Commun. Math. Phys. 274, 1–30 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0272-9
Communications in
Mathematical Physics
Boson Stars as Solitary Waves Jürg Fröhlich1 , B. Lars G. Jonsson1,2 , Enno Lenzmann3,4 1 Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland.
E-mail:
[email protected];
[email protected] 2 Division of Electromagnetic Engineering, School of Electrical Engineering, Royal Insitute of Technology,
SE-100 44 Stockholm, Sweden
3 Department of Mathematics, ETH Zurich, 8092 Zurich, Switzerland 4 Department of Mathematics, MIT, Cambridge, MA 02139, USA.
E-mail:
[email protected] Received: 14 December 2005 / Accepted: 11 December 2006 Published online: 19 June 2007 – © Springer-Verlag 2007
Abstract: We study the nonlinear equation −∆ + m 2 − m ψ − (|x|−1 ∗ |ψ|2 )ψ on R3 , i∂t ψ = which is known to describe the dynamics of pseudo-relativistic boson stars in the meanfield limit. For positive mass parameters, m > 0, we prove existence of travelling solitary waves, ψ(t, x) = eitµ ϕv (x − vt), for some µ ∈ R and with speed |v| < 1, where c = 1 corresponds to the speed of light in our units. Due to the lack of Lorentz covariance, such travelling solitary waves cannot be obtained by applying a Lorentz boost to a solitary wave at rest (with v = 0). To overcome this difficulty, we introduce and study an appropriate variational problem that yields the functions ϕv ∈ H1/2 (R3 ) as minimizers, which we call boosted ground states. Our existence proof makes extensive use of concentration-compactness-type arguments. In addition to their existence, we prove orbital stability of travelling solitary waves ψ(t, x) = eitµ ϕv (x − vt) and pointwise exponential decay of ϕv (x) in x. 1. Introduction In this paper and its companion [4], we study solitary wave solutions — and solutions close to such — of the pseudo-relativistic Hartree equation 1 ∗ |ψ|2 ψ on R3 . −∆ + m 2 − m ψ − (1.1) i∂t ψ = |x| Here ψ(t, x) is a complex-valued wave field, and the symbol ∗ stands for √ convolution √ on R3 . The operator −∆ + m 2 − m, which is defined via its symbol k 2 + m 2 − m in Fourier space, is the kinetic energy operator of a relativistic particle of mass, m ≥ 0, and the convolution kernel, |x|−1 , represents the Newtonian gravitational potential in appropriate physical units.
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J. Fröhlich, B. L. G. Jonsson, E. Lenzmann
As recently shown by Elgart and Schlein in [3], Eq. (1.1) arises as an effective dynamical description for an N -body quantum system of relativistic bosons with two-body interaction given by Newtonian gravity. Such a system is a model system for a pseudorelativistic boson star. That is, we √ consider a regime, where effects of special relativity (accounted for by the operator −∆ + m 2 − m) become important, but general relativistic effects can be neglected. The idea of a mathematical model of pseudo-relativistic boson stars dates back to the works of Lieb and Thirring [10] and of Lieb and Yau [11], where the corresponding N -body Hamiltonian and its relation to the Hartree energy functional H(ψ) = 2E(ψ) are discussed, with E(ψ) defined in (1.3), below. Let us briefly recap the state of affairs concerning Eq. (1.1) itself. With help of the conserved quantities of charge, N (ψ), and energy, E(ψ), given by N (ψ) = |ψ|2 dx, (1.2)
R3
1 1 1 2 ∗ |ψ|2 |ψ|2 dx, ψ −∆ + m − m ψ dx − (1.3) E(ψ) = 2 R3 4 R3 |x| results derived so far can be summarized as follows (see also Fig. 1 below). – Well-Posedness: For any initial datum ψ0 ∈ H1/2 (R3 ), there exists a unique solution ψ ∈ C0 [0, T ); H1/2 (R3 ) ∩ C1 [0, T ); H−1/2 (R3 ) , (1.4)
for some T > 0, where Hs (R3 ) denotes the inhomogeneous Sobolev space of order s. Moreover, we have global-in-time existences (i. e., T = ∞) whenever the initial datum satisfies the condition N (ψ0 ) < Nc , (1.5) N = Nc
E
I
II
0
N
ground states at rest
III
E = − 12 mN
Fig. 1. Qualitative diagram for the boson star Eq. (1.1) with positive mass parameter m > 0. Here N = N (ψ0 ) and E = E(ψ0 ) denote charge and energy for the initial condition ψ0 ∈ H1/2 (R3 ). In region I, all solutions are global in time and the (unboosted) ground states are minimizers of E(ψ) subject to fixed N (ψ0 ) = N with 0 < N < Nc . If N exceeds Nc , the energy E can attain values below − 21 m N . As shown in [5] for 3 spherically symmetric ψ0 ∈ C∞ c (R ) that belong to region III, we have in fact blow-up of ψ(t) within a finite time. Finally, the qualitative behavior of solutions with initial conditions in region II seems to be of indefinite nature
Boson Stars as Solitary Waves
3
where Nc > 4/π is some universal constant; see [7] for a detailed study of the Cauchy problem for (1.1) with initial data in Hs (R3 ), s ≥ 1/2. – Solitary Waves: Due to the focusing nature of the nonlinearity in (1.1), there exist solitary wave solutions, which we refer to as solitary waves, given by ψ(t, x) = eitµ ϕ(x),
(1.6)
where ϕ ∈ H1/2 (R3 ) is defined as a minimizer of E(ψ) subject to N (ψ) = N fixed. Any such minimizer, ϕ(x), is called a ground state and it has to satisfy the corresponding Euler-Lagrange equation 1 ∗ |ϕ|2 )ϕ = −µϕ, (1.7) −∆ + m 2 − m ϕ − |x| for some µ ∈ R. An existence proof of ground states, for 0 < N (ϕ) < Nc and m > 0, can be found in [11]. The method used there is based on rearrangement inequalities that allow one to restrict one’s attention to radial functions, which simplifies the variational calculus. But in order to extend this existence result to so-called boosted ground states, i. e., x in (1.6) is replaced by x −vt and Eq. (1.7) acquires the additional term, i(v · ∇)ϕ, we have to employ concentration-compactness-type methods; see Theorem 2.1 and its proof, below. 3 – Blow-Up: Any spherically symmetric initial datum, ψ0 ∈ C∞ c (R ), with 1 E(ψ0 ) < − mN (ψ0 ) 2
(1.8)
leads to blow-up of ψ(t) in a finite time, i. e., we have that limtT ψ(t) H1/2 = ∞ holds, for some T < ∞. We remark that (1.8) implies that the smallness condition (1.5) cannot hold. See [5] for a proof of this blow-up result.1 In physical terms, finite-time blow-up of ψ(t) is indicative of “gravitational collapse” of a boson star modelled by (1.1); the constant Nc appearing in (1.5) may then be regarded as a “Chandrasekhar limit mass.” We now come to the main issues of the present paper which focuses on existence and properties of travelling solitary waves for (1.1). More precisely, we consider solutions of the form ψ(t, x) = eitµ ϕv (x − vt) (1.9) with some µ ∈ R and travelling velocity, v ∈ R3 , such that |v| < 1 holds (i. e., below the speed of light in our units). We point out that, since Eq. (1.1) is not Lorentz covariant, solutions such as (1.9) cannot be directly obtained from solitary waves at rest (i. e., we set v = 0) and then applying a Lorentz boost. To circumvent this difficulty, we plug the ansatz (1.9) into (1.1). This yields 1 ∗ |ϕv |2 ϕv = −µϕv , −∆ + m 2 − m ϕv + i(v · ∇)ϕv − (1.10) |x| which is an Euler-Lagrange equation for the following minimization problem i Ev (ψ) := E(ψ) + ψ(v · ∇)ψ dx = min! subject to N (ψ) = N . 2 R3
(1.11)
1 In [5] the energy functional, E(ψ), is shifted by + 1 mN (ψ). Thus, condition (1.8) reads E(ψ ) < 0 in 0 2
[5].
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J. Fröhlich, B. L. G. Jonsson, E. Lenzmann
We refer to such minimizers, ϕv ∈ H1/2 (R3 ), as boosted ground states throughout this paper. Indeed, we will prove existence of boosted ground states when |v| < 1 and 0 < N < Nc (v) holds, as well as non-existence when N ≥ Nc (v); see Theorem 2.1, below. Our existence proof rests on concentration-compactness arguments which for our √ problem need some technical modifications, due to the pseudo-differential operator −∆ + m 2 . Apart from existence of boosted ground states, we are also concerned with properties such as “orbital stability” and exponential decay of ϕv (x) in x; see Theorems 3.1 and 4.1, below. We remark that both properties rely crucially on the positivity of the mass parameter, i. e., we have m > 0 in (1.1). By contrast, it is shown, for instance, in [5] that (resting) solitary waves become unstable when m = 0, due to nearby initial data leading to blow-up solutions. In a companion paper [4], we will explore the effective dynamics of (slowly) travelling solitary waves in an external potential; see also Sect. 5 for a short summary of these result. The plan of this paper is as follows: – In Sect. 2, we set-up the variational calculus for problem (1.11) and we prove existence of boosted ground states, ϕv ∈ H1/2 (R3 ), for 0 < N (ϕv ) < Nc (v) and |v| < 1, as well as their nonexistence if N (ϕv ) ≥ Nc (v); see Theorem 2.1, below. – Section. 3 addresses “orbital stability” of travelling solitary waves ψ(t, x) = eitµ ϕv (x − vt); see Theorem 3.1, below. – In Sect. 4, we derive pointwise exponential decay and regularity of boosted ground states; see Theorem 4.1, below. – In Sect. 5, we sketch the main result of [4] describing the effective dynamics of travelling solitary waves in an external potential. – In App. A–C, we collect and prove several technical statements which we refer to throughout this text. Notation. Lebesgue spaces of complex-valued functions on R3 will be denoted by L p (R3 ), with norm · p and 1 ≤ p ≤ ∞. We define the Fourier transform for f ∈ S(R3 ) (i. e., Schwartz functions) according to 1 (F f )(k) = f (k) = f (x)e−ik·x dx, (1.12) (2π )3/2 R3 where F extends to S (R3 ) (i. e., the space of tempered distributions) by duality. For s ∈ R, we introduce the operator (1 − ∆)s via its multiplier (1 + |k|2 )s in Fourier −1 [(1 + |k|2 )s F f ]. Likewise, we define the operator space, i. e., we set (1 − ∆)s f = F √ 2 −∆ + m through its multiplier |k|2 + m 2 in Fourier space. We employ Sobolev spaces, Hs (R3 ), of fractional order s ∈ R defined by Hs (R3 ) := f ∈ S (R3 ) : (1 − ∆)s/2 f ∈ L2 (R3 ) (1.13) and equipped with the norm f Hs := (1 − ∆)s/2 f 2 . Since we exclusively deal with R3 , we often write L p and Hs instead of L p (R3 ) and s H (R3 ) in what follows. A further abbreviation we use is given by f dx := f (x) dx. (1.14) R3
R3
Boson Stars as Solitary Waves
5
We equip L2 (R3 ) with a complex inner product, ·, · , defined as f, g := f¯g dx. R3
(1.15)
Operator inequalities (in the sense of quadratic forms) are denoted by A ≤ B, which means that ψ, Aψ ≤ ψ, Bψ holds for all ψ ∈ D(|A|1/2 ) ⊆ D(|B|1/2 ), where A and B are self-adjoint operators on L2 (R3 ) with domains D(A) and D(B), respectively. Finally, we remark that we employ the notation v · ∇ = 3k=1 vk ∂xk , where v ∈ R3 is some fixed vector. 2. Existence of Boosted Ground States We consider the following minimization problem: E v (N ) := inf Ev (ψ) : ψ ∈ H1/2 (R3 ), N (ψ) = N ,
(2.1)
where N (ψ) is defined in (1.2), and N > 0, v ∈ R3 , with |v| < 1, denote given parameters. Furthermore, we set Ev (ψ) :=
i 1 ψ, −∆ + m 2 − m ψ + ψ, (v · ∇)ψ 2 2 1 1 − ∗ |ψ|2 |ψ|2 dx. 3 4 R |x|
(2.2)
Any minimizer, ϕv ∈ H1/2 (R3 ), for (2.1) has to satisfy the corresponding Euler-Lagrange equation given by 1 ∗ |ϕv |2 ϕv = −µϕv , −∆ + m 2 − m ϕv + i(v · ∇)ϕv − |x|
(2.3)
with some Lagrange multiplier, −µ ∈ R, where this sign convention turns out to be convenient for our analysis. In what follows, we refer to such minimizers, ϕv , for (2.1) as boosted ground states, since they give rise to moving solitary waves ψ(t, x) = eitµ ϕv (x − vt),
(2.4)
for (1.1) with travelling speed v ∈ R3 with |v| < 1. Concerning existence of boosted ground states, we have the following theorem which generalizes a result derived in [11] for minimizers of (2.1) with v = 0. Theorem 2.1. Suppose that m > 0, v ∈ R3 , and |v| < 1. Then there exists a positive constant Nc (v) depending only on v such that the following holds. i) For 0 < N < Nc (v), problem (2.1) has a minimizer, ϕv ∈ H1/2 (R3 ), and it satisfies (2.3), for some µ ∈ R. Moreover, every minimizing sequence, (ψn ), for (2.1) with 0 < N < Nc (v) is relatively compact in H1/2 (R3 ) up to translations, i. e., there exists a sequence, (yk ), in R3 and a subsequence, (ψn k ), such that ψn k (· + yk ) → ϕv strongly in H1/2 (R3 ) as k → ∞, where ϕv is some minimizer for (2.1). ii) For N ≥ Nc (v), no minimizer exists for problem (2.1), even though E v (N ) is finite for N = Nc (v).
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J. Fröhlich, B. L. G. Jonsson, E. Lenzmann
Remarks. 1) It has been proved in [11] that (2.1) with v = 0 has a spherically symmetric minimizer, which can be chosen to be real-valued and nonnegative. But the proof given in [11] crucially relies on symmetric rearrangement arguments that allow to restrict to radial functions in this special case. For v = 0, such methods cannot be used and a general discussion of (2.1) needs a fundamental change of methods. Fortunately, it turns out that the concentration-compactness method introduced by P.-L. Lions in [12] is tailor-made for studying (2.1). To prove Theorem 2.1, we shall therefore proceed along √ the lines of [12]. But — due to the presence of the pseudodifferential operator −∆ + m 2 in (2.2) — some technical modifications have to be taken into account and they are worked out in detail in App. A. 2) A corresponding existence result for boosted ground states can also be derived when −1/|x| in (2.2) is replaced by some other attractive two-body potential, e. g., a Yukawa type potential Φ(x) = −e−µ|x| /|x| with µ > 0. But then minimal L2 -norm of minimizers may also arise, i. e., the condition N > N∗ (v; Φ) enters for some N∗ (v; Φ) > 0. 3) We do not know whether we have uniqueness of minimizers (up to phase and translation) for problem (2.1). Even the simpler case, where one assumes v = 0, has not been settled yet.
2.1. Setting up the Variational Calculus. Before we turn to the proof of Theorem 2.1, we collect and prove some preliminary results. First one easily verifies that Ev (ψ) is real-valued (by using, for instance, Plancherel’s theorem for the first two terms in (2.2)). Moreover, the inequality √ 1
∗ |ψ|2 |ψ|2 dx ≤ Sv ψ, ( −∆ + iv · ∇)ψ ψ, ψ , (2.5) R3 |x| which is proven in App. B, ensures that Ev (ψ) is well-defined on H1/2 (R3 ). As stated in Lemma B.1, inequality (2.5), with |v| < 1, has an optimizer, Q v ≡ 0, which satisfies √
−∆ Q v + i(v · ∇)Q v −
1 ∗ |Q v |2 Q v = −Q v |x|
(2.6)
and yields the best constant, Sv , in terms of Sv =
2 . Q v , Q v
(2.7)
Correspondingly, we introduce the constant, Nc (v), by Nc (v) :=
2 . Sv
(2.8)
By Lemma B.1, we also have the bounds Nc ≥ Nc (v) ≥ (1 − |v|)Nc ,
(2.9)
where Nc (v = 0) = Nc > 4/π is, of course, the same constant that appeared in Sect. 1. We now state our first auxiliary result for (2.1).
Boson Stars as Solitary Waves
7
Lemma 2.1. Suppose that m ≥ 0, v ∈ R3 , and |v| < 1. Then the following inequality holds:
N √ ψ, −∆ + iv · ∇ ψ − m N 2Ev (ψ) ≥ 1 − (2.10) Nc (v) for all ψ ∈ H1/2 (R3 ) with N (ψ) = N . Here Nc (v) is the constant introduced in (2.8) above. Moreover, we have that E v (N ) ≥ − 21 m N for 0 < N ≤ Nc (v) and E v (N ) = −∞ for N > Nc (v). Finally, any minimizing sequence for problem (2.1) is bounded in H1/2 (R3 ) whenever 0 < N < Nc (v). Proof (of Lemma 2.1). Let the assumption on m√and v stated above be satisfied. Estimate √ (2.10) is derived by noting that −∆ + m 2 ≥ −∆ and using inequality (2.5) together with the definition of Nc (v) in (2.8). Furthermore, that E v (N ) ≥ − 21 m N for N ≤ Nc (v) is a consequence of (2.10) itself. To see that E v (N ) = −∞ when N > Nc (v), we recall from Lemma B.1 that there exists an optimizer, Q v ∈ H1/2 (R3 ), with N (Q v ) = Nc (v), for inequality√(2.5). Using that Q v turns (2.5) into an equality and noticing that √ −∆ + m 2 − m ≤ −∆ holds, a short calculation yields E v (N ) ≤ Ev (λQ v )
m=0
=−
1 λ2 (λ2 − 1) ∗ |Q v |2 |Q v |2 dx. 4 R3 |x|
(2.11)
For N > Nc (v), we can choose λ > 1 which implies that the right-hand side is strictly negative and, in addition, by L 2 -norm preserving rescalings, Q v (x) → a 3/2 Q v (ax) with a > 0, we find that E v (N ) ≤ Ev (λa 3/2 Q v (a·)) = aEv (λQ v ) → −∞, (2.12) m=0
m=0
with λ > 1 fixed and as a → ∞. Thus, we deduce that E v (N ) = −∞ when N > Nc (v). To see the H1/2 (R3 )-boundedness of any minimizing √ √ sequence, (ψn ), with 0 < N < Nc (v), we note that −∆ + iv · ∇ ≥ (1 − |v|) −∆ holds. Hence we see that √ supn ψn , −∆ψn ≤ C < ∞, thanks to (2.10). This completes the proof of Lemma 2.1. As a next step, we derive an upper bound for E v (N ), which is given by the nonrelativistic ground state energy, E vnr (N ), defined below. Here the positivity of the mass parameter, m > 0, is essential for deriving the following estimate. Lemma 2.2. Suppose that m > 0, v ∈ R3 , and |v| < 1. Then we have that E v (N ) ≤ −
1 1 − 1 − v 2 m N + E vnr (N ), 2
(2.13)
where E vnr (N ) is given by E vnr (N ) := inf Evnr (ψ) : ψ ∈ H1 (R3 ), N (ψ) = N , Evnr (ψ)
√ 1 1 − v2 1 ∗ |ψ|2 |ψ|2 dx. := |∇ψ|2 dx − 4m 4 R3 |x| R3
(2.14) (2.15)
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J. Fröhlich, B. L. G. Jonsson, E. Lenzmann
Proof (of Lemma 2.2). To prove (2.13), we pick a spherically symmetric function, φ ∈ H1 (R3 ) with N (φ) = N , and we introduce the one-parameter family φλ (x) := eiλv·x φ(x) = eiλ|v|z φ(x), with λ > 0.
(2.16)
Here and in what follows, we assume (without loss of generality) that v is parallel to the z-axis, i. e., v = |v|ez . One checks that
i λv 2 φλ , (v · ∇)φλ = − N, 2 2
(2.17)
using the fact that φ, ∇φ = 0 holds, by spherical symmetry of φ(x). Hence, we find that
i|v| 1 φλ , ∂ z φλ Ev (φλ ) = φλ , −∆ + m 2 − m φλ + 2 2 1 1 2 2 − ∗ |φλ | |φλ | dx 4 R3 |x|
1 1
1 2 2 φλ , −∆ + m − m φλ − v λN − ∗ |φ|2 |φ|2 dx = 2 4 R3 |x| =: A + B. (2.18) To estimate A in (2.18), we recall the operator inequality
−∆ + m 2 ≤
1 (−∆ + m 2 + λ2 ), 2λ
(2.19)
which follows from the elementary inequality 2ab ≤ a 2 + b2 . Thus, we are led to
1 1 1 φλ , (−∆ + m 2 + λ2 )φλ − m N − v 2 λN 4λ 2 2 1 1 2 2 1 2 2 = λ v N + φ, −∆φ + (m + λ )N − m N − v 2 λN . 4λ 2 2
A≤
(2.20)
By minimizing the upper bound (2.20) with respect √ to λ > 0, which is a matter of elementary calculations, we obtain with λ∗ = m/ 1 − v 2 the estimate 1 Ev (φλ∗ ) ≤ − 1 − 1 − v 2 m N 2 √ 1 1 1 − v2 + φ, −∆φ − ∗ |φ|2 |φ|2 dx 4m 4 R3 |x| 1 = − 1 − 1 − v 2 m N + Evnr (φ). 2
(2.21)
Next, we remark that Evnr (ψ) is the energy√ functional for the non-relativistic boson star problem with mass parameter m v = m/ 1 − v 2 . Indeed, it is known from [8] that Evnr (ψ) subject to N (ψ) = N has a spherically symmetric minimizer, φ∗ ∈ H1 (R3 ), with E vnr (N ) = Evnr (φ∗ ) < 0, (2.22) which completes the proof of Lemma 2.2.
Boson Stars as Solitary Waves
9
By making use of Lemma 2.2, we show that the function E v (N ) satisfies a strict subadditivity condition. This is essential to the discussion of (2.1) when using concentrationcompactness-type methods. Lemma 2.3. Suppose that m > 0, v ∈ R3 , and |v| < 1. Then E v (N ) satisfies the strict subadditivity condition E v (N ) < E v (α) + E v (N − α) (2.23) whenever 0 < N < Nc (v) and 0 < α < N . Here Nc (v) is the constant of Lemma 2.1. Moreover, the function E v (N ) is strictly decreasing and strictly concave in N , where 0 < N < Nc (v). Remarks. 1) Condition m > 0 is necessary for (2.23) to hold. To see this, note that if m = 0 then Ev (ψλ ) = λEv (ψ) holds, where ψλ = λ3/2 ψ(λx) and λ > 0. This leads to the conclusion that E v (N ) is either 0 or −∞ when m = 0. 2) The fact that E v (N ) is strictly concave will be needed in our companion paper [4] when making use of the symplectic structure associated with the Hamiltonian PDE (1.1). More precisely, the strict concavity of E v (N ) will enable us to prove the nondegeneracy of the symplectic form restricted to the manifold of solitary waves. nr (N ) < 0 holds, Proof (of Lemma 2.3). By Lemma 2.2 and the fact that E vnr (N ) ≤ E v=0 by (2.22), we deduce that
E v (N ) < −
1 1 − 1 − v2 m N . 2
(2.24)
Next, we notice the following scaling behavior: E v (N ) = N ev (N ),
(2.25)
where ev (N ) :=
1
i −∆ + m 2 − m ψ + ψ, (v · ∇)ψ 2 2 N 1 ∗ |ψ|2 |ψ|2 dx . − (2.26) 4 R3 |x| inf
ψ∈H1/2 , ψ 22 =1
ψ,
This shows that ev (N ) is strictly decreasing, provided that we know that we may restrict the infimum to elements such that 1 (2.27) ∗ |ψ|2 |ψ|2 dx ≥ c > 0 R3 |x| holds for some c. Suppose now that (2.27) were not true. Then there exists a minimizing sequence, (ψn ), such that 1 ∗ |ψn |2 |ψn |2 dx → 0, as n → ∞. (2.28) 3 R |x| But on account of the fact that (cf. App. C)
ψ, −∆ + m 2 − m ψ + i ψ, (v · ∇)ψ ≥ − 1 − 1 − v 2 m N ,
(2.29)
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J. Fröhlich, B. L. G. Jonsson, E. Lenzmann
we conclude that E v (N ) = N ev (N ) ≥ −
1 1 − 1 − v2 m N , 2
(2.30)
which contradicts (2.24). Thus ev (N ) is strictly decreasing. Returning to (2.25) and noting that ev (N ) < 0 holds, by (2.24), we deduce that E v (ϑ N ) < ϑ E v (N ), for ϑ > 1 and 0 < N < Nc (v).
(2.31)
By an argument presented in [12], this inequality leads to the strict subadditivity condition (2.23). Finally, we show that E v (N ) is strictly decreasing and strictly concave on the interval (0, Nc (v)). To see that E v (N ) = N ev (N ) is strictly decreasing, we notice that ev (N ) is strictly decreasing and negative. Furthermore, we remark that ev (N ) = inf{linear functions in N } has to be a concave function. Therefore it follows that E v (N ) = N ev (N ) is strictly concave, since the left- and right-derivatives, D ± E v (N ), exist and are found to be strictly decreasing, by using that ev (N ) is concave and strictly decreasing. 2.2. Proof of Theorem 2.1. We now come to the proof of Theorem 2.1 and we suppose that m > 0, v ∈ R3 , and |v| < 1 holds. Proof of Part i) Let us assume that 0 < N < Nc (v),
(2.32)
where Nc (v) is the constant defined in (2.8). Furthermore, let (ψn ) be a minimizing sequence for (2.1), i. e., lim Ev (ψn ) = E v (N ), with ψn ∈ H1/2 (R3 ) and N (ψn ) = N for all n ≥ 0. (2.33)
n→∞
By Lemma 2.1, we have that E v (N ) > −∞ and that (ψn ) is a bounded sequence in H1/2 (R3 ). We now apply the following concentration-compactness lemma. Lemma 2.4. Let (ψn ) be a bounded sequence in H1/2 (R3 ) such that N (ψn ) = R3 |ψn |2 dx = N for all n ≥ 0. Then there exists a subsequence, (ψn k ), satisfying one of the three following properties: i) Compactness: There exists a sequence, (yk ), in R3 such that, for every > 0, there exists 0 < R < ∞ with |ψn k |2 dx ≥ N − . (2.34) |x−yk | 0, there exist two bounded sequences, (ψk1 ) and (ψk2 ), in H1/2 (R3 ) and k0 ≥ 0 such that, for all k ≥ k0 , the following properties hold: ψn − (ψ 1 + ψ 2 ) ≤ δ p (), for 2 ≤ p < 3, (2.35) k k k p with δ p () → 0 as → 0, and |ψk1 |2 dx − α ≤ and R3
R3
|ψk2 |2 dx − (N − α) ≤ ,
dist (supp ψk1 , supp ψk2 ) → ∞, as k → ∞. Moreover, we have that
lim inf ψn k , T ψn k − ψk1 , T ψk1 − ψk2 , T ψk2 ≥ −C(), k→∞
where C() → 0 as → 0 and T := v ∈ R3 , |v| < 1.
(2.36) (2.37)
(2.38)
√ −∆ + m 2 − m + i(v · ∇) with m ≥ 0 and
Remark. We refer to App. A for the proof of Lemma 2.4. Part i) and ii) are standard, but part iii) requires some technical arguments, due to the presence of the pseudo-differential operator T . Invoking Lemma 2.4, we conclude that a suitable subsequence, (ψn k ), satisfies either i), ii), or iii). We rule out ii) and iii) as follows. Suppose that (ψn k ) exhibits property ii). Then we conclude that 1 ∗ |ψn k |2 |ψn k |2 dx = 0, (2.39) lim 3 k→∞ R |x| by Lemma A.1. But as shown in the proof of Lemma 2.3, this implies E v (N ) ≥ −
1 1 − 1 − v2 m N , 2
(2.40)
which contradicts (2.24). Hence ii) cannot occur. Let us suppose that iii) is true for (ψn k ). Then there exists α ∈ (0, N ) such that, for every > 0, there are two bounded sequences, (ψk1 ) and (ψk2 ), with α − ≤ N (ψk1 ) ≤ α + , (N − α) − ≤ N (ψk2 ) ≤ (N − α) + ,
(2.41)
for k sufficiently large. Moreover, inequality (2.38) and Lemma A.2 allow us to deduce that E v (N ) = lim Ev (ψn k ) ≥ lim inf Ev (ψk1 ) + lim inf Ev (ψk2 ) − r (), k→∞
k→∞
k→∞
(2.42)
where r () → 0 as → 0. Since (ψk1 ) and (ψk2 ) satisfy (2.41), we infer E v (N ) ≥ E v (α + ) + E v (N − α + ) − r (),
(2.43)
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J. Fröhlich, B. L. G. Jonsson, E. Lenzmann
using that E v (N ) is decreasing in N . Passing to the limit → 0 and by continuity of E v (N ) in N (recall that E v (N ) is a concave function on an open set), we deduce that E v (N ) ≥ E v (α) + E v (N − α)
(2.44)
holds for some 0 < α < N . This contradicts the strict subadditivity condition (2.23) stated in Lemma 2.3. Therefore iii) is ruled out. By the discussion so far, we conclude that there exists a subsequence, (ψn k ), such that i) of Lemma 2.4 is true for some sequence (yk ) in R3 . Let us now define the sequence k := ψn k (· + yk ). ψ
(2.45)
k ) is a bounded sequence in H1/2 (R3 ), we can pass to a subsequence, still Since (ψ k ), such that (ψ k ) converges weakly in H1/2 (R3 ) to some ϕv ∈ H1/2 (R3 ) denoted by (ψ k → ϕv strongly in L p (R3 ) as k → ∞, for as k → ∞. Moreover, we have that ψ loc 2 ≤ p < 3, thanks to a Rellich-type theorem for H1/2 (R3 ) (see, e. g., [9, Theorem 8.6] for this). But on account of the fact k |2 dx ≥ N − , |ψ (2.46) |x| 0 and suitable R = R() < ∞, we conclude that k → ϕv strongly in L p (R3 ) as k → ∞, for 2 ≤ p < 3. ψ
(2.47)
Next, by the Hardy–Littlewood–Sobolev and Hölder’s inequality, we deduce that 1 1 k |2 |ψ k |2 dx − ∗ |ψ ∗ |ϕv |2 |ϕv |2 dx R3 |x| R3 |x| k 312/5 + ϕv 312/5 ψ k − ϕv 12/5 . ≤ C ψ (2.48) k converges strongly to ϕv in L12/5 (R3 ), as k → ∞, and From (2.47), we have that ψ therefore 1 1 k |2 |ψ k |2 dx = ∗ |ψ ∗ |ϕv |2 |ϕv |2 dx. (2.49) lim k→∞ R3 |x| R3 |x| Moreover, we have that k ) ≥ Ev (ϕv ) ≥ E v (N ), E v (N ) = lim Ev (ψ
(2.50)
T (ψ) := ψ, −∆ + m 2 ψ + i ψ, (v · ∇)ψ ,
(2.51)
k→∞
since the functional
is weakly lower semicontinuous on H1/2 (R3 ), see Lemma A.4 in App. A. Thus, we have proved that ϕv ∈ H1/2 (R3 ) is a minimizer for (2.1), i. e., we have E v (N ) = Ev (ϕv ) and N (ϕv ) = N . To prove the relative compactness of minimizing sequences in H1/2 (R3 ) (up to transk ) = lations), we notice that there has to be equality in (2.50), which leads to limk→∞ T (ψ T (ϕv ). By Lemma A.4, this fact implies a posteriori that k → ϕv strongly in H1/2 (R3 ) as k → ∞, ψ which completes the proof of part i) of Theorem 2.1.
(2.52)
Boson Stars as Solitary Waves
13
Proof of Part ii) To complete the proof of Theorem 2.1, we address its part ii). Clearly, no minimizer exists if N > Nc (v), since in this case we have that E v (N ) = −∞, by Lemma 2.1. Next, we show that E v (N ) = − 21 m N holds if N = Nc (v), which can be seen as follows. We take an optimizer, Q v ∈ H1/2 (R3 ), for inequality (2.5); see Lemma B.1 and recall that N (Q v ) = Nc (v). Then √
1 1 (λ) E v (N ) ≤ Ev (Q (λ) Q v , −∆ + m 2 − −∆ Q (λ) (2.53) − mN, v )= v 2 2 (λ)
(λ)
for N = Nc (v), where Q v (x) := λ3/2 Q v (λx) with λ > 0, so that N (Q v ) = N (Q v ) = Nc (v). Using Plancherel’s theorem and by dominated convergence, we deduce that √ (λ) (λ)
2 v (k)|2 λ2 k 2 + m 2 − λ|k| dk Q v , −∆ + m − −∆ Q v = |Q R3
→ 0 as λ → ∞.
(2.54)
Thus, we conclude that E v (N ) ≤ − 21 m N for N = Nc (v). In combination with the estimate E v (N ) ≥ − 21 m N for N ≤ Nc (v) taken from Lemma 2.1, this shows that 1 E v (N ) = − m N for N = Nc (v). 2
(2.55)
Finally, we prove that there does not exist a minimizer for (2.1) with N = Nc (v). We argue by contradiction as follows. Suppose that ϕv ∈ H1/2 (R3 ) is a minimizer for (2.1) √ with √ N = Nc (v). For ϕv ∈ H1/2 (R3 ), ϕv ≡ 0, and m > 0, we use ϕv , −∆ + m 2 ϕv > ϕv , −∆ ϕv to obtain 1 1 1 − m N = Ev (ϕv )m>0 > Ev (ϕv )m=0 − m N ≥ − m N , 2 2 2
(2.56)
which is a contradiction. Here we use Lemma 2.1 to estimate Ev (ϕv )|m=0 ≥ 0 for N (ϕv ) = Nc (v). Hence no minimizer exists for (2.1) if N ≥ Nc (v). This completes the proof of Theorem 2.1. 3. Orbital Stability The purpose of this section is to address “orbital stability” of travelling solitary waves ψ(t, x) = eitµ ϕv (x − vt),
(3.1)
where ϕv ∈ H1/2 (R3 ) is a boosted ground state. By the relative compactness of minimizing sequences (see Theorem 2.1) and by using a general idea presented in [2], we are able to prove the following abstract stability result. Theorem 3.1. Suppose that m > 0, v ∈ R3 , |v| < 1, and 0 < N < Nc (v). Let Sv,N denote the corresponding set of boosted ground states, i. e., Sv,N := ϕv ∈ H1/2 (R3 ) : Ev (ϕv ) = E v (N ), N (ϕv ) = N , which is non-empty by Theorem 2.1.
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J. Fröhlich, B. L. G. Jonsson, E. Lenzmann
Then the solitary waves given in (3.1), with ϕv ∈ Sv,N , are stable in the following sense. For every > 0, there exists δ > 0 such that inf
ϕv ∈Sv,N
ψ0 − ϕv H1/2 ≤ δ implies that sup inf
t≥0 ϕv ∈Sv,N
ψ(t) − ϕv H1/2 ≤ .
Here ψ(t) denotes the solution of (1.1) with initial condition ψ0 ∈ H1/2 (R3 ). Remark. It is an interesting and open question whether uniqueness (modulo phase and translation) of boosted ground states holds, i. e., we have that Sv,N is of the form {eiγ ϕv (· − y) : γ ∈ R, y ∈ R3 }, for some fixed ϕv ∈ Sv,N . Proof (of Theorem 3.1). Let m and v satisfy the given assumptions. Since we have N < Nc (v) ≤ Nc , we can choose δ > 0 sufficiently small such that inf φ∈Sv,N ψ0 − φ H1/2 ≤ δ guarantees that N (ψ0 ) < Nc . By the global well-posedness result for (1.1) derived in [7], we have that the corresponding solution, ψ(t), exists for all times t ≥ 0. Thus, taking supt≥0 is well-defined. Let us now assume that orbital stability (in the sense defined above) does not hold. Then there exists a sequence on initial data, (ψn (0)), in H1/2 (R3 ) with inf ψn (0) − ϕ H1/2 → 0, as n → ∞,
ϕ∈Sv,N
(3.2)
and some > 0 such that inf ψn (tn ) − ϕ H1/2 > , for all n ≥ 0,
ϕ∈Sv,N
(3.3)
for a suitable sequence of times (tn ). Note that (3.2) implies that N (ψn (0)) → N as n → ∞. Since N < Nc by assumption, we can assume — without loss of generality — that N (ψn (0)) < Nc holds for all n ≥ 0, which guarantees (see above) that the corresponding solution, ψn (t), exists globally in time. Next, we consider the sequence, (βn ), in H1/2 (R3 ) that is given by βn := ψn (tn ).
(3.4)
By conservation of N (ψ(t)) and of Ev (ψ(t)), whose proof can be done along the lines of [7] for the conservation of E(ψ(t)), we have that N (βn ) = N (ψn (0)) and Ev (βn ) = Ev (ψn (0)), which, by (3.2), implies lim Ev (βn ) = E v (N ) and
n→∞
lim N (βn ) = N .
n→∞
Defining the rescaled sequence √ n := an βn , where an := N /N (βn ), β and using the fact (βn that n H1/2 ≤ C|1 − an | → 0, as n → ∞. βn − β 1/2 By continuity of Ev : H (R3 ) → R, we deduce that n ) = E v (N ) and N (β n ) = N , for all n ≥ 0. lim Ev (β
(3.5)
(3.6)
) has to be bounded in H1/2 (R3 ), by virtue of Lemma 2.1, we infer
n→∞
(3.7) (3.8)
n ) is a minimizing sequence for (2.1) which, by Theorem 2.1 part i), has to Therefore (β n k ), that strongly converges in H1/2 (R3 ) (up to translations) to contain a subsequence, (β some minimizer ϕ ∈ Sv,N . In particular, inequality (3.3) cannot hold when βn = ψn (tn ) n . But in view of (3.7), this conclusion is easily extended to the sequence is replaced by β (βn ) itself. Thus, we are led to a contradiction and the proof of Theorem 3.1 is complete.
Boson Stars as Solitary Waves
15
4. Properties of Boosted Ground States Concerning fundamental properties of boosted ground states given by Theorem 2.1, we have the following result. Theorem 4.1. Let m > 0, v ∈ R3 , |v| < 1, and 0 < N < Nc (v). Then every boosted ground state, ϕv ∈ H1/2 (R3 ), of problem (2.1) satisfies the following properties. i) ϕv ∈ Hs (R3 ) for all s ≥ 1/2. √ ii) The corresponding Lagrange multiplier satisfies µ > (1 − 1 − v 2 )m. Moreover, we have pointwise exponential decay, i. e., |ϕv (x)| ≤ Ce−δ|x|
(4.1)
holds for all x ∈ R3 , where δ > 0 and C > 0 are suitable constants. iii) For v = 0, the function ϕv (x) can be chosen to be radial, real-valued, and strictly positive. Remarks. 1) By part i) and Sobolev embeddings, any boosted ground state is smooth: ϕv ∈ C∞ (R3 ). Moreover, we have that ϕv ∈ L1 ∩ L∞ , due to part ii). 2) Part iii) follows from the discussion presented in [11], except for the strict positivity which we will show below. 3) For a more precise exponential decay estimate for ϕv (x), see Lemma C.1 in App. C. Proof (of Theorem 4.1). Part i): We rewrite the Euler-Lagrange equation (3.1) for ϕv as (H0 + λ)ϕv = F(ϕv ) + (λ − µ)ϕv , for any λ ∈ R, where H0 := −∆ + m 2 − m + i(v · ∇),
F(ϕv ) :=
1 ∗ |ϕv |2 ϕv . |x|
(4.2)
(4.3)
By [7, Lemma 3], we have that F : Hs (R3 ) → Hs (R3 ) for all s ≥ 1/2 (F is indeed locally Lipschitz). Thus, the right-hand side in (4.2) belongs to H1/2 (R3 ). Since H0 is bounded from below, we can choose λ > 0 sufficiently large such that (H0 + λ)−1 exists. This leads to ϕv = (H0 + λ)−1 F(ϕv ) + (λ − µ)ϕv . (4.4) Noting that (H0 +λ)−1 : Hs (R3 ) → Hs+1 (R3 ), we see that ϕv ∈ H3/2 (R3 ). By repeating the argument, we conclude that ϕv ∈ Hs (R3 ) for all s ≥ 1/2. This proves part i). Part ii): The exponential decay follows from Lemma C.1, provided that the Lagrange multiplier, −µ, satisfies −µ < − 1 − 1 − v 2 m, (4.5) which means that −µ lies strictly below the essential spectrum of H0 ; see App. C. To prove (4.5), we multiply the Euler-Lagrange equation by ϕ v and integrate to obtain 1 1 ∗ |ϕv |2 |ϕv |2 dx = −µN . (4.6) 2E v (N ) − 2 R3 |x| Using the upper bound (2.24) for E v (N ), we conclude that −µN < − 1 − 1 − v 2 m N ,
(4.7)
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J. Fröhlich, B. L. G. Jonsson, E. Lenzmann
which proves (4.5). Part iii): For the sake of brevity, we write ϕ(x) := ϕv=0 (x). By [11] problem (2.1), with v = 0, has a minimizer that equals its symmetric-decreasing rearrangement, i. e., ϕ(x) = ϕ ∗ (x). In particular, ϕ(x) is a spherically symmetric, real-valued, nonincreasing function with ϕ(x) ≥ 0. It remains to show that ϕ(x) > 0 holds. To see this, we put λ = µ in (4.4), which is possible by the proof of ii), and we obtain −1 ϕ= −∆ + m 2 − m + µ F(ϕ). (4.8) √ By using functional calculus for the self-adjoint operator −∆ + m 2 on L2 (R3 ) with domain H1 (R3 ), we find that ∞ √ −1 2 −∆ + m 2 − m + µ = e−tµ e−t ( −∆+m −m) dt. (4.9) 0
Referring to the explicit formula (C.10) for v = 0, we see that the integral kernel, √ 2 −m) −t ( −∆+m e (x, y), is strictly positive. In view of (4.8), (4.9), and the fact that F(ϕ) ≥ 0, we conclude that ϕ(x) > 0 holds for almost every x ∈ R3 . But since ϕ(x) is a nonincreasing and continuous function, we deduce that ϕ(x) > 0 has to be true for all x ∈ R3 . This completes the proof of Theorem 4.1. 5. Outlook Our analysis presented so far serves as a basis for the upcoming work in [4] which explores the effective motion of travelling solitary waves in an external potential. More precisely, we consider 1 i∂t ψ = ∗ |ψ|2 ψ on R3 . −∆ + m 2 − m ψ + V ψ − (5.1) |x| Here the external potential V : R3 → R is assumed to be a smooth, bounded function with bounded derivatives. Note that its spatial variation introduces the length scale ext = ∇V −1 ∞.
(5.2)
In addition, another length scale, sol , enters through the exponential decay of ϕv (x), i. e., we have that sol = δ −1 , (5.3) where δ > 0 is the constant taken from Theorem 4.1. On intuitive grounds, one expects that if we have that sol ext (5.4) holds, then solutions, ψ(t, x), of (5.1) that are initially close to some ϕv (x) should approximately behave like point-particles, at least on a large (but possibly finite) interval of time. We now briefly sketch how this heuristic picture of point-particle behavior of solitary waves is addressed by rigorous analysis in [4]. There we introduce a nondegeneracy assumption on the linearized operator L1 0 L := (5.5) 0 L2
Boson Stars as Solitary Waves
17
acting on L2 (R3 ; R2 ) with domain H1 (R3 ; R2 ), where L 1 ξ :=
1 1 ∗ ϕ2 ξ − 2 ∗ (ϕξ ) ϕ, −∆ + m 2 − m + µ0 ξ − |x| |x|
L 2 ξ :=
1 ∗ ϕ 2 ξ. −∆ + m 2 − m + µ0 ξ − |x|
(5.6) (5.7)
Here ϕ(x) = ϕv=0 (x) is some ground state at rest, whose corresponding Lagrange multiplier we denote by µ0 ∈ R. Note that we can assume that ϕv=0 (x) is spherically symmetric and real-valued, see Theorem 4.1. The nondegeneracy condition introduced in [4] now reads 0 ∂x1 ϕ ∂x2 ϕ ∂x3 ϕ ker(L) = span , , , . (5.8) ϕ 0 0 0 Under this kernel assumption, we then construct in [4] (by an implicit-function-type argument) a map (v, µ) → ϕv,µ , so that ϕv,µ ∈ H1/2 solves Eq. (1.7) and (v, µ) belongs to the small neighborhood around (0, µ0 ). The main result proven in [4] can now be sketched as follows. We consider suitable external potentials of the form V (x) := W (x). (5.9) Furthermore, let ϕv0 ,µ0 with |v0 | 1 be given and choose 1 so that (5.4) holds. Then for any initial datum, ψ0 (x), such that |||ψ0 − eiϑ0 ϕv0 ,µ0 (· − a0 )||| ≤ , for some ϑ0 ∈ R and a0 ∈ R3 ,
(5.10)
where ||| · ||| is some weighted Sobolev norm, the corresponding solution, ψ(t, x), of (5.1) can be uniquely written as ψ(t, x) = eiϑ [ϕv,µ (x − a) + ξ(t, x − a)], for 0 ≤ t < C −1 .
(5.11)
Here |||ξ ||| = O() holds and the time-dependent functions {ϑ, a, v, µ} satisfy equations of the following form: N˙ = O( 2 ), ϑ˙ = µ − V (a) + O( 2 ), (5.12) 2 a˙ = v + O( ), γ (µ, v)v˙ = −∇V (a) + O( 2 ), where N = N (ϕv,µ ). The term γ (µ, v) can be viewed as an “effective mass” which takes relativistic effects into account. Finally, we remark that the proof of (5.11) and (5.12) makes extensive use of the Hamiltonian formulation of (5.1) and its associated symplectic structure restricted to the manifold of solitary waves. Moreover, assumption (5.8) enables us to derive additional properties of ϕv (x), for |v| 1, such as cylindrical symmetry with respect to the v-axis, which is of crucial importance in the analysis presented in [4]. A. Variational and Pseudo-Differential Calculus In this section of the appendix, we collect and prove results needed for our variational and pseudo-differential calculus.
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A.1. Proof of Lemma 2.4. Let (ψn ) be a bounded sequence in H1/2 (R3 ) with ψn 22 = N for all n. Along the lines of [12], we define the sequence, (Q n ), of Lévy concentration functions by Q n (R) := sup |ψn |2 dx, for R ≥ 0. (A.1) y∈R3 |x−y| 0 be given. Suppose that ξ, φ ∈ C∞ (R3 ) with 0 ≤ φ, ξ ≤ 1 such that ξ(x) ≡ 1 for 0 ≤ |x| ≤ 1,
ξ(x) ≡ 0 for |x| ≥ 2,
(A.4)
φ(x) ≡ 0 for 0 ≤ |x| ≤ 1,
φ(x) ≡ 1 for |x| ≥ 2.
(A.5)
Furthermore, we put ξ R (x) := ξ(x/R) and φ R (x) := φ(x/R), for R > 0, and we introduce (A.6) ψk1 := ξ R1 (· − yk )ψn k and ψk2 := φ Rk (· − yk )ψn k . As shown in [12, Proof of Lemma III.1], there exists
and a sequence, (Rk ), with
R1 () → ∞, as → 0,
(A.7)
Rk → ∞, as k → ∞,
(A.8)
such that (ψk1 ) and (ψk2 ) satisfy (2.36) and (2.37) in Lemma 2.4. Moreover, we have that
R3
|ψn k − (ψk1 + ψk2 )|2 dx ≤ 4,
(A.9)
for k sufficiently large. By [9, Theorem 7.16], we see that ψk1 and ψk2 defined in (A.6) are bounded in 1/2 H (R3 ). More precisely, using the technique of the proof given there and the explicit formula √ | f (x) − f (y)|2 dx dy, for f ∈ H1/2 (R3 ), (A.10) f, −∆ f = (const.) 4 3 3 |x − y| R ×R we deduce that
g f H1/2 ≤ C g ∞ + ∇g ∞ f H1/2 .
(A.11)
1 1 ψk1 H1/2 ≤ C 1 + and ψk2 H1/2 ≤ C 1 + , R1 Rk
(A.12)
Thus, we find that
Boson Stars as Solitary Waves
19
for some constant C = C(M), where M = supk≥0 ψn k H1/2 < ∞. Thus, (ψk1 ) and (ψk2 ) are bounded sequences in H1/2 (R3 ). This fact together with Hölder’s and Sobolev’s inequalities leads to ψn − (ψ 1 + ψ 2 ) ≤ δ p (), for 2 ≤ p < 3, k k k p
(A.13)
where δ p () → 0 as → 0. This proves (2.35) in Lemma 2.4. It remains to show property (2.38) in Lemma 2.4. Since
lim inf ψn k , (−m)ψn k − ψk1 , (−m)ψk1 − ψ 2 , (−m)ψk2
(A.14)
≥ −m N + m(α − ) + m(N − α − ) ≥ −2m → 0, as → 0,
(A.15)
k→∞
we observe that it suffices to prove the claim
lim inf ψn k , Aψn k − ψk1 , Aψk1 − ψk2 , Aψk2 ≥ −C(), k→∞
(A.16)
for some constant C() → 0 as → 0, where A :=
−∆ + m 2 + i(v · ∇) + λ,
(A.17)
with m ≥ 0, v ∈ R3 , |v| < 1, and λ > 0 is some constant so that √ A ≥ (1 − |v|) −∆ + λ ≥ λ > 0.
(A.18)
In view of (A.14), adding any fixed λ can be done without loss of generality. Next, we recall definition (A.6) and rewrite the left-hand side in (A.16) as follows
lim inf ψn k , (A − ξk Aξk − φk Aφk )ψn k ,
(A.19)
ξk (x) := ξ R1 (x − yk ) and φk (x) := φ Rk (x − yk ).
(A.20)
k→∞
where Using commutators [X, Y ] := X Y − Y X , we find that A − ξk Aξk − φk Aφk = A(1 − ξk2 − φk2 ) − [ξk , A]ξk − [φk , A]φk √ √ √ √ = A(1 − ξk2 − φk2 ) A − A[ A, (ξk2 + φk2 )] −[ξk , A]ξk − [φk , A]φk . (A.21) Note that
√
A > 0 holds, due to A > 0. By applying Lemma A.3, we obtain [ξk , A]
L2 →L2
[φk , A]
L2 →L2
≤ C ∇ξk ∞ ≤
C , R1
(A.22)
≤ C ∇φk ∞ ≤
C . Rk
(A.23)
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J. Fröhlich, B. L. G. Jonsson, E. Lenzmann
To estimate the remaining commutator in (A.21), we use (A.58) in the proof of Lemma A.3 to find that ∞ √ √ 1 2 [ A, (ξ 2 + φ 2 )] 2 2 ≤ C 1 + 1 2 2 ds s (A.24) k k L →L R1 Rk (s + A) L →L 0 ∞ √ 1 s 1 ≤C + ds (A.25) R1 Rk (s + λ)2 0 1 1 . (A.26) + ≤C R1 Rk Returning to (A.19) and using that ψn k H1/2 ≤ C, we conclude, for k large, that √ √ ψn k , (A − ξk Aξk − φk Aφk )ψn k ≥ Aψn k , (1 − ξk2 − φk2 ) Aψn k 1 1 (A.27) −C + R1 Rk 1 1 , (A.28) ≥ −C + R1 Rk since (1 − ξk2 − φk2 )(x) ≥ 0 when k is sufficiently large. Finally, we note that Rk → ∞ as k → ∞ as well as R1 () → ∞ as → 0 holds, which leads to
(A.29) lim inf ψn k , (A − ξk Aξk − φk Aφk )ψn k ≥ −C() → 0, as → 0. k→∞
The proof of Lemma 2.4 is now complete.
A.2. Technical Details for the Proof of Theorem 2.1. Lemma A.1. Let (ψn ) satisfy the assumptions of Lemma 2.4. Furthermore, suppose that there exists a subsequence, (ψn k ), that satisfies part ii) of Lemma 2.4. Then 1 ∗ |ψn k |2 |ψn k |2 dx = 0. lim k→∞ R3 |x| Remark. A similar statement can be found in [12] in the context of other variational problems. For the sake of completeness, we present its proof for the situation at hand. Proof (of Lemma A.1). Let (ψn k ) be a bounded sequence in H1/2 (R3 ) such that |ψn k |2 dx = N , for all k ≥ 0, (A.30) R3
and assume that (ψn k ) satisfies part ii) in Lemma 2.4, i. e., |ψn k |2 dx = 0, for all R > 0. lim sup
(A.31)
For simplicity, let ψk := ψn k . We introduce f δ (x) := |x|−1 χ (x){|x|−1 ≥δ} , with δ > 0,
(A.32)
k→∞ y∈R3 |x−y| 0 and δ > 0, let gδR (x) := min{ f δ (x), R}, f δR (x) := max{ f δ (x) − R, 0}χ (x){|x|≤R} + f δ (x)χ (x){|x|>R} .
(A.34) (A.35)
Notice that f δ ≤ gδR χ{|x|≤R} + f δR holds. In view of (A.33), this leads to 1 ∗ |ψk |2 |ψk |2 dx ≤ δC + |ψk (x)|2 |ψk (y)|2 R3 |x| R 3 ×R 3 × gδR (x − y)χ (x − y){|x−y|≤R} dx dy
+ ψk 48/3 f δR 2 =: δC + I + I I,
(A.36)
using Young’s inequality and that f δR ∈ L2 (R3 ). By our assumption on (ψk ), we find that 2 |ψk (x)| dx |ψk (y)|2 dy → 0, as k → ∞. I ≤R R3
|x−y|≤R
Furthermore, we have that I I ≤ C f δR 2 ,
(A.37)
by Sobolev’s inequalities and the fact that (ψk ) is bounded in H1/2 (R3 ). Thus, we obtain 0≤
R3
1 ∗ |ψk |2 |ψk |2 dx ≤ δC + C f δR 2 + r (k), for all δ, R > 0, |x|
(A.38)
where r (k) → 0 as k → ∞. Since f δR 2 → 0 as R → ∞, for each fixed δ > 0, the assertion of Lemma A.1 follows by letting R → ∞ and then sending δ to 0. Lemma A.2. Suppose that > 0. Let (ψn ) satisfy the assumptions of Lemma 2.4 and let (ψn k ) be a subsequence that satisfies part iii) with sequences (ψk1 ) and (ψk2 ). Then, for k sufficiently large,
1 1 2 2 − ∗ |ψn k | |ψn k | dx ≥ − ∗ |ψk1 |2 |ψk1 |2 dx 3 |x| R3 |x| R 1 − ∗ |ψk2 |2 |ψk2 |2 dx − r1 (k) − r2 (), R3 |x| where r1 (k) → 0 as k → ∞ and r2 () → 0 as → 0.
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J. Fröhlich, B. L. G. Jonsson, E. Lenzmann
Proof (of Lemma A.2). Let > 0 and suppose that (ψn k ), (ψk1 ), and (ψk2 ) satisfy the assumptions stated above. Introducing βk := ψn k − (ψk1 + ψk2 )
(A.39)
and expanding the squares, we find that 1 1 2 2 ∗ |ψn k | |ψn k | dx = ∗ |ψk1 |2 |ψk1 |2 dx 3 3 |x| |x| R R 4 1 22 2 2 ∗ |ψk | |ψk | dx + In , (A.40) + R3 |x| n=0
where I0 =
I1 = I2 =
I3 = I4 =
1 22 1 1 2 ∗ |ψk | |ψk | dx + 4 ∗ (Re ψ¯ k1 ψk2 ) (Re ψ¯ k1 ψk2 ) dx 2 3 3 |x| |x| R R 1 1 2 1 2 +4 ∗ |ψk | (Re ψ¯ k ψk ) dx R3 |x| 1 ∗ |ψk2 |2 (Re ψ¯ k1 ψk2 ) dx, (A.41) +4 3 |x| R 1 ∗ |ψk1 + ψk2 |2 )(Re β¯k (ψk1 + ψk2 )) dx, 4 (A.42) 3 R |x| 1 ∗ (Re β¯k (ψk1 + ψk2 )) (Re β¯k (ψk1 + ψk2 )) dx 4 R3 |x| 1 ∗ |ψk1 + ψk2 |2 |βk |2 dx, (A.43) +2 3 |x| R 1 ∗ |βk |2 (Re β¯k (ψk1 + ψk2 )) dx, 4 (A.44) 3 |x| R 1 ∗ |βk |2 |βk |2 dx. (A.45) R3 |x|
To estimate I0 , we notice that if k is sufficiently large then ψk1 and ψk2 have disjoint supports receding from each other, i. e., dk := dist (supp ψk1 , supp ψk2 ) → ∞, as k → ∞;
(A.46)
see the proof of Lemma 2.4 in Sect. A.1. Thus, the last three terms of the right-hand side in (A.41) equal 0 if k is large, since ψ¯ k1 ψk2 = 0 a. e. if k is sufficiently large. Also by (A.46), we infer 1 |ψk2 (y)|2 dx dy |ψk1 (x)|2 3 3 |x − y| R ×R χ{|x−y|≥dk } (x − y) 2 = |ψk1 (x)|2 |ψk (y)|2 dx dy 3 3 |x − y| R ×R 1 2 2 2 −1 C ≤ ψk 2 ψk 2 |x| χ{|x|≥dk } (x)∞ ≤ → 0, as k → ∞, (A.47) dk
Boson Stars as Solitary Waves
23
using Young’s inequality. Thus we have shown that |I0 | ≤ r1 (k) → 0, as k → ∞.
(A.48)
The remaining terms I1 –I4 are controlled by the Hardy-Littlewood-Sobolev inequality and Hölder’s inequality as follows: |I1 | ≤ C( ψk1 312/5 + ψk2 312/5 ) βk 12/5 ,
(A.49)
|I2 | ≤ C( ψk1 212/5 + ψk2 212/5 ) βk 212/5 ,
(A.50)
|I3 | ≤ |I4 | ≤
C( ψk1 12/5 C β 412/5 .
+ ψk2 12/5 ) βk 312/5 ,
(A.51) (A.52)
We notice that ψk1 12/5 and ψk2 12/5 are uniformly bounded, by Sobolev’s inequality and the H1/2 -boundedness of these sequences. Furthermore, we have that βk 12/5 ≤ r2 () → 0, as → 0,
(A.53)
by part iii) of Lemma 2.4. Hence we conclude that |I1 + · · · + I4 | ≤ r2 () → 0, as → 0, which proves Lemma A.2.
(A.54)
A.3. Commutator Estimate. An almost identical result is needed in [5], but we provide its proof again. √ Lemma A.3. Let m ≥ 0, v ∈ R3 , and define Av := −∆ + m 2 + i(v · ∇). Furthermore, suppose that f (x) is a locally integrable and that its distributional gradient, ∇ f , is an L∞ (R3 ) vector-valued function. Then we have that [Av , f ] L2 →L2 ≤ Cv ∇ f ∞ , for some constant Cv that only depends on v. Remark. This result can be deduced by means of Calderón–Zygmund theory for singular integral operators and its consequences for pseudo-differential operators (see, e. g., [13, Sect. VII.3]). We give an elementary proof which makes good use of the spectral theorem, enabling us to write the commutator in a convenient way. Proof (of Lemma A.3). Since [i(v · ∇), f ] = iv · ∇ f holds, we have that [i(v · ∇), f ] L2 →L2 ≤ |v| ∇ f ∞ .
(A.55)
Thus, it suffices to prove our assertion for A := Av=0 , i. e., A := p 2 + m 2 , where p = −i∇.
(A.56)
Since A is a self-adjoint operator on L2 (R3 ) (with domain H1 (R3 )), functional calculus (for measurable functions) yields the formula 1 ∞ 1 ds A−1 = . (A.57) √ 2 π 0 s A +s
24
J. Fröhlich, B. L. G. Jonsson, E. Lenzmann
Due to this fact and A = A2 A−1 , we obtain the formula √ 1 ∞ s ds [A, f ] = [A2 , f ] 2 . 2 π 0 A +s A +s
(A.58)
Clearly, we have that [A2 , f ] = [ p 2 , f ] = p · [ p, f ] + [ p, f ] · p, which leads to √ 1 ∞ s ds [A, f ] = p · [ p, f ] + [ p, f ] · p 2 . (A.59) π 0 p2 + m 2 + s p + m2 + s Moreover, since [ p, f ] = −i∇ f holds, we have that 1 2 , [ p, f ] L 2 →L 2 ≤ ∇ f ∞ . (A.60) p2 + m 2 + s s 3 Hence we find, for arbitrary test functions ξ, η ∈ C∞ c (R ), that √ ∞
ds s [ p, f ] · p η ξ, p2 + m 2 + s p2 + m 2 + s 0 √ ∞
s ds ≤ [ p, f ]ξ, p η ( p 2 + m 2 + s)2 0 √ ∞
p s ds 1 + ξ, , [ p, f ] · η p2 + m 2 + s p2 + m 2 + s 0 √ ∞ p s ds ≤ [ p, f ]ξ 2 η 2 2 2 2 ( p + m + s) 0 ∞ p ds . +2 ξ 2 ∇ f ∞ η √ 2 s( p + m 2 + s) 2 0
Evaluation of the s-integrals yields (A.61) ≤ C ∇ f ∞ ξ 2 √
p p 2 +m 2
η2 ≤ C ∇ f ∞ ξ 2 η 2 .
(A.61)
(A.62)
The same estimate holds if [ p, f ] · p is replaced by p · [ p, f ] in (A.61). Thus, we have found that ξ, [A, f ]η ≤ C ∇ f ∞ ξ 2 η 2 , for ξ, η ∈ C∞ (R3 ), (A.63) c 3 2 3 with some constant C independent of m. Since C∞ c (R ) is dense in L (R ), the assertion 2 for the L -boundedness of [A, f ] now follows. This completes the proof of Lemma A.3.
A.4. Lower Semicontinuity. Lemma A.4. Suppose that m > 0, v ∈ R3 , with |v| < 1. Then the functional
T (ψ) := ψ, −∆ + m 2 ψ + ψ, i(v · ∇)ψ is weakly lower semicontinuous on H1/2 (R3 ), i. e., if ψk ψ weakly in H1/2 (R3 ) as k → ∞, then lim inf T (ψk ) ≥ T (ψ). k→∞
Moreover, if limk→∞ T (ψk ) = T (ψ) holds, then ψ k → ψ strongly in H1/2 (R3 ) as k → ∞.
Boson Stars as Solitary Waves
25
Proof (of Lemma A.4). Assume that m > 0, v ∈ R3 , with |v| < 1 holds. By Fourier transform and Plancherel’s theorem, we have that
(k)|2 |ψ k 2 + m 2 − (v · k) dk. (A.64) T (ψ) = R3
We notice that
c1 (|k| + m) ≤
k 2 + m 2 − (v · k) ≤ c2 (|k| + m),
(A.65)
for √ some suitable constants c1 , c2 > 0, where the lower bound follows from the inequality k 2 + m 2 ≥ (1 − δ)|k| + δm, with 0 < δ < 1, and the fact that |v| < 1 holds. Thus, ψ T := T (ψ) (A.66) defines a norm that is equivalent to · H1/2 . Consequently, the notion of weak and strong convergence for these norms coincide. Finally, by (A.64), we identify ψ T with the taken with respect to the integration measure L 2 -norm of ψ
k 2 + m 2 − (v · k) dk. (A.67) dµ = The assertion of Lemma A.4 now follows from corresponding properties of the L2 (R3 , µ)-norm; see, e. g., [9, Theorem 2.11] for L p (, µ)-norms, where is a measure space with positive measure, µ, and 1 < p < ∞. B. Best Constant and Optimizers for Inequality (2.5) Lemma B.1. For any v ∈ R3 with |v| < 1, there exists an optimal constant, Sv , such that
1 √ ∗ |ψ|2 |ψ|2 dx ≤ Sv ψ, −∆ + iv · ∇ ψ ψ, ψ (B.1) R3 |x| holds for all ψ ∈ H1/2 (R3 ). Moreover, we have that Sv =
2 , Q v , Q v
(B.2)
where Q v ∈ H1/2 (R3 ), Q v ≡ 0, is an optimizer for (B.1) and it satisfies √ 1 ∗ |Q v |2 Q v = −Q v . −∆ Q v + i(v · ∇)Q v − |x|
(B.3)
In addition, the following estimates hold: Sv=0
0 and β > 0. In particular, it is sufficient to show that Iv (α = 1, β = 1) is finite and attained so that Sv = −Iv (1, 1). (B.7) In fact, we will show that all minimizing sequences for I (1, 1) are relatively compact in H1/2 (R3 ) up to translations. In turn, this relative compactness implies that all minimizing sequences for problem (B.5) are relatively compact in H1/2 (R3 ) up to translations and rescalings: For any minimizing sequence, (ψn ), for (B.5), there exist sequences, {(yk ), (ak ), (bk )}, with yk ∈ R3 , 0 = ak ∈ C, 0 = bk ∈ R, such that (B.8) ak ψn k bk (· + yk ) → Q v strongly in H1/2 (R3 ) as k → ∞, along a suitable subsequence, (ψn k ), and Q v minimizes (B.5). First we show that I (α, β) is indeed finite. The Hardy–Littlewood–Sobolev inequality implies √ 1 ∗ |ψ|2 |ψ|2 dx ≤ C |ψ|2 26/5 = C ψ 412/5 ≤ C ψ, −∆ψ ψ, ψ , (B.9) R3 |x| √ where we use Sobolev’s inequality ψ 23 ≤ C ψ, −∆ ψ in R3 and Hölder’s inequa√ √ lity. Since ψ, −∆ψ ≤ (1 − |v|)−1 ψ, ( −∆ + iv · ∇)ψ , we deduce that I (α, β) ≥ −Cαβ > −∞,
(B.10)
for some constant C. On the other hand, we have that
I (α, β) < 0,
(B.11)
since R3 (|x|−1 ∗ |ψ|2 )|ψ|2 dx = 0 when ψ ≡ 0. Next, we show that Iv (1, 1) is attained. Let (ψn ) be a minimizing sequence for Iv (1, 1). In order to invoke Lemma 2.4, we notice that R3 |ψn |2 dx = 1 and that (ψn ) is √ √ bounded in H1/2 (R3 ), since ψ, ( −∆ + iv · ∇)ψ is equivalent to ψ, −∆ψ when |v| < 1, by (A.65) with m = 0. Let us suppose now that case ii) of Lemma 2.4 occurs. Referring to Lemma A.1, we conclude that I (1, 1) = 0 holds, which contradicts (B.11). Next, let us assume that dichotomy occurs for a subsequence of (ψn ), i. e., property iii) of Lemma 2.4 holds. Using Lemma A.2 and the lim inf-estimate stated in iii) of Lemma 2.4 and by taking the limit → 0, we conclude that Iv (1, 1) ≥ Iv (α, β) + Iv (1 − α, 1 − β),
(B.12)
for some α ∈ (0, 1) and β ∈ [0, 1]. On the other hand, we have the scaling behaviour Iv (α, β) = αβ Iv (1, 1) < 0,
(B.13)
Boson Stars as Solitary Waves
27
which follows from (B.6) and rescaling ψ(x) → aψ(bx) with a, b > 0. Combining (B.12) with (B.13) we get a contradiction. Therefore dichotomy for minimizing sequences is ruled out. In summary, we see that any minimizing sequence, (ψn ), for Iv (1, 1) contains a subsequence, (ψn k ), with a sequence of translations, (yk ), satisfying property i) of Lemma v strongly 2.4. Similarly to the proof of Theorem 2.1, we conclude that ψn k (· + yk ) → Q 1/2 3 1/2 3 in H (R ) as k → ∞, where Q v ∈ H (R ) is a minimizer for Iv (1, 1). To show that the best constant, Sv , is given by (B.2) with Q v minimizing (B.5) and v . Since satisfying (B.3), let us denote the minimizer constructed above for I (1, 1) by Q v also minimizes the unconstrained problem (B.5), it has to satisfy the corresponding Q Euler-Lagrange equation which reads as follows: √
v + iv · ∇ Q v − −∆ Q
2 1 v |2 Q v = 0, v + Q ∗ |Q Sv |x|
(B.14)
√ v , ( −∆ + iv · ∇) Q v = 1 and Q v , Q v = 1 holds. By putting where we use that Q √ −1/2 Q v = 2Sv Q v , we see that Q v minimizes (B.5) and satisfies (B.2). Moreover, we have that Q v , Q v = 2/Sv holds. Finally, we turn to the estimates for Sv stated in Lemma B.1. That Sv=0 < π/2 holds follows from the√appendices in [11, 7].√To see that Sv ≤ (1 − |v|)−1 Sv=0 is true, we use the estimate −∆ ≤ (1 − |v|)−1 ( −∆ + iv · ∇). Moreover, it is known from the discussion in [7] that if v = 0 the minimizer, Q v=0 , for (2.2) can be chosen to be radial (by symmetric rearrangement). This implies that Q v=0 , ∇ Q v=0 = 0, which leads to Sv=0 ≤ Sv . C. Exponential Decay In this section, we prove pointwise exponential decay for solutions, ϕ ∈ H1/2 (R3 ), of the nonlinear equation
1 ∗ |ϕ|2 ϕ = −µϕ. −∆ + m 2 − m ϕ + i(v · ∇)ϕ − |x|
(C.1)
Clearly, ϕ(x) is an eigenfunction for the Schrödinger type operator H = H0 + V, where H0 :=
1 ∗ |ϕ|2 ). −∆ + m 2 − m + i(v · ∇) and V := − |x|
(C.2)
(C.3)
By using the bootstrap argument for regularity (presented in the proof of Theorem 4.1)), we have that ϕ ∈ Hs (R3 ) for all s ≥ 1/2, which shows in particular that ϕ is smooth. Investigating the spectrum of H0 we find that σ (H0 ) = σess (H0 ) = [Σv , ∞), where the bottom of the spectrum is given by Σv = ( 1 − v 2 − 1)m.
(C.4)
(C.5)
28
J. Fröhlich, B. L. G. Jonsson, E. Lenzmann
To see this, we remark that the function f (k) = (k 2 + m 2 )1/2 − m − v · k
(C.6)
√ √ obeys f (k) ≥ ( 1 − v 2 )m − m with equality for k = (mv/ 1 − v 2 ). We have the following result. Lemma C.1. Suppose that m > 0, v ∈ R3 , and |v| < 1. Furthermore, let ϕ ∈ H1/2 (R3 ) Σv +µ √ be a solution of (C.1) with −µ < Σv . Then, for every 0 < δ < min m, , there 1−v 2 exists 0 < C(δ) < ∞ such that |ϕ(x)| ≤ Ce−δ|x| holds for all x ∈ R3 . Proof (of Lemma C.1). We rewrite (C.1) as follows : ϕ = −(H0 + µ)−1 V ϕ,
(C.7)
where H0 and V are defined in (C.3). Note that (H0 + µ)−1 exists, since we have that µ ∈ σ (H0 ) holds, by the assumption that −µ < Σv . We consider the Green’s function, G µ (x − y), given by 1 (x − y), (C.8) G µ (x − y) = F −1 √ k2 + m2 − m − v · k + µ √ where F : S → S denotes the Fourier transform. Since 1/ k 2 + m 2 . . . does not belong to L1 (R3 ), we cannot use Payley–Wiener type theorems directly to deduce pointwise exponential decay for G µ (z) in |z|. To overcome this difficulty, we first notice that ∞ ∞ √ 2 2 (H0 + µ)−1 = e−tµ e−t H0 dt = e−t (µ−m) e−t ( p +m −v· p) dt, (C.9) 0
0
by self-adjointness of H0 and functional calculus. Here and in what follows, we put p = −i∇ √ for convenience. By using the explicit formula for the Fourier transform of exp{−t k 2 + m 2 } (see e. g., [9]) in R3 and by analytic continuation, we obtain from (C.9) the formula ∞ t m t 2 + (z + itv)2 dt. (C.10) e−t (µ−m) 2 K G µ (z) = Am 2 2 t + (z + itv) 0 Here K 2 (z) stands for the modified Bessel function of the third kind, and Am > 0 denotes some constant. Notice that w = t 2 + (z + itv)2 = (1 − v 2 )t 2 + z 2 + 2itv · z
(C.11)
is a complex number with |arg w| < π/2. Next we analyze G µ (z) for |z| ≤ 1/m and for |z| > 1/m separately. From [1] we recall the estimate C |K 2 (mw)| ≤ , for |arg w| < π/2, (C.12) |w|2
Boson Stars as Solitary Waves
29
which implies that G µ (z) with |z| ≤ 1/m satisfies the bound ∞ t |G µ (z)| ≤ C e−t (µ−m) 2 2 (1 − v )t + |z|2 + 2itv · z 0 K 2 m (1 − v 2 )t 2 + |z|2 + 2itv · z dt ∞ t ≤C e−t (µ−m) dt. 2 )t 2 + |z|2 ]2 [(1 − v 0
(C.13)
Since µ − m ≥ 0, the t-integral is finite for z = 0 and we obtain C , for |z| ≤ 1/m, |z|2 √ √ where we use that |a + ib| ≥ |a| and | a + ib| ≥ |a| holds for a, b ∈ R. To estimate G µ (z) for |z| > 1/m, we use the bound |G µ (z)| ≤
(C.14)
e−mw e−m|Re w| ≤ C |K 2 (mw)| ≤ C , for |arg w| < π/2 and |w| > 1, (C.15) |w|2 |w|2 √ taken from [1]. By means of the inequality a 2 + b2 ≥ (1 − )|a| + |b|, for any 0 < ≤ 1, we proceed to find that ∞ √ t 2 e−t (µ−m+(1−) 1−v m) dt, (C.16) |G µ (z)| ≤ Ce−m|z| 2 [(1 − v )t 2 + |z|2 ]2 0 for |z| > 1/m. The assumption on µ allows us to choose ∈ (0, 1] such that the exponent in the t-integral is nonpositive. The best is given by Σv + µ ∈ (0, 1], (C.17) = min 1, √ m 1 − v2 and hence |G µ (z)| ≤ Ce−m|z| ≤C
0
∞
t [(1 − v 2 )t 2
+ |z|2 ]2
e−m|z| , for |z| > 1/m. |z|2
dt (C.18)
Combining now (C.14) and (C.18), we see that |G µ (z)| ≤ C
e−m|z| , for z ∈ R3 , |z|2
(C.19)
where is given by (C.17) and C is some constant. This shows that G µ (z) exhibits exponential decay; in particular, we have that G µ ∈ L p (R3 ) if 1 ≤ p < 3/2. Returning to (C.7), we notice that ϕ(x) = − G µ (x − y)V (y)ϕ(y) dy. (C.20) R3
30
J. Fröhlich, B. L. G. Jonsson, E. Lenzmann
Moreover, the function V (x) = −(|x|−1 ∗ |ϕ|2 )(x) obeys V ∈ C0 (R3 ) and
lim V (x) = 0,
|x|→∞
(C.21)
since f ∗ g is a continuous function vanishing at infinity, provided that f ∈ L p and g ∈ L p with 1/ p + 1/ p = 1 and p > 1; see, e. g., [9]. Here we note that, e. g., |x|−1 ∈ L2 (R3 ) + L4 (R3 ) and in particular |ϕ|2 ∈ L4/3 (R3 ) ∩ L2 (R3 ) since ϕ ∈ Hs (R3 ) for all s ≥ 1/2 (cf. beginning of App. C). Using (C.20), (C.19) and (C.21), the claimed pointwise exponential decay of ϕ(x) follows from a direct adaptation of an argument by Slaggie and Wichmann for exponential decay of eigenfunctions for Schrödinger operators; see, e. g., [6] for a convenient exposition of this method. This completes the proof of Lemma C.1. Acknowledgement. The authors are grateful to I. M. Sigal and M. Struwe for useful discussions, as well as the referee for some helpful comments. Lastly, E. L. also thanks D. Christodoulou.
References 1. Abramowitz, M., Stegun, I.A.: ed.: Handbook of mathematical functions with formulas, graphs, and mathematical tables. New York: Dover Publications Inc., 1992, Reprint of the 1972 edition 2. Cazenave, T., Lions, P.-L.: Orbital stability of standing waves for some nonlinear Schrödinger equations. Commun. Math. Phys. 85, 549–561 (1982) 3. Elgart, A., Schlein, B.: Mean field dynamics of Boson Stars. Comm. Pure Appl. Math. 60(4), 500– 545 (2006) 4. Fröhlich, J., Jonsson, B.L.G., Lenzmann, E.: Effective dynamics for Boson stars. Nonlinearity 32, 1031– 1075 (2007) 5. Fröhlich, J., Lenzmann, E.: Blow-up for nonlinear wave equations describing Boson Stars. http://arxiv.org/list/math-ph/0511003, 2006 to appear in Comm. Pure Appl. Math. 6. Hislop, P.D.: Exponential decay of two-body eigenfunctions: A review. In: Proceedings of the Symposium on Mathematical Physics and Quantum Field Theory (Berkeley, CA, 1999), Volume 4 of Electron. J. Differ. Equ. Conf., pages 265–288 (electronic), San Marcos, TX, 2000. Southwest Texas State Univ. 7. Lenzmann, E.: Well-posedness for semi-relativistic Hartree equations of critical type. http://arXiv.org/list/math.AP/0505456, 2005, to appear in Mathematical Physics, Analysis, and Geometry 8. Lieb, E.H.: Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation. Stud. Appl. Math. 57, 93–105 (1977) 9. Lieb, E.H., Loss, M.: Analysis, Volume 14 of Graduate Studies in Mathematics. A Providence, RI: Amer. Math. Soc., second edition, 2001 10. Lieb, E.H., Thirring, W.: Gravitational collapse in quantum mechanics with relativistic kinetic energy. Ann. Physics 155(2), 494–512 (1984) 11. Lieb, E.H., Yau, H.-T.: The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics. Commun. Math. Phys. 112, 147–174 (1987) 12. Lions, P.-L.: The concentration-compactness principle in the calculus of variations. The locally compact case, Part I. Ann. Inst. Henri Poincaré, 1(2), 109–145 (1984) 13. Stein, E.M.: Harmonic Analysis. Princeton, NJ: Princeton University Press, 1993 Communicated by H.-T. Yau
Commun. Math. Phys. 274, 31–64 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0271-x
Communications in
Mathematical Physics
Unoriented WZW Models and Holonomy of Bundle Gerbes Urs Schreiber, Christoph Schweigert, Konrad Waldorf Fachbereich Mathematik, Schwerpunkt Algebra und Zahlentheorie, Universität Hamburg, Bundesstraße 55, D–20146 Hamburg, Germany. E-mail:
[email protected];
[email protected];
[email protected] Received: 24 January 2006 / Accepted: 26 July 2006 Published online: 13 June 2007 – © Springer-Verlag 2007
Abstract: The Wess-Zumino term in two-dimensional conformal field theory is best understood as a surface holonomy of a bundle gerbe. We define additional structure for a bundle gerbe that allows to extend the notion of surface holonomy to unoriented surfaces. This provides a candidate for the Wess-Zumino term for WZW models on unoriented surfaces. Our ansatz reproduces some results known from the algebraic approach to WZW models. manche meinen lechts und rinks kann man nicht velwechsern werch ein illtum Ernst Jandl [Jan95] Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Bundle Gerbes with Jandl Structures . . . . . . . . . . . . . 2.1 Bundle gerbes and stable isomorphisms . . . . . . . . . 2.2 Jandl structures . . . . . . . . . . . . . . . . . . . . . . 2.3 Classification of Jandl structures . . . . . . . . . . . . . 2.4 Local data . . . . . . . . . . . . . . . . . . . . . . . . . 3. Holonomy of Gerbes with Jandl Structure . . . . . . . . . . . 3.1 Double coverings, fundamental domains and orientations 3.2 Unoriented surface holonomy . . . . . . . . . . . . . . .
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K.W. is supported with scholarships by the German Israeli Foundation (GIF) and by the Rudolf und Erika Koch–Stiftung.
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3.3 Holonomy in local data . . . . . . . . . 3.4 Examples . . . . . . . . . . . . . . . . 4. Gerbes and Jandl Structures in WZW models 4.1 Oriented and orientable WZW models . 4.2 Unoriented WZW models . . . . . . . . 4.3 Crosscaps and the trivial line bundle . . 4.4 Examples of target spaces . . . . . . . .
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1. Introduction Wess-Zumino-Witten (WZW) models are one of the most important classes of (twodimensional) rational conformal field theories. They describe physical systems with (non-abelian) current symmetries, provide gauge sectors in heterotic string compactifications and are the starting point for other constructions of conformal field theories, e.g. the coset construction. Moreover, they have played a crucial role as a bridge between Lie theory and conformal field theory. It is well-known that for the Lagrangian description of such a model, a WessZumino term is needed to get a conformally invariant theory [Wit84]. Later, the relation of this term to Deligne hypercohomology has been realized [Gaw88] and its nature as a surface holonomy has been identified [Gaw88, Alv85]. More recently, the appropriate differential-geometric object for the holonomy has been identified as a hermitian U (1) bundle gerbe with connection and curving [CJM02]. Already the case of non-simply connected Lie groups with non–cyclic fundamental group, such as G := Spin(4n)/Z2 × Z2 shows that gerbes and their holonomy are really indispensable, even when one restricts one’s attention to oriented surfaces without boundary. The original definition of the Wess-Zumino term as the integral of a three form H over a suitable three-manifold cannot be applied to such groups; moreover, it could not explain the well-established fact that to such a group two different rational conformal field theories that differ by “discrete torsion” can be associated. Bundle gerbes will be central for the problem we address in this paper. A long series of algebraic results indicate that the WZW model can be consistently considered on unorientable surfaces. Early results include a detailed study of the abelian case [BPS92] and of SU (2) [PSS95b, PSS95a]. Sewing constraints for unoriented surfaces have been derived in [FPS94]. Already the abelian case [BPS92] shows that not every rational conformal field theory that is well-defined on oriented surfaces can be considered on unoriented surfaces. A necessary condition is that the bulk partition function is symmetric under exchange of left and right movers. This restricts, for example, the values of the Kalb-Ramond field in toroidal compactifications [BPS92]. Moreover, if the theory can be extended to unoriented surfaces, there can be different extensions that yield inequivalent correlation functions. This has been studied in detail for WZW theories based on SU (2) in [PSS95b, PSS95a]; later on, this has been systematically described with simple current techniques [HS00, HSS99]. Unifying general formulae have been proposed in [FHS+ 00]; the structure has been studied at the level of NIMreps in [SS03]. Aspects of these results have been proven in [FRS04] combining topological field theory in three-dimensions with algebra and representation theory in modular tensor categories. As a crucial ingredient, a generalization of the notion of an algebra with involution, i.e. an algebra together with an algebra-isomorphism to the opposed algebra, has been
Unoriented WZW Models and Holonomy of Bundle Gerbes
33
identified in [FRS04]; the isomorphism is not an involution any longer, but squares to the twist on the algebra. An algebra with such an isomorphism has been called a Jandl algebra. A similar structure, in a geometric setting, will be the subject of the present article. The success of the algebraic theory leads, in the Lagrangian description, to the quest for corresponding geometric structures on the target space. From previous work [BCW01, HSS02, Bru02] it is clear that a map k : M → M on the target space with the additional property that k ∗ H = −H will be one ingredient. Examples like the Lie group S O(3), for which two different unoriented WZW models with the same map k are known, already show that this structure does not suffice. We are thus looking for an additional structure on a hermitian bundle gerbe which allows to define a Wess-Zumino term, i.e. which allows to define holonomy for unoriented surfaces. For a general bundle gerbe, such a structure need not exist; if it exists, it will not be unique. In the present article, we make a proposal for such a structure. It exists whenever there are sufficiently well-behaved stable isomorphisms between the pullback gerbe k ∗ G and the dual gerbe G ∗ . If one thinks about a gerbe as a sheaf of groupoids, the formal similarity to the Jandl structures in [FRS04] becomes apparent, if one realizes that the dual gerbe plays the role of the opposed algebra. For this reason, we term the relevant structure a Jandl structure on the gerbe. We show that the Jandl structures on a gerbe on the target space M, if they exist at all, form a torsor over the group of flat equivariant hermitian line bundles on M. As explained in Sect. 4.3, this group always contains an element L k−1 of order two. We show that two Jandl structures that are related by the action of L k−1 provide amplitudes that just differ by a sign that depends only on the topology of the worldsheet. Such Jandl structures are considered to be essentially equivalent. We finally show that a Jandl structure allows to extend the definition of the usual gerbe holonomy from oriented surfaces to unoriented surfaces. We derive formulae for these holonomies in local data that generalize the formulae of [GR02, Alv85] for oriented surfaces. To give a concrete impression of a Jandl structure, we write out the local data of a Jandl structure for a given gerbe G on the target space M. To this end, we first recall the local data of a hermitian bundle gerbe in a good open cover {Vi }i∈I of M: we have a 2-form Bi for each open set Vi , a 1-form Ai j on each intersection Vi ∩ V j and a U (1)valued function gi jk on each triple intersection Vi ∩ V j ∩ Vk . They are required to satisfy the following constraints:
A jk
−1 g jkl · gikl · gi jl · gi−1 jk = 1, − Aik + Ai j + dlog gi jk = 0,
−d Ai j + B j − Bi = 0. To write down the local data of a Jandl structure for a given involution k : M → M in a succinct manner, we make the simplifying assumption that we have a cover {Vi }i∈I that is invariant under k, k(Vi ) = Vi , and that is still good enough to provide local data. The local data of a Jandl structure then consist of a U (1)-valued function ji : Vi → U (1) for each open subset, a U (1)-valued function ti j : Vi ∩V j → U (1) on two-fold intersections and a 1-form Wi ∈ 1 (Vi ). They relate the pullbacks of the gerbe data under k to the local data of the dual gerbe as follows:
34
U. Schreiber, C. Schweigert, K. Waldorf
k ∗ Bi = −Bi + dWi , k ∗ Ai j = −Ai j − dlog(ti j ) + W j − Wi , −1 k ∗ gi jk = gi−1 jk · t jk · tik · ti j .
The local data of a Jandl structure are required to be equivariant under k in the sense that k ∗ Wi = Wi − dlog( ji ), k ∗ ti j = ti j · j −1 j · ji , k ∗ ji = ji−1 . It should be appreciated that the functions ti j are not transition functions of some line bundle; as we will explain in Sect. 2.4, they are rather the local data describing an isomorphism of line bundles appearing in the Jandl structure. The notion of a Jandl structure naturally explains algebraic results for specific classes of rational conformal field theories. It is well-known that both the Lie group SU (2) and its quotient S O(3) admit two Jandl structures that are essentially different (i.e. that do not just differ by a sign depending on the topology of the surface). In the case of SU (2), this is explained by the fact that two different involutions are relevant: g → g −1 and g → zg −1 , where z is the non-trivial element in the center of SU (2). Indeed, since SU (2) is simply-connected, we have a single flat line bundle and hence for each involution only two Jandl structures which are essentially the same. The two involutions of SU (2) descend to the same involution of the quotient S O(3). The latter manifold, however, has fundamental group Z2 and thus twice as many equivariant flat line bundles as SU (2). The different Jandl structures of S O(3) are therefore not explained by different involutions on the target space but rather by the fact that one involution admits two essentially different Jandl structures. Needless to say, there remain many open questions. A discussion of surfaces with boundaries is beyond the scope of this article. The results of [FRS04] suggest, however, that a Jandl structure leads to an involution on gerbe modules. Most importantly, it remains to be shown that, in the Wess-Zumino-Witten path integral for a surface , the holonomy we introduced yields amplitudes that take their values in the space of conformal blocks associated to the complex double of , which ensures that the relevant chiral Ward identities are obeyed. To this end, it will be important to have a suitable reformulation of Jandl structures at our disposal. Indeed, the holonomy we propose in this article also arises as the surface holonomy of a 2-vector bundle with a certain 2-group; this issue will be the subject of a separate publication. 2. Bundle Gerbes with Jandl Structures 2.1. Bundle gerbes and stable isomorphisms. In preparation of the following sections, in this section we define an equivalence relation on the set of stable isomorphisms between two fixed bundle gerbes. To this end, we first set up the notation concerning bundle gerbes and stable isomorphisms. We mainly adopt the formalism used by Murray and collaborators, see [CJM02] for example, as well as by Gaw¸edzki and Reis [GR02]. Definition 1. A hermitian U (1) bundle gerbe G with connection and curving over a smooth manifold M consists of the following data: a surjective submersion π : Y → M, a hermitian line bundle p : L → Y [2] with connection, an associative isomorphism ∗ ∗ ∗ L ⊗ π23 L −→ π13 L µ : π12
(1)
Unoriented WZW Models and Holonomy of Bundle Gerbes
35
of hermitian line bundles with connection over Y [3] , and a 2-form C ∈ 2 (Y ) which satisfies π2∗ C − π1∗ C = curv(L). (2) Here Y [ p] denotes the p-fold fiber product of π : Y → M, which is a smooth manifold since π is a surjective submersion. For example π12 : Y [3] → Y [2] is the projection on the first two factors. Remark 1. From now on we will use the following conventions: the term line bundle refers to a hermitian line bundle with connection, and an isomorphism of line bundles refers to an isomorphism of hermitian line bundles with connection. Accordingly, we refer to Definition 1 by the term gerbe. The 2-form C is called curving, and the isomorphism µ is called multiplication. One can show that there is a unique 3-form H ∈ 3 (M) with π ∗ H = dC; this 3-form is called the curvature of the gerbe and is denoted by H = curv(G). To each gerbe G, we associate the dual gerbe G ∗ . It has the same surjective submersion π : Y → M, but the dual line bundle L ∗ → Y [2] with multiplication ∗ ∗ ∗ ∗ ∗ ∗ L ⊗ π23 L −→ π13 L , (µ∗ )−1 : π12
(3)
and the negative curving −C. Accordingly, the curvature of the dual gerbe satisfies curv(G ∗ ) = −curv(G).
(4)
Even more, the classes of G and the one of G ∗ in Deligne hypercohomology are inverses. For a smooth map f : N → M and a pullback diagram Yf
f˜
πf
N
/Y π
f
/M
,
(5)
∗ π f : Y f → N is a surjective submersion, and together with the line bundle f˜ L over ∗ ˜∗ ˜∗ Y [2] f , the multiplication f µ and the curving f C, we have defined a gerbe f G. If f : M → M is a diffeomorphism, Y f is canonically isomorphic to Y , such that f˜ = idY and π f = f −1 ◦ π . The curvature of the pullback gerbe is
curv( f ∗ G) = f ∗ curv(G).
(6)
Remark 2. As we did in the last paragraph, whenever there is a map f˜ : Y f → Y , we will use the same letter for the induced map on higher fiber products. Definition 2. A trivialization T = (T, τ ) of a gerbe G is a line bundle T → Y , together with an isomorphism τ : L ⊗ π2∗ T −→ π1∗ T (7) of line bundles over Y [2] , which is compatible with the isomorphism µ of the gerbe.
36
U. Schreiber, C. Schweigert, K. Waldorf
We call a gerbe G trivial, if it admits a trivialization. A choice of a trivialization T gives the 2-form C − curv(T ) ∈ 2 (Y ), which descends to a unique 2-form ρ ∈ 2 (M) with π ∗ ρ = C − curv(T ). This 2-form satisfies dρ = H , so the curvature H of a trivial gerbe is an exact form. If there are two trivializations T1 = (T1 , τ1 ) and T2 = (T2 , τ2 ) of the same gerbe G, one obtains an isomorphism α := τ1−1 ⊗ τ2∗ : π1∗ (T1 ⊗ T2∗ ) −→ π2∗ (T1 ⊗ T2∗ ),
(8)
Y [2] .
of line bundles over From the compatibility condition between the multiplication µ and both τ1 and τ2 the cocycle condition ∗ ∗ ∗ π23 α ◦ π12 α = π13 α
(9)
follows. Such an isomorphism determines a unique descent line bundle N → M with connection together with an isomorphism ν : π ∗ N → T1 ⊗T2∗ [Bry93]. The two 2-forms ρ1 and ρ2 coming from the two trivializations are related by ρ2 = ρ1 + curv(N ). Definition 3. Let G and
G
(10)
be two gerbes. A stable isomorphism A : G −→ G
consists of a line bundle A → Z over the fiber product Z :=
(11) Y
× M Y with curvature
curv(A) = p ∗ C − p ∗ C,
(12)
α : p ∗ L ⊗ p ∗ L ∗ ⊗ π2∗ A −→ π1∗ A
(13)
and an isomorphism of line bundles over Z [2] , which is compatible with the multiplications µ and µ of both gerbes. Here p and p denote the projections from Z to Y and to Y respectively. Since the pullbacks of the curvings C and C to Z differ by a closed 2-form, the curvatures of stably isomorphic gerbes, defined by the differential of C, are equal. Definition 4. Let G and G be two gerbes, and A1 and A2 two stable isomorphisms from G to G . A morphism β : A1 =⇒ A2 (14) is an isomorphism β : A1 → A2 of line bundles over Z , which is compatible with α1 and α2 in the sense that the diagram p ∗ L ⊗ p ∗ L ∗ ⊗ π2∗ A1
α1
/ π ∗ A1 1
1⊗1⊗π2∗ β
p ∗ L ⊗ p ∗ L ∗ ⊗ π2∗ A2
α2
π1∗ β
(15)
/ π ∗ A2 1
of isomorphisms of line bundles over Z [2] commutes. The definition of such a morphism of stable isomorphisms already appeared in [Ste00]. We call two stable isomorphisms equivalent, if there is a morphism between them. This defines an equivalence relation on the set of stable isomorphisms between two fixed gerbes G and G .
Unoriented WZW Models and Holonomy of Bundle Gerbes
37
2.2. Jandl structures. Recall that for a group K acting on a manifold M by diffeomorphisms k : M → M, a K -equivariant structure on a line bundle L → M is a family ϕ k k∈K of isomorphisms ϕ k : k ∗ L −→ L
(16)
of line bundles, which respect the group structure of K in the sense that ϕ 1 : L → L is the identity, and the multiplication law ϕ k1 k2 = ϕ k2 ◦ k2∗ ϕ k1
(17)
is satisfied. Remember that according to our convention in Remark 1 all line bundles have connections, and all isomorphisms of line bundles preserve them. In this article, we only consider the group K = Z2 for the sake of simplicity. Let G be a gerbe over M and let K = Z2 act on M. Denote the action of the non-trivial element k by k : M → M. Assume that there is a stable isomorphism A = (A, α) : k ∗ G → G ∗ . Recall that in this particular situation, A is a line bundle over the space Z = Yk × M Y , where Yk := Y and πk := k −1 ◦ π as in our discussion of the pullback of G by a diffeomorphism k. We still denote the projections from Z to Y and to Yk by p and p respectively. Define the surjective submersion π Z := π ◦ p : Z → M. As k 2 = id M , the permutation map (18) k˜ : Z −→ Z : (yk , y) −→ (y, yk ) gives the following commuting diagram: Z
k˜
πZ
M
/Z πZ
k
(19)
/M
Furthermore, since also k˜ 2 = id Z , we even have a lift of the action of K into Z . Definition 5. A Jandl structure on G is a collection J = (k, A, ϕ) consisting of • a smooth action of K = Z2 on M, where we denote the non-trivial element and the diffeomorphism associated to that non-trivial element by k : M → M. • a stable isomorphism of gerbes A = (A, α) : k ∗ G → G ∗ . • a K -equivariant structure ϕ := ϕ k on the line bundle A, which is compatible with the stable isomorphism A in the sense that the diagram p ∗ L ⊗ p ∗ L ⊗ π2∗ A
α
1
1⊗1⊗π2∗ ϕ
p ∗ L ⊗ p ∗ L ⊗ k ∗ π2∗ A
/ π∗ A π1∗ ϕ
k∗α
of isomorphisms of line bundles over Z [2] commutes.
/ k∗π ∗ A 1
(20)
38
U. Schreiber, C. Schweigert, K. Waldorf
We can immediately deduce a necessary condition for the existence of a Jandl structure for a given gerbe G, namely the condition that the gerbes k ∗ G and G ∗ are stably isomorphic. Since the curvatures of stably isomorphic gerbes are equal, this in turn demands k ∗ H = −H (21) for the curvature H = curv(G) of G. In particular, there will be gerbes on manifolds with involution which do not admit a Jandl structure. Definition 6. Two Jandl structures J and J on the same gerbe G are equivalent, if the following conditions are satisfied: • the actions are the same, i.e. k and k are the same diffeomorphisms, • there is a morphism β : A ⇒ A of stable isomorphisms in the sense of Definition 4 such that • β : A → A is even an isomorphism of K -equivariant line bundles on Z . Next, we show that Jandl structures behave well under the pullback of gerbes along a smooth map f : N → M. Let J = (k, A, ϕ) be a Jandl structure on G. Assume that there is an action of K = Z2 on N by a diffeomorphism g, such that the diagram f
N
/M
g
(22)
k
N
/M
f
commutes. Consider the pullback of G by f as discussed before, and define Z f := (Y f )g × N Y f
(23)
and the permutation map g˜ : Z f → Z f . Then f˜
Zf
}} }} } } ~} }
Zf
/Z k˜
g˜
f˜
/Z
πZ
πZ f πZ f
{{ {{ { { { }{ N
N
g
f
πZ
(24)
f /M } }} }}k } ~}} /M
is a cube with commuting faces. It follows that f ∗ A := ( f˜∗ A, f˜∗ α) is a stable isomorphism from g ∗ f ∗ G to f ∗ G ∗ . Furthermore, f˜∗ ϕ is a K -equivariant structure on f˜∗ A, where K acts by g. ˜ In summary, f ∗ J := (g, f˜∗ A, f˜∗ ϕ) defines a pullback Jandl structure on
f ∗ G.
(25)
Unoriented WZW Models and Holonomy of Bundle Gerbes
39
2.3. Classification of Jandl structures. If a gerbe G admits a Jandl structure, it is natural to ask, how many inequivalent choices exist. So we are interested in the set Jdl(G, k) of equivalence classes of Jandl structures J = (k, −, −) with a fixed action of K = Z2 via k. This will be crucial in the discussion of the unoriented WZW model in Sect. 4. To approach this task, we first investigate the set Hom(G, G ) of equivalence classes of stable isomorphisms between G and G . We start by recalling the following Lemma 1 ([CJM02]). (i) If N → M is a flat line bundle and A = (A, α) is a stable isomorphism, then N .A := (A ⊗ π Z∗ N , α ⊗ 1) is also a stable isomorphism. (ii) If A1 = (A1 , α1 ) and A2 = (A2 , α2 ) are two stable isomorphisms, then there is a unique flat line bundle N → M such that A1 and N .A2 are equivalent as stable isomorphisms. Proof. For the first part we note that because N is flat, A and A ⊗ π Z∗ N have the same curvature, so that (12) is satisfied. For the second part, we use the isomorphism α1−1 ⊗ α2∗ : π1∗ (A1 ⊗ A∗2 ) −→ π2∗ (A1 ⊗ A∗2 )
(26)
which satisfies the cocycle condition because of the compatibility of α1 and α2 with µ and µ . This determines a unique line bundle N → M with connection together with an isomorphism ν : π Z∗ N → A1 ⊗ A∗2 . Because (12) requires the curvatures of both A1 and A2 to be the same, N is flat. Now ν determines an isomorphism A1 → A2 ⊗ π Z∗ N , which is a morphism A1 ⇒ N .A2 .
We denote the group of isomorphism classes of flat line bundles over M by Pic0 (M). It is a subgroup of the Picard group Pic(M) of isomorphism classes of hermitian line bundles with connection over M. Lemma 2. The set Hom(G, G ) of equivalence classes of stable isomorphisms is a torsor over the flat Picard group Pic0 (M). Proof. We will (a) define the action and show that it is (b) transitive and (c) free. (a) We act [N ].[A] := [N .A], where the right-hand side was defined in Lemma 1 (i). This definition is independent of the choice of representatives N and A: an isomorphism N → N gives an isomorphism N .A → N .A, which in fact is a morphism of stable isomorphisms N .A ⇒ N .A. On the other hand, a morphism A ⇒ A of stable isomorphisms induces a morphism N .A ⇒ N .A . Because N .A is defined using the group structure on the group of isomorphism classes of line bundles with connection, it respects the group structure on Pic0 (M), and hence defines an action. (b) The transitivity follows directly from Lemma 1 (ii). (c) Let [A] be an element in Hom(G, G ), let N be a flat line bundle and let us assume that N .A and A are equivalent, in particular A ⊗ π Z∗ N is isomorphic to A. Since N is unique by Lemma 1 (ii), it is the trivial line bundle. Hence the action is free.
This lemma allows us to make use of the flat Picard group Pic0 (M). Remember that line bundles are, according to our convention in Remark 1, line bundles with connection.
40
U. Schreiber, C. Schweigert, K. Waldorf
It is well understood [Bry93], that the Picard group Pic(M) of isomorphism classes of line bundles fits into the exact sequence 0
/ Pic(M)
/ H1 (M, U (1))
curv
/ 2 (M) .
(27)
In particular this means Pic0 (M) ∼ = H1 (M, U (1)). This cohomology group can be computed using the universal coefficient theorem 0
/ Ext(H0 (M), U (1))
/ H1 (M, U (1))
/ Hom(H1 (M), U (1))
/ 0 . (28)
If M is connected, the Ext-group is trivial and we obtain Pic0 (M) ∼ = Hom(π1 (M), U (1)).
(29)
An equivariant version of Lemma 2 applies to Jandl structures. We denote the group of isomorphism classes of flat K -equivariant line bundles by Pic0K (M) and call it the flat K -equivariant Picard group. In this equivalence relation isomorphisms are isomorphisms of equivariant line bundles with connection. Theorem 1. The set Jdl(G, k) of equivalence classes of Jandl structures on G with involution k is a torsor over the flat K -equivariant Picard group Pic0K (M). Proof. (a) We first describe the action of a flat line bundle N over M with equivariant structure ν on a Jandl structure J = (k, A, ϕ). According to diagram (19), π Z∗ ν : π Z∗ N → k˜ ∗ π Z∗ N is a K -equivariant structure on π Z∗ N . Now, by taking the tensor product of A and π Z∗ N as K -equivariant line bundles, we obtain an equivariant structure ϕ ⊗ π Z∗ ν on the line bundle of N .A. So we define N .J := (k, N .A, ϕ ⊗ π Z∗ ν). Since Z [2] π2
π1∗ π Z∗ ν
Z
π1
/Z πZ
πZ
π2∗ π Z∗ ν.
(30)
/M
(31)
commutes, we have = This shows that condition (20) for Jandl structures is satisfied for N .J . The arguments in the proof of Lemma 2 (a) apply here too and show that this defines an action on equivalence classes. (b) Let two equivalence classes of Jandl structures be represented by J1 and J2 . We already know from Lemma 1 (ii) that there is a flat line bundle N → M together with an isomorphism β : A1 → A2 ⊗ π Z∗ N , which is a morphism of stable isomorphism β : N .A1 ⇒ A2 . We have to show that there is an equivariant structure on N such that β is an isomorphism of equivariant line bundles. Remember that we defined N by a descent isomorphism α1−1 ⊗ α2∗ in (26). Because the equivariant structures on A1 and A2 are compatible with α1 and α2 respectively due to the property (20) of Jandl structures, the descent isomorphism is an isomorphism of equivariant line bundles. Thus N is an equivariant line bundle, and β is an isomorphism of equivariant line bundles.
Unoriented WZW Models and Holonomy of Bundle Gerbes
41
(c) Let J = (k, A, ϕ) represent a Jandl structure on G, and let N be a flat line bundle over M with equivariant structure ν, such that N .J and J are equivalent. It follows from Lemma 2 that N is the trivial line bundle. Furthermore, π Z∗ ν is the trivial equivariant structure on π Z∗ N , so that ν is the trivial equivariant structure on N.
For an action of a discrete group K on M, an equivariant version of the sequence (27) is derived in [Gom03], namely 0
/ H1 (M, U (1)) K
/ Pic K (M)
curv
/ 2 (M) K .
(32)
Here, H1K (M, U (1)) is the equivariant cohomology of M, i.e. the cohomology of the associated Borel space. In particular, we get for flat equivariant line bundles Pic0K (M) ∼ = H1K (M, U (1)).
(33)
2.4. Local data. Let G be a gerbe over M and V = {Vi }i∈I be a good open cover of M. Let MV be the disjoint union of all the Vi ’s. The p-fold fiber product of MV over M is just the disjoint union of all p-fold intersections of the Vi ’s. Recall from [CJM02] how to extract local data from G: A choice of local sections si : Vi → Y gives a fiber preserving map s : MV → Y by (x, i) → si (x). Pull back the line bundle L → Y [2] with its connection ∇ along s to a line bundle on the double intersections, and choose local sections σi j : Vi ∩ V j → s ∗ L. Pull back the isomorphism µ of the gerbe, too. Then define local data, namely smooth functions gi jk : Vi ∩ V j ∩ Vk → U (1), real-valued 1-forms Ai j ∈ 1 (Vi ∩ V j ) and 2-forms Bi ∈ 2 (Vi ) by the following relations: ∗ ∗ ∗ σi j ⊗ π23 σ jk = gi jk · π13 σik , (34) s ∗ µ π12 1 s ∗ ∇(σi j ) = Ai j ⊗ σi j , (35) i Bi = si∗ C. (36) ˇ These local data give elements g, A, B in the Cech-Deligne double complex for the cover V, and the cochain (g, A, B) satisfies the Deligne cocycle condition D (g, A, B) = (1, 0, 0) ,
(37)
or equivalently in components
A jk
−1 · gi jl · gi−1 g jkl · gikl jk = 1, − Aik + Ai j + dlog gi jk = 0,
(39)
−d Ai j + B j − Bi = 0.
(40)
dBi = H |Vi ,
(41)
(38)
Furthermore, it satisfies where the 3-form H is the curvature of the gerbe. The dual gerbe and the pullback gerbe f ∗ G along some map f : N → M can be conveniently expressed in local data as follows: by choosing the same si and the dual
42
U. Schreiber, C. Schweigert, K. Waldorf
sections σi∗j , one gets (g −1 , −A, −B) = −(g, A, B) as local data of G ∗ . Furthermore, if we induce a cover { f −1 Vi }i∈I of N , and choose the pullback sections f ∗ si and f˜∗ σi j , then we obtain ( f ∗ g, f ∗ A, f ∗ B) = f ∗ (g, A, B) as local data of f ∗ G. We next need to derive local data of trivializations and stable isomorphisms. So, let T = (T, τ ) be a trivialization of G. Since T is a line bundle over Y , we can pull it back with s : MV → Y to a line bundle over the open subsets, and choose local sections σi : Vi → s ∗ T . We also pull back the isomorphism τ to an isomorphism s ∗ τ : s ∗ L ⊗ π2∗ s ∗ T −→ π1∗ s ∗ T . Then we obtain smooth functions h i j : Vi ∩ V j → U (1) by s ∗ τ σi j ⊗ π2∗ σ j = h i j · π1∗ σi .
(42)
(43)
Let be the connection of T . It defines connection 1-forms Mi ∈ 1 (Vi ) by s ∗ (σi ) =
1 Mi ⊗ σi . i
(44)
ˇ The local data h and M are again elements in the Cech-Deligne double complex. Now the compatibility of τ and µ in Definition 2 is equivalent to −1 gi jk = h i j · h ik · h jk ,
and the condition, that the isomorphism τ respect connections, is equivalent to Ai j = −dlog h i j + M j − Mi .
(45)
(46)
Furthermore, the local 2-form ρ = Bi + dMi coincides with the 2-form ρ obtained from Definition 2. The last three properties of h and M are equivalent to the Deligne coboundary equation (47) (g, A, B) = (1, 0, ρ) + D (h, M) . Now consider a stable isomorphism A : G → G of gerbes over M. With respect to the good open cover {Vi }i∈I we may have chosen local sections si , σi j and si , σij to get local data (g, A, B) and (g , A , B ) of G and G respectively. We construct a map s˜ : MV −→ Y × M Y : (x, i) −→ (si (x), si (x)),
(48)
and pull the line bundle A → Y × M Y of the stable isomorphism together with its connection back to MV. We also pull back the isomorphism α and get an isomorphism s˜ ∗ α : s ∗ L ⊗ s ∗ L ∗ ⊗ π2∗ s˜ ∗ A −→ π1∗ s˜ ∗ A.
(49)
Then we choose local sections σi : Vi → s˜ ∗ A. We obtain local data in the form of smooth functions ti j : Vi ∩ V j → U (1) and connection 1-forms Wi ∈ 1 (Vi ) by the following relations: s˜ ∗ α σi j ⊗ σi∗j ⊗ π2∗ σ j∗ = ti j · π1∗ σi , (50) s˜ ∗ (σi ) =
1 Wi ⊗ σi . i
(51)
Unoriented WZW Models and Holonomy of Bundle Gerbes
43
Note that the functions ti j are not transition functions of some bundle but are defined by the isomorphism α. ˇ These local data t and W are elements in the Cech-Deligne double complex. The compatibility of α with the isomorphisms µ and µ of both gerbes as isomorphisms of hermitian line bundles with connection according to Definition 3 is equivalent to −1 gi jk · gi−1 jk = t jk · tik · ti j ,
Ai j −
Ai j
= −dlog(ti j ) + W j − Wi ,
(52) (53)
while the condition (12) on the curvature of A is equivalent to Bi − Bi = dWi .
(54)
The three last equations are in turn equivalent to the Deligne coboundary equation (55) (g, A, B) − g , A , B = D (t, W ) . This formalism of local data reproduces results on bundle gerbes and their stable isomorphisms, for example Lemma 1 (ii). Consider again two gerbes G and G , and now two stable isomorphisms A1 and A2 both from G to G . We may have extracted local data (t1 , W1 ) of A1 and (t2 , W2 ) of A2 such that Eq. (55) holds for both. It follows D(t · t −1 , W − W ) = (1, 0, 0),
(56)
which is the Deligne cocycle condition for a flat hermitian line bundle over M. This is the bundle N constructed in Lemma 1 (ii). We are now in a position to derive the local data of a Jandl structure J = (k, A, ϕ) on a gerbe G. Recall that k : M → M is the action of the non-trivial element of K = Z2 acting on M, in particular k 2 = id M . We simplify the situation by considering an open cover V = {Vi }i∈I of M, which is invariant under k, i.e. k(Vi ) = Vi , and which is still good enough to enable us to extract local data. The generalization to other covers is straightforward, but makes the notation somewhat more cumbersome. Recall further that A is a stable isomorphism from k ∗ G → G ∗ . Let (t, W ) be local data of A, obtained by pulling back the line bundle A → Z by s˜ : MV → Z from Eq. (48) and choosing local sections σi : Vi → s˜ ∗ A. As we derived for the local data of the dual gerbe and the pullback gerbe, Eq. (55) here appears as k ∗ (g, A, B) = −(g, A, B) + D(t, W ),
(57)
k ∗ Bi = −Bi + dWi , k ∗ Ai j = −Ai j − dlog(ti j ) + W j − Wi ,
(58) (59)
−1 k ∗ gi jk = gi−1 jk · t jk · tik · ti j .
(60)
or equivalently:
Now recall that a part of a Jandl structure is a K -equivariant structure ϕ : k ∗ A → A on A. By pullback with s˜ , we obtain s˜ ∗ ϕ : k ∗ s˜ ∗ A −→ s˜ ∗ A.
(61)
44
U. Schreiber, C. Schweigert, K. Waldorf
Now, because σi is a section of s˜ ∗ A, k ∗ σi = σi ◦k is a section of k ∗ s˜ ∗ A on the same patch Vi , since the latter is invariant under k. This allows us to extract a local U (1)-valued function ji : Vi → U (1), defined by s˜ ∗ ϕ(σi ) = ji · σi ◦ k.
(62)
The compatibility of ϕ with α in the sense of diagram (20) is equivalent to k ∗ (t, W ) = (t, W ) − D ( j) ,
(63)
k ∗ Wi = Wi − dlog( ji ),
(64)
or in turn equivalently ∗
k ti j = ti j ·
j −1 j
· ji .
(65)
By definition of an equivariant structure, the K = Z2 group law (17) is satisfied. In terms of local data, this is equivalent to k ∗ ji = ji−1 .
(66)
In summary, the Jandl structure J = (k, A, ϕ) gives rise to local data (t, W ) and j which satisfy the following three conditions: k ∗ (g, A, B) = −(g, A, B) + D(t, W ), k ∗ (t, W ) = (t, W ) − D ( j) , k ∗ ji = ji−1 .
(67) (68) (69)
Again, using local data, we can reproduce results on Jandl structures like Theorem 1. In detail, let J be a Jandl structure on G with local data (t, W ) and j. Let N be a flat K -equivariant hermitian line bundle over M with transition functions n i j : Vi ∩ V j → U (1) and local connection 1-forms Ni ∈ 1 (Vi ) with D(n, N ) = (1, 0, 0).
(70)
The equivariant structure on N determines smooth functions νi : Vi → U (1) with k ∗ (n, N ) = (n, N ) − D(ν)
(71)
(t , W ) := (t, W ) + (n, N ), j := j · ν
(72) (73)
and k ∗ ν = ν −1 . Then,
are local data of the Jandl structure N .J . Indeed, Eq. (67) is satisfied because of the Deligne cocycle condition (70). Compute k ∗ (t , W ) = k ∗ (t, W ) + k ∗ (n, N ) = (t, W ) − D ( j) + (n, N ) − D(ν) = (t , W ) − D( j ),
(74)
this is Eq. (68), and the last equation (69) for j is just a consequence from the conditions on j and ν.
Unoriented WZW Models and Holonomy of Bundle Gerbes
45
Let now J and J be two Jandl structures on G with local data (t, W ), j and respectively; (75) (n, N ) := (t, W ) − (t , W )
(t , W ), j
are the local data of the flat descent line bundle N , and using Eq. (67), we get its cocycle condition D(n, N ) = (1, 0, 0). (76) Now compute k ∗ (n, N ) = k ∗ (t, W ) − k ∗ (t , W ) = (t, W ) − D( j) − (t , W ) + D( j ) = (n, N ) − D(ν),
(77)
where we defined ν := j · j −1 . Hence, N and k ∗ N are isomorphic as hermitian line bundles with connection via an isomorphism represented by ν. By definition, we have k ∗ ν = ν −1 , this means, that ν is a K -equivariant structure. 3. Holonomy of Gerbes with Jandl Structure 3.1. Double coverings, fundamental domains and orientations. Let us first recall the setup that allows to define holonomy around closed oriented surfaces. This is a gerbe G over M and a closed oriented surface together with a smooth map φ : → M. Following [CJM02], we pull back G along φ to a gerbe over . For dimensional reasons, φ ∗ G is trivial. As explained in Sect. 2.1, a trivialization T determines a 2-form ρ ∈ 2 (), while another trivialization T determines a 2-form ρ = ρ + curv(N ). Since curv(N ) defines an integral class in cohomology, we have ρ = ρ mod 2π Z. (78)
So the integral is independent of the choice of a trivialization up to 2π Z, and admits therefore the following Definition 7. The holonomy of G around the closed oriented surface φ : → M is defined as holG (φ, ) := exp i ρ ∈ U (1). (79)
We state three important properties of this definition: • The dual gerbe has inverse holonomy, holG (φ, ) = holG ∗ (φ, )−1 .
(80)
• If A : G → G is a stable isomorphism, we have holG (φ, ) = holG (φ, ).
(81)
¯ we denote the same manifold with the opposite orientation; then we obtain • By ¯ −1 . holG (φ, ) = holG (φ, )
(82)
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U. Schreiber, C. Schweigert, K. Waldorf
Obviously, the orientation on is essential for this definition. In this section we will define the holonomy around unoriented or even unorientable surfaces. The most important property of this definition will be, that it reduces to Definition 7 if is orientable and an orientation is chosen. One of the main tools will be an orientation covering. Let be a smooth manifold (without orientation). ˆ → with an Definition 8. An orientation covering of is a double covering pr : ˆ such that the canonical involution σ : ˆ → ˆ is orientationoriented manifold , reversing. Recall three basic properties of orientation coverings (some of them can be found for example in [BG88]): • It is unique up to orientation-preserving diffeomorphisms of covering spaces. ˆ → ˆ preserves fibers and permutes the the sheets. • The canonical involution σ : ˆ is connected if and only if is not • Under the assumption that is connected, orientable. ˆ we will from now on refer to this unique orientation Due to the first point, by ˆ M)σ,k we denote the space cover. Let k : M → M be an involution on M. By C ∞ (, ˆ → M for which the diagram of smooth maps φˆ : ˆ
φˆ
σ
ˆ
/M k
φˆ
/M
(83)
commutes in the category of smooth manifolds (neglecting orientations). Let be orientable. Lemma 3. An orientation on defines a bijection ˆ M)σ,k −→ C ∞ (, M). C ∞ (,
(84)
ˆ consists of two disjoint copies of with opposite orienProof. Since is orientable, ˆ in the covering pr : ˆ → . tations. An orientation on is a global section or : → ˆ → M be a map. Define its image as φ := φˆ ◦ or. On the other hand, Now let φˆ : ˆ separately given a map φ : → M, we define the preimage φˆ on the two sheets of ˆ ˆ as φ|or() := φ and φ|σ or() := k ◦ φ respectively.
If is not orientable or no orientation of is chosen, we will make use of the following generalization of an orientation. ˆ is a submanifold F ⊂ ˆ possibly with Definition 9. A fundamental domain for in (piecewise smooth) boundary, satisfying the following two conditions as sets: (i) F ∩ σ (F) = ∂ F, ˆ (ii) F ∪ σ (F) = .
Unoriented WZW Models and Holonomy of Bundle Gerbes
47
Fig. 1. The construction of a fundamental domain by local orientations for a dual triangulation
This is a generalization of an orientation on in the sense that any orientation on ˆ which in turn defines a fundamental domain, namely gives a global section or : → ˆ F := or(), one of the two copies of in . We show the existence of such a fundamental domain for an arbitrary closed surface by an explicit construction, which we will also use in Sect. 3.3. Let U = {Ui }i∈I be ˆ One can think of such an open cover of , which admits local sections ori : Ui → . sections as local orientations. Choose a dual triangulation T of , subordinate to the cover U, together with a subordinating map i : T → I . So, for each face f ∈ T there is an index i( f ) with f ⊂ Ui( f ) , as well as for each edge e ∈ T and for each vertex v ∈ T . Because we have a dual triangulation, each vertex is trivalent. Consider a common edge e = f 1 ∩ f 2 of two faces f 1 and f 2 . We call the edge e orientation-preserving, if ori( f1 ) (e) = ori( f2 ) (e), (85) otherwise we call it orientation-reversing. So the set of edges splits in a set E of orientation-preserving, and a set E¯ of orientation-reversing edges. If v is a vertex, the number of orientation-reversing edges ending in v must be even, and since we started with a dual triangulation, it is either zero or two. Hence, the edges in E¯ form non-intersecting closed lines in , Define the subset
F := ori( f ) ( f ) (86) f ∈T
ˆ and endow it with the subspace topology. The boundary of F is exactly the union of of the preimages of orientation-reversing edges under the covering map,
∂F = pr −1 (e), (87) e∈ E¯
and hence a disjoint union of piecewise smooth circles. This shows that F is a subˆ with piecewise smooth boundary. It satisfies the two properties of a manifold of fundamental domain, and hence shows the existence of such a fundamental domain. ˆ The following observation will Let now F be any fundamental domain for in . be essential.
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U. Schreiber, C. Schweigert, K. Waldorf
Fig. 2. The orientation on ∂ F
Lemma 4. The quotient ∂ F := ∂ F/σ is a 1-dimensional oriented closed submanifold of . Proof. We act with σ on property (i) of the fundamental domain F: σ (∂ F) = σ (F ∩ σ (F)) = F ∩ σ (F) = ∂ F.
(88)
ˆ without fixed This shows that σ restricts to an involution on ∂ F. Since σ acts on points, the quotient ∂ F/σ is a submanifold of , and as ∂ F is closed, so is the quotient. ˆ induces an orientation on F. Because σ is orientation-reversing, The orientation of the orientation of σ (F) is opposite to the one induced on σ (F) as a submanifold of ˆ Hence, ∂ F and ∂(σ (F)) are equal as sets as well as oriented submanifolds. Thus σ . preserves the orientation on ∂ F.
3.2. Unoriented surface holonomy. The setup for the definition of holonomy around closed unoriented surfaces is • a gerbe G over a smooth manifold M with Jandl structure J = (k, A, ϕ) • a closed surface ˆ M)σ,k . • a map φˆ ∈ C ∞ (, ˆ along φ, ˆ The idea of the definition is the following: Pull back the gerbe G to 2 ˆ choose a trivialization and determine the 2-form ρˆ ∈ () as in Definition 7. Choose ˆ The integral a fundamental domain F for in . exp i ρˆ (89) F
is independent neither of the choice of the trivialization – which enters in ρˆ – nor of the choice of the fundamental domain F. The Jandl structure, however, allows to correct (89) by a boundary term in such a way that the holonomy becomes well-defined. We will now give a detailed definition of this boundary term, and then show that it gives rise to a well-defined holonomy. Recall that a gerbe G consists of the following data: a surjective submersion π : Y → M, a line bundle L → Y [2] , an isomorphism µ, and a 2-form C ∈ 2 (Y ). Recall that
Unoriented WZW Models and Holonomy of Bundle Gerbes
49
the pullback gerbe φˆ ∗ G consists of a pullback Yφ
φ˜
/Y
πφ
ˆ
π
φˆ
/M
,
(90)
the pullback line bundle φ˜ ∗ L, isomorphism φ˜ ∗ µ and 2-form φ˜ ∗ C. Accordingly, a trivialization T of φˆ ∗ G is a line bundle T → Yφ together with an isomorphism τ : φ˜ ∗ L ⊗ πφ ∗2 T −→ πφ ∗1 T
(91)
ˆ with of line bundles over Yφ[2] . It determines a 2-form ρˆ ∈ 2 () πφ∗ ρˆ = φ˜ ∗ C − curv(T ).
(92)
Due to the commutativity of diagram (83), φˆ ∗ J = (σ, φ˜ ∗ A, φ˜ ∗ ϕ) is a Jandl structure on φˆ ∗ G. Recall that part of the data is a line bundle φ˜ ∗ A → Z φ over the space Z φ := (Yφ )σ ׈ Yφ , and an isomorphism φ˜ ∗ α : p ∗ φ˜ ∗ L ⊗ p ∗ φ˜ ∗ L ∗ ⊗ πφ ∗2 φ˜ ∗ A −→ πφ ∗1 φ˜ ∗ A
(93)
of line bundles over Z φ[2] , where p and p are the projections in Zφ p
p
πφ
Yφ σ
/ Yφ
σ ◦πφ
/ ˆ
.
(94)
Further, the action of K by σ lifts to Z φ via the permutation map σ˜ , and φˆ ∗ J contains an K -equivariant structure φ˜ ∗ ϕ on φ˜ ∗ A. Combining the trivialization with the Jandl structure, we define a line bundle R := φ˜ ∗ A ⊗ p ∗ T ∗ ⊗ p ∗ T ∗
(95)
over Z φ . In addition, we define an isomorphism r := φ˜ ∗ α −1 ⊗ p ∗ τ ∗ ⊗ p ∗ τ ∗ : πφ ∗1 R −→ πφ ∗2 R
(96)
of line bundles over Z φ[2] . The compatibility of τ and α with the isomorphism µ of G guarantees the cocycle condition πφ ∗23r ◦ πφ ∗12 r = πφ ∗13r
(97)
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U. Schreiber, C. Schweigert, K. Waldorf
ˆ together with an over Z φ[3] , hence R determines a unique descent line bundle Rˆ → , ∗ ˆ isomorphism π Z R → R. We shall compute the curvature of these bundles, namely φ
(95) curv (R) = φ˜ ∗ curv (A) − p ∗ curv (T ) − p ∗ curv (T ) (12)
= p ∗ (φ˜ ∗ C − curv(T )) + p ∗ (φ˜ ∗ C − curv(T ))
(92)
=
p ∗ πφ∗ ρˆ
+
p ∗ πφ∗ ρˆ
(99) (100)
(94)
= π Z∗ φ (σ ∗ ρˆ + ρ). ˆ
Hence the curvature of Rˆ is
(98)
(101)
ˆ = σ ∗ ρˆ + ρ. curv( R) ˆ
(102) ˆ The next step is to define the σ -equivariant structure on R. Note that the canonical permutation of tensor products is an equivariant structure on p ∗ T ∗ ⊗ p ∗ T ∗ , since the permutation map σ˜ exchanges p and p . Together with the equivariant structure φ˜ ∗ ϕ on φ˜ ∗ A, the tensor product (95) is the tensor product of two equivariant line bundles. By definition of a Jandl structure ϕ is compatible with α, which means that the descent isomorphism r is an isomorphism of equivariant line bundles. Hence, also the descent ˆ is endowed with an equivariant structure. bundle Rˆ over It is a standard fact [Gom03, Bry00], that if K is discrete and acts freely, ˆ defines a unique line bundle Q on the quotient a K -equivariant line bundle Rˆ → ˆ /K = . ˆ Now choose a fundamental domain F of in . Definition 10. The holonomy of the gerbe G with Jandl structure J around the unoriented closed surface is defined as ˆ ) := exp i holG ,J (φ, ρˆ · hol Q (∂ F)−1 . (103) F
In this definition, the compensating term hol Q (∂ F) is the holonomy of the line bundle Q around the one-dimensional closed oriented submanifold ∂ F. Theorem 2. The holonomy defined in Definition 10 depends neither on the choice of the fundamental domain F nor on the choice of the trivialization T . Proof. Let F be another fundamental domain. We define the set B := Int(F) ∩ σ (Int(F )),
(104)
where Int denotes the interior. As the intersection of two open sets, B is open and hence ˆ It contains those parts of F, which are not contained in F (cf. a submanifold of . Fig. 3). Because we excluded the boundaries of F and F , we have B ∩ σ (B) = ∅,
(105)
ˆ with image B. such that there is a unique section or B : pr(B) → From Fig. 3, we have ˆ ρˆ = ρˆ − ρˆ + ρˆ = ρˆ − curv( R), F
F
B
σ (B)
F
B
(106)
Unoriented WZW Models and Holonomy of Bundle Gerbes
51
Fig. 3. The difference between two fundamental domains
since σ is orientation-reversing. By Stokes’ theorem, the exponential of the integral of the curvature of Rˆ over B is nothing but the holonomy of that line bundle around ∂ B. Thus, ˆ = hol ˆ (∂ B)−1 = hol Q (pr(∂ B))−1 . exp −i curv( R) R B
This is the term which is compensated by the boundary term, which is hol Q (∂ F )−1 = hol Q (∂ F)−1 · hol Q (pr(∂ B)). In summary
exp i
F
ρˆ · hol Q
(∂ F )−1
= exp i ρˆ · hol Q (∂ F)−1 ,
(107)
(108)
F
i.e. the holonomy is independent of the choice of the fundamental domain. Now let T = (τ , T ) be another trivialization of φˆ ∗ G. As discussed in Sect. 2.1, ˆ together with an isomorphism ν : π ∗ N ⊗ T → T , such there is a line bundle N → φ that the 2-forms ρˆ and ρˆ are related by ρˆ = ρˆ + curv(N ).
(109)
For the line bundle Rˆ defined in (95) this means R = R ⊗ π Z∗ σ ∗ N ⊗ π Z∗ N ,
(110)
and its descent line bundle Rˆ is Rˆ = Rˆ ⊗ σ ∗ N ⊗ N .
(111)
This is an equation of σ -equivariant line bundles, where Rˆ and Rˆ obtain equivariant structures from the Jandl structure as described before, and K := σ ∗ N ⊗ N carries the canonical σ -equivariant structure by permuting the order in the tensor product. Hence, Eq. (111) pushes into the quotient, namely Q = Q ⊗ K¯ .
(112)
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U. Schreiber, C. Schweigert, K. Waldorf
The holonomy of the descent bundle K¯ satisfies hol K¯ (∂ F) = hol N (∂ F) = holσ ∗ N (∂ F).
(113)
This finally means ρˆ · hol Q (∂ F)−1 exp i F (109) = exp i ρˆ + curv(N ) · hol Q⊗ K¯ (∂ F)−1 F (113) = exp i ρˆ · hol N (∂ F) · hol N (∂ F)−1 · hol Q (∂ F)−1 F ρˆ · hol Q (∂ F)−1 , = exp i
(114) (115) (116)
F
thus the holonomy is independent of the choice of the trivialization.
The following lemma asserts that the definition of holonomy is compatible with the definition of equivalence of Jandl structures. Lemma 5. The holonomy of a gerbe G with Jandl structure J only depends on the equivalence class of J . Proof. Let J = (k, A, ϕ) and J = (k, A , ϕ ) be two equivalent Jandl structures on G. It is shown in Theorem 1 that there is a unique flat equivariant line bundle N on M, such that N .A ∼ = A as equivariant line bundles. Because the action of Pic0K (M) is free, and A and A are isomorphic, N is the trivial equivariant line bundle. Remember the definition of the bundle R → Z in Eq. (95). For the two Jandl structures we get ˆ Since N is the R = R ⊗ π Z∗ N , and hence the descent bundles Rˆ = Rˆ ⊗ N over . trivial equivariant line bundle, Rˆ and Rˆ are isomorphic as equivariant line bundles, and thus define isomorphic line bundles Q and Q over . Isomorphic line bundles have the same holonomies, so Definition 10 is independent of the equivalence class of J .
An important condition for any notion of unoriented surface holonomy is its compatibility with ordinary surface holonomy for oriented surfaces: Theorem 3. If is orientable, for any choice of an orientation, the holonomy defined in Definition 10 reduces to the ordinary holonomy defined in Definition 7, ˆ ) = holG (φ, ), holG ,J (φ,
(117)
where φ and φˆ are related by the bijection of Lemma 3. In particular, if G admits a Jandl structure, the holonomy of G does not depend on the orientation. ˆ be a choice of an orientation on . Then F := or() is a Proof. Let or : → fundamental domain with empty boundary ∂ F = ∅. Choose a trivialization T of φˆ ∗ G to 2 ˆ obtain the 2-form ρˆ ∈ (). Then the left-hand side is equal to exp i or() ρ, ˆ because ∗ ∗ ˆ ˆ of Theorem 2. Because φ and φ correspond to each other, or φ G is the same gerbe as φ ∗ G, and or ∗ T is a trivialization with 2-form ρ = or ∗ ρ. ˆ Thus, the right-hand side is equal to exp i ρ and therefore equals the ordinary holonomy.
Unoriented WZW Models and Holonomy of Bundle Gerbes
53
3.3. Holonomy in local data. Let {Vi }i∈I be an open cover of M. To avoid notation, we assume that it is invariant under k and still good enough to admit all the local sections necessary to extract local data (g, A, B) of the gerbe G and (t, W, j) of the Jandl strucˆ →M ture J , as we explained in Sect. 2.4. We pull back the cover {Vi }i∈I along φˆ : −1 and obtain a cover {Uˆ i }i∈I with Uˆ i := φˆ (Vi ), together with pullback local data. Next, choose local data (h, M) of the trivialization T of the pullback gerbe and a 2-form ˆ so that ρˆ ∈ 2 (), φˆ ∗ g, φˆ ∗ A, φˆ ∗ B = 1, 0, ρˆ + D (h, M) (118) ˆ holds. Following the definition of the bundle R → Z in Eq. (95), the bundle Rˆ → has local data (119) (r, R) := φˆ ∗ (t, W ) − σ ∗ (h, M) − (h, M); the condition that Rˆ descends is equivalent to the Deligne cocycle condition D(r, R) = (1, 0),
(120)
which follows from Eqs. (118) and (67). ˆ M)σ,k , the pullback cover is invariant under σ . Because φˆ is an element of C ∞ (, Hence it projects to a cover of with open sets Ui := pr(Uˆ i ). Choose local sections ˆ and a dual triangulation T of , subordinate to the cover {Ui }i∈I , together ori : Ui → with a subordinating map i : T → I . As we did in Sect. 3.1 we choose the fundamental domain
F := ori( f ) ( f ), (121) f ∈T
where the f ’s are the faces of the triangulation. We now introduce three abbreviations. Let ωi2 ∈ 2 (Uˆ i ), ωi1j ∈ 1 (Uˆ i ∩ Uˆ j ) and ωi jk : Uˆ i ∩ Uˆ j ∩ Uˆ k → U (1) be some local data. First we denote the integral over a face f by ⎞ ⎛ 2 1 ⎠ I f (ω, ω1 , ω2 ) := exp ⎝i ωi( ωi( f) + i f )i(e) ·
v∈∂e
ori( f ) ( f )
e∈∂ f
ε( f,e,v)
ωi( f )i(e)i(v) (ori( f ) (v)),
ori( f ) (e)
(122)
where ε( f, e, v) ∈ {1, −1} indicates whether v is the end or the starting point of the edge e with respect to the orientation ori( f ) . Second, we denote the integral of some local data ωi1 ∈ 1 (Uˆ i ) and ωi j : Uˆ i ∩ Uˆ j → U (1) along an edge e of a face f by ε( f,e,v) 1 ωi(e) ωi(e)i(v) (ori( f ) (v)). (123) Ie, f (ω, ω1 ) := exp i · ori( f ) (e)
v∈∂e
Recall that the set of edges in T splits into the set E of orientation-preserving edges and the set E¯ of orientation-reversing edges. For an orientation-preserving edge e ∈ f 1 ∩ f 2 we have Ie, f1 (ω, ω1 ) = Ie, f2 (ω, ω1 )−1 , (124)
54
U. Schreiber, C. Schweigert, K. Waldorf
while for an orientation-reversing edge Ie, f1 (ω, ω1 ) = Ie, f2 (σ ∗ ω, σ ∗ ω1 )
(125)
holds. In the latter case, since e is orientation-reversing, we have either ori(e) (e) = ori( f1 ) (e) or ori(e) (e) = ori( f2 ) (e), so that we can write just Ie (ω, ω1 ), where for f the choice of the face with the coinciding orientation is understood. Third, if v is a vertex of an edge e, we define for some smooth function ωi : Uˆ i → U (1) ε( f,e,v) Iv,e, f (ω) := ωi(v) (ori( f ) (v)). (126) ¯ we call Now if v is the common vertex of two orientation-reversing edges e1 , e2 ∈ E, v orientation-preserving, if ori(e1 ) (v) = ori(e2 ) (v) and orientation-reversing otherwise. Let us denote the set of orientation-reversing vertices by V¯ . If v is such a vertex, we just write Iv (ω) instead of Iv,e, f (ω), where for e the choice of the edge as well as for f the face with the coinciding orientation is understood. Now the first factor in the holonomy formula (103) is ⎞ ⎛ (127) exp i ρˆ = exp ⎝i φˆ ∗ Bi( f ) + dMi( f ) ⎠ . F
f ∈T
ori( f ) ( f )
Following [CJM02], by using Stoke’s theorem, Eq. (118) and our abbreviations, we end up with exp i ρˆ = I f (φˆ ∗ g, φˆ ∗ A, φˆ ∗ B) · Ie, f (h, M)−1 . (128) F
f ∈T
f ∈T e∈∂ f
Here the second factor collects the boundary contributions that appear in the application of Stokes’ theorem. Let us assume for the moment that is oriented, and all sections ori coincide with the global orientation restricted to Ui . In this situation, we have only orientation preserving edges, and each of them appears twice in the second factor. Since the contributions are inverse by (124), the second factor vanishes. We obtain the local holonomy formula expressed only by the local data of the gerbe, as it appeared originally in [Alv85]. If is not oriented, the second factor still consists of two contributions for each ¯ which are orientation-reversing edge e ∈ E, Ie, f1 (h, M) · Ie, f2 (h, M) = Ie (h · σ ∗ h, M + σ ∗ M). Hence, in the general case, the second factor of (128) is Ie, f (h, M)−1 = Ie (h · σ ∗ h, M + σ ∗ M)−1 . f ∈T e∈∂ f
(129)
(130)
e∈ E¯
For the second factor of the holonomy formula (103) we have to compute the holonomy of the descent line bundle Q around ∂ F. Note that
ori(e) (e) (131) Eˆ¯ := e∈ E¯
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Fig. 4. Assignment of local data. The middle layer shows and the subordinated indices; the top and lower ˆ layer show parts of the two sheets of
is a fundamental domain of ∂ F in ∂ F with boundary consisting of the preimages of the orientation-reversing vertices v ∈ V¯ . Now the holonomy of Q around ∂ F is equal to ˆ¯ where at the boundary points the equivariant structure of the holonomy of Rˆ around E, ˆ R is used; this is Ie (r, R) · Iv (φˆ ∗ j). (132) hol Q (∂ F) = e∈ E¯
v∈V¯
Since e is orientation-reversing, Ie (r, R) = Ie (φˆ ∗ t · σ ∗ h −1 · h −1 , φˆ ∗ W − σ ∗ M − M) = Ie (φˆ ∗ t, φˆ ∗ W ) · Ie (h · σ ∗ h, M + σ ∗ M)−1 .
(133) (134)
The second factor of (134) cancels (130) so that all the local data coming from the trivialization drops out. It remains ˆ = holG ,J (, φ) I f (φˆ ∗ g, φˆ ∗ A, φˆ ∗ B) · Ie (φˆ ∗ t, φˆ ∗ W )−1 · Iv (φˆ ∗ j), (135) f ∈T
e∈ E¯
v∈V¯
depending only on the local data of the gerbe and of the Jandl structure. We visualize this formula in Fig. 4.
3.4. Examples. In the next two subsections we will apply the general formula (135) to some examples of surfaces , and we will simplify the situation considerably by starting with the pullback gerbe φˆ ∗ G which allows us to choose a triangulation adapted to .
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Fig. 5. Klein bottle
Fig. 6. Klein bottle with a dual triangulation
Fig. 7. A fundamental domain for the Klein bottle in its double covering
Fig. 8. The real projective plane
3.4.1. Klein bottle Think of the Klein bottle as a rectangle with the identifications of the boundary indicated by arrows as in Fig. 5. The identification by the vertical arrows is orientation-preserving, while the one by the horizontal arrows is orientation-reversing. A dual triangulation is shown in Fig. 6. Note that this is a triangulation with only one face. We choose a local section from that face into the double cover, and define the fundamental domain F as its image, as indicated in Fig. 7. Here we dropped the arrows, but the identifications are still to be understood, so that both points labelled by v are identified. This means that we can choose the local orientations of the edges such that the orientation-reversing edges form a closed line, as indicated by the thick line. So there is no orientation-reversing vertex, and the local datum j of the Jandl structure is not relevant for the holonomy around the Klein bottle. 3.4.2. The real projective plane We proceed in the same way as for the Klein bottle, so think of the real projective plane RP 2 as a two-gon with the identification on the boundary indicated by arrows in Fig. 8. The identification is orientation-reversing. An example of a dual triangulation is for example shown in Fig. 9. Now we choose local sections from these two faces into the double cover, for example as shown in Fig. 10. ˆ and v is an orientation-reversing Note that here the thick line is not a closed line in , vertex. According to the local holonomy formula (135) here the local datum j of the Jandl structure enters in the holonomy.
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Fig. 9. A dual triangulation of the real projective plane with two faces
Fig. 10. A fundamental domain of the real projective plane in its double covering
4. Gerbes and Jandl Structures in WZW models 4.1. Oriented and orientable WZW models. In the following we are concerned with Lie groups M, and we will use the following notation. The left multiplication with a group element h is denoted by lh : M → M, and the map which assigns to h the inverse group element h −1 is denoted by Inv : M → M. The left invariant Maurer-Cartan form is denoted by θ , and the right invariant form by θ¯ . We call a gerbe G over M left invariant, if it is stably isomorphic to the gerbe lh∗ G for each h ∈ M, and similar for right and bi-invariance. A WZW model is a theory of maps φ : → M from a worldsheet into a target space M, which is a Lie group together with additional structure, called the background fields. It assigns to each map φ an amplitude, i.e. a number in U (1), as the weight of this map in a path integral. To be more precise: Definition 11. An oriented WZW model consists of a compact connected Lie group M, which is equipped with an Ad-invariant metric g = −, − on its Lie algebra and a bi-invariant gerbe G. It assigns an amplitude Aortd g,G (φ, ) := exp (iSkin (φ)) · holG (, φ)
(136)
to a map φ : → M from a closed oriented conformal worldsheet to M, where the kinetic term is ∗ 1 φ θ ∧ φ ∗ θ . (137) Skin (φ) := 2 Note that the conformal structure and the orientation on determine the Hodge star. In [Wit84] Witten discussed this theory for M = SU (2), which is an example for a compact, simple, connected and simply-connected Lie group. In this particular situation, the holonomy can be written as the exponential of the Wess-Zumino term, (138) holG (, φ) = exp i φ˜ ∗ H , B
so that we can express the amplitudes as Aortd g,G (φ, ) = exp(iSWZW (φ))
(139)
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with the action functional
φ˜ ∗ H .
SWZW (φ) := Skin (φ) +
(140)
B
Here B is a 3-dimensional manifold with boundary , φ˜ is an extension of φ on B, and H is the curvature of the gerbe G. Witten observed two symmetries of the WZW model on the type of Lie groups he considered. The first is translation symmetry: the action functional SWZW (φ) is invariant under the translation φ → lh ◦ φ. The associated conserved Noether current is given by J (φ) := −(1 + )φ ∗ θ ,
(141)
which is a 1-form on with values in the Lie algebra of M. To obtain this conserved, non-abelian current, Witten derived a specific relative normalization of the kinetic and the Wess-Zumino term, which was also adapted here. The second symmetry Witten observed is the invariance of the action functional SWZW (φ) under what he called parity transformation: reverse the orientation on and replace φ by φ¯ := Inv ◦ φ. Accordingly, the conserved current J (φ) for and the one ¯ the manifold with reversed orientation, namely for , ¯ = (1 − )φ ∗ θ¯ , J¯(φ)
(142)
are often called equivalent. Note that here the right invariant Maurer-Cartan form appears. In that sense, the parity transformation exchanges left and right movers. We now want to generalize this equivalence to any compact connected Lie group M. It is a simple consequence of the properties of the holonomy of G, that the parity symmetry ortd ¯ Aortd (143) g,G (φ, ) = A g,G (Inv ◦ φ, ) holds, if the gerbes Inv∗ G and G ∗ are stably isomorphic. Note that this is a condition on the gerbe G. It should not come as a surprise that in Witten’s discussion there is no such condition: Lemma 6. If G is a bi-invariant gerbe over a compact, simple, connected and simply connected Lie group, then Inv∗ G and G ∗ are stably isomorphic. Proof. Because stably isomorphic gerbes have the same curvatures, the curvature H of the bi-invariant gerbe G is a bi-invariant 3-form. It is a theorem by Cartan that on compact, simple, connected, simply connected Lie groups M the space of bi-invariant 3-forms is the span of the canonical 3-form ν, which satisfies Inv∗ ν = −ν. Hence Inv∗ G and G ∗ have the same curvature. Because the set of stable isomorphism classes of gerbes of the same curvature form a torsor over H2 (M, U (1)) [GR02], which here is the trivial group, the gerbes Inv∗ G and G ∗ are stably isomorphic.
We now give an even more general definition of parity transformations of a target space M with metric g and gerbe G. Definition 12. A parity transformation map is an isometry k : M → M of the metric g of order two, such that k ∗ G and G ∗ are stably isomorphic. We denote the set of parity transformation maps by P(M, g, G).
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Consider an oriented WZW model with target space M, Ad-invariant metric g and bi-invariant gerbe G. If k ∈ P(M, g, G) is a parity transformation map, we obtain the parity symmetry ortd ¯ Aortd (144) g,G (φ, ) = A g,G (k ◦ φ, ). We already discussed that k = Inv is a parity transformation map in the sense of Definition 12, if the gerbes Inv∗ G and G ∗ are stably isomorphic. However, for oriented WZW models on compact connected Lie groups there are more such parity transformation maps. Because the gerbe G is supposed to be bi-invariant, we try an ansatz k := lh ◦ Inv for some group element h ∈ M. The condition k 2 = id M restricts h to be an element of the center Z (M). So, the set P(M, g, G) of parity transformation maps for a compact connected Lie group M and a bi-invariant gerbe G, such that G ∗ is stably isomorphic to Inv∗ G, contains at least {l z ◦ Inv | z ∈ Z (M)} ⊂ P(M, g, G).
(145)
In particular, P(M, g, G) is not empty in the situation we are interested in. As a preparation for the unoriented case, we now relate parity symmetry to the ˆ Start with an oriented WZW model on together with a parity orientation cover : transformation map k. Let φ : → M be a map. By Lemma 3, there is a unique map ˆ M)k,σ . Once we have the orientation cover ˆ and the map φ, ˆ we may forget φˆ ∈ C ∞ (, their origin, in particular the orientation on . Then we may give the following Definition 13. An orientable WZW model consists of a compact connected Lie group M, which is equipped with an Ad-invariant metric g on its Lie algebra, a bi-invariant gerbe G and a parity transformation map k ∈ P(M, g, G). To a closed orientable conˆ M)k,σ , the following amplitude Aorble (φ, ˆ ) formal surface and a map φˆ ∈ C ∞ (, g,G is assigned. Choose any orientation on , and obtain a map φ : → M by Lemma 3. Define ortd ˆ (146) Aorble g,G (φ, ) := A g,G (φ, ). The amplitude is well-defined: if we had chosen the other orientation, we would get the same amplitudes, due to the fact that k is a parity transformation map and satisfies Eq. (144).
4.2. Unoriented WZW models. In the last section we gave the definition of an orientable ˆ M)k,σ makes use WZW model. The derivation of the amplitude of a map φˆ ∈ C ∞ (, of the existence of an orientation on both in the kinetic term and in the holonomy term. In this section, we want to overcome this obstruction. ˆ for a Let us first discuss the kinetic term. We want to define the kinetic term Skin (φ) ∞ k,σ ˆ M) in such a way that if is orientable, it reduces to the kinetic map φˆ ∈ C (, term Skin (φ) of the corresponding map φ. Note that ˆ := L(φ)
1 ∗ φˆ θ ∧ φˆ ∗ θ 2
(147)
ˆ which satisfies is a 2-form on , ˆ = −L(φ). ˆ σ ∗ L(φ)
(148)
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ˆ defines a 2-density Lden (φ) ˆ [BT82, BG88] on . The This property tells us that L(φ) integral of a 2-density over a surface is defined without respect to the orientability of this surface, so we define ˆ := Skin (φ)
ˆ Lden (φ).
(149)
To make the integral (149) more explicit, choose a triangulation T of , and for each ˆ where U f is some open neighborhood of f face f ∈ T a local section or f : U f → , in . By definition of the integral of a density, ˆ = ˆ Skin (φ) L(φ). (150) f ∈T
or f ( f )
One immediately checks that this definition is independent of the choice of the local sections: if one chooses for one face f the other orientation, namely σ (or f ), the corresponding term in the sum (150), ∗ ˆ ˆ ˆ L(φ) = − σ L(φ) = L(φ), (151) σ (or f ( f ))
or f ( f )
or f ( f )
gives the same contribution. It is also independent of the choice of the triangulation. Furthermore, if is orientable, we can choose a triangulation with a single face f = ˆ = Skin (φ), which was precisely our requirement on Skin (φ). ˆ and get Skin (φ) We have already discussed in Sect. 3 how to define surface holonomies for an arbitrary ˆ M)k,σ : we have to choose a Jandl structure J closed surface with a map φˆ ∈ C ∞ (, ˆ ) is defined in Definition 10 in such a way that if is orientable, on G. Then holG ,J (φ, it coincides by Theorem 3 with holG (φ, ). Remember that a necessary condition on the existence of a Jandl structure J = (k, −, −) was that the gerbes k ∗ G and G ∗ are stably isomorphic. We already have encountered this condition for the orientable WZW model, so that it does not come as an additional restriction. This leads us to the following Definition 14. An unoriented WZW model consists of a compact connected Lie group M, which is equipped with an Ad-invariant metric g on its Lie-algebra and a bi-invariant gerbe G with Jandl structure J , whose action of Z2 on M is a parity transformation map k ∈ P(M, g, G). To a closed conformal surface and a map ˆ M)k,σ the amplitude φˆ ∈ C ∞ (, ˆ ) := exp iSkin (φ) ˆ · holG ,J (φ, ˆ ) Aunor ( φ, (152) g,G ,J is assigned. According to the definition of both factors, if is orientable, we have orble ˆ ˆ Aunor g,G ,J (φ, ) = A g,G (φ, ).
(153)
If is even oriented, by Eq. (146) we have ortd ˆ Aunor g,G ,J (φ, ) = A g,G (φ, ).
(154)
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4.3. Crosscaps and the trivial line bundle. In the following two sections we use the classification of Jandl structures to classify unoriented WZW models with a fixed gerbe G and a fixed parity transformation map k ∈ P(M, g, G). By Theorem 1, the set of equivalence classes of Jandl structures of G with the action of K = Z2 on M defined by k is a torsor over the flat K -equivariant Picard group Pic0K (M). In this section we discuss a special element of this group. On any manifold, there is the trivial line bundle L 1 := M × C with the trivial hermitian metric and the trivial connection, which is flat. It represents the unit element of the flat Picard group Pic0 (M). Recall the following facts concerning equivariant line bundles [Gom03]. There are two obstructions for a given line bundle to admit equivariant structures: the first depends on the bundle and the group action, namely that k∗ L ⊗ L ∗ ∼ = L 1,
(155)
which is still to be understood as an equation of hermitian line bundles with connection. 2 (K , U (1)). Now, The second obstruction is a class in the group cohomology group HGrp if both obstructions are absent, the possible equivariant structures are parameterized by 1 (K , U (1)) which is just the group of one-dimensional the group cohomology group HGrp characters of K . In our case K = Z2 we have 1 (K , U (1)) = Z2 , HGrp
(156)
2 (K , U (1)) HGrp
(157)
= 0,
so that the second obstruction vanishes, and every line bundle L, which satisfies the remaining obstruction (155) admits exactly two K -equivariant structures. In particular L 1 itself satisfies (155). We exhibit its two equivariant structures explicitly. Remember from Sect. 2.2, that we have to choose an isomorphism ϕ : k∗ L 1 → L1
(158)
of line bundles, such that ϕ ◦ k ∗ ϕ = id L 1 . So both choices are either ϕ1 = id M×C or ϕ−1 : (x, z) → (x, −z). We denote L 1 together with the equivariant structure ϕ1 by L 1K . It represents the unit element of Pic0K (M). We denote L 1 together with the equivariant K . Note that L K ⊗ L K = L K as equivariant line bundles. Hence structure ϕ−1 by L −1 −1 −1 1 it represents a non-trivial element of order two in Pic0K (M). The whole construction is completely independent of M, so Pic0K (M) always contains at least these two elements. As a consequence, if a gerbe G admits a Jandl structure K .J is another, inequivalent Jandl structure on G. We will now investigate J , then L −1 the difference between the corresponding unoriented WZW models. We work with local data, so let {Vi }i∈I be a good open cover of M. Choose all the sections that have been introduced in Sect. 2.4, and extract local data (t, W ), j of the Jandl structure J . We also explained how to extract a local datum νi : Vi → U (1) from an equivariant structure on a line bundle over M. The local datum of L 1K is the constant K is the constant global function global function ν1 = 1, and the local datum of L −1 ν1 = −1. According to the definition of the action of Pic0K (M) on Jdl(G, k), the local data of K L −1 .J are (t, W ) and − j. Now observe the occurrences of the local datum j in the local holonomy formula (135): it appears for each orientation-reversing vertex v ∈ V¯ .
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Following our example in Sect. 3.4.2, this happens in the presence of a crosscap. We conclude that the amplitudes of both unoriented WZW models with Jandl structures J K .J differ by a sign for each crosscap in . and L −1 4.4. Examples of target spaces. We would like to discuss three examples of target spaces, namely the Lie groups SU (2), S O(3), where the Ad-invariant metric on their Lie algebras is given by their Killing forms, and the two-dimensional torus T 2 = S 1 × S 1 with the euclidean scalar product. The gerbes are supposed to be bi-invariant. 4.4.1. The Lie group SU (2) Following our general discussion, the actions of Z2 on SU (2) we have to consider are given by k : g → g −1 and k : g → −g −1 , where −1 ∈ Z (SU (2)) is the non-trivial element in the center. The same maps were considered in [HSS02, Bru02, BCW01]. Fix a bi-invariant gerbe G over SU (2). Up to stable isomorphism, this is G = G0⊗n , where G0 is the basic gerbe over SU (2) [Mei02]. By Lemma 6, both k’s are parity transformation maps. The set Jdl(G, k) is a torsor over Pic0K (SU (2)) by Theorem 1. In order to compute the group of equivariant flat line bundles, we first observe Pic0 (M) = Hom(π1 (M), U (1)) = 0,
(159)
since SU (2) is simply connected. So up to isomorphism there is only one flat line bundle, the trivial one. Hence there are exactly two inequivalent Jandl structures for each map k and each bi-invariant gerbe G; this is in agreement with the results of [PSS95a, PSS95b]. 4.4.2. The Lie group S O(3) The center of S O(3) is trivial, so that we have only one action to consider, namely by k : g → g −1 . Let G be a bi-invariant gerbe over S O(3), such that k ∗ G and G ∗ are stably isomorphic. Such gerbes for example are constructed up to stable isomorphism in [GR03]. We have to investigate the group Pic0K (S O(3)) of flat equivariant line bundles. Again we first consider the group Pic0 (S O(3)) of flat line bundles and classify equivariant structures on them. By π1 (S O(3)) = Z2 we have Hom(π1 (S O(3)), U (1)) = Hom(Z2 , U (1)) = Z2 ,
(160)
so there are - up to isomorphism - two flat line bundles. We will give them explicitly: As S O(3) is the quotient of SU (2) by q : g → −g, the two flat line bundles over S O(3) K . correspond to the two equivariant flat line bundles over SU (2), namely L 1K and L −1 K ˜ Clearly, L 1 descends to the trivial flat line bundle L 1 → S O(3), which admits equivariant structures, more precisely, according to the discussion in Sect. 4.3, there are two K descends to a non-trivial flat line bundle L ˜ −1 → S O(3), and we have to of them. L −1 ask whether it admits equivariant structures, which is equivalent to the condition that (161) d L˜ −1 := k ∗ L˜ −1 ⊗ L˜ ∗−1 ∼ = L˜ 1 . Now d L˜ −1 is a flat line bundle, and hence either isomorphic to L˜ −1 or to L˜ 1 . Because Pic0 (S O(3)) is a group of order two, we have L˜ −1 ⊗ L˜ −1 = L˜ 1 . The assumption d L˜ −1 ∼ = L˜ −1 would therefore mean k ∗ L˜ −1 ∼ = L˜ 1 , which is a contradiction since L˜ 1 is ∗ ˜ the trivial bundle and k L −1 is not. Hence (161) is true, and L˜ ∗−1 admits two equivariant structures. All together, there are four equivariant flat line bundles over S O(3) and hence four Jandl structures on G; again, this is in agreement with [PSS95a, PSS95b].
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4.4.3. The two-dimensional Torus T 2 For dimensional reasons, all gerbes over T 2 are trivial and have curvature H = 0. This allows us to discuss an example with a parity transformation map k, which is not of the form k = l z ◦ Inv but simply the identity map k = id. This allows us to make contact with [BPS92]. Now let G be a bi-invariant gerbe over T 2 . The set Jdl(G, id) is a torsor over Pic0K (T 2 ) by Theorem 1, which is isomorphic to H1K (T 2 , U (1)) by Eq. (33). The Borel space associated to the trivial K -action is TK2 = EZ2 × T 2 . With EZ2 = RP ∞ we have H1K (T 2 , U (1)) = H 1 (TK2 , U (1)) = H1 (RP ∞ , U (1)) ⊕ H1 (T 2 , U (1)) = Z2 ⊕ U (1) ⊕ U (1) = Z2 ⊕ T 2 . We now assume that the gerbe G admits a Jandl structure J = (id, A, ϕ). In particular, A = (A, α) is a stable isomorphism from G to G ∗ . Recall that a gerbe G consists of the following data: a surjective submersion π : Y → M, a line bundle L → Y [2] , an isomorphism µ, and a 2-form C ∈ 2 (Y ). Recall further that here A is a line bundle over Z = Y [2] , and both projections p and p from Z to Y coincide with π2 , π1 : Y [2] → Y . The condition on the curvature of A in Definition 3 now reads curv(A) = π1∗ C + π2∗ C.
(162)
Furthermore, since for all gerbes the curving C satisfies −π2∗ C + π1∗ C = curv(L), we have 2π2∗ C = curv(A) − curv(L), (163) which is an equation of 2-forms on Y [2] . On the right-hand side we have a closed 2-form which defines an integral class in cohomology. Since π2 is a surjective submersion, also 2C defines a class in H2 (Y, Z). Because the gerbe G is trivial, we can choose a trivialization T and obtain the 2-form B ∈ 2 (M) as in Definition 7, which satisfies π ∗ B = C + curv(T ) and dB = H = 0. Usually one chooses T such that B is constant, then it is nothing but the Kalb-Ramond “B-Field”. Because π is also a surjective submersion it follows that 2B defines a class in H2 (M, Z). Thus we have derived the quantization condition that the B-Field has half integer valued periods. This condition was originally found in [BPS92] by an analysis of the bulk spectrum of right and left movers. References [Alv85]
Alvarez, O.: Topological quantization and cohomology. Commun. Math. Phys. 100, 279– 309 (1985) [BCW01] Bachas, C., Couchoud, N., Windey, P.: Orientifolds of the 3-sphere. JHEP 12, 003 (2001) [BG88] Berger, M., Gostiaux, B.: Differential Geometry: Manifolds, Curves, and Surfaces. Volume 115 of Graduate Texts in Mathematics, Berlin-Heidelberg-New York: Springer, 1988 [BPS92] Bianchi, M., Pradisi, G., Sagnotti, A.: Toroidal compactification and symmetry breaking in openstring theories. Nucl. Phys. B 376, 365–386 (1992) [Bru02] Brunner, I.: On orientifolds of WZW models and their relation to geometry. JHEP 01, 007 (2002) [Bry93] Brylinski, J.-L.: Loop spaces, Characteristic Classes and Geometric Quantization. Volume 107 of Progress in Mathematics, Basel: Birkhäuser, 1993 [Bry00] Brylinski, J.-L.: Gerbes on complex reductive Lie Groups. http://arxiv.org/list/math/0002158, 2000 [BT82] Bott, R., Tu, L.W.: Differential Forms in Algebraic Topology, Volume 82 of Graduate Texts in Mathematics, Berlin-Heidelberg-New York: Springer, 1982
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[CJM02] Carey, A.L., Johnson, S., Murray, M.K.: Holonomy on D-Branes. J. Geom. Phys. 52(2), 186– 216 (2002) [FHS+ 00] Fuchs, J., Huiszoon, L.R., Schellekens, A.N., Schweigert, C., Walcher, J.: Boundaries, crosscaps and simple currents. Phys. Lett. B 495(3–4), 427–434 (2000) [FPS94] Fioravanti, D., Pradisi, G., Sagnotti, A.: Sewing constraints and non-orientable open strings. Phys. Lett. B 321, 349–354 (1994) [FRS04] Fuchs, J., Runkel, I., Schweigert, C.: TFT construction of RCFT correlators ii: unoriented world sheets. Nucl. Phys. B 678(3), 511–637 (2004) [Gaw88] Gaw¸edzki, K.: Topological Actions in two-dimensional Quantum Field Theories. In: Nonperturbative Quantum Field Theory, London: Plenum Press, 1988 [Gom03] Gomi, K.: Equivariant smooth deligne cohomology. Osaka J. Math. 42(2), 309–337 (2003) [GR02] Gaw¸edzki, K., Reis, N.: Wzw branes and gerbes. Rev. Math. Phys. 14(12), 1281–1334 (2002) [GR03] Gaw¸edzki, K., Reis, N.: Basic gerbe over non-simply connected compact groups. J. Geom. Phys. 50(1–4), 28–55 (2003) [HS00] Huiszoon, L.R., Schellekens, A.N.: Crosscaps, boundaries and t-duality. Nucl. Phys. B 584(3), 705–718 (2000) [HSS99] Huiszoon, L.R., Schellekens, A.N., Sousa, N.: Klein bottles and simple currents. Phys. Lett. B 470(1), 95–102 (1999) [HSS02] Huiszoon, L.R., Schalm, K., Schellekens, A.N.: Geometry of wzw orientifolds. Nucl. Phys. B 624(1–2), 219–252 (2002) [Jan95] Jandl, E.: Lechts und rinks. Munich: Luchterhand Literaturverlag, 1995 [Mei02] Meinrenken, E.: The basic gerbe over a compact simple lie group. Enseign. Math., II. Sér. 49 (3–4), 307–333 (2002) [PSS95a] Pradisi, G., Sagnotti, A., Stanev, Y.S.: The open descendants of nondiagonal su(2) wzw models. Phys. Lett. B 356, 230–238 (1995) [PSS95b] Pradisi, G., Sagnotti, A., Stanev, Y.S.: Planar duality in su(2) wzw models. Phys. Lett. B 354, 279– 286 (1995) [SS03] Sousa, N., Schellekens, A.N.: Orientation matters for nimreps. Nucl. Phys. B 653(3), 339– 368 (2003) [Ste00] Stevenson, D.: The Geometry of Bundle Gerbes, PhD thesis, University of Adelaide, http:// arxiv.org/list/math.DG/0004117, 2000 [Wit84] Witten, E.: Nonabelian bosonization in two dimensions. Commun. Math. Phys. 92, 455–472 (1984) Communicated by M.R. Douglas
Commun. Math. Phys. 274, 65–80 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0275-6
Communications in
Mathematical Physics
Mass Under the Ricci Flow Xianzhe Dai1,2, , Li Ma3, 1 Department of Mathematics, University of California, Santa Barbara, CA 93106, USA.
E-mail:
[email protected] 2 Chern Institute of Mathematics, Tianjin, China 3 Department of Mathematical Science, Tsinghua University, Peking 100084, People’s Republic of China.
E-mail:
[email protected] Received: 25 January 2006 / Accepted: 25 January 2007 Published online: 6 June 2007 – © Springer-Verlag 2007
Abstract: In this paper, we study the change of the ADM mass of an ALE space along the Ricci flow. Thus we first show that the ALE property is preserved under the Ricci flow. Then, we show that the mass is invariant under the flow in dimension three (similar results hold in higher dimension with more assumptions). A consequence of this result is the following. Let (M, g) be an ALE manifold of dimension n = 3. If m(g) = 0, then the Ricci flow starting at g can not have Euclidean space as its (uniform) limit. 1. Introduction Ricci flow is an important geometric evolution equation in Riemannian Geometry. It was introduced by R. Hamilton in 1982 (see [8]) and used extensively by him to prove some outstanding results on 3-manifolds and 4-manifolds. Recently it has been used spectacularly by G. Perelman [14] to study the geometrization conjecture on 3-manifold. The flow has also been very useful in the study of pinching results and the metric smoothing process. As a natural geometric tool, Ricci flow should be used to study properties of physically meaningful objects such as mass, entropy, etc. In this paper we would like to understand the behavior of mass under the Ricci flow. In general relativity, isolated gravitational systems are modeled by spacetimes that asymptotically approach Minkowski spacetime at infinity. The spatial slices of such spacetimes are then the so-called asymptotically flat or asymptotically Euclidean (AE in short) manifolds. That is, Riemannian manifolds (M n , g) such that M = M0 ∪ M∞ (for simplicity we deal only with the case of one end; the case of multiple ends can be n dealt with similarly) with M0 compact and M∞ R − B R (0) for some R > 0 so that in the induced Euclidean coordinates the metric satisfies the asymptotic conditions gi j = δi j + O(r −τ ), ∂k gi j = O(r −τ −1 ), ∂k ∂l gi j = O(r −τ −2 ). Partially supported by NSF and NSFC.
(1.1)
The research is partially supported by the National Natural Science Foundation of China 10631020 and
SRFDP 20060003002.
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Here τ > 0 is the asymptotic order and r is the Euclidean distance to a base point. The total mass (the ADM mass) of the gravitational system can then be defined via a flux integral [1, 11], 1 m(g) = lim (∂i gi j − ∂ j gii ) ∗ d x j . (1.2) R→∞ 4ωn S R Here ωn denotes the volume of the (n − 1)-sphere and S R the Euclidean sphere with radius R centered at the base point. By [3], when the scalar curvature is integrable and τ > n−2 2 , the mass m(g) is well defined and independent of the coordinates at infinity, and therefore is a metric invariant. The famous Positive Mass Theorem, proved by Schoen-Yau [19] (later Witten gave an elegant spinor proof [20]), says that the mass m(g) ≥ 0 if the scalar curvature is nonnegative (and the manifold is spin). Moreover, m(g) = 0 if and only if M is the Euclidean space. There is also the notion of asymptotically locally Euclidean, or ALE, manifolds. For our purpose we will use the following characterization of the ALE property of a complete non-compact Riemannian manifold (M, g). Namely we use the curvature decay condition |Rm|(x) = O(d(x)−(2+τ ) )
(1.3)
for some τ > 0 as d(x) → ∞, and the volume growth condition V ol(Br , g) ≥ V r n
(1.4)
for some constant V > 0. Here Rm is the Riemannian curvature tensor of the metric g, and d(x) is the distance function from the base point. According to [3, Theorem (1.1)], (1.3), (1.4) imply that (M, g) is asymptotically lon cally Euclidean. Namely, M = M0 ∪M∞ with M0 compact and M∞ R − B R (0) / , n where ⊂ O(n) is a finite group acting freely on R − B R (0), so that the asymptotic conditions (1.1) hold. For an ALE manifold (M, g), the mass m(g) can be defined by (1.2) again, except that S R should be taken as the distance sphere or, equivalently, the quotient of the Euclidean sphere by . An ALE manifold (M, g) is actually AE if the asymptotic volume ratio µ = 1. Here µ = lim V (Br , g)/ n r n , r →∞
(1.5)
where n is the volume of the unit ball in the standard n-dimensional Euclidean space Rn . As we mentioned, we would like to investigate the behavior of the mass m(g) under the Ricci flow. Recall that the Ricci flow is a family of evolving metrics g(t) such that ∂ g(t) = −2Rc(g(t)), ∂t
(1.6)
on M with g(0) = g, where Rc(g(t)) is the Ricci tensor of the metric g(t). To make sure that the mass is well defined under this evolving flow g(t), we first need to show that the ALE property is preserved along the flow. Our main result is the following Theorem 1. Let g(t), 0 ≤ t ≤ T < +∞, be a Ricci flow on M with (M, g(0)) being an ALE (AE resp.) manifold of dimension n. Assume that g(t) has uniformly bounded sectional curvature. Then
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A) The ALE (AE resp.) property is preserved along the flow. B) The integrability condition R ∈ L 1 is also preserved along the flow provided (4.25) and (4.26) hold, which is the case if either R = O(r −q ), q > n and τ > n−4 2 or }. τ > max{n − 4, n−2 2 C) Under the conditions above, the mass m(t) = m(g(t)) is well defined, and m (t) = Ri d S i , (1.7) Sr →∞
where d S i = 4ω1 n ∗ d xi and Ri denotes the covariant derivative of R. Furthermore, n−2 if R = O(r −q ) for q > n and τ > n−2 2 , or τ > max{n − 3, 2 }, the mass is invariant under the Ricci flow. In particular, the mass is invariant in dimension 3 (τ > 21 ). D) Assume that the initial metric g(0) = g satisfies the additional decay condition ∂k ∂l ∂ p gi j = O(r −τ −3 ),
(1.8)
then the mass is invariant under the Ricci flow if n ≤ 6 (and τ > n−2 2 ) or if τ > n−2 max{n − 4, 2 }. That the ALE property is preserved will be obtained by using the maximum principles of Ecker-Huisken [5] and W.X.Shi [15], Cf. also P.Li and S.T.Yau [12]. To compute the changing rate of the mass of the evolving metric g(t), the key part is to get a decay estimate of the spatial derivative of the scalar curvature function R(x, t) of the metric g(t) at infinity, which is furnished by Shi’s local gradient estimate [15]. Theorem 2. Let g(t), 0 ≤ t < +∞, be a Ricci flow on M with uniformly bounded sectional curvature. Assume further that each g(t) is ALE and g(t) converges uniformly to an ALE metric g∞ as t goes to infinity. Then lim m(g(t)) = m(g∞ ).
t→∞
The notion of uniform convergence is introduced by using the space Mτ of [11], see Definition 13. A direct consequence of Theorems 1 and 2 is the following Corollary 3. Let (M, g) be an ALE manifold of dimension n and of asymptotic order τ > max{n − 3, n−2 2 }. If m(g) = 0, then the Ricci flow starting at g can not converge uniformly to a Euclidean space. Note that the Ricci flow preserves nonnegative scalar curvature [8]. This raises the prospect of proving the positive mass theorem of Schoen and Yau via Ricci flow, which we intend to investigate elsewhere. On the other hand, one can also look for applications of our results by combining with the positive mass theorem. For example, one sees that there are no complete non-compact Riemannian manifolds satisfying the hypothesis of the Main Theorem in [15] in dimension 3 by using the long time convergence result of [15], Theorems 1 and 2, and the positive mass theorem, see also [4, 7]. Let us explain why Theorem 1 comes so natural. We begin by recalling some basic facts about Ricci flow on complete non-compact Riemannian manifold (M, g) with bounded sectional curvature K 0 . Let g(t) be a family of the metrics evolving under the Ricci flow on M with initial data g, 0 ≤ t ≤ T < +∞. We shall write by ∇g(t) and Ri jkl (t) the Riemannian connection and Riemannian curvature tensor of g(t) respectively. Hamilton proved in [8] that
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the asymptotic volume ratio µ(t) = µ(g(t)) of (1.5) is a constant under the Ricci flow with bounded curvature and nonnegative Ricci curvature, where |Rm| → 0 at infinity on the complete non-compact Riemannian manifold. This result tells us that if µ = 1 at t = 0, then the Ricci flow can not have Euclidean space as its limit. It is well known that Ricci flow smoothes out the metric. W.X.Shi [16] showed that there exists a positive constant T > 0 such that for any integer α ≥ 0 and any 0 < t ≤ T , there exist constants c(n, K 0 ), c(n, K 0 , T ) and c(n, , K 0 , α, t) such that e−c(n,K 0 )t g ≤ g(t) ≤ ec(n,K 0 )t g, |∇g − ∇g(t) | ≤ c(n, K 0 )t, |Ri jkl | ≤ c(n, K 0 , T ), and α Ri jkl (t)| ≤ c(n, K 0 , α, t). |∇g(t)
All these facts will be implicitly used in this paper. It is clear that the volume growth condition (1.4) is preserved along the Ricci flow. In Sect. 18 in [8], Hamilton further proved that if the curvature Rm → 0 as s → +∞ for the initial metric, where s is the distance function to a fixed point of the metric g, the same is true for each g(t). So it is very natural for one to expect that if the curvature of the initial metric has decay at infinity, then the same is true for the evolving metric g(t). With this understanding, one would like to know the change of the mass, a natural invariant of the metric, under the flow. Finally, we refer the reader to [6] for related discussions under the (worldsheet) RG flow. Also, in a recent preprint [13], a different approach is used to study these questions that we address here. Throughout this paper we will denote by C, c various constants depending only on dimension. 2. Preliminaries In this section we briefly introduce some facts on Ricci flow. We shall use notations from [9]. Let M be a manifold of dimension n, g(t) a family of metrics evolving by Ricci flow (1.6). Then the curvature tensor evolves by the equation ∂ Rm = Rm + Rm ∗ Rm, ∂t
(2.9)
where Rm ∗ Rm denotes a quadratic expression of the curvature tensor. It follows then ∂ |Rm|2 = |Rm|2 − 2|∇ Rm|2 + Rm ∗ Rm ∗ Rm, ∂t which yields ∂ |Rm|2 ≤ |Rm|2 + C|Rm|3 . ∂t
(2.10)
The evolution equation for the scalar curvature is much simpler, and one has ∂ R = R + 2|Rc|2 . ∂t
(2.11)
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Now let X be a point in M. Let Y = {Ya }, 1 ≤ a ≤ n be a frame at X . In local coordinates X = {x i }, we have Ya = yai ∂/∂ x i . Let gab = g(Ya , Yb ), and let ∇ba = ybi ∂/∂ yai be the vector fields tangent to the fibers of the frame bundle. Write by Da the vector field on the frame bundle F(M) which is the lift of the vector Ya at Y ∈ F(M). Then we have j
Da = yai [∂ x i − ikj yb ∂/∂ ybk ], where ikj ’s are the Christoffel symbols of the connection. Under the Ricci flow, we can define the evolving orthonormal frame on M such that ∂t Fai = g i j R jk Fak , where (g i j ) is the inverse matrix of (gi j ). Then we set Dt = ∂t + Rab g bc ∇cb . Note that Dt gab = 0. This says that Dt is the unique tangent vector field to the orthonormal bundle. Choose a metric on F(M) such that Da , ∇cb are an orthonormal basis. Then we can see that Dt − ∂t is a space-like vector orthonormal to the orthonormal frame bundle. A useful fact for us is that for a smooth function u on M × (0, T ), we have (Dt − )Da u = Da (∂t − )u.
(2.12)
We now recall Hamilton’s argument [8]. Assume that a K -bounded smooth function u satisfies the heat equation u t = u, in M
(2.13)
with |Du|2 ≤ δ at t = 0. Then we have by (2.12), Dt Da u = Da u, and thus ∂t |Du|2 = |Du|2 − 2|D 2 u|2 . By the maximum principle of Shi [15] we have |Du|2 (x, t) ≤ δ.
(2.14)
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Let F = t|D 2 u|2 + |Du|2 . Then by a direct computation, we have ∂t F ≤ F − (1 − cK t)|D 2 u|2 . Hence, by the maximum principle of Shi [15] again we get for t ≤ 1/cK , F(x, t) ≤ δ 2 , which implies that √ |D 2 u| ≤ δ/ t. Note that | u|2 ≤ n|D 2 u|2 . Using the heat equation we obtain that for t ≤ 1/cK , √ √ |u t | ≤ nδ/ t. Therefore, we have
Take δ ≤
√
√ √ |u(x, t) − u(x, 0)| ≤ 2 nδ t. K 2 and assume that lim u(x, 0) = 0,
x→∞
in M
uniformly. Then we can conclude using an iteration argument that for any t ∈ [0, T ], lim u(x, t) = 0,
x→∞
in M
uniformly. In fact, Hamilton [8] showed that for any δ > 0 and for any bounded smooth function u 0 ∈ C 1 (M) with lim x→∞ u 0 (x) = 0, one can find a bounded smooth solution u(x, t) to the heat equation such that u 0 (x) ≤ u(x, 0) and |Du|2 (x, t) ≤ δ on M ×[0, T ]. 3. ALE is Preserved In this section we study the ALE property under the Ricci flow and show that it is preserved. The question can be reduced to the study of the non-negative solutions to the heat equation u t = u, in M
(3.15)
with initial data u(0) = u 0 , where = g(t) is the Laplacian operator of the family of metrics g(t). We assume that u 0 has a decay O(d(x)−σ ) for some σ > 0. Theorem 4. Let g(t) be the solution of the Ricci flow (1.6) over [0, T ]. Assume that g(t) has uniform curvature bound |Rm(g(t))| ≤ K . Then non-negative solutions to (3.15) have the same decay rate as the initial data u 0 . The main tools here are the maximum principles, especially the maximum principle of [5, Theorem 4.3]. For the reader’s convenience, we quote the result here (the superscript ‘t’ is put in here to emphasize the t-dependence from the metric g(t)).
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Theorem 5 (Ecker-Huisken). Suppose that the complete non-compact manifold M n with Riemannian metric g(t) satisfies the uniform volume growth condition volt (Brt ( p)) ≤ exp k(1 + r 2 )
(3.16)
for some point p ∈ M and a uniform constant k > 0 for all t ∈ [0, T ]. Let w be a function on M × [0, T ] which is smooth on M × (0, T ] and continuous on M × [0, T ]. Assume that w and g(t) satisfy i) the differential inequality ∂ w − t w ≤ a · ∇w + bw, ∂t
(3.17)
where the vector field a and the function b are uniformly bounded sup M×[0,T ] | a| ≤ α1 , sup M×[0,T ] | b| ≤ α2
(3.18)
for some constants α1 , α2 < ∞; ii) the initial data w( p, 0) ≤ 0
(3.19)
for all p ∈ M; iii) the growth condition 0
T
exp −α3 d t ( p, y)2 |∇w|2 (y)dµt dt < ∞
(3.20)
M
for some constant α3 > 0; iv) bounded variation condition in metrics
∂
sup M×[0,T ] g(t)
≤ α4 ∂t
(3.21)
for some constant α4 < ∞. Then, we have w≤0
(3.22)
on M × [0, T ]. In our situation, with the metric g(t) coming from the Ricci flow (1.6), the condition (3.21) is clearly satisfied by the uniform curvature bound. The uniform volume growth condition (3.16) also follows immediately from the volume comparison theorem via the curvature bound. The differential inequality will be coming from a modification of the solution of the heat equation (3.15). To see that the coefficients are uniformly bounded per (3.18) requires the following lemma.
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Lemma 6. Let g(t) be the solution of the Ricci flow (1.6) over [0, T ] with g(0) = g being an ALE. Assume that g(t) has uniform curvature bound |Rm(g(t))| ≤ K . Then, for sufficiently large R, there is a smooth positive function f on M such that f (x) = C0 >> 1, for x ∈ B R ; c d t (x) ≤ f (x) ≤ Cd t (x) for x ∈ M − B R . Moreover f ≥ C0 , |∇ t f | ≤ C1 , | t f | ≤ C2 . Proof. Since (M, g) is ALE, we have coordinates at infinity, which we denote by x. Let |x| be the Euclidean distance function. Choose a smooth increasing function φ(s) on R such that φ(s) = C0 = R − 1, if s ≤ R − 1; φ(s) = s, if s ≥ R, and |φ | ≤ 1,
|φ | ≤ 2.
We define our function f to be f (x) = φ(|x| ). Then clearly f ≥ C0 . Since the metrics g(t) are all equivalent and g(0) = g is ALE, we can use the Euclidean norm in estimating |∇ t f | and | t f |. Then |∇ t f | = |φ ||∇ t |x| | ≤ C2 . Similarly the estimate | t f | ≤ C2 follows from the coordinate expression of the Laplacian 1 ∂ t = √ det g(t) ∂ xi and the known estimate for g(t).
det g(t)g i j (t)
∂ ∂x j
We now suppress the superscript ‘t’ with the understanding that all covariant derivatives and the Laplacian are taken with respect to g(t). In our application of Theorem 5 to the proof of Theorem 4, we will let w = f σ u, where σ > 0. Then the growth condition (3.20) follows from the gradient bound (2.14), which implies that |∇w| ≤ C(T ) f σ +1 . We now turn to the proof of Theorem 4.
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Proof. For simplicity, we assume that (M, g0 ) is an ALE with one end. Let u 0 (x) = O(d(x)−σ ) as d(x) → ∞, where the distance function is with respect to a fixed point o in M. Choose a global smooth positive function f (x) on M as in Lemma 6 and let h(x) = f (x)σ . Set w(x, t) = h(x)u(x, t). Then, by a direct computation we have that wt = hu t , wi = h i u + hu i , and w = hu + 2∇h∇u + h u. Hence, (∂t − )w = Bw − 2∇ log h∇w, where B(x, t) = 2|∇h|h 2−h h . Note that the coefficients B and ∇ log h are uniformly bounded by Lemma 6. In particular, |B| ≤ b. Since 2
w(x, 0) = d(x)σ u 0 (x) ≤ D < +∞, and (∂t − )(w − Detb ) ≤ B(w − Detb ) − 2∇ log h∇(w − Detb ), we have by the maximum principle of Ecker-Huisken, Theorem 5, (see also the proofs of Theorem 18.2 [8] and Theorem 4.3 in [5], see also [15]) that there exists a uniform constant C1 > 0 such that max MT w ≤ C1 , where MT = M × [0, T ]. This implies the desired decay for u(x, t).
We are now in a position to prove the first part of Theorem 1. In fact, since |Rm(g0 )| ≤ C0 d(x)−σ for d(x) >> 1, where σ = 2 + τ , we can choose a bounded smooth function u 0 , which dominates the function |Rm(g0 )|2 such that it is C0 d(x)−2σ for d(x) >> 1 and has bounded gradient. Let u be the solution of the heat equation as above. Then under Ricci flow, we have from (2.10) and the uniform curvature bound, ∂t |Rm|2 ≤ |Rm|2 + C K |Rm|2 while ∂t (eC K t u) = (eC K t u) + C K eC K t u. Therefore, we have by the maximum principle of Shi [15] and Theorem 4 that |Rm|2 ≤ eC K t u ≤ eC K t d(x)−2σ , on M. Thus, under the Ricci flow, the ALE property is preserved.
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Remark 7. That the ALE property is preserved does not follow from [10, Remark 0.11], as was claimed there. The same is true for AE, as we have the following analog of a theorem of Hamilton [8]. Corollary 8. Let g(t), 0 ≤ t ≤ T , be a Ricci flow on M with uniformly bounded sectional curvature. Assume further that g(0) is ALE. Then the asymptotic volume ratio µ(t) = µ(g(t)) is constant along the Ricci flow. Proof. If (M, g) is ALE, it follows in [3] that M = M0 ∪ M∞ nfrom the characterization with M0 compact and M∞ R − B R (0) / , where ⊂ O(n) is a finite group n acting freely on R − B R (0), so that the asymptotic conditions (1.1) hold, where the asymptotic coordinates come from the projection of the Euclidean coordinates under Rn − B R (0) → Rn − B R (0) / . Therefore, (M, g) has the asymptotic volume ratio µ=
1 . ||
Since ALE is preserved along Ricci flow by Theorem 1, we deduce that µ(t) is a constant by the continuity of µ(t), as shown in the following lemma. Lemma 9. Let g(t), 0 ≤ t ≤ T , be a Ricci flow on M with uniformly bounded scalar curvature. Assume that the asymptotic volume ratio µ(t) = µ(g(t)) are well defined. Then µ(t) is a continuous function of t. Proof. The volume of a ball changes according to the formula d R(g(t))dvg(t) . V (Br , g(t)) = − dt Br
(3.23)
Hence |
d V (Br , g(t))| ≤ K V (Br , g(t)), dt
where K denotes the uniform bound on the scalar curvature. Therefore, e−K (t−t0 ) V (Br , g(t0 )) ≤ V (Br , g(t)) ≤ e K (t−t0 ) V (Br , g(t0 )), from which the continuity follows.
4. The Changing Rate of Mass For the mass to be well defined, one needs the integrability condition R ∈ L 1 in addition to the requirement that the asymptotic order τ > n−2 2 [2, 17]. We have seen that the ALE property is preserved along the Ricci flow. We now examine the integrability condition. One thing that in particular guarantees the integrability is the decay condition R = O(r −q ), We have
q > n.
(4.24)
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Theorem 10. Let g(t) be the solution of the Ricci flow (1.6) over [0, T ] with uniformly bounded curvature. Assume that g(0) is ALE with asymptotic order τ > 0 and its scalar curvature satisfy the decay condition R(0) = O(r −q ), q > 0. Then the scalar curvature of g(t) satisfies the decay condition
R(t) = O(r −q ), q = min{q, 2(τ + 2)}. Proof. This is similar to our proof of the ALE property. The scalar curvature satisfies the evolution equation ∂ R = R + 2|Rc|2 . ∂t By the assumption and our result on the ALE property, we have |Rc|2 = O(r −2(τ +2) ). Let f be the function in Lemma 6 and w = f q R. Then (∂t − )w ≤ Bw + 2q ∇ log f ∇w + C,
where f q |Rc|2 ≤ C. Hence, by the argument in the proof of ALE property, we have max w ≤ C1 . MT
In particular, when the order of decay of the initial scalar curvature q > n and the asymptotic order τ > n−4 2 , then the order of decay of the evolving scalar curvature also satisfies q > n. In general, without assuming the ALE conditions, we show that under the natural condition T |Rc|2 < +∞, (4.25) 0
the property R ∈
L1
M
is preserved under the Ricci flow if the decay condition R(t) = O(r −σ )
(4.26)
holds uniformly for some σ ≥ n − 2 and all t ∈ [0, T ]. We remark that both conditions (4.25) and (4.26) are always true for 0 < T < +∞ if the initial metric is ALE, provided the order σ = 2+τ ≥ n −2 (i.e. τ ≥ n −4) and 2σ > n in the curvature decay condition (1.3), as it follows from our result on ALE property and Theorem 10. Theorem 11. Let g(t) be the solution of the Ricci flow (1.6) over [0, T ] with uniformly bounded curvature. Assume that the conditions (4.25) and (4.26) hold. Then the property R ∈ L 1 is preserved under the Ricci flow. Proof. Recall that on M, R = Rt − 2|Rc|2 . Let p = 1 + with small > 0. Let φ be a non-negative cut-off function such that 0 ≤ φ ≤ 1 on M, φ = 1 on Br (o), φ = 0 outside B2r (o), and |∇φ|2 ≤ 4φ/r 2 .
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Then
dt φ 2 |R| p−1 (Rt − 2|Rc|2 ) sgnR M 0 t = dt φ 2 |R| p−1 ( R) sgnR M 0 t = −2 dt φ|R| p−1 < ∇φ, ∇ R > sgnR M 0 t − ( p − 1) dt φ 2 |R| p−2 |∇ R|2 M 0 t 2 2( p − 1) t ≤ dt |∇φ|2 |R| p − dt φ 2 |∇(|R| p/2 )|2 p−1 0 p2 M M 0 t 2 2( p − 1) t p ≤ dt φ|R| − dt φ 2 |∇(|R| p/2 )|2 ( p − 1)r 2 0 p2 M M 0 t 2 2C T r n−2− pσ →0 ≤ dt φ|R| p ≤ 2 ( p − 1)r 0 p−1 M t
as r → ∞. Here we have used the decay condition R = O(r −σ ) for some σ > n − 2. By direct computation, we have
t
φ 2 |R| p−1 (Rt − 2|Rc|2 ) sgnR M 0 t t = −2 dt φ 2 |R| p−1 |Rc|2 sgnR + dt φ 2 |R| p−1 Rt sgnR M M 0 0 t 1 t d 2 p−1 2 = −2 dt φ |R| |Rc| sgnR + dt φ 2 |R| p p dt M M 0 0 1 t 2 p + dt φ |R| R p 0 M t 1 1 dt φ 2 |R| p−1 |Rc|2 sgnR + φ 2 |R| p (t) − φ 2 |R| p (0) = −2 p M p M 0 M 1 t dt φ 2 |R| p R. + p 0 M dt
Hence, we have
1 dt φ 2 |R| p−1 |Rc|2 sgnR + φ 2 |R| p (t) p M M 0 1 t 1 φ 2 |R| p (0) + dt φ 2 |R| p R − p M p 0 M 2C T r n−2− pσ ≤ . p−1
−2
t
Mass Under the Ricci Flow
77
Sending r → +∞, we get t 1 1 dt |R| p−1 |Rc|2 sgnR + |R| p (t) − |R| p (0) −2 p M p M M 0 1 t dt |R| p R ≤ 0. + p 0 M Sending p → 1, we have that t t dt |Rc|2 sgnR + |R|(t) − |R|(0) + dt |R|R ≤ 0. −2 M
0
M
M
M
0
That is,
|R(t)| − M
|R(0)| ≤ M
T
2
dt
R +2 M
0
which implies that R ∈ L 1 for each t > 0.
T
|Rc|2 ,
dt 0
M
Using a similar argument, we can show that the property |Rm| ∈ L p ( p ≥ 1) is preserved under the Ricci flow with bounded curvature. We now look at the change of mass under the Ricci flow. Let S be a hypersurface in M. Without loss of generality, we can assume that M is oriented. We can take the local frame Fa such that F1 , ..., Fn−1 are tangent to S and Fn = ν is orthogonal to S at X . Let ωa be a local frame dual to Fa . Then the area form on S is d S = ω1 ∧ ... ∧ ωn−1 . Let f ba = ∂/∂t ωa (Fb ). Then we have f ba = −ωa (∂/∂t Fb ), which is a decay term of the same order as R jk = O(r −σ ), and ∂/∂t ωa = f ba ωb . Hence, ∂/∂t d S = O(r −σ ). It is also clear that ∂/∂t g(Fn , Fa ) = O(r −σ ). Hence ∂/∂t d S i = O(r −σ ).
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Therefore,
∂ ∂ gi j, j − g j j,i d S i m (t) = ∂t Sr →∞ ∂t ∂ gi j, j − g j j,i + d Si ∂t Sr →∞
and the second term in the above equation is zero. So, under the flow, we have m (t) = −2 (Ri j, j − R j j,i )d S i . Sr →∞
By using the contracted first Bianchi identity 2Ri j, j = Ri , we have that
m (t) = −2 =
Sr →∞
1 Ri − Ri d S i 2
Ri d S i .
Sr →∞
Note that, by using the local gradient estimate of Shi (see Theorem 13.1 in [8]), we have that Ri = O(|x|−σ ) for σ = 2 + τ . Hence, we have m (t) = 0 when n = 3. The same is true if τ > n − 3 for any dimension n ≥ 3. Similarly, if the initial metric satisfies the additional decay condition 1.8, then one has a better estimate Ri = O(|x|−σ ) for σ = 3 + τ . Therefore, m (t) = 0 provided n−2 τ + 3 > n − 1. This will be the case if n ≤ 6 and τ > n−2 2 , or τ > min{n − 4, 2 }. On the other hand, if R = O(r −q ), q > n, then Theorem 10 applies and we once again have m (t) = 0. Combining the results in Sect. 3, and 4, we have proved Theorem 1. Let us now make an observation. Using the divergence theorem we have 1 m (t) = Rdvg(t) . 4ωn Br →∞ Comparing this with the formula R = Rt − 2|Rc|2 , and d dt we obtain that m (t) =
1 4ωn
Rdvg(t) = Br
Rt dvg(t) − Br
R 2 dvg(t) , Br
(Rt − 2|Rc|2 )dvg(t) Br →∞
1 r →∞ 4ωn
= lim
d dt
Br
This yields the following result.
|Rc|2 dvg(t) . (4.27)
R 2 dvg(t) − 2
Rdvg(t) + Br
Br
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Proposition 12. Under the Ricci flow for ALE metrics with (4.25) we have d 2|Rc|2 − R 2 dvg(t) , Rdvg(t) = dt M M provided that n = 3 or τ > n − 3. 5. Uniform Convergence We now turn our attention to Theorem 2. First, we introduce the notion of uniform conk,q vergence in our context. To this end, we now discuss the weighted Sobolev spaces Wτ and a certain related space Mτ on an ALE space M, see [2, 11]. q For q ≥ 1 and τ ∈ R, the weighted Lebesque space L τ (M) consists of locally integrable functions u on M for which the norm 1/q −τ q −n uq,τ = |r u| r dvol M k,q
is finite. For nonnegative integer k, the weighted Sobolev space Wτ (M) is the set of u q for which |∇ i u| ∈ L τ (M) for 0 ≤ i ≤ k, with the norm uq,k,τ =
k
∇ i uq,τ .
i=0 ∞ metrics g on M such that, in For τ > n−2 2 , we define Mτ to be the set of all C some asymptotic coordinates,
gi j − δi j ∈ Wτ1,q (M),
R(g) ∈ L 1 (M).
(5.28)
We equip Mτ with the norm gMτ = gi j − δi j W 1,q + R(g) L 1 . τ
(5.29)
Now, consider the Ricci flow (M, g(t)) which we assume to exist for all time 0 ≤ t < ∞. Furthermore, suppose that each (M, g(t)) is ALE of asymptotic order τ > n−2 2 and that the scalar curvature R(t) of g(t) is integrable on M (so that the ADM mass is well defined), i.e. g(t) ∈ Mτ . Definition 13. We say that g(t) converges uniformly to g∞ ∈ Mτ as t → ∞ if g(t) converges to g∞ in Mτ . i.e., lim g(t) − g∞ Mτ = 0.
t→∞
We now prove Theorem 2. Proof. This follows from the argument of [11, Lemma 9.4]. The key here is the following identity, first observed in [17, 2]. In terms of the asymptotic coordinate, R(g) = ∂ j (∂i gi j − ∂ j gii ) + O(r −2τ −2 ), where the O(r −2τ −2 ) is controlled by the Wτ -norm of g. 1,q
(5.30)
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Let η be a cut off function which is identically 1 for large r and 0 for r ≤ 1 and inside. Then, by the Divergence Theorem 1 η(∂i gi j − ∂ j gii ) ∗ d x j m(g) = lim R→∞ 4ωn S R −η∇ ∗ β + β, ∇η dvol, = M
where β = (∂i gi j − ∂ j gii )∂ j is the mass density vector. Theorem 2 now follows from the formula above and (5.30). Acknowledgements. Part of the work is done while both authors were visiting the Nankai Institute of Mathematics, Tianjin, China. We would like to thank Nankai Institute and its Director Weiping Zhang for the hospitality. The first author thanks Gary Horowitz and Rick Ye for interesting discussions and Gary for bringing the reference [6] to his attention.
References 1. Arnowitt, S., Deser, S., Misner, C.: Coordinate invariance and energy expressions in general relativity. Phys. Rev. 122, 997–1006 (1961) 2. Bartnik, R.: The mass of an asymptotically flat manifold. Comm. Pure Appl. Math. 39, 661–693 (1986) 3. Bando, S., Kasue, A., Nakajima, H.: On a construction of coordiantes at infinity on manifold with fast curvature decay and maximal volume growth. Invent. Math. 97, 313–349 (1989) 4. Chen, B., Zhu, X.: Complete Riemannian manifolds with pointwise pinched curvature. Invent. Math. 140(2), 423–452 (2000) 5. Ecker, K., Huisken, G.: Interior estimates for hypersurfaces moving by mean curvature. Invent. Math. 105, 547–569 (1991) 6. Gutperle, M., Headrick, M., Minwalla, S., Schomerus, V.: Spacetime Energy Decreases under Worldsheet RG Flow. JHEP 0301, 073 (2003) 7. Greene, R., Petersen, P., Zhu, S.: Riemannian manifolds of faster-than-quadratic curvature decay. Internat. Math. Res. Notices 9, 363–377 (1994) 8. Hamilton, R.: The formation of Singularities in the Ricci flow. Surveys in Diff. Geom. 2, 7–136 (1995) 9. Hamilton, R.: The Harnack estimate for the Ricci flow. J. Diff. Geom. 37, 225–243 (1993) 10. Kapovitch, V.: Curvature bounds via Ricci smoothing. Illinois J. Math. 49(1), 259–263 (2005) 11. Lee, J.M., Parker, T.: The Yamabe problem. Bull. AMS 17, 37–91 (1987) 12. Li, P., Yau, S.T.: On the parabolic kernel of the Schrodinger operators. Acta. Math. 158, 153–201 (1986) 13. Oliynyk, T., Woolgar, E.: Asymptotically Flat Ricci Flows. http://arxiv.org/list/math.DG/0607438, 2006 14. Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. http:// arxiv.org/list/math.DG/math.DG/0211159, 2002 15. Shi, W.X.: Ricci deformation of the metric on complete noncompact Riemannian manifolds. J. Diff. Geom. 30, 303–394 (1989) 16. Shi, W.X.: Deforming the metric on complete Riemannian manifolds. J. Diff. Geom. 30, 223–301 (1989) 17. Schoen, R.: Variational theory for the total scalar curvature functional for Riemannian metrics and related topics. In: Topics in Calculus of Variations, LNM 1365, Berlin-Heidelberg-New York: Springer-Verlag, 1989 18. Schoen, R., Yau, S.T.: Lectures on Differential Geometry. Cambridge, MA: International Press (1994) 19. Schoen, R., Yau, S.T.: On the proof the positive mass conjecture in general relativity. Commun. Math. Phys. 65, 45–76 (1979) 20. Witten, E.: A new proof of the positive energy theorem. Commun. Math. Phys. 80, 381–402 (1981) Communicated by G.W. Gibbons
Commun. Math. Phys. 274, 81–122 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0274-7
Communications in
Mathematical Physics
Translation-Invariance of Two-Dimensional Gibbsian Point Processes Thomas Richthammer Mathematisches Institut der Universität München, Theresienstraße 39, D-80333 München, Germany. E-mail:
[email protected] Received: 7 March 2006 / Accepted: 11 January 2007 Published online: 6 June 2007 – © Springer-Verlag 2007
Abstract The conservation of translation as a symmetry in two-dimensional systems with interaction is a classical subject of statistical mechanics. Here we establish such a result for Gibbsian particle systems with two-body interaction, where the interesting cases of singular, hard-core and discontinuous interaction are included. We start with the special case of pure hard core repulsion in order to show how to treat hard cores in general. 1. Introduction Gibbsian processes were introduced by R. L. Dobrushin (see [D1] and [D2]), O. E. Lanford and D. Ruelle (see [LR]) as a model for equilibrium states in statistical physics. (For general results on Gibbs measures on a d-dimensional lattice we refer to the books of H.-O. Georgii [G], B. Simon [Sim] and Y. G. Sinai [Sin], which cover a wide range of phenomena.) The first results concerned existence and uniqueness of Gibbs measures and the structure of the set of Gibbs measures related to a given potential. The question of uniqueness is of special importance, as the non-uniqueness of Gibbs measures can be interpreted as a certain type of phase transition occurring within the particle system. A phase transition occurs whenever a symmetry of the potential is broken, so it is natural to ask, under which conditions symmetries are broken or conserved. The answer to this question depends on the type of the symmetry (discrete or continuous), the number of spatial dimensions and smoothness and decay conditions on the potential (see [G], Chapters 6.2, 8, 9 and 20). It turns out that the case of continuous symmetries in two dimensions is especially interesting. The first progress in this case was achieved by M. D. Mermin and H. Wagner, who showed for special two-dimensional lattice models that continuous internal symmetries are conserved ([MW] and [M]). In [DS] R. L. Dobrushin and S. B. Shlosman established conservation of symmetries for more general potentials which satisfy smoothness and decay conditions, and C.-E. Pfister improved this in [P]. Later also continuum systems were considered: S. Shlosman obtained results
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for continuous internal symmetries ([Sh]), while J. Fröhlich and C.-E. Pfister treated the case of translation of point particles ([FP1] and [FP2]). All these results rely on the smoothness of the interaction, but in [ISV] D. Ioffe, S. Shlosman and Y. Velenik were able to relax this condition. Considering a lattice model they showed that continuous internal symmetries are conserved, whenever the interaction can be decomposed into a smooth part and a part which is small with respect to L 1 -norm, using a perturbation expansion and percolation theory. We generalised this to a point particle setting ([Ri1]). Here we will investigate the conservation of translational symmetry for non-smooth, singular or hard-core potentials in a point particle setting. While we treat non-smoothness by generalising ideas used in [Ri1], we will give an approach to singular potentials which is different from the one given in [FP1] and [FP2]. The advantage of our approach is that the integrability condition (2.13) of [FP2] is simplified and relaxed and the case of hard-core potentials can easily be included. Thus we are able to show the conservation of translational symmetry for the pure hard core model, for example. In Sect. 2 we will first confine ourselves to this special case of pure hard core repulsion. The corresponding result (Theorem 1) is of interest on its own and its proof shows how to deal with hard cores in the general case. For this general case we then define a suitable class of potentials (Definition 1), give some sufficient conditions for potentials to belong to that class (Lemmas 1 and 2) and state the general result obtained (Theorem 2). The precise setting is then given in Sect. 3. The proofs of the lemmas from Sects. 2 and 3 are relegated to Sect. 4. In Sects. 5 and 7 we will give the proofs of Theorems 1 and 2 respectively. The proofs of the corresponding lemmas are relegated to Sects. 6 and 8 respectively. In the proof of the general case arguments of the special case have to be modified and refined by new concepts and ideas at several instances. So for sake of clarity we will repeat arguments from the proof of Theorem 1 in the proof of Theorem 2 whenever necessary.
2. Result We consider particles in the plane R2 without internal degrees of freedom. The chemical potential − log z of the system is given via an activity parameter z > 0. The interaction between particles is modelled by a translation-invariant pair potential U , i.e. a measurable function U : R2 → R := R ∪ {∞}, which is assumed to be symmetric in that U (x) = U (−x) for all x ∈ R2 . The potential of two particles x1 , x2 ∈ R2 is then given by U (x1 − x2 ). We first consider the particular case of pure hard core repulsion, where the size and the shape of the hard core are given by a norm |.|h on R2 . The corresponding pure hard-core potential Uhc is defined by Uhc (x) :=
∞ 0
for for
|x|h ≤ 1 |x|h > 1.
Theorem 1. Let z > 0 be an activity parameter, |.|h be a norm on R2 and Uhc be the corresponding pure hard-core potential. Then every Gibbs measure corresponding to Uhc and z is translation-invariant.
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83
The proof of Theorem 1, which is given in Sect. 5, will show how to deal with hard cores in the general case presented below. In order to describe a class of potentials for which translational symmetry is conserved we will define important properties of sets, functions and potentials. A set A ⊂ R2 is called symmetric if A = −A. We call U a standard potential if U is a measurable, symmetric pair potential and its hard core K U := {U = +∞} is bounded. Usually the hard core will be empty, {0} or a disc, but in our setup we are able to treat fairly general hard cores. For a given function ψ : R2 → R+ we say that a standard potential U has ψ-dominated derivatives on the set A if ∂i2 U (x + tei ) ≤ ψ(x)
for all x ∈ A, t ∈ [−1, 1] s.t. x + tei ∈ A
for i = 1, 2. Here e1 = (1, 0), e2 = (0, 1) and ∂i is the partial derivative in direction ei . The above definition is meant to imply that these derivatives exist. In the context of ψ-domination we will use the notion of a decay function, which is defined to satisfy ψ < ∞
and
ψ(x)|x|2 d x < ∞.
This definition of course does not depend on the choice of norm |.|, but for sake of definiteness let |.| be the maximum norm on R2 . If U is a potential, z is an activity parameter and X0 is a set of boundary conditions, we say that the triple (U, z, X0 ) is admissible if all conditional Gibbs distributions corresponding to U and z with boundary condition taken from X0 are well defined, see Definition 2 in Sect. 3.3. Important examples are the cases of superstable potentials with tempered boundary configurations and nonnegative potentials with arbitrary boundary conditions, see Sect. 3.4. For admissible (U, z, X0 ) the set of Gibbs measures GX0 (U, z) corresponding to U and z with full weight on configurations in X0 is a well defined object. Finally we need bounded correlations: For admissible (U, z, X0 ) we call ξ ∈ R a Ruelle bound if the correlation function of every Gibbs measure µ ∈ GX0 (U, z) is bounded by powers of ξ in the sense of (3.3) in Sect. 3.3. Definition 1. Let (U, z, X0 ) be an admissible triple with Ruelle bound ξ , where U : R2 → R is a translation-invariant standard potential. We say that U is smoothly approximable if there is a decomposition of U into a smooth part U¯ and a small part u in the following sense: We have a symmetric, compact set K ⊃ K U , a decay function ψ and measurable symmetric functions U¯ , u : K c → R such that U = U¯ − u and u ≥ 0 on K c , U¯ has ψ-dominated derivatives on K c , 2 2 U u(x)|x| ˜ d x < ∞ and λ (K \ K ) + Kc
where u˜ := 1 − e−u ≤ u ∧ 1.
Kc
u(x) ˜ dx
0 and k ≥ 0 such that |U (x)| ≤ k/|x|4+ for large |x|. Our main result is now the following: Theorem 2. Let (U, z, X0 ) be admissible with Ruelle bound, where U : R2 → R is a translation-invariant standard potential. If U is smoothly approximable then every Gibbs measure µ ∈ GX0 (U, z) is translation-invariant. For a generalisation of the above result to the case of particles with inner degrees of freedom, i.e. Gibbsian systems of marked particles, we refer to [Ri2]. 3. Setting 3.1. State space. We will use the notations N := {0, 1, . . .}, R+ := [0, ∞[, R := R ∪ {+∞}, r1 ∨ r2 := max{r1 , r2 } and r1 ∧ r2 := min{r1 , r2 }
for r1 , r2 ∈ R.
On R2 we consider the maximum norm |.| and the Euclidean norm |.|2 . For > 0 the -enlargement of a set A ⊂ R2 is defined by A := {x + x : x ∈ A, |x |2 < }.
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85
The state space of a particle is the plane R2 . The Borel-σ -algebra B2 on R2 is induced by any norm on R2 . Let B2b be the set of all bounded Borel sets and λ2 be the Lebesgue measure on (R2 , B2 ). Integration with respect to this measure will be abbreviated by d x := dλ2 (x). Often we consider the centred squares r := [−r, r [2 ⊂ R2
(r ∈ R+ ).
We also want to consider bonds between particles. For a set X we denote the set of all bonds in X by E(X ) := {A ⊂ X : # A = 2}. A bond will be denoted by x x := {x, x }, where x, x ∈ X such that x = x . For a bond set B ⊂ E(X ) (X, B) is an (undirected) graph, and we set X,B
x ←→ x
:⇔
∃ m ∈ N, x0 , . . . , xm ∈ X : x = x0 , x = xm , xi−1 xi ∈ B for all 1 ≤ i ≤ m.
This connectedness relation is an equivalence relation on X whose equivalence classes are called the B-clusters of X . Let X,B C X,B (x) := {x ∈ X : x ←→ x } and C X,B () := C X,B (x ) x ∈X ∩
denote the B-clusters of a point x and a set respectively. Primarily we are interested in the case X = R2 . On the corresponding bond set E(R2 ) we consider the σ -algebra F E(R2 ) := {{x1 x2 ∈ E(R2 ) : (x1 , x2 ) ∈ M} : M ∈ (B2 )2 }. Every symmetric function u on R2 can be considered a function on E(R2 ) via u(x x ) := u(x − x ). 3.2. Configuration space. A set of particles X ⊂ R2 is called finite locally finite
if # X < ∞, and if #(X ∩ ) < ∞ for all ∈ B2b ,
where # denotes the cardinality of a set. The configuration space X of particles is defined as the set of all locally finite subsets of R2 , and its elements are called configurations of particles. For X, X¯ ∈ X let X X¯ := X ∪ X¯ . For X ∈ X and ∈ B2 let X := X ∩ (restriction of X to ), X := {X ∈ X : X ⊂ } (set of all configurations in ) and N (X ) := # X (number of particles of X in ). The counting variables (N )∈B2 generate a σ -algebra on X, which will be denoted by FX. For ∈ B2 let FX , be the σ -algebra on X obtained by restricting FX to X , −1 FX, be the σ -algebra on X obtained from FX and let FX, := e , by the restriction
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mapping e : X → X , X → X . The tail σ -algebra or σ -algebra of the events far from the origin is defined by FX,cn . FX,∞ := n≥1
Let ν be the distribution of the Poisson point process on (X, FX), i.e. 1 −λ2 () d x1 . . . d xk f ({xi : 1 ≤ i ≤ k}), ν(d X ) f (X ) = e k! k k≥0
for any FX, -measurable function f : X → R+ , where ∈ B2b . For ∈ B2b and X¯ ∈ X, let ν (.| X¯ ) be the distribution of the Poisson point process in with boundary condition X¯ , i.e. ¯ ν(d X ) f (X X¯ c ) ν (d X | X ) f (X ) = for any FX-measurable function f : X → R+ . It is easy to see that ν is a stochastic kernel from (X, FX,c ) to (X, FX). The configuration space of bonds E is defined to be the set of all locally finite bond sets, i.e. E := {B ⊂ E(R2 ) : #{x x ∈ B : x x ⊂ } < ∞ for all ∈ B2b }. On E the σ -algebra FE is defined to be generated by the counting variables N E : E → N, B → #(E ∩ B) (E ∈ F E(R2 ) ). For a countable set E ∈ E one can also consider the Bernoulli-σ -algebra B E on E E := P(E) ⊂ E, which is defined to be generated by the family of sets ({B ⊂ E : e ∈ B})e∈E . Given a family ( pe )e∈E of reals in [0, 1] the Bernoulli measure on (E E , B E ) is defined as the unique probability measure for which the events ({B ⊂ E : e ∈ B})e∈E are independent with probabilities ( pe )e∈E . It is easy to check that the inclusion (E E , B E ) → (E, FE) is measurable. Thus any probability measure on (E E , B E ) can trivially be extended to (E, FE). 3.3. Gibbs measures. Let U : R2 → R be a potential and z > 0 an activity parameter. For finite configurations X, X ∈ X we consider the energy terms U (x1 − x2 ) and W U (X, X ) := U (x1 − x2 ). H U (X ) := x1 x2 ∈E(X )
x1 ∈X x2 ∈X
) The last definition can be extended to infinite configurations X whenever W U (X, X 2 2 converges as ↑ R through the net Bb . The Hamiltonian of a configuration X ∈ X in ∈ B2b is given by HU (X ) := H U (X ) + W U (X , X c ) = U (x1 − x2 ), x1 x2 ∈E (X )
where E (X ) := {x1 x2 ∈ E(X ) : x1 x2 ∩ = ∅}.
Translation-Invariance of Two-Dimensional Gibbsian Point Processes
The integral U,z ¯ ( X ) := Z
87
U ν (d X | X¯ ) e−H (X ) z # X
is called the partition function in ∈ B2b for the boundary condition X¯ c ∈ X. In order to ensure that the above objects are well defined and the partition function is finite and positive we need the following definition: Definition 2. A triple (U, z, X0 ) consisting of a potential U : R2 → R, an activity parameter z > 0 and a set of boundary conditions X0 ∈ FX,∞ is called admissible if for all X¯ ∈ X0 and ∈ B2b the following holds: W U ( X¯ , X¯ c ) has a well defined value U,z ¯ in R and Z ( X ) is finite. If (U, z, X0 ) is admissible, ∈ B2b and X¯ ∈ X0 then W U (X , X¯ c ) ∈ R is well defined for every X ∈ X, because X0 ∈ FX,∞ implies X X¯ c ∈ X0 . As a consequence U,z ¯ ( X ) is well defined. Furthermore by definition it is finite and the partition function Z by considering the empty configuration one can show that it is positive. The conditional Gibbs distribution γU,z (.| X¯ ) in ∈ B2b with boundary condition X¯ ∈ X0 is thus well defined by U 1 U,z ¯ γ (A| X ) := U,z ν (d X | X¯ ) e−H (X ) z # X 1 A (X ) for A ∈ FX. ¯ Z (X ) γU,z is a probability kernel from (X0 , FX0 ,c ) to (X, FX). Let GX0 (U, z) := {µ ∈P1 (X, FX) : µ(X0 ) = 1 µ(A|FX,c ) =
γU,z (A|.)
and µ-a.s. ∀ A ∈ FX, ∈ B2b }
be the set of all Gibbs measures corresponding to U and z with whole weight on boundary conditions in X0 . It is easy to see that for any probability measure µ ∈ P1 (X, FX) such that µ(X0 ) = 1 we have the equivalence µ ∈ GX0 (U, z)
⇔
(µ ⊗ γU,z = µ ∀ ∈ B2b ).
So for every µ ∈ GX0 (U, z), f : X → R+ measurable and ∈ B2b we have µ(d X ) f (X ) = µ(d X¯ ) γU,z (d X | X¯ ) f (X ).
(3.1)
If we consider a fixed potential and a fixed activity we will omit the dependence on U U,z . As a consequence of (3.1) the hard core K U of a and z in the notations γU,z and Z potential U implies that particles are not allowed to get too close to each other, i.e. for admissible (U, z, X0 ) and µ ∈ GX0 (U, z) we have µ({X ∈ X : ∃x, x ∈ X : x = x , x − x ∈ K U }) = 0.
(3.2)
For admissible (U, z, X0 ) and a Gibbs measure µ ∈ GX0 (U, z) we define the correlation function ρ U,µ by U U ¯ ρ U,µ (X ) = e−H (X ) µ(d X¯ ) e−W (X, X )
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for any finite configuration X ∈ X. If there is a ξ = ξ(U, z, X0 ) ≥ 0 such that ρ U,µ (X ) ≤ ξ # X
for all finite X ∈ X and all µ ∈ GX0 (U, z),
(3.3)
then we call ξ a Ruelle bound for (U, z, X0 ). Actually we need this bound on the correlation function in the following way: Lemma 3. Let (U, z, X0 ) be admissible with Ruelle bound ξ . For every Gibbs measure µ ∈ GX0 (U, z) and every measurable f : (R2 )m → R+ , m ∈ N we have = µ(d X ) f (x1 , . . . , xm ) ≤ (zξ )m d x1 . . . d xm f (x1 , . . . , xm ). (3.4) x1 ,...,xm ∈X
We use = as a shorthand notation for a multiple sum such that the summation indices are assumed to be pairwise distinct. 3.4. Superstability and admissibility. Now we will discuss some conditions on potentials which imply that (U, z, X0 ) is admissible and has a Ruelle bound whenever the set of boundary conditions X0 is suitably chosen. Apart from purely repulsive potentials such as the pure hard-core potential considered in Theorem 1 we also want to consider superstable potentials in the sense of Ruelle, see [R]. Therefore let r := r + [−1/2, 1/2[2 ⊂ R2
(r ∈ Z2 )
be the unit square centred at r and let Z2 (X ) := {r ∈ Z2 : N r (X ) > 0} be the minimal set of lattice points such that the corresponding squares cover the configuration. A potential U : R2 → R is called superstable if there are real constants a > 0 and b ≥ 0 such that for all finite configurations X ∈ X, H U (X ) ≥ [a N r (X )2 − bN r (X )]. r ∈Z2 (X )
U is called lower regular if there is a decreasing function : N → R+ with ∞ such that W U (X, X ) ≥ −
r ∈Z2 (X ) s∈Z2 (X )
r ∈Z2
(|r |)
0 and U : R2 → R be a translation-invariant pair potential. (a) If U is purely repulsive, i.e. U ≥ 0, then (U, z, X) is admissible with Ruelle bound ξ := 1. (b) If U is superstable and lower regular then (U, z, Xt ) is admissible and admits a Ruelle bound. The first assertion is a straightforward consequence of the fact that all energy terms are nonnegative. For the second assertion see [R]. 3.5. Conservation of translational symmetry. Every τ ∈ R2 gives a translation on the configuration space X via gτ (X ) := X + τ := {x + τ : x ∈ X }. We say that a measure µ on (X, FX) is τ-invariant if µ ◦ gτ−1 = µ, and µ is translationinvariant if it is τ-invariant for every τ ∈ R2 . The following lemma gives a sufficient condition for the conservation of τ-symmetry. Lemma 5. Let (U, z, X0 ) be admissible, where U : R2 → R is a translation-invariant potential. If for all cylinder events D ∈ FX,m (m ∈ N) and all Gibbs measures µ ∈ GX0 (U, z) we have µ(D + τ) + µ(D − τ) ≥ µ(D),
(3.5)
then every Gibbs measure µ ∈ GX0 (U, z) is τ-invariant. We further note that R2 is generated by the set {τi ei : 0 ≤ τi < 1/2, i ∈ {1, 2}}, so we only have to consider translations of this special form in order to establish translationinvariance of a set of Gibbs measures. 3.6. Concerning measurability. We will consider various types of random objects, all of which have to be shown to be measurable with respect to the considered σ -algebras. However we will not prove measurability of every such object in detail. Instead we will now give a list of operations that preserve measurability. Lemma 6. Let X, X ∈ X, B, B ∈ E, x ∈ R2 and p ∈ be variables, where (, F) is a measurable space. Let f : × R2 → R and g : × E(R2 ) → R be measurable. Then the following functions of the given variables are measurable with respect to the considered σ -algebras: f ( p, x ), X ∩ X , X ∪ X , X \ X , X + x, (3.6) x ∈X
g( p, b ), B ∩ B , B ∪ B , B \ B , B + x,
(3.7)
b ∈B
inf f ( p, x ), {x ∈ X : f ( p, x ) = 0}, C X,B (x), E(X ),
x ∈X
the number of different clusters of (X, B).
(3.8) (3.9)
Using this lemma and well known theorems, such as the measurability part of Fubini’s theorem, we can check the measurability of all objects considered.
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4. Proof of the Lemmas from Sections 2 and 3 4.1. Smoothly approximable potentials: Lemma 2. Let (U, z, X0 ), ξ , ψ, K˜ and U˜ (in case (b)) be as in the formulation of Lemma 2. By compactness of K U we can choose an > 0 such that the -enlargement K := (K U ) of the hard core K U has the property c := 1/(zξ ) − λ2 (K \ K U ) > 0. In case (a) let U1 := U and in case (b) let U1 := U˜ . Let R ≥ 1 such that
K ∪ K˜ ⊂ R
and furthermore
cR
2U˜ (x)|x|2 d x
0 and f δ : R → R+ be a symmetric smooth probability density with support in the |.|2 -disc B2 (δ), e.g. f δ (x) :=
1 2 2 −1 1 B2 (δ) (x)e−(1−|x|2 /δ ) , cδ
where cδ :=
e−(1−|x|2 /δ 2
B2 (δ)
2 )−1
d x.
Then
d x f δ (x )U (x − x )
U2 (x) := U ∗ f δ (x) :=
is a smooth approximation of U on C. By continuity of U and compactness of C a sufficiently small δ guarantees |U2 (x) − U (x)| < c :=
c 4λ2 (C)
for x ∈ C.
Let g : R2 → [0, 1] be a smooth symmetric function such that g = 0 on R and g = 1 on cR+1 . Now we can define U¯ , u : K c → R by U¯ := (1 − g)(U2 + c ) + gU1
and
u := U¯ − U.
It is easy to verify that the constructed objects have all the properties described in Definition 1 in both cases (a) and (b). 4.2. Property of the Ruelle bound: Lemma 3. For every n ∈ N, every measurable g : Xn → R+ and every X¯ ∈ X0 we have
νn (d X | X¯ )
f (x1 , . . . , xm ) g(X )
x1 ,...,xm ∈X n
=
=
n m
d x1 . . . d xm f (x1 , . . . , xm )
νn (d X | X¯ ) g({x1 , . . . , xm }X ).
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Combining this with (3.1), the definition of the conditional Gibbs distribution and the definition of the correlation function we get = µ(d X ) f (x1 , . . . , xm ) =
x1 ,...,xm ∈X n
µ(d X¯ )
=
n
m
1 U,z ¯ Z (X ) n
νn (d X | X¯ )
=
f (x1 , . . . , xm ) e−Hn (X ) z # X n U
x1 ,...,xm ∈X n
d x1 . . . d xm f (x1 , . . . , xm ) z m ρ U,µ ({x1 , . . . , xm }).
Now we use (3.3) to estimate the correlation function by the Ruelle bound ξ . Letting n → ∞ the assertion follows from the monotone limit theorem. 4.3. Sufficient condition: Lemma 5. The lemma can be shown exactly as Proposition (9.1) in [G] and we will only outline the proof: We first note that (X, FX) is a standard Borel space, which follows from [DV], Theorem A2.6.III. Hence the point particle version of Theorem (7.26) in [G] implies that every Gibbs measure can be decomposed into extremal Gibbs measures. Thus without loss of generality we may assume µ to be extremal. Suppose now that µ is not τ-invariant, i.e. µ ◦ gτ−1 = µ, which also implies µ ◦ gτ = µ. As the extremality of µ implies the extremality of µ ◦ gτ−1 and µ ◦ gτ , the point particle version of Theorem (7.7) guarantees the existence of sets A− , A+ ∈ FX,∞ such that µ ◦ gτ−1 (A− ) = 0, µ ◦ gτ (A+ ) = 0 and µ(A− ) = µ(A+ ) = 1. Hence for A := A− ∩ A+ we have µ ◦ gτ (A) + µ ◦ gτ−1 (A) = 0 < 1 = µ(A). On the other hand by assumption (3.5) we know that µ ◦ gτ + µ ◦ gτ−1 ≥ µ on the algebra of all cylinder events. By the monotone class theorem this inequality even holds on all of FX, which contradicts the above inequality. 4.4. Measurability: Lemma 6. Details concerning measurability of functions of point processes can be found in [DV, K or MKM], for example. The first part of (3.6) is the measurability part of Campbell’s theorem. For the rest of (3.6) it suffices to observe that for ∈ B2b we have 1{x=x ∈} , N (X \ X ) = N (X ) − N (X ∩ X ), N (X ∩ X ) = x∈X x ∈X
N (X + x) =
1 (x + x) and N (X ∪ X ) = N (X ) + N (X \ X ).
x ∈X
For (3.7) we can argue similarly. For c ∈ R, ∈ B2b , x ∈ R2 and L ∈ F E(R2 ) , inf f ( p, x ) < c ⇔ 1{ f ( p,x ) 0. Let |.|h be a norm on R2 and U := Uhc the corresponding pure hard-core potential. As U is purely repulsive we know that (U, z, X) is admissible with Ruelle bound ξ := 1 by Lemma 4, part (a). Let K := K U and > 0. If we choose sufficiently small we have cξ := λ2 (K \ K U )
0 in order to control probabilities close to 0. As all the above objects are fixed for the whole proof we will ignore dependence on them in our notations.
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5.2. Generalised translation. Let n > n and X ∈ X. We consider the bond set K n := {x1 x2 ∈ E(X ) : x1 x2 ∩ n = ∅, x1 − x2 ∈ K }. Every time we use this notation it will be clear from the context which configuration X it refers to. Note that K n is finite as X is locally finite and K is bounded. For a bounded set ∈ B2b let rn,X () = sup{|y | : y ∈ C X,K n ()} denote the range of the corresponding K n -cluster. In the following lemma we consider the case = n , where n ∈ N is the number fixed in Sect.5.1. Lemma 7. We have sup µ(d X ) rn,X (n ) < ∞. n>n
By the Chebyshev inequality we therefore can choose an integer R > n , such that for every n > n we have δ for G n := {X ∈ X : rn,X (n ) < R} ∈ FX. µ(G n ) ≥ 1 − 2 For n > R we define the functions q : R+ → R,
Q : R+ → R, r : R × R+ → R
1 , 1 ∨ (s log(s)) k q(s ) r (s, k) := ds , (s∨0)∧k Q(k) q(s) :=
τn : R → R
and
Q(k) :=
by
k
q(s)ds, 0
τn (s) := τ r (s − R, n − R).
For a sketch of the graph of τn see Fig. 1. Some important properties of τn are the following: τn (s) = τ for s ≤ R, τn (s) = 0 for s ≥ n and τn is decreasing.
(5.3)
For X ∈ X and x ∈ X we define an,X (x) to be the point of C X,K n (x) with maximal |.|-distance to the origin. (If there is more than one such point we choose the maximal one with respect to the lexicographic order for the sake of definiteness.) Then (5.3) implies |an,X (x)| ≥ |x|
and
τn (|an,X (x)|) = min{τn (|x |) : x ∈ C X,K n (x)}.
The transformation Tn0 : R2 → R2 , Tn0 (x) := x + τn (|x|)e1 can also be viewed as a transformation on X, such that every point x of a configuration X is translated the distance τn (|x|) in direction e1 . We would like to use this generalised translation Tn0 as a tool for our proof just as in [FP1] and [FP2].
Fig. 1. Graph of τn
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T. Richthammer
5.3. Good configurations. In order to deal with the hard core we will replace the above translation Tn0 by a transformation Tn : X → X which is required to have the following properties: (1) For X ∈ X the transformed configuration X˜ = Tn (X ) is constructed by translating every x ∈ X a certain distance tn,X (x) in direction e1 . We note that we do not require the particles to be translated independently. (2) Particles in the inner region n −1 are translated by τ e1 , and particles in the outer region n c are not translated at all. (3) Tn is bijective, the density of the transformed process with respect to the untransformed process under the measure ν can be calculated explicitly and we have a suitable estimate on this density. (4) The Hamiltonian HUn (X ) is invariant under Tn , i.e. particles within hard core distance remain within hard core distance and particles at larger distance remain at larger distance. Property (2) implies that the translation of the chosen cylinder event D is the same as the transformation of D by Tn . Properties (3) and (4) imply that the density of the transformed process with respect to the untransformed process under the measure µ can be estimated. Therefore a transformation with these properties seems to be a good tool for proving (3.5). However, in general it is difficult to construct a transformation with all the given properties. For example properties (2) and (4) cannot both be satisfied if X is a configuration of densely packed hard-core particles. If n > R and X ∈ G n then such a situation can not occur, and by Lemma 7 this is the case with high probability. Similar problems arise for the other properties, so we will content ourselves with a transformation satisfying the above properties only for configurations X from a set of good configurations 3
G n := X ∈ G n : i (n, X ) < 1 ∈ FX. (5.4) i=1
The functions i (n, X ) will be defined whenever we want good configurations to have a certain property. In Lemma 13 we then will prove that the set of good configurations G n has probability close to 1 when n is big enough. Up to that point we consider a fixed n ≥ R + 1. 5.4. Modifying the generalised translation. With a view to properties (1) and the second part of (2) we define the transformation Tn : X → X by k k + τn,X e1 : 1 ≤ k ≤ m(X )} = {x + tn,X (x)e1 : x ∈ X } Tn (X ) := X cn ∪ {Pn,X k k for every X ∈ X, where m(X ) := # X n , {Pn,X : 1 ≤ k ≤ m(X )} = X n , τn,X is k the translation distance of Pn,X and the translation distance function tn,X : X → R is k ) := τ k defined by tn,X (x) := 0 for x ∈ X cn and tn,X (Pn,X n,X for 1 ≤ k ≤ m(X ). We k k . In order to are left to identify the points Pn,X of X and their translation distances τn,X k k if it simplify notation we will omit the dependence on X in m(X ), tn,X , Pn,X and τn,X
Translation-Invariance of Two-Dimensional Gibbsian Point Processes
95
Fig. 2. Every point Pnk is translated by τnk e1
Fig. 3. Construction of tnk
is clear which configuration is considered. In our construction we would like to ensure that the points Pnk are ordered in a way such that 0 =: τn0 ≤ τn1 ≤ . . . ≤ τnm .
(5.5)
This relation will be an important tool for showing the bijectivity of the transformation as required in property (3) of the last subsection. As required in (4) we also would like to have x1 , x2 ∈ X, x1 − x2 ∈ K ⇒ tn,X (x1 ) = tn,X (x2 ), x1 , x2 ∈ X, x1 − x2 ∈ / K ⇒ (x1 + tn,X (x1 )e1 ) − (x2 + tn,X (x2 )e1 ) ∈ / K.
(5.6) (5.7)
With these properties in mind we will now give a recursive definition of Pnk and τnk for k a fixed configuration X ∈ X using a translation distance function tnk := tn,X : R2 → R in each step. In the kth construction step (1 ≤ k ≤ m) let tnk := tn0 ∧ m Pni ,τni = tnk−1 ∧ m Pnk−1 ,τnk−1 , 0≤i 21 m x ,t (x) := t + h x ,t f K (x − x ) + ∞ 1{ f K (x−x )=1} else, where
h x ,t := |τn (|x | − c K ) − t|.
Note that the first case in the definition of m x ,t has been introduced in order to bound the slope of m x ,t . In Sect. 6.2 we will show important properties of this auxiliary function, but for the moment we will content ourselves with the intuition given by Fig. 4. Using Lemma 6 one can show that all the above objects are measurable with respect to the considered σ -algebras. In the rest of this section we will convince ourselves that the above construction has indeed all the required properties. Lemma 8. The construction satisfies (5.5), (5.6) and (5.7). Lemma 9. For good configurations X ∈ G n we have (Tn X − τ e1 )n −1 = X n −1
and (Tn X )n c = X n c .
(5.8)
Lemma 10. The transformation Tn : X → X is bijective. Actually in the proof of Lemma 10 we construct the inverse of Tn . This is needed in the proof of Lemma 11, where we will show for every X¯ ∈ X that νn (.| X¯ ) is absolutely −1 continuous with respect to νn (.| X¯ ) ◦ T−1 n with density ϕn ◦ Tn , where ϕn (X ) :=
m(X )
1 + ∂1 t k (P k ) . n,X n,X
(5.9)
k=1
The proof will also show that definition (5.9) makes sense νn ( . | X¯ )-a.s., in that the considered derivatives exist. Lemma 11. For every X¯ ∈ X and every FX-measurable function f ≥ 0, dνn (.| X¯ ) f. dνn (.| X¯ ) ( f ◦ Tn · ϕn ) =
(5.10)
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¯ n and ϕ¯n be defined Considering (3.5) we also need the backwards translation. So let T analogously to the above objects, where now e1 is replaced by −e1 . The previous lemmas ¯ n is not the apply analogously to this deformed backwards translation. We note that T inverse of Tn . 5.5. Final steps of the proof. From (3.1) and Lemma 11 we deduce µ(Tn (D ∩ G n )) U 1 ¯ = µ(d X ) νn (d X | X¯ ) 1Tn (D∩G n ) (X ) z # X n e−Hn (X ) ¯ Z n ( X ) 1 ¯ = µ(d X ) νn (d X | X¯ ) Z n ( X¯ ) 1Tn (D∩G n ) ◦ Tn (X ) z #(Tn X )n e−Hn (Tn X ) ϕn (X ). U
By Lemma 10 Tn is bijective, by (5.8) #(Tn X )n = # X n and by (5.6) and (5.7) we have HUn (Tn X ) = HUn (X ). Hence the above integrand simplifies to 1 D∩G n (X ) z # X n e−Hn (X ) ϕn (X ), U
¯ n . So and we have an analogous expression for the backwards transformation T ¯ n (D ∩ G n )) + µ(Tn (D ∩ G n )) − µ(D ∩ G n ) µ(T U 1 ¯ = µ(d X ) νn (d X | X¯ ) 1 D∩G n (X ) z # X n e−Hn (X ) ¯ Z n ( X )
× ϕ¯n (X ) + ϕn (X ) − 1]. We note that for X ∈ G n we have 1
1
ϕ¯n (X ) + ϕn (X ) ≥ 2 (ϕ¯ n (X )ϕn (X )) 2 ≥ 2 e− 2 ≥ 1, where we have used the arithmetic-geometric-mean inequality in the first step and the following estimate in the second step: Lemma 12. For X ∈ G n we have log ϕ¯n (X ) + log ϕn (X ) ≥ −1.
(5.11)
¯ n (D ∩ G n )) + µ(Tn (D ∩ G n )) ≥ µ(D ∩ G n ). µ(T
(5.12)
Hence we have shown that
In (5.12) we would like to replace D ∩ G n by D, and for this we need G n to have high probability: Lemma 13. If n ≥ R + 1 is chosen big enough, then µ(G cn ) ≤ δ.
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For the proof of Theorem 1 we choose such an n ≥ R + 1. Because of D ∈ FX,n −1 and (5.8) we have ∀ X ∈ D ∩ Gn :
(Tn X − τ e1 )n −1 ∈ D,
i.e. Tn X ∈ D + τ e1 ,
and an analogous result for the backwards transformation. Hence Tn (D ∩ G n ) ⊂ D + τ e1
¯ n (D ∩ G n ) ⊂ D − τ e1 . T
and
Using these inclusions and Lemma 13 we deduce from (5.12), µ(D − τ e1 ) + µ(D + τ e1 ) ≥ µ(D) − δ. δ > 0 was chosen to be an arbitrary positive real, so we get the estimate (3.5) by taking the limit δ → 0. Now the claim of the theorem follows from Lemma 5. 6. Proof of the Lemmas from Section 5 6.1. Cluster bounds: Lemma 7. For n > n and X ∈ X we want to estimate rn,X (n ). For any path x0 , . . . , xm in the graph (X, K n ) such that x0 ∈ n we have |xm | ≤ |x0 | +
m
|xi − xi−1 | ≤ n + mc K .
i=1
By considering all possibilities for such paths we obtain rn,X (n ) ≤ n +
=
1{x0 ∈n } mc K
m≥1 x0 ,...,xm ∈X
m
1{xi xi−1 ∈K n } .
i=1
Using the hard core property (3.2) and Lemma 3 we get Rn := µ(d X )rn,X (n ) − n ≤
µ(d X )
1{x0 ∈n } mc K
x0 ,...,xm ∈X
m≥1
≤
=
1 K \K U (xi − xi−1 )
i=1
(zξ )m+1
m
d x0 . . . d xm 1{x0 ∈n } mc K
m≥1
m
1 K \K U (xi − xi−1 ).
i=1
By (5.1) we can estimate the integrals over d xi in the above expression beginning with i = m. This gives m times a factor cξ and the integration over d x0 gives an additional factor λ2 (n ) = (2n )2 . Thus Rn ≤ (2n )2 zξ c K
m(cξ zξ )m < ∞,
m≥1
where the last sum is finite because cξ zξ < 1.
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6.2. Properties of the auxiliary function . Let f : I → R be a function on an interval. f is called 1/2-Lipschitz-continuous if 1 |r − r | for all r, r ∈ I. 2 f is called piecewise continuously differentiable if it is continuous and if | f (r ) − f (r )| ≤
∃ countable and closed M ⊂ I :
f is continuously differentiable on I \ M.
As M is closed, the connected components of I \ M are countably many intervals. For a strictly monotone piecewise continuously differentiable transformation f on R we can apply the Lebesgue transformation theorem: The derivative f is well defined λ1 -a.s. and for every B1 -measurable function g ≥ 0 we have g(x )d x . (6.1) g( f (x))| f (x)|d x = The above properties are inherited as follows: Lemma 14. Let f 1 , f 2 : I → R be functions on an interval I . (a) If f 1 and f 2 are 1/2-Lipschitz-continuous, then so is f 1 ∧ f 2 . (b) If f 1 and f 2 are piecewise continuously differentiable, then so is f 1 ∧ f 2 . For the proof of these easy facts we refer to [Ri2]. A function f : R2 → R is called 1/2e1 -Lipschitz-continuous or piecewise continuously e1 -differentiable if for all r2 ∈ R the function f (., r2 ) is 1/2-Lipschitz-continuous or piecewise continuously differentiable respectively. Lemma 15. For x ∈ R2 and t ∈ R the function τn (|.|) ∧ m x ,t is 1/2-e1 -Lipschitz-continuous and piecewise continuously e1 -differentiable. For details of the proof we again refer to [Ri2]. Basically Lemma 15 follows from Lemma 14. The only difficulty is to show the continuity of τn (|.|) ∧ m x ,t , which might be a problem because of the jump to infinity of m x ,t in case of h x ,t c f ≤ 1/2. But if x ∈ ∂{m x ,t < ∞} = ∂{ f K (. − x ) < 1} then x − x is contained in the closure of K . Hence |x − x | ≤ c K , which implies |x | − c K ≤ |x|. As τn is decreasing we obtain τn (|x|) ≤ τn (|x | − c K ) ≤ t + h x ,t ≤ m x ,t (x) by definition of h x ,t , which implies the claimed continuity. 6.3. Properties of the construction: Lemma 8. We will first investigate monotonicity and regularity properties of tnk and Tnk : Lemma 16. For X ∈ X and k ≥ 0, tnk is 1/2-e1 -Lipschitz-continuous and piecewise cont. e1 -differentiable, Tnk
is ≤e1 -increasing and bijective.
(6.2) (6.3)
Proof. tnk is the minimum of finitely many functions of the form τn (|.|) ∧ m x ,t , where x ∈ R2 and t ∈ R. Hence (6.2) is an immediate consequence of Lemmas 15 and 14. Statement (6.2) implies that Tnk is e1 -continuous and ≤e1 -increasing, and hence bijective. This shows (6.3).
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For (5.5) it suffices to observe that for every 2 ≤ k ≤ m we have τnk = tnk (Pnk ) = tnk−1 (Pnk ) ∧ m Pnk−1 ,τnk−1 (Pnk ) ≥ τnk−1 . This follows from the definition of τnk and tnk , from tnk−1 (Pnk ) ≥ τnk−1 by the definition of Pnk−1 and from m x ,t ≥ t. For (5.6) and (5.7) let x1 , x2 ∈ X . Without loss of generality j we may suppose that x1 = Pn and x2 = Pni , where 0 ≤ i ≤ j. Here Pn0 is interpreted j to be any point of X cn . We first observe that Pn ∈ i := {x ∈ R2 : tni (x) ≥ τni } and j ∀ x ∈ (Pni + K ) ∩ i : tn (x) = tni (x) ∧ m Pnk ,τnk (x) = τni . (6.4) i≤k≤ j
This holds as tni (x) ≥ τni by definition of i , m Pnk ,τnk ≥ τni by (5.5) and m Pni ,τni (x) = τni by j
j
j
j
j
x ∈ Pni +K . If Pn −Pni ∈ K , then Pn ∈ (Pni +K )∩i , so (6.4) implies τn = tn (Pn ) = τni , j j / K . We have Pn ∈ i \ (Pni + K ) and which shows (5.6). For (5.7) suppose Pn − Pni ∈ j j j τn = tn (Pn ) by definition, so it suffices to show j
Tn (i \ (Pni + K )) = i \ (Pni + K ) + τni e1 .
(6.5)
In order to show this we fix r ∈ R. Continuity of tni (., r ) implies tni = τni on ∂i (., r ). j j Just as in the proof of (6.4) it follows that tn = τni on ∂i (., r ). But Tn (., r ) is increasing, continuous and bijective by (6.3), so j
Tn (i ) = i + τni e1 , and combining this with (6.4) we are done. 6.4. Properties of the deformed translation: Lemma 9. The following lemma shows how to estimate the translation distances τnk . Lemma 17. For X ∈ X and k ≥ 0 we have
τnk ≤ tn0 (Pnk ) for all k ≥ k, τnk
≥
tn0 (an,X (Pnk ))
if X ∈ G n .
(6.6) (6.7)
Proof. Assertion (6.6) follows from the definition of Pnk and from tnk ≤ tn0 . For the proof of (6.7) let X ∈ G n . We first would like to show that X,K n
∀x, x ∈ X : |x| ≤ |x |, x ←→ x ⇒ |τn (|x| − c K ) − τn (|x |)|c f ≤ 1/2.
(6.8)
Defining 1 (n, X ) :=
x,x ∈X
1{|x|≤|x |} 1
X,K n
{x ←→x }
2 4 τn (|x| − c K ) − τn (|x |) c2f ,
(6.9)
we have 1 (n, X ) < 1 by definition of the set G n of good configurations in (5.4) and by X ∈ G n . Hence every summand of 1 is less than 1, which implies (6.8). We now can prove (6.7) by induction on k. For k = 0 we have equality if the right-hand side is
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defined to be 0. For the inductive step k − 1 → k let i ≤ k − 1. By (6.6), the inductive hypothesis and (6.8) we have 0 ≤ τn (|Pni | − c K ) − τni c f ≤ τn (|Pni | − c K ) − τn (|an,X (Pni )|) c f ≤ 1/2, / K . Thus so h Pni ,τni c f ≤ 1/2. Therefore m Pni ,τni (Pnk ) = ∞ whenever Pnk − Pni ∈ τnk = tnk (Pnk ) = tn0 (Pnk ) ∧ m Pni ,τni (Pnk ) ≥ tn0 (an,X (Pnk )), i s such that s > R and s < n we have s ∧n q(t − R) q(s − R) 0 ≤ r (s − R, n − R)−r (s − R, n− R) = dt ≤ (s − s) Q(n− R) R∨s Q(n − R) by the monotonicity of q. Defining n¯ := n + c K and R¯ := R + c K we thus have τn (x, x ) ≤ 1{x∈n¯ } τ 2 (|x | − |x| + c K )2 q
¯ 2 q(|x| − R) ¯ 2 Q(n¯ − R)
for x, x ∈ R2 ,
(6.22)
using the substitution s := |x | and s := |x| − c K . (If s ≤ R or s ≥ n then τn (x, x ) = 0.) The following relations will give us control over the relevant terms of the right-hand side of (6.22). We first observe that ¯ 2 ≤ 16 R¯ 2 + 32Q(n¯ − R) ¯ for n¯ ≥ 2 R. ¯ d x q(|x| − R) (6.23) q
n¯
Indeed, writing s := |x| we obtain n¯
¯ 2 d x q(|x| − R)
≤ 16 R¯ 2 + 32
≤
2 R¯
ds 8s + 0
n− ¯ R¯
q(s)ds
≤
n− ¯ R¯ R¯
2 ¯ ds 8(s + R)q(s)
¯ 16 R¯ 2 + 32Q(n¯ − R).
0
In the first step we used q ≤ 1, and in the second step R¯ ≤ s and sq(s) ≤ 2. We observe lim Q(n) = ∞, which is a consequence of log log n ≤ Q(n) for n > 1. Therefore by n→∞ (6.23), ¯ 2 q(|x| − R) lim c(n) = 0 for c(n) := dx . (6.24) ¯ 2 n→∞ Q(n¯ − R) n¯ Finally, for x0 , . . . , xm ∈ R2 such that xi − xi−1 ∈ K we have |xi − xi−1 | ≤ c K , so (|xm | − |x0 | + c K )2 ≤ (m + 1)2 c2K .
(6.25)
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Now we will use the ideas of the proof of Lemma 7. For X ∈ X we can estimate the summands of 1 (n, X ) by considering all paths x0 , . . . , xm in the graph (X, K n ) connecting x = x0 and x = xm . By (6.22) and (6.25) we can estimate 1 (n, X ) by a constant c times
=
(m + 1)2
1{x0 ∈n¯ }
x0 ,...,xm ∈X
m≥0
m ¯ 2 q(|x0 | − R) 1{xi xi−1 ∈K n } . ¯ 2 Q(n¯ − R) i=1
Using Lemma 3 we can thus proceed as in the proof of Lemma 7: µ(d X ) 1 (n, X ) ≤ zξ c (m + 1)2 (cξ zξ )m c(n). m≥0
Likewise,
µ(d X )2 (n, X ) ≤ 2zξ τ 2 c(n).
Finally, we can estimate 3 (n, X ) by a constant c times
(m + 1)2
=
1{x0 ∈n¯ }
x0 ,...,xm ∈X
m≥0
×
m
1{xi xi−1 ∈K n }
i=1
¯ 2 q(|x0 | − R) ¯ 2 Q(n¯ − R)
1 K (x − x0 ) +
x ∈X,x =xi ∀ i
m
1 K (x j − x0 ) .
j=1
The second sum in the brackets can be estimated by m. As above µ(d X ) 3 (n, X ) ≤ zξ c (m + 1)2 (cξ zξ )m c(n)(zξ cξ + m). m≥0
In the bounds on the expectations of 1 and 3 the sums over m are finite by (5.1). Collecting all estimates and using (6.24) we thus find that µ(d X )
3 i=1
i (n, X ) ≤
δ 2
for sufficiently large n, and µ(G cn ) ≤ δ follows from the high probability of G n , the Chebyshev inequality and the definition of G n in (5.4). 7. Proof of Theorem 2: Main Steps 7.1. Basic constants. Let (U, z, X0 ) be admissible with Ruelle bound ξ , where U : R2 → R is a translation-invariant, smoothly approximable standard potential. We choose K , ψ, U¯ and u according to Definition 1. W.l.o.g. we may assume 0 ∈ K , U¯ = U and u = 0 on K . We then let > 0 so small that 1 . (7.1) u(x)d ˜ x< cξ := λ2 (K \ K U ) + zξ Kc
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In addition to the function f K and the constants c K and c f introduced in Sect. 5.1 we also define 2 u(x)|x| ˜ d x and cψ := ψ ∨ d x ψ(x)(|x|2 ∨ 1). (7.2) cu := Kc
These constants are finite by our assumptions. Finally, we fix a Gibbs measure µ ∈ GX0 (U, z), a cylinder event D ∈ FX,n −1 , where n ∈ N, a translation distance τ ∈ [0, 1/2], the translation direction e1 and a real δ > 0. 7.2. Decomposition of µ and the bond process. For n ∈ N and X ∈ X we consider the bond set E n (X ) := E n (X ) = {x1 x2 ∈ E(X ) : x1 x2 ∩ n = ∅}. On (E E n (X ) , B E n (X ) ) we introduce the Bernoulli measure πn (.|X ) with bond probabilities (u(b)) ˜ b∈E n (X )
where
u(b) ˜ := 1 − e−u(b) ,
using the shorthand notation u(x1 x2 ) := u(x1 − x2 ) for x1 , x2 ∈ R2 . We note that 0 ≤ u(b) ˜ < 1 for all b ∈ E n (X ) as 0 ≤ u < ∞. As remarked earlier πn (.|X ) can be extended to a probability measure on (E, FE). For all D ∈ FE, πn (D|.) is FXmeasurable, so πn is a probability kernel from (X, FX) to (E, FE). Lemma 22. Let n ∈ N. We have
µ ⊗ νn (G n ) = 1 and µ(G n ) = 1 for G n := {X ∈ X0 :
u(b) ˜ < ∞}.
b∈E n (X )
For X ∈ G n the Borel-Cantelli lemma implies that every bond set is finite πn (.|X )-a.s., so πn ({B}|X ) = 1, B⊂E n (X )
where the summation symbol We have πn ({B}|X ) = u(b) ˜ b∈B
indicates that the sum extends over finite subsets only.
(1 − u(b)) ˜ = e−Hn (X ) u
b∈E n (X )\B
(eu(b) − 1),
b∈B
so for every X ∈ G n the Hamiltonian Hu n (X ) is finite, and thus the decomposition of the potential gives a corresponding decomposition of the Hamiltonian ¯
HUn (X ) = HUn (X ) − Hu n (X ). Using (3.1) we conclude that for every FX ⊗ FE-measurable function f ≥ 0, 1 µ(d X¯ ) f (X, B) dµ ⊗ πn f = νn (d X | X¯ ) Z n ( X¯ ) B⊂E n (X ) U¯ (eu(b) − 1). ×z # X n e−Hn (X ) b∈B
(7.3)
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Here by Lemma 22 on both sides we have X ∈ G n with probability one, thus the equality follows from the above decomposition. If f does not depend on B at all, the integral on the left-hand side of (7.3) is just the µ-expectation of f , as πn is a probability kernel, and from the right-hand side we learn that the perturbation u of the smooth potential U¯ can be encoded in a bond process B such that the perturbation affects only those pairs of particles with x1 x2 ∈ B. On (E E n (X ) , B E n (X ) ) we denote the counting measure concentrated on finite bond sets by πn (.|X ). Again πn can be considered as a probability kernel from (X, FX) to (E, FE). For all FE-measurable functions f ≥ 0 we have f (B). πn (d B|X ) f (B) = B⊂E n (X )
7.3. Generalised translation. First of all, we need to augment each bond set B by additional bonds between all particles that are close to each other. That is, for n > n , X ∈ X and B ⊂ E n (X ) we introduce the K -enlargement of B by B+ := B ∪ {x1 x2 ∈ E n (X ) : x1 − x2 ∈ K }. We then consider the range of the B+ -cluster of ∈ B2b , rn,X,B+ () = sup{|x | : x ∈ C X,B+ ()}. Lemma 23. We have sup µ ⊗ πn (d X, d B) rn,X,B+ (n ) < ∞. n>n
By the Chebyshev inequality we therefore can choose an integer R > n , such that for every n > n the event G n := {(X, B) ∈ X × E : rn,X,B+ (n ) < R, B ⊂ E n (X ) finite} ∈ FX ⊗ FE has probability
µ ⊗ πn (G n ) ≥ 1 − δ/2.
For n > R we define the functions q, Q, r and τn exactly as in Sect. 5.2. For X ∈ X, B ∈ E n (X ) and x ∈ X we define an,X,B+ (x) to be a point of C X,B+ (y) such that |an,X,B+ (x)| ≥ |x|, τn (|an,X,B+ (x)|) = min{τn (|x |) : x ∈ C X,B+ (y)} and an,X,B+ (x) is a measurable function of x, X and B. 7.4. Good configurations. In order to deal with the hard core and the perturbation encoded in the bond process, we will introduce a transformation Tn : X × E → X × E which is required to have the following properties: (1) Whenever B is a set of bonds between particles in X , the transformed configura˜ = Tn (X, B) is constructed by translating every particle x ∈ X by a tion ( X˜ , B) certain distance tn,X,B (x) in direction e1 , and by translating bonds along with the corresponding particles.
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(2) Particles in the inner region n −1 are translated by τ e1 , and particles in the outer region n c are not translated at all. (3) Particles connected by a bond in B are translated the same distance. (4) Tn is bijective, and the density of the transformed process with respect to the untransformed process under the measure ν ⊗ πn can be calculated explicitly. ¯ ¯ (5) We have suitable estimates on this density and on HUn ( X˜ ) − HUn (X ). For the last assumption we need particles within hard core distance to remain within hard core distance and particles at larger distance to remain at larger distance. Property (2) implies that the translation of the chosen cylinder event D is the same as the transformation of D by Tn . Properties (3)-(5) are chosen with a view to the right-hand side of (7.3): If Tn has these properties then the density of the transformed process with respect to the untransformed process under the measure µ ⊗ πn can be estimated. We will content ourselves with a transformation satisfying the above properties only for (X, B) from a set of good configurations 5
G n := (X, B) ∈ G n : i (n, X, B) < 1/2 ∈ FX ⊗ FE.
(7.4)
i=1
The functions i (n, X, B) will be defined whenever we want good configurations to have a certain property. In Lemma 28 we then will prove that the set of good configurations G n has probability close to 1 when n is big enough. Up to that point we consider a fixed n ≥ R + 1. 7.5. Modifying the generalised translation. The construction of the deformed translation Tn will go along the same lines as the corresponding construction in Sect. 5.4. However, here we also have to consider bonds between particles, and by property (3) from the last section we know that we have to translate not just particles, but whole B-clusters. For a rigorous recursive definition of Tn (X, B) we first consider the case that B is a 0 0 finite subset of E n (X ). Let tn,X,B := τn (|.|), Cn,X,B be the B-cluster of the outer region 0 0 c n , m = m(X, B) the number of different B-clusters of X \ Cn,X,B and τn,X,B := 0. In the kth construction step (1 ≤ k ≤ m) let k−1 k 0 := tn,X,B ∧ m x,τ k−1 = tn,X,B ∧ m x,τ i , tn,X,B k−1 x∈Cn,X,B
n,X,B
0≤i 0. As we show below, there is a multiplicative function closely related to the determinant but which has a global logarithm (if k > 0). However, as we also show below, there is a determinant function, the ‘adiabatic determinant’ in this sense, provided the group G −∞ sus(2k) is ‘dressed’ by replacing it by an extension with respect to a star product, of which G −∞ sus(2k) is the principal term. This extension is homotopically trivial, i.e. still gives a classifying space for K-theory.
146
R. Melrose, F. Rochon k More precisely, consider the space sus(2n) (X ; E)[[ε]] of formal power series ∞
k aµ εµ , aµ ∈ sus(2n) (X ; E)
µ=0
(see (2.9) for the definition) equipped with the star-product ⎛ ⎞ ∞ ∞ (A ∗ B)(u) = ⎝ aµ ε µ ⎠ ∗ bν εν µ=0
=
∞ ∞ µ=0 ν=0
ν=0
⎞ ∞ p iε p ⎠ ⎝ ω(Dv , Dw ) aµ (v)bν (w) 2 p p! ⎛
εµ+ν
p=0
(5) v=w=u
∗ (X ; E)[[ε]], where ω is the standard symplectic form on R2n . This for A, B ∈ sus(2n) gives a corresponding group −∞ G −∞ sus(2n) (X ; E)[[ε]] = {Id +Q; Q ∈ sus(2n) (X ; E)[[ε]], −∞ 0 (X ; E)[[ε]], (Id +Q) ∗ (Id +P) = Id ∈ sus(2n) (X ; E)[[ε]]} ∃ P ∈ sus(2n)
with group law given by the star-product (5). Then G −∞ sus(2n) (X ; E) is a retraction of −∞ G sus(2n) (X ; E)[[ε]]. Our first main result is the following. Theorem 1. There is a multiplicative ‘adiabatic’ determinant function ∗ deta : G −∞ sus(2n) (X ; E)[[ε]] −→ C ,
deta (A ∗ B) = deta (A) deta (B) ∀ A, B ∈ G −∞ sus(2n) (X ; E)[[ε]], which generates H1 (G −∞ sus(2n) (X ; E)[[ε]]). This is proven in §3 by considering a corresponding determinant for mixed isotropic operators and taking the adiabatic limit. Given a (locally trivial) fibration of compact manifolds Z
M
(6)
φ
B and a family of elliptic 2n-suspended operators
k (M/B; E, F) D ∈ sus(2n)
with vanishing numerical index, one can construct an associated determinant line bundle Deta (D) → B as described in §3, the definition being in terms of (a slightly extended notion of) principal bundles; a related construction can be found in [17]. More generally,
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this construction can be extended to a fully elliptic family of product-suspended operators (see the Appendix and §2 for the definition)
k,k D ∈ psus(2n) (M/B; E, F)
with vanishing numerical index. Our second result is to relate this determinant line bundle with Quillen’s definition via Bott periodicity. Let D0 ∈ 1 (M/B; E, F) be a family of elliptic operators with vanishing numerical index. Define, by recurrence for n ∈ N, the fully elliptic product-suspended families by
∗ itn − τn Dn−1 1,1 ∈ psus(2n) Dn (t1 , . . . , tn , τ1 , . . . , τn ) = (M/B; 2n−1 (E ⊕ F)), Dn−1 itn + τn where 2n−1 (E ⊕ F) is the direct sum of 2n−1 copies of E ⊕ F. In §5 we prove Theorem 2 (Periodicity of the determinant line bundle). For each n ∈ N, there is an isomorphism Deta (Dn ) ∼ = Det(D0 ) as line bundles over B. In §6, we investigate the counterpart of the eta invariant for the determinant of Theorem 1. After extending the definition given in [13] to product-suspended operators, we relate this invariant (denoted here ηsus ) to the extension of the original spectral definition of Atiyah, Patodi and Singer given by Wodzicki [21]. Namely consider ηz (A) =
sgn(a j )|a j |−z ,
(7)
j
where the a j are the eigenvalues of A in order of increasing |a j | repeated with multiplicity. Theorem 3. If A ∈ 1 (X ; E) is an invertible self-adjoint elliptic pseudodifferential 1,1 operator and A(τ ) = A + iτ ∈ psus (X ; E) is the corresponding product-suspended family then ηsus (A(τ )) = regz=0 ηz (A) = η(A)
(8)
is the regularized value at z = 0 of the analytic extension of (7) from its domain of convergence. The eta invariant for product-suspended operators is, as in the suspended case discussed in [13], a log-multiplicative functional
k,k l,l ηsus (AB) = ηsus (A) + ηsus (B), A ∈ psus (X ; E), B ∈ psus (X ; E).
Finally, in §7, we show (see Theorem 4) that in the appropriate context, this eta invariant can be interpreted as the logarithm of the determinant of Theorem 1. To discuss these results, substantial use is made of various classes of pseudodifferential operators, in particular product-type suspended operators and mixed isotropic operators. An overview of the various classes used in this paper is given in §2 and some of their properties are discussed in the Appendix.
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1. Determinant Line Bundle Quillen in [18] introduced the determinant line bundle for a family of ∂ operators. Shortly after, Bismut and Freed in [4] and [3] generalized the definition to Dirac operators. We will show here that this is induced by the Fredholm determinant, as a representation of the group G −∞ . To do so we need to slightly generalize the standard notion of a principal bundle. 1.1. Bundles of groups. Definition 1.1. Let G be a topological group (possibly infinite dimensional). Then a fibration G → B over a compact manifold B with typical fibre G is called a bundle of groups with model G if its structure group is contained in Aut(G), the group of automorphisms of G. The main example of interest here is the bundle of smoothing groups, with fibre G −∞ (Z b ) on the fibres of a fibration (6). In this case the group is smooth and the bundle inherits a smooth structure. Definition 1.2. Let φ : G −→ B be a bundle of groups with model G, then a (right) principal G-bundle is a smooth fibration π : P −→ B with typical fibre G together with a continuous (or smooth) fibrewise group action h : Pb × Gb ( p, g) −→ p · g −1 ∈ Pb which is continuous (or smooth) in all variables, locally trivial and free and transitive on the fibres. An isomorphism of principal G-bundles is an isomorphism of the total spaces which intertwines the group actions. The fibre actions combine to give a continuous map from the fibre product P × B G = {( p, g) ∈ P × G; π( p) = φ(g)} −→ P. Definition 1.2 is a generalization of the usual notion of a principal bundle for a group G in the sense that a principal G-bundle π : P −→ B is naturally a principal G-bundle for the trivial bundle of groups G = G × B → B. Any bundle of groups G → B is itself a principal G-bundle and should be thought of as the trivial principal G-bundle. Thus a principal G-bundle P → B is trivial, as a principal G-bundle, if it is isomorphic as a principal G-bundle to G. 1.2. Classifying principal bundles. Lemma 1.3. If G has a topological classifying sequence of groups G −→ E G −→ B G
(1.1)
(so E G is weakly contractible) which is a Serre fibration, G is a bundle of groups modelled on G with structure group H ⊂ Aut(E G, G), the group of automorphisms of E G restricting to automorphisms of G, then, principal G-bundles over compact bases are classified up to G-isomorphism by homotopy classes of global sections of a bundle G(B G) of groups with typical fibre B G.
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Proof. The assumption that the structure group of G is a subgroup of Aut(E G, G) allows the bundle of groups G to be extended to a bundle of groups with model E G. Namely taking an open cover of X by sets over which G is trivial, the fibres may be extended to E G, the transition maps then extend to the larger fibres and the cocycle condition continues to hold. Denote the resulting bundle of groups, G(E G) ⊃ G, with typical fibre E G. The quotient bundle G(B G) = G(E G)/G is a bundle of groups with typical fibre B G and structure group Aut(E G, G) acting on B G. Similarly, any (right) principal G-bundle, P, has an extension to a principal G(E G)bundle, P(E G), P(E G)x = Px × G(E G)x /Gx , ( p, e) ≡ ( pg −1 , eg −1 ). Since the group E G is, by hypothesis, weakly contractible, and the base is compact, the extended bundle P(E G) has a continuous global section. As in the case of a traditional principal bundle, the quotient of this section by the fibrewise action of G gives a section of G(B G). Since all sections of a bundle with contractible fibre are homotopic, the section of G(B G) is well-defined up to homotopy. Bundles isomorphic as principal G bundles give homotopic sections and the construction can be reversed as in the standard case. Namely, given a continuous section u : B −→ G(B G) we may choose a ‘good’ open cover, {Ui } of B, so that each of the open sets is contractible and G is trivial over them. By assumption, the sequence (1.1) is a Serre fibration, and the fibre is weakly contractible, so it follows that u lifts to a global section u˜ : B −→ G(E G). The subbundle, given by the fibres G ⊂ E G in local trivializations, is well-defined and patches to a principal G bundle from which the given section can be recovered.
1.3. Associated bundles. As in the usual case there is a notion of a vector bundle associated to a principal G-bundle. Suppose given a fixed (real or complex) vector space V and a smooth bundle map r : G × V → B × V which is a family of representations, rb : Gb × V → V of the Gb . Then, from P and r, one can form the associated vector bundle P ×r V with fibre (P ×r V )b = Pb × V / ∼b , where ∼b is the equivalence relation ( pg, rb (g −1 , v)) ∼b ( p, v).
1.4. Det(P). Consider again the fibration of closed manifolds (6) and let D ∈ m (M/B; E), D : C ∞ (M; E + ) → C ∞ (M; E − )
(1.2)
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be a family of elliptic operators parameterized by the base B. Then G −∞ (M/B; E + ) B with fibres
G −∞ (Z b ; E b+ ) = Id +Q; Q ∈ −∞ (Z b ; E + (b)), Id +Q b is invertible
is a bundle of groups, with model G −∞ . To the family D we associate the bundle G −∞
P(D)
(1.3)
B of invertible perturbations of D by smoothing operators where the fibre at b is Pb (D) = Db + Q b ; Q b ∈ −∞ (Z b ; E + , E − ), Db + Q b is invertible . The assumption that the numerical index vanishes implies that Pb (D) is non-empty. In fact, for each b ∈ B, the group G −∞ (Z b ; E + (b)) acts freely and transitively on the right on Pb (D) to give P(D) the structure of a principal G −∞ (M/B; E + )-bundle. On the other hand, the Fredholm determinant gives a smooth map det : G −∞ −→ C∗ ∼ = GL(1, C) which restricts to a representation in each fibre. Thus the construction above gives a line bundle associated to the principal bundle (1.3); for the moment we denote it Det(P). 1.5. Quillen’s definition. Proposition 1.4. For an elliptic family of pseudodifferential operators of order m > 0 with vanishing numerical index, the determinant line bundle of Quillen, Det(D), is naturally isomorphic to the line bundle, Det(P), associated to the bundle (1.3) and the determinant as a representation of the structure group. Proof. First we recall Quillen’s definition (following Bismut and Freed [3]). Since it extends readily we consider a pseudodifferential version rather than the original context of Dirac operators. So, for a fibration as in (6), let D be the smooth family of elliptic pseudodifferential operators of (1.2). We also set E ± (b) = E Z , Eb± = C ∞ (Z b , E ± (b)) b and consider the infinite dimensional bundles E ± over B. By assumption, Db has vanishing numerical index. Choosing inner products on E ± and a positive smooth density on the fibres of M allows the adjoint D ∗ of D to be defined. Then, for each b ∈ B, Db∗ Db : Eb+ −→ Eb+ and Db Db∗ : Eb− −→ Eb− have a discrete spectrum with nonnegative eigenvalues. They have the same positive eigenvalues with Db an isomorphism of the corresponding eigenspaces. Given λ > 0, the sets Uλ = b ∈ B; λ is not an eigenvalue of Db∗ Db
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− + are open and H[0,λ) ⊂ E + and H[0,λ) ⊂ E − , respectively spanned by the eigenfunctions ∗ ∗ of Db Db and of Db Db with eigenvalues less than λ, are bundles over Uλ of the same − + ⊕ H[0,λ) is a superbundle to which we dimension, k = k(λ). Now, H[0,λ) = H[0,λ) associate the local determinant bundle − + Det(H[0,λ) ) = (∧k H[0,λ) )−1 ⊗ (∧k H[0,λ) ). − + A linear map P : H[0,λ) → H[0,λ) induces a section − + det(P) = ∧m P : ∧m H[0,λ) −→ ∧m H[0,λ)
(1.4)
of Det(H[0,λ) ). + For 0 < λ < µ, H[0,µ) = H[0,λ) ⊕ H(λ,µ) over Uλ ∩ Uµ , where H(λ,µ) = H(λ,µ) ⊕ − − + H(λ,µ) and H(λ,µ) and H(λ,µ) are respectively the local vector bundles spanned by the eigenfunctions of Db∗ Db and Db Db∗ with associated eigenvalues between λ and µ. Thus, + if D(λ,µ) denotes the restriction of D to H(λ,µ) , then (1.4) leads to transition maps φλ,µ : Det(H[0,λ) ) s −→ s ⊗ det(D(λ,µ) ) ∈ Det(H[0,µ) ) over Uλ ∩ Uµ . The cocycle conditions hold over triple intersections and the resulting bundle, which is independent of choices made (up to natural isomorphism), is Quillen’s determinant bundle, Det(D). Let Q b ∈ −∞ (Z b ; E + , E − ), for b ∈ U ⊂ B open, be a smooth family of perturbations such that Db + Q b is invertible; it therefore gives a section of P over U. The associated bundle Det(P) is then also trivial over U with U b −→ (Db + Q b , 1) being a non-vanishing section. For λ > 0, let P[0,λ) be the projection onto H[0,λ) , and − − + + denote by P[0,λ) and P[0,λ) the projections onto H[0,λ) and H[0,λ) respectively. Then, on − + U ∩ Uλ for λ large enough, P[0,λ) (Db + Q b )P[0,λ) is invertible, and one can associate to the section Db + Q b of P the isomorphism FU ,λ : Det(P) [(Db + Q b , c)] −→
− + det(P[0,λ) (Db + Q b )P[0,λ) ) det(A(Q b , λ))c ∈ Det(D), (1.5)
− + ) is defined by (1.4), where det(P[0,λ) (Db + Q b )P[0,λ) − + Q b P[0,λ) )−1 (Db + Q b ) ∈ G −∞ A(Q b , λ) = (Db + P[0,λ) b ,
(1.6)
and det(A(Q b , λ)) ∈ C∗ is the determinant defined on G −∞ b . The map FU ,λ induces a global isomorphism of the two notions of determinant bundle since it is independent of choices. Indeed, it is compatible with the equivalence relation − + ∼b in the sense that for each g ∈ G −∞ (Z b ; E + ) such that both P[0,λ) (Db + Q b )P[0,λ) − + and P[0,λ) (Db + Q b )g P[0,λ) are invertible, FU ,λ ((D + Q b )g, det(g −1 )c) = FU ,λ ((D + Q b ), c).
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It is also compatible with increase of λ to µ in that φλ,µ ◦ FU ,λ = FU ,µ on U ∩ Uλ ∩ Uµ . This is readily checked − + (Db + Q b )P[0,λ) ) det(A(Q b , λ))c] φλ,µ ◦ FU ,λ (Db + Q b , c) = φλ,µ [det(P[0,λ)
− + + = det(A(Q b , λ)) det(P[0,λ) (Db + Q b )P[0,λ) )⊗ det(D(λ,µ) )c − − + + = det(A(Q b , λ)) det(P[0,µ) (Db + P[0,λ) Q b P[0,λ) )P[0,µ) )c
− + (Db + Q b )P[0,µ) )× = det(A(Q b , λ)) det(P[0,µ)
− − + + det((Db + P[0,µ) Q b P[0,µ) )−1 (Db + P[0,λ) Q b P[0,λ) ))c
− + (Db + Q b )P[0,µ) )c = det(A(Q b , µ)) det(P[0,µ)
= FU ,µ (Db + Q b , c).
(1.7)
1.6. Metric on Det(P). The Quillen metric has a rather direct expression in terms of the definition of the determinant bundle as Det(P). Namely, if (Db + Q b ) is a section of P over the open set U ⊂ B, then
1 |(Db + Q b , 1)| Q = exp − ζb (0) , (1.8) 2 where ζb is the ζ -function associated to the self-adjoint positive elliptic operator (Db + Q b )∗ (Db + Q b ) as constructed by Seeley [19]. When Ab = Id +Rb ∈ G −∞ with b + + Rb : H[0,λ) → H[0,λ) for some λ > 0, Proposition 9.36 of [2], adapted to this context, shows that |(Db + Q b )Ab | Q = | det(Ab )| |Db + Q b | Q ,
(1.9)
but then by continuity the same formula follows in general. Moreover, in the form (1.8), Quillen’s metric generalizes immediately to the case of an arbitrary family of elliptic pseudodifferential operators with vanishing numerical index.
1.7. Primitivity. Lemma 1.5. The determinant bundle is ‘primitive’ in the sense that there is a natural isomorphism Det(P Q) Det(P) ⊗ Det(Q)
(1.10)
for any elliptic families Q ∈ m (M/B; E, F), P ∈ m (M/B; F, G) of vanishing numerical index. Proof. Let P and Q denote the principal bundles of invertible smoothing perturbations of P and Q. Let Pb + Rb and Q b + Sb be local smooth sections over some open set U. Certainly L b = (Pb + Rb )(Q b + Sb ) is a local section of the principal bundle for P Q and (L b , 1) as a local section of Det(P Q) may be identified with the product of the sections
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(Pb + Rb , 1) and (Q b , Sb , 1) as a section of Det(P) ⊗ Det(Q). Changing the section of P to (Pb + Rb )gb modifies the section L b to L b gb , gb = (Q b + Sb )−1 gb (Q b + Sb ). Since det(gb ) = det(gb ), the identification is independent of choices of sections and hence is global and natural. Later, it will be convenient to restrict attention to first order elliptic operators. This is not a strong restriction since for k ∈ Z, let D ∈ k (M/B; E, F) be a smooth family of elliptic pseudodifferential operators with vanishing numerical index. Let M/B ∈ 2 (M/B; F) be an associated family of Laplacians, so that M/B + Id is a family of invertible operators. k−1
Corollary 1.6. The family D = ( M/B + Id)− 2 D ∈ 1 (M/B; E, F) has determinant bundle isomorphic to the determinant bundle of D. 2. Classes of Pseudodifferential Operators Since several different types, and in particular combinations of types, of pseudodifferential operators are used here it seems appropriate to quickly review the essentials. 2.1. m (X ; E, F). On a compact manifold without boundary the ‘traditional’ algebra (so consisting of ‘classical’ operators) may be defined in two steps using a quantization map. The smoothing operators acting between two bundles E and F may be identified as the space −∞ (X ; E, F) = C ∞ (X 2 ; Hom(E, F) ⊗ R ).
(2.1)
Here Hom(E, F)x,x = E x ⊗ Fx is the ‘big’ homomorphism bundle and = π R∗ is the lift of the density bundle from the right factor under the projection π R : X 2 −→ X. The space m (X ; E, F) may be identified with the conormal sections, with respect to the diagonal, of the same bundle m (X ; E, F) = Iclm (X 2 , Diag; Hom(E, F) ⊗ R ).
(2.2)
More explicitly Weyl quantization, given by the inverse fibre Fourier transform from T ∗ X to T X, qg : ρ −m C ∞ (T ∗ X ; π ∗ hom(E, F)) a −→ −n (2π ) χ exp (iv(x, y) · ξ )a(m(x, y), ξ )dξ dg y ∈ m (X ; E, F) T∗X
(2.3)
is surjective modulo −∞ (X ; E, F). Here a Riemann metric, g, is chosen on X and used to determine a small geodesically convex neighbourhood U of the diagonal in X 2 which is identified as a neighbourhood U of the zero section in T X by mapping (x, y) ∈ U to m(x, y), the mid-point of the geodesic joining them in X and to v(x, y) ∈ Tm(x,y) X, the tangent vector to the geodesic at that mid-point in terms of the length parameterization of the geodesic from y to x. The cut-off χ ∈ Cc∞ (U ) is taken to be identically equal
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to 1 in a smaller neighbourhood of the diagonal. Connections on E and F are chosen and used to identify Hom(E, F) over U with the lift of hom(E, F) to U , dξ is the fibre density from g on T ∗ X and dg y is the Riemannian density on the right (in the y variable). The symbol a is a classical symbol of order k on T ∗ X realized as ρ −k a , where a ∈ C ∞ (T ∗ X ) with T ∗ X the compact manifold with boundary arising from the radial compactification of the fibres of T ∗ X and ρg = |ξ |−1 g outside a compact set in T ∗ X is a boundary defining function for that compactification. Then qg (a) ∈ −∞ (X ; E, F) if and only if a ∈ C˙∞ (T ∗ X ) is a smooth function vanishing to all orders on the boundary of T ∗ X , i.e. is a symbol of order −∞. This leads to the short exact ‘full symbol sequence’ σg
−∞ (X ; E, F) −→ ∞ (X ; E, F) −→ C ∞ (S ∗ X ; hom(E, F))[[ρ, ρ −1 ]] (2.4) with values in the Laurent series in ρ (i.e. formal power series in ρ with finite factors of ρ −1 ). The leading part of this is the principal symbol σm
m−1 (X ; E, F) −→ m (X ; E, F) −→ C ∞ (S ∗ X ; hom(E, F) ⊗ Rm ),
(2.5)
where Rm is the trivial bundle with sections which are homogeneous of degree m over T ∗ X \0. Pseudodifferential operators act from C ∞ (X ; E) to C ∞ (X ; F) and composition gives a filtered product,
m (X ; F, G) ◦ m (X ; E, F) ⊂ m+m (X ; E, G),
(2.6)
which induces a star product on the image spaces in (2.4), a g b = ab +
∞
B j (a, b),
(2.7)
j=1
where the B j are smooth bilinear differential operators with polynomial coefficients on T ∗ X lowering total order, in terms of power series, by j. The leading part gives the multiplicativity of the principal symbol. m m 2.2. sus( p) (X ; E, F). There is a natural Fréchet topology on (X ; E, F), corres∞ ponding to the C topology on the symbol and the kernel away from the diagonal. Thus, smoothness of maps into this space is well-defined. The p-fold suspended operators are a subspace p m m ∞ sus( (2.8) R ; (X ; E, F) p) (X ; E, F) ⊂ C
in which the parameter-dependence is symbolic (and classical). In terms of the identification (2.2) this reduces to m sus( p) (X ; E, F) = p M 2 p FR−1p Icl, (X × R , Diag ×{0}; Hom(E, F) ⊗ ) , M = m + . (2.9) R S 4
Here we consider conormal distributions on the non-compact space X 2 × R p but with respect to the compact submanifold Diag ×{0}; the suffix S denotes that they are to be
Periodicity
155
Schwartz at infinity and then the inverse Fourier transform is taken in the Euclidean variables R p giving the ‘symbolic’ parameters. The shift of m to M is purely notational. These kernels can also be expressed directly as in (2.3) with a replaced by (2.10) a ∈ ρ −m T ∗ X × R p ; π ∗ hom(E, F) . Composition, mapping and symbolic properties are completely analogous to the ‘unsuspended’ case. Note that we use the abbreviated notation for suffixes sus(1) = sus . If D is a first order elliptic differential operator acting on a bundle on X then D +iτ ∈ 1 (X ; E) is elliptic in this sense and invertible, with inverse in −1 (X ; E), if D is sus sus self-adjoint and invertible. However this is not the case for general (elliptic self-adjoint) D ∈ 1 (X ; E); we therefore introduce larger spaces which will capture these operators and their inverses.
m,m 2.3. psus( p) (X ; E, F). By definition in (2.9), before the inverse Fourier transform is taken, the singularities of the ‘kernel’ are constrained to Diag ×{0} ⊂ X 2 × R p . For product-type (really partially-product-type corresponding to the fibration of X × R p with base R p ) the singularities are allowed to fill out the larger submanifold
X 2 × {0} ⊃ Diag ×{0}.
(2.11)
Of course they are not permitted to have arbitrary singularities but rather to be conormal with respect to these two, nested, submanifolds
m,m psus( p) (X ; E, F) = M M 2 p 2 (X × R , X × {0}, Diag ×{0}; Hom(E, F) ⊗ ) , FR−1p Icl, R S
M =m+
p p n , M = m + − . 4 4 2
(2.12)
The space of classical product-type pseudodifferential operators is discussed succinctly in an appendix below. Away from Diag ×{0} the elements of the space on the right are just classical conormal distributions at {0} × R p , so if χ ∈ C ∞ (X 2 ) vanishes near the diagonal (or even just to infinite order on it)
m,m −m ∞ C (R p × X 2 ; Hom(E, F) ⊗ R ) K ∈ psus( p) (X ; E, F) =⇒ χ K ∈ ρ
(2.13)
is just a classical symbol in the parameters depending smoothly on the variables in X 2 . Conversely, if χ ∈ C ∞ (X 2 ) has support sufficiently near the diagonal then the kernel is given by a formula as in (2.3), χ exp (iv(x, y) · ξ )a(m(x, y), ξ, τ )dξ dg y , χ K = (2π )−n −m
a ∈ (ρ )
T∗X −m ∞
(ρ )
C (S; π ∗ hom(E, F)), S = [T ∗ X × R p ; 0T ∗ X × ∂R p ]. (2.14)
Here the space on which the ‘symbols’ are smooth functions (apart from the weight factors) is the same compactification as in (2.10) but then blown up (in the sense of [11]) at the part of the boundary (i.e. infinity) corresponding to finite points in the cotangent bundle. Then ρ is a defining function for the ‘old’ part of the boundary and ρ for the
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new part, produced by the blow-up. Conversely (2.14) and (2.13) together (for a partition of unity) define the space of kernels. From the general properties of blow-up, if ρ ∈ C ∞ (T ∗ X × R p ) is a defining function for the boundary then ρ = ρ ρ after blow-up. From this it follows easily that m,m m sus( p) (X ; E, F) ⊂ psus( p) (X ; E, F).
(2.15)
Again these ‘product suspended’ operators act from S(X × R p ; E) to S(X × R p ; F) and have a doubly-filtered composition m 1 ,m
m 2 ,m
m 1 +m 2 ,m 1 +m 2
1 2 psus( p) (X ; F, G) ◦ psus( p) (X ; E, F) ⊂ psus( p)
(X ; E, G).
(2.16)
The symbol map remains, but now only corresponds to the part of the amplitude in (2.14) at ρ = 0,
σm
m−1,m m,m m,m psus( p) (X ; E, F) −→ psus( p) (X ; E, F) −→ Spsus( p) (X ; E, F)
(2.17)
with
m,m (X ; E, F) = C ∞ ([S(T ∗ × R p ); 0 × S p−1 ]; hom(E, F) ⊗ Rm,m ) Spsus(d)
the space of smooth sections of a bundle over the sphere bundle corresponding to T ∗ X × R p , blown up at the image of the zero section and with Rm,m a trivial bundle capturing the weight factors. The other part of the amplitude corresponds to a more global ‘symbol map’ called here the ‘base family’,
βm
m,m −1 m,m ∞ p−1 psus( ; m (X ; E, F) ⊗ Rm ), p) (X ; E, F) −→ psus( p) (X ; E, F) −→ C (S
(2.18) taking values in pseudodifferential operators on X depending smoothly on the parameters ‘at infinity’, i.e. in S p−1 with the appropriate homogeneity bundle (over S p−1 ). These two symbol maps are separately surjective and jointly surjective onto pairs satisfying the natural compatibility condition σm (βm (A)) = σm (A)∂
(2.19)
that the symbol family, restricted to the boundary of the space on which it is defined, is the symbol family of the base family. An operator in this product-suspended class is ‘fully elliptic’ if both its symbol and its base family are invertible. If it is also invertible then its inverse is in the corresponding space with opposite orders. An elliptic suspended operator is automatically fully elliptic when considered as a product-suspended operator using (2.15).
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m 2.4. iso(2n,) (Rn ). The suspension variables for these product-suspended operators are purely parameters. However, for the adiabatic limit constructions here, on which the paper relies heavily, we use products which are non-local in the parameters. In the trivial case of X = {pt} we are dealing just with symbols above and the corresponding non-commutative product is the ‘isotropic’ algebra of operators on symbols on R2n , as operators on Rn , for any n. This is variously known as the Weyl algebra or the Moyal product (although both often are taken to mean slightly different things). The isotropic pseudodifferential operators of order k act on the Schwartz space S(Rn ) and, using Euclidean Weyl quantization, may be identified with the spaces ρ −k C ∞ (R2n ). Thus, in terms of their distributional kernels on R2n , this space of operators is given by essentially the same formula as (2.3),
qW : ρ −k C ∞ (R2n ) b −→ qW (b)(t, t ) = (2π )−n
Rn
ei(t−t )·τ b
t + t −k , τ dτ ∈ iso (Rn ). 2
(2.20)
This map is discussed extensively in [10]. In this case qW , with inverse σW , is an isomorphism onto the algebra and restricts to an isomorphism of the ‘residual’ algebra −∞ iso (Rn ) = qW (S(R2n )). The corresponding star product is the Moyal product. The full product on symbols on R2n may be written explicitly as a ◦ω b(ζ ) = π −2n eiξ ·ξ +iη·η +2iω(ξ ,η ) a(ζ + ξ )b(ζ + η)|ωξ |n |ωη |n , (2.21) R8n
where the integrals are not strictly convergent but are well defined as oscillatory integrals. Here ω is the standard symplectic form on R2n . By simply using linear changes of variables, it may be seen that this product and the more general ones in which ω is replaced by an arbitrary non-degenerate antisymmetric bilinear form on R2n are all isomorphic. In fact the product depends smoothly on ω as an antisymmetric bilinear form, even as it becomes degenerate. When ω ≡ 0 the product reduces to the pointwise, commutative, product of symbols. In fact it is not necessary to assume that the underlying Euclidean space is even dimensional for this to be true; of course in the odd-dimensional case the form cannot be non-degenerate and correspondingly there is always at least one ‘commutative’ variable. The adiabatic limit here corresponds to replacing the standard symplectic form ω by ω and allowing ↓ 0. As already noted, this gives a family of products on the classical symbol spaces which is smooth in and is the commutative product at = 0. We denote m the resulting smooth family of algebras by iso(2n,) (Rn ).
m,m 2.5. iso(2n,) (X ; E, F). Now, we may replace the parameterized product on the productsuspended algebra by ‘quantizing it’ as in (2.21), in addition to the composition in X itself. For the ‘adiabatic’ choice of ω this induces a one parameter family of quantized products m 1 ,m
m 2 ,m
m 1 +m 2 ,m 1 +m 2
1 2 [0, 1] × psus(2n) (X ; F, G) × psus(2n) (X ; E, F) −→ psus(2n)
(X ; E, G). (2.22)
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The suspended operators still form a subalgebra. The Taylor series as ↓ 0 is given by ∞ (i)k ω(D , D )A(v)B(w) . (2.23) (A ◦ B)(u) ∼ v w k 2 k! v=w=u k=0
A more complete discussion of product suspended operators and the mixed isotropic product may be found in the appendix.
m,m 2.6. psus(2n) (X ; E, F)[[]]. This is the space of formal power series in with coeffi
m,m (X ; E, F). The product (2.23) projects to induce a product cients in psus(2n) m 1 ,m
m 2 ,m
m 1 +m 2 ,m 1 +m 2
1 2 (X ; F, G)[[]] × psus(2n) (X ; E, F)[[]] −→ psus(2n) psus(2n)
(X ; E, G)[[]] (2.24)
which is consistent with the action on formal power series
m,m psus(2n) (X ; E, F)[[]] A : C ∞ (X ; E)[[]] −→ C ∞ (X ; F)[[]].
3. Adiabatic Determinant Let E −→ X be a complex vector bundle over a compact manifold X. Consider the infinite dimensional group −∞ G −∞ sus(2n) (X ; E) = {Id +Q; Q ∈ sus(2n) (X ; E), Id +Q is invertible}
of invertible 2n-suspended smoothing perturbations of the identity. A naive notion of determinant would be given by using the 1-form d log d(A) = Tr sus(2n) (A−1 d A), where Tr sus(2n) (B) =
1 Tr(B(t, τ ))dtdτ (2π )2n R2n
is the regularized trace for suspended operators as defined in [13]. The putative determinant is then given by
1
dγ d(A) = exp Tr sus(2n) γ −1 ds , (3.1) ds 0 where γ : [0, 1] → G −∞ sus(2n) (X ; E) is any smooth path such that γ (0) = Id and γ (1) = A. Although d(A) is multiplicative, it is topologically trivial, in the sense that for any smooth loop γ : S1 → G −∞ sus(2n) (X ; E), one has
1 −1 dγ −1 dγ ds = ds dt dτ = 0. (3.2) Tr sus(2n) γ Tr γ ds (2π )2n R2n S1 ds S1 So this is not a topological analogue of the usual determinant.
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159
3.1. Isotropic determinant. To obtain a determinant which generates the 1-dimensional cohomology, we instead use the isotropic quantization of § B. At the cost of slightly deforming the composition law on G −∞ sus(2n) (X ; E), this determinant will be multiplicative as well. Notice first that because of the canonical identification −∞,−∞ −∞ psus(2n) (X ; E) = sus(2n) (X ; E)
there is no distinction between G −∞ (X ; E) and the group −∞,−∞ G −∞,−∞ psus(2n) (X ; E) = {Id +Q; Q ∈ psus(2n) (X ; E), Id +Q is invertible},
so in this context, we can interchangeably think in terms of suspended or productsuspended operators. For ∈ [0, 1], we use the ◦ -product of Theorem 5 to define the group 0,0 −∞ G −∞ iso(2n,) (X ; E) = {Id +Q; Q ∈ sus(2n) (X ; E), ∃ P ∈ psus(2n) (X ; E),
P ◦ (Id +Q) = (Id +Q) ◦ P = Id}. (3.3) For = 0, we have the canonical group isomorphism −∞ G −∞ iso(2n,0) (X ; E) = G sus(2n) (X ; E). −∞ so that On the other hand, for > 0, the group G −∞ iso(2n,) (X ; E) is isomorphic to G it is possible to transfer the Fredholm determinant to it.
Proposition 3.1. For > 0, there is a natural multiplicative determinant ∗ det (A) : G −∞ iso(2n,) (X ; E) → C
defined for A ∈ G −∞ iso(2n,) (X ; E) by
det (A) = exp
1 0
dγ ds , Tr γ −1 ◦ ds
where γ : [0, 1] → G −∞ iso(2n,) (X ; E) is any smooth path with γ (0) = Id and γ (1) = A so d log det (A) = Tr (A−1 ◦ d A). Proof. To show that det is well-defined and multiplicative, it suffices to show that it reduces to the Fredholm determinant under a suitable identification of G −∞ iso(2n,) (X ; E) −∞ −∞ with G . From Appendix C it follows that G iso(2n,) (X ; E) acts on S(X × Rn ; E). Fix a Riemannian metric g on X and a Hermitian metric h on E. Let ∈ 2 (X ; E) be the corresponding Laplace operator. Then consider the mixed isotropic operator n ∂2 2 2 = + (X ; E), − 2 + ti ∈ iso(2n,) ∂t i i=1
160
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R. Melrose, F. Rochon
∂2 2 is the harmonic oscillator on Rn . As an operator acting on − + t i=1 i ∂t 2
n
i
S(X × Rn ; E), has a positive discrete spectrum. Let {λk }k∈N be the eigenvalues, in non-decreasing order, with corresponding eigensections f k = λk f k ,
f k ∈ S(X × Rn ; E)
such that { f k }k∈N is an orthonormal basis of L2 (X × Rn ; E). This gives an algebra isomorphism −∞ (X ; E) A −→ f i , A f j L2 ∈ −∞ , F : iso(2n,) −∞ . Under these and a corresponding group isomorphism F : G −∞ iso(2n,) (X ; E) → G isomorphisms, one has
Tr (A) = Tr(F (A)) and consequently det (Id +A) = det Fr (F (Id +A)).
(3.4)
3.2. Asymptotics of det . Now, for any δ > 0 we can consider the group of sections, 2n ∞ −∞ (R2n × X )); G −∞ iso ([0, δ] × R × X ; E) = {A ∈ C ([0, δ]; Id + 2n A() ∈ G −∞ iso, (R × X ; E) ∀ ∈ [0, δ]}. (3.5)
Proposition 3.2. The determinant with respect to the ◦ product defines 2n ∞ d et : G −∞ iso ([0, δ] × R × X ; E) −→ C ((0, δ])
which takes the form n−1 k−n 2n ak (A) F (A) ∀ A ∈ G −∞ d et(A)() = exp iso ([0, δ] × R × X ; E)
(3.6)
(3.7)
k=0
where 2n ∞ F : G −∞ iso ([0, δ] × R × X ; E) A −→ F (A) ∈ C ([0, δ]),
(3.8)
2n ∞ functions and the a only depend and ak : G −∞ k iso ([0, δ] × R × X ; E) −→ C are C on the Taylor series of A.
Proof. Since the group is open (for each ∈ [0, 1] and also for the whole group) the tangent space at any point is simply C ∞ ([0, δ]; −∞ (R2n × X ; E)). With the usual identifications for a Lie group the form A−1 ◦ d A therefore takes values in C ∞ ([0, δ]; −∞ (R2n × X ; E)). On the other hand, the trace functional is not smooth down to = 0. In fact it is rescaled by a factor of −n . Thus, d log det (A) = Tr (A−1 ◦ d A) ∼
∞ k=0
is −n times a smooth function.
αk k−n
(3.9)
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161
For any smooth map 2n f : S1 → G −∞ iso ([0, δ] × R × X ; E),
the integral S1 f ∗ d log det (A) also has an asymptotic expansion S1
f ∗ d log det (A) ∼
∞
ck k−n , ck =
k=0
S1
αk ∈ C.
(3.10)
On the other hand, by (3.4), this is a winding number so cannot depend on . Hence ck = αk = 0 for k = n. (3.11) S1
So, for k = n, αk is exact and then (3.7) follows directly by integration along any path 2n γ : [0, 1] → G −∞ iso ([0, δ] × R × X ; E) with γ (0) = Id and γ (1) = A. The range space is path-connected, so
1
ak (A) =
γ ∗ αk , k < n
0
is independent of the path and well-defined.
3.3. Star product. The restriction map at = 0 −∞ 2n R : G −∞ iso ([0, δ] × R × X ; E) −→ G sus(2n) (X ; E)
(3.12)
2n × X ; E) be the is surjective. From this it follows that if we let G˙ −∞ iso ([0, δ] × R subgroup of those elements which are equal to the identity to infinite order at = 0 then the quotient 2n 2n ˙ −∞ G −∞ iso ([0, δ] × R × X ; E)/G iso ([0, δ] × R × X ; E) = −∞ (R2n × X ; E)[[]] (3.13) G −∞ sus(2n) (X ; E) +
is the obvious formal power series group, namely with invertible leading term and arbitrary smoothing lower order terms. The composition law is the one induced by the ◦ -product. Since the higher order terms in amount to an affine extension of the leading part, this formal power series group is also a classifying group for odd K-theory.
k,k Definition 3.3. We denote by psus(2n) (X ; E)[[ε]], k, k ∈ R ∪ {−∞}, the space of formal series ∞
aµ ε µ
µ=0
k,k with coefficients aµ ∈ psus(2n) (X ; E), where ε is a formal parameter.
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k,k l,l For A ∈ psus(2n) (X ; F, G)[[ε]] and B ∈ psus(2n) (X ; E, F)[[ε]] the ∗-product
k+l,k +l (X ; E, G)[[ε]] is A ∗ B ∈ psus(2n) ⎞ ⎛ ∞ ∞ µ ν aµ ε ⎠ ∗ bν ε A ∗ B(u) = ⎝ µ=0
=
∞ ∞ µ=0 ν=0
⎛
ν=0
⎞ ∞ pε p i ⎠, εµ+ν ⎝ ω(Dv , Dw ) p A(v)B(w) 2 p p! v=w=u p=0
where u, v, w ∈ R2n . Since this is based on the asymptotic expansion (C.3) of Appendix B, its associativity follows immediately from the associativity of the ◦ -product. 3.4. Adiabatic determinant. This product is consistent with that of the quotient group in (3.13), so Lemma 3.4. The quotient group G −∞ sus(2n) (X ; E)[[]] of (3.13) is canonically isomorphic to −∞ −∞ G −∞ sus(2n) (X ; E)[[ε]] = {(Id +Q); Q ∈ sus(2n) (X ; E)[[ε]], ∃ P ∈ sus(2n) (X ; E)[[ε]] 0 (X ; E)[[ε]]}. such that (Id +Q) ∗ (Id +P) = Id ∈ sus(2n)
We can now prove Theorem 1 stated in the Introduction. Theorem 1. The functional 2n ∗ F0 (A) : G −∞ iso ([0, δ] × R × X ; E) → C
induces a multiplicative determinant deta on the formal power series group G −∞ sus(2n) (X ; E)[[ε]] in the sense discussed above, i.e. it is a smooth multiplicative function which generates H1 . Proof. From (3.9),
F0 (A) = exp
γ
αn ,
(3.14)
where γ : [0, 1] → G −∞ sus(2n) (X ; E)[[ε]] is any smooth path with γ (0) = Id and γ (1) = A. In the expansion (3.9), the only non-trivial cohomological contribution comes from αn . Since det corresponds to the Fredholm determinant under the identification of −∞ the integral of α along a generator of the fundamental G −∞ n iso(2n,) (X ; E) with G group is ±2πi. Thus, the determinant induced by F0 (A) has the desired topological behavior.
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For the multiplicativity, from (3.9), Tr((AB)−1 d(AB)) = Tr(B −1 A−1 d AB + B −1 A−1 Ad B) = Tr(A−1 d A) + Tr(B −1 d B),
(3.15)
where the ∗-product is used to compose elements and define the inverses. From the ε-expansion of (3.15), αn (A ∗ B) = αn (A) + αn (B). As a consequence, the determinant defined in (3.1) is multiplicative.
(3.16)
This determinant can be used to define the determinant line bundle of a fully elliptic k,k (M/B; E, F)[[ε]] of fibrewise product 2n-suspended pseudodiffamily D ∈ psus(2n) ferential operators on a fibration (6). Full ellipticity here corresponds to ellipticity of the leading term D0 and its invertibility for large values of the parameters. Assume k,k (Z b ; E b , Fb )[[ε]] can be perturbed in addition that for each b ∈ B, Db ∈ psus(2n) −∞ by Q b ∈ sus(2n) (Z b ; E b , Fb )[[ε]] to be invertible, where invertibility is equivalent to invertibility of the leading term. Then over the manifold B, consider the bundle of invertible smoothing perturbations with fibres −∞ Pb (D) = {Db + Q b ; Q b ∈ sus(2n) (Z b ; E b , Fb )[[ε]], −k,−k ∃ P ∈ psus(2n) (Z b ; Fb , E b )[[ε]], P ∗ (Db + Q b ) = (Db + Q b ) ∗ P = Id}. (3.17)
Let G −∞ sus(2n)
/ G −∞ (M/B; E) sus(2n)
(3.18)
φ
B be the bundle of groups with fibre at b ∈ B,
−∞ G −∞ sus(2n) (Z b ; E b )[[ε]] = {Id +Q; Q ∈ sus(2n) (Z b ; E b )[[ε]], 0,0 ∃ P ∈ psus(2n) (Z b ; E b )[[ε]]P ∗ (Id +Q) = (Id +Q) ∗ P = Id}. (3.19)
Then P(D) is a principal G −∞ sus(2n) (M/B; E)[[ε]]-bundle in the sense of Definition 1.2. Definition 3.5. The adiabatic determinant line bundle associated to the family D of product 2n-suspended elliptic pseudodifferential operators is Deta (D) = P(D) ×deta C induced by the adiabatic determinant as representation of the bundle of groups (3.18). 4. Periodicity of the Numerical Index In the next section, we establish a relation between the determinant line bundles of a family of standard elliptic pseudodifferential operators and the determinant line bundle just defined for families of 2n-suspended operators. Here we consider the corresponding question for the numerical index.
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4.1. Product-suspended index. A product-suspended operator
m,m P ∈ psus(k) (Z ; E, F)
is fully elliptic if both its symbol in the usual sense and its base family are invertible. Here the base family, elliptic because of the invertibility of the symbol, is parameterized by Sk−1 . As a family of operators over Rk , P has a families index. Since by assumption the family is invertible at, and hence near, infinity the family defines an index class in compactly-supported K-theory Z k even 0 k (4.1) ind(P) ∈ K (R ) = {0} k odd. Thus by choosing a generator (i.e. Bott element) in K 0 (R2n ) a product 2n-suspended family has a numerical index which we will denote indsus(n) (since it only arises for even numbers of parameters). The families index of Atiyah and Singer does not apply directly to this setting although it does apply if the operator is in the ‘suspended’ subspace (and so in particular m = m.) Using the properties of the suspended eta invariant we will show in §9 that the suspended index can be expressed in terms of the ‘adiabatic’ η invariant discussed below. Namely, suppose a linear decomposition R2n = R × R2n−1 is chosen in which the variables are written τ and ξ. Then, for some R ∈ R, P(τ, ξ ) is invertible for |τ | ≥ R for all ξ ∈ R2n−1 . Furthermore, by standard index arguments we may find a family of smoothing operators, A, of compact support in (τ, ξ ) such that P = P + A is invertible for all τ ≤ R. Then 1 indsus(n) (P) = − ηa(n-1) (P τ =R ) − ηa(n-1) (P τ =R ) . (4.2) 2 4.2. Periodicity. Here we show that there is a ‘Bott map’ from ordinary pseudodifferential operators into product-type suspended operators which maps the usual index to the suspended index (although most of the proof is postponed until later). Thus if D ∈ 1 (Z ; E, F) is an elliptic operator then
it − τ D∗ 1,1 2 ˆ , Dˆ ∈ psus(2) (Z ; E ⊕ F) (4.3) R (t, τ ) −→ D(t, τ ) = D it + τ is an associated twice-suspended fully elliptic operator. In [14], such a family is realized explicitly as the indicial family of a product-suspended cusp operator. The ellipticity of Dˆ follows from the fact that
∗ 0 D D + t2 + τ 2 2,2 ∈ psus(2) (Z ; E ⊕ F) (4.4) Dˆ ∗ Dˆ = 0 D D∗ + t 2 + τ 2 is an elliptic family which is invertible for t 2 + τ 2 > 0. Definition 4.1. Given an elliptic operator D ∈ 1 (Z ; E, F), we define by recurrence 1,1 on n ∈ N0 , elliptic product-suspended operators Dn ∈ psus(2n) (Z ; 2n−1 (E ⊕ F) by
∗ Dn−1 itn − τn Dn (t1 , . . . , tn , τ1 , . . . , τn ) = Dn−1 itn + τn with D0 = D.
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Lemma 4.2. If D is elliptic then Dn is a totally elliptic product 2n-suspended operator for all n and indsus(n) (Dn ) = ind(D). Proof. Both the ordinary index and the n-suspended index (on fully elliptic 2n-suspended operators) are homotopy invariant. Since the map D −→ Dn maps invertible operators to invertible operators it follows that ind(D) = 0 implies indsus(n) (Dn ) = 0. Indeed, ind(D) = 0 means there exists a smoothing operator Q ∈ −∞ (Z ; E, F) such that D + Q is invertible. Then (D + s Q)n is a homotopy of fully elliptic 2n-suspended operators which is invertible for s = 1 so indsus(n) (Dn ) = 0. The actual equality of the index is proved below in §9, using (4.2). 5. Periodicity of the Determinant Line Bundle 5.1. Adiabatic determinant bundle. Returning to the setting of a fibration with compact fibres, φ : M → B, as in (6), let D ∈ 1 (M/B; E, F) be a family of elliptic pseudodifferential operators with vanishing numerical index. From Lemma 4.2 (the part that is already proved), the suspended index of the fully elliptic family Dn ∈ 1,1 psus(2n) (M/B; E, F), given by Definition 4.1, also vanishes. Thus the fibres −∞ Ppsus(2n) (Dn )b = {Dn,b + Q b ; Q b ∈ sus(2n) (Z b ; 2n−1 (E b ⊕ Fb ))[[ε]], −1 ∃ ( Dˆ n,b + Q b )−1 ∈ psus(2n) (Z b ; 2n−1 (E b ⊕ Fb ))[[ε]]} (5.1) n−1 (E ⊕ F))[[ε]]are non-empty and combine to give a principal-G −∞ sus(2n) (M/B; 2 bundle as in (3.17). Since we have defined an adiabatic determinant on these groups we have an associated determinant bundle
Det sus(2n) (D) = Deta (Dn ) = Ppsus 2n (D) ×deta C.
(5.2)
5.2. Isotropic determinant bundle. One can make a different, but similar, construction using the isotropic quantization of Dn . 1,1 Definition 5.1. For > 0, let Dˆ n ∈ iso(2n,) (M/B; 2n−1 (E ⊕ F)) be the isotropic quantization of Dn as in Appendix C, so giving an operator on S(Rn ×X ; 2n−1 (E b ⊕Fb )).
As discussed earlier for families of standard elliptic operators, there are two equivalent definitions of the determinant line bundle for Dˆ n . Namely, Quillen’s spectral definition or as an associated bundle to the principal bundle of invertible perturbations. In the latter n−1 (E ⊕ F))[[ε]]-bundle has fibre case, the principal G −∞ iso(2n,) (M/B; 2 −∞ (Z b ; 2n−1 (E b ⊕ Fb )), Piso(2n,) ( Dˆ en )b = { Dˆ 2n,b + Q b ; Q b ∈ iso(2n,) −1,−1 ∃ ( Dˆ 2n,b + Q b )−1 ∈ iso(2n,) (Z b ; 2n−1 (E b ⊕ Fb ))}. (5.3)
Note that this fibre is non-empty as soon as the original family D has vanishing numerical index. Indeed, we know that Dn then has vanishing suspended index and hence has an invertible perturbation by a smoothing operator (in the suspended sense). The isotropic product is smooth down to = 0, where it reduces to the suspended product (pointwise in the parameters). Thus such a perturbation is invertible with respect to the isotropic product for small > 0. Since these products are all isomorphic for > 0, it follows that perturbations as required in (5.3) do exist.
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Proposition 5.2. Let D ∈ 1 (M/B; E, F) be an elliptic family with vanishing numerical index, then for each n ∈ N0 and > 0, the determinant line bundle Det( Dˆ n ) is naturally isomorphic to the determinant line bundle Det(D). Proof. The proof is by induction on n ∈ N0 starting with the trivial case n = 0. We proceed to show that Det( Dˆ n+1 ) ∼ = Det( Dˆ n ). In Quillen’s definition of the determinant line bundle, only the eigenfunctions of the low eigenvalues are involved and the strategy is to identify the eigensections of the low eigenvalues of Dˆ n with those of Dˆ n+1 . The isotropic quantization of the polynomial τn2 + tn2 , is the harmonic oscillator, Hn , so D ˆ ∗ Dˆ n,b + H − 0 ˆ∗ ˆ n+1 n,b Dn+1,b Dn+1,b = D ˆ n,b Dˆ ∗ + H + . (5.4) 0 n+1 n,b are positive, with the smallest being simple. The eigensections The eigenvalues of Hn+1 ∗ ∗ with small eigenvalues are of the form of Dˆ n+1,b Dˆ n+1,b and Dˆ n+1,b Dˆ n+1,b
0 ϕn+1 ⊗ f + ( f b ) = , − ( f b ) = , (5.5) 0 ϕn+1 ⊗ Dˆ n,b f ∗ D ˆ n,b with eigenvalue less than 2. Note also that where f is an eigenfunction of Dˆ n,b ˆ on such an eigenfunction, Dn+1,b acts as ∗ D ˆ∗ iCn+1 n,b + ( f ) = − ( f ) (5.6) b b D ˆ n,b iCn+1 ∗ ϕ since Cn+1 n+1 = 0. For 0 < λ < 2, consider the open set
Uλ = {b ∈ B; λ is not an eigenvalue of Db∗ Db }.
(5.7)
+,k ∗ D ˆ k,b Let H[0,λ) denote the vector bundle over Uλ spanned by the eigenfunctions of Dˆ k,b
−,k with eigenvalues less than λ. Let H[0,λ) denote the vector bundle over Uλ spanned by ∗ the eigenfunctions of Dˆ k,b Dˆ k,b with eigenvalues less than λ. Then there are natural identifications ±,n ±,n+1 ± FU±,n ,λ : H[0,λ) f b −→ ( f b ) ∈ H[0,λ) .
(5.8)
Thus, directly from Quillen’s definition of the determinant bundle Dˆ n+1,b and Dˆ n,b have isomorphic determinant line bundles. 5.3. Adiabatic limit of Det( Dˆ n ). m (M/B; E, F) is a family of fully elliptic operators Proposition 5.3. If P ∈ psus(2n) with vanishing numerical index then the bundle over B × [0, 1] with fibre −∞ Pb, = Q ∈ sus(2n) (Z b , E b ); −m −1 (Z b ; Fb , E b ) for the isotropic product (5.9) ∃ (P + Q) ∈ psus(2n)
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is a principal G-bundle for the bundle of groups with fibre −∞ −∞ (Z b , E b ) = Id +A, A ∈ sus(2n) (Z b , E b ); Gsus(2n), −∞ (Z b , E b ) for the isotropic product ∃ (Id +A)−1 = Id +B, B ∈ sus(2n)
(5.10)
and the associated determinant bundle defined over > 0 extends smoothly down to = 0 and at = 0 is induced by the adiabatic determinant. Proof. This is just the smoothness of the ‘rescaled’ determinant (i.e. with the singular terms removed) down to = 0. We will now complete the proof of Theorem 2 in the Introduction which we slightly restate as Theorem 2 (Periodicity of the determinant line bundle). Let D ∈ 1 (M/B; E, F) be an elliptic family with vanishing numerical index, then for n ∈ N and > 0, Deta (Dn ) ∼ = Det( Dˆ n ) ∼ = Det(D). Proof. The existence of the second isomorphism follows from Proposition 5.2. The first follows from Proposition 5.3. 6. Eta Invariant In [13] a form of the eta invariant was discussed for elliptic and invertible once-suspended families of pseudodifferential operators. Applied to the spectral family (on the imaginary axis) of a self-adjoint invertible Dirac operator this new definition was shown to reduce to the original definition, of Atiyah, Patodi and Singer in [1] of the eta invariant of a single operator. Here, the definition in [13] is shown to extend to (fully) elliptic, invertible, product-suspended families. In §9 it is further extended to such productsuspended families in any odd number of variables. The extension to single-parameter product-suspended operators allows us to apply the definition to A + iτ, τ ∈ R, for A ∈ 1 (X ; E) an invertible elliptic selfadjoint pseudodifferential operator and check that this reduces to the spectral definition, now as given by Wodzicki ([21]). Again the extended (and below also the ‘adiabatic’) eta invariant gives a log-multiplicative function for invertible families η(AB) = η(A) + η(B)
(6.1)
and this allows us to show quite directly that the associated τ invariant is a determinant in the sense discussed above.
m,m (X ; E) is a product-suspended family it 6.1. Product-suspended eta. If B ∈ psus satisfies
∂N m−N ,m −N B(τ ) ∈ psus (X ; E) ∀ N ∈ N0 . ∂τ N
(6.2)
This implies that for N large, say N > dim X + m, the differentiated family takes values in operators of trace class on L 2 .
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Proposition 6.1. For any m, m ∈ Z, if N ∈ N is chosen sufficiently large then,
N ∂ m,m (X ; E) =⇒ Tr E B(τ ) ∈ C ∞ (R p ) B ∈ psus ∂τ N
(6.3)
has a complete asymptotic expansion (possibly with logarithms) as τ → ±∞ and the coefficient of T 0 in the expansion as T → ∞ FB,N (T ) =
T −T
Tr(B) = LIM FB,N (T ), T →∞
N tN t1 ∂ ... Tr E B(s) ds dt N . . . dt1 ∂s N 0 0
(6.4)
is independent of the choice of N and defines a trace functional Z,Z Z,Z Tr : psus (X ; E) −→ C, Tr([A, B]) = 0 ∀ A, B ∈ psus (X ; E)
which reduces to
(6.5)
−∞,−∞ Tr E (B(τ )) dτ ∀ A ∈ psus (X ; E). (6.6) R As already noted, ∂sN B(s) is a continuous family of trace class operators as soon
Tr(B) =
Proof. as N > dim X + m. Then (6.3) is a continuous function and further differentiation again gives a continuous family of trace class operators so the trace is smooth. To see that this function has a complete asymptotic expansion we appeal to the discussion of the structure of the kernels of such product-suspended families in Appendix B. −n−1,0 It suffices to consider the trace of a general element B ∈ psus (X ; E). Since the ∞ 2 kernels form a module over C (X ) we can localize in the base variable (not directly in the suspended variable since that has global properties). Localizing near a point away from the diagonal gives a classical symbol in the suspending variable with values in the smoothing operators. Since the trace is the integral over the diagonal this makes no contribution to (6.3). Thus it suffices to suppose that B is supported in the product of a coordinate neighbourhood with itself over which the bundle E is trivial. Locally (see (2.3)) the kernel is given by Weyl quantization of a product-type symbol so the trace becomes the integral of the sum of the diagonal terms and hence we need only consider 1 a(x, ξ, τ )d xdξ , (6.7) (2π )n where a is compactly supported in the base variables x. Now by definition, a is a smooth function, with compact support, on the product Rn × [R × Rn ; ∂(R × {0})]. Thus we can further localize the support of a on this blown up space. There are three essentially different regions, corresponding to the part of the boundary which arises from the radial compactification, the part arising from the blow up and the corner. The first of these regions corresponds to a true suspended family, as considered in [13]. In this region the variable |ξ | dominates, and |τ | ≤ C|ξ | on the support so the integral takes the form 1 1 φ(r τ )r n+1 f (x, ω, r, r τ )r −n−1 d xdωdr (2π )n 0 1 = sφ(R) f (x, ω, Rs, R)d xdωd R, s = 1/τ. (6.8) 0
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Here, φ has compact support and f (with the factor of r n+1 representing the order −n−1) is smooth. The result is smooth in s = 1/τ, which corresponds to a complete asymptotic expansion with only non-negative terms. The second region corresponds to boundedness of the variable ξ with the function being a classical symbol (by assumption of order at most 0) in τ so integration simply gives a symbol 1 a(x, ξ, τ )d xdξ. (6.9) (2π )n The third region is the most problematic. Here the two boundary faces of the compactification are defined by r = 1/|ξ | and |ξ |/τ and with polar variables ω = ξ/|ξ |. Thus the integral takes the form r n+1 f (x, ω, r, s/r )r −n−1 d xdωdr ∈ C ∞ ([0, 1)s ) + (log s)C ∞ ([0, 1)s ), (6.10) where f is smooth and with compact support near 0 in the last two variables. This is a simple example of the general theorem on pushforward under b-fibrations in [12], or the ‘singular asymptotics lemma’ of Brüning and Seeley (see also [9]) and is in fact a type of integral long studied as an orbit integral. In any case the indicated regularity follows and this proves the existence of a complete asymptotic expansion, possibly with single logarithmic terms. It follows that the integral in (6.4) also has a complete asymptotic expansion as T → ∞; where in principle there can be factors of (log T )2 after such integration. Thus the coefficient of T 0 does exist, and defines Tr(B). Now if N is increased by one in the definition, the additional integral gives the same formula (6.4) except that a constant of integration may be added by the first integral. After N additional integrals, this adds a polynomial, so the result is changed by the integral over [−T, T ] of a polynomial. This is an odd polynomial, so has no constant term in its expansion at infinity. Thus the definition of Tr(B) is in fact independent of the choice of N . The trace identity follows directly from (6.4), since if B = [B1 , B2 ], then any derivative is a sum of commutators between operators with order summing to less than −n and the trace of such a term vanishes. Thus applied to a commutator (6.4) itself vanishes. Using this trace functional on product-suspended operators we extend the domain of the eta invariant. Proposition 6.2. The eta invariant for any fully elliptic, invertible element A ∈ m,m psus (X ; E) defined using the regularized trace 1 ˙ A˙ = ∂ A Tr(A−1 A), πi ∂τ is a log-multiplicative functional, in the sense of (6.1). η(A) =
(6.11)
Proof. Certainly (6.11) defines a continuous functional on elliptic and invertible productsuspended families. The log-multiplicativity, (6.1), follows directly since if B is another invertible product-suspended family then ∂(AB) ˙ + B −1 B˙ = B −1 A−1 AB ∂τ ˙ = η(A). and the trace identity shows that Tr(B −1 A−1 AB) (AB)−1
(6.12)
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6.2. η(A + iτ ) = η(A). To relate this functional on product-suspended invertible operators to the more familiar eta invariant for self-adjoint elliptic pseudodifferential operators we rewrite the definition in a form closer to traditional zeta regularization, starting with the regularized trace. Consider the meromorphic family t+−z of tempered distributions with support in [0, ∞). This family has poles only at the positive integers, with residues being derivatives of the delta function at the origin. For Re z sufficiently positive and non-integral, t+−z can be paired with the function FB,N (t) in (6.4), since this is smooth and of finite growth at infinity. This pairing gives a meromorphic function in Re z > C, with poles only at the natural numbers since the poles of t+−z are associated with the behaviour at 0, where FB,N is smooth. In fact this pairing g(z) = T+−z−1 , FB,N (T )
(6.13)
extends to be meromorphic in the whole complex plane. Indeed, dividing the pairing into two using a cut-off ψ ∈ Cc∞ ([0, ∞)) which is identically equal to 1 near 0, g(z) = T+−z−1 , ψ(T )FB,N (T ) + T+−z−1 , (1 − ψ(T )))FB,N (T ),
(6.14)
the first term is meromorphic with poles only at z ∈ N and the poles of the second term arise from the terms in the asymptotic expansion of FB,N (T ). Notice that there is no pole at z = 0 for the first term since the residue of T+−z−1 at z = 0 is a multiple of the delta function and FB,N (0) = 0. The pole at z = 0 for the second term arises exactly from the coefficient of T 0 in the asymptotic expansion so we see that Tr(B) = resz=0 g(z).
(6.15)
Any terms ak (log T )k for k ∈ N, in the expansion do not contribute to the residue since they integrate to regular functions at z = 0 plus multiples of z −k .
m,m Proposition 6.3. For B ∈ psus (X ; E) and any N > m − dim X − 1, the regularized trace is the residue at z = 0 of the meromorphic continuation from Re z >> 0, z ∈ / Z, of (t + i0) N −z (t − i0) N −z (−1) N +1 N , Tr E (∂t B(t) . (6.16) + (N − z) . . . (1 − z)(−z) 1 + e−πi z 1 + eπi z
Proof. Consider the identity t+−z−1 =
d N +1 −z+N 1 t . (N − z) . . . (1 − z)(−z) dt N +1 +
(6.17)
After inserting this into (6.14), integration by parts is justified (since (6.17) holds in the sense of distributions on the whole real line, supported in [0, ∞)), and shows that
N
N 1 ∂ ∂ g(z) = t+−z+N , (−1) N +1 Tr E B (t)−Tr B (−t), E N (N −z) . . . (1−z)(−z) ∂t ∂t N (6.18) where the pairings are defined, and holomorphic, for Re z large and z non-integral. Using the identity t+z =
(t + i0)z (t − i0)z + , 1 − e2πi z 1 − e−2πi z
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(6.18) becomes g(z) = D(t, z) =
(−1) N +1 D(t, z), Tr E (N − z) . . . (1 − z)(−z)
∂N B (t), ∂t N
(t + i0) N −z (t − i0) N −z + 1 − e2πi(N −z) 1 − e−2πi(N −z) (−t + i0) N −z (−t − i0) N −z + (−1) N . + (−1) N 2πi(N −z) 1−e 1 − e−2πi(N −z)
Now, (−t − i0)−z = eπi z (t + i0)−z so
eπi z 1 (t + i0) N −z + D(t, z) = 1 − e−2πi z 1 − e2πi z
1 e−πi z (t − i0) N −z , + + 1 − e2πi z 1 − e−2πi z which reduces (6.20) to (6.16).
(6.19)
(6.20)
This allows us to prove a result of which Theorem 3 in the introduction is an immediate corollary. Theorem 3. If A ∈ 1 (X ; E) is a self-adjoint elliptic and invertible pseudodifferential operator then η(A + iτ ), defined through (6.11) reduces to the (regularized) value at z = 0 of the analytic continuation from Re z >> 0 of sgn(a j )|a j |−z , (6.21) j
where the a j are the eigenvalues of A, in order of increasing |a j | repeated with multiplicities. Proof. With A(τ ) = A + iτ the eta invariant defined by (6.11) reduces to 1 1 η(A + iτ ) = Tr (A + iτ )−1 = resz=0 h(z), π π
(6.22)
where h(z) is the function (6.16) with B(t) = (A + it)−1 . Computing the N th derivative ∂N −1 (A + iτ ) = i(−1) N +1 N !(τ − i A)−N −1 . (6.23) ∂τ N The trace is therefore given, for any N > n, by
N ∂ −1 tr E = i(−1) N +1 N ! (A + iτ ) (τ − ia j )−N −1 . (6.24) N ∂τ j
This converges uniformly with its derivatives so can be inserted in the pairing (6.16) and the order exchanged. Thus a N (z)i(−1) N +1 N ! h(z) = lim τ N −z (τ − ia j )−N −1 dτ ↓0 R+i 1 + e−πi z j N +1 N! a N (z)i(−1) + lim τ N −z (τ − ia j )−N −1 dτ, (6.25) ↓0 R−i 1 + eπi z j
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where a N (z) =
(−1) N +1 . (N − z) . . . (1 − z)(−z)
Each of these contour integrals is actually independent of > 0 for smaller than the minimal |a j |. By residue computation, in the first sum by moving the contour to infinity in the upper half plane and in the second by moving the contour into the lower half plane ±2πi (N −z)···(1−z) e∓πi z/2 |a j |−z ±a j > 0 N! . (6.26) τ N −z (τ − ia j )−N −1 dτ = 0 ±a j < 0 R±i Inserting this into (6.25) shows that η(A + iτ ) is the residue at z = 0 of 1 sgn(a j )|a j |−z . z cos(π z/2)
(6.27)
j
By definition, the usual eta invariant, η(A), is the value at z = 0 of the continuation of the series in (6.27). This series is the analytic continuation of the trace of an entire family of classical elliptic operators of complex order −z (namely A−z ( + − − ), where ± are the projections onto the span of positive and negative eigenvalues) which can have only a simple pole at z = 0. In fact, here, it is known that there is no singularity, i.e. the residue vanishes. Even without invoking this we conclude the desired equality, since the explicit meromorphic factor in (6.27) is odd in z, so a pole in the continuation of the series would not affect the residue. 7. Universal η and τ Invariants That the differential of the eta invariant of a family of self-adjoint Dirac operators is a multiple of the first (odd) Chern class of the index, in odd cohomology of the base, of the family is well-known. In the case of the suspended eta invariant discussed in [13] and above, we show that the η invariant is, in appropriate circumstances, the logarithm of a determinant, which is to say a multiplicative function giving the first odd Chern class. Initially we show this in the context of classifying spaces for K-theory, then in the geometric context of (2n + 1)-fold suspended odd elliptic families. Consider again the algebra of once-suspended isotropic pseudodifferential operators of order 0 on Rn , with values in smoothing operators on a compact manifold X. This can be identified with the smooth functions on R2n+1 × X 2 and the subspace (7.1) I+ = A ∈ C ∞ (R2n+1 × X 2 ); A ∼ = 0 in {t ≤ 0} ∩ S2n × X 2 , is a subalgebra. Here, t is the suspending parameter and equality is in the sense of Taylor series at infinity on the compactified Euclidean space. Thus the subalgebra is just the sum of the smoothing ideal (identified with the functions vanishing to infinite order everywhere at the boundary) and the subalgebra of functions vanishing in t < 0. In fact I+ is also an ideal. We consider the corresponding group G+ = {B = Id +A, A ∈ I+ , B −1 = Id +B , B ∈ I+ }.
(7.2)
Periodicity
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Now we may use the suspending variable t to identify the upper half-sphere {t > 0} ∩ S2n of the boundary of R2n+1 with R2n , {t > 0} ∩ S2n [(t, x, ξ )] −→ (X, ) = (x/t, ξ/t) ∈ R2n .
(7.3)
The inverse image under pull-back of S(R2n ) is then naturally identified with {a ∈ C ∞ (S2n ); a = 0 in t < 0}, where S2n is the boundary of the radial compactification of R2n+1 . This allows the space of formal power series S(R2n )[[t]] to be identified with the formal power series at the boundary of the subspace of C ∞ (R2n+1 ) consisting of the functions vanishing in t < 0. The same identifications carry over to the case of functions valued in the smoothing operators and so gives a short exact sequence of algebras −∞ (Rn × X ) sus
/ I+
/ S(R2n × X 2 )[[t]] .
(7.4)
Lemma 7.1. In (7.4), the product induced on the quotient is the standard product (valued in smoothing operators on X ) on R2n (i.e. the ‘Moyal product’). n ˆ Bˆ ∈ 0 Proof. Let A, B ∈ I+ be the symbols of two operators A, psus(1) (R × X ). Then the asymptotic expansion at infinity of the symbol of Aˆ Bˆ is given by the standard product ∞ 1 k ˆ ∼ σ ( Aˆ B) (D D − D D ) A(t, x, ξ )B(t, y, η) . (7.5) x η y ξ k!(2i)k k=0
x=y,η=ξ
Under the map (7.3), the asymptotic expansion (7.5) becomes an asymptotic expansion at {t > 0} ∩ S2n ⊂ R2n+1 , ∞ 1 1 k ˆ ∼ σ ( Aˆ B) (D D − D D ) A(t, t X, t)B(t, tY, t) (7.6) . X Y k!(2i)k t 2k X =Y,=
k=0
Thus, if A(t, t X, t) ∼
∞ ∞ 1 1 a (X, ), B(t, t X, t) ∼ bk (X, ) k tk tk k
(7.7)
k
are the asymptotic expansions of A and B at {t > 0} ∩ S2n ⊂ R2n+1 , then 1 1 k ˆ ∼ (D D − D D ) a (X, )b (Y, ) σ ( Aˆ B) X Y l m X =Y,= k!(2i)k t 2k+l+m k,l,m≥0
(7.8) ˆ at {t > 0} ∩ S2n ⊂ R2n+1 . But the right-hand is the asymptotic expansion of σ ( Aˆ B) side is precisely the standard product on S(R2n × X 2 )[[ε]] with ε = t12 . Corresponding to this exact sequence of algebras is the exact sequence of groups consisting of the invertible perturbations of the identity n −∞ G −∞ (Rn × X )[[t]]. sus (R × X ) −→ G+ −→ G
(7.9)
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Theorem 4. In this ‘delooping’ sequence, the first group is classifying for even K-theory, the central group is (weakly) contractible and the quotient is (therefore) a classifying group for odd K-theory; the eta invariant, defined as in (6.11), η : G+ −→ C
(7.10)
restricts to twice the index on the normal subgroup and eiπ η = deta is the adiabatic determinant on G −∞ (Rn × X )[[t]]. Proof. As a first step in the proof we consider the behaviour of the regularized trace. Lemma 7.2. The regularized trace Tr on the central algebra in (7.4) restricts to the integrated trace on the smoothing subalgebra and
∂b Tr b2n d X d (7.11) = ∂t R2n for any b ∈ I+ , where bk is the term of order k in the formal power series of the image in (7.4). Proof. When the parameter t is fixed, an element b ∈ I+ is actually a smoothing operator, since the asymptotic behavior on the surface where t is constant is determined by the equatorial sphere t = 0 at infinity. Thus the definition, from (6.4), of Tr(b) for any element b ∈ I+ may be modified by dropping all N integrals, i.e. we may take N = 0. Indeed, taking N > 0 and then integrating results in the case N = 0, plus a polynomial which, as noted earlier, does not affect the result. Carrying out the last integral by the fundamental theorem of calculus,
˙ = LIM Tr(b) b(T, x, ξ )d xdξ − b(−T, x, ξ )d xdξ , (7.12) T →∞
R2n
R2n
where LIM stands for the constant term in the asymptotic expansion. The second term in (7.12) corresponds to t < 0 where b is rapidly decreasing so does not contribute to the asymptotic expansion. Now, making the scaling change of variable in (7.3), transforms (7.12) to ˙ = LIM T −2n ˜ Tr(b) b(T, X, )d X d, (7.13) T →∞
R2n
where b˜ is the transformed function. Thus (7.13) picks out the term of homogeneity 2n ˜ This gives exactly (7.11). (in T ) in the formal expansion of b. 1 Now, by definition, the eta invariant is πi Tr(a −1 a). ˙ It follows directly that restricted to the smoothing subgroup this lies in 2Z. Thus D = eiπ η does indeed descend to the quotient group in (7.9). This group is connected, so to check that it reduces to the ‘adiabatic’ determinant defined earlier we only need check the variation formula, both being 1 on the identity. Along a curve a(s),
1 d d a˙ da d −1 da η(a(s)) = − a −1 a −1 a˙ = Tr (a ) . (7.14) Tr a −1 ds πi ds ds dt ds
Periodicity
175
Thus the identity (7.11) shows that
d a˜ d η(a(s)) = Tr (a(s) ˜ )2n , ds ds
(7.15)
where a˜ is the image of a in the third group in (7.9). The identity term in a does not affect the argument since it is annihilated by d/ds. Since the right hand side of (7.15) is the variation formula for the logarithm of the adiabatic determinant this proves the theorem. 8. Geometric η and τ Invariants Returning to the ‘geometric setting’ of a fibration (6) with compact fibres, consider a m,m totally elliptic family A ∈ psus (M/B; E, F). Although we allow for operators between different bundles here, (6.11) is still meaningful as a definition of the eta invariant if A is invertible. Consider the principal bundle, of the type discussed above, G −∞ sus (M/B; E)
A
(8.1)
ν
B
with fibre −∞ −m,−m Ab = A + Q; Q ∈ sus (Z b ; E b , Fb ),(A + Q)−1 ∈ psus (Z b ; Fb , E b ) . (8.2) Proposition 8.1. The eta invariant, defined by (6.11), is a smooth function on A such that for the fibre action of the structure group at each point η(A(Id +L)) = η(A) + 2 ind(Id +L)
(8.3)
τ = eiπ η : B −→ C∗
(8.4)
so projects to
which represents the first odd Chern class of the index bundle of the family A. In particular this result applies to an elliptic, self-adjoint, family of pseudodifferential operators of order 1 by considering the spectral family. Proof. That η : A −→ C is well-defined follows from the discussion above as does the multiplicativity (8.3). Thus, τ is well-defined as a function on B and it only remains to check the topological interpretation. Note that the fibre of A is non-empty at each point of the base. In fact it is always possible to find a global smoothing perturbation to make the family invertible, although only when the families index vanishes is this possible with a smoothing perturbation of compact support in the parameter space. Thus, in complete generality, it is possible to choose a smooth map Q + : R −→ −∞ (M/B; E, F) such that Q + (t) = 0 for t > 0,
(A(t) + Q(t))−1 ∈ −m,−m (M/B; E, F) ∀ t ∈ R.
(8.5)
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This follows directly from the fact that the index bundle, over R× B, is trivial for t 1. Thus the short exact sequence of groups π∞
−∞ −∞ G −∞ (M/B; E) sus (M/B; E) −→ G +,sus (M/B; E) −→ G
(8.8)
is the ‘delooping sequence’ for G −∞ (M/B; E). In particular the central group is weakly contractible and we may consider the enlarged principal bundle G −∞ +,sus (M/B; E)
A+
(8.9)
B −∞ defined by replacing G −∞ sus above by G +,sus . The existence of Q + shows that this bundle is trivial, i.e. has a global section
q : B −→ A+ which induces a ‘classifying bundle map’ −∞ −∞ (M/B; E), q(A ˜ b + Q b ) = (Ab + Q +,b )−1 (Ab + Q b ) ∈ G+,sus (Z b ; E b ). q˜ : A −→ G+,sus
Now, the definition and basic properties of the eta invariant given by (6.11) are quite insensitive to the enlargement of A to A+ and so still define a smooth function η+ : −∞ A+ −→ C. The same is true for the group G+,sus (M/B; E), defining the corresponding −∞ function η˜ : G+,sus (M/B; E) −→ C and the discussion of multiplicativity shows that ˜ η = η+ ◦ q ◦ ν + η˜ ◦ q.
(8.10)
From the fundamental theorem of calculus, ∗ d log det, iπ d η˜ = π∞
(8.11)
so we conclude from (8.10) that τ = eiπ η = eiπ η
+ ◦q◦π
(π∞ q) ˜ ∗ det
(8.12)
defines the same cohomology class as the determinant on the classifying group, i.e. the first odd Chern class of the index bundle.
Periodicity
177
9. Adiabatic η We may further extend the discussion above by replacing the once-product-suspended spaces by (2n + 1)-times product-suspended spaces using the isotropic quantization in 2n of the variables, as in Theorem 5 applied to a decomposition R2n+1 = R × R2n with the standard symplectic form used on R2n . Let A[[]] be the principal bundle of invertible perturbations for the family A with respect to the star product from (C.3).
m,m Proposition 9.1. If A ∈ psus(2n+1) (M/B; E, F) is a fully elliptic family and (6.11) is used, with the product interpreted as the parameter-dependent product of Theorem 5 for the symplectic form on R2n then the resulting eta invariant on the bundle of smoothing perturbations has an asymptotic expansion as ↓ 0 which projects to
η : A[[]] −→ −n C[[]]
(9.1)
which has constant term the adiabatic eta invariant ηa(n) : A[[]] −→ C
(9.2)
which generates the first odd Chern class of the index bundle. Proof. This is essentially a notational extension of the results above.
In particular (4.2) is a consequence of this result and Bott periodicity. Namely, given an 2n product-suspended family we may always choose a smoothing family, analogous to Q + in (8.5) which is Schwartz in the second 2n − 1 variables and in the first is Schwartz at −∞ and of the form Q 0 + Q with Q Schwartz at +∞ and Q 0 constant in the first variable (and Schwartz in the remainder). By Bott periodicity, the even index of the family is the odd K-class on R2n−1 × B given by the product (A(t) + Q 0 )A(t)−1 for t large. Then (4.2) follows by an elementary computation and the proof of Lemma 4.2 follows directly. Appendix A: Symbols and Products By choice of a quantization map, spaces of pseudodifferential operators on a compact manifold can be identified, modulo smoothing operators, with the appropriate spaces of symbols on the cotangent bundle as in (2.3). It is important to discuss, and carefully distinguish between, several classes of such symbols and operators. To prepare for this we describe here classes of product-type symbols for a pair of vector spaces; subsequently this is extended to the case of vector bundles. For a real vector space V, the space of classical symbols of order 0 on V is just C ∞ (V ), the space of smooth functions on the radial compactification. In terms of any 1 Euclidean metric on V, ρ(v) = (1 + |v|2 )− 2 is a defining function for the boundary of V and the space of symbols of any complex order z on V is S z (V ) = ρ −z C ∞ (V ).
(A.1)
If W is a second real vector space then we may consider the radial compactification V × W and corresponding symbol spaces S z (V × W ). The natural projection πW : V × W −→ W does not extend to a map from V × W to W and correspondingly classical symbols on W do not generally lift to be classical symbols on V × W. Rather
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V → V × W may be considered as an embedded submanifold, simply the closure (of the preimage in V × W ) of V × {0}. On the other hand there is certainly a smooth projection from V × W to W ; the smooth functions, S 0 (V ; S 0 (W )) = S 0 (W ; S 0 (V )) = C ∞ (V × W ))
(A.2)
on this space are symbols on V with values in the symbols on W (or vice-versa). The main space we wish to consider here has some properties between these two compactifications of V × W. Namely, in terms of radial (real) blow-up (in the sense of [11]), we set V
V × W = [V × W ; ∂(V × {0})].
(A.3)
This manifold with corners has two boundary faces (unless one or both of the factors is one-dimensional in which case either or both of the boundary hypersurfaces may have two components). We use a superscript V to refer to the new boundary hypersurface produced by the blow-up in (A.3). Lemma A.1. The projection πW : V × W −→ W extends to a smooth map π W : V V × W −→ W which is a fibration (with fibres which are manifolds with boundary) and in terms of Euclidean metrics on V and W the functions
ρV (v, w) =
1 + |w|2 1 + |v|2 + |w|2
21
1
and ρr (v, w) = (1 + |w|2 )− 2
extend from V × W to be smooth functions on V V × W and are defining functions for the two boundary faces. Proof. To check the first statement of the lemma, notice that the projection V × W → V has a smooth extension pW : V × W \ ∂(V × {0}) → W which is a fibration with typical fibre given by V. Blowing up the submanifold ∂(V ×{0}) in V × W exactly allows us to extend pW to a fibration π W : V V × W −→ W with typical fibre given by V . Indeed, in V × W near the submanifold ∂(V × {0}), we can consider the generating functions (i.e. everywhere containing a coordinate system) vˆ =
v 1 w , σV = , w = = σV w. 1 1 |v| 2 (1 + |v| ) 2 (1 + |v|2 ) 2
The blow up amounts to introducing polar coordinates r V = (σV2 + w 2 ) 2 , (ϕ, θˆ ) = ( 1
σV w , ) rV rV
Periodicity
179
so that the blow-down map is given locally by V
V × W = [V × W ; ∂(V × {0})] (v, ˆ r V , ϕ, θˆ ) −→ (v, ˆ σV = r V ϕ, w = r V θˆ ).
In these polar coordinates, and for r V > 0, the fibration pW is given by ˆ r V , ϕ, θˆ ) = pW (v,
θˆ
ϕ
, 1 1 (ϕ 2 + |θˆ |2 ) 2 (ϕ 2 + |θˆ |2 ) 2
∈ W,
(A.4)
where we have used the identification of W with the upper half-sphere which is the closure of the image W w −→ (
1 (1 + |w|2 )
1 2
,
w 1
(1 + |w|2 ) 2
) ∈ {(a, b) ∈ R × W ; a ≥ 0, a 2 + |b|2 = 1}.
Thus, pW extends to r V = 0 to give the desired fibration. 1 It follows from this that a defining function for the boundary of W such as (1+|w|2 )− 2 lifts from W to be smooth and to define the ‘old’ boundary hypersurface, the one not 1 produced by the blow up. Now (1 + |w|2 + |v|2 ) 2 is a smooth boundary defining function on V × W . It therefore lifts under the blow up in (A.3) to be the product of defining functions for both boundary hypersurfaces and so
ρV (v, w) =
1 + |w|2 1 + |v|2 + |w|2
21
is a boundary defining function for the new boundary produced by the blow-up.
Now we define general spaces of ‘partial-product’ symbols by
S z,z (V V × W ) = ρrz ρVz C ∞ (V V × W ).
(A.5)
Directly from this definition,
S z,z (V V × W ) · S ζ,ζ (V V × W ) = S z+ζ,z +ζ (V V × W ).
(A.6)
Two of the ‘remainder’ classes have simpler characterizations. Namely S −∞,z (V V × W ) = C˙∞ (W ; S z (V )), S −∞,−∞ (V V × W ) = C˙∞ (V × W ) = S(V × W ).
(A.7)
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Appendix B: Product Suspended Operators We can now introduce a generalization of the ‘suspended’ algebra considered in [13] and in [15] (an algebra similar to the suspended algebra was already introduced by Shubin in [20]). The d-fold suspended pseudodifferential algebra on a compact manifold X may be viewed as a space of smooth maps from Rd into k (X ; E, F) in which the parameters (which we think of as the base variables for a fibration) appear as ‘symbolic variables’. The inverse Fourier transform identifies the suspended space k d ˇk sus(d) (X ; E, F) ⊂ (R × X ; E, F)
directly, as is done in [13], with the elements which are translation-invariant in Rd and have convolution kernels vanishing rapidly at infinity, with all derivatives, in these variables; this space may also be defined directly as in (2.3). The subspace of smoothing operators is −∞ sus(d) (X ; E, F) = S(Rd × X 2 ; Hom(E, F) ⊗ R )
in terms of the Schwartz space. Then the finite-order operators may be specified, up to smoothing terms, by Weyl quantization as qg : ρ −k C ∞ (Rd × T ∗ X ; π ∗ hom(E, F)) a −→ k (2π )−n χ eiv(x,y)·ξ a(m(x, y), ζ, ξ )dξ dg ∈ sus(d) (X ; E, F), T∗X
(B.1)
where the symbol space is compactified in the joint fibre Rd × Tx∗ X. The resulting full symbol sequence is as in (2.4) except that the formal power series have coefficients on the sphere bundle of R p × T ∗ X ; the parameters do not affect the operators B j , acting on T ∗ X, appearing in the product. If A ∈ 1 (X ; E) is a first order pseudodifferential operator and τ is the suspension 1 1 variable for sus(1) (X ; E), then A + iτ is not in general an element of sus(1) (X ; E). 1 In fact, A + iτ ∈ sus(1) (X ; E) if and only if A is a differential operator. Similarly, for A ∈ 1 (X ; E, F), the operator
it + τ A∗ (B.2) A it − τ 1 is in sus(2) (X ; E ⊕ F) if and only if A is a differential operator. This restriction to differential operators is unfortunate since the operator A +iτ arises in the alternative definition of the eta invariant as described in Sect. 6, while in Sect. 5 the operator (B.2) is used to implement Bott periodicity for determinant line bundles. For these reasons, and others, we pass to the wider context of product-suspended operators. We first need to enlarge the space of symbols as in Appendix A. Identifying X with the zero section of T ∗ X , consider the blown-up space X
Rd × T ∗ X = [Rd × T ∗ X ; ∂Rd × X ],
where Rd × T ∗ X is the radial compactification of Rd × T ∗ X fibre by fibre and Rd × X ⊂ Rd × T ∗ X
(B.3)
Periodicity
181
is the closure of Rd × X in Rd × T ∗ X . In terms of a Riemannian metric g and the Euclidean metric on Rd , Lemma A.1 generalizes directly to Lemma B.1. The projection Rd × T ∗ X → T ∗ X extends to a smooth map π T ∗ X : X Rd × T ∗ X −→ T ∗ X which is a fibration with typical fibre Rd , and the smooth functions 1
ρsus (v, w) =
(1 + |w|2 ) 2
1
(1 + |v|2 + |w|2 )
1 2
, ρr (v, w) = (1 + |w|2 )− 2 , v ∈ Rd , w ∈ T ∗ X,
define the two boundary faces. Proof. This results from the invariance of the construction in Appendix A under those linear transformations of V × W which leave V invariant, so Lemma A.1 extends to the case of a vector bundle. For z, z ∈ C, the space of (partially) product-type symbols with values in a vector bundle over X is then
−z ∞ X d S z,z ( X Rd × T ∗ X ; U ) = ρr−z ρsus C ( R × T ∗ X ; U ).
(B.4) 1
On Rd × X × X , consider the boundary defining function ρτ (τ ) = (1 + |τ |2 )− 2 . Let E and F be smooth complex vector bundles on X. For z ∈ C set
−∞,z (X ; E, F) = ρτ−z C ∞ (Rd × X × X ; Hom(E, F) ⊗ R X ), psus(d)
(B.5)
where R X = π3∗ X, π3 being the projection on the third factor, and X being the bundle of densities on X. This is the space of smoothing operators (defined as usual through their kernels) on X depending symbolically on d parameters; which we identify as the product-suspended operators of order −∞ on X. Definition B.2. The general spaces of product d-suspended pseudodifferential operators of order k, k ∈ Z acting from S(Rd × X ; E) to S(Rd × X ; F) is
k,k −∞,k psus(d) (X ; E, F) = qg (S k,k ( X Rd × T ∗ X ; hom(E, F))) + psus(d) (X ; E, F),
where qg is the Weyl quantization (B.1) applied to these more general symbol spaces. We limit attention to integral orders here only because it is all that is needed. Pseudodifferential operators are included in the product-suspended operators k,0 (X ; E, F), k (X ; E, F) ⊂ psus(d)
being independent of the parameters. For integers l ≤ k, l ≤ k , there are inclusions
l,l k,k (X ; E, F) ⊂ psus(d) (X ; E, F). psus(d)
Furthermore, as we will see below in Theorem 5, product d-suspended operators compose in the expected way
k,k l,l k+l,k +l psus(d) (X ; E, F) ◦ psus(d) (X ; G, E) ⊂ psus(d) (X ; G, F).
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Suspended operators are particular instances of product suspended operators, k,k k sus(d) (X ; E, F) ⊂ psus(d) (X ; E, F), k ∈ Z,
and −∞,−∞ −∞ (X ; E, F) = psus(d) (X ; E, F). sus(d)
Product d-suspended pseudodifferential operators are intimately related with the algebra of product-type operators introduced in [16]. More precisely, consider the projection φ : Rd × X → Rd
(B.6)
as a fibration. If E and F are smooth complex vector bundles on X, then as discussed in [16], to such a fibration one can associate the space of product-type pseudodifferential operators of order (k, k )
k,k d φ− p (R × X ; E, F)
acting from Cc∞ (Rd × X ; E) to C ∞ (Rd × X ; F). Given τ ∈ Rd , let Tτ : Rd × X → Rd × X denote the translation in the first factor Tτ (t, x) = (t − τ, x). We can consider the product-type pseudodifferential operators which are translation-invariant in the Euclidean variable, that is, satisfying Tτ∗ (A f ) = ATτ∗ f, ∀ τ ∈ Rd , f ∈ Cc∞ (Rd × X ; E).
(B.7)
In terms of the Schwartz kernel K A of A, this means that K A acts by convolution in the first factor A f (x, t) = K A (t − s, x, x ) f (x , s)ds, Rd
X
is a density in the x variable. Now one can ask in addition that this convolution
where K A kernel decay to all orders at infinity
K A ∈ Cc−∞ (Rd × X 2 ; Hom(E, F) ⊗ R X ) + S(Rd × X 2 ; Hom(E, F) ⊗ R X ). (B.8) This leads to the following characterization of product d-suspended operators. Lemma B.3. Fourier transformation in the suspension variables ˆ ( A(τ ) f )(x) = e−itτ K A (t, x, x ) f (x )dt, τ ∈ Rd X
Rd
is an isomorphism of the space of translation-invariant product-type pseudodifferential operators satisfying (B.8) onto the d-parameter product-suspended pseudodifferential operators; it preserves products.
Periodicity
183
Proof. Modulo small changes of notation, this is the same as for suspended operators. One advantage of the alternative definition through Lemma B.3 is that the Fredholm theory for product d-suspended operators follows almost immediately from the corresponding Fredholm theory for product-type operators. Indeed, the principal symbol map and the base family map for product-type operators gives via the inclusion (using the ˇ psus(d) (X ; E, F) ⊂ k,k (Rd × X ; E, F) a corresponding inverse Fourier transform) φ− p symbol map and base family map for product d-suspended operators. For the convenience of the reader, we will define these directly without referring to product-type operators. Of the two boundary faces of X R p × T ∗ X , the ‘old’ boundary, or really its blow-up, Bσ = [S(Rd × T ∗ X ); S(Rd ) × X ] with X being the zero section of T ∗ X, carries the replacement for the usual principal symbol. In terms of a quantization map as above, this is given by the restriction of the full symbol a ∈ S m,m (Rd T ∗ X ; E, F) of an operator A = qg (a) to this boundary face,
m,m m,m σm,m : psus(d) (X ; E, F) −→ Spsus(d) (X ; E, F)
(B.9)
with
m,m Spsus(d) (X ; E, F) = C ∞ (Bσ ; hom(E, F) ⊗ N −m ⊗ Nff−m ),
where N is the normal bundle to Bσ and and Nff is the normal bundle of the ‘new’ boundary, which is canonically identified with the normal bundle to the boundary of Bσ . Both are trivial bundles. This corresponds to the multiplicative short exact sequence
σm,m
m−1,m m,m m,m 0 −→ psus(d) (X ; E, F) −→ psus(d) (X ; E, F) −→ Spsus(d) (X ; E, F)→0.
(B.10)
k,k (X ; E, F) is elliptic if its principal A product d-suspended operator A ∈ psus(d) symbol σm,m (A) is invertible. Ellipticity alone does imply that the family is Fredholm for each value of the parameter but, as for product-type operators, it does not suffice to allow the construction of a parametrix modulo Schwartz-smoothing errors. There is a second symbol map which takes into account the behavior of the operator for large values of the suspension parameters. Let Bsus ⊂ X Rd × T ∗ X denote the ‘new’ boundary, which is the ‘front face’ produced by the blow up. The fibration of Lemma B.1 gives a canonical identification of Bsus with S(Rd ) × T ∗ X . Thus, the restriction map (using a boundary defining function ρsus for Bsus ) becomes
R : S k,k ( X Rd × T ∗ X ; hom(E, F)) a −→ ρm a ∈ C ∞ (S(Rd ); S k,k (T ∗ X ; hom(E, F))). sus
Bsus
k,k
(B.11)
−∞,k
Given A = qg (a1 ) + A2 ∈ psus(d) (X ; E, F) with A2 ∈ psus(d) (X ; E, F) and
a1 ∈ S k,k ( X Rd × T ∗ X ; hom(E, F)) the base family is defined by k A2 ) B ∈ C ∞ (S(Rd ); m (X ; E, F)). L(A) = qg (R(a1 )) + ρsus sus
(B.12)
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Proposition B.4. The base family (B.12) is independent of choices and corresponds to the multiplicative short exact sequence
k,k −1 k,k 0 −→ psus(d) (X ; E, F) −→ psus(d) (X ; E, F) −→ C ∞ (S(Rd ); k (X ; E, F)) −→ 0, L
(B.13) so
m,m k,k L(A ◦ B) = L(A) ◦ L(B), A ∈ psus(d) (X ; E, F), B ∈ psus(d) (X ; G, E).
Proof. The fact that there is a short exact sequence is essentially by definition of L . The fact that L is a homomorphism follows by very simple ‘oscillatory testing’. Namely, if k,k u ∈ C ∞ (X ; E) and A ∈ psus( p) (X ; E, F) then Au ∈ ρτ−k C ∞ (R p × X ; F) and L(A)u = ρτk Au ∂ R p ∈ C ∞ (S p−1 × X ; F).
(B.14)
k,k Definition B.5. The joint symbol J (A) of an operator A ∈ psus(d) (X ; E, F) is the combination of its principal symbol and its base family J (A) = (σ (A), L(A)) where σ (L(A)) = σ (A) B . σ
An operator A is said to be fully elliptic if its joint symbol is invertible. The important feature that motivates the introduction of product-suspended operators (as opposed to suspended operators) is the following lemma. 1,1 Lemma B.6. If A ∈ 1 (X ; E) then the one-parameter family τ −→ A + iτ ∈ psus(1) (X ; E) and if B ∈ 1 (X ; E, F), then the two-parameter family
B∗ 1,1 ˆ τ ) = it + τ ∈ psus(2) (X ; E ⊕ F). (t, τ ) −→ B(t, B it − τ
Moreover if A is self-adjoint and elliptic (respectively B is elliptic) then A + iτ (respecˆ is fully elliptic. tively B) In fact, it suffices that all the eigenvalues of the symbol of A have a nonvanishing real part for A + iτ to be fully elliptic. Proof. Fix a quantization qg . In the first case a ∈ ρ −1 C ∞ (T ∗ X ; π ∗ hom(E)) exists such that (A − qg (a)) ∈ −∞ (X ; E). Then −∞,1 (X ; E), a + iτ ∈ S 1,1 ( X R × T ∗ X ; E) and A + iτ − qg (a + iτ ) ∈ psus(1) 1,1 which shows that A + iτ ∈ psus(1) (X ; E). The symbol of A + iτ is invertible if σ (A)
has no eigenvalues in iR and its base family is ±i Id at the two components of ∂Rτ . Thus A + iτ is fully elliptic. In the second case, choose b ∈ ρ −1 C ∞ (T ∗ X ; π ∗ hom(E, F)) such that B − qg (b) ∈ −∞ (X ; E).
Periodicity
Then
185
it + τ bˆ = b
b∗ it − τ
∈ S 1,1 ( X R2 × T ∗ X ; E, F)
ˆ ∈ −∞,1 (X ; E, F), which shows that Bˆ ∈ 1,1 (X ; E, F). To see and Bˆ − qg (b) psus(2) psus(1) that Bˆ is fully elliptic when B is elliptic, consider the invertible operator
∗ 0 B B + t2 + τ 2 + 1 2,2 ∗ ˆ ˆ ∈ psus(2) Q = B B+1= (X ; E ⊕ F). 0 B B∗ + t 2 + τ 2 + 1 Then −2,−2 (Q −1 Bˆ ∗ ) Bˆ − Id E⊕F = −Q −1 ∈ psus(2) (X ; E ⊕ F),
ˆ −1 = J (Q −1 Bˆ ∗ ) exists, which shows that Bˆ is fully elliptic. so that J ( B)
Appendix C: Mixed Isotropic Operators Next we proceed to the ‘parameter quantization’ of these spaces of product suspended operators. That is, we introduce a new product depending on the choice of an antisymmetric form on R p . These products are used above in the identification of the determinant bundle, as constructed in the product 2n-suspended case, with the determinant bundle as introduced by Quillen. To do so we use an adiabatic limit, with a parameter which passes from the quantized to the unquantized case discussed above; for the isotropic algebra itself such degenerations are treated in [7] and as shown there implements Bott periodicity. So, to introduce these spaces we simply combine (2.3) and its Euclidean analogue (2.20). Note that the quantization map will be global in the Euclidean variables but can only be local near the diagonal in the manifold. In defining these spaces we use the formula for the action of an operator by Weyl quantization in (2.21). Proposition C.1. Let X be a compact manifold E and F complex bundles over X , then for any p ∈ N combining (2.21) with the operator product gives a smooth family of associative products m 1 ,m
m 2 ,m
m 1 +m 2 ,m 1 +m 2
1 2 2 (Rn ) × psus(2n) (X ; F, G) × psus(2n) (X ; E, F) −→ psus(2n)
(X ; E, G). (C.1)
This follows by combining essentially standard treatments of the composition of pseudodifferential operators with those of the ‘isotropic’ operators on Rn . We are especially interested in the ‘adiabatic limit’ where the general ω is replaced by ω for a fixed antisymmetric form. The cases which occur above are where p is even and ω is non-degenerate, or where p is odd and ω has maximal rank. In this case we state the corresponding corollary of the result above (see also [10] and [7]). Theorem 5. For any fixed antisymmetric form on R p , the composition (C.1) induces a smooth 1-parameter family of quantized products
k,k l,l k+l,k +l [0, 1] × psus( p) (X ; F, G) × psus( p) (X ; E, F) −→ psus( p) (X ; E, G)
(C.2)
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and as ↓ 0 there is a Taylor series expansion A ◦ B(u) ∼
∞ (−i)k k=0
2k k!
ω(Dv , Dw )k A(v)B(w)v=w=u ,
(C.3)
in particular, when = 0 the product reduces to the usual parameterized product of suspended operators.
References 1. Atiyah, M.F., Patodi, V.K., Singer, I.M.: Spectral asymmetry and Riemannian geometry. II. Math. Proc. Camb. Phils. Soc. 78(3), 405–432 (1975) 2. Berline, N., Getzler, E., Vergne, M.: Heat kernels and Dirac operators. Springer-Verlag, Berlin (1992) 3. Bismut, J.-M., Freed, D.: The analysis of elliptic families, II. Commun. Math. Phys. 107, 103–163 (1986) 4. Bismut, J.-M., Freed, D.: The analysis of elliptic families: Metrics and connections on determinant bundles. Commun. Math. Phys. 106, 159–176 (1986) 5. Bott, R., Seeley, R.: Some remarks on the paper of Callias. Commun. Math. Phys. 62, 235–245 (1978) 6. Dai, X., Freed, D.S.: and determinant lines. J. Math. Phys. 35(10), 5155–5194 (1994) 7. Epstein, C.L., Melrose, R.B.: The Heisenberg algebra, index theory and homology. This became [8] without Mendoza as coauthor 8. Epstein, C.L., Melrose, R.B., Mendoza, G.: The Heisenberg algebra, index theory and homology. In preparation 9. Grieser, D., Gruber, M.J.: Singular asymptotics lemma and push-forward theorem. In: Approaches to singular analysis (Berlin, 1999), Oper. Theory Adv. Appl., Vol. 125, Basel: Birkhäuser, 2001, pp. 117– 130 10. Hörmander, L.: The Weyl calculus of pseudo-differential operators. Comm. Pure Appl. Math. 32, 359–443 (1979) 11. Melrose, R.B.: Analysis on manifolds with corners. In preparation 12. Melrose, R.B.: Calculus of conormal distributions on manifolds with corners. Internat. Math. Res. Notices 1992(3), 51–61 (1992) 13. Melrose, R.B.: The eta invariant and families of pseudodifferential operators. Math. Res. Lett. 2(5), 541–561 (1995) 14. Melrose, R.B., Rochon, F.: Boundaries, eta invariant and the determinant bundle. Preprint, http://arxiv.org/ list/math.DG/0607480, 2006 15. Melrose, R.B., Rochon, F.: Families index for pseudodifferential operators on manifolds with boundary. IMRN (22), 1115–1141 (2004) 16. Melrose, R.B., Rochon, F.: Index in K-theory for families of fibred cusp operators. K-Theory 37, 25–104 (2006) 17. Pressley, A., Segal, G.: Loop groups. Oxford Science publications, Oxford Univ. Press, Oxford (1986) 18. Quillen, D.: Determinants of Cauchy-Riemann operators over a Riemann surface. Funct. Anal. Appl. 19, 31–34 (1985) 19. Seeley, R.T.: Complex powers of an elliptic operator. Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966), Providence, R.I.: Amer. Math. Soc., 1967, pp. 288–307 20. Shubin, M.A.: Pseudodifferential operators and spectral theory. Berlin-Heidelberg-New York: SpringerVerlag, 1987, Moscow, Nauka: 1978 21. Wodzicki, M.: Spectral asymmetry and zeta functions. Invent. Math. 66, 115–135 (1982) Communicated by L. Takhtajan
Commun. Math. Phys. 274, 187–216 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0261-z
Communications in
Mathematical Physics
Fast Soliton Scattering by Delta Impurities Justin Holmer, Jeremy Marzuola, Maciej Zworski Mathematics Department, Evans Hall, University of California, Berkeley, CA 94720, USA. E-mail:
[email protected] Received: 9 August 2006 / Accepted: 22 November 2006 Published online: 31 May 2007 – © Springer-Verlag 2007
Abstract: We study the Gross-Pitaevskii equation with a repulsive delta function potential. We show that a high velocity incoming soliton is split into a transmitted component and a reflected component. The transmitted mass (L 2 norm squared) is shown to be in good agreement with the quantum transmission rate of the delta function potential. We further show that the transmitted and reflected components resolve into solitons plus dispersive radiation, and quantify the mass and phase of these solitons. 1. Introduction We study the Gross-Pitaevskii equation (NLS) with a repulsive delta function potential (q > 0) i∂t u + 21 ∂x2 u − qδ0 (x)u + u|u|2 = 0 (1.1) u(x, 0) = u 0 (x). As initial data we take a fast soliton approaching the impurity from the left: u 0 (x) = eivx sech(x − x0 ), v 1, x0 0.
(1.2)
Because of the homogeneity of the problem this covers the case of the general soliton profile Asech(Ax). The quantum transmission rate at velocity v is given by the square of the absolute value of the transmission coefficient, see (2.2) below, Tq (v) = |tq (v)|2 =
v2
v2 . + q2
(1.3)
For the soliton scattering the natural definition of the transmission rate is given by Tqs (v) = lim
t→∞
u(t)x>0 2L 2 u(t)2L 2
=
1 lim u(t)x>0 2L 2 , 2 t→∞
(1.4)
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provided that the limit exists. We expect that it does and that for fixed q/v, there is a σ > 0 such that Tqs (v) = Tq (v) + O(v −σ ), as v → +∞. (1.5) Based on the comparison with the linear case (see (2.21) below) and the numerical evidence [9] we expect (1.5) with σ = 2. Towards this heuristic claim we have Theorem 1. Let δ satisfy 23 < δ < 1. If u(x, t) is the solution of (1.1) with initial condition (1.2) and x0 ≤ −v 1−δ , then for fixed q/v, 3 1 v2 |u(x, t)|2 d x = 2 + O(v 1− 2 δ ), as v → +∞, (1.6) 2 2 x>0 v +q uniformly for |x0 | + v −δ ≤ t ≤ (1 − δ) log v. v We see that by taking δ very close to 1, we obtain an asymptotic rate just shy of v −1/2 . More precisely, we show that there exists v0 = v0 (q/v, δ), diverging to +∞ as δ ↑ 1 and q/v → +∞, such that for fixed q/v, if v ≥ v0 , then 1 3 v 2 2 ≤ cv 1− 2 δ . |u(x, t)| d x − 2 2 + q2 v x>0 The constant c appearing here is independent of all parameters (q, v, and δ). We have conducted a numerical verification of Theorem 1 – see Fig. 2. It shows that the approximation given by (1.6) is very good even for velocities as low as ∼ 3, at least for def
0.6 ≤ α = q/v ≤ 1.4. A more elaborate numerical analysis will appear in our forthcoming paper [9]. Our second result shows that the scattered solution is given, on the same time scale, by a sum of a reflected and a transmitted soliton, and of a time decaying (radiating) term – see the fourth frame of Fig. 1. This is further supported by a forthcoming numerical study [9]. In previous works in the physics literature (see for instance [2]) the resulting waves were only described as “soliton-like”. Theorem 2. Under the hypothesis of Theorem 1 and for |x0 | + 1 ≤ t ≤ (1 − δ) log v, v we have, as v → +∞,
3 u(x, t) = u T (x, t) + u R (x, t) + O L ∞ (t − |x0 |/v)−1/2 + O L 2x (v 1− 2 δ ), x u T (x, t) = eiϕT ei xv+i(AT −v 2
2 )t/2
u R (x, t) = eiϕ R e−i xv+i(A R −v 2
A T sech(A T (x − x0 − tv)),
2 )t/2
A R sech(A R (x + x0 + tv)),
(1.7)
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Fig. 1. Numerical simulation of the case q = v = 3, x0 = −10, at times t = 0.0, 2.7, 3.3, 4.0. Each frame is a plot of amplitude |u| versus x
Fig. 2. A plot of the numerically obtained transmission Tqs (v) versus velocity v for five values of α = q/v = 0.6, 0.8, 1.0, 1.2, 1.4. The dashed lines are the corresponding theoretical v → +∞ asymptotic values given by 1/(1 + α 2 ).
where A T = (2|tq (v)| − 1)+ , A R = (2|rq (v)| − 1)+ , and ϕT = arg tq (v) + ϕ0 (|tq (v)|) + (1 − A2T )|x0 |/2v, ϕ R = arg rq (v) + ϕ0 (|rq (v)|) + (1 − A2R )|x0 |/2v,
(1.8)
190
J. Holmer, J. Marzuola, M. Zworski 1 0.9 0.8 0.7
reflection rate
0.6 0.5 0.4 transmission rate
0.3 0.2
soliton reflection soliton transmission
0.1 0
0
0.5
1
1.5
2
2.5
3
def
Fig. 3. Comparison of linear and nonlinear scattering coefficients as functions of α = q/v.
∞
ϕ0 (ω) = 0
ζ sin2 π ω log 1 + dζ. 2 2 cosh π ζ ζ + (2ω − 1)2
Here tq (v) and rq (v) are the transmission and reflection coefficients of the delta-potential (see (2.2)). When 2|tq (v)| = 1 or 2|rq (v)| = 1 the first error term in (1.7) is modified to 1 ((log(t − |x0 |/v))/(t − |x0 |/v)) 2 ). OL ∞ x Here and later we use the standard notation k a a ≥ 0, a+k = 0 a < 0.
(1.9)
This asymptotic description holds for v greater than some threshold depending on q/v and δ, as in Theorem 1. The implicit constant in the O L 2x error term is entirely independent of all parameters (q, v, and δ), although the implicit constant in the O L ∞ x error term depends upon q/v, or more precisely, the proximity of |tq (v)| and |rq (v)| to 1 2. A comparison of the transmission and reflection coefficients (1.3) of the δ potential, and of the soliton transmission and reflections coefficients appearing in (1.7), is shown in Fig. 3. Scattering of solitons by delta impurities is a natural model explored extensively in the physics literature – see for instance [2, 8], and references given there. The heuristic insight that at high velocities “linear scattering” by the external potential should dominate the partition of mass is certainly present there. In the mathematical literature the dynamics of solitons in the presence of external potentials has been studied in high velocity or semiclassical limits following the work of Floer and Weinstein [6], and Bronski and Jerrard [1] – see [7] for recent results and a review of the subject. Roughly speaking, the soliton evolves according to the classical motion of a particle in the external potential. That is similar to the phenomena in other settings, such as the motion of the Landau-Ginzburg vortices.
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The possible novelty in (1.6) and (1.7) lies in seeing quantum effects of the external potential strongly affecting soliton dynamics. As shown in Fig. 2, Theorem 1 gives a very good approximation to the transmission rate already at low velocities. Figure 1 shows time snapshots of the evolution of the soliton, and the last frame suggests the soliton resolution (1.7). We should stress that the asymptotic solitons are resolved at a much larger time – see [9]. The proof of the two theorems, given below in §3–4, proceeds by approximating the solution during the “interaction phase” (the interval of time during which the solution significantly interacts with the delta potential at the origin) by the corresponding linear flow. This approximation is achieved, uniformly in q, by means of Strichartz estimates established in §2. The use of the Strichartz estimates as an approximation device, as opposed to say, energy estimates, is critical since the estimates obtained depend only upon the L 2 norm of the solution, which is conserved and independent of v. Thus, v functions as an asymptotic parameter; larger v means a shorter interaction phase and a better approximation of the solution by the linear flow. Theorem 2 combines this analysis with the inverse scattering method. The delta potential splits the incoming soliton into two waves which become single solitons. 2. Scattering by a Delta Function Here we present some basic facts about scattering by a delta-function potential on the real line. Let q ≥ 0 and put Hq = −
1 d2 + q δ0 (x). 2 dx2
We define special solutions, e± (x, λ), to (Hq − λ2 /2)e± = 0, using notation given in (1.9): 0 0 e± (x, λ) = tq (λ)e±iλx x± + (e±iλx + rq (λ)e∓iλx )x∓ , (2.1) where tq and rq are the transmission and reflection coefficients: tq (λ) =
iλ q , rq (λ) = . iλ − q iλ − q
(2.2)
They satisfy two equations, one standard (unitarity) and one due to the special structure of the potential: (2.3) |tq (λ)|2 + |rq (λ)|2 = 1, tq (λ) = 1 + rq (λ). We use the representation of the propagator in terms of the generalized eigenfunctions– see for instance the notes [16] covering scattering by compactly supported potentials. The resolvent Rq (λ) = (Hq − λ2 /2)−1 , def
has kernel given by Rq (λ)(x, y) =
1 e+ (x, λ)e− (y, λ)(x − y)0+ + e+ (y, λ)e− (x, λ)(x − y)0− . iλtq (λ)
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This gives an explicit formula for the spectral projection, and hence the Schwartz kernel of the propagator: ∞ 1 2 e−itλ /2 e+ (x, λ)e+ (y, λ) + e− (x, λ)e− (y, λ) dλ. (2.4) exp(−it Hq ) = 2π 0 The propagator for Hq is described in the following Lemma 2.1. Suppose that ϕ ∈ L 1 and that supp ϕ ⊂ (−∞, 0]. Then exp(−it Hq )ϕ(x) = 0 , exp(−it H0 )(ϕ ∗ τq )(x)x+0 + (exp(−it H0 )ϕ(x) + exp(−it H0 )(ϕ ∗ ρq )(−x))x− (2.5)
where 0 ρq (x) = −q exp(q x)x− , τq (x) = δ0 (x) + ρq (x).
(2.6)
Proof. All we need to do is to combine (2.1) and (2.4). Using the support property of ϕ we compute, ϕ(y)e+ (y, λ)dy = rq (−λ)ϕ(−λ) ˆ + ϕ(λ), ˆ ϕ(y)e− (y, λ)dy = tq (−λ)ϕ(−λ), ˆ so that
1 exp(−it Hq )ϕ x>0 = 2π
∞ 0
e−itλ
2 /2
tq (λ)eiλx (rq (−λ)ϕ(−λ) ˆ + ϕ(λ)) ˆ ˆ dλ + (rq (λ)eiλx + e−iλx )tq (−λ)ϕ(−λ)
1 2 iλx = e−itλ /2 tq (λ)ϕ(λ)e ˆ dλ 2π R = exp(−it H0 )(τq ∗ ϕ)(x), τq (λ) = tq (λ),
where we used the fact that rq (−λ)tq (λ) + rq (λ)tq (−λ) = 0. Similarly, using rq (−λ)rq (λ) + tq (−λ)tq (λ) = 1, we have ∞
1 2 iλx −iλx ˆ dλ e−itλ /2 ϕ(λ)e + rq (λ)ϕ(λ)e ˆ exp(−it Hq )ϕ x 0. Then u L p
r [0,T ] L x
≤ cϕ L 2 + c f L p˜
r˜ [0,T ] L x
The constant c is independent of q and T . Moreover, in (2.7), we can take f (x, t) = g(t)δ0 (x) and, on the right-hand side of (2.9), replace f L p˜ L r˜ with g 4 . [0,T ] x
3 L [0,T ]
def
Proof. We put Uq (t) = exp(−it Hq ), so that Uq (t) is a unitary group on L 2 (R). For ϕ ∈ L 1 (R) we have, using Lemma 2.1, 0 Uq (t)(ϕx± ) L ∞ Uq (t)ϕ L ∞ ≤ ±
≤
±
0 0 U0 (t) L 1 →L ∞ ((ϕx± ) ∗ τq L 1 + (ϕx± ) ∗ ρq L 1 )
(2.10)
1 ≤√ (1 + 2ρq L 1 )ϕ L 1 π |t| 3 ϕ L 1 . ≤√ π |t|
By the Riesz-Thorin interpolation theorem (see for instance [10, Theorem 7.1.12]) we have 1 1 − 21 1− r2 + = 1. , 1 ≤ r ≤ 2, (2.11) U (t) L r →L r ≤ C|t| r r The estimate (2.9) with f ≡ 0 reads U (t)g L tp L r ≤ Cg L 2 (R) , x
which by duality is equivalent to
U (−s)F(s)ds
R
L 2 (R )
≤ CF
p
L t L rx
.
(2.12)
The two equivalent estimates together give ((2.12) is applied with p , r replaced by p, ˜ r˜ – it is easily checked that (2.8) still holds)
U (t − s)F(s)ds ≤ F L p˜ L r˜ .
R
p
L t L rx
t
x
Putting F(s) = 1l[0,t] (s) f (s, x) we obtain (2.9) for u 0 = 0. Hence it suffices to prove (2.12).
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Put def
T F(x) =
R
U (−s)F(s, x)ds,
and note that T ∗ g(s, x) := U (s)g(x). The estimate (2.12) is equivalent to T ∗ T G, F L 2 ≤ CG t,x
p
L t L rx
F
which is the same as ≤ CG G(t), U (t − s)F(s) dtds R R
p
L t L rx
p
,
L t L rx
F
p
L t L rx
.
(2.13)
To obtain (2.13) from (2.11) we apply the Hardy-Littlewood-Sobolev inequality which says that if K a (t) = |t|−1/a and 1 < a < ∞ then K a ∗ F L α (R) ≤ CF L β (R) ,
1 1 1 = − 1 + , 1 < β < α, α β a
see for instance [10, Theorem 4.5.3]. We apply it with 1 2 1 1− , α = p, β = p , = a 2 r which is the admissibility condition (2.8).
We now turn to the large velocity asymptotics of the linear flow exp(−it Hq ). Proposition 2.3. Let θ ∈ C ∞ (R) be bounded, together will all of its derivatives. Let ϕ ∈ S(R), v > 0, and suppose supp[θ (•)ϕ(• − x0 )] ⊂ (−∞, 0]. Then for 2|x0 |/v ≤ t ≤ 1, e−it Hq [ei xv ϕ(x − x0 )] = t (v)e−it H0 [ei xv ϕ(x − x0 )] −it H0
+ r (v)e + e(x, t),
[e
−i xv
(2.14)
ϕ(−x − x0 )]
where, for any k ≥ 0, e(·, t) L 2 ≤
1 ∂x [θ (x)ϕ(x − x0 )] L 2 v ck + xk ϕ(x) H k (tv)k + 4(1 − θ (x))ϕ(x − x0 ) L 2x .
In §3, Proposition 2.3 will be applied with θ (x) a smooth cutoff to x < 0, and ϕ(x) = sechx with x0 = −v 1−δ 0. Before proving Proposition 2.3, we need the following
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Lemma 2.4. Let ψ ∈ S(R) with supp ψ ⊂ (−∞, 0]. Then 0 e−it Hq [ei xv ψ(x)](x) = e−it H0 [ei xv ψ(x)](x) x− −it H0
(2.15)
+ t (v)e [e ψ(x)](x) x+0 −it H0 −i xv + r (v)e + e(x, t),
i xv
[e
0 ψ(−x)](x) x−
where e(x, t) L 2x ≤
1 ∂x ψ L 2 v
uniformly in t. Proof of Lemma 2.4. By (2.5) with ϕ(x) = ei xv ψ(x), e(x, t) = [e−it H0 (ϕ ∗ (τ − t (v)δ0 ))(x)] x+0 0 + [e−it H0 (ϕ ∗ (ρ − r (v)δ0 ))(−x)] x− ,
and thus it suffices to show e−it H0 (ϕ ∗ (τ − t (v)δ0 ))(x) L 2x ≤
1 ∂x ψ L 2x v
(2.16)
and e−it H0 (ϕ ∗ (ρ − r (v)δ0 ))(x) L 2x ≤
1 ∂x ψ L 2x . v
The proofs of these two estimates are similar, so we only carry out the proof of (2.16). By unitarity of e−it H0 and Plancherel’s identity, ˆ − v)(t (λ) − t (v)) 2 . eit H0 [ϕ ∗ (τ − t (v)δ0 )](x) L 2x = ψ(λ L λ
(2.17)
Since t (λ) − t (v) =
−iq(λ − v) (iλ − q)(iv − q)
we have |t (λ) − t (v)| ≤ (λ − v)/v. Using this to estimate the right-hand side of (2.17) and applying Plancherel’s identity again yields (2.16). Proof of Proposition 2.3. Apply (2.15) to ψ(x) = θ (x)ϕ(x − x0 ) to obtain 0 e−it Hq [ei xv ϕ(x − x0 )](x) = e−it H0 [ei xv ϕ(x − x0 )](x) x−
(2.18)
+ t (v)e−it H0 [ei xv ϕ(x − x0 )](x) x+0 0 + r (v)e−it H0 [e−i xv ϕ(−x − x0 )](x) x− + e1 (x, t) + e2 (x, t),
where e1 (x, t) is as in Lemma 2.4 and (putting f (x) = ei xv (1 − θ (x))ϕ(x − x0 )) 0 e2 (x, t) = + e−it Hq f (x) − e−it H0 f (x) x−
0 − t (v)e−it H0 f (x) x+0 − r (v)e−it H0 [ f (−x)](x) x− .
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By Lemma 2.4, e1 (x, t) L 2x ≤
1 ∂x [θ (x)ϕ(x − x0 )] L 2x v
uniformly for all t, and by unitarity of the linear flows, e2 (x, t) L 2x ≤ 4(1 − θ (x))ϕ(x − x0 ) L 2x also uniformly in all t. Now restrict to the time interval 2|x0 |/v ≤ t ≤ 1. By (2.18), it remains to show that ck xk ϕ(x) Hxk , x0 (tv)k e−it H0 [ei xv ϕ(x − x0 )](x) L 2
≤
(2.19)
The second of these is in fact equivalent to the first, since for any function g(x), e−it H0 [g(−x)](x) = e−it H0 [g(x)](−x). Now we establish (2.19). Since [ei•v ϕ(• − x0 )]ˆ(λ) = e−i x0 (λ−v) ϕ(λ ˆ − v),
e
−it H0
[e
i xv
1 ϕ(x − x0 )](x) = 2π
ˆ − v) dλ ei xλ e−i x0 (λ−v) e−itλ /2 ϕ(λ 1 2 2 = e−itv /2 ei xv ˆ dλ. eiλ(x−x0 −tv) e−itλ /2 ϕ(λ) 2π 2
By k applications of integration by parts in λ, e
iλ(x−x0 −tv) −itλ2 /2
e
ϕ(λ) ˆ dλ =
i x − x0 − tv
k
eiλ(x−x0 −tv) ∂λk [e−itλ
2 /2
ϕ(λ)] ˆ dλ.
Since 2|x0 |/v ≤ t, we have −x0 − tv < 0 and thus |x − x0 − tv| ≥ | − x0 − tv| ≥ tv/2 for x < 0. Hence 0 |e−it H0 [ei xv ϕ(x − x0 )](x)|2 d x −∞
ck iλ(x−x0 −tv) k −itλ2 /2
e ≤ ∂λ [e ϕ(λ)] ˆ dλ (2.20)
2 k (tv) Lx
ck
k −itλ2 /2 = [e ϕ(λ)] ˆ
2
∂ λ k Lλ (tv) from which the result follows by applying the Leibniz product rule and the Plancherel identity once again (and using that t ≤ 1).
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Remark. Suppose that u(x, t) = e−it Hq [ei xv ψ(x)], ψ ∈ S(R), supp ψ ⊂ (−∞, 0), ψ L 2 = 1. Then for t 1 and as v → +∞, ∞ |u(x, t)|2 d x = 0
v2 1 . +O v2 + q 2 v2
(2.21)
In fact using (2.5) and an estimate similar to (2.20) we see that for t ≥ 1, ∞ |u(x, t)|2 d x = e−it H0 ((ei•v ψ) ∗ τq )x+0 2 = (ei•v ψ) ∗ τq 22 + O(v −∞ ) 0
1 ˆ − v)/(iλ − q)2 + O(v −∞ ) iλψ(λ 2π λ2 1 ˆ − v)|2 dλ + O(v −∞ ). |ψ(λ = 2π |λ−v|≤√v λ2 + q 2 =
An expansion in powers of (λ − v)/v gives (2.21). 3. Soliton Scattering In this section, we prove Theorem 1. We recall the notation for operators from Sect. 2 and introduce short-hand notation for the nonlinear flows: • H0 = − 21 ∂x2 . The flow e−it H0 is termed the “free linear flow”. • Hq = − 21 ∂x2 + qδ0 (x). The flow e−it Hq is termed the “perturbed linear flow”. • NLSq (t)ϕ, termed the “perturbed nonlinear flow” is the evolution of initial data ϕ(x) according to the equation i∂t u + 21 ∂x2 u − qδ0 (x)u + |u|2 u = 0. • NLS0 (t)ϕ, termed the “free nonlinear flow” is the evolution of initial data ϕ(x) according to the equation i∂t h + 21 ∂x2 h + |h|2 h = 0. From Sect.1 we recall the form of the initial condition: u 0 (x) = ei xv sech(x − x0 ), v 1, x0 ≤ −v 1−δ , 23 < δ < 1, and we put u(x, t) = NLSq (t)u 0 (x). We begin by outlining the scheme, and will then supply the details. The O notation always means the L 2x difference, uniformly on the time interval specified, and up to a multiplicative factor that is independent of q, v, and δ (any such dependence will be exhibited explicitly). Phase 1 (Pre-interaction). Consider 0 ≤ t ≤ t1 , where t1 = |x0 |/v − v −δ so that x0 + vt1 = −v 1−δ . The soliton has not yet encountered the delta obstacle and propagates according to the free nonlinear flow u(x, t) = e−itv
2 /2
eit/2 ei xv sech(x − x0 − vt) + O(qe−v
1−δ
), 0 ≤ t ≤ t1 .
(3.1)
The analysis here is valid provided v is greater than some absolute threshold (independent 1−δ of q, v, or δ). But if we further require that v be sufficiently large so that v −3/2 ev ≥ α 1−δ ≤ v −1/2 ≤ v −δ/2 . This is the error that arises in the (recall α = q/v), then qe−v main argument of Phase 2 below. Phase 2 (Interaction). Let t2 = t1 + 2v −δ and consider t1 ≤ t ≤ t2 . The incident soliton, beginning at position −v 1−δ , encounters the delta obstacle and splits into a transmitted
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component and a reflected component, which by time t = t2 , are concentrated at positions v 1−δ and −v 1−δ , respectively. More precisely, at the conclusion of this phase (at t = t2 ), u(x, t2 ) = t (v)e−it2 v + r (v)e + O(v
2 /2
−it2
− 12 δ
eit2 /2 ei xv sech(x − x0 − vt2 )
v 2 /2
e
it2 /2 −i xv
e
(3.2)
sech(x + x0 + vt2 )
).
This is the most interesting phase of the argument, which proceeds by using the following three observations: • The perturbed nonlinear flow is approximated by the perturbed linear flow for t1 ≤ t ≤ t2 . • The perturbed linear flow is split as the sum of a transmitted component and a reflected component, each expressed in terms of the free linear flow of soliton-like waveforms. • The free linear flow is approximated by the free nonlinear flow on t1 ≤ t ≤ t2 . Thus, the soliton-like form of the transmitted and reflected components obtained above is preserved. The brevity of the time interval [t1 , t2 ] is critical to the argument, and validates the approximation of linear flows by nonlinear flows. Phase 3 (Post-interaction). Let t3 = t2 + (1 − δ) log v, and consider [t2 , t3 ]. The transmitted and reflected waves essentially do not encounter the delta potential and propagate according to the free nonlinear flow, u(x, t) = e−itv +e
2 /2
eit2 /2 ei xv NLS0 (t − t2 )[t (v)sech(x)](x − x0 − tv)
−itv 2 /2
(3.3)
eit2 /2 e−i xv NLS0 (t − t2 )[r (v)sech(x)](x + x0 + tv)
3
+ O(v 1− 2 δ ),
t2 ≤ t ≤ t3 .
This is proved by a perturbative argument that enables us to evolve forward a time (1−δ) log v at the expense of enlarging the error by a multiplicative factor of e(1−δ) log v = 3 v 1−δ . The error thus goes from v −δ/2 at t = t2 to v 1− 2 δ at t = t3 . Now we turn to the details. 3.1. Phase 1. Let u 1 (x, t) = NLS0 (t)u 0 (x) and u(x, t) = NLSq (t)u 0 (x). Let w = u − u 1 . Recall that t1 = |x0 |/v − v −δ so that x0 + vt1 = −v 1−δ . Note that u 1 (x, t) = e−itv
2 /2
eit/2 ei xv sech(x − x0 − tv).
We will need the following perturbation lemma. Lemma 3.1. If ta < tb , tb − ta ≤ c1 , and w(·, ta ) L 2 + qu 1 (0, t) L ∞ ≤ 1, then [t ,t ] a b
w L ∞
2 [ta ,tb ] L x
≤ c2 (w(·, ta ) L 2x + qu 1 (0, t) L ∞ ), [t ,t ] a b
where the constants c1 and c2 depend only on constants appearing in the Strichartz estimates and are, in particular, independent of q and v.
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Proof. w solves i∂t w + ∂x2 w − qδ0 (x)w = −|w + u 1 |2 (w + u 1 ) + |u 1 |2 u 1 + qδ0 (x)u 1 = − |w|2 w − (2u 1 |w|2 + u¯ 1 w 2 ) − (2|u 1 |2 w + u 21 w) ¯ +qδ0 (x)u 1 . cubic
quadratic
linear
From this equation, w is estimated using Proposition 2.2. For the cubic nonlinear term we take p˜ = r˜ = 6/5 and estimate by Hölder as |w|2 w L 6/5
6/5 [ta ,tb ] L x
≤ (tb − ta )1/2 w2L 6
6 [ta ,tb ] L x
w L ∞
2 [ta ,tb ] L x
.
Since complex conjugates become irrelevant in the estimates, both quadratic terms are treated identically. In Proposition 2.2, we take p˜ = r˜ = 6/5 and estimate by Hölder as u 1 w 2 L 6/5
6/5 [ta ,tb ] L x
≤ (tb − ta )1/2 w2L 6 L 6 u 1 L ∞ L 2x [ta ,tb ] [ta ,tb ] x √ 1/2 2 ≤ 2(tb − ta ) w L 6 L 6 . [ta ,tb ] x
For the linear terms (both of the form u 21 w), we take p˜ = r˜ = 6/5 in Proposition 2.2 and estimate as u 21 w L 6/5
6/5 [ta ,tb ] L x
≤ (tb − ta )1/2 w L 6
6 [ta ,tb ] L x
≤ 2(tb − ta )2/3 w L 6
u 1 L 6
6 [ta ,tb ] L x
6 [ta ,tb ] L x
u 1 L ∞
2 [ta ,tb ] L x
.
The delta term is estimated by the concluding sentence of Proposition 2.2 as qu(0, t) L 4/3
[ta ,tb ]
≤ q(tb − ta )3/4 u(0, t) L ∞ . [t ,t ] a b
Since tb −ta ≤ 1, collecting the above estimates we have (taking w X = w L ∞ L 2x + [ta ,tb ] w L 6 L 6 ), [ta ,tb ] x
. w X ≤ cw(·, ta ) L 2x + c(tb − ta )1/2 (w X + w2X + w3X ) + cqu(0, t) L ∞ [t ,t ] a b
Provided (tb − ta )1/2 ≤ 1/(2c) above, the linear term on the right can be absorbed by the left as . w X ≤ 2cw(·, ta ) L 2x + 2c(tb − ta )1/2 (w2X + w3X ) + 2cqu(0, t) L ∞ [t ,t ] a b
Continuity of w X (tb ) as a function of tb shows that provided 2c(tb − ta )1/2 (4cw(·, ta ) L 2 + 4cqu 1 (0, t) L ∞ ) ≤ 1/2, the above estimate implies [t ,t ] a b
w X ≤ 4cw(·, ta ) L 2x + 4cqu(0, t) L ∞ , [t ,t ] a b
concluding the proof.
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Now we proceed to apply Lemma 3.1. The constants c1 and c2 will, for convenience of exposition, be taken to be c1 = 1 and c2 = 2. Let k ≥ 0 be the integer such that k ≤ t1 < k + 1. (Note that k = 0 if the soliton starts within a distance v of the origin, i.e. −v − v 1−δ ≤ x0 ≤ −v 1−δ , and the inductive analysis below is skipped.) Apply Lemma 3.1 with ta = 0, tb = 1 to obtain (since w(·, 0) = 0) w L ∞
2 [0,1] L x
≤ 2qu 1 (0, t) L ∞ ≤ 2qsech(x0 + v). [0,1]
Apply Lemma 3.1 again with ta = 1, tb = 2 to obtain w L ∞
2 [1,2] L x
≤ 2(w(·, 1) L 2x + qu 1 (0, t) L ∞ ) [1,2] ≤ 22 qsech(x0 + v) + 21 qsech(x0 + 2v).
We continue inductively up to step k, and then collect all k estimates to obtain the following bound on the time interval [0, k]: w L ∞
2 [0,k] L x
≤ 2q
k
2k− j sech(x0 + jv).
j=1
The estimate sechα ≤ 2e−|α| reduces matters to bounding 2k qe x0 +v
k−1
2− j e jv
j=0
and, after summing the geometric series, we obtain w L ∞
2 [0,k] L x
≤ c2k e x0 +v
(2−1 ev )k − 1 ≤ cqe x0 +kv , 2−1 ev − 1
where the last inequality requires 2−1 ev ≥ 2. Finally, applying Lemma 3.1 on [k, t1 ], w L ∞
2 [0,t1 ] L x
≤ c(qe x0 +kv + qsech(x0 + t1 v)) ≤ cqe−v
1−δ
.
As a consequence, (3.1) follows. 3.2. Phase 2. We shall need a lemma stating that the free nonlinear flow is approximated by the free linear flow, and that the perturbed nonlinear flow is approximated by the perturbed linear flow. Both estimates are consequences of the corresponding Strichartz estimates (Proposition 2.2). Crucially, the hypotheses and estimates of this lemma depend only on the L 2 norm of the initial data ϕ. Below, (3.5) is applied with ϕ(x) = u(x, t1 ), and u(x, t1 ) L 2x = u 0 L 2 is independent of v; thus v does not enter adversely into the analysis. Lemma 3.2. If ϕ ∈ L 2 and 0 < tb such that tb < c1 ϕ−4 , then L2 NLS0 (t)ϕ − e−it H0 ϕ L ∞
≤ c2 tb ϕ3L 2 ,
NLSq (t)ϕ − e−it Hq ϕ L ∞
≤ c2 tb ϕ3L 2 ,
2 [0,tb ] L x 2 [0,tb ] L x
1/2
(3.4)
1/2
(3.5)
where c1 and c2 depend only on constants appearing in the Strichartz estimates. In particular, they are independent of q.
Fast Soliton Scattering by Delta Impurities
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Proof. Estimate (3.4) is in fact a special case of (3.5) obtained by taking q = 0. Let h(t) = NLSq (t)ϕ so that i∂t h + 21 ∂x2 h − qδ0 (x)h + |h|2 h = 0 with h(x, 0) = ϕ(x). Let us define 2 6 6 X = L∞ [0,tb ] L x ∩ L [0,tb ] L x ,
with the natural norm, • X . We apply Proposition 2.2 with, in the notation of that proposition, u(t) = h(t) − e−it Hq ϕ, f = −|h|2 h, p = r = 6, p˜ = r˜ = 6/5, and then again with p = ∞, r = 2, p˜ = r˜ = 6/5, to obtain h(t) − e−it Hq ϕ X ≤ c|h|2 h L 6/5
6/5 [0,tb ] L x
.
The generalized Hölder inequality, 1 1 1 1 = + + , p q1 q2 q3
h 1 h 2 h 3 p ≤ h 1 q1 h 2 q2 h 3 q3 ,
applied with h j = h, p = 6/5 and q1 = q2 = 6, q3 = 2, gives h(t) − e−it Hq ϕ X ≤ Ch2L 6
6 [0,tb ] L x
h L 2
1/2
≤ Ctb h2L 6
6 [0,tb ] L x
2 [0,tb ] L x
h L ∞
2 [0,tb ] L x
(3.6)
1/2
≤ Ctb h3X . Another application of the homogeneous Strichartz estimate shows that e−it Hq ϕ X ≤ Cϕ L 2 , and consequently, 1/2
h X ≤ cϕ L 2 + ctb h3X . 1/2
By continuity of h X (tb ) in tb , if ctb (2cϕ L 2 )2 ≤ 1/2, h X ≤ 2cϕ L 2 . Substituting into (3.6) yields the result.
Now we proceed to apply Lemma 3.2. Set t2 = t1 + 2v −δ , and apply (3.5) on [t1 , t2 ] to obtain u(·, t) = NLSq (t − t1 )[u(·, t1 )] = e−i(t−t1 )Hq [u(·, t1 )] + O(v −δ/2 ). By combining this with (3.1), u(·, t) = e−it1 v
2 /2
eit1 /2 e−i(t−t1 )Hq [ei xv sech(x − x0 − t1 v)] + O(v −δ/2 ).
(3.7)
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By Proposition 2.3 with θ (x) = 1 for x ≤ −1 and θ (x) = 0 for x ≥ 0, ϕ(x) = sech(x), and x0 replaced by x0 + t1 v, e−i(t2 −t1 )Hq [ei xv sech(x − x0 − vt1 )](x) = t (v)e−i(t2 −t1 )H0 [ei xv sech(x − x0 − vt1 )](x) + r (v)e−i(t2 −t1 )H0 [e−i xv sech(x + x0 + vt1 )](x)
(3.8)
+ O(v −1 ). By combining (3.7), (3.8) and (3.4), u(·, t) = t (v)e−it1 v
2 /2
+ r (v)e−it1 v
eit1 /2 NLS0 (t2 − t1 )[ei xv sech(x − x0 − vt1 )](x)
2 /2
eit1 /2 NLS0 (t2 − t1 )[e−i xv sech(x + x0 + vt1 )](x)
+ O(v −δ/2 ). By noting that NLS0 (t2 − t1 )[ei xv sech(x − x0 − t1 v)] = e−i(t2 −t1 )v
2 /2
ei(t2 −t1 )/2 ei xv sech(x − x0 − t2 v)
and NLS0 (t2 − t1 )[e−i xv sech(x + x0 + t1 v)] = e−i(t2 −t1 )v
2 /2
ei(t2 −t1 )/2 e−i xv sech(x + x0 + t2 v),
we obtain (3.2). 3.3. Phase 3. Let t3 = t2 + (1 − δ) log v. Label u tr (x, t) = e−itv
2 /2
eit2 /2 ei xv NLS0 (t − t2 )[t (v)sech(x)](x − x0 − tv)
for the transmitted (right-traveling) component and u ref (x, t) = e−itv
2 /2
eit2 /2 e−i xv NLS0 (t − t2 )[r (v)sech(x)](x + x0 + tv)
for the reflected (left-traveling) component. By Appendix A, for each k ∈ N, there is a constant c(k) > 0 and an exponent σ (k) > 0 such that u tr (x, t) L 2
x0
+ |u tr (0, t)| + |u ref (0, t)| ≤
c(k)(log v)σ (k) v k(1−δ)
(3.9)
uniformly on the time interval [t2 , t3 ]. We shall need the following perturbation lemma, again a consequence of the Strichartz estimates.
Fast Soliton Scattering by Delta Impurities
203
Lemma 3.3. Let w = u − u tr − u ref . If ta < tb , tb − ta ≤ c1 , and w(·, ta ) L 2x + then
c(k)q(log v)σ (k) ≤ 1, v k(1−δ)
w L ∞
2 [ta ,tb ] L x
≤ c2 w(·, ta ) L 2x
c(k)q(log v)σ (k) + v k(1−δ)
.
The constants c1 , c2 depend only on constants appearing in the Strichartz estimates and are in particular independent of q and v. Proof. We write the equation satisfied by w: i∂t w + 21 ∂x2 w − qδ0 (x)w = − |w + u tr + u ref |2 (w + u tr + u ref ) + |u tr |2 u tr + |u ref |2 u ref + qδ0 (x)u tr − qδ0 (x)u ref = − |w|2 w − (2(u tr + u ref )|w|2 + (u¯ tr + u¯ ref )w 2 ) ¯ − (2|u tr + u ref |2 w + (u tr + u ref )2 w) − (u 2tr u¯ ref + 2u ref |u tr |2 + u 2ref u¯ tr + 2u tr |u ref |2 ) + qδ0 (x)u tr − qδ0 (x)u ref , tr−delta
tr−ref interaction
ref−delta
and we estimated w using Proposition 2.2. The cubic, quadratic, and linear in w terms on the first line are estimated exactly as was done in the proof of Lemma 3.1. For the “tr − ref interaction terms” (taking u ref |u tr |2 as a representative example), we apply Proposition 2.2 with p˜ = 4/3, r˜ = 1 and estimate as u ref |u tr |2 L 4/3
1 [ta ,tb ] L x
u tr L 2x =
≤ c(tb − ta )3/4 u tr L ∞
2 [ta ,tb ] L x
u ref u tr L ∞
2 [ta ,tb ] L x
,
(3.10)
√ 2|t (v)| by mass conservation for the free nonlinear flow, and
u tr u ref L ∞
2 [ta ,tb ] L x
≤ u tr u ref L ∞
2 [ta ,tb ] L x0
u tr L ∞
2 [ta ,tb ] L x0
.
Now ∞ = NLS0 (t)[r (v)sech](x) L ∞ L ∞ u ref L ∞ t Lx t x
1/2
1/2
≤ NLS0 (t)[r (v)sech](x) L ∞ L 2 ∂x NLS0 (t)[r (v)sech](x) L ∞ L 2 t
x
t
x
≤c ∞ ≤ c. by mass and energy conservation of the free nonlinear flow. Similarly, u tr L ∞ t Lx By this and (3.9), the above yields
u tr u ref L ∞
[ta ,tb
2 ]Lx
≤
c(k)(log v)σ (k) . v k(1−δ)
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J. Holmer, J. Marzuola, M. Zworski
Thus, by (3.10),
c(k)(log v)σ (k) (3.11) v k(1−δ) and similarly for all other “tr−ref interaction” terms. Now we address the “tr−delta” and “ref−delta” terms (working with qδ0 (x)u tr as the representative of both). By Proposition 2.2, we estimate as u ref |u tr |2 L 4/3
1 [ta ,tb ] L x
qu tr (0, t) L 4/3
[ta ,tb ]
≤
≤ c(tb − ta )3/4 qu tr (0, t) L ∞ . [t ,t ] a b
By (3.9),
(log v)σ (k) . (3.12) [ta ,tb ] v k(1−δ) Collecting (3.11), (3.12), and the estimates for cubic, quadratic, and linear terms in w (as exposed in Lemma 3.1), we have, with w X = w L ∞ L 2x + w L 6 L 6 , qu tr (0, t) L 4/3
≤ c(k)q(tb − ta )3/4
[ta ,tb ]
[ta ,tb ] x
w X ≤ cw(·, ta ) L 2 + c(tb − ta )1/2 (w X + w2X + w3X ) +
c(k)q(log v)σ (k) . v k(1−δ)
If c(tb − ta )1/2 ≤ 21 , then the first-order w-term on the right side can be absorbed by the left, giving w X ≤ 2cw(·, ta ) L 2 + 2c(tb − ta )1/2 (w2X + w3X ) + By continuity of w X (tb ) in tb , if 2c(tb − ta )1/2 4cw(·, ta ) L 2
2c(k)q(log v)σ (k) . v k(1−δ)
4c(k)q(log v)σ (k) + v k(1−δ)
≤
1 2
we have w X ≤ 4cw(·, ta ) L 2 + completing the proof.
4c(k)q(log v)σ (k) , v k(1−δ)
Assume that α = q/v has been fixed. Choose k = k(δ) large so that k(1 − δ) ≥ 3. Then the coefficient appearing in Lemma 3.3 is bounded by q (log v)σ (k) c(k)q(log v)σ (k) ≤ c(k) . v v2 v k(1−δ) Now take v sufficiently large in terms of q/v and k (thus in terms of δ) so that the above is bounded by v −1 . Now we implement Lemma 3.3. For convenience of exposition, we take c1 = 1, c2 = 2. Let be the integer such that < (1 − δ) log v < + 1. We then apply Lemma 3.3 successively on the intervals [t2 , t2 + 1], . . . , [t2 + − 1, t2 + ] as follows. Applying Lemma 3.3 on [t2 , t2 + 1], we obtain w(·, t) L ∞
2 [t2 ,t2 +1] L x
≤ 2(w(·, t2 ) L 2x + v −1 ).
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205
Applying Lemma 3.3 on [t2 + 1, t2 + 2] and combining with the above estimate, w(·, t) L ∞
2 [t2 +1,t2 +2] L x
≤ 22 w(·, t2 ) L 2x + (22 + 2)v −1 .
Continuing up to the th step and then collecting all of the above estimates, w(·, t) L ∞
2 [t2 ,t3 ] L x
≤ 2 w(·, t2 ) L 2x + (2 + · · · + 2)v −1 .
Since w(·, t2 ) L 2x ≤ v −δ/2 and 2 ≤ v 1−δ , 3
w(·, t) L ∞
2 [t2 ,t3 ] L x
≤ cv 1− 2 δ ,
(3.13)
thus proving (3.3). Now we complete the proof of the main theorem and obtain (1.6). By (3.13) and (3.9), u(·, t) − u tr (·, t) L 2
x>0
≤ w(·, t) L 2
x>0
+ u ref (·, t) L 2
x>0
3
≤ cv 1− 2 δ .
(3.14) 3
Since u tr (·, t) L 2x = t (v), (3.9) implies u tr (·, t) L 2 = t (v) + O(v 1− 2 δ ), which x>0 combined with (3.14) gives (1.6) and proves Theorem 1. 4. Resolution of Outgoing Waves In this section, we prove Theorem 2. We note that the proof of Theorem 1 presented in §3 in fact provided a more complete long-time description of the solution: u(x, t) = e−itv
2 /2
eit2 /2 ei xv NLS0 (t − t2 )[t (v)sech](x − x0 − tv)
(4.1)
−itv 2 /2 it2 /2 −i xv
NLS0 (t − t2 )[r (v)sech](x + x0 + tv) 3 |x0 | + v −δ ≤ t ≤ c(1 − δ) log v, + O L 2x (v 1− 2 δ ), v where t (v), r (v) are defined in (2.2) and NLS0 (t)ϕ denotes the solution to the NLS equation i∂t h + 21 ∂x2 h + |h|2 h = 0 (without potential) and initial data h(x, 0) = ϕ(x). It thus suffices to obtain the resolution of NLS0 (t − t2 )[t (v)sech] and NLS0 (t − t2 )[r (v)sech] into solitons plus radiation decaying in L ∞ x . By the phase invariance of the free nonlinear flow t (v) NLS0 (t − t2 )[|t (v)|sech] NLS0 (t − t2 )[t (v)sech] = |t (v)| +e
e
e
and similarly for NLS0 (t − t2 )[r (v)sech]. Since 0 ≤ |t (v)|, |r (v)| ≤ 1, we apply asymptotics (B.1) proved of Appendix B using the inverse scattering method. When |t (v)| or |r (v)| is equal to 1/2 we use the result of [12] recalled in (B.2). The result obtained by these substitutions differs from that stated in Theorem 2 by a factor of 1 − A2T −δ exp i ·v (4.2) 2 for u T (x, t), owing to the fact that t2 = |x0 |/v + v −δ . But (4.2) differs from 1 by ∼ v −δ , and thus omitting it only introduces a discrepancy of v −δ in both L 2x and L ∞ x . There is a similar inconsequential disparity in the u R (x, t) part.
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Appendix A. Spatial Localization of the Free Nonlinear Propagation Let ϕ ∈ S and
i∂t h + 21 ∂x2 h + |h|2 h = 0, h(x, 0) = ϕ(x).
(A.1)
Notational conventions. We denote ∂x by ∂ hereafter. The x and t dependence of h(x, t) will be routinely dropped. The constants c(k), σ (k), and the polynomials gγ (t) that appear below may change (enlarge) from one line to the next without comment. The constants c(k) depend on the fixed function ϕ ∈ S. The solution h satisfies conservation of mass and conservation of energy, which means that the integrals E0 = |h|2 d x, E 2 = − (|∂h|2 − |h|4 )d x, R
R
are independent of time t. Since h2L ∞ ≤ h L 2 ∂h L 2 , we have h4L 4 ≤ h3L 2 ∂h L 2 and it follows from the E 2 and E 0 conservation that ∂h L 2 ≤ c, where c depends on ϕ L 2 and ∂ϕ L 2 . In fact, there are an infinite number of conserved integrals, E k , with integrands defined inductively as follows: 1 fk + f j1 f j2 , (A.2) f 0 = |h|2 , f k+1 = h∂ h j1 + j2 =k−1
see [17, §8] for a proof of this fact (rescaling time and putting κ = 2 produces an agreement with our slightly different convention). The inductive definition of f k and the Sobolev embedding theorem can now be used to show that, for ≥ 2, E 2 = (−1) |∂ h(x)|2 d x + O((1 + h Hx−1 )2+2 ), (A.3) R
and hence for ≥ 0, we have
∂ h L 2 ≤ c(),
(A.4)
where c() depends upon Sobolev norms of the initial data ϕ of at most order . We now elaborate on how to obtain (A.3). An inductive argument using (A.2) shows that for k ≥ 0, f k is of the form f k = h∂ k h¯ + h p(2 j + 1, k − 2 j), (A.5) j≥1, 2 j≤k
where p(n, m) indicates a linear combination of terms of degree n and cumulative order m, or more precisely terms of the form ˜ α1 + · · · + αn = m ∂ α1 h˜ ∂ α2 h˜ · · · ∂ αn h,
(A.6)
¯ To prove (A.3) for ≥ 2, one uses (A.5) for k = 2 and it only and h˜ is either h or h. remains to verify that for any n ≥ 4 and m ≤ 2 − 2, (A.7) p(n, m) d x ≤ hnH −1 . We now show this. Note that in (A.6), we may assume without loss of generality that α1 ≤ α2 ≤ · · · ≤ αn .
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Case 1. αn ≤ − 1. It follows that α j ≤ − 2 for all j ≤ n − 2 and αn−1 ≤ − 1. We estimate as: ⎛ ⎞ n−2 ∂ α1 h˜ ∂ α2 h˜ · · · ∂ αn h˜ d x ≤ ⎝ ∂ α j h L ∞ ⎠ ∂ αn−1 h L 2 ∂ αn h L 2 ≤ chnH −1 j=1
by Sobolev embedding. Case 2. αn ≥ . In this case, we begin by integrating by parts to obtain ˜ ∂ −1 h˜ d x. (−1)αn −+1 ∂ αn −+1 (∂ α1 h˜ · · · ∂ αn−1 h)
(A.8)
The Leibniz rule expansion is ˜ = ∂ αn −+1 (∂ α1 h˜ · · · ∂ αn−1 h)
˜ cµ ∂ µ1 +α1 h˜ · · · ∂ µn−1 +αn−1 h,
(A.9)
where the sum is over (n − 1)-tuples µ such that µ1 + · · · + µn−1 = αn − + 1 and cµ is some constant depending on µ. By adding the α and µ constraints, we obtain that (µ1 + α1 ) + · · · + (µn−1 + αn−1 ) ≤ − 1 and thus there is at most one index j∗ (1 ≤ j∗ ≤ n − 1) such that µ j∗ + α j∗ = − 1 and for all remaining j (1 ≤ j ≤ n − 1, j = j∗ ) we have µ j + α j ≤ − 2. (If no such j∗ exists, take j∗ to be any fixed index 1 ≤ j∗ ≤ n − 1.) By substituting (A.9) into (A.8), we estimate as ∂ α1 h˜ ∂ α2 h˜ · · · ∂ αn h˜ d x ⎛ ⎞ n−1 ≤⎝ ∂ µ j +α j h L ∞ ⎠ ∂ µ j∗ +α j∗ h L 2 ∂ −1 h L 2 ≤ chnH −1 j=1, j= j∗
again by Sobolev embedding. This concludes the proof of (A.7), thus (A.3), and thus (A.4). Using that the commutator [(x + it∂), i∂t + 21 ∂ 2 ] = 0 and some integration by parts manipulations, we have the pseudoconformal conservation law: t 2 2 4 4 |(x + it∂)h(x, t)| d x −t |h(x, t)| d x + s |h(x, s)| d xds = |xϕ(x)|2 d x. x
x
0
x
x
From this, (A.4) for = 0, 1, and the Gagliardo-Nirenberg estimate hL44 ≤ hL32 ∂h L 2 , we have xh L 2 ≤ ct, where c depends on xϕ L 2 , ϕ L 2 , and ∂ϕ L 2 . We want to show that more generally, for each k ∈ Z, k ≥ 0, we have x α ∂ β h L 2 ≤ c(k)tσ (k)
for α + β = k, α, β ≥ 0, α, β ∈ Z.
(A.10)
Here c(k) is a constant depending on k and weighted Sobolev norms of the initial data ϕ (up to order 2k), and σ (k) is a positive exponent depending upon k. We are not concerned
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J. Holmer, J. Marzuola, M. Zworski
with obtaining the optimal value of σ (k); the mere fact that the bound in (A.10) is power-like in t, as opposed to exponential in t, suffices for our purposes. In our proof, both c(k) and σ (k) will be increasing with k, and will go to +∞ as k → +∞. Let 0 = ∂ and 1 = (x + it∂). Note that both operators have the commutator property [ j , (i∂t + 21 ∂ 2 )] = 0, j = 0, 1. (A.11) We first claim that for each k ≥ 0, there exists a constant c(k) > 0 and an exponent σ (k) > 0 such that j1 j2 · · · jk h L 2 ≤ c(k)tσ (k) for all j1 , . . . jk ∈ {0, 1}.
(A.12)
When we wish to consider a composition of the form j1 j2 · · · jk and do not care to report whether each operator in the composition is 0 or 1 , we will instead write the composition as k . We prove (A.12) by induction on k . When k = 0, (A.12) is just the mass conservation law. Suppose that (A.12) holds for 0, . . . , k − 1; we aim to prove it holds for k. The main ingredient (in addition to the inductive hypothesis) is (A.4). Fix j1 , . . . jk ∈ {0, 1}, and apply the operator j1 · · · jk to the equation, pair with −i j1 · · · jk h, integrate in x, take twice the real part, and appeal to (A.11) to obtain (A.13) ∂t j1 · · · jk h2L 2 = 2 Re i j1 · · · jk |h|2 h j1 · · · jk h d x. Note that ¯ = ∂ F(h, h) ¯ h + F(h, h) ¯ 0 h 0 F(h, h)h and ¯ = it∂ F(h, h) ¯ h + F(h, h) ¯ 1 h. 1 F(h, h)h Both of these product rules take the form ¯ h + F(h, h) ¯ h, |h|2 h = g(t)∂ F(h, h) where g(t) is a polynomial in t of degree ≤ 1. Thus we see that j1 · · · jk |h|2 h = |h|2 j1 · · · jk h + gγ (t)∂ γ1 |h|2 γ2 h, γ1 +γ2 =k γ2 ≤k−1
where gγ (t) is a polynomial in t. Substituting into (A.13), we obtain two terms: the first is zero since it is the real part of a purely imaginary number; the second is estimated by the Hölder inequality to obtain: | ∂t j1 · · · jk h2L 2 | ≤ c(k)tσ (k) ×
sup ∂ j |h|2 L ∞ j≤k−1
sup h L 2 j1 · · · jk h L 2 . j
j≤k−1
By Sobolev embedding estimates, (A.4), and the induction hypothesis, we have | ∂t j1 · · · jk h2L 2 | ≤ c(k)tσ (k) j1 · · · jk h L 2 from which (A.12) follows.
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Now to deduce (A.10) from (A.12), we just note that since x = 1 − it0 , there are polynomials g j (t) such that the following relation holds: xα∂β = g j (t) j1 · · · jα+β . j∈{0,1}α+β
Let us now consider the application of (A.10) to obtain (3.9) in the Phase 3 analysis. We have x0 + tv ≥ v 1−δ for t ≥ t2 . If x < 0, then v k(1−δ) ≤ (x0 + tv)k ≤ |x − x0 − tv|k . Thus v k(1−δ) u tr (x, t) L 2
x