonic crystals there are no free charges, no electric current, and no magnetization, i.e., ρ = 0, j = 0 and M = 0. Differentiating (A.3) w.r.t t and substituting ∂t B via (A.2) yields E − ∇(∇ · E) = ∂t2 E + ∂t2 P,

(A.5)

where we additionally used the rule ∇ × ∇ × u = −u + ∇(∇ · u). For polarized light, i.e., in the one-dimensional situation (A.5) simplifies into ∂x2 E = ∂t2 E + ∂t2 P.

(A.6)

In order to close (A.6) the polarization P has to be expressed in terms of E. Equation (1.1) can be obtained by the following modelling. We split P into an instantaneous part Pinst and into a non-instantaneous part Pnon . For the instantaneous part Pinst we choose the linear constitutive law Pinst = (s −1)E with s(x) = s(x +1) a one-periodic function. In the simplest model for the non-instantaneous part Pnon the nuclei of the atoms are fixed and the centers of the electrons move like a nonlinear oscillator. This simple modeling finally leads to a system ∂x2 E = s∂t2 E + ∂t2 Pnon , Pnon = Nj=1 P j , ∂t2 P j

+ ω2j P j

+ r j P j |P j | = d j E, 2

(A.7) (A.8) (A.9)

with constants ω j , r j , d j , and where N is the number of different kinds of molecules. In our modeling dissipation is neglected. The argument to come to (1.1) is as follows. For E = E 0 eiωt , Eq. (A.9) possesses solutions P j = P0 j eiωt with − ω2 P0 j + ω2j P0 j + r j P0 j |P0 j |2 = d j E 0 .

(A.10)

For photonic crystals the parameters ω2j , r j , and d j depend periodically on x. For small E 0 (A.10) can be solved w.r.t. P0 j , i.e., we have a constitutive law P0 j (x, ω) = α j (x, ω)E 0 j (x, ω) + β j (x, ω)E 03 j (x, ω) + · · · .

(A.11)

For ω in the optical window the changes in α j (x, ω) are negligible w.r.t. ω, i.e., the relation (A.11) is modeled independently of ω, or equivalently ω =: ω˜ j is fixed. Then multiplying (A.10) by eiωt yields the relation (−ω˜ 2j + ω2j + r j |P j |2 )P j = d j E,

840


which can be inverted for small E, i.e., dj Pj = 2 E − rj ω j − ω˜ 2j

dj 2 ω j − ω˜ j

3 |E|2 E + · · · .

(A.12)

Next we replace ∂t2 P in (A.7) via (A.8) and (A.9), i.e., ∂x2 E = s∂t2 E +

N

(d j E − ω2j P j − r j P j |P j |2 ).

(A.13)

j=1

Using (A.12) to replace P j in (A.13) yields (1.1) when terms of order O(|P j |5 ) are neglected. We do not claim that this modeling is the physically most realistic modeling of photonic crystals, but we claim that with the same arguments which are used in the derivation of the other simplified models for the description of photonic crystals our starting Eq. (1.1) can be derived, too. Since we are interested in real-valued solutions we have E|E|2 = E 3 . Our Theorem 1.1 guarantees that in this modeling an infinitely extended photonic crystal can be designed which can be used as perfect optical storage, where the breather solution of Theorem 1.1 stands for a one in the digital encoding of information. There is no radiation and the information will be stored for all times. In reality the photonic crystals have finite size such that radiation of the pulse will be present. Although much smaller than the dispersion which comes from the periodic structure, dissipation cannot be neglected in the long term. But the strongest obstruction in practice is that the pulse will move with a fraction of the velocity of light if the underlying carrier wave has not precisely the wave number with the horizontal tangency. Hence, w.r.t. this possible application our result so far is mainly of theoretical interest. References 1. Bambusi, D., Paleari, S., Penati, T.: Existence and continuous approximation of small amplitude breathers in 1D and 2D Klein–Gordon lattices. Preprint 2009 2. Bambusi, D., Penati, T.: Continuous approximation of breathers in one and two dimensional DNLS lattices. Nonlinearity 23, 143–157 (2010) 3. Birnir, B., McKean, H.P., Weinstein, A.: The rigidity of sine-Gordon breathers. Comm. Pure Appl. Math. 47(8), 1043–1051 (1994) 4. Busch, K., Schneider, G., Tkeshelashvili, L., Uecker, H.: Justification of the Nonlinear Schrödinger equation in spatially periodic media. ZAMP, 57, 1–35 (2006) 5. Busch, K., von Freyman, G., Linden, S., Mingaleev, S.F., Theshelashvili, L., Wegener, M.: Periodic nanostructures for photonics. Phys. Rep. 444, 101–202 (2007) 6. Denzler, J.: Nonpersistence of breather families for the perturbed sine Gordon equation. Commun. Math. Phys. 158(2), 397–430 (1993) 7. Eastham, M.S.P.: The spectral theory of periodic differential equations. Edinburgh: Scottish Academic Press, 1973 8. Eckmann, J.-P., Wayne, C.E.: The nonlinear stability of front solutions for parabolic partial differential equations. Commun. Math. Phys. 161(2), 323–334 (1994) 9. Feynman, R.P., Leighton, R.B., Sands, M.: The Feynman lectures on physics. Vol. 2: Mainly electromagnetism and matter. Reading, MA-London: Addison-Wesley Publishing Co., Inc., 1964 10. Groves, M.D., Mielke, A.: A spatial dynamics approach to three-dimensional gravity-capillary steady water waves. Proc. Roy. Soc. Edinburgh Sect., A 131, 83–136 (2001) 11. Groves, M.D., Schneider, G.: Modulating pulse solutions for a class of nonlinear wave equations. Commun. Math. Phys. 219(3), 489–522 (2001)


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12. Groves, M.D., Schneider, G.: Modulating pulse solutions for quasilinear wave equations. J. Diff. Eq. 219(1), 221–258 (2005) 13. Groves, M.D., Schneider, G.: Modulating pulse solutions to quadratic quasilinear wave equations over exponentially long length scales. Commun. Math. Phys. 278(3), 567–625 (2008) 14. Guckenheimer, J., Holmes, P.: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Applied Mathematical Sciences, 42. New York: Springer-Verlag, 1983 15. Haragus, M., Schneider, G.: Bifurcating fronts for the Taylor-Couette problem in infinite cylinders. Z. Angew. Math. Phys. 50(1), 120–151 (1999) 16. Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Springer Lecture Notes in Mathematics Vol. 840, Berlin-Heidelberg-NewYork:Springer, 1981 17. James, G., Noble, P.: Breathers on diatomic Fermi-Pasta-Ulam lattices. Physica D 196(1–2), 124–171 (2004) 18. James, G., Sirr, Y.: Center manifold theory in the context of infinite one-dimensional lattices. The FermiPasta-Ulam problem, Lecture Notes in Phys. Vol. 728, Berlin-Heidelberg-New York: Springer, 2008, pp. 208–238 19. James, G., Sanchez-Rey, B., Cuevas, J.: Breathers in inhomogeneous nonlinear lattices: an analysis via center manifold reduction. Rev. Math. Phys. 21(1), 1–59 (2009) 20. MacKay, R.S., Aubry, S.: Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators. Nonlinearity 7, 1623–1643 (1994) 21. Kirchgässner, K.: Wave solutions of reversible systems and applications. J. Diff. Eq. 45, 113–127 (1982) 22. Lescarret, V., Blank, C., Chirilus-Bruckner, M., Chong, C., Schneider, G.: Standing modulating pulse solutions for a nonlinear wave equation in periodic media. Nonlinearity 22(8), 1869–1898 (2009) 23. Ntinos, A.A.: Lengths of instability intervals of second order periodic differential equations. Quart. J. Math. Oxford 27, 387–394 (1976) 24. Pankov, A.: Periodic nonlinear Schrödinger equation with application to photonic crystals. Milan J. Math. 73, 259–287 (2005) 25. Pelinovsky, D.E., Kevrekidis, P.G., Frantzeskakis, D.J.: Persistence and stability of discrete vortices in nonlinear Schrödinger lattices. Physica D 212, 20–53 (2005) 26. Pelinovsky, D., Schneider, G.: Justification of the coupled-mode approximation for a nonlinear elliptic problem with a periodic potential. Applicable Analysis 86(8), 1017–1036 (2007) 27. Pelinovsky, D., Schneider, G., MacKay, R.S.: Justification of the lattice equation for a nonlinear elliptic problem with a periodic potential. Commun. Math. Phys. 284(3), 803–831 (2008) 28. Reed, M., Simon, B.: Methods of modern mathematical physics. IV. Analysis of operators. New YorkLondon: Academic Press, 1978 29. Vanderbauwhede, A., Iooss, G.: Center manifold theory in infinite dimensions. In: Dynamics reported: expositions in dynamical systems, Berlin: Springer, 1992, pp. 125–163 Communicated by P. Constantin


Communications in


Quaternionic Kähler Detour Complexes and N = 2 Supersymmetric Black Holes D. Cherney1 , E. Latini1,2 , A. Waldron1 1 Department of Mathematics, University of California, Davis, CA 95616, USA.

E-mail: [email protected]; [email protected]; [email protected]

2 INFN, Laboratori Nazionali di Frascati, CP 13, I-00044 Frascati, Italy. E-mail: [email protected]

Received: 18 March 2010 / Accepted: 6 August 2010 Published online: 5 January 2011 – © The Author(s) 2011. This article is published with open access at Springerlink.com

Abstract: We study a class of supersymmetric spinning particle models derived from the radial quantization of stationary, spherically symmetric black holes of four dimensional N = 2 supergravities. By virtue of the c-map, these spinning particles move in quaternionic Kähler manifolds. Their spinning degrees of freedom describe mini-superspace-reduced supergravity fermions. We quantize these models using BRST detour complex technology. The construction of a nilpotent BRST charge is achieved by using local (worldline) supersymmetry ghosts to generate special holonomy transformations. (An interesting byproduct of the construction is a novel Dirac operator on the superghost extended Hilbert space.) The resulting quantized models are gauge invariant field theories with fields equaling sections of special quaternionic vector bundles. They underly and generalize the quaternionic version of Dolbeault cohomology discovered by Baston. In fact, Baston’s complex is related to the BPS sector of the models we write down. Our results rely on a calculus of operators on quaternionic Kähler manifolds that follows from BRST machinery, and although directly motivated by black hole physics, can be broadly applied to any model relying on quaternionic geometry. Contents 1. 2. 3. 4. 5.

6.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . Detour Complexes . . . . . . . . . . . . . . . . . . . . . . . . Special Geometry . . . . . . . . . . . . . . . . . . . . . . . . N = 2 Supersymmetric Black Holes and Quaternionic Geometry HyperKähler Sigma Model . . . . . . . . . . . . . . . . . . . 5.1 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Charges . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . Quaternionic Kähler, N = 4, d = 1 SUGRA . . . . . . . . . . . 6.1 Rigid Lefschetz–Verbitsky model . . . . . . . . . . . . . .

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D. Cherney, E. Latini, A. Waldron

6.2 Gauged Lefschetz–Verbitsky model . 6.3 Dirac quantization . . . . . . . . . . . 7. BRST and the Geometry of Ghosts . . . . 8. A Quaternionic Geometric Calculus . . . . 9. The Quaternionic Kähler Detour Complex 10. Conclusions . . . . . . . . . . . . . . . .

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859 859 860 862 865 868

1. Introduction The main result of this paper is a detour complex for quaternionic Kähler manifolds. In physics language, this amounts to a gauge theory of higher (quaternionic) “forms” on these manifolds. To be precise, we utilize special holonomy to split the tangent bundle of a 4n-dimensional quaternionic Kähler manifold M into a product of rank 2 and 2n vector bundles H and E [1], TM ∼ = E ⊗ H, and present an equation of motion and gauge invariances for sections of ∧E (or, more generally, ∧E ⊗ H ). The results of the paper will appeal to multiple audiences including: (i) Those readers interested in the differential geometry of quaternionic Kähler spaces. (ii) Readers studying various supersymmetric quantum mechanical and spinning particle models in quaternionic Kähler and hyperKähler backgrounds (such as gravitational instanton moduli spaces [2], Hitchin’s moduli space of stable Higgs bundles [3,4], geometric Langlands theory [5] and hypermultiplet moduli spaces [6], to name a few). (iii) Readers looking for applications of the BRST detour quantization of orthosymplectic constraint algebras developed for applications to higher spin systems in [7], on which these results heavily rely. (iv) Readers wanting to apply our results to supergravity (SUGRA) black hole quantization since, remarkably, the mathematical structure presented above is exactly what is called for when studying the minisuperspace quantization of N = 2 SUGRA black holes [8–11]. (In particular, wavefunctions valued in ∧E describe the fermionic degrees of freedom of these models.) Therefore the paper is structured so that any of these readerships can easily extract the information they need. In Sect. 2, we introduce the notion of a detour complex, beginning with simple examples. We then generalize our previous results on Kähler detour complexes to hyperKähler manifolds. This result follows immediately from an isomorphism between super Lie algebras of geometric operators mapping Dolbeault and Lefschetz operators on Kähler forms to their hyperKähler analogues acting on sections of ∧E. We then explain a main difficulty solved in this paper: the construction of a geometric detour complex for quaternionic Kähler manifolds is seemingly obstructed by the higher rank of the analogous geometric super algebra. This problem is overcome in later sections by understanding the key rôle played by the BRST superghosts in the description of quaternionic geometry. The main requisite geometric data is presented in Sect. 3 together with our notations and conventions. In Sect. 4 we review the relationship between quaternionic Kähler spinning particles and four dimensional black holes; the original motivation for this work. The latter can be described by a spinning particle model coming from the minisuperspace reduction of N = 2 supergravities [8,9]. The “BPS” conditions of this spinning particle model (i.e., requiring solutions for which the local fermion supersymmetry transformations vanish) equal the reduction of the analogous conditions in the four dimensional SUGRA.

Quaternionic Kähler Detour Complexes and N = 2 Supersymmetric Black Holes

845

Since those conditions amount to the attractor mechanism [12–14] for four dimensional supersymmetric black holes, the quantized spinning particle model is an excellent laboratory for studying these objects.1 In particular, it allows a minisuperspace analysis of the Ooguri–Strominger–Vafa conjecture [19] and the relationship between black hole wave functions and vacuum selection in string theory [20]. This equivalence between the attractor flow equation and supersymmetric geodesic motion was observed in [8,9,21]. The introduction of BRST techniques to solve what could be stated as a purely geometrical problem suggests the presence of an underlying gauge invariant physical model. This is indeed the case. The first of the relevant models is a hyperKähler supersymmetric quantum mechanics. This model can be enhanced to include quaternionic Kähler backgrounds once its four worldline supersymmetries are gauged. This yields a supersymmetric spinning particle model consistent in any quaternionic Kähler manifold. We describe these models in Sects. 5 and 6, respectively. Sections 3, 8 and 9 can in principle be read by geometers in isolation from the other more physical sections. In Sect. 8, we give a calculus of geometric operators acting on sections of ∧E. Although, we were motivated to write these operators for quantum mechanical BRST reasons, the results themselves are purely geometric. They form the basic building blocks of our quaternionic detour complex. They also place in a much more general setting the Dirac, Dirac–Fueter and detour operator employed some time ago by Baston [24]. Finally our main result is given in Sect. 9, orchestrating all the previous results to build a gauge invariant, higher “form” quantum field theory on quaternionic Kähler manifolds. It relies on the construction of a nilpotent BRST charge given in Sect. 7 achieved by utilizing the supersymmetry ghosts to generate special holonomy transformations. An interesting byproduct of this computation is a novel Dirac operator on the BRST superghost Hilbert space. Aside from providing an explicit quantization of the fermion modes of minisuperspace N = 2 supersymmetric black holes, our quaternionic detour complex has many potential further applications and generalizations. In particular, it is closely related to the twistor methods of [25]. Also, in some sense, the model is a higher spin theory, so the methods of Vasiliev may be applicable to writing interactions for infinite towers of these quantum fields (see [26,27] for an excellent review of these methods). Given the existence of the underlying SUGRA theory, this is a very tantalizing possiblity. These and other directions for future work are discussed in the conclusions. 2. Detour Complexes The simplest example of a geometrical detour complex is given by the superalgebra, on any Riemannian manifold M, generated by the exterior derivative d and the codifferential δ: {δ, d} = .

(1)

Here, the right hand side is the form Laplacian which is a central element of this algebra. These operators act on differential forms ∈ (∧M), which may be viewed as wavefunctions of an N = 2 supersymmetric quantum mechanical model [28], with the Hamiltonian and (δ, d) the two supercharges. Gauging the corresponding worldline 1 A very useful introduction to BPS black holes and the attractor mechanism is [10,11] (the formulation in [16–18] also fits our viewpoint well).

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translation and supersymmetries yields a spinning particle (or 1-dimensional SUGRA) model which can be quantized using BRST machinery. In mathematical terms this amounts to computing the Lie algebra cohomology of the superalgebra (1). However, when defining Lie algebra cohomology for superalgebras, some care is needed [29]. In physics terms this amounts to choices of vacua/polarizations for commuting superghosts [30,31]. It turns out that a distinguished choice exists such that the cohomology is neatly arranged in terms of gauge invariances, Bianchi identities and the equations of motion of a gauge invariant field theory. In a higher spin setting this was first observed in the context of an unfolded formulation and what is called the “twisted adjoint representation” [37,38]. (Very recently the unfolding technique has been shown to be equivalent to the BRST one [32]. The idea of studying worldline descriptions of higher spin systems, via detour and path integral quantization has also been analyzed in [33] and [34,35].) In [36] we used a split choice of ghost polarization2 to construct detour complexes from constraint algebras. (For systems with anti-commuting ghosts, this method reproduces known results [44,45] for totally symmetric higher spin fields). The term “detour complex” was chosen because the BRST technology produced complexes of the type studied recently by conformal geometers, the main idea being to connect standard complexes and their duals by (typically higher order in derivatives) detour operators [46–49]. For the simplest case of the de Rham complex, the detour machinery yields a cohomology neatly encapsulated by the complex. d d d δ δ δ · · · −→ M −→ M −→ M → · · · · · · → M −→ M −→ M −→ · · · . ⏐ ⏐ δd

The self-adjoint detour operator δd encodes the equations of motion δd A = 0 of a p-form gauge field A and connects the standard de Rham complex to its dual. These incoming and outgoing complexes encode the gauge and gauge for gauge symmetries, and Bianchi as well as Bianchi for Bianchi identities of p-form electromagnetism. A more sophisticated example is that of the Kähler detour complex; on these manifolds the exterior derivative and codifferential decompose into Dolbeault operators and their duals [50,51] ¯ d = ∂ + ∂,

∗

δ = ∂ ∗ + ∂¯ ,

subject to the superalgebra 1 ¯ ∂¯ ∗ }. = {∂, 2 In addition, an sl(2) Lefschetz algebra acts on the Dolbeault cohomology of a Kähler manifold M. This corresponds to the R symmetry algebra of the above N = 4 superalgebra ∗ ∗ ∂ ∂ ∂¯ ∂¯ , ¯ = , ,L = ¯ , ∂ ∂ −∂ ∗ −∂ ∗ [H, ] = −2, [H, L] = 2L, [, L] = H. {∂, ∂ ∗ } =

Differential forms on a Kähler manifold are bigraded by their holomorphic and antiholomorphic degrees ( p, q) in terms of which the eigenvalues of the operator H are 2 The technique of split ghost polarizations is equivalent to the twisted adjoint representation of [37,38]. It has also been employed in [39–43].


847

p + q − 21 dim M. The operator maps ( p, q) to ( p − 1, q − 1)-forms by contracting with the Kähler form and the operator L is its dual. The Kähler analog of p-form electromagnetism [52] follows by a detour complex treatment of the spinning particle3 model obtained by gauging worldline translations, supersymmetries and the R-symmetry . Nilpotentcy of Q = ∂ ∂∂p + ∂¯ ∂∂p¯ acting on polynomials in Grassmann even variables p, p¯ with coefficients in ∧M yields the left-hand side of the complex. ∂

M

M

∂

∂¯

M

∂

···

∂¯

∗ ∂¯

M

M

∂∗

M

∂¯

∂

∂

∂∗

∂∗

···

M

∂∗

∗ ∂¯

∂¯

∂∗

¯∗

¯∗

M −−−→ M

∂

M

∂

G

∂

∂¯

M

M

∗ ∂¯

∗ ∂¯

∂∗

∂¯

Upon fixing a dimension for M and a bi-grading ( p, q) this incoming complex becomes the Hodge diamond from complex manifold theory. It may be interpreted as gauge (and gauge for gauge) invariances of the “long” or detour operator G. Explicitly, gauge invariance reads A → A + ∂α + ∂¯ α. ¯ Clearly the equations ∂ ∂¯ A = 0 are invariant, yet potentially over or underdetermined. Taking the Kähler trace yields the desired equations of motion ∂ ∂¯ A = 0. However, the operator ∂ ∂¯ is not self-adjoint and so does not naturally connect the “incoming” Dolbeault complex with the “outgoing” dual complex depicted on the right hand side above. The self adjoint operator √ I1 (2 L) ¯ ∗ ∗ ¯ ¯ G = : I0 (2 L) ( − 2∂∂ − 2∂ ∂ ) + 2 √ (∂ ∂ + L ∂ ∗ ∂¯ ) : L √

∗

found in [52] gives an equivalent equation of motion G A = 0. Here : • : denotes normal ordering of • by form degree and the functional dependence on L through the modified Bessel functions of the second kind is analytic at the origin. In the special case that M is hyperKähler, replacing differential forms by sections of ∧E gives another representation of the above N = 4 supersymmetry algebra: The 3 Supersymmetric mechanics on Kähler manifolds have been extensively studied in [53–56 and 57–60].

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tangent bundle T M for 4n-dimensional manifolds M with quaternionic holonomy splits into a product of vector bundles TM ∼ = H ⊗ E, of rank 2 and 2n, respectively. The connection on a hyperKähler manifold acts on sections X α and X A of H and E, respectively, as ∇ X α = d X α + ωβα X β ,

∇ X A = d X A + BA X B ,

where the one-form BA is sp(2n)-valued. Writing the Levi-Civita connection as ∇ α A in a basis for H ⊗ E, there are sp(2) doublets of exterior derivatives and codifferentials acting on ∧E via dα : X A1 ... Ak → ∇ α[A1 X A2 ...Ak+1 ] , δ α : X A1 ... Ak → k∇α A X A A1 ...Ak−1 , in the index notation explained in Sect. 3. They obey the N = 4 algebra {dα , dβ } = 0 = {δ α , δ β }, 1 {δ α , dβ } = − δαβ , 2 where is the Bochner Laplacian ∇μ ∇ μ . Only an sp(2) subalgebra of the so(2, 2) R-symmetry of this N = 4 superalgebra acts non-trivially in this hyperKähler representation. The non-trivial R-symmetries are built from the sp(2n) invariant tensor J , g : X A1 ...Ak → J [A1 A2 X A3 ...Ak+2 ] , N : X A1 ... Ak → k X A1 ... Ak , tr : X A1 ...Ak → k(k − 1) J AB X B A A1 ...Ak−2 , and obey the algebra [tr, N] = 2 tr,

The dictionary dα ↔

[tr, g] = 4(N − n), [N, g] = 2 g, α [δ α , N] = δ α , [N, d ] = dα , [δ α , g] = 2 dα . [tr, dα ] = 2 δ α ,

∗ ∂ ¯∗ ¯∂ , δ α ↔ −∂ −∂ ,

g ↔ 2L, tr ↔ 2,

between the Kähler and hyperKähler representations of the N = 4 superalgebra allows the Kähler detour complex to be translated directly to a hyperKähler one. In particular, nilpotence of the operator Q = dα ∂ ∂pα on polynomials in the Grassmann even variables p α with coefficients in (∧E) gives gauge and gauge for gauge invariances of the over-determined, Maxwell like, and Einstein versions of the hyperKähler equations of motion dα dα A = 0 ⇒ tr dα dα A = 0 ⇔ G A = 0, √ I1 ( g tr) (dα dα tr + g δ α δ α ) :, G = : I0 ( g tr) ( + 2 dα δ α ) − 2 √ g tr


849

for gauge fields A ∈ (∧E). Explicitly, the gauge invariance reads A → A + dα αα . The equation of motion dα dα A = 0 was first generalized to the more complicated quaternionic Kähler case by Baston [24], and later recovered in the context of BPS, N = 2 supersymmetric black hole systems in [25]. The main result of this paper is to further extend this generalization to the full “Einstein” equations of motion G A = 0 in the quaternionic Kähler setting. It relies on a trio of geometric operators (one of which is Baston’s original second order operator) transforming as a triplet under sp(2)R-symmetries. We now present the basic geometric data on quaternionic Kähler manifolds needed for this paper. 3. Special Geometry HyperKähler and quaternionic Kähler manifolds in dimension 4n and signature (2n, 2n) enjoy sp(2n) and sp(2) ⊗ sp(2n) holonomy, respectively.4 In either case, this implies that the tangent bundle splits into a product of vector bundles [6] TM ∼ =H⊗E of rank 2 and 2n, respectively. Therefore, we denote curved and flat indices by μ, ν, . . . and m, n, . . . respectively, and decompose tangent space indices as m = α A, where A = 1, . . . , 2n and α = 1, 2 label the fundamental representations of sp(2n) and sp(2), respectively. The invariant so(2n, 2n) metric decomposes this way as ηmn = εαβ J AB , where εαβ and J AB are the sp(2) and sp(2n) invariant, antisymmetric tensors. This allows for all indices to be raised and lowered independently. For example, v A ≡ J AB v B , v α ≡ vβ εβα and εα β = δβα = −εβ α . Note that we use an uphill convention. The action of the connection on sections of H and E, respectively, is given by ∇ X α = d X α + ωβα X β ,

∇ X A = d X A + BA X B ,

where both ωαβ and AB are symmetric. On hyperKähler manifolds, only the latter is non-zero. This may be extended to arbitrary tensor products of sections of H and E in the obvious way. For the purposes of calculations involving such products, we specify this action by introducing representations of the sp(2n) and sp(2) subalgebras of the full local Lorentz algebra so(2n, 2n). The generators of these algebras are represented as operators T AB and t αβ , indexed by symmetric pairs of indices, that act on sp(2n) and sp(2) indices by T AB X C = J C A X B + J C B X A , t αβ X γ = γ α X β + γβ X α .

(2)

4 The maximally split signature corresponds to paraquaternionic holonomy – all our results apply to general signatures, this choice being a matter of notational convenience.

850


These operators satisfy [T AB , T C D ] = J C A T B D + J C B T AD + J D A T BC + J D B T AC , αβ γ δ t ,t = εγ α t βδ + εγβ t αδ + εδα t βγ + εδβ t αγ , their extension to higher tensors is by the usual Leibnitz rule, and thus ∇=d+

1 α β 1 A B ω t + T . 2 β α 2 B A

Throughout this paper, the symbol ∇ will refer to this definition. The final geometric ingredient needed here is the Riemann tensor. As a result of special holonomy it has the decomposition [6] Rα A β B γ C δ D = ε(α|γ | εβ)δ J AB JC D + εαβ εγ δ [J(A|C| J B)D + ABC D ].

(3)

Hence, the commutator of covariant derivatives on sections of H and E follows from: [∇ Aα , ∇ Bβ ] φCγ = J B A εγ (α φCβ) + εβα JC(A φ B)γ + εβα D ABC φ Dγ . This specifies an action on higher rank tensors which can be succinctly expressed in terms of the operators 1 1 C [∇ Aα , ∇ Bβ ] = J B A tαβ + εβα T AB + D T ABC D . 2 2 The tensor ABC D is totally symmetric and will appear only seldomly in this paper since it cannot couple to the antisymmetric sections of ∧E which appear in our models. The terms proportional to the constant are present only on quaternionic Kähler manifolds and vanish for the hyperKähler case.5 Finally, note that the Ricci and scalar curvatures are Rmn = −(n + 2)ηmn and R = −4n(n + 2). 4. N = 2 Supersymmetric Black Holes and Quaternionic Geometry Breitenlohner, Maison and Gibbons [60] showed that Kaluza–Klein reduction along a single isometry of a four dimensional, curved space non-linear sigma models coupled to Maxwell fields 1 1 I (4) 4 A ∗ B ∗ J J S=− d x −g R + gAB (ϕ)dϕ ∧ dϕ + F ∧ MIJ F + NIJ F , 2 2 (where A, B = 1, . . . , n S the number of scalar fields and I, J = 1, . . . , n V the number of vector fields) yields a three dimensional curved space non-linear sigma model 1 3 S=− d x −g R + gμν (φ)dφ μ ∧ ∗ dϕ ν . 2 The metric gμν on the moduli space of the three dimensional non-linear sigma model (4) depends on that of the four dimensional sigma model gAB as well as the couplings MIJ and NIJ of the Maxwell field strengths F I to the four dimensional scalars ϕ A . We refer 5 Note that these are not proportional to η r [m ηn]s – the constant curvature Riemann tensor – since general quaternionic Kähler manifolds are not constant curvature.


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to the original paper [60] for the precise formulæ. Suffice it to say, that the n S scalars in four dimensions are enlarged to a set of n S + 2n V + 2 scalars coming from the dilaton, dualized graviphoton, Maxwell Kaluza–Klein scalar modes and dualized three Maxwell fields. They span the moduli space M of the three dimensional sigma model, and in this paper we will be primarily interested in the case that dim M = 4n. In particular when the original four dimensional theory is the bosonic sector of N = 2 SUGRA, the four dimensional scalar moduli space is a Kähler manifold and its image under dimensional reduction is a (para)quaternionic Kähler manifold. This correspondence is known as the c-map [61–64]. When the reduction isometry is generated by a timelike Killing vector, solutions of the three dimensional sigma model correspond to stationary solutions of the four dimensional theory. If we make the additional assumption of spherical symmetry of the three dimensional stationary slices ds 2 = N 2 (ρ)dρ 2 + r 2 (ρ)(dθ 2 + sin2 θ dϕ 2 ), solutions then derive from a one dimensional action 1 S=− dρ N + N −1 (r 2 − r 2 φ μ gμν φ ν ) , 2 where primes denote ρ-derivatives. This model can be interpreted as a relativistic particle moving in a cone metric dr 2 − r 2 dφ μ gμν dφ ν , over the quaternionic Kähler moduli space M. Classical solutions separate into radial motion and geodesics on the moduli space M. Of these, the extremal black hole solutions of the original four-dimensional theory are necessarily in correspondence with lightlike geodesics [60]; the radial quantization of static, spherically symmetric black holes in Einstein and Einstein-Maxwell gravity has been studied in [65–70]. The consequences of the four dimensional local supersymmetry of the underlying N = 2 SUGRA can be incorporated in this minisuperspace approximation by computing the dimensional reduction of the supersymmetry transformations (see [8,9]). BPS states follow by requiring that the transformations of the fermions vanish. This requirement splits into a radial condition dr = N dρ, as well as the BPS conditions of a (worldline) locally supersymmetric extension of a relativistic, massless particle moving in the moduli space M. Indeed, imposing r = N on the constraint N 2 = r 2 −r 2 φ μ gμν φ ν implied by the N -variation of the above action yields r 2 φ μ gμν φ ν = 0. Therefore we can reinterpret r 2 = 1/e as the inverse einbein of a massless relativistic particle moving in M. The coupling of this particle to worldline fermions θ Ai = (θ A∗ , θ A ) is determined by requiring that their supersymmetry variations coincide with those obtained by dimensional reduction of the four dimensional SUGRA variations. This leads to a one dimensional SUGRA with action principle 1 ◦μ 1 i ∇θiA ◦ν i jA B x gμν x + θ A + e θ A θi B θ θ j . S = dt 2e 2 dt 4

852


In this formula ◦

x μ ≡ x˙ μ − V μ αA θ Ai ψiα , is the supercovariantized tangent vector and ψiα are worldline gravitini; the gauge fields for the four local worldline supersymmetries. The BRST quantization of this supersymmetric spinning particle model is a central focus of this paper. 5. HyperKähler Sigma Model We now construct a supersymmetric, non-linear sigma model in a 4n-dimensional, hyperKähler target space (M, gμν ). The field content of the model consists of bosonic worldline embedding coordinates x μ (t), and fermionic spinning degrees of freedom θ Ai (t). Their dynamics are governed by the simple action A 1 μ ν i ∇θi S= dt x˙ gμν x˙ + θ A . (4) 2 dt The (rigid) symmetries of the model are 1. Worldline translations: δx μ = ξ x˙ μ ,

δθ Ai = ξ θ˙Ai .

(5)

λi j = λ ji .

(6)

2. Sp(2)R-symmetry: δθ Ai = λi j θ A j , 3. N = 4 supersymmetry: δx μ = V μ αA θ Ai εiα , μ Vm

μ V Aα

Dθ Ai = −x˙ μ Vμ αA εαi .

(7)

Here = are the inverse written with split flat indices and D is the i i B μ covariant variation: Dθ A ≡ δθ A − δx μ A θ Bi . On functions of x μ it equals δx ρ ∇ρ ; it obviates the requirement to vary covariantly constant quantities. In this regard it helps to observe that δ = D when varying scalars (such as the action). To see explicitly that the action (4) is supersymmetric, we note the identities vielbeine6

∇δx μ D x˙ μ = , dt ∇ A B θ A = δx μ x˙ ν Rμν BA θiB = δx Cα x˙αD BC D, D θi . dt i

(8)

Variations linear in fermions cancel by virtue of the first identity, but there are potentially ∇ cubic fermion terms proportional to 21 θ Ai [D, dt ]θiA . Using the second identity we see C A B that these vanish since ABC D θi θ j θk ≡ 0. 6 The vielbeine/orthonormal frames, denoted V m obey μ

Vμ αA Vν αA = −gμν ,

β

β

Vμ αA V μ B = −δ BA δα .

Special holonomy dictates that in addition to these identities for Vμ αA (jocularly, the “zweimalhalbsovielbein”) it is also true that: 1 1 β β V(μ αA Vν) A = − gμν δα , V(μ αA Vν) αB = − gμν δ BA . 2 2n


853

5.1. Quantization. To quantize the model we write it in first order form 1 i A 1 (1) μ μν ˙ S = dt pμ x˙ + θ A θi − πμ g πν , 2 2 where πμ = pμ + θ Ai μ BA θiB , and directly impose the canonical commutation relations dictated by the Darboux form of the first order kinetic terms: [ pμ , x ν ] = −iδμν ,

j

{θ Ai , θ B } = −i i j J AB .

(9)

We introduce a Fock representation on a vacuum state |0 as7 ηA ∂ i , pμ |0 = 0 = θ A → |0. −i ∂η∂ A ∂η A The fermionic anticommutator (9) implies {

∂ , η B } = δ AB , ∂η A

so the creation operators η A produce Fock states which may be identified with sections of the bundle ∧E: (∧E) ≡ φ A1 ...Ak (x)η A1 · · · η Ak |0 ≡ |φ A1 ...Ak .

(10)

The form of πμ in the action above may be understood in terms of this representation; in general the covariant momentum is πμ = pμ −

i Pμ mn M mn , 2

where M mn generate the local Lorentz algebra [M mn , M r s ] = M ms ηnr − M ns ηmr + M nr ηms − M mr ηns . For hyperKähler manifolds the spin connection acts as Pmn M mn = AB T AB , where T AB , defined in (2), generate sp(2n). On ∧E one may alternatively represent sp(2n) by bilinears in the spinning degrees of freedom; T AB ≡ −2η(A

∂ ∂η B)

(11)

acts identically on to the operator introduced in (2). This explains the form of πμ ; acting on ∧E-valued states it produces the covariant derivative8 i πμ = pμ − μAB T AB = −i∇μ . 2 7 The positive definite quantum mechanical inner product for the spinning degrees of freedom is defined † by taking η A = ∂ A . ∂η 8 As usual for first quantized models, π π = ∇ ∇ because π does not see the open index of π . μ ν μ ν μ ν

854


5.2. Charges. Our next task is to write down charges generating the symmetries (5)–(7). At the quantum level these are subject to ordering ambiguities which we resolve by relating symmetry charges and geometric operations. Firstly, we expect the Hamiltonian – the generator of worldline translations – to correspond to the Laplacian ≡ ∇μ ∇ μ : −2H = . This is true so long as we adopt the quantum ordering H=

i 1 π Aα π Aα − Aα BA π Bα , 2 2

παA ≡ V μ αA πμ .

The four supercharges transform as a doublets under the sp(2) holonomy subalgebras as well as under a Lefschetz-Verbitsky sp(2) algebra which we introduce below. They are built from the sp(2n) contraction of the spinning degrees of freedom θ Ai with the covariant momenta. On states they act as A η ∇α A dα i ≡ Qα ≡ , δα −∇αA ∂η∂ A where, again, the operator ordering is chosen based on the natural geometric action: |∇α[A1 φ A2 ...Ak+1 ] dα . = Q iα = δα |k∇α A φ A A2 ...Ak The operator dα : k E → H ⊗ k+1 E, belongs to a sequence of Dirac operators introduced by Baston in a study of quaternionic complexes [24]. Indeed the operators dα and δ α are analogous to the Dolbeault operators on forms, but they act on (∧E) instead of (∧T M). Next, we present the R-symmetry charges generating (6). They can be derived from geometric grounds alone as follows: Firstly observe that since we deal with wavefunctions (10), there is no prohibition on adding anti-symmetric E-tensors with differing number of indices. The state in (10) is in fact an eigenstate of the number or “index” operator N = ηA

∂ . ∂η A

(12)

The invariant tensor J AB allows us to construct two further bilinears, tr =

∂ ∂ , ∂η A ∂η A

g = η AηA.

(13)

These act on states as suggested by their names; the operator tr removes a pair of indices by tracing with the invariant tensor J AB : tr |φ A1 ...Ak = k(k − 1)|φ A AA3 ...Ak . Conversely, its adjoint, g adds a pair of indices by multiplying by J AB and antisymmetrizing: g |φ A1 ...Ak = |J[A1 A2 φ A3 ...Ak+2 ] .


We arrange these generators in a symmetric 2 × 2 matrix g N−n ij f = . N−n tr

855

(14)

These are precisely the charges corresponding to the R-symmetries (6) and obey the sp(2) algebra [ f i j , f kl ] = ki f

jl

+ k j f il + li f

jk

+ l j f ik .

We note that one may view this representation of sp(2) as the Howe dual of the representation of sp(2n) generated by T AB (i.e., sp(2) and sp(2n) are the commutants of one another in so(2n, 2n)). In an equation [ f i j , T AB ] = 0. Moreover, the quadratic Casimirs of these two algebras are related by c = g tr − N(N − 2n − 2) =

1 ij 1 f f i j + n(n + 2) = − T AB T AB . 2 2

(15)

The above geometric operators are closely related to the so(4, 1) Verbitsky algebra acting on differential forms on hyperKähler manifolds. (An elegant description of this algebra from a supersymmetric quantum mechanical viewpoint is given in [51].) In fact {g, N, tr} generate an sp(2) subalgebra of so(4, 1) corresponding to writing d x μ as d xαA and studying Verbitsky transformations which do not act on the H -index α. Alternatively, we may view this algebra as a generalization of the Lefschetz subalgebra that acts on forms on a Kähler manifold. Henceforth we adopt the hybrid designation “Lefschetz–Verbitsky algebra”. After some calculation we find9 1 ij εαβ , [ f i j , Q kα ] = 2 k(i Q αj) , 2 (16) [ f i j , f kl ] = ki f jl + k j f il + li f jk + l j f ik , [, f i j ] = 0 = [, Q iα ]. j

{Q iα , Q β } =

5.3. Summary. The hyperKähler sigma model presented in this section (and summarized in Fig. 1) provides a geometric representation of the algebra {Q I , Q J } = J I J D, with J the invariant rank two tensor of so(2, 2). This algebra belongs to the family of orthosymplectic algebras for which the BRST detour quantization procedure [36] was developed. The most general R-symmetry of this algebra is so(2, 2), with generators R I J acting as [R I J , Q K ] = 2JK [I Q J ] . 9 It is interesting to note that this algebra is an Inönü–Wigner contraction of the osp(2|2) superalgebra where the bosonic sp(2) and so(1, 1) blocks are generated by f i j and H respectively while Q iα belong to off diagonal fermionic blocks. The rescaling of osp(2|2) generators H → λ2 H and Q iα → λ Q iα , and the limit λ → ∞ recovers the algebra above.

856


Fig. 1. Geometric data for the quantized hyperKähler sigma model

Upon breaking the index I = iα , so that J I J = αβ i j , a Howe dual pair of sp(2) α(i j) i subalgebras generated by R(αβ)i and R α are readily identified. In our hyperKähler sigma model, only the Lefschetz–Verbitsky sp(2) part of the R-symmetry algebra acts α(i j) non-trivially and is identified by R α → f i j . The model we have written down makes sense also on a quaternionic Kähler manifold. The geometric interpretations of the charges and wavefunctions is unaltered. What does change however is the algebra of charges which is no longer a super Lie algebra, but receives deformations from the non-vanishing sp(2) holonomy of a quaternionic Kähler manifold. Fortunately however, these deformations produce a first class constraint algebra. Therefore local, or spinning particle models can be constructed by gauging supersymmetries. These are the subject of the next section. 6. Quaternionic Kähler, N = 4, d = 1 SUGRA Upon replacing the hyperKähler target space with a quaternionic Kähler one, it is no longer possible to maintain the rigid N = 4 supersymmetry algebra (16). However, by requiring the algebra to hold only weakly we may instead study local symmetries. There are various choices for first class algebras built from the generators H , Q iα and f i j . Gauging the Hamiltonian H yields a model which is worldline reparameterization independent—generally a desirable feature. Local, N = 4, worldline supersymmetry is achieved by gauging the supercharges Q iα . Thereafter, one can also consider gauging some combination of R symmetry generators. From a spinning particle perspective gauging {H, Q iα } and {H, Q iα , f i j } might seem most natural. In general the choice depends on the particular physical or geometric application one has in mind. Also, in general, when quantizing a first class constraint algebra, one needs to keep in mind what quantization


857

procedure will be employed. Possibly the simplest choice is a naïve Dirac quantization where one attempts to impose the constraints directly as operator relations on the physical Hilbert space. Often however, this is not the most interesting choice, and far more can be learned from a BRST approach. In this section we construct the classical spinning particle models corresponding to the {H, Q iα } and {H, Q iα , f i j } gaugings. In the remainder of the paper, we will be primarily concerned with the BRST quantization of the former of these. In particular we show, motivated by ideas from higher spin theories, that gauging only a single R symmetry generator tr within a BRST detour setting produces a gauge invariant quantum field theoretical model on quaternionic Kähler spaces. The first step is to introduce Lagrange multipliers (gauge fields) for each constraint Constraints

Gauge Fields

H ≈0 Q iα ≈ 0

Lapse N Gravitini ψiα

H ≈0 Q iα ≈ 0 f ij ≈ 0

Lapse N Gravitini ψiα Yang–Mills Ai j

In this one-dimensional setting, these gauge fields have no dynamics. The charges Q iα and f i j are the same as those of the hyperKähler sigma model in Sect. 5, while we add curvature corrections to the Hamiltonian H reflecting that the background is now quaternionic Kähler. These are determined by ensuring that the algebra of charges is first class. Let us give details for each model separately. 6.1. Rigid Lefschetz–Verbitsky model. Gauging only the Q iα and H yields a model with rigid Lefschetz–Verbitsky symmetries. Since we work in a quaternionic Kähler target space as described in Sect. 3 the connection ∇ now is both sp(2) and sp(2n)-valued. There are two easy methods to compute the (second order) action and its symmetries. The first is to start with the sigma model action (4) and to proceed using the Noether method, whose first step couples the gravitini to the supersymmetry current/charges Q iα . This computation is analogous to the one employed by Bagger and Witten [6] to compute matter couplings to N = 2, d = 4 SUGRA. Alternatively, we can begin with a first order action given by the sum of the standard symplectic current dt{ pμ x˙ μ + 21 θ Ai θ˙iA } and the product of Lagrange multipliers (N , ψiα ) with their corresponding constraint. Thereafter, a Legendre transformation yields the second order action. The results are equivalent and we find 1 ◦μ 1 i ∇θiA N i ◦ν jA B x gμν x + θ A + θ θ θ θj , S = dt (17) 2N 2 dt 4 A iB which enjoys symmetries:

858


1. Local worldline reparameterizations: δx μ = ξ x˙ μ ,

δθ Ai = ξ θ˙Ai ,

δN =

d(ξ N ) dt

δψiα =

d(ξ ψiα ) . dt

2. Sp(2)R-Rigid symmetry: δθ Ai = λi j θ A j ,

δψαi = λi j ψα j .

3. Local N = 4 supersymmetry: δx μ = V μ αA θ Ai εiα , 1 ◦ Dθ Ai = − x μ Vμ αA εαi , N δ N = ψαi εiα , Dψαi =

N i A j ∇εαi + θ θ ε . dt 2 A j α

In these formulæ, D is again the covariant variation, but just like the connection ∇, it β too is now sp(2) covariant so that, for example, Dψαi = δψαi − δx μ ωμ α ψβi . Also, we have introduced the supercovariant tangent vector ◦

x μ ≡ x˙ μ − V μ αA θ Ai ψiα . To verify invariance of this action, notice that the supercovariant tangent vector transforms as i ∇θ δN ◦ μ 1 ◦ ◦μ A α x + V μ αA x ν V [μ αA V ν] βA εβi ψiα . Dx = ε − θ Ai ψiα + 2N dt i N ∇εαi dt is shorthand for the two fermion gravitini variations. The last ◦ x ν A[μν] so do not contribute to the variation of the bosonic matter

Here ψiα ≡ Dψαi −

terms are of the form 1 x◦ 2 kinetic term 2N , while the leading term perfectly ensures the kinetic terms vary into 1 ◦ ◦ μ 1 i ∇θiA ∇ 1 ◦α 1 i i x μ x + θA − x A ψα + θ A D, θiA . (18) δ = 2N 2 dt N 2 dt

These cancel the variation of the four point fermi coupling to the Riemann tensor. This relies on the quaternionic Kähler analog of the identity (8) which yields δx x˙ times the Riemann tensor for the commutator of covariant worldline derivatives and variations. ◦ Trading x˙ for x yields exactly the terms required to cancel the variation of the lapse N multiplying the four point coupling. A final point worth stressing is that the parameter is not fixed by the requirement of local supersymmetry in one dimension. In dimension four, coupling N = 2 SUGRA to matter fixes the scalar curvature in terms of Newton’s constant κ [6]. (This follows by requiring variations of the Einstein–Hilbert and Rarita–Schwinger terms to cancel at order κ 0 in the Noether procedure.) Both these terms are absent in our one dimensional model.


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6.2. Gauged Lefschetz–Verbitsky model. To gauge the Lefschetz–Verbitsky sp(2) symmetry we need only replace the covariant derivative ∇ in (17) by its sp(2) covariantization A

∇ defined by A

∇v i ∇ vi ≡ + Ai j v j . dt dt Therefore the gauged action reads ⎧ ⎫ A ⎨ 1 ◦ ⎬ A N i 1 ∇ θi ◦ x μ gμν x ν + θ Ai + θ A θi B θ j A θ jB , S = dt ⎩ 2N ⎭ 2 dt 4

(19)

which differs from (17) by a Lagrange multiplier term 21 θ Ai Ai j θ j A (so the gauge field Ai j is a unit weight, worldline tensor density or volume form). In addition to the new local Lefschetz–Verbitsky symmetry, j

δθ Ai = λij θ A ,

δψαi = λij ψαj ,

δ Ai j = λ˙ i j + 2 Ak(i λk , j)

the supersymmetry transformations are modified to read δx μ = V μ αA θ Ai εiα , 1 ◦ Dθ Ai = − x μ Vμ αA εαi , N δ N = ψαi εiα , A

N i A j ∇ εαi + θ θ ε , = dt 2 A j α δ Ai j = 0.

Dψαi

These results and other gaugings follow easily from the canonical analysis of the next section. 6.3. Dirac quantization. To perform a canonical analysis and Dirac quantization of Lefschetz–Verbitsky model we first note that the symplectic structure the rigid ! dt pμ x˙ μ + 21 θ Ai θ˙iA implies the same Fock space structure as in the hyperKähler case (see in particular formulæ (9)–10)). The Dirac Hilbert space is therefore again sections of the antisymmetric sp(2n) tensor bundle ∧E. The (quantized) supercharges Q iα and Lefschetz–Verbitsky generators take the same form as in the analysis of the hyperKähler sigma model in Sect. 5.2. The Hamiltonian H receives a curvature correction term (implied by the four-fermi term in the action (17) proportional to the lapse N ). Again these charges may all be quantized with orderings obtained by ensuring that the quantum algebra of constraints is first class. The Dirac quantization of the model then amounts simply to imposing the conditions H = Q iα = 0 on wavefunctions valued in (∧E). (The gauged Lefschetz–Verbitsky model incurs the additional constraint f i j = 0.) We pay little attention to an analysis of this quantum system because it suffers a certain deficiency which we now explain, and will remedy in the next section by means of a BRST analysis:

860


On a quaternionic Kähler manifold we must remember that the spin connection has both sp(2n) and sp(2) valued parts which couple naturally to the respective generators T AB and tαβ . However, from the spinning degrees of freedom θ Ai of this model, we can only build a representation of the sp(2n) generators T AB . On the one hand, this seems sufficient because acting on ∧E-sections, we still have iπμ = ∇μ . But, acting with a supersymmetry generator Q iα introduces an sp(2) index α, and we seem to have no way, in the spinning particle model context, to obtain further covariant derivatives acting correctly on α. A geometer might consider constructing supersymmetry-like operators built from the covariant derivative by fiat (and in fact, the geometric calculus Sect. 8 of this paper can be taken on its own and read this way). However, there is a very natural physical mechanism to introduce additional spinning degrees of freedom that can represent the sp(2) generators tαβ . In fact, this is precisely what BRST quantization of the model does. 7. BRST and the Geometry of Ghosts The one dimensional quaternionic Kähler spinning particle model enjoys local worldline supersymmetry and reparameterization invariances. This implies that they form a first class algebra (even though the supercharges do not commute with the Hamiltonian unlike those in the hyperKähler sigma model where they generate genuine symmetries). In this section we present the nilpotent, quantum, BRST charge for this algebra. Again, unlike the hyperKähler model, this constraint algebra is higher rank; it does not form a Lie algebra. This means that, in principle, we need to resort to homological perturbation methods to construct the BRST charge. (The reader may consult [15] for a detailed account of the analysis of gauge theories using BRST techniques and in particular the construction of a nilpotent BRST charge for higher rank algebras.) Although standard, such a computation is rather involved, so instead we present a solution relying on the underlying quaternionic geometry. The general structure of the BRST charge we search for is given by expanding it in powers of the worldline reparameterization ghost c and its antighost b represented ∂ as ∂c , Q BRST = c D + Q − M

∂ . ∂c

(20)

If our constraint algebra were a Lie algebra (as it is in the hyperKähler case), the operator D would be the worldline Hamiltonian and Q the contraction of the supercharges with commuting supersymmetry ghosts ciα . However, since we have a higher rank constraint algebra, we must add terms with higher powers of ghosts and antighosts. We determine these by making a simple geometric ansatz for Q and then requiring nilpotency of Q BRST . The key geometric idea is that ghosts and antighosts can be used to represent the sp(2) special holonomy generators. The quantized commuting superghosts ciα and superantighosts bαi with algebra β

[bαi , c j ] = δ ij δαβ j

j

(21)

allow formation of bilinears cαi bβ − cβ bαi that generate a faithful representation of so(2, 2), the R-symmetry algebra of our first class constraint superalgebra, on the ghosts


861

(and/or antighosts). Specializing to the Howe dual subalgebras generated by f i j = −2c(iα b j)α , gh

αβ

β)

tgh = −2ci(α b i ,

(22)

we obtain representations of the Lefschetz–Verbitsky and H -bundle special holonomy sp(2) algebras, respectively. (We will discuss the precise definition of the superghost Hilbert space at the end of this section, but for now concentrate on building a nilpotent BRST charge.) This means that we can solve the problem of the covariant momentum operator πμ discussed in the previous section—namely that it was not covariantized with respect to the sp(2) holonomy—by using the above ghost representation for tαβ . So we now construct a covariant momentum operator i i μ ≡ pμ − μ BA T AB − ωμ αβ tαβ , 2 2

(23)

which acts on both E and H bundles. (In some sense, the ghosts play the rôle of frames for the bundle H .) In turn we introduce BRST-extended supersymmetry charges θ Ai V μ αA μ and consider the ansatz A i η Q ≡ iciα V μ α A μ ∂ ∂η A

for the form of Eq. (20). Before proceeding, it is worth noting that we have actually found a new Dirac operator: Reunifying sp(2) and sp(2n) indices as a single so(2n, 2n) index m = Aα and forming the combination γ m = cαi η A ∂η∂ A , i

we find a Clifford algebra {γ m , γ n } = M ηmn , 1 M ≡ cαi cαi . 2 Since the covariant momentum (23) acts as the covariant derivative, a Dirac-type operator follows Q = γ m ∇m .

(24)

Returning to our BRST charge computation, a simple Weitzenbock-like calculation10 shows Q2 = MD,

(25)

where the BRST-extended Hamiltonian is 1 n gh ij 2 D = − ( f i j + f gh )( f i j + f i j ) − (n + 2). 4 2 10 Note that the computation of the term coupling the curvature to two Dirac matrices relies heavily on γ m being a composite built from ghosts and spinning degrees of freedom.

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In this expression, = + 41 (T 2 + t 2 ) is a quaternionic Kähler Lichnerowicz wave operator, which will be introduced in Sect. 8. It satisfies [, Q] = 0. Further, since f i j gh ij and f gh obey [ f i j + f i j , ckα Q kα ] = 0 and the latter commutes with11 M, we have the following identities [D, M] = [Q, D] = [Q, M] = Q2 − MD = 0.

(26)

These immediately imply that the BRST charge (20) is nilpotent. The form of this BRST charge is exactly suited to the detour quantization methods of [36]. To that end we next specify our choice of ghost vacuum. We represent the ghost algebra (21) in a Fock representation by splitting the ghosts and antighosts into derivatives and power series coordinate coefficients. The choice of vacuum is determined by splitting the Verbitsky–Lefschetz doublets as ∂ ∂ ciα = z α , biα = − p α . (27) ∂ pα ∂z α Therefore we may view (z α , p α ) as creation operators for symmetric H -bundle indices. So states in the superghost extended Hilbert space are sections of A1 t (∧E ⊗ (H )⊗2 ) ≡ φ A1 ...Ak βα11 ...β · · · η Ak z α1 · · · z αs pβ1 · · · pβt |0 ...αs (x) η (β ...β )

= |φ[A1 ...Ak ] (α11 ...αts ) = ⎧ ⎨

k

⎩

"

⊗

t #$

%

.

⊗ $ %" # s

In the Young diagram notation the column denotes antisymmetrized E-indices while the rows are symmetrized H -indices. We now have a well-defined BRST cohomology. Before analyzing it via BRST detour methods, we take a short geometric excursion to develop a quaternionic calculus of the various operators that will appear in those results. 8. A Quaternionic Geometric Calculus On a d-dimensional Einstein manifold the Riemann tensor decomposes as Rμνρσ =

2 (gμρ gνσ − gνρ gμσ ) + Wμνρσ . (d − 1)(d − 2) $ %" # $ %" # Constant Curvature

Weyl

The special constant curvature case—when the Weyl tensor vanishes—enjoys many distinguishing properties, including a Lichnerowicz wave operator which commutes with generalized gradient and divergence operators acting on tensors of very general types. Comparing this formula with the one for the quaternionic Kähler Riemann tensor in (3) we see that the totally symmetric tensor ABC D plays a rôle similar to the Weyl tensor;12 11 In fact, linear combinations of the ghost bilinears mentioned below Eq. (21) are precisely those which commute with M. 12 In fact, in four dimensions it plays the rôle of the anti-self dual Weyl tensor [1,24].


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if we could somehow find a “regime” in which it did not contribute we might be able to analyze quaternionic Kähler geometry along lines similar to the constant curvature case. In fact, exactly such a regime does exist, namely sections of the product of ∧E with the tensor bundle T H (with sections being arbitrary H -tensors) (E ⊗ T H ) φ[A1 ...Ak ] α1 ...αs ,

φ[A1 ...Ak ]

∈

∈

the idea being that antisymmetry in sp(2n) indices prevents the totally symmetric tensor ABC D from contributing. In particular, the central operations will be the quaternionic generalizations of the Dolbeault operators α d : (E ⊗ T H ) −→ (E ⊗ T H )⊗2 δα α1 ...αs

→

α φ α1 ...αs ∇[A 1 A2 ...Ak+1 ]

k∇ αA φ A [A2 ...Ak ] α1 ...αs

.

These operators are motivated by the quantized supersymmetry charges of the previous sections, but are more general since they can act on arbitrary H -tensors. For computations, it is often useful to adopt a hybrid E-index free notation where φ[A1 ...Ak ] α1 ...αs → α1 ...αs = φ A1 ...Ak α1 ...αs η A1 · · · η Ak , dα = η A ∇ αA , δ α = −∇ α A

∂ , ∂η A

and the Grassmann variables η A play the rôle of the anticommuting differentials d x μ employed in the theory of differential forms. The non-dynamical Lefschetz–Verbitsky charges g N−n ij f = N − n tr act exactly as described in 5.2 on the antisymmetric E-indices (with the same expressions in terms of η’s), namely adding or removing pairs of antisymmetrized indices using the invariant tensor J AB or counting indices. In terms of these dα , δ α obey a very elegant algebra 1 {dα , dβ } = − g t αβ , 2 1 αβ 1 α β {d , δ } = ε ( − c) − t αβ (N − n), 2 2 1 {δ α , δ β } = − tr t αβ , 2

(28)

where c is again the Lefschetz–Verbitsky sp(2) Casimir operator of (15). These formulæ can be repackaged even more simply by noticing that the operator 2 1 1 T = T AB T AB = + T 2 + t 2 , with t 2 = tαβ t αβ 4 4

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commutes with dα and δ α . This is an extremely important result, so we shall call a quaternionic Kähler Lichnerowicz wave operator. Its existence validates our claim that by studying the bundle ∧E ⊗ T H , quaternionic Kähler geometry could be made to mimic its constant curvature counterpart. Specialized to totally symmetric H -tensors, the operators (dα , δ α ) coincide with the action of the BRST-extended supersymmetry charges in Sect. 7, therefore we adopt the suggestive notation dα Qiα = , δα and call these operators generalized supercharges. We may now unify the algebra (28) as j

{Qiα , Qβ } =

1 & − 1 f i j tαβ , εαβ i j 2 2

with & ≡ − 1 f i j f i j − 1 tαβ t αβ − n (n + 2). 4 4 2 It is interesting to note that these formulæ enjoy a complete symmetry when all H -indices α, β, . . . are exchanged with their Lefschetz–Verbitsky counterparts i, j, . . .. This symmetry appears more starkly when we compute the products of generalized supercharges j

Qiα Qβ =

1 & − 1 f i j tαβ − 1 εαβ bi j − 1 i j bαβ , εαβ i j 4 4 2 2

where we have defined the bilinears bi j ≡ Q(iα Q j)α ,

i bαβ ≡ Qi(α Qβ) .

Observe that, since the generalized supercharges form sp(2) doublets under Lefschetz– Verbitsky and H -symmetries [ f i j , Qkα ] = ki Qαj + k j Qiα ,

[tαβ , Qiγ ] = εγ α Qiβ + εγβ Qiα ,

the six charge bilinears bαβ and bi j form two adjoint sp(2) triplets. This leads one to wonder whether these operators form a pair of sp(2) algebras when commuted among themselves. This question is particularly pressing when we observe that the operator dα dα + g, coincides with that introduced by Baston in his construction of quaternionic analogues of Dolbeault cohomology on quaternionic Kähler manifolds. In fact, this operator is one of a triplet of operators Bi j = bi j + f i j which we shall call Baston operators. This structure of R-symmetry groups represented in terms of bilinears in supercharges has appeared before [24]. For example, for differ¯ − 2∂δ − ential forms on a Kähler manifold, bilinears in the Dolbeault operators {δ δ, ¯ ∂ ∂} ¯ obey an sp(2) Lie algebra (up to an overall factor of the central form Laplacian 2∂¯ δ, on the right-hand side of commutators). Also a similar phenomenon holds for more


865

Fig. 2. The quaternionic Kähler calculus

general orthosymplectic algebras [7]. Moreover, the Kähler result immediately implies the same algebra for the bi j on hyperKähler manifolds. In the more general quaternionic Kähler case one no longer finds a Lie algebra built from bi j but instead the following rather interesting deformation thereof:13 & − 2) B j)l) + B j)l) ( & − 2) − f j)l) (bαβ t αβ + 1 t 2 ) . [Bi j , Bkl ] = (i(k ( 2 The Weyl ordering on the right-hand side is necessary because (as opposed to the qua& is not central. Note ternionic Kähler Lichnerowicz wave operator ) the operator that the operators bαβ + tαβ obey an analogous algebra, thanks to the aforementioned symmetry between H -indices and Lefschetz–Verbitsky ones. The main formulæ of this section are summarized in Fig. 2. We now orchestrate these geometric results with our BRST detour techniques to construct our main result, a gauge invariant quaternionic Kähler quantum field theory. 9. The Quaternionic Kähler Detour Complex The BRST detour quantization formalism presented in [36], takes as its input a BRST charge of the form (20), together with a representation of the underlying constraint algebra acting on sections of a bundle over some manifold M, and outputs a classical field 13 It would be interesting to investigate whether the last terms in this formula can be absorbed by replacing & with the BRST Hamiltonian. Of course, this could only be the case specializing to the BRST the operator superghost Hilbert space of the previous section.

866


theory on M. The equation of motion, gauge invariances, and Bianchi identities are concisely summarized in a detour complex Q Q Q Q Gauge Gauge Equations of Bianchi/Noether −→ −→ · · · . · · · −→ parameters −→ fields motion/currents identities ⏐ D−Q M −1 Q ⏐ The · · · on the ends of the complex describe any gauge for gauge symmetries and their accompanying Bianchi for Bianchi identities. The models described by the above complex depend on towers of gauge fields (possibly infinitely many for the case when the constraint algebra contains Grassmann odd generators). There are cases when these towers of gauge fields have a simple geometric interpretation (including the quaternionic Kähler models described here–see our conclusions for a discussion of this point). These towers of gauge fields arise because the physical cohomology retains a dependence on certain bilinears in ghosts. Generically it is desirable to remove this ghost dependence; this can be achieved by gauging further combinations of R symmetries (the “ghostbusting” procedure of [36]). This leads to more standard physical models with equations of motion and local invariances of the form ( + · · · )A = 0,

δ A = Dα,

where is typically the Laplace operator, A denotes some type of gauge field, and the operator D generates its gauge invariance. The · · · ’s stand for terms required for the equation of motion to be gauge invariant. The operator + · · · can be expressed in a simple “Labastida” form (a name which refers to its origin in the theory of higher spin theories) or equivalently as a self-adjoint “Einstein operator” (this name was chosen since the linearized Einstein tensor is one of the simplest examples). The latter form immediately implies a gauge invariant action principle. Let us now apply these results to the model at hand, we focus on the main formulæ, referring the reader to the articles [36] for detailed derivations of the underlying methodology. Firstly the “long operator” D − QM −1 Q can be defined as acting on wavefunctions (y) ∈ ∧E[y] built from polynomials in a commuting bilinear in superghosts y = 2z α pα with coefficients in (∧E) (because this space forms the ghost number zero kernel of the operator M). Explicitly it yields a gauge invariant equation of motion gh

Bi j f i j = 0,

(29) gh

where, acting on functions of only y, the operators f i j have the simple expression gh fi j

y = −2(y∂ y + 1)

− 2(y∂ y + 1) 4(y∂ y2 + 2∂ y )

.

This model is but a stepping stone to our theory of interest, obtained by also gauging the Lefschetz–Verbitsky generator tr. This choice may seem ad hoc, but is well known in the higher spin literature (for example, it is necessary to obtain the linearized Einstein tensor in the case of a spin 2 theory). In particular it removes all dependence of the


867

physical cohomology on the ghost bilinear y. The physical gauge fields now take values in ∧E only. In fact, gauging the R-symmetry tr amounts to restricting the y dependence of (y) in the detour complex to √ I1 ( ytr) = √ ϕ, ϕ ∈ ∧E, ytr and pushing the long operator in (29) past the operator-valued Bessel function yields the very simple “Labastida” equation of motion tr (dα dα + g) ϕ = 0.

(30)

In particular, notice that this equation factorizes as the product of tr with the operator discovered long ago by Baston [24]. In fact this gauge theory, on a quaternionic Kähler manifold mimics the higher form ( p, q)-form Kähler Electromagnetism theory presented in [52] (observe the correspondence between the Dolbeault bilinear ∂ ∂¯ and the Baston operator dα dα + g). The Labastida equation of motion enjoys the Maxwell like gauge invariance δϕ = dα ξα , thanks to the identity

dα dα + g dβ ξβ = 0, first uncovered by Baston [24]. In fact the Labastida equation of motion has further gauge for gauge symmetries and accompanying Bianchi for Bianchi identites. These are most easily displayed by writing the Labastida equation of motion in a form following from the variation of an action. This is achieved by constructing the self-adjoint Einstein operator14 √ √

I1 ( g tr) I1 ( g tr) G = : √ : tr dα dα + g = δ α δ α + tr g : √ : = G∗ , 2 g tr 2 g tr in terms of which the Labastida equation of motion is equivalent to the “Einstein” equation of motion Gϕ = 0. The Einstein operator has the compact, and manifestly self-adjoint expression

√ G = : I0 ( g tr) dα δ α + δ α dα + 2 (N − n) −2

√ I1 ( g tr) √ g tr

(dα dα + g) tr + g (δ α δ α + tr) :

In all the above formulæ, normal ordering denoted by : • : puts all factors of g and tr to the far left and right, respectively and we have restored the dependence on the scalar 14 The derivation of this result is described in [36,42,43] and amounts to composing the long operator with the Bessel series to balance its appearance on the right in [42,43] and fixing y-independent representatives of coker (y + g).

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curvature through so that the → 0 hyperKähler limit is manifest. It is important to note that this operator acts on sections of ∧E of arbitrary degree. Therefore, the equation of motion we write down is really the generating function for the equations valid at any degree and in arbitrary dimensions, this is what necessitates the operator-valued Bessel functions. Given the Einstein operator, we can now express the equations of motion, gauge and gauge for gauge invariances, Bianchi and Bianchi for Bianchi identities neatly in a single complex D D F F · · · −→ ∧E ⊗ H −→ · · · · · · −→ ∧E ⊗ H −→ · · · . ⏐ ⏐ G

(31)

Here the operators D and F are closely related to the Dirac and Dirac–Fueter operators introduced by Baston [24]. Explicitly, they act on sections of ∧E ⊗ H as D : φ A1 ...Ak α1 ...αs → s∇[Aα 1 φ A2 ...Ak+1 ]α α1 ...αs−1 ,

(32) α2 ...αs+1 ) 1 A F : φ A1 ...Ak α1 ...αs → k∇ (α . A φ A1 ...Ak−1 ' s A1 · · · η Ak z · · · z ∈ ∧E ⊗ H , In an index free notation where = k,s φ αA11...α α1 αs ...Ak η we may simply write D = η A ∇α A

∂ ∂ = dα , ∂z α ∂z α

F = z α ∇ αA

∂ = zα δα . ∂η A

Both these operators are nilpotent by virtue of the algebra (28) and the identity t αβ ψαβγ1 ···γs = 0. Moreover, (dα dα + g) D = 0 = F (δ α δ α + tr), verify the veracity of the complex (31). The incoming complex with differential D can be viewed as the quaternionic generalization of the Dolbeault complex [24], while the outgoing complex with differential F is its dual (i.e. the Dirac–Fueter type operator F is a codifferential). Physically they encode gauge invariances and Bianchi identities. The Einstein operator G gives the detour connecting the two complexes and, physically, the equations of motion. Notice also, that it can connect the equations of motion at any degree in ∧E or H , so gauge potentials are generic sections of ∧E ⊗ H . The mathematical elegance of this model is perhaps surprising, but even more remarkable is its rôle as the arena for a minisuperspace quantization of N = 2 supersymmetric black holes. We further discuss this and other possible applications of our theory in the conclusions. 10. Conclusions The results presented in this paper rely on an analogy between (i) differential forms on a Kähler manifold, (ii) tensors on a constant curvature manifold and (iii) the bundle ∧E ⊗ T H over a quaternionic manifold obtained by splitting its tangent bundle using the sp(2n) ⊗ sp(2) special holonomy and then taking antisymmetric sections of the sp(2n) part E


869

Fig. 3. A map of the physical models encountered in this paper

along with arbitrary H -tensors. The analogy with Kähler differential forms holds because the natural geometric operators on this bundle are in correspondence with the Dolbeault operators and the generators of the Lefschetz symmetry of Dolbeault cohomology. There is a relation to constant curvature manifolds because, acting on sections of ∧E, only the covariantly constant part of the quaternionic Kähler Riemann tensor contributes. This means that the properties of the geometric operators we have studied are algebraically similar to the Lichnerowicz wave operator and the set of geometric operators that commute with it on a constant curvature manifold. In fact a main result of this paper is the geometric calculus of operators, including a central wave operator, acting on (∧E ⊗T H ). Remarkably, this seemingly purely mathematical structure was motivated by a study of supersymmetric black holes in four dimensional spacetime. The route from four dimensional black holes to local quantum field theories on quaternionic Kähler manifolds is sketched in Fig. 3. It began with N = 2 SUGRA in four dimensions. Reducing along an isometry and specializing to spherical symmetry led to a spinning model with four local worldline supersymmetries. Thanks to the c-map this spinning particle moves in a quaternionic Kähler manifold. Moreover, fermionic degrees of freedom were retained in order that the BPS conditions of the spinning particle model corresponded to the reduced ones of the four dimensional SUGRA, and therefore in turn to the linear evolution equations of the attractor mechanism. We then studied the quantization of this model through BRST detour methods. This led to the gauge invariant equation of motion (29). Let us make a few remarks on this model. Given a 4n-dimensional quaternionic Kähler manifold, it is always possible to find a 4n + 4 dimensional hyperKähler manifold whose metric is a quaternionic cone over the

870


original 4n-dimensional model [71–75]. In the work [71], the dimensionally reduced supersymmetry parameters of the four dimensional SUGRA were shown to correspond to the extra four coordinates required to build a 4n + 4 dimensional hyperKähler cone over the quaternionic Kähler, stationary, spherically symmetric, black hole moduli space. However, in BRST quantization the ghosts correspond to the local gauge parameters, in particular the superghosts play the rôle of the supersymmetry parameters. Hence, the model (29), where we made no additional gaugings to eliminate ghosts, really should be viewed as a model on the hyperKähler cone. This explains the third signpost on the roadmap 3. The next stop on the roadmap was motivated by ideas from higher spin models. In particular, our aim was to write down a model where all ghosts had been eliminated from the physical cohomology. Based on ideas coming from our earlier work on orthosymplectic constraint algebras, we suspected that gauging the Lefschetz–Verbitsky trace operator would lead to a gauge invariant quantum field theory generalizing both p-form electromagnetism and ( p, q)-form Kähler electromagnetism to quaternionic Kähler manifolds. This hunch was correct and led to the model (31). Interestingly enough, it could have been the case that this choice of route would lead to a model that did not describe supersymmetric black holes. However, it is clear that in fact the quaternionic Kähler model does so, and in a fascinating way. Examining the Labastida form of the equation of motion (30) we see that it is a product of the Baston operator and the Lefschetz–Verbitsky trace operator. As shown in [25], by explicitly constructing the quaternionic Penrose transform underlying Baston’s quaternionic generalization of the Dolbeault complex, at least in the scalar sector of ∧E, zero modes of the Baston operator correspond to supersymmetric black hole states. We suspect that within BRST quantization, this picture can be extended to a general correpsondence with the Baston complex. In this case, solutions to our quaternionic Kähler electromagnetism theory would fall into two classes: 1. BPS solutions in the kernel of dα dα + g. 2. Solutions whose non-vanishing image under dα dα + g lies in the kernel of tr. This explains the last signpost of the roadmap (3). Clearly our work opens many avenues for further study: Firstly, since our BRST quantization methods produce a gauge theory on the hyperKähler cone and furthermore rely on a polarization where one fourier transforms over half the ghost variables (alias quaternionic cone coordinates), there should exist a rather direct relationship between BRST quantization and the quaternionic twistor methods of [25]. Secondly, our quaternionic Kähler higher form electromagnetism may provide an interesting arena for further studies of minisuperspace black hole quantization. One might hope that constructing interactions for this abelian gauge theory could lead to a far more detailed understanding of these theories (perhaps along the lines of the multicentered configuration and attractor flow trees—“third quantization” [76]). This might sound extremely ambitious, since higher spin interactions are fraught with inconsistencies. However, it is possible that some of the methods of Vasiliev, who has constructed three point higher spin interaction using a combination of unfolding techniques (which are closely related to our BRST framework) and Chern–Simons like equations of motions based on a star product, could solve this problem. Also, we cannot help but remark that whenever two seemingly disparate fields (such as higher spin interactions and four dimensional black hole physics) turn out to be related, oftentimes the flow of new ideas is


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bidirectional. In fact, we suspect that higher quantum corrections to N = 2 supergravities in four dimensions could even have implications for possible higher spin interactions. Finally, another topic that is worth further investigation is the novel Dirac operator in (24). This operator acts on the BRST superghost Hilbert space; in the context of this paper it was merely a tool for constructing a nilpotent BRST charge. However, we suspect that it might have a distinguished rôle to play. In particular, it would be fascinating to compute the Witten index of this operator. Given that it was built from a supersymmetric quantum mechanical model, standard quantum methods may suffice for this. Acknowledgements. A.W. would like to thank Andy Neitzke and Boris Pioline for an early collaboration on this work, as well as many absolutely invaluable discussions. We would also like to thank Fiorenzo Bastianelli, Roberto Bonezzi, Olindo Corradini, Dmitry Fuchs, Carlo Iazeolla and Albert Schwarz for useful discussions and comments. 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.

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Communications In Mathematical Physics - Volume 283

Communications In Mathematical Physics - Volume 270

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Communications In Mathematical Physics - Volume 294

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Communications In Mathematical Physics - Volume 274

Communications in Mathematical Physics - Volume 239

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Communications in Mathematical Physics - Volume 211

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Communications In Mathematical Physics - Volume 298

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