Commun. Math. Phys. 250, 1–21 (2004) Digital Object Identifier (DOI) 10.1007/s00220-004-1137-0
Communications in
Mathematical Physics
On the Structure of KMS States of Disordered Systems Stephen Dias Barreto1 , Francesco Fidaleo2 1
Department of Mathematics, Padre Conceicao College of Engineering, Verna Goa 403 722, India. E-mail:
[email protected] 2 Dipartimento di Matematica, Universit`a di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Roma, Italy. E-mail:
[email protected] Received: 18 March 2003 / Accepted: 12 March 2004 Published online: 27 July 2004 – © Springer-Verlag 2004
Abstract: We study the set of KMS states of spin systems with random interactions. This is done in the framework of operator algebras by investigating Connes and Borchers –invariants of W ∗ –dynamical systems. In the case of KMS states exhibiting a property of invariance with respect to the spatial translations, some interesting properties emerge naturally. Such a situation covers KMS states obtained by infinite–volume limits of finite–volume Gibbs states, that is the quenched disorder. This analysis can be considered as a step towards fully understanding the very complicated structure of the set of temperature states of quantum spin glasses, and its connection with the breakdown of the symmetry for replicas.
1. Introduction Traditionally, spin glasses have been studied as spin systems with random interactions. Extensive investigations on the existence of the thermodynamic limit have been made, and the equilibrium statistical mechanics of such systems has been studied, see [6, 8, 16, 27, 28, 30, 42] and the literature cited therein. It is expected that a spin glass exhibits phase transition at temperatures below a critical temperature Tc = Tf , where Tf is the freezing temperature of the spin glass. There is thus a breakdown of ergodicity, and each equilibrium state can be decomposed into its ergodic components with the same free energy, see e.g. [15, 16, 28], see also [41] for a very recent rigorous result. One of the main problems with the spin glass is to characterize and label the ergodic components. Because of the complicated random structure, this becomes a formidable task. In an attempt to understand this complex behaviour of a quantum spin glass, we study its temperature (i.e. KMS) states. This is done in the framework of operator algebras by modelling the spin glass as a quantum spin system with random interactions. Our analysis applies to the more general situation of disordered spin models, the spin glasses being the most interesting examples among them.
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S.D. Barreto, F. Fidaleo
We start by using a standard procedure (see e.g. [13], Sect. IV.6.γ ), and define the C∗ appropriate C ∗ –algebra of observables as A := A⊗L∞ (, µ). Here, A := Mn (C) Zd
is the algebra of observables of a spin system on the lattice Zd , whereas, in order to take into account the disorder, the probability space (, µ) is the sample space for the coupling constants, the latter being random variables on it. Further, we also assume that the space translations act ergodically on (, µ) as measure preserving transformations. In this way, the time evolution and the spatial translations can be appropriately defined on the whole algebra A as mutually commuting groups of automorphisms, see below. It is well–known that the set of temperature states of such a system is a very intricate one. Roughly speaking, the presence of multiple phases is connected to the existence of some temperature state ϕ such that the GNS representation πϕ generates a von Neumann algebra with non–trivial centre Zπϕ .1 However, we always have πϕ I ⊗ L∞ (, µ) ⊂ Zπϕ for the temperature states under consideration. Thus, in order to understand the structure of KMS states of a disordered system, it seems to be natural to decompose any such state with respect to the image πϕ I ⊗ L∞ (, µ) of the common algebra L∞ (, µ) as a first step. This allows us to investigate the Borchers invariant B of the von Neumann algebra πϕ (A) generated by the GNS representation of the KMS state ϕ.2 Our main results can be summarized as follows. We start by proving a result which is of interest in itself (Theorem 4.4). Namely, consider a C ∗ –algebra B equipped with a sequence {αn }n∈N of ∗–automorphisms, and a state ϕ left invariant by the αn . If the GNS representation πϕ of ϕ acts on a separable Hilbert space, the support–projection of ϕ in the bidual B∗∗ is central and the sequence {αn }n∈N is aymptotically Abelian with respect to ϕ, then the type III0 component is absent in πϕ (B) . Coming back to our set–up, we consider locally normal KMS states ϕ on A, invariant with respect to the spatial translations.3 Such a situation naturally arises for KMS states obtained by infinite–volume limits of finite–volume Gibbs states, that is it covers quenched disorder. For such states, the Borchers invariant B (πϕ (A) ) coincides up to a normalization depending on the temperature, with the Arveson spectrum sp(τ ω ) of the random evolution {τtω }t∈R at fixed ω ∈ , the latter being almost surely independent of ω, see [4]. Further, let the centre Zπϕ of the GNS representation of ϕ coincide with L∞ (, µ). This situation can be considered the natural substitute of “pure thermodynamical phase” in the disordered case. In the last case, we conclude that there exists a unique λ ∈ (0, 1] such that the GNS representation πϕ of ϕ is made of type IIIλ factors almost surely. Such result is in accordance with the standard fact that the physically relevant quantities do not depend on the disorder. Possible applications are presented by an example discussed in some detail in Sect. 5. We end with Sect. 6 containing some general considerations on the KMS boundary condition for quantum disordered spin systems, taking into account known results relative to the classical case. The paper is supplemented by some appendices containing technical results relative to the direct integral decomposition of unbounded selfadjoint operators, the explicit description of their spectra, and finally some useful properties of Connes and Borchers –invariants of W ∗ –dynamical systems. 1
For the Gelfand–Naimark–Segal (GNS for short) constuction see [39], Sect. I.9. The Borchers invariant B is the natural generalization of the Connes invariant to W ∗ –dynamical systems with non–trivial centres, see [7, 11, 32]. 3 In our setting, “locally normal” simply means that ϕ is normal when restricted to I ⊗ L∞ (, µ), see Sect. 2. 2
KMS States of Disordered Systems
3
2. Preliminaries In the present paper we deal only with von Neumann algebras with separable preduals unless specified otherwise. Besides, all representations of C ∗ –algebras are understood to be on separable Hilbert spaces. We start with the usual spin algebra A :=
C∗
Mj
,
j ∈Zd
where Mi = Mn (C) for every site i ∈ Zd , and the C ∗ –completion is defined w.r.t. the unique C ∗ –cross norm. We consider also a standard measure space (, µ) based on a compact separable space , and a Borel probability measure µ. The group Zd of the spatial translations is supposed to act on the probability space (, µ) by measure preserving ergodic transformations {Tx }x∈Zd . A one–parameter random group of automorphisms (t, ω) ∈ R × → τtω ∈ Aut(A)
(2.1)
is acting on A. It is supposed to be strongly continuous in the time variable for each fixed ω ∈ , and jointly strongly measurable. Consider, for a localized element A, the strongly measurable function fA,t (ω) := τtω (A). We get fA,t L∞ (,µ;A) ≡ esssup τtω (A) A = A A , ω∈
where the last equality follows as τtω
is isometric. We assume further that τ acts locally. Namely, if A is a localized element of A, then the function fA,t ∈ L∞ (, µ; A) belongs to the C ∗ –subalgebra A ⊗ L∞ (, µ).4 The group {αx }x∈Zd of spatial translations acts in a natural way on A by shifting the observables on the lattice. Finally, we assume the commutation rule τtTx ω αx = αx τtω
(2.2)
for each x ∈ Zd , ω ∈ , and t ∈ R. Such a picture arises in a natural way in the study of disordered systems (see e.g. [5, 13]), and more precisely when one considers spin systems with random Hamiltonians. The last one is the natural framework of spin glasses, see [6, 15, 16, 27] and the references cited therein. For example, one could start from a net {H (ω)} ⊂Zd , the
being finite subsets of Zd , of local random Hamiltonians, which is made up of (A )s.a. – valued measurable functions arising from finite–range interactions, and satisfying the equivariance condition H +x (ω) = αx (H (T−x ω)) .
(2.3)
If this is the case, we assume that the one–parameter random automorphisms group (2.1) will arise from the infinite–volume limit of the corresponding ones relative to the local random Hamiltonians satisfying (2.3), see [10], Sect. 6.2, and [4], Sect. 3. In order to avoid trivial situations, we suppose that the random automorphisms group (2.1) 4
This fact would follow from ergodicity, for random systems arising from finite–range interactions described in Sects. 3 and 4 of [4], provided that the action of Zd on the sample space (, µ) is ergodic as described above. It is trivially satisfied for almost all (quantum versions of) models present in literature (see e.g. [6, 8, 27, 28, 30, 42]) as that in Sect. 5, taking into account that the addenda in (5.1) commute with each other.
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is non–trivial. Notice that all the conditions listed above are automatically satisfied in many cases of interest, see for example the pivotal model treated in Sect. 5. The following result concerns the almost surely independence of the Arveson spectrum for random systems. This is precisely Theorem 5.3 of [4]. For the reader’s convenience, we report here such a result by filling a slight gap in the original proof.5 Theorem 2.1. Under the above assumptions, there exists a measurable set F ⊂ of full measure, and a closed set ⊂ R such that ω ∈ F implies sp(τ ω ) = . Proof. We get by [32], Proposition 8.1.9, s ∈ R |fˆ(s)| ≤ τfω sp(τ ω ) = f ∈L1 (R)
where “ˆ” stands for (inverse) Fourier transform, and +∞ τfω (A) := f (t)τtω (A) dt , −∞
the integral being understood in the Bochner sense. By a standard density argument, we can reduce the situation to a dense set {fk }k∈N ⊂ L1 (R). Define k (ω) := τfωk . It was shown in [4] that the functions k are measurable and invariant. By ergodicity, they are constant almost everywhere. Let {Nk }k∈N be null subsets of such that, for each k ∈ N c and ω ∈ Nk , k (ω) = k ∞ . c ω0 Consider F := k∈N Nk , and take := sp(τ ), where ω0 is any element of F . As an immediate consequence of this, we have that F is a measurable set of full measure, and ω ∈ F implies sp(τ ω ) = .
In order to study the temperature states of such a disordered system, we define A := A ⊗ L∞ (, µ), ∗ where the above C ∗ –tensor product is uniquely determined as any commutative C – d algebra is nuclear. The following local structure ([9]) A finite subsets of Z of A is inherited from the corresponding one of A by defining for finite subsets of Zd ,
A = A ⊗ L∞ (, µ) . Notice that, by identifying A with a closed subspace of L∞ (, µ; A), each element A ∈ A is uniquely represented by a measurable essentially bounded function ω → A(ω) with values in A. The group Zd of all the space translations is naturally acting on the C ∗ –algebra A as
Further, define on
ax (A)(ω) := αx (A(T−x ω)) .
(2.4)
A , bounded,
tt (A)(ω) := τtω (A(ω)) .
(2.5)
5 Compare with the analogous result ([23], Th´eor`eme III.1) concerning the spectrum of a one–dimensional random discretized Schr¨odinger operator.
KMS States of Disordered Systems
5
Proposition 2.2. Under the above assumptions, {tt }t∈R extends to a one–parameter group of automorphisms of A satisfying t t a x = a x tt ,
t ∈ R, x ∈ Zd .
(2.6)
Proof. Let A be a localized element of A. Then, A(ω) = ak (ω)Ak , k
where the above sum is finite, the ak are complex valued measurable essentially bounded functions, and {Ak } ⊂ A , for some bounded region ⊂ Zd . By (2.5), we get ak (ω)τtω (Ak ) . tt (A)(ω) = k
) are measurable Under the assumptions listed above, the functions ω → τtω (Ak
A → A. We essentially bounded functions in A ⊗ L∞ (, µ) ≡ A. Hence, tt :
compute
tt (A) A ≡ esssup τtω (A(ω)) A = esssup A(ω) A ≡ A A . ω∈
ω∈
Namely, tt is a ∗–preserving isometric homomorphism of the dense set
A of A,
{tt }t∈R
into A with inverse t−t . Further, t0 = id and tt1 +t2 = tt1 tt2 . Therefore, extends to a one–parameter group of automorphisms of all of A. Taking into account (2.2), it readily follows that {tt }t∈R and {ax }x∈Zd commute with each other.
In order to take into account the effects of the disorder, it is natural to start by considering A as describing the algebra of observables on which the space translations, described by {ax }x∈Zd , as well as the time evolution given by {tt }t∈R act, see [5, 13]. Notice that A contains copies A⊗I and I ⊗L∞ (, µ) of A and L∞ (, µ) respectively, denoted by an abuse of notation, also by A and L∞ (, µ). Let π be a representation of A. We easily get π(L∞ (, µ)) ⊂ Zπ . Suppose that π is normal when restricted to L∞ (, µ). The last condition simply means that π is locally normal as each local algebra A is finite–dimensional, see [38]. In such a situation, there exists a measure ν on , absolutely continuous w.r.t. µ, such that π(L∞ (, µ)) ∼ L∞ (, ν) .
(2.7)
Equivalently, we can find a measurable set Eν ⊂ (precisely the “support” of ν) uniquely determined by ν modulo null sets, such that one can choose for ν ν(E) = µ(E ∩ Eν ) . As π is locally normal, we have also π(A) = π(A ⊗ C()) , that is π(A ⊗ C()) is a weakly dense separable C ∗ –subalgebra of π(A) .
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S.D. Barreto, F. Fidaleo
We can consider the subcentral decomposition of the restriction of π to the separable C ∗ –subalgebra A ⊗ C(), w.r.t. π(L∞ (, µ)) ≡ π(C()) , see [39], Theorem IV 8.25. We obtain ⊕ π= πω µ(dω) (2.8)
on Hπ =
⊕
Hω µ(dω)
where, for ω ∈ Eνc , the complement of the support Eν of ν, πω is the trivial representation on the trivial Hilbert space Hω ≡ {0}. The measurable field {πω }ω∈ of representations of A ⊗ C() is uniquely determined by its restriction on A. This follows from the fact that for each A ∈ A and f ∈ L∞ (, µ), we have ⊕ π(A ⊗ f ) = f (ω)πω (A ⊗ I )µ(dω) .
Now, if
M := π(A) =
⊕
Mω ν(dω) ,
then for almost all ω ∈ ,
Mω = πω (A ⊗ L∞ (, µ)) ≡ πω (A ⊗ I ) . If ϕ is a locally normal state on A, then there exists ([39], Prop. IV.8.34) a ∗–weak measurable field {ϕω }ω∈ of positive forms on A such that, for each A ∈ A, ϕ(A) = ϕω (A(ω))µ(dω) , (2.9)
the function ω → A(ω) being the representative of A in L∞ (, µ; A). Consider the GNS representation πϕ relative to ϕ. It is straightforward to check that, for almost all ω ∈ , πϕω is unitarily equivalent to the restriction of πω to A ⊗ I ∼ A, where πϕω is the GNS representation of ϕω , and πω is the representation occurring in (2.8) relatively the decomposition of πϕ . Finally, we show that A is asymptotically Abelian w.r.t. spatial translation under any locally normal state. Proposition 2.3. Let ϕ be a locally normal state on A. Then, for each A, B ∈ A we have, lim ϕ [ax (A), B]∗ [ax (A), B] = 0 . |x|→+∞
Proof. As the elementary tensors are total in norm, we prove the assertion for such a total set. Let f, g ∈ L∞ (, µ) and A, B ∈ A. Taking into account
ϕω µ(dω) = ϕ(1) = 1,
we obtain by (2.4) and (2.9), ϕ [ax (A ⊗ f ), B ⊗ g]∗ [ax (A ⊗ f ), B ⊗ g] f (T−x ω)g(ω)2 ϕω [αx (A), B]∗ [αx (A), B] µ(dω) =
2 2 ≤ f ∞ g ∞ ϕω µ(dω) [αx (A), B] 2 −→ 0 ,
as A is asymptotically Abelian in norm w.r.t. the spatial translations.
KMS States of Disordered Systems
7
For most of the models considered in the literature, the time evolution (2.1) exhibits better regularity properties. For example, it could be jointly continuous. In the last situation, {tt }t∈R given by (2.5) leaves A ⊗ C() globally stable. Further, the space describing the disorder could have a natural local structure. As the disorder is kept fixed during the analysis (we should treat only states ϕ such that µϕI ⊗C() ≺ µ), we have chosen A ⊗ L∞ (, µ) instead of A ⊗ C() as the algebra of observables. We also have disregarded the possible local structure of the disorder. In this way, the information about the disorder is directly encoded into the algebra of observables, yet we can treat more general models.6
3. General Properties of Temperature States In the present section we study some general properties of KMS states on the algebra of observables. We start by recalling the definition of the KMS boundary condition. be the space consisting of the (analytic continuation of the) Fourier transLet D forms of functions in the class D, the latter being the space made of all infinitely often differentiable compactly supported functions in R. A state ϕ on the C ∗ –algebra B satisfies the KMS boundary condition at inverse temperature β ∈ R\{0}, w.r.t the group of automorphisms {τt }t∈R if (i) t → ϕ(Aτt (B)) is a continuous function for every A, B ∈ B, (ii) ϕ(Aτt (B))f (t − iβ) dt = ϕ(τt (B)A)f (t) dt whenever f ∈ D. For the equivalent characterizations of the KMS boundary condition, the main results about KMS states, and finally the connections with Tomita theory of von Neumann algebras, see [10, 18, 34, 35, 40] and the references cited therein. Let ϕ be a KMS state on B w.r.t. {τt }t∈R at inverse temperature β which are kept fixed during the analysis. Consider all positive selfadjoint operators T ηZπϕ such that ϕ ∈ DT 1/2 and T 1/2 ϕ = 1.7 Consider the state ϕT on B given by ϕT := (πϕ ( · )T 1/2 ϕ , T 1/2 ϕ ) .
(3.1)
6 Consider the following modification of the model described by (5.1), whose formal Hamiltonian is given by
H =−
1 f (Ji,j )(i − j )σ (i)σ (j ) , 2 d
(2.10)
i,j ∈Z
where f is a real valued bounded Borel non-continuous function on K, and , µ are given in (5.2). Then, the model is essentially the same as that in Sect. 5, but the time evolution associated to (2.10) does not preserve the continuous A–valued functions. One can reduce oneself to the situation of Sect. 5 by taking the pull–back measure by f . In this way, the C ∗ –algebra of observables changes drastically, without affecting the substance of the model described by (2.10). Furthermore, if we consider any very simple disordered spin system on Z2 subjected to an external random two–value magnetic field described by a chessboard configuration (i.e. consisting of two points), the disorder variables have no local structure. 7 For a closed operator T with polar decomposition T = V |T |, being affiliated to a von Neumann algebra M means that both V and (1 + T ∗ T )−1 belong to M. In such a situation we write T ηM.
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S.D. Barreto, F. Fidaleo
Proposition 3.1. There is a one–to–one correspondence T → ϕT between (i) the set {T } of positive operators affiliated to the centre Zπϕ such that ϕ ∈ DT 1/2 and T 1/2 ϕ = 1, (ii) the set {ϕT } of KMS states normal w.r.t. ϕ.8 Proof. By rescaling, we can suppose that β = 1. Put M := πϕ (B) . Let T be as above, then ϕT defined by (3.1) is a KMS state on B by Prop. 5.3.29 of [10]. Consider the cyclic projection P ∈ M onto the subspace [MT 1/2 ϕ ]. Then P ∈ Zπϕ as it coincides with the central support of T . The induction map ρ(X) = P X[MT 1/2 ϕ ] realizes the searched map of πϕ (B) = M onto πϕT (B) = MP . Namely, ϕT is normal w.r.t. ϕ. Suppose now that ψ is a KMS state on B normal w.r.t. ϕ. Let ρ be the normal map of πϕ (B) onto πψ (B) such that, for each A ∈ B, ρ(πϕ (A)) = πψ (A). Let now f ∈ D, and X = πϕ (A), Y = πϕ (B). We get ϕ ρ(Xσt (Y ))ψ , ψ f (t − i) dt ≡ ψ(Aτt (B))f (t − i) dt ϕ = ψ(τt (B)A)f (t) dt ≡ ρ(σt (Y )X)ψ , ψ f (t) dt, where σ ϕ is the modular group relative to the extension ( · ϕ , ϕ ) of ϕ to all of M (which we are also denoting by ϕ). As πϕ (B) is dense in M w.r.t. the strong operator topology, we conclude by a standard density argument that (ρ( · )ψ , ψ ) is also a KMS state on M w.r.t. σ ϕ . Then, by [10], Prop. 5.3.29, there exists a positive operator T as above such that ψ = ϕT . The uniqueness follows again by Prop. 5.3.29 of [10].
Now we specialize the above analysis to the case under consideration. Proposition 3.2. Let ϕ be a locally normal t–KMS state on A at inverse temperature β. Consider the field {ϕω }ω∈ of positive forms on A in (2.9) arising from the subcentral decomposition w.r.t. πϕ (L∞ (, µ)). Then, for almost all ω ∈ , the form ϕω is τ ω –KMS at the same inverse temperature β. Proof. As tt does not leave in general A ⊗ C() globally stable, we cannot directly apply Theorem A.13 of [37] to the situation under consideration. So we give a direct Taking into account (2.9), proof of the result. Namely, let ϕ be a KMS state and f ∈ D. we get by the Fubini Theorem, µ(dω)h(ω) dt[ϕω (Aτtω (B))f (t − iβ) − ϕω (τtω (B)A)f (t)] = 0 for each h ∈ L∞ (, µ), and A, B ∈ A.9 Then, for each A, B ∈ A there exists a measurable set A,B ⊂ of full measure such that on A,B , (3.2) ϕω (Aτtω (B))f (t − iβ) dt = ϕω (τtω (B)A)f (t) dt . 8 Recall that a representation π of the C ∗ –algebra B is normal w.r.t. another representation π if 1 2 there exists a normal ∗–epimorphism ρ : π2 (B) → π1 (B) such that ρ ◦π2 = π1 . The state ϕ ∈ S (B) is normal w.r.t. ψ ∈ S (B) if the GNS representation πϕ is normal w.r.t. πψ . 9 Notice that ϕ could be trivial on a set of positive measure. This does not affect the proof as we will ω recover that ϕω should be the trivial KMS functional on such a set.
KMS States of Disordered Systems
9
We choose a countable dense set A0 ⊂ A. Then (3.2) is satisfied for each A, B ∈ A0 , on the set 0 := A,B ⊂ of full measure. Fix A, B ∈ A and choose sequences A,B∈A0
{An }, {Bn } in A0 , converging to A, B respectively. We get by the Lebesgue Dominated Convergence Theorem, ϕω (Aτtω (B))f (t − iβ) dt = lim ϕω (An τtω (Bn ))f (t − iβ) dt n = lim ϕω (τtω (Bn )An )f (t) dt = ϕω (τtω (B)A)f (t) dt, n
which is the KMS boundary condition for all ϕω such that ω ∈ 0 .
In what follows, we give a description of KMS states which are normal w.r.t. a fixed locally normal KMS state in terms of fields of positive operators. Let ϕi , i = 1, 2, be KMS states on A, normal w.r.t. a fixed locally normal KMS state ϕ. Consider the subcentral decomposition (2.8) of the GNS representation πϕ w.r.t. L∞ (, µ). By Proposition 3.1, Proposition 3.2 and Proposition A.2, there exist measurable fields {Ti (ω)}ω∈ , i = 1, 2, of positive operators affiliated to Zπϕω . Suppose that Ei , i = 1, 2, is the complement of the measurable set consisting of all ω ∈ for which Ti (ω) = 0. It is straightforward to verify that (i) πϕ1 ≺ πϕ2 ⇒ ν(E1 \E2 ) = 0, (ii) πϕ1 ◦ πϕ2 ⇒ ν(E1 ∩ E2 ) = 0, where ≺ and ◦ mean the usual containment and disjointness of representations respectively, and ν is the measure appearing in (2.7). The reverse implications can be obtained by looking at each fibre. 4. Invariant Temperature States and Their Spectral Properties In the present section we study the structure of states on A which are invariant w.r.t. the action {ax }x∈Zd of spatial translations (invariant for short). Roughly speaking, the last property means that the state under consideration satisfies the invariance property only “in average”.10 Then we specialize our analysis to the case of KMS states. The last ones enjoy some nice structural properties we are going to analyze. We start by summarizing some basic properties of locally normal states (not necessarily KMS) invariant w.r.t. spatial translations. Theorem 4.1. Let ϕ be a locally normal invariant state on A. Consider the decomposition appearing in (2.9). Then (i) ϕω ◦ αx = ϕT−x ω for all x ∈ Zd , (ii) ϕω (I ) = 1, (iii) Zπϕω ∼ = ZπϕT ω for all x ∈ Zd , x
where the above equalities, as well as the unitary equivalence, are satisfied almost everywhere. Proof. For each locally normal invariant state ϕ on A and each A ∈ A, we obtain by (2.4), (2.9), and after an elementary change of variable, 10 Such an invariant state ϕ on A is described by a class of states {ϕ } ω ω∈ on the spin algebra A exhibiting a natural equivariance property, see Theorem 4.1.
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S.D. Barreto, F. Fidaleo
ϕ ◦ ax (A) = = ≡
=
ϕω (αx (A(T−x ω)))µ(dω) ϕTx ω (αx (A(ω)))µ(dω) ϕTx ω ◦ αx ((A(ω)))µ(dω) ϕω ((A(ω)))µ(dω),
where the last equality is obtained as ϕ is supposed to be invariant. Namely, by Theorem IV.8.34 of [39], ϕω ◦ αx = ϕT−x ω almost surely for each fixed x ∈ Zd . By identifying the GNS triplet (πϕω ◦αx , Hϕω ◦αx , ϕω ◦αx ) relative to ϕω ◦ αx with (πϕω ◦ αx , Hϕω , ϕω ), we have also πϕω ◦αx ∼ = πϕT−x ω almost surely. As Zd is countable, (i) and (iii) are satisfied for all x ∈ Zd simultaneously, on a common measurable set of full measure. By (i), it is easy to show that the measurable function ω → ϕω (I ) is an invariant one, then (ii) follows by ergodicity.
Consider for α ∈ {∞} ∪ {1, 2, . . . } ∪ {λ∞ , λ0 , λ1 , . . . }, the Abelian von Neumann algebras L∞ (Eα , να ) defined as follows. For α ∈ {∞} ∪ {1, 2, . . . }, (En , νn ) is the countable set En = n of cardinality n, equipped with the counting measure νn (the symbol ∞ corresponds to the denumerable cardinality). For α = λn , (Eλn , νλn ) is the disjont union [0, 1] ∪ n equipped with the measure νλn made of the Lebesgue measure λ on [0, 1], and the counting measure on n (the value n = 0 corresponds to L∞ ([0, 1], λ)). As a corollary to Theorem 4.1, we have Corollary 4.2. Let ϕ be as in Theorem 4.1. Then there exists a unique α ∈ {∞} ∪ {1, 2, . . . } ∪ {λ∞ , λ0 , λ1 , . . . } such that Zπϕω ∼ L∞ (Eα , να ) almost surely. Proof. Consider the sets α ⊂ consisting of ω such that Zπϕω ∼ L∞ (Eα , να ). Such sets are measurable ([31]) and give a disjoint countable partition of . Moreover, each α is invariant w.r.t. the action of spatial translations by (iii) of the cited theorem. The assertion follows by ergodicity.
We cannot conclude that Zπϕω is almost surely of a unique multiplicity class.11 However, for locally normal invariant KMS states we have that Zπϕω is almost surely of infinite multiplicity, see Theorem 4.6. We report a lemma which is crucial for the proof of the results which follow. Consider a sequence {αn }n∈N of ∗–automorphisms of a C ∗ –algebra B, together with a state ϕ on B which is invariant for {αn }. Let {Un } be the covariant implementation of {αn } relative to the GNS triplet (πϕ , Hϕ , ϕ ) corresponding to ϕ. Denote by α˜ n := adUn the corresponding automorphism on M := πϕ (B) . 11 Notice that there are uncountable many Abelian von Neumann algebras acting on separable Hilbert spaces, up to unitary equivalence.
KMS States of Disordered Systems
11
Lemma 4.3. Suppose that ϕ is separating for πϕ (B) .12 Then lim ϕ [αn (A), B]∗ [αn (A), B] = 0 n→∞
for every A, B ∈ B implies
(4.1)
lim [α˜ n (X), Y ]ξ = 0
n→∞
for every X, Y ∈ πϕ (B) and ξ ∈ Hϕ . Proof. As ≡ ϕ is cyclic for M , it is enough to show that n → ∞ implies [α˜ n (X), Y ] → 0. Let ε > 0 and X, Y ∈ M1 be fixed. Then there exist X , Y ∈ M \{0} such that (X − X ) < ε,
(Y − Y ) < ε .
Further, we can find by the Kaplansky Density Theorem, A, B ∈ B with πϕ (A) ≤ 1, πϕ (B) ≤ 1, such that (X − πϕ (A)) < 1 ∧ 1/ Y ε , (Y − πϕ (B)) < 1 ∧ 1/ X ε . We get [X, α˜ n (Y )] − πϕ ([A, αn (B)]) ≤ X α˜ n (Y ) − πϕ (A)α˜ n (πϕ (B)) + α˜ n (Y )X − α˜ n (πϕ (B))πϕ (A) . As both the terms of the r.h.s. of the above inequality are estimated in the same way, we consider only the first one. We get Xα˜ n (Y ) − πϕ (A)α˜ n (πϕ (B)) ≤ α˜ n−1 (X − πϕ (A))(Y − Y ) + Y α˜ n−1 (X − πϕ (A)) + πϕ (A)(α˜ n (Y − πϕ (B))) <2ε + Y 1 ∧ 1/ Y ε + 1 ∧ 1/ X ε ≤ 4ε which leads to the assertion.
As a direct consequence, we have the following result describing the structure of the von Neumann algebras generated by GNS representations associated to modular asymptotically Abelian states on C ∗ –algebras. Such a result can be applied immediately to the model under consideration, and yet wider applications are possible.13 Theorem 4.4. For a given C ∗ -algebra B and its ∗–automorphisms {αn }n∈N , let a state ϕ of B satisfy the following conditions: (i) Hϕ is a separable Hilbert space, (ii) ϕ is separating for πϕ (B) , (iii) Eq. (4.1) for every A, B ∈ B, (iv) ϕ is αn –invariant for all n ∈ N, where (πϕ , Hϕ , ϕ ) is the GNS triplet of ϕ. Then πϕ (B) does not contain type I∞ , II∞ and III0 components. 12 Under such a condition, the state ( · , ) is a KMS state (at inverse temperature 1) on π (B) ϕ ϕ ϕ w.r.t. the modular automorphisms group constructed from ϕ . 13 We are indebted to one of the referees for the following presentation of our main results.
12
S.D. Barreto, F. Fidaleo
Proof. By (iv), Lemma 4.3 and Theorem 13 of [25], it is enough to show that the type III0 component is absent. Let E be the central projection corresponding to the type III0 component of πϕ (B) which is well–defined by (i), see [36]. Let {Un }n∈N be the unitary implementation of the αn which exists by (iv). Then [E, Un ] = 0. Indeed, En = Un EUn∗ is also a central projection and πϕ (B) En = Un (πϕ (B) )E)Un∗ must be of type III0 , hence En ≤ E. The same argument implies 1 − En ≤ 1 − E. Then En = E. Hence, if E = 0, then Eϕ = 0 by (ii) and the state given for B ∈ B, by ϕE (B) := (πϕ (B)Eϕ , ϕ )/(Eϕ , ϕ ) satisfies the same conditions as ϕ. By Lemma 4.3, on applying and Theorem 2 of [19], we conclude that Exp (B (πϕ (A) E)) = σ (ϕ E)\{0} . Proposition C.2 then implies that the spectrum of ϕ E is contained in the two point set consisting of 0 and 1. Since ϕ is injective and commute with the central projection E, the point 0 is not contained as an isolated point in the spectrum of its restriction to the range of E. Hence ϕE must be a trace, a contradiction to the fact that E is the support of the type III0 component.
We are ready to apply the previous results to our situation. Proposition 4.5. Let ϕ be a locally normal invariant KMS state on A at inverse temperature β ∈ R\{0}. We have for the Borchers invariant, B (πϕ (A) ) = −β sp(τ ω ) almost surely. Proof. We know that the space and time translations commute with each other, see (2.6). Hence, as a consequence of Lemma 4.3 and Prop. 2.3, on applying Theorem 2 of [19],14 we conclude that Exp (B (πϕ (A) )) = σ (ϕ )\{0} . The assertion follows by Theorem 2.1, together with Corollary B.2, taking into
account that in our situation sp(τ ω ) = − β1 σ (ln ϕω ), see [4], Prop. 5.5. Proposition 4.6. Let ϕ be a locally normal invariant KMS state on A. Then only type IIIλ factors, λ ∈ (0, 1], can appear in its central decomposition. Proof. Theorem 4.4 says that type III0 component, and also an infinite semifinite component never appears in the factor decomposition of πϕ (A) . Further, also type II1 is avoided as we are assuming that {tt }t∈R is non–trivial. The proof follows as the finite– dimensional component is trivially excluded.
As an immediate consequence, we have the following Corollary 4.7. Let ϕ be as in Proposition 4.6. Suppose that Zπϕ ∼ L∞ (, µ). Then there exists a unique λ ∈ (0, 1] such that πϕω (A) are type IIIλ factors almost surely. Proof. The above condition means that Mω ≡ πϕω (A) are factors almost surely. Then by Proposition 4.5, B (M) = (Mω ) almost surely, where is Connes –invariant ([11]). The proof follows by Proposition 4.6.
14
See also the quite similar result in [2], Theorem 1.
KMS States of Disordered Systems
13
5. Some Applications to Disordered Models In order to explain natural applications of the previous results to concrete disordered systems, we specialize our analysis to the quantum model described by the following random Hamiltonian.15 We start with models arising from the formal Hamiltonian 1 Ji,j (i − j )σ (i)σ (j ) . (5.1) H =− 2 d i,j ∈Z
Here, σ (i) is the spin operator with two eigenvalues acting on the i th site (i.e. the Pauli matrix σz ), the Ji,j are independent identically distributed random variables with common distribution p(dξ ) on the real line, and finally (i) ≥ 0 determines the range of the interaction. To avoid technical complications, we restrict ourselves to finite–range interactions: supp < +∞, and small disorder: supp p =: K is a compact subset of the real line. The (anti)ferromagnetic models correspond to the cases (K ⊂ R− ) K ⊂ R+ , whereas spin glass models correspond to symmetric p. In this situation, K , µ(dω) = p(dJi,j ) . (5.2) = Zd ×Zd
i,j ∈Zd
It is a routine matter to verify that the spatial translations x ∈ Zd which shift each random variable Ji,j by −x, act by measure preserving ergodic transformations Tx of . It is a standard fact that, for a fixed realization of the couplings {Ji,j }, the spin algebra A admits KMS states at each inverse temperature β > 0 w.r.t. the time evolution associated to the Hamiltonian (5.1), see e.g. [10]. We analyze in some detail the uniqueness case.16 Namely, suppose that for a fixed β > 0, the Ising type model under consideration admits a unique KMS state, say ϕω , almost surely. Lemma 5.1. Under the above assumptions, the map ω ∈ → ϕω ∈ S(A) is ∗–weak measurable. Further, it satisfies almost surely the condition of equivariance ϕω ◦ αx = ϕT−x ω
(5.3)
w.r.t. the spatial translations, simultaneously. Proof. Consider on A ⊗ C() the states ϕ , ⊂ Zd finite, which are given by extensions to all of A⊗C(), of the states corresponding to the finite–volume Gibbs ensemble (with any chosen boundary condition) associated to the random Hamiltonian (5.1). By compactness, the net {ϕ } has a subsequence converging to the state ϕ. The latter is a KMS state on A⊗C() w.r.t. the random time evolution as that given in (2.1), determined by (5.1), such that the measure corresponding to ϕI ⊗C() is precisely µ. Further, it is invariant w.r.t. the space translations {ax }x∈Zd given on A ⊗ C() as in (2.4). Looking at each fibre, by Proposition 3.2 and Theorem 4.1 we have a decomposition of ϕ into a direct integral of states which, by uniqueness, coincides with the ϕω almost everywhere. Then, the map ω → ϕω is measurable and satisfies (5.3) almost surely w.r.t. µ.
15 The analysis presented in this section applies mutatis mutandis to models described by more general local Hamiltonians {H } ⊂Zd , coming from finite–range interactions as outlined in Sect. 2. 16 The hypotheses of Lemma 5.1 are satisfied if the quantum model under consideration admits some critical temperature. The situation is well clarified for many classical disordered models (see e.g. [28]), contrary to the quantum situation where, in the knowledge of the authors, there are few rigorous results concerning this point. However, it is expected also that quantum disordered systems exhibit critical temperatures at the high temperature regime.
14
S.D. Barreto, F. Fidaleo
From now on, we restrict ourselves to the situation where there is only one KMS state ϕω on A at inverse temperature β > 0 which is kept fixed during the analysis. In this case, the structure of the locally normal KMS states on A ≡ A ⊗ L∞ (, µ) is well understood.17 To this end, we have the following Proposition 5.2. For the model under consideration, under the above assumptions there is a one–to–one correspondence f → ϕf between positive normalized equivalence classes of L1 –functions on the sample space (, µ) and locally normal KMS states on A at inverse temperature β, where the state ϕf is given by ϕf (A) = f (ω)ϕω (A(ω))µ(dω) , A ∈ A, (5.4)
and ϕω is, almost surely, the unique KMS state on A at the same β. Proof. Taking into account Lemma 5.1, the field {ϕω } is measurable, then the state given in (5.4) is well–defined, and gives by construction a locally normal KMS state at inverse temperature β. Further, different functions give rise to different states. Now, let us fix a locally normal KMS state ϕ on A. By Proposition 3.2, there exists a measurable field {ψω } of KMS forms on A such that ϕf (A) = ψω (A(ω))µ(dω) .
By uniqueness of the KMS state ϕω , we obtain ψω = ψω (I )ϕω almost surely. The function f we are looking for is precisely f (ω) := ψω (I ). This concludes the proof.
In a situation such as the one just described above, there is a unique locally normal KMS state ϕ on A which is translation invariant. Further, there exists a unique λ > 0 such that ϕ is a direct integral of IIIλ factors almost surely. This immediately leads us to the following Proposition 5.3. Under the above assumptions, the following assertions hold true for the model under consideration. (i) There exists a unique locally normal a–invariant KMS state ϕ on A at inverse temperature β. Such a state is given by ϕ(A) = ϕω (A(ω))µ(dω) , A ∈ A,
where ϕω is, almost surely, the unique KMS state on A at the same β. (ii) The von Neumann algebra πϕ (A) generated by the GNS representation associated with the unique locally normal translation invariant KMS state ϕ on A is a direct integral of IIIλ factors for a fixed λ ∈ (0, 1]. Proof. The first part follows by ergodicity (Theorem 4.1), whereas the latter readily follows from Corollary 4.7. Consider the inverse temperature β > 0 such that the model under consideration admits a unique β–KMS state on A w.r.t. the time evolution associated to the Hamiltonian (5.1) at fixed values of couplings Jij , almost surely. For such β (or more generally 17
4.7.
Most of the forthcoming analysis also applies to the case when Zπϕ ∼ L∞ (, µ), see Corollary
KMS States of Disordered Systems
15
in the situation described in Corollary 4.7), the fact that πϕω (A) is almost surely a type IIIλ factor for a fixed λ ≡ λβ ∈ (0, 1] cannot be directly proven by using the standard results in [2, 33]. This is because the ϕω do not satisfy any natural invariance condition. We end the section by briefly describing what happens in the “multiple phase” regime. After taking the infinite–volume limit along various subsequences nk ↑ Zd , we will find, in general different, locally normal translation invariant t–KMS states on A at fixed inverse temperature β. Fix one such state ϕ. Then, one recovers a ∗–weak measurable field {ϕω }ω∈ ⊂ S(A) of τ ω –KMS states satisfying the equivariance property (5.3). According to Proposition 3.1, the set of the t–KMS states ϕT ∈ S(A), locally normal w.r.t. ϕ, have the form ϕT (A) = πϕω (A(ω))T (ω)1/2 ϕω , T (ω)1/2 ϕω H µ(dω) .
ϕω
Here, (πϕω , Hϕω , ϕω ) is the GNS representation of ϕω , {T (ω)}ω∈ is a measurable field of closed densely defined operators on Hϕω affiliated to the (isomorphic) centres Zϕω respectively, satisfying ϕω ∈ DT (ω)1/2 almost surely, and µ(dω) = 1. We refer the reader to Sect. 3 for further details.
T (ω)1/2 ϕω 2Hϕ
ω
6. Comments The present section is devoted to some comments relative to temperature states of quantum disordered spin systems, taking into account known results for the classical situation. Collecting together the results of Proposition 3.2 and Theorem 4.1, we recover for KMS states of A, all (the generalizations to the quantum case of) the properties which characterize the metastates. Metastates naturally arise when one considers thermodynamic limits of finite–volume Gibbs states on the skew space of joint configurations of spins and couplings which exist by compactness, that is for quenched disorder.18 The measure on such a skew space satisfies the following properties. Its marginal distribution of the couplings is the given probability measure describing the disorder, whereas the conditional distribution of the spins, given the couplings, is some infinite–volume Gibbs state almost surely. Such a field of infinite–volume Gibbs states satisfies an equivariant property as in Theorem 4.1, that is it gives an Aizenman–Wehr metastate, see [1], Sect. 5. For recent results on metastates, we refer the reader to [28–30] and the references cited therein. It can happen that the measure so obtained is not necessarily jointly Gibbsian, see [14, 22]. Therefore, the KMS boundary condition for the algebra A generated by spin and disorder variables w.r.t. the natural time evolution {tt }t∈R allows to recover quantum metastates after direct integral decomposition, the last one assuming the meaning of the quantum counterpart of the classical procedure of conditioning w.r.t. the disorder variables. The phenomenon that infinite–volume limits of finite–volume Gibbs states might be non-Gibbsian, can be partially explained in purely a mathematical setting as follows. Without any specific mention of the disordered systems, a quite similar approach was adopted in [21] (see also [3]) in order to study the statistical mechanics of semi–quantum lattice systems, that is systems whose quasi–local algebras of observables have non–trivial centres, then including commutative systems. Many of disordered systems treated in this paper fall into the framework of [21, 3], provided that the algebra generated by the 18 We refer the reader to [17] for some considerations relative to quenched and annealed disorder without referring to the local structure of the models under consideration.
16
S.D. Barreto, F. Fidaleo
disorder variables also has a local structure made of (commutative) finite–dimensional algebras.19 The last condition is satisfied for the model in Sect. 5 under the additional condition |K| < +∞, K given in (5.2). In such a situation, A := A ⊗ C() and the states ϕ under consideration are those such that the measures µϕI ⊗C() on determined by ϕI ⊗C() are absolutely continuous w.r.t. µ. In order to describe semi–quantum systems, an alternative approach was pursued in [21]. Namely, the algebra A is embedded as a subalgebra, into a quasi–local simple C ∗ – algebra F in a canonical way. Further, a Umegaki conditional expectation ε : F → A is constructed. In such a way, (invariant) states ϕ on A are lifted by ε, to (invariant) states ψϕ := ϕ ◦ ε on F. It was shown that the Gibbs condition for the state ψϕ on F, which turns out to be equivalent to the KMS condition for ψϕ , is equivalent to the Gibbs condition for the invariant state ϕ on A. Obviously, the KMS condition for invariant states ϕ on A is weaker than the corresponding condition for lifted states ψϕ on F, hence than the Gibbs condition for ϕ or equivalently for ψϕ .20 For semi–quantum systems (i.e. most of quantum disordered models under consideration) there is then a non–trivial condition which does not refer to the local structure, the KMS condition, which is weaker than the Gibbs condition. On the other end, for such models, limit points of finite–volume Gibbs states are KMS states, thanks to Prop. 5.3.25 of [10]. Summarizing, infinite-volume limits of finite-volume Gibbs states on the algebra A describing the spin and disorder variables should satisfy the KMS boundary condition, but they could be not necessarily infinite–volume Gibbs states. The last fact is well–known for classical disordered systems. Further, in the classical case there seems to be no characterization for such a “weak Gibbsianess” which does not refer to the local structure of the observable algebra, see [14, 22, 26] for the physical interpretation of such a phenomenon, and for further details. Therefore, the KMS boundary condition for the algebra generated by spin and disorder variables seems to be exactly tailored in order to describe the analog of the weak Gibbsianess in the quantum case. Moreover, it does not refer to the local structure of the observable algebra. The investigation of conditions which allow to seek the joint Gibbsian states among the weak Gibbsian (i.e. KMS) ones for quantum disordered systems, as well as the study of such a phenomenon for specific models, is a very interesting research program which is beyond the aims of the present paper. Appendix A. Decomposition of Closed Opertators By a measurable field of closed densely defined linear operators on ⊕ H= Hω dµ(ω)
we mean a field ω → T (ω) of closed linear operators on Hω with dense domain DT (ω) such that all functions ω → (T (ω)x(ω), y(ω))Hω are measurable for all measurable 19 In general, the approach followed in [21, 3] is suitable for (semi–)classical systems only if the net {H } of local Hamiltonians are assigned a–priori and not recovered by dynamical considerations. This is due to the fact that, contrary to the quantum case, the description of the equilibrium statistical mechanics of some relevant classical systems (i.e. the lattice gases, [24]) has no direct dynamical meaning. 20 The fact that the KMS condition is weaker than the Gibbs condition is a standard mathematical phenomenon which can be easily understood by considering the extreme situation of commutative systems, where the KMS condition is trivially satisfied for all states.
KMS States of Disordered Systems
17
fields of vectors {x(ω)} ⊂ DT (ω) , and {y(ω)} ⊂ Hω , ( · , · )Hω being the inner product on Hω . In such a situation, we define
⊕
T :=
T (ω) dµ(ω)
(A.1)
on the domain
DT := {x(ω)} ⊂
DT (ω) {x(ω)} measurable, 2 T (ω)x(ω) Hω dµ(ω) < +∞ .
(A.2)
Proposition A.1. The linear operator T on H defined by (A.1), with domain given by (A.2), is densely defined and closed. Proof. For each x ∈ H with x(ω) ∈ DT (ω) , we define z ∈ DT as z(ω) := (1 ∨ T (ω)x(ω) Hω )−1 x(ω) . It is easy to verify that each z belongs to DT .21 But the set of z(ω) arising from such z is total in Hω for each ω ∈ , that is DT is dense in H. Let now xn → x, T xn → y. There exist subsequences xnk (ω) → x(ω) and T xnk (ω) → y(ω) almost everywhere. Then, for almost all ω ∈ , x(ω) ∈ DT (ω) and T (ω)x(ω) = y(ω). Further, T (ω)x(ω) 2Hω dµ(ω) = y 2 ,
that is x ∈ DT and T x = y.
A densely defined closed operator T is said to be decomposable if it commutes with all diagonalizable operators. Let T = V |T | be the polar decomposition of T . It is straighforward to verify that T is decomposable iff V and (1 + T ∗ T )−1 both commute with the algebra of diagonalizable operators, see also [12]. We leave to the reader the proof of the following Proposition A.2. Let T be a densely defined closed decomposable operator. Then there exists a measurable field {T (ω)}ω∈ of densely defined closed operators such that ⊕ T (ω) dµ(ω) . T =
Let ϕ be a normal semifinite faithful (n.s.f. for short) weight on the von Neumann algebra M, and consider any subcentral decomposition ⊕ Mω dµ(ω) M=
of M. To such a decomposition, there corresponds a decomposition ⊕ ϕω dµ(ω) ϕ=
of the weight ϕ into n.s.f. weights, as described in Appendix A of [37]. 21
We are assuming without loss of generality, that the Borel measure µ is finite.
18
S.D. Barreto, F. Fidaleo
Proposition A.3. In the above situation, we have for the corresponding Tomita operators
⊕
ϕ =
⊕
ϕω dµ(ω) ,
Jϕ =
Jϕω dµ(ω) .
Proof. As ϕ and Jϕ commute with every element in the centre Z(M) of M ([35], Sect. 10), they are decomposable. The proof relative to the explicit decomposition of ϕ and Jϕ is analogous to that in [20], Lemma 3.3.
Notice that the above situation covers the analogous one described in Theorem A.13 of [37], relative to lower semicontinuous KMS weights on separable C ∗ –algebras.
Appendix B. On the Spectrum of Decomposable Operators The following proposition describes the spectrum of a decomposable selfadjoint operator in terms of the spectra of operators living on the fibres. ⊕ Proposition B.1. Let T = T (ω) dµ(ω) be selfadjoint. Then σ (T ) =
c σ (T (ω)) E ⊂ measurable , µ(E ) = 0 .
ω∈E
Proof. Let λ ∈ P(T ), then there exists a measurable set E such that µ(E c ) = 0 and (i) λ ∈ P(T (ω)), (ii) RT (ω) (λ) ≤ K for each ω ∈ E. As in the selfadjoint case we have RT (ω) (λ) = dist(λ, σ (T (ω)))−1 for the resolvent RT (ω) (λ), we conclude that λ ∈ Int sion follows analogously.
P(T (ω)) . The reverse inclu-
ω∈E
Corollary B.2. Let T be as in Proposition B.1. Suppose that for some measurable F of full measure, there exists a closed set C ⊂ R such that σ (T (ω)) = C for each ω ∈ F . Then σ (T ) = C. Proof. Let E be any measurable set of full measure. We have in such a situation,
ω∈F
and the proof follows.
σ (T (ω)) ⊂
ω∈E
σ (T (ω))
KMS States of Disordered Systems
19
Appendix C. On the Borchers Spectrum In the present section we deal with arbitrary von Neumann algebras unless it is specified otherwise. The Borchers spectrum B (α) ([7]) of an action g ∈ G → αg ∈ Aut(M) of a locally compact group G on a von Neumann algebra M is defined as sp(α e ), B (α) := e
where e runs over all invariant projections with central support z(e) = I . The Connes spectrum (α)([11]) is analogously defined by requiring that e merely runs over all non-zero invariant projections. As B (σ ϕ ) does not depend on the modular group σ ϕ corresponding to any n.s.f. weight ϕ on M ([19], Prop. 1), we can define B (M) := B (σ ϕ ) . Proposition C.1. For a von Neumann algebra M we have Exp (B (M)) = S(M)\{0}, where S(M) is Connes S–invariant ([11]). Proof. The proof is analogous to that of part (iii) in Prop. 1 of [19], taking into account that ϕv (x) := ϕ(vxv ∗ ) is a n.s.f. weight on M for each n.s.f. weight ϕ, and each isometry of M with left support e = vv ∗ in the centralizer M ϕ , see [34], Sect. 2.21.
It is almost immediate to show that M semifinite implies the triviality of B (M). Conversely in the separable case, the triviality of the Borchers spectrum implies that only type III0 factors and/or semifinite factors can appear in the factor decomposition of M. The last facts are summarized in the following Proposition C.2. Let M be a von Neumann algebra. (i) If M is semifinite, then B (M) = {0}. (i) Suppose that M has a separable predual and let ⊕ M= Mω dµ(ω) σ (Z(M))
be its factor decomposition. If B (M) = {0}, then µ(E c ) = 0, where E is the measurable set consisting of ω ∈ σ (Z(M)) such that Mω is either semifinite or type III0 . Proof. By Theorem 8.9.4 in [32], M semifinite implies B (M) = {0}. Conversely, suppose that M has separable predual. Let B (M) = {0}, and consider the set F in σ (Z(M)) consisting of ω ∈ σ (Z(M)) such that Mω is type IIIλ , λ ∈ (0, 1]. Such a set is measurable, see [36]. Suppose that F has strictly positive measure. Taking into account the definition of the Arveson spectrum of an action (see [32]) together with Proposition B.1, we obtain
B (M) ⊃ (Mω )
⊃
{E|µ(E c )=0} ω∈E
{E|µ(E c )=0} ω∈E∩F
which is a contradiction.
(Mω ) {0}
20
S.D. Barreto, F. Fidaleo
Acknowledgements. We thank R. Longo and L. Zsid´o for some helpful suggestions. We are grateful to the anonymous referees whose suggestions contributed to improve the presentation of the paper.
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29. Newman, M.N., Stein, D.L.: Thermodynamic chaos and the structure of short–range spin glasses. In: Mathematical aspects of spin glasses and neural networks, eds. A. Bovier, P. Picco, Basel-Boston-Berlin: Birkh¨auser, 1998, pp. 243–287 30. Newman, M.N., Stein, D.L.: Ordering and broken symmetry in short-ranged spin glasses. Available at http://arXiv:cond–mat/0301403 v2, 2003 31. Nielsen, O.A.: Direct integral theory. New York-Basel: Marcel Dekker, 1980 32. Pedersen, G.K.: C ∗ –algebras and their automorphism groups. London: Academic Press, 1979 33. Størmer, E.: Spectra of states, and asymptotically Abelian C ∗ –algebras. Commun. Math. Phys. 28, 279–294 (1972) 34. Strˇatilˇa, S.: Modular theory in operator algebras. Tunbridge Wells-Kent: Abacus Press, 1981 35. Strˇatilˇa, S., Zsid´o, L.: Lectures on von Neumann algebras, Tunbridge Wells, Kent: Abacus Press, 1979 36. Sutherland, C.E.: Crossed products, direct integrals and Connes’ classification of type III-factors. Math. Scand. 40, 209–214 (1977) 37. Sutherland, C.E.: Cartan subalgebras, transverse measures and non–type–I Plancherel formulae. J. Funct. Anal. 60, 281–308 (1985) 38. Takesaki, M.: On the conjugate space of operator algebra. Tohˆoku Math. J. 10, 194–203 (1958) 39. Takesaki, M.: Theory of operator algebras I. Berlin-Heidelberg-New York: Springer, 1979 40. Takesaki, M., Winnink, M.: Local normality in quantum statistical mechanics. Commun. Math. Phys. 30, 129–152 (1973) 41. Talagrand, M.: On the meaning of Parisi’s functional order parameter. C. R. Acad. Sci. Paris, Ser. I 337, 625–628 (2003) 42. Young, A.P. (ed.): Spin glasses and random fields. Singapore: World Scientific, 1997 Communicated by J.L. Lebowitz
Commun. Math. Phys. 250, 23–45 (2004) Digital Object Identifier (DOI) 10.1007/s00220-004-1138-z
Communications in
Mathematical Physics
Intersection Numbers of Twisted Cycles Associated with the Selberg Integral and an Application to the Conformal Field Theory Katsuhisa Mimachi1 , Masaaki Yoshida2 1 2
Department of Mathematics, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo 152-8551, Japan. E-mail:
[email protected] Department of Mathematics, Kyushu University, Ropponmatsu, Fukuoka 810-8560, Japan. E-mail:
[email protected]
Received: 2 June 2003 / Accepted: 6 February 2004 Published online: 27 July 2004 – © Springer-Verlag 2004
Dedicated to Professor Kazuhiko Aomoto on the occasion of his 65th birthday Abstract: Intersection numbers of twisted (or loaded) cycles associated with the Selberg integral are studied. In particular, the self-intersection number of the cycle which is invariant under the action of the symmetric group is expressed by the product of trigonometric functions. This formula reproduces the four-point correlation functions in the conformal field theory calculated by Dotsenko-Fateev in [3]. In our study, a compact non-singular model (Terada model) of the configuration space of n + 3 points on the real projective line and a q-analogue of the Chu-Vadermonde formula for the hypergeometric series play a crucial role. Intersection numbers of the corresponding cocycles are also studied.
Contents 1. Intersection Numbers of Twisted Cycles . . . . . 2. Low Dimensional Cases . . . . . . . . . . . . . 2.1 1-dimensional case . . . . . . . . . . . . . 2.2 2-dimensional case . . . . . . . . . . . . . 2.3 3-dimensional case . . . . . . . . . . . . . 3. n-Dimensional Case . . . . . . . . . . . . . . . . 4. Proof of Theorem 1 . . . . . . . . . . . . . . . . 5. An Application to the Conformal Field Theory . . 6. Intersection Numbers of Twisted Cocycles . . . . 6.1 Intersection numbers . . . . . . . . . . . . 6.2 Selberg integral case . . . . . . . . . . . . 6.3 Reciprocity relation for the Selberg integral
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24 26 26 27 29 30 35 39 40 40 42 44
This is a revised version of “Intersection numbers of twisted cycles and the correlation functions of the conformal field theory”, Kyushu Univ. preprint series 2002-23.
24
K. Mimachi, M. Yoshida
Introduction Let L be a local system (locally constant sheaf) determined by the integrand u(t) =
n i=1
tiα (1 − ti )β
(ti − tj )2γ
1≤i<j ≤n
of the Selberg integral [9]. In the locally finite twisted homology group Hnlf (T , L) with coefficients in L, there is a unique cycle C (up to a constant factor) which is invariant under the action of the symmetric group Sn acting as the permutations of the coordinates of T , where T is the complement of the singular locus of u(t): T = {t = (t1 , . . . , tn ) ∈ Cn | ti = tj (i = j ), ti = 0, 1}. The main purpose of the present paper is to give an expression of the self-intersection number C • C as the product of trigonometric functions in α, β and γ (Theorem 1). To derive it, we make use of a smooth compactification, constructed in [12], of the configuration space of the colored n + 3 distinct points on the real projective line; each of the (n + 2)!/2 open cells becomes an n-polyhedron which is called the Terada model of dimension n or simply the Terada-n. (This is named after the pioneering work [10] on the configuration spaces.) When we evaluate the self-intersection number C • C, the adjacency relation of the Terada-n’s and the coding of their faces used in [12] plays a crucial role. Note that the Terada-n is known as the Stasheff polytope or the associahedron in the study of the homotopy (n + 2)-associativity. As an application of the formula in Theorem 1, we show that the coefficients of the four-point correlation function in the conformal field theory calculated by DotsenkoFateev in [3] are identified with the intersection numbers of the twisted cycles associated with the conformal blocks. This is a generalization of our previous work in [8], where only the special cases n = 1, 2 are discussed. The n = 1, 2 cases are studied by Dotsenko-Fateev in their first paper [2]. Our study of the Terada-n in Sect. 3 is also used when we derive an explicit formula of the self-intersection number of the twisted cocycle associated with the integrand of the Selberg integral (Theorem 2). The intersection number is expressed as the product of linear forms in α, β and γ . The quadratic relation of hypergeometric integrals stated in [7] (p. 219) together with Theorems 1 and 2 leads to the reciprocity relation for the Selberg integral. 1. Intersection Numbers of Twisted Cycles Let u(t) = i fi (t)αi be a multivalued function on T ⊂ Cn , where αi ∈ R and T is the complement of the singular locus ∪i {fi (t) = 0} in Cn . Let L be the local system (locally constant sheaf) defined by u: the sheaf consisting of the local solutions of dL = Lω for ω = du(t)/u(t). Let Hn (T , L) be the nth homology group with coefficients in L, Hnlf (T , L) the nth locally finite homology group with coefficients in L. Elements of these twisted homology groups are called twisted cycles or loaded cycles. Under some genericity conditions on the functions fi and the exponents αi , we have the isomorphism, called the regularization, reg : Hnlf (T , L) −→ Hn (T , L),
Intersection Numbers of Twisted Cycles
25
which is the inverse of the natural map Hn (T , L) → Hnlf (T , L). To define intersection numbers for , ∈ Hnlf (T , L), we regularize one of them and compute the intersection number of the consequent cycles. Actually, the intersection form • : Hnlf (T , L) × Hnlf (T , L) −→ C is the Hermitian form defined by (, ) −→ • =
aρ aσ
ρ, σ
It (ρ, σ )vρ (t)vσ (t)/|u|2
t∈ρ∩σ
for , ∈ Hnlf (T , L), if reg and are represented by reg =
aρ ρ ⊗ v ρ ,
=
ρ
aσ σ ⊗ vσ ,
σ
where aρ , aσ ∈ C, each ρ or σ is an n-simplex, vρ or vσ a section of L on ρ or σ , the complex conjugation, and Ix (ρ, σ ) the topological intersection number of ρ and σ at x. The value • of the intersection form for , ∈ Hnlf (T , L) is called the intersection number of and . Here we assumed αi ∈ R, however, the following argument and Theorem 1 remain valid for αi ∈ C if we introduce the dual sheaf L∨ of L and defined the intersection form as
−
Hnlf (T , L) × Hnlf (T , L∨ ) −→ C. We refer the reader to [8] for more detail. Let n u(t) = tiα (1 − ti )β
(tj − ti )2γ
1≤i<j ≤n
i=1
be a multivalued function defining the local system L, where we assume α, β, γ ∈ R\Z and some genericity condition on α, β, γ . Let us define, for σ ∈ Sn , a loaded n-cycle Cσ = Dσ ⊗ uDσ (t), where Dσ is a bounded domain {(t1 , . . . , tn ) ∈ Rn | 0 < tσ (1) < · · · < tσ (n) < 1} with the standard orientation, and uDσ (t) is the standard loading defined in Remark 1 below. (The orientation of Dσ adopted in [8] is different from the present one.) Then C :=
Cσ
σ ∈Sn
is a generator of the Sn -invariant subspace Hnlf (T , L)Sn . The following main theorem gives an evaluation of the self-intersection number of C:
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K. Mimachi, M. Yoshida
Theorem 1. Jn (α, β, γ ) : = C 2 = C • C = n!
n j =1
1 − e(γ ) 1 − e(α + β + (n + j − 2)γ ) (1 − e(α + (j − 1)γ ))(1 − e(β + (j − 1)γ )) 1 − e(j γ )
√ n n −1 s(α + β + (n + j − 2)γ )s(γ ) = n! , 2 s(α + (j − 1)γ )s(β + (j − 1)γ )s(j γ ) j =1
√
where e(A) = exp(2π −1A) and s(A) = sin(π A). Remark 1. If each factor fi (t) of u(t) is defined over R, and D is a domain of the real manifold TR (the real locus of T ), then it is convenient to load D with a section uD (t) = ( i fi (t))αi i
of L on D, and to make a loaded cycle D ⊗ uD (t), where i = ± is so determined that
i fi (t) is positive on D, and the argument of i fi (t) is assigned to be zero. This choice of a section is said to be standard. In this paper, we adopt only the standard loading. Thus, we frequently omit the loading and denote just the topological cycles for simplic−−→ ity. For example, when u(t) = t α (1 − t)β , we denote by (0, 1) to express (0, 1) ⊗ u(t), −−−→ and (1, ∞) for (1, ∞) ⊗ t α (t − 1)β . In the next section, as a warm-up, we study low dimensional cases. 2. Low Dimensional Cases 2.1. 1-dimensional case. The function defining the local system L on T = C\{0, 1} is u(t) = t α (t − 1)β , where α, β, α + β ∈ R\Z. Then, the self-intersection number J1 = {0 < t < 1} • {0 < t < 1} can be expressed as the sum of local contributions at the barycenters of all the possible faces of the closure {0 < t < 1} : √ 1 −1 s(α + β) 1 e(α)e(β) − 1 −1 − − =− = , (2.1) dα dβ (e(α) − 1)(e(β) − 1) 2 s(α)s(β) where da = e(a) − 1,
√ e(a) = exp(2π −1a),
and
s(a) = sin(π a).
Indeed, a regularization reg C ∈ H1 (T , L) of C = (0, 1) ∈ H1lf (T , L) can be given by 1 1 −−−−−→ reg C = S( ; 0) + [ , 1 − ] − S(1 − ; 1) ⊗ u(t), dα dβ where the symbol S(a ; z) stands for the positively oriented circle centered at the point z with starting and ending point a, is a small positive number and the argument of each
Intersection Numbers of Twisted Cycles
27 C˜
reg C
1
0 Figure 1
factor of u(t) on the oriented circle S( ; 0) or S(1 − ; 1) is defined so that arg t takes values from 0 to 2π on S( ; 0), and arg(1 − t) from 0 to 2π . On the other hand, the other C can be deformed into the sine-like curve C˜ as is described in Fig. 1. Thus the number of the intersection points of the supports of reg C and of C˜ is three, and we have the expression (2.1) of the intersection number C • C. Remark 2. In general, let a loaded cycle Y supported by |Y | be given. If all the hypersurfaces (with non-trivial exponents) touching the cycle |Y | cross normally, then it is known [5] that the self-intersection number Y 2 is equal to the sum of the local contributions at the barycenters of all of the faces of the closure of |Y |. In what follows, we use this fact without any mention of the construction of regY and of the deformation of Y . 2.2. 2-dimensional case. The function defining the local system L is u(t) = (t2 − t1 )2γ i=1,2 tiα (1 − ti )β . In the complex (t1 , t2 )-space T ∼ = C2 , we consider the lines {t1 = 0}, {t2 = 0}, {t1 = 1}, {t2 = 1}, {t1 = t2 }, with exponents α, α, β, β, 2γ , respectively. Since the singular locus of u(t) is not normally crossing, we blow up the (t1 , t2 )-space centered at the two triple singular points (t1 , t2 ) = (0, 0) and (1, 1), and obtain a surface T ; let π : T → T ( C2 ) be the projection. As a result, the strict transform of the two bounded chambers of R2 defined by 0 < t1 < t2 < 1 and 0 < t2 < t1 < 1 are the pentagons—truncated triangles—bounded by the seven lines (Fig. 2), where the numerals along the segments indicate the corresponding exponents. Then the intersection number {0 < t1 < t2 < 1} • {0 < t1 < t2 < 1} is given by the sum 1 1 1 1 1 + + + + dα dβ d2β+2γ d2γ d2α+2γ 1 1 1 1 1 1 1 1 1 1 + + + + + dα dβ dβ d2β+2γ d2β+2γ d2γ d2γ d2α+2γ d2α+2γ dα
1+
of the local contributions at the barycenters of all the possible faces of the pentagon π −1 {0 < t1 < t2 < 1}, where A denotes the closure of A.
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K. Mimachi, M. Yoshida
t2
π −1 {0 < t1 < t2 < 1}
2γ
α 2γ
β
β π
2γ + 2β t1
α
α α
2γ + 2α
β
β
Figure 2
On the other hand, the intersection number {0 < t1 < t2 < 1} • {0 < t2 < t1 < 1} is equal to e(γ ) d2γ
π −1 {0 < t1 = t2 < 1} • π −1 {0 < t2 = t1 < 1} .
Since the boundary of the segment π −1 {0 < t1 = t2 < 1} consists of two points with exponents 2α + 2γ and 2β + 2γ , we have
π −1 {0 < t1 = t2 < 1} • π −1 {0 < t2 = t1 < 1} = −1 −
1 1 − . d2α+2γ d2β+2γ
Consequently, we obtain J2 = {0 < t1 < t2 < 1} • ({0 < t1 < t2 < 1} + {0 < t2 < t1 < 1}) 2 dα+β+γ dα+β+2γ dγ = dα dα+γ dβ dβ+γ d2γ 1 s(α + β + γ )s(α + β + 2γ )s(γ ) . =− 4 s(α)s(α + γ )s(β)s(β + γ )s(2γ ) In this 2-dimensional case, the Terada model is just a pentagon, and the adjacency of the two pentagons Tσ = π −1 {t|0 < tσ (1) < tσ (2) < 1} (σ ∈ S2 ), which are piecewiselinearly isomorphic to the Terada-2, is quite simple. In the higher dimensional cases, the Terada models are not this simple; studies of their combinatorial properties will play a crucial role.
Intersection Numbers of Twisted Cycles
29
Figure 3
2.3. 3-dimensional case. The function defining the local system L is u(t) = (tj − ti )2γ tiα (1 − ti )β . 1≤i<j ≤3
1≤i≤3
In the complex t = (t1 , t2 , t3 )-space T ∼ = C3 , we consider the planes {ti = 0}, {ti − 1 = 0}, (1 ≤ i ≤ 3) and {ti − tj = 0} (1 ≤ i < j ≤ 3), with exponents α, β and 2γ , respectively. Let π : T → T be the minimal blow-up of T along the non-normally crossing loci of the union of these planes. Then the strict transform Tσ = π −1 {0 < tσ (1) < tσ (2) < tσ (3) < 1} is the polyhedron bounded by six pentagons and three rectangles. It is piecewise-linearly isomorphic to the Terada model in dimension 3 (Terada-3 for short). In order for the reader to recall the shape of the polygon Tσ , we often call it “the Terada-3 Tσ .” See Fig. 3, where Tid is illustrated. Here the suffix id corresponds to the identity element id of the symmetric group S3 . The boundary of the Terada-3 Tid = π −1 {0 < t1 < t2 < t3 < 1} consists of the six penatagons π −1 {0 = t1 < t2 < t3 < 1},
π −1 {0 < t1 = t2 < t3 < 1},
π −1 {0 < t1 < t2 = t3 < 1},
π −1 {0 < t1 < t2 < t3 = 1},
π −1 {0 = t1 = t2 = t3 },
π −1 {t1 = t2 = t3 = 1},
with exponents α, 2γ , 2γ , β, 6γ + 3α, 6γ + 3β, the three rectangles π −1 {0 = t1 = t2 < t3 < 1}, π −1 {0 < t1 = t2 = t3 < 1}, π −1 {0 < t1 < t2 = t3 = 1}, with exponents 2γ + 2α, 6γ , 2γ + 2β, 21 edges and 14 vertices.
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K. Mimachi, M. Yoshida
Let us see the adjacency of the Terada-3’s Tσ for σ ∈ S3 . We can readily see that π −1 {0 < t2 < t1 < t3 < 1} is adjacent to Tid through the pentagon π −1 {0 < t1 = t2 < t3 < 1}, and that π −1 {0 < t3 < t2 < t1 < 1} is adjacent to Tid through the rectangle π −1 {0 < t1 = t2 = t3 < 1}. Similarly, π −1 {0 < t2 < t3 < t1 < 1} is adjacent to Tid through the edge π −1 {0 < t1 < t2 = t3 < 1} ∩ π −1 {0 < t1 = t2 = t3 < 1}. Thus the value J3 Tσ = Tid • 3! σ ∈S3
is equal to the sum 2 e(3γ ) −1 π {0 < t1 = t2 = t3 < 1} Tid2 − d6γ 2 e(γ ) −1 π {0 < t1 = t2 < t3 < 1} + d2γ 2 e(γ ) −1 π {0 < t1 < t2 = t3 < 1} + d2γ 2 e(γ ) e(3γ ) −1 π {0 < t1 = t2 < t3 < 1} ∩ π −1 {0 < t1 = t2 = t3 < 1} − d2γ d6γ 2 e(γ ) e(3γ ) −1 π {0 < t1 < t2 = t3 < 1} ∩ π −1 {0 < t1 = t2 = t3 < 1} . − d2γ d6γ Every summand—the self-intersection number of each face—will be calculated in the next section. If we admit the results there (see Example 3), we have J3 = − =−
3!dα+β+2γ dα+β+3γ dα+β+4γ dγ2 dα dα+γ dα+2γ dβ dβ+γ dβ+2γ d2γ d3γ 3! s(α + β + 2γ )s(α + β + 3γ )s(α + β + 4γ )s(γ )2 . 8 s(α)s(α + γ )s(α + 2γ )s(β)s(β + γ )s(β + 2γ )s(2γ )s(3γ )
3. n-Dimensional Case Let u(t) =
n
tiα (1 − ti )β
i=1
(tj − ti )2γ
1≤i<j ≤n
be a multivalued function defining the local system L. In the complex t = (t1 , . . . , tn )space T ∼ = Cn , we consider the hyperplanes {ti = 0}, {ti − 1 = 0} (1 ≤ i ≤ n),
and
{ti − tj = 0} (1 ≤ i < j ≤ n)
with exponents α, β, and 2γ , respectively. Let π : T → T be the minimal blow-up of T along the non-normally crossing loci of the union of these hyperplanes (cf. [11]). Each strict transformation Tσ = π −1 {0 < tσ (1) < · · · < tσ (n) < 1}
Intersection Numbers of Twisted Cycles
31
of the closure of the chamber defined by 0 < tσ (1) < · · · < tσ (n) < 1,
σ ∈ Sn
in Rn is piecewise-linearly isomorphic to the Terada model of dimension n (Terada-n for short). In what follows, we fix t0 and tn as t0 = 0,
tn = 1,
where for notational simplicity, we set n = n + 1, and for 0 ≤ i < i + k ≤ n with (i, i + k) = (0, n ), we denote by (i · · · i + k) the hyperface π −1 {t0 < t1 < · · · < ti = · · · = ti+k < · · · < tn < tn } of Tid . Then the hyperfaces of the Terada-n Tid = π −1 {t0 < t1 < · · · < tn < tn } are known ([12]) to be (0 · · · i ), (1 · · · i + 1), . . . , (n − i + 1 · · · n )
i = 1, . . . , n.
For a given family Si (1 ≤ i ≤ k) of sets, if any pair Si and Sj for 1 ≤ i < j ≤ k satisfies the condition either 1. Si ∩ Sj = ∅, or 2. Si Sj or Si Sj , then we say that the family satisfies the condition (DC), which is named after disjoint/contained. Each (n − k)-face of Terada-n Tid can be uniquely expressed as the intersection of k hyperfaces, say, H1 = (a1 · · · b1 ), H2 = (a2 · · · b2 ), . . . , Hk = (ak · · · bk ), where the family of the sets {a1 · · · b1 }, {a2 · · · b2 }, . . . , {ak · · · bk } enjoys the condition (DC).
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K. Mimachi, M. Yoshida
Example 1. (1) When n = 1, the Terada-1 Tid = {0 < t1 < 1} is the segment with the boundary consisting of the two points (01) = {0 = t1 } and (11 ) = {t1 = 1}. (2) When n = 2, the Terada-2 Tid = π −1 {0 < t1 < t2 < 1} is the pentagon bounded by the five edges (01) = π −1 {0 = t1 } ∩ Tid , (12) = π −1 {t1 = t2 } ∩ Tid , (22 ) = π −1 {t2 = 1} ∩ Tid , (012) = π −1 {0 = t1 = t2 } ∩ Tid , (122 ) = π −1 {t1 = t2 = 1} ∩ Tid , and with the five vertices (01) ∩ (22 ), (01) ∩ (012), (12) ∩ (012), (12) ∩ (122 ), (22 ) ∩ (122 ). (3) When n = 3, the boundary of the Terada-3 Tid = π −1 {0 < t1 < t2 < t3 < 1} consists of the 2-dimensional faces (01) = π −1 {0 = t1 } ∩ Tid , (12) = π −1 {t1 = t2 } ∩ Tid , (23) = π −1 {t2 = t3 } ∩ Tid , (33 ) = π −1 {t3 = 1} ∩ Tid , (012) = π −1 {0 = t1 = t2 } ∩ Tid , (123) = π −1 {t1 = t2 = t3 } ∩ Tid , (233 ) = π −1 {t2 = t3 = 1} ∩ Tid , (0123) = π −1 {0 = t1 = t2 = t3 } ∩ Tid , (1233 ) = π −1 {t1 = t2 = t3 = 1} ∩ Tid , the 1-dimensional faces (01) ∩ (23), (01) ∩ (0123), . . . , etc. and 0-dimensional faces (01) ∩ (23) ∩ (0123), (01) ∩ (012) ∩ (0123), . . . , etc. Since the hyperplanes {0 = ti },
1≤i≤k
and
{ti = tj },
1≤i<j ≤k
pass through the (n − k)-simplex 0 = t1 = · · · = tk < tk+1 < · · · < tn < 1, the exponent of the hyperface (01 · · · k) for 1 ≤ k ≤ n is kα + k(k − 1)γ . Similarly, the exponent of the hyperface (p · · · p + k) for 1 ≤ p < k + p ≤ n is k(k + 1)γ , and that of (n − k + 1 · · · n ) for 1 ≤ k ≤ n is kβ + k(k − 1)γ . For a hyperface (a · · · b) with exponent λ, we set [a · · · b] =
1 1 = . e(λ) − 1 dλ
(3.1)
Intersection Numbers of Twisted Cycles
33
Since the hypersurfaces in question are normally crossing, Theorem of [5] implies Lemma 1. The self-intersection number Tid2 of the Terada-n Tid equals the sum of (−)n
n
[a1 · · · b1 ][a2 · · · b2 ] · · · [ak · · · bk ]
k=0
for all the possible sets {ai , . . . , bi } (1 ≤ i ≤ k) satisfying the condition (DC), where 0 ≤ ai < bi ≤ n and (ai , bi ) = (0, n ). The summand for k = 0 is regarded as 1. Remark 3. The sequence a, · · · , b of [a · · · b] is always the consecutive one a, a + 1, a +2, · · · , b−1, b; recall the convention of the symbol (a · · · b) made for the hyperfaces in the beginning of this section. In what follows, we take this convention without further comment. Example 2. (1) When n = 1, Tid2 = −1 − [01] − [11 ]. (2) When n = 2, Tid2 =1 + [01] + [12] + [22 ] + [012] + [122 ]
+ [01][22 ] + [01][012] + [12][012] + [12][122 ].
On the other hand, we have Lemma 2. If a Terada-n Tσ for some σ ∈ Sn is adjacent to Tid through a face F = ∩i Hi , where Hi = (pi · · · pi + qi ) and 1 ≤ pi < pi + qi ≤ n, then qi −1 qi (qi +1)/2 (−1) e(γ ) [pi · · · pi + qi ] F • F. Tid • Tσ = i
Here the self-intersection number F • F is given by the sum of (−1)dimF [a1 · · · b1 ][a2 · · · b2 ] · · · [ak · · · bk ] for all the possible sets {aj , . . . , bj } (0 ≤ aj ≤ n, 1 ≤ bj ≤ n , 1 ≤ j ≤ k) such that {pi · · · pi + qi } and {aj · · · bj } (1 ≤ j ≤ k) satisfy the condition (DC) and that (aj , bj ) = (0, n ). The product for k = 0 is regarded as 1. Remark 4. Not all of the Terada-n’s of the form π −1 {0 < tσ (1) < · · · < tσ (n) < 1} for σ ∈ Sn are adjacent to Tid . Indeed, when n = 4, exactly π −1 {0 < t2 < t4 < t1 < t3 < 1}
and π −1 {0 < t3 < t1 < t4 < t2 < 1}
are not adjacent to Tid . A combinatorial reasoning of this fact will be discussed elsewhere in the future. Example 3. (1) When n = 2, we have π −1 {0 < t1 < t2 < 1} • π −1 {0 < t2 < t1 < 1} = e(γ )[12](12) • (12) = e(γ )[12]{−1 − [012] − [122 ]}.
34
K. Mimachi, M. Yoshida
(2) When n = 3, we have π −1 {0 < t1 < t2 < t3 < 1} • π −1 {0 < t2 < t1 < t3 < 1} = e(γ )[12](12) • (12), where (12) • (12) = 1 + [012] + [0123] + [123] + [1233 ] + [33 ] + [012][0123] + [012][33 ] + [0123][123] + [123][1233 ] + [1233 ][33 ], and π −1 {0 < t1 < t2 < t3 < 1} • π −1 {0 < t3 < t1 < t2 < 1} = −e(γ )e(3γ )[12][123]{(12) ∩ (123)} • {(12) ∩ (123)}, where
{(12) ∩ (123)} • {(12) ∩ (123)} = −1 − [0123] − [1233 ].
Combination of Lemma 1 and Lemma 2 yields that Jn = Tid • Tσ n! σ ∈Sn
can be expressed as the sum of (−)n (−)qi e(γ )qi (qi +1)/2 [pi · · · pi + qi ] [aj · · · bj ] i
j
n
for 1 ≤ pi < pi + qi ≤ n, 0 ≤ aj < bj ≤ such that (aj , bj ) = (0, n ), and that the sets {pi , . . . , pi + qi }, {aj , . . . , bj } satisfy the condition (DC). Note that, if there is, in the summands, a term of the form (−)qi e(γ )qi (qi +1)/2 [pi · · · pi + qi ] [aj · · · bj ], 1≤i≤l
j
then there is, in the summands, also a term of the form (−)qi e(γ )qi (qi +1)/2 [pi · · · pi + qi ] [pi · · · pi + qi ] [aj · · · bj ] . 1≤i≤l i=k1 ,... ,kt
i=k1 ,... ,kt
Thus we have l
t=0 1≤k1 <···
×
[pi · · · pi + qi ]
i=k1 ,... ,kt
qi
(−) e(γ )
qi (qi +1)/2
1≤i≤l i=k1 ,... ,kt
[pi · · · pi + qi ] [aj · · · bj ] j
l pi · · · pi + qi [aj · · · bj ], = i=1 i
j
j
Intersection Numbers of Twisted Cycles
35
where we set p · · · p + q = [p · · · p + q](1 + (−)q e(γ )(q+1)q/2 ) =
−1 q+1 1 − (−)q e(γ )( 2 )
. (3.2)
Hence we obtain Proposition 1. (−)n
Jn = pi · · · pi + qi [aj · · · bj ]. n! i
j
Here the sets {pi , . . . , pi + qi } for 1 ≤ pi < pi + qi ≤ n and {aj , . . . , bj } for 0 = aj or bj = n with (aj , bj ) = (0, n ) satisfy the condition (DC), and the term for the empty set is regarded as 1. Example 4. −J1 = 1 + [01] + [11 ], 1 J2 = 1 + [01] + [012](1 + [01] + 12) 2 + 12 + [122 ](1 + 12 + [22 ]) + [22 ] + [01][22 ]. Summing up the right-hand side of the equality in Proposition 1 in this general case is not so simple. In the next section, we carry out this summation procedure. 4. Proof of Theorem 1 A (DC)-monomial in 0, 1, . . . , n is the product pi · · · pi + qi [0 · · · bj ] [aj · · · n ] i
j
(4.1)
j
for a (DC)-set consisting of {pi , . . . , pi + qi }, {0, . . . , bj } and {aj , . . . , n }, where 1 ≤ pi < pi + qi ≤ n, 1 ≤ aj , bj ≤ n. The length of the (DC)-monomial (4.1) is the maximum of the cardinalities of the sequences {pi , · · · , pi + qi }, {0 · · · bj } and {aj · · · n }. A (DC)-monomial for a subsequence {a, . . . , b} of {0, . . . , n } is defined similarly. For example, the (DC)-monomial for the sequence {a, . . . , b} with 0 < a < n is the product (4.1) without the second product-factor j [0 · · · bj ]. Example 5. When n = 4, [0123]23123[44 ],
12341234[12344 ]
are monomials of length 4 and 5. Denote by a · · · b the sum of all the possible (DC)-monomials in {a, . . . , b} for 1 ≤ a < b ≤ n, and put Xn = 1 · · · n,
n ≥ 2.
36
K. Mimachi, M. Yoshida
For instance, X2 = 1 + 12, X3 = (1 + 123)(1 + 12 + 23), X4 = (1 + 1234)(1 + 12 + 23 + 34 + 1234 + (1 + 12 + 23)123 + (1 + 23 + 34)234). Similarly, put [[01 · · · k]] :=[01 · · · k] × {the sum of all the possible (DC)-monomials in 0, 1, . . . , k of length at most k}, [[n − k + 1 · · · n ]] :=[n − k + 1 · · · n ] × {the sum of all the possible (DC)-monomials in n − k + 1, . . . , n of length at most k}. Then one can readily see that
(−)n
Jn = n!
[[0 · · · i]]i + 1 · · · n − j [[n − j + 1 · · · n ]].
i,j ≥0, i+j ≤n
Evaluating the form i + 1 · · · n − j by (3.2), we see that the value of this form is equal to Xn−i−j . Let Y (k, n) be the sum of the (DC)-monomials in 1, . . . , n of length exactly k with Y (1, n) = 1, and X(k, n) be the sum of the (DC)-monomials in 1, . . . , n of length at most k with X(1, n) = 1. Then we have Xn = X(n, n),
X(k, n) =
k
Y (j, n),
Y (n, n) = 1 · · · n X(n − 1, n)
j =1
and
X(n, n) = Y (n, n) + X(n − 1, n).
Moreover we have Lemma 3. (i) n
g (2) + (−)n X(n − 1, n) = (1 − g)n , (g; g)n (ii) Y (n, n) =
(−)n+1 (1 − g)n , (g; g)n
(iii)
where (A; q)n =
n
n g (2) X(n, n) = (1 − g)n , (g; g)n
i=1 (1 − Aq
i−1 )
and g = e(γ ).
n ≥ 2,
Intersection Numbers of Twisted Cycles
37
Proof. We proceed by induction on n. We note that X(n − 1, n) = X(n − 1, n − 1) +
n−2
1 · · · k X(k − 1, k) X(n − k, n − k)
k=2
+ 1 · · · n − 1 X(n − 2, n − 1)X(1, 1). Hence, by the induction hypothesis, we have X(n − 1, n) = (1 − g)n
n−1 (−)k+1 k=1
= (1 − g)
ng
n−k g( 2 ) (g; g)k (g; g)n−k
(n2) + (−)n (g; g)n
.
Here the last equality follows from the equation n k=0
n−k (−1)n−k g ( 2 ) = 0, (g; g)k (g; g)n−k
which is given by the q-binomial theorem (cf. [4]) (x; q)n =
n k=0
n−k (q; q)n q ( 2 ) (−x)n−k (q; q)k (q; q)n−k
with x = 1 and q = g. This proves (i). Hence, Y (n, n) = 1 · · · n X(n − 1, n) implies (ii), and X(n, n) = Y (n, n) + X(n − 1, n) implies (iii). This completes the proof. Next, we have Lemma 4. (i) Ak := [[0 · · · k]] =
(g − 1)k , (a; g)k (g; g)k
k ≥ 1,
(ii) Bk := [[n − k + 1 · · · n ]] =
(g − 1)k , (b; g)k (g; g)k
k ≥ 1,
where a = e(α), b = e(β) and g = e(γ ). Proof. We first note that Ak = [0 · · · k] kp=1 Ak−p Xp . We next recall the q-Chu-Vandermonde formula (see [4]) −n c (c/b; q)n q ,b ; q, q n = . (4.2) 2 ϕ1 c b (c; q)n Letting c → ∞, we have n i (q −n ; q)i (b; q)i q −(2i ) q = b−n , − (q; q)i b
0≤i≤n
38
K. Mimachi, M. Yoshida
and then, by the change of b → a −1 q −n+1 , we have
i (q −n ; q)i (a −1 q −n+1 ; q)i i q −(2) −aq 2n−1 = a n q n(n−1) . (q; q)i
0≤i≤n
On the other hand, we have k
Ak−p Xp = (g − 1)k
p=1
=
k p=1
(g − 1)k (a; g)k (g; g)k
k p=1
p (−1)p g ( 2 ) (g; g)p (a; g)k−p (g; g)k−p
(a −1 g 1−k ; g)p (g −k ; g)p −(p) g 2 (−ag 2k−1 )p , (g; g)p
since p (ag k−p ; g)p (a −1 g 1−k ; g)p g ( 2 ) (−ag k−p )p 1 = = . (a; g)k−p (a; g)k (a; g)k
Therefore, we have k
Ak−p Xp =
p=1
(g − 1)k (a k g k(k−1) − 1). (a; g)k (g; g)k
(4.3)
This completes the proof of (i). The equality (ii) follows from the same argument.
We return to the proof of Theorem 1. Since the equality (4.3) can be rewritten as m i=0
m (g − 1)m a m g 2( 2 ) Am−i Xi = (a; g)m (g; g)m
(X0 := 1),
we have j n j =0 i=0
Aj −i Xi Bn−j =
j n (g − 1)j a j g 2(2)
Bn−j (a; g)j (g; g)j −n −1 1−n (g − 1)n g ,b g 2n−1 ; g, abg = ϕ 2 1 a (b; g)n (g; g)n j =0
=
(g − 1)n (abg n−1 ; g)n , (b; g)n (g; g)n (a; g)n
where the last equality follows from the q-Chu-Vandermonde formula (4.2). This completes the proof of Theorem 1.
Intersection Numbers of Twisted Cycles
39
5. An Application to the Conformal Field Theory This section is devoted to deriving the four-point correlation functions calculated by Dotsenko-Fateev in the second paper [3] of a series of their papers. The coefficients of the four-point functions are identified with the intersection numbers of twisted cycles associated with the conformal blocks. The cases of n = 1 and 2 have been studied in our previous paper [8]. Let n u(t) = (tj − ti )2g tia (ti − 1)b (ti − z)c 1≤i<j ≤n
i=1
be a multivalued function defining the local system L on T = {t = (t1 , . . . , tn ) ∈ Cn | ti = tj (i = j ), ti = 0, 1, z}, and the exponents a, b, c, g ∈ R\Z satisfy some genericity conditions. Let V = 1≤i≤l CCi ⊂ Hnlf (T , L) be a subspace invariant under the action of the fundamental group π1 (z, C\{0, 1}). If the loaded cycle Ci is expressed as ρ aiρ ρ ⊗ uρ (t), where each ρ is an n-simplex in T , and uρ (t) a section of L on ρ, then we define a function Ii (z) by the integral aiρ uρ (t)dt1 · · · dtn . ρ
ρ
Let Ih = ( Ci • Cj )1≤i,j ≤l be the intersection matrix. Then, the Hermitian form F (z, z) = (Ih−1 )ij Ii (z)Ij (z) 1≤i,j ≤l
is invariant under the action of π1 (z, C\{0, 1}). Note that the symmetric group Sn acts on T as permutations of the coordinates, and so that it acts also on Hnlf (T , L). We take V to be the subspace Hnlf (T , L)Sn consisting of Sn -invariant elements. To fix an idea, from now on, we assume z is real and 0 < z < 1. Let us define Ck by Ck =
σ k (t) ⊗ uk (t) ,
0 ≤ k ≤ n,
σ ∈Sn
where k = {t | 0 < t1 < · · · < tk < z, 1 < tk+1 < · · · < tn } and uk =
1≤i<j ≤n
(tj − ti )2g
k i=1
tia (1 − ti )b (z − ti )c
n i=k+1
tia (ti − 1)b (ti − z)c ,
40
K. Mimachi, M. Yoshida
with the argument of each factor of uk zero. Then C0 , . . . , Cn form a basis of Hnlf (T , L)Sn . Note that Ck is equal to the domain n {(t1 , . . . , tk ) | 0 < tσ (1) < · · · < tσ (k) < z } k σ ∈Sk
×
{(tk+1 , . . . , tn ) | 1 < tσ (k+1) < · · · < tσ (n) < ∞ }
σ ∈Sn−k
standardly loaded (see Remark 1) with u(t). Therefore, Theorem 1 implies that n Jk (a, c, g)Jn−k (b, −a − b − c − 2(n − 1)g, g), Ck • Ck = k where Jk is defined in Theorem 1. On the other hand, since C0 , . . . , Cn are mutually disjoint, among the intersection numbers Ci • Cj (0 ≤ i, j ≤ n), only the self-intersection numbers Ck • Ck (0 ≤ k ≤ n) are not zero. Hence, if we set uk dt1 · · · dtn , Ik (z) = n! k
then we have
F (z, z) = n−k j =1
1 n!
2 √ −1
n n k k=0 j =1
s(a + (j − 1)g)s(c + (j − 1)g)s(jg) s(a + c + (k + j − 2)g)s(g)
s(−a − b − c − 2(n − 1)g + (j − 1)g)s(b + (j − 1)g)s(jg) × |Ik |2 . s(−a − c − 2(n − 1)g + (n − k + j − 2)g)s(g)
This coincides with (3.19) in [3], the four-point correlation function in the conformal field theory, up to a constant multiplicative factor.
6. Intersection Numbers of Twisted Cocycles 6.1. Intersection numbers. We begin this subsection with a review on the intersection numbers for twisted cocycles. We refer the reader to [1] and [6] for more details. Let u(t) = i fi (t)αi be a multivalued function on T ⊂ Cn , where αi ∈ C, and T the complement of the singular locus ∪i {fi (t) = 0} in Cn . Here we assume that each Di = {fi (t) = 0} is a nonsingular hypersurface such that D = ∪Di is normally crossing. The form ω = du(t)/u(t) is an integrable 1-form holomorphic outside D. The covariant derivative ∇ω = d + ω∧ defines the de Rham cohomology group H n (T , ∇ω ) = {ξ ∈ E n (T ) | ∇ω ξ = 0}/∇ω E n−1 (T ),
Intersection Numbers of Twisted Cycles
41
where E n (T ) is the space of smooth n-forms on T . Similarly, Hcn (T , ∇ω ) is defined by the compactly supported forms. Under the genericity condition that αi ∈ / Z and / Z, we have the isomorphism i αi ∈ ιω : H n (T , ∇ω ) −→ Hcn (T , ∇ω ), which is the inverse of the natural homomorphism Hcn (T , ∇ω ) → H n (T , ∇ω ). Then the intersection number H n (T , ∇ω ) × H n (T , ∇−ω ) −→ C is defined by H (T , ∇ω ) × H (T , ∇−ω ) (ϕ, ψ) −→ ϕ, ψω = n
ιω (ϕ) ∧ ψ ∈ C.
n
T
Let ϕ be an n-form holomorphic outside D. At each point p ∈ D through which pass s hypersurfaces Di1 (p) , . . . , Dis (p) , we assume that ϕ admits an expression ϕ=
dfis (p) dfi1 (p) ∧ ··· ∧ ∧ {a holomorphic (n − s)-form} fi1 (p) fis (p) + {a holomorphic n-form}.
Let us regard ϕ as an n-form representing an element of H n (X, ∇ω ) and also that of H n (X, ∇−ω ). Then the self-intersection number ϕ, ϕω is evaluated as √ ϕ, ϕω = (2π −1)n p
1 , αi1 (p) · · · αin (p)
where the summation is taken for every point p ∈ D through which pass n hypersurfaces Di1 (p) , . . . , Din (p) , and along which ϕ has poles. For instance, the self-intersection number of dt1 . . . dtn t1 · · · tn (1 − t1 − · · · − tn ) associated with the function u(t) = t1α1 · · · tnαn (1 − t1 − · · · − tn )α0 is given by n √ 1 (2π −1)n i=0
√ α0 + · · · + αn = (2π −1)n . α0 · · · αn αj
0≤j ≤n j =i
This is the sum of the contributions from the vertices of the simplex. tj ≥ 0 (1 ≤ j ≤ n), t1 + · · · + tn ≤ 1. In general, the evaluation of the self-intersection number of a form is the sum of the contributions at every intersection of n pole divisors.
42
K. Mimachi, M. Yoshida
6.2. Selberg integral case. In the case of u(t) =
n
tiα (1 − ti )β
(tj − ti )2γ ,
ω=
1≤i<j ≤n
i=1
du u
we have the following. Theorem 2. The self-intersection number φ, φω of dt1 ∧ · · · ∧ dtn φ = n ∈ H n (T , ∇ω ) t (1 − t ) i i=1 i is equal to n √ (2π −1)n j =1
α + β + (n + j − 2)γ . (α + (j − 1)γ )(β + (j − 1)γ )
Proof. Since the singular locus D = ∪1≤i<j ≤n {ti − tj = 0} ∪1≤i≤n {ti = 0} ∪1≤i≤n {ti − 1 = 0} T → T which appeared in is not normally crossing, we consider the blow-up π : Sects. 2 and 3, and follow the recipe in §6.1 for ω˜ = π ∗ ω,
φ˜ = π ∗ φ.
Because the form φ˜ has no poles along the strict transforms of the hyperplanes ti −tj = 0, the form φ˜ has poles of type dt1 dtn ∧ ··· ∧ t1 tn exactly at vertices (of the Terada-n’s) which √ do not belong to two Terada-n’s. Thus the ˜ φ ˜ ω˜ equals (2π −1)n n! times the sum of the local contribution intersection number φ, at every vertex of Tid not touching another Terada-n Tσ for σ ∈ Sn \{id}. In case n = 2, relevant vertices are (in Fig. 4, they are marked by bold dots) (012) ∩ (01), (01) ∩ (22 ), (22 ) ∩ (122 ). Thus the sum of the contribution from these vertices is given by 1 1 1 1 1 1 (α + β + γ )(α + β + 2γ ) 1 + + = . 2α + 2γ α α β β 2β + 2γ 2 α(α + γ )β(β + γ ) In case n = 3, relevant vertices are (0123)∩(012)∩(01), (012)∩(01)∩(33 ), (01)∩(33 )∩(233 ), (33 )∩(233 )∩(1233 ). See Fig. 5, where these vertices are marked by bold dots. The five other Terada-3’s touch Tid through two pentagons, a rectangle and two edges; they are painted in gray. Thus the sum of the contribution from these four vertices is
Intersection Numbers of Twisted Cycles
43
α β
Tid 2β + 2γ Tσ 2γ
2α + 2γ Figure 4 2β + 2γ (233 )
3β + 6γ
β α
(1233 )
(33 )
(01) 2γ 2α + 2γ
(012)
(12)
6γ (123)
2γ (23)
3α + 6γ (0123)
Figure 5
1 1 1 1 1 1 1 1 1 1 1 1 + + + 3α + 6γ 2α + 2γ α 2α + 2γ α β α β 2β + 2γ β 2β + 2γ 3β + 6γ 1 (α + β + 2γ )(α + β + 3γ )(α + β + 4γ ) = . 3! α(α + γ )(α + 2γ )β(β + γ )(β + 2γ ) In the general n-dimensional case, relevant vertices are (01) ∩ (012) ∩ · · · ∩ (01 · · · k) ∩ (k + 1 · · · n ) ∩ · · · ∩ (n − 1nn ) ∩ (nn ),
44
K. Mimachi, M. Yoshida
k = 0, . . . , n. Thus the sum of the contribution from the relevant vertices is given by n
k
i=1 (iα + i(i − 1)γ )
k=0
1 n−k
j =1 (jβ
+ j (j − 1)γ )
,
which is equal to n 1 n 1 n−k k k n! i=1 (α + (i − 1)γ ) j =1 (β + (j − 1)γ ) k=0 n n n 1 1 j =n−k+1 (β + (j − 1)γ ) n = k k n! j =1 (β + (j − 1)γ ) i=1 (α + (i − 1)γ ) k=0
β − (−n) − n + 1 n k 1 1 γ k n = , α n! j =1 (β + (j − 1)γ ) k=0 k! γ k where (α)n = α(α + 1) · · · (α + n − 1). Moreover, the Chu-Vandermonde formula (a special case of Gauss’ summation formula) for the hypergeometric function 2 F1 : (γ − β)n −n, β ;1 = 2 F1 γ (γ )n implies that
β n (−n)k − − n + 1 γ α k=0 k! γ k
n k
=
+ β + (n − 1 + j − 1)γ ) n . j =1 (α + (j − 1)γ )
j =1 (α
This completes the proof of Theorem 2. 6.3. Reciprocity relation for the Selberg integral. Denote by Sα,β,γ the Selberg integral (0,1)n
n
tiα (1 − ti )β
i=1
1≤i<j ≤n
dt1 · · · dtn |ti − tj |2γ n . i=1 ti (1 − ti )
The quadratic relation of hypergeometric integrals stated in [7] (p.219) yields
Sα,β,γ × S−α,−β,−γ = C • C × φ, φω ,
(6.1)
where C = Cσ is the cycle in Theorem 1, and φ is the form defined in Theorem 2. We would like to call the identity (6.1) the reciprocity relation for the Selberg integral; because this is an honest generalization of the reciprocity relation B(α, β)B(−α, −β) =
1 − e(α + β) 2π i(α + β) · (1 − e(α))(1 − e(β)) αβ
(6.2)
Intersection Numbers of Twisted Cycles
for the Beta integral
45
1
dt . t (1 − t) 0 The reciprocity relation (6.2) for the Beta integral can be derived by the relation B(α, β) =
t α (1 − t)β
(α)(β) (α + β) and the reciprocity relation for the Gamma function: √ e(α/2) 2π −1 π (α)(−α) = · or (α)(1 − α) = , 1 − e(α) α s(α) √ where e(α) = exp(2π −1a) and s(a) = sin(π a). (Note that this can be thought of confluent version of (6.2).) In the same way, the evaluation of Sα,β.γ due to Selberg [9]: 1 1 n ··· tiα−1 (1 − ti )β−1 |ti − tj |2γ dt1 · · · dtn B(α, β) =
0
0 i=1
1≤i<j ≤n
n (α + (j − 1)γ )(β + (j − 1)γ )(j γ + 1) = (α + β + (n + j − 2)γ )(γ + 1)
(6.3)
j =1
and (6.3) reassure the identity (6.1). Thus the identities (6.1) and (6.3) tell that Theorem 1 implies Theorem 2, and vice versa. We dared in this paper to give proofs of both Theorems, because we believe that our combinatorial-geometric proofs are interesting in their own sake. Acknowledgement. The second author thanks Joichi Kaneko for useful comments on the Selberg integral.
References 1. Cho, K., Matsumoto, K.: Intersection theory for twisted cohomologies and twisted Riemann’s period relations I. Nagoya Math J. 139, 67–86 (1995) 2. Dotsenko, VI.S., Fateev, V.A.: Conformal algebra and multipoint correlation functions in 2D statistical models. Nuc. Phys. B240[FS12], 312–348 (1984) 3. Dotsenko, VI.S., Fateev, V.A.: Four-point correlation functions and operator algebra in 2D conformal invariant theories with central charge C ≤ 1. ibid., B251[FS13], 691–734 (1985) 4. Gasper, G., Rahman, M.: Basic Hypergeometric Series. Cambridge: Cambridge Univ. Press, 1990 5. Kita, M., Yoshida, M.: Intersection theory for twisted cycles II. Math. Nach. 168, 171–190 (1994) 6. Matsumoto, K.: Intersection numbers for logarithmic k-forms. Osaka J. Math. 35, 873–893 (1998) 7. Matsumoto, K., Yoshida, M.: Recent progress of interesection theory for twisted (co)homology groups. Adv. Studies in Pure Math. 27, 217–237 (2000) 8. Mimachi, K., Yoshida, M.: Intersection numbers of twisted cycles and the correlation functions of the conformal field theory. Commun. Math. Phys. 234, 339–358 (2003) 9. Selberg, A.: Bemerkninger om et multiplet integral. Norske Mat. Tidsskr. 26, 71–78 (1944) 10. Terada, T.: Fonctions hyperg´eom´etriques F1 et fonctions automorphes I. J. Math. Soc. Japan 35, 451–475 (1983) 11. Tsuchiya, T., Kanie, Y.: Fock space representations of the Virasoro algebra–Intertwining operators. Publ. RIMS. Kyoto Univ. 22, 259–327 (1986) 12. Yoshida, M.: The democratic compactification of configuration spaces of point sets on the real projective line. Kyushu J. Math. 50, 493–512 (1996) Communicated by L. Takhtajan
Commun. Math. Phys. 250, 47–63 (2004) Digital Object Identifier (DOI) 10.1007/s00220-004-1142-3
Communications in
Mathematical Physics
A Reduction Theorem for Highest Weight Modules over Toroidal Lie Algebras Ivan Dimitrov1 , Vyacheslav Futorny2 , Ivan Penkov3 1
Department of Mathematics, Yale University, 10 Hillhouse Ave., New Haven, CT 06520, USA. E-mail:
[email protected] 2 Instituto de Matem´atica e Estat´ıstica, Universidade de S˜ao Paulo, Caixa Postal 66281, CEP 05315-970 S˜ao Paulo, Brasil. E-mail:
[email protected] 3 Department of Mathematics, University of California, Riverside, CA 92521, USA. E-mail:
[email protected] Received: 22 June 2003 / Accepted: 26 January 2004 Published online: 27 July 2004 – © Springer-Verlag 2004 ◦
Abstract: Let g =g [t1±1 , . . . , tn±1 ] ⊕ D ⊕ z be a toroidal Lie algebra corresponding to ◦
a semisimple Lie algebra g. We describe all Borel subalgebras of g which contain the ◦
◦
◦
Cartan subalgebra h =h ⊕D ⊕z, where h is a fixed Cartan subalgebra of g. We show that each such Borel subalgebra b determines a parabolic decomposition g = n− ⊕ k ⊕ n+ , where k is a proper toroidal subalgebra of g and b ⊂ k ⊕ n+ . Our first main result is that, for any weight λ which does not vanish on z ∩ [k, k], an arbitrary subquotient of the Verma module Mg (λ) is induced from its k ⊕ n+ –submodule of n+ –invariant vectors. This reduces the study of subquotients of Mg (λ) to the study of subquotients of Verma modules over k. We then introduce categories Ok and Og and their respective blocks Og (ν) and Ok (ν) corresponding to a central charge ν which is nonzero on z ∩ [k, k]. Our second main result is that the functors of induction and n+ –invariants are mutually inverse equivalences of the category Ok (ν) and the full subcategory of Og (ν) whose objects are generated by their n+ –invariants.
Introduction Toroidal Lie algebras are universal central extensions of iterated loop algebras with n > 1 variables and are a most natural generalization of affine Lie algebras. They were introduced by R. Moody, S. Rao and T.Yokonuma in [MRY] and since then have attracted considerable mathematical attention. Toroidal Lie algebras have also appeared in mathematical physics. T. Inami, H. Kanno, T. Ueno and C.-S. Xiong, [IKUX], have shown that a toroidal Lie algebra with n = 2 arises as a symmetry algebra in a four dimensional nonlinear sigma model with a Wess–Zumino like term introduced by S. Donaldson, [D], much in the same way as affine Lie algebras arise as symmetry algebras in the Wess– Zumino–Witten model. T. Inami, H. Kanno and T. Ueno, [IKU], have generalized this construction to arbitrary n.
48
I. Dimitrov, V. Futorny, I. Penkov
A natural construction of representations of a toroidal Lie algebra is via induction from parabolic subalgebras whose Levi components are proper toroidal subalgebras. This construction has been carried out in special cases and the resulting modules have been studied by S. Berman, Y. Billig, B. Cox, V. Futorny and I. Kashuba, see [BB, BC, CF, FK]. Furthermore, in several of these works a natural phenomenon of reduction to modules over a proper toroidal subalgebra has been singled out. It generalizes the reduction to an affine or Heisenberg subalgebra known to exist for highest weight modules for nonstandard Borel subalgebras of affine Lie algebras, see [C and F1]. The purpose of the present paper is to study arbitrary highest weight modules over toroidal Lie algebras and to understand the reduction phenomenon in general. No general definition of a Borel subalgebra has appeared in the literature on toroidal algebras before. Our first observation is that since a standard Cartan subalgebra h acts semisimply on a toroidal Lie algebra g (i.e. g has a root decomposition with respect to h), the definition of a Borel subalgebra of a Lie algebra with a root decomposition given in [DP1 and DP2] applies to g. We give a complete combinatorial description of all Borel subalgebras of g containing a fixed h. In particular, for n > 1, there are infinitely many nonconjugate Borel subalgebras (this was already noticed in [CF]). Furthermore, using the combinatorics of the root system of g, we assign to each Borel subalgebra a proper toroidal subalgebra k ⊂ g. (As k can be finite dimensional, we allow n to equal 0 in the definition of a toroidal Lie algebra.) In this way we obtain a parabolic decomposition g = n− ⊕ k ⊕ n+ with b ⊂ k ⊕ n+ . While k = h for a generic b, for special choices of b the subalgebra k may equal almost any toroidal subalgebra of g (for a more detailed discussion see Sect. 2). The first main result of the present paper is that, if k is infinite dimensional and the restriction of λ onto the center z of g intersected with [k, k] is nonzero, then any subquotient M of the Verma module Mg (λ) is isomorphic to the g–module induced from + the k ⊕ n+ –submodule M n ⊂ M consisting of all n+ –invariant vectors in M. This + reduces the study of M to the study of the k–module M n , and is a vast generalization of previously obtained results in [BC, CF and FK] . Our theorem implies that, if a Borel subalgebra b yields an infinite dimensional toroidal subalgebra k, then only very special b–highest weight modules or their submodules are not parabolically induced from k: for those the highest weight λ must vanish on the part of the center of g contained in [k, k]. In fact, explicit vertex operator constructions of such modules have been given by S. Berman and Y. Billig, [BB], and it has been shown that they can be integrable with finite dimensional weight spaces. Moreover, we may speculate that the generic Borel subalgebras, i.e. those for which k is finite dimensional and hence our result does not apply, yield mostly irreducible Verma modules. This may deserve further study, however it is beyond the scope of the present paper. Our reduction result yields a natural equivalence of categories. We define categories Og and Ok of g– and k–modules respectively, which admit an ascending filtration whose associated quotients are isomorphic to subquotients of Verma modules. We then consider their respective blocks Og (ν) and Ok (ν) of modules with a central charge ν. The second main result of the paper is that, for any ν which does not vanish on z ∩ [k, k], the functors of parabolic induction and the functor of n+ –invariants are mutually inverse equivalences of Ok (ν) and the full subcategory of Og (ν) whose objects are generated by their n+ –invariants. Any irreducible object of Og (ν) is also an object of the latter subcategory, but as we show, there are reducible modules in Og (ν) which are not generated by their n+ –invariants.
Reduction Theorem for Highest Weight Modules over Toroidal Lie Algebras
49
Notational Conventions. The ground field is C and all Lie algebras and their representations are automatically assumed to be defined over C. Some real vector spaces are also considered but this is clearly indicated. The superscript ∗ stands for dual space, U (·) stands for enveloping algebra, spanA stands for span over a submonoid A ⊂ C. We put R+ := {r ∈ R | r ≥ 0} and Z+ := Z ∩ R+ . Finally, a cone in a real vector space W is by definition an R+ –invariant submonoid C ⊂ W . 1. Triangular Decompositions and Highest Weight Modules Let W be a finite dimensional real vector space. For any subset P ⊂ W \{0}, a decomposition P = P + P − is called a triangular decomposition of P , if the cone C + ⊂ W generated by P + ∪(−P − ) (or, equivalently, its opposite cone C − ) contains no nontrivial vector subspace of W , cf. [DP2]. Let F = {{0} = Fn ⊂ Fn−1 ⊂ . . . ⊂ F1 ⊂ F0 = W } be a flag of maximal length in W , where n = dim W . We call F oriented if, for each i, one of the connected components of Fi \Fi+1 is labeled by + and the other one is labeled by −, i.e. Fi \Fi+1 = (Fi \Fi+1 )+ (Fi \Fi+1 )− . An oriented flag F determines the triangular decomposition P ± := P ∩ (∪i (Fi \Fi+1 )± ) of any subset P ⊂ W \{0}. It is easy to see that if P is finite, every triangular decomposition of P can be determined by an oriented flag F for which P ∩ F1 = ∅. This is no longer true if P is infinite. We now introduce an invariant of a triangular decomposition. For a subset M ⊂ W , let M denote the closure of M in the natural topology of W . Given a triangular decomposition P = P + P − , set P0 := (C + ∩ C − ) ∩ P . The following statements are an exercise on closed convex sets. Proposition 1. Let P = P + P − be a triangular decomposition of P ⊂ W \{0}. (i) For any oriented flag of maximal length F = {{0} = Fn ⊂ Fn−1 ⊂ . . . ⊂ F1 ⊂ F0 = W } which determines the decomposition P = P + P − , we have P0 ⊂ F1 . (ii) There exists an oriented flag of maximal length F = {{0} = Fn ⊂ Fn−1 ⊂ . . . ⊂ F1 ⊂ F0 = W } which determines the decomposition P = P + P − and such that P0 = F1 ∩ P . Let g be a (complex) Lie algebra with a fixed finite dimensional abelian selfnormalizing subalgebra h ⊂ g for which g decomposes as g = h ⊕ (⊕α∈h∗ \{0} gα ),
(1)
where gα := {g ∈ g | [h, g] = α(h)g for every h ∈ h}. The set := {α ∈ h∗ \{0} | gα = 0} is the root system of g. A Borel subalgebra b of g containing h is by definition any subalgebra of the form b = h ⊕ (⊕α∈+ gα ) for some triangular decomposition = + − of in the real vector space spanR . We put b± := ⊕α∈± gα . The triangular decomposition = + − corresponding to b defines also the subalgebra gb := h ⊕ (⊕α∈0 gα ) of g. A g-module V is a weight module (with respect to h) if V = ⊕µ∈h∗ V µ , where V µ := {v ∈ V | h · v = µ(h)v for all h ∈ h}. The support of V is the set supp V := {µ ∈ h∗ | V µ = 0}. Any submodule or quotient of a weight module is a weight module. The enveloping algebra U (g) is a weight module under the adjoint action of g with supp U (g) = Q := spanZ . For λ ∈ h∗ denote by v λ the one dimensional b–module on which h acts via λ and b+ acts by zero. The g–module Mg (λ) := U (g) ⊗U (b) v λ is the Verma module with b–highest weight λ. Note that Mg (λ) is a weight module which is isomorphic to U (b− )
50
I. Dimitrov, V. Futorny, I. Penkov
as a b− –module. Furthermore, Mg (λ) admits a unique maximal proper g–submodule, and hence, a unique irreducible quotient module Lg (λ). A highest weight module with b–highest weight λ is defined as an arbitrary nonzero quotient of Mg (λ). 2. Toroidal Lie Algebras Let be a finitely generated free abelian group (written additively), and let A() denote the group algebra of , i.e. A() = { γ ∈ cγ t γ | cγ = 0 for finitely many γ }, where t is a formal variable. Set D := Hom(, C). Given a finite dimensional reductive (possi◦ ◦ bly abelian) Lie algebra g, the following commutation relations endow g ⊗ A() ⊕ D with a Lie algebra structure:
[g ⊗ t γ , g ⊗ t γ ] = [g, g ] ⊗ t γ +γ , [d, g ⊗ t γ ] = d(γ )g ⊗ t γ , [d, d ] = 0, ◦
where g, g ∈ g, γ , γ ∈ , and d, d ∈ D. Let ϕ be a nondegenerate symmetric ◦ ◦ invariant bilinear form on g, and let z := C ⊗Z . The toroidal Lie algebra t(g, , ϕ) := ◦
◦
g ⊗A()⊕D⊕z is defined as the central extension of g ⊗A()⊕D via the commutation relation
[g ⊗ t γ + d, g ⊗ t γ + d ] = [g, g ] ⊗ t γ +γ + d(γ )g ⊗ t γ − d (γ )g ⊗ t γ +δγ ,−γ ϕ(g, g )1 ⊗ γ . ◦
Example 1. Let g be simple and = Zn . Then we can identify A() with C[t1±1 , . . ., tn±1 ] so that t (k1 ,... ,kn ) = t1k1 . . . tnkn for (k1 , . . . , kn ) ∈ Zn , and D ∼ , . . ., = span{t1 ∂t∂1
|t1 =...=tn =1 ◦ ∂ g tn ∂tn }. Furthermore, let κ be the Killing form of . The toroidal algebra |t1 =...=tn =1 ◦ ◦ ◦ g[n] := t(g, Zn , κ) is the “standard” toroidal Lie algebra, cf. [MRY]. Finally, let h ◦ ◦ be a Cartan subalgebra of g, ϕ = κ ◦ ◦ and = Zn . In this case t(h, Zn , ϕ) is a |h×h ◦
Heisenberg subalgebra of g[n].
Proposition 2. ◦ ◦ (i) If g is simple, then g[n] is isomorphic to the universal central extension of ◦
g ⊗ C[t1± , . . . , tn± ] ⊕ D. ◦
◦
◦
◦
◦
g z g z (ii) If g = (⊕m i=1 i ) ⊕ is reductive with simple components i and center , then ◦
◦
g t(g, , ϕ) ∼ = ((⊕m i=1 t( i , i , ϕ
◦
◦
|gi ×gi
◦
)) ⊕ t(z, 0 , ϕ ◦
◦
|z×z
))/J,
where i are copies of , and J is the central ideal generated by all differences of corresponding elements in i and j for 1 ≤ i = j ≤ m. Sketch of the proof. Statement (i) follows from the fact that any universal central ◦ ◦ extension of the Zn -graded Lie algebra g ⊗ C[t1±1 , . . . , tn±1 ] is isomorphic to g ⊗ C[t1±1 , . . . , tn±1 ] ⊕ C[t ±1 ,... ,tn±1 ] /dC[t1±1 , . . . , tn±1 ] (see [K]). Statement (ii) is a simple 1 exercise.
Reduction Theorem for Highest Weight Modules over Toroidal Lie Algebras
51 ◦
The term toroidal Lie algebra is often used for a consecutive extension of g ⊗ C[t1±1 , . . . , tn±1 ] by an infinite dimensional center, and then by a (finite or infinite dimensional) derivations algebra. In the present paper we consider the “minimal nontrivial” ◦ extensions of g ⊗ C[t1±1 , . . . , tn±1 ] defined above. ◦
◦
Unless explicitly stated otherwise, we will assume that g is reductive. Fix g = ◦
◦
◦
t(g, , ϕ). If h is a Cartan subalgebra of g, then h := h ⊕ D ⊕ z is a Cartan subalgebra ◦
◦
of g (here and below we identify h with h ⊗ C ⊂ g), and g admits a root decomposition ◦
(1). To describe the root system of g, notice that ⊂ h∗ ⊕ D∗ as z is in the center of g. Since is canonically embedded into D∗ = (Hom(, C))∗ , we will denote the elements of and their images in D∗ by the same letters. Then ◦
= ( + ) ∪ (\{0}), ◦
◦
(2)
◦
as gα+γ = gα ⊗ t γ and gγ = h ⊗ t γ for α ∈ , γ ∈ . In what follows we will also identify with the subset 1 ⊗Z of z. This enables us to evaluate on linear functions defined on z. With the conventions above γ ∈ may stand for three different things but the meaning of γ will be clear from the context. Namely, when we write γ alone or x ⊗ t γ we understand γ as an element of , when we write α + γ we understand γ as an element of D∗ , and when we write ϕ(γ ) we understand γ as an element of z. 3. The Decomposition g = n− ⊕ k ⊕ n+ Let b be a Borel subalgebra of g corresponding to a triangular decomposition = + − . Set 0 := ∩ (0 ∪ {0}) and note that 0 is a subgroup of . Indeed, according to Proposition 1(ii), 0 = ∩ F1 for a suitable hyperplane F1 ⊂ spanR . Hence 0 = ∩ (0 ∪ {0}) = ∩ (( ∩ F1 ) ∪ {0}) = (( ∩ F1 ) ∩ ) ∪ {0} = ∩ F1 , ◦
i.e. 0 equals the subgroup ∩ F1 of . For every α ∈ , (α + ) ∩ 0 ⊂ α + D∗ is ◦
◦
either empty or a translation of 0 . Denote by k the set of all roots α ∈ for which ◦ ◦ ◦ (α + ) ∩ 0 ⊂ α + D∗ is a translation of 0 , and let k := h ⊕ (⊕ ◦ gα ). α∈k
◦
◦
Proposition 3. The Lie algebra k is a reductive subalgebra of g. Furthermore, gb = ◦ h ⊕ (⊕α∈0 gα ) is a proper toroidal subalgebra of g isomorphic to t(k, 0 , ϕ ◦ ◦ ). |k×k
◦
◦
◦
◦
Proof. We verify first that k is a subalgebra of g. Indeed, if α, β ∈ k and α+β ∈ , then α + γ1 ∈ F1 and β + γ2 ∈ F1 , for some γ1 , γ2 ∈ , and hence (α + β) + (γ1 + γ2 ) ∈ F1 , ◦ ◦ ◦ ◦ ◦ i.e. α + β ∈ k . Furthermore, k = −k , and therefore k ⊂ g is reductive. The fact that ϕ ◦
◦
|k×k
is nondegenerate follows from the fact that ϕ
◦
◦
|h×h
and ϕ
◦
◦
|gα ×g−α
◦
, for α ∈ ,
are nondegenerate. Note also that gb is a proper subalgebra of g as 0 is a proper subset of . ◦ To establish an isomorphism between gb and t(k, 0 , ϕ ◦ ◦ ), fix a system of simple ◦
|k×k
◦
roots {α1 , . . . , αs } of k , and for every 1 ≤ i ≤ s fix a nonzero element g ±αi ∈ g±αi .
52
I. Dimitrov, V. Futorny, I. Penkov ◦
Furthermore, for every 1 ≤ i ≤ s, fix γi ∈ such that αi +γi ∈ 0 . Denote by k the subalgebra of gb generated by gi±αi ⊗ t γi for 1 ≤ i ≤ s and h. Then there exists a unique Lie ◦
◦
algebra isomorphism θ : k → k with θ (g ±αi ) = g ±αi ⊗ t γi and such that θ ◦
|zk ⊗C⊕D ⊕z
◦
◦
is the identity, where zk denotes the center of k. The reader will check immediately that ◦
the extension : t(k, 0 , ϕ ◦
◦
|k×k
◦
) → gb of θ defined by (g ⊗ t γ ) := θ(g) ⊗ t γ for
g ∈ k and γ ∈ 0 is an isomorphism.
When the Borel subalgebra b of g is given, we will denote gb by k, and the root system ◦
0 of k by k . We have k = (k + 0 ) ∪ (0 \{0}). In particular, 0 = 0 precisely when k is infinite dimensional. Set also n± = ⊕α∈± \k gα . Then g = n− ⊕ k ⊕ n+ . ◦
It is a natural problem to determine all pairs (0 , k) arising from Borel subalgebras b of g as in Proposition 3 above. We will now reduce this problem to a purely combinatorial question about root systems. Although we do not answer the latter, we show that the set of such pairs is rather large. An example of a pair which does not arise from a Borel subalgebra of g is given below, see Example 2. Assume b is determined by the partition = + − . Fix a hyperplane F1 ⊂ spanR corresponding to = + − as in Proposition 1(ii). Let F1 = ker τ , where τ : spanR → R is a nonzero linear ◦
function. Notice that spanR = (spanR ) ⊕ (spanR ), and hence τ = τ1 + τ2 , where ◦
τ1 : spanR → R and τ2 : spanR → R are the restrictions of τ to spanR and ◦ ◦ ◦ spanR respectively. Then 0 = ker τ1 ∩ and k = τ2−1 (τ1 ()) ∩ . Moreover, τ1 () is a discrete subgroup of R of rank d := rk − rk 0 . There are three possible cases. ◦
◦
(i) 0 = . Then k = τ2−1 ({0}) for a nonzero τ2 , i.e. the root system of k is the ◦
◦
intersection of with a proper subspace of spanR . ◦
(ii) rk = 1 and rk 0 = 0. Then k = ∅.
◦
(iii) None of the above holds for and 0 . In this case finding all possible pairs (0 , k) ◦
◦
is equivalent to finding all root subsystems of of the form τ2−1 ( ) ∩ , where is a subgroup of R of rank d. ◦
◦
◦
Example 2. 1. Let be a basis of , 0 ⊂ and let k be the root subsystem of ◦
◦
◦
with basis 0 . Then k = ker τ2 ∩ for an appropriate τ2 . Hence the pair (0 , k) arises from a Borel sublagebra of g unless 0 = and 0 = , or rk = 1, rk 0 = 0 and 0 = ∅. ◦ ◦ 2. Let rk 0 = rk −1 = 0. Then τ1 () ∼ = Z and the condition on k is k = τ2−1 (Z)∩ ◦
◦
◦
for some τ2 . In particular, if g is simple of type B2 with = {±ε1 ± ε2 , ±ε1 , ±ε2 }, ◦ ◦ ◦ then τ2−1 (τ1 ()) ∩ can be any proper subsystem of . Thus k = sl(2) ⊕ sl(2) (when ◦ ◦ ◦ ◦ ◦ = sl(2) or k = h. However, if g is simple of type C3 , the reader k = {±ε1 ± ε2 }), k ∼ ◦
◦
will check that the subsystem consisting of the long roots cannot be realized as k .
Reduction Theorem for Highest Weight Modules over Toroidal Lie Algebras
53 ◦
◦
◦
Furthermore, it is not difficult to verify that for any subalgebra k of g the pair (0 , k) arises from a Borel subalgebra of g whenever rk − rk 0 is large enough. Finally, note that the construction of k admits natural iteration. Indeed, the Borel subal− gebra b determines a triangular decomposition 0 = + 0 0 , i.e. a Borel subalgebra of k. By applying the above construction to k and its Borel subalgebra k ∩ b, we obtain a toroidal subalgebra k2 of k. Continuing the process, we obtain a strictly descending sequence of toroidal Lie algebras h = ks ⊂ . . . ⊂ k1 = k ⊂ g
(3)
together with a fixed Borel subalgebra ki ∩ b of ki for each 1 ≤ i ≤ s. Example 3. Here is an example of two Borel subalgebras in g = sl(3)[3] for which k is ◦
the same but the corresponding sequences (3) differ. Let = {±α, ±β, ±(α + β)} and let δ1 , δ2 , δ3 be fixed generators of . Define the linear function √ √ τ : spanR → R by putting τ (α) = 0, τ (β) = 2, τ (δ1 ) = 0, τ (δ2 ) = 1, τ (δ3 ) = 3. It determines a partition = −− ++ , where := ker τ ∩ and ++ := {ξ ∈ | τ (ξ ) > 0}, −− := −++ . Clearly, = (Zδ1 \{0}) ∪ (±α + Zδ1 ). Consider the following two − + − triangular decompositions of : := ( )+ 1 ( )1 and := ( )2 ( )2 , + + where ( )1 := {nδ1 , ±α + nδ1 | n > 0} ∪ {α}, ( )2 := {nδ1 | n > 0} ∪ (α + Zδ1 ), ± + ±± ( )± . One checks immediately that ( )− i := −( )i . Finally, set ()i := i + − = ()i ()i for i = 1, 2 are two triangular decompositions of . In both cases 0 = and k ∼ = sl(2)[1] is an affine Lie algebra with roots . The two triangular decompositions of produce respectively a standard and an imaginary Borel subalgebra of k. More precisely, the sequences corresponding to the decompositions − + − = ()+ 1 ()2 and = ()2 ()2 are respectively h⊂k∼ = sl(2)[1] ⊂ g
(4)
and ◦
h ⊂ k2 ∼ = t(h, Zδ1 , κ
◦
◦
|h×h
)⊂k∼ = sl(2)[1] ⊂ g.
(5)
4. On the Structure of Subquotients of Mg (λ) ◦
For the rest of the paper b is a fixed Borel subalgebra of g = t(g, , ϕ) corresponding to the triangular decomposition = + − . We have g = n− ⊕ k ⊕ n+ , where n± := ⊕α∈(± \k ) gα . For λ ∈ h∗ , set Dk (λ) := λ + spanZ k and put Mk := ⊕µ∈Dk (λ) M µ for any subquotient M of Mg (λ). In this section we assume that 0 = 0, i.e. that k is infinite dimensional, and establish a relation between submodules of Mg (λ) and respective submodules of Mk (λ). We ◦
consider first the case when g is semisimple as here the result is very natural and easy to state. ◦
Theorem 1. Let g be semisimple, k be infinite dimensional and N be a nontrivial submodule of Mg (λ). The following assertions hold:
54
I. Dimitrov, V. Futorny, I. Penkov
(i) Nk = 0. (ii) If λ(0 ) = 0, then the canonical g–module homomorphism U (g) ⊗U (k⊕n+ ) Nk → N is an isomorphism. The following corollary is immediate. ◦
Corollary 1. Let g be semisimple and k be infinite dimensional. (i) For any λ the Verma module Mg (λ) is irreducible if and only if the Verma module Mk (λ) is irreducible. (ii) If λ(0 ) = 0, then Lg (λ) ∼ = U (g) ⊗U (k⊕n+ ) Lk (λ). The proof of Theorem 1 uses induction on the Lie algebra g which requires a similar ◦ statement for reductive g as well. Therefore, in the rest of the section we will state and ◦ prove an analog of Theorem 1 for reductive g. Theorem 1 is a particular case of Theorem 2 below. ◦ ◦ ◦ ◦ ◦ Let g = gss ⊕ z be a reductive Lie algebra with semisimple part gss and center z. For λ ∈ h∗ , put ν := λ|z and ◦
Z − (ν) := span{z ⊗ t γ ∈ n− | z ∈ z and λ(γ ) = 0}. ◦
− ⊕Z − (ν), where the splitting of t(h, , ϕ Then n− = n ◦
◦
◦
◦
|h×h
◦
)∩n− is compatible with the
− ⊕k⊕n+ . One can consider U (˜ splitting h = hss ⊕ z. Set g˜ := n g) as a left U (g)–module − with trivial action of Z (ν), and define the g–module M (λ) as U (˜g) ⊗U (b) v λ . g ◦
Theorem 2. Let g be reductive, k be infinite dimensional and N be a nontrivial submodg (λ). The following assertions hold. ule of M (i) Nk = 0. (ii) If λ(0 ) = 0, then the canonical g–module homomorphism U (˜g) ⊗U (k⊕n+ ) Nk → N is an isomorphism. The following corollary is immediate. ◦
Corollary 2. Let g be reductive and k be infinite dimensional. (i) For any λ the module M g (λ) is irreducible if and only if the Verma module Mk (λ) is irreducible. (ii) If λ(0 ) = 0, then Lg (λ) ∼ = U (˜g) ⊗U (k⊕n+ ) Lk (λ). Proof of Theorem 2. Fix an oriented flag F of maximal length which determines the triangular decomposition = + − as in Proposition 1 (ii). Let F1 = ker τ , i.e. α ∈ is a root of k if and only if τ (α) = 0, cf. Sect. 3. Put b± = ⊕α∈± gα , and let 1λ ∈ v λ be a fixed nonzero vector. ◦
◦
◦
Consider a decomposition g = ⊕p gp , where each gp is either a simple Lie algebra ◦
or is the one dimensional abelian Lie algebra C z for some z ∈ z with ϕ(z, z) = 0. As a first step we show that it suffices to establish the theorem for each of the algebras gp :=
Reduction Theorem for Highest Weight Modules over Toroidal Lie Algebras ◦
t(gp , , ϕ
◦
◦
◦
|gp ×gp
◦
55 ◦
) (cf. Proposition 2 (ii)). Set g¯ p := ⊕i =p gi , g¯ p := t(g¯ p , , ϕ
◦
◦
|g¯ p ×g¯ p
)
and assume that Theorem 2 is true for g1 and, by induction, also for g¯ 1 . Let X ∈ N be a nonzero weight vector written as X = ( i ui vi wi ) · 1λ , where ui ∈ U (n− ∩ g1 ), vi ∈ U (k ∩ g1 ), wi ∈ U (b− ∩ g¯ 1 ) and the vectors wi · 1λ are linearly independent. g1 (λ|h∩g ). Set Let Mi denote the g1 –module generated by wi · 1λ . Then Mi ∼ = M 1 Xi := (ui vi wi ) · 1λ . The g1 –module Ni generated by Xi is a submodule of Mi . By Theorem 2 (i) applied to the g1 –modules N1 and M1 , there exists u¯ ∈ U (g1 ) for which u¯ · X1 = (uu ¯ 1 v1 ) · (w1 · 1λ ) is a nonzero weight vector in (N1 )k∩g1 . But then u¯ · X = 0 and, clearly, u¯ · X ∈ Nk∩g1 . Repeating the above argument for g¯ 1 instead of g1 and u¯ · X instead of X, we prove that Nk = 0. If we furthermore assume that λ(0 ) = 0, Theorem 2 (ii) applied to the g1 –modules Ni and Mi implies that vi · (wi · 1λ ) ∈ Nk∩g1 , and hence X belongs to the g1 –module generated by ⊕i (Ni )k∩g1 . A similar argument for g¯ 1 implies that N is generated by Nk . ◦ It now remains to prove the theorem when g is simple or one dimensional. First we ◦
consider the case when g = Cz for some z with ϕ(z, z) = 0. Let X ∈ N be a nonzero vector. Write X = ( i ui vi ) · 1λ , where ui ∈ U (n− ) are distinct monomials in {z ⊗ t γ | λ(γ ) = 0} and vi ∈ U (k). We will show that vi ∈ Nk for every i. If z ⊗ t γ occurs in some of the monomials ui , then for any r ∈ Z+ we have −γ r r ui v i · 1 λ , (z ⊗ t ) · X = r!(−ϕ(z, z)λ(γ )) i
where ui is the monomial ui with r terms equal to z ⊗ t γ erased whenever at least r such terms occur in ui , and ui = 0 otherwise. (In this case U (n− ) is commutative so the procedure of erasing terms from monomials is not ambiguous.) Taking r to be the maximal degree in which z ⊗ t γ occurs in a monomial ui , we obtain the nonzero vector 1 (z ⊗ t −γ )r · X = uj vj · 1λ ∈ N, X := r!(−ϕ(z, z)λ(γ ))r j
where vj equals vi for some i and uj is obtained from ui by erasing all terms equal to
z ⊗ t γ . The same procedure applied to X yields the nonzero vector 1 −γ r ) ·X = uk vk · 1λ ∈ N, X := (z ⊗ t (r )!(−ϕ(z, z)λ(γ ))r k
where vk equals vi for some i and uk is obtained from ui by erasing all terms equal to z ⊗ t γ and z ⊗ t γ . Continuing in this way, we obtain that vi0 · 1λ ∈ N for some i0 . Then we set X¯ := X − (ui0 vi0 ) · 1λ and show by an easy induction argument that vi · 1λ ∈ N ◦ for every i. This completes the proof in the case when g is abelian. Note that in this case the condition on λ is not needed for assertion (ii). ◦ For the rest of the proof g is a simple Lie algebra and we fix an indivisible element γ ∈ 0 , which has the additional property λ(γ ) = 0, whenever λ(0 ) = 0. (An element γ ∈ 0 is indivisible if lγ ∈ for l ∈ Q implies l ∈ Z.) The proof will be carried out in two separate cases: when rk 0 ≥ rk − 1 and when rk 0 < rk − 1.
56
I. Dimitrov, V. Futorny, I. Penkov ◦
◦
◦
Case 1. rk 0 ≥ rk − 1. Fix a basis B of g consisting of root vectors of g and of a ◦
◦
◦
basis of h with the property that for each h ∈ h ∩ B there are corresponding elements ◦
e, f ∈ B so that {e, h, f } forms a canonical sl(2)–basis. Then the set B = {x ⊗ t γ ∈ ◦ b− | x ∈ B , γ ∈ } is a basis of b− . For x ∈ B, let rt(x) denote the corresponding root. Assume N µ = 0 and fix a nonzero vector X ∈ N µ . Write X = ( i ui vi ) · 1λ , where − ui ∈ U (n ) are distinct monomials in the basis B and vi ∈ U (k). We are going to prove that (i) Nk = 0; (ii) if λ(0 ) = 0, then vi ∈ Nk for every i. To prove (ii) it is enough to show that vi0 · 1λ ∈ Nk for some i0 for which ui0 has maximal degree (here degree refers to the canonical filtration of U (n− )) among all ui . Indeed, then we will consider X − ui0 vi0 · 1λ = ( i =i0 ui vi ) · 1λ , for which there are fewer monomials ui whose degree equals the degree of ui0 , and will complete the argument by induction. Fix a set of generators γ1 , . . . , γn of so that γ1 , . . . , γn−1 ∈ 0 . Recall that ◦
Q = spanZ and notice that τ (Q) = Zτ (γn ) + τ () is a discrete subset of R. Furthermore, fix a linear order on − with the property that β ≺ α whenever one of the following conditions holds: – for some i α is a root of ki and β is not a root of ki ; – there is no such i, but (α, γ ) < (β, γ ). Finally, fix a linear order on B in a compatible way, i.e. x ≺ y whenever rt(x) ≺ rt(y). The proof of (i) and (ii) will be carried out by induction on τ (λ − µ). Denote by TX− the set of roots α of n− for which there exists a root vector x ∈ gα ∩ B which occurs in a monomial ui of maximal degree, and let TX0 be the set of roots β of k for which there exists a root vector y ∈ gβ ∩ B which occurs in some vi . Set TX := TX− ∪ TX0 . Fix a root α0 ∈ TX− that cannot be written as a nonnegative integer combination of roots from (TX ∪ {γ })\{α0 } (since TX is a finite set, such a root always exists). Now we consider two subcases. ◦
◦
◦
◦
Subcase 1. α0 ∈ . Then α0 = α 0 + γ0 for some α 0 ∈ and γ0 ∈ . Let f ∈ B be ◦ ◦ the root vector corresponding to α 0 and let e ∈ B be the root vector corresponding to ◦ −α 0 . Set h := [e, f ]. We claim that the vector X := (e ⊗ t −γ0 +Kγ ) · X is nonzero for large enough K ∈ Z+ . Indeed, assume that f ⊗ t γ0 occurs in u1 = x1 . . . xp , where xi ∈ B ∩ n− and α0 is among rt(x1 ), . . . , rt(xp ). Then (e ⊗ t −γ0 +Kγ ) · (u1 v1 · 1λ ) −γ 0 ( i=1 x1 . . . xi−1 [e ⊗ t +Kγ , xi ]xi+1 . . . xp v1 ) · 1λ (x1 . . . xj −1 xj +1 . . . xp )((lϕ(e, f )) (h ⊗ t Kγ )v1 ) · 1λ + X p
= = ,
(6)
γ0 where j is the minimal index for which xj = f ⊗t and there are exactly l such indices. Now write X as X = ( j uj vj ) · 1λ , where u1 := x1 . . . xj −1 xj +1 . . . xp , see (6). We claim that v1 = (lϕ(e, f )) (h ⊗ t Kγ )v1 , i.e. that u1 can only be obtained from commuting vectors of the form e ⊗ t −γ0 +Kγ with f ⊗ t γ0 in (6). Indeed, this is a consequence of
Reduction Theorem for Highest Weight Modules over Toroidal Lie Algebras
57
the choice of α0 : rt(e ⊗ t −γ0 +Kγ ) = −α0 + Kγ . On one hand, commuting e ⊗ t −γ0 +Kγ consecutively with vectors among x1 , . . . , xp different from f ⊗ t γ0 cannot result in a root vector whose corresponding root belongs to . On the other hand, commuting e ⊗ t −γ0 +Kγ consecutively with at least two vectors among x1 , . . . , xp of which at least one equals f ⊗ t γ0 will result in a root vector of n− whose corresponding root β has a product (β, γ ) larger than all such products corresponding to the vectors xi . This proves that X = 0, and clearly X ∈ N µ with τ (λ − µ ) < τ (λ − µ). Hence, by induction, Nk = 0. If, furthermore, λ(0 ) = 0, the induction assumption implies 1/(lϕ(e, f )) v1 = ((h ⊗ t Kγ )v1 ) · 1λ ∈ N . If h ⊗ t Kγ ∈ k2 , then h ⊗ t −Kγ · (((h ⊗ t Kγ )v1 ) · 1λ ) = −Kϕ(h, h)λ(γ )v1 · 1λ ∈ N , which shows that v1 · 1λ ∈ N . The proof is complete in this subcase. If h ⊗ t Kγ ∈ k2 , we use Theorem 2 for the proper subalgebra k instead of g, its Borel subalgebra b ∩ k, the corresponding Verma module Mk (λ) and its submodule N ∩ Mk (λ). A computation similar to the one above shows that v1 · 1λ ∈ N . Subcase 2. α0 = γ0 ∈ . Fix a vector h⊗t γ0 which occurs in a monomial ui of maximal ◦
degree and consider the vectors e and f of B corresponding to h, see the convention ◦ ◦ ◦ about the choice of B . Denote the root of g corresponding to e by α. Let q be the small◦
− . Then, similarly to Subcase 1, consider est integer for which α + qγ0 is a root of n +Kγ −qγ 0 X := (f ⊗ t ) · X and write X = ( j u1 v1 ) · 1λ . There are two possibilities: α + (q − 1)γ0 is a root either of n+ or of k. An argument similar to the one used in Subcase 1 shows that, for large K,
– if α + (q − 1)γ0 is a root of n+ , then we have (after possible relabeling) u1 = (f ⊗ t −(q−1)γ0 +Kγ )x1 . . . xi−1 xi+1 . . . xp and v1 = −2l v1 , where i is the minimal index for which xi = h ⊗ t γ0 and l is the number of indices r with xr = h ⊗ t γ0 ; – if α + (q − 1)γ0 is a root of k, then we have (after possible relabeling) u1 = x1 . . . xi−1 xi+1 . . . xp and v1 = (−2l) (f ⊗ t −(q−1)γ0 +Kγ )v1 , where i is the minimal index for which xi = h ⊗ t γ0 and l is the number of indices r with xr = h ⊗ t γ0 . The proof is completed by induction as in Subcase 1. Case 2. rk 0 < rk − 1. In this case the set {τ (γ ) | γ ∈ } is a dense subset of R. Fix a nonzero vector X ∈ N and write X = ( i ui vi ) · 1λ , where ui ∈ U (n− ) are linearly ◦
◦
◦
independent and vi ∈ U (k). Furthermore, fix α ∈ and a nonzero e ∈ gα . We element will prove first that N contains nonzero vectors of the form ( i ui vi ) · 1λ , where ui are (distinct) monomials in {e ⊗ t γ ∈ n− | γ ∈ } and all vi are among vi . ◦
◦
◦
As in Case 1 we work with a basis B of g consisting of root vectors of g and a basis of ◦
◦
◦
h. However, the basis for h will be fixed later in the proof. Note that, since h is abelian, ◦
U (t(h, , ϕ
◦
◦
◦
|h×h
) ∩ n− ) is simply the symmetric algebra of t(h, , ϕ
◦
◦
|h×h ◦
) ∩ n− , and
hence all definitions below are independent of the choice of a basis of h. Consider a ◦
linear order on B such that ◦
◦
◦
◦ ◦
– if x1 and x2 ∈ B are root vectors with corresponding roots β 1 and β 2 , then (α, β 1 ) < ◦ ◦
(α, β 2 ) implies x1 ≺ x2 ,
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I. Dimitrov, V. Futorny, I. Penkov ◦
◦
◦ ◦
◦
– if x1 ∈ B is a root vector with corresponding root β 1 , and x2 ∈ B ∩h, then (α, β 1 ) < 0 ◦ ◦
implies x1 ≺ x2 and (α, β 1 ) ≥ 0 implies x2 ≺ x1 .
◦
◦
We set B := {x ⊗ t γ ∈ b− | x ∈ B } and extend the order on B to a linear order ◦ on B, i.e. if x1 ≺ x2 ∈ B , then x1 ⊗ t γ ≺ x2 ⊗ t γ ∈ B. Any monomial u = z1 . . . zl in B is written so that if zi ≺ zj , then i < j . The inverse lexicographical order on the monomials in B has the following property: if zl−i ≺ zl −i , where i is the minimal nonnegative integer for which zl−i and zl −i correspond to different ele◦
ments of B , then u = z1 . . . zl ≺ z1 . . . zl = u . Also, if u = u u, with u = 1, then u ≺ u. Let X = ( i ui vi ) · 1λ ∈ N µ , where ui are distinct monomials in B. For u = (x1 ⊗ t γ1 ) . . . (xl ⊗ t γl ) we define the height of a monomial u in B as ◦ ◦◦ ◦ ◦ ◦ ◦ ht(u) := j (α − β j , α), where xj ∈ gβ j , assuming that g0 := h. We are going to prove the existence of ( i ui vi ) · 1λ ∈ N as above by induction on the maximal height of the monomials u1 , u2 , . . . . If the maximal height is 0, there is nothing to prove. Otherwise, assume that the monomial u1 has maximal height and is, in addition, maximal (with respect to the inverse lexicographical order) among all monomials of maximal height. ◦
Let u1 = x1 . . . xk . . . xl , where k is the largest index for which the vector z of B corre◦ ◦ ◦ ◦ sponding to xk does not equal e. We now choose a root vector e ∈ gα with (α, α ) > 0 ◦
◦
such that [e , z] = 0, which is possible as g is simple. At this point, if z ∈ h, we fix a ◦
basis h1 , h2 , . . . of h such that [e , hi ] = 0 for i ≥ 2. Since {τ (γ ) | γ ∈ } is dense in R, ◦ there exists γ ∈ such that τ (rt(x)) < −(τ (α ) + τ (γ )) < 0 for any x ∈ B entering a monomial ui . It is not difficult to check that (e ⊗ t γ +Kγ ) · X = j uj vj , where vj are among vi and the height of each of the elements uj is smaller than the height of u1 . This proves the existence of a vector in N as required. We may furthermore assume that ◦ all monomials ui are of the same degree. Indeed, consider h and f ∈ g so that {e, h, f } is a canonical sl(2)–basis. Then, repeating the procedure above, we can find a nonzero vector X = ( i ui vi ) · 1λ for which ui are monomials in {f ⊗ t γ ∈ n− | γ ∈ }. Now the monomials ui are necessarily of the same degree. Assume that X = ( i ui vi ) · 1λ ∈ N , where ui are (distinct) monomials of the same degree d in {e ⊗ t γ ∈ n− | γ ∈ }. As in Case 1 we need to prove that Nk = 0, and, if λ(0 ) = 0, then vi ∈ Nk . The proof will be carried by induction on the maximal degree of a monomial ui . Set X := {γ ∈ | e ⊗ t γ occurs in a monomial ui }. Let γ0 ∈ X be the element with the following properties: τ (γ0 ) = minγ ∈ τ (γ ) and if τ (γ0 ) = τ (γ ) for some γ ∈ X , then γi − γ0 ∈ + . This simply means that, for any integer K and any γ ∈ X , [f ⊗ t −γ0 +Kγ , e ⊗ t γ ] is either in n+ or in k ∩ b− . Assume furthermore, that u1 = x1 . . . xl , where exactly p of the vectors x1 , . . . , xl equal e ⊗ t γ0 , and x1 is one of them. Consider X := (f ⊗ t −γ0 +Kγ ) · X. The reader will verify easily that if X = ( i ui vi ) · 1λ , then, after a possible relabeling, u1 = x2 . . . xl , v1 = p(h ⊗ t Kγ )v1 and all ui are polynomials in {e ⊗ t γ ∈ n− | γ ∈ } of degree d − 1. An obvious induction argument shows that Nk = 0. In order to prove that v1 ∈ Nk we proceed as in the last paragraph of the proof of Case 1, Subcase 1. ◦
◦
◦
Example 4. Let g be simple, 0 = and k = h. In this case Theorem 1 has been proved in [FK]. The other previously treated cases are:
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59
◦
◦
– rk = 1, arbitrary g and arbitrary k, [CFM], ◦ ◦ ◦ – rk = 2, rk 0 = 1, arbitrary g and k = g, [CF], ◦
◦
◦
◦
◦
◦
– rk = 2, rk 0 = 1, g = sl(2) and arbitrary k (i.e., k = g or k = h), [FK]. Example 3 revisited. In the notations of Example 3, 0 = Zδ1 is the lattice corresponding both to k and k2 . Starting with the first Borel subalgebra we can apply Theorem 1 under the condition that λ(δ1 ) = 0. This reduces the study of subquotients of Mg (λ) to the study of subquotients of Verma modules of k ∼ = sl(2)[1] for a standard Borel subalgebra, which are known to have a rich and interesting structure. For the second Borel subalgebra we can apply the reduction theorem twice under the same condition. (Note that in general one needs a stronger condition to apply a second reduction.) This shows that the structure of the subquotients of Mg (λ) is much simpler in this case, as all Verma modules of the Heisenberg algebra k2 with nonzero highest weight are irreducible. 5. Equivalence of Categories In this final section we introduce an analog of the category O for toroidal Lie algebras and extend Theorems 1 and 2 to appropriate equivalences of categories. ◦ Consider first the case when g is semisimple. Let Og denote the full subcategory of the category g–mod whose objects M admit an ascending g–module filtration {Mi } such that the quotients Mi /Mi+1 are isomorphic to subquotients of Verma modules. The category Ok is defined analogously. Since z acts via a character on each subquotient of a highest weight module, z acts locally finitely on any object of Og or Ok . Both categories Og and Ok are closed under countable direct sums. Let Og (ν) and Ok (ν) be the full subcategories of Og and Ok respectively, such that their objects have the property that z acts via the homomorphism ν : z → C on all quotients of their defining filtrations. The category Og is then the full subcategory of countable dimensional g–modules in the direct product of categories ×ν∈z∗ Og (ν) whose objects are countably dimensional vector spaces over C. A similar statement holds for Ok as well. Induction provides a functor I : Ok → Og ,
N → U (g) ⊗U (k⊕n+ ) N,
where n+ acts trivially on N . The canonical image of N in U (g) ⊗U (k⊕n+ ) N consists of n+ –invariants and, hence, I (N ) is generated by its n+ –invariants. Clearly, the restriction of I on Ok (ν), which we also denote by I , is a well defined functor I : Ok (ν) → Og (ν). For ν(0 ) = 0 we will explicitly construct a functor adjoint to I . We start with the following lemma. +
Lemma 1. Let M be a subquotient of Mg (λ). If λ(0 ) = 0, then M n = Mk , where + M n are the n+ –invariants of M and Mk is as in Sect. 4. Proof. It suffices to prove the lemma for quotients of Mg (λ). Let M = Mg (λ)/K. By + definition, (Mg (λ)/K)k = Mg (λ)k /Kk and, by Theorem 1, Mg (λ)n = Mg (λ)k and + + + + K n = Kk . Hence, it is enough to show that (Mg (λ)/K)n = Mg (λ)n /K n . + + + The inclusion (Mg (λ)/K)n ⊃ Mg (λ)n /K n is obvious. The opposite inclusion + follows from Theorem 1. Indeed, assume to the contrary, that 0 = m ∈(Mg (λ)/K)n ∈
60
I. Dimitrov, V. Futorny, I. Penkov +
+
Mg (λ)n /K n . Then m generates a proper submodule M of M with the property that Mk ∩ Mk = 0. The preimage M of M in Mg (λ) has the property that Mk = Kk , which is in contradiction with Theorem 1 (ii) as M = K. Lemma 1 implies now that for ν(0 ) = 0 we have a well defined functor R : Og (ν) → Ok (ν),
+
M → M n .
+
Indeed, in order to see that M n is an object of Ok (ν) we consider a filtration of M whose associated quotients are subquotients of Verma g–modules. The consecutive application of Lemma 1 to the quotients Mi /Mi−1 ensures that the associated quotients of + + the filtration {Min } of M n are subquotients of Verma k–modules. + If ν(0 ) = 0, it is not true in general that, for any object M of Og (ν), R(M) = M n is an object of Ok (ν). Here is an example. Example 5. Let g = sl(2)[1], let b be an imaginary Borel subalgebra, i.e. k is the Heisenberg algebra, and let λ = 0. The module Mg (0) is well understood, see [F2]. Denote by N the maximal proper submodule of Mk (0), let K be the g–submodule of Mg (0) generated by N and set M := Mg (0)/K. A direct computation shows that M n+ contains + + a k–submodule isomorphic to n− . Therefore, if M n admitted a filtration {(M n )i } as − − required, n would have nontrivial k ∩ b–invariants. However, the k–module n has no + k ∩ b–invariants, and hence M n is not an object of the category Ok (0). ◦
Theorem 3. Let g be semisimple and ν(0 ) = 0. (i) The functor I : Ok (ν) → Og (ν) is a left adjoint to the functor R : Og (ν) → Ok (ν), i.e. R ◦ I is naturally isomorphic to the identity functor on Ok (ν). (ii) The functors R and I are mutually inverse equivalences of Ok (ν) and the full subcategory Ogk (ν) of Og (ν) whose objects are generated as g–modules by their n+ – invariants. Proof. (i) Let N be an object of Ok (ν) with a filtration {Ni }. Clearly, there is a canonical injection of k–modules +
: N → (U (g) ⊗U (k⊕n+ ) N )n . To see that is a surjection, assume the contrary. Then for some minimal i there is a + + nonzero vector m ∈ Min := (U (g) ⊗U (k⊕n+ ) Ni )n , m ∈ (Ni ). Hence the projection + of m in Mi /Mi−1 is a nontrivial n –invariant vector such that m ∈ (Mi /Mi−1 )k . This contradicts Lemma 1, therefore is an isomorphism. The above argument shows that R ◦ I is naturally isomorphic to the identity functor on the objects of Ok (ν). A straightforward checking shows that this is true on morphisms too. To show (ii), notice that for any object M of Og (ν) there is a canonical morphism of g–modules + : U (g) ⊗U (k⊕n+ ) M n → M. We claim that is injective. To see this it suffices to prove that is injective on + U (g) ⊗U (k⊕n+ ) Min for every i, {Mi } being the defining filtration on M. The latter is established by a simple inductive argument based on Theorem 1. Finally, in order to assert that is surjective one has to assume that M is an object of Ogk (ν). This implies (ii) modulo some details that the reader will easily fill in.
Reduction Theorem for Highest Weight Modules over Toroidal Lie Algebras
61
Here is an explicit example of an object in Og (ν) which is not an object of Ogk (ν). Example 6. Let g = sl(2)[1], cf. Example 1. Set e(i) := e ⊗ t i , f (i) := f ⊗ t i , h(i) = h ⊗ t i , for i ∈ Z. Here {e, f, h} is a standard basis of sl(2) and κ is normalized so that κ(h, h) = 2. Then the explicit commutation relations of sl(2)[1] are as follows: [e(i), f (j )] = h(i + j ) + iδi,−j c, [h(i), e(j )] = 2e(i + j ), [h(i), f (j )] = −2f (i + j ), [h(i), h(j )] = 2iδi,−j c, [e(i), e(j )] = [f (i), f (j )] = 0, [d, e(i)] = ie(i), [d, h(i)] = ih(i), [d, f (i)] = if (i), [c, e(i)] = [c, h(i)] = [c, f (i)] = [c, d] = 0. Fix the Borel subalgebra b = h ⊕i>0 Ch(i) ⊕i Ce(i). For the corresponding triangular decomposition g = n− ⊕ k ⊕ n+ we have n− = ⊕i Cf (i), k = ⊕i Ch(i) ⊕ Cc ⊕ Cd and n+ = ⊕i Ce(i). First we construct an indecomposable k⊕n+ –module M0 of length 2. Let C[s1 , s2 , ...] be the ring of polynomials in infinitely many variables {si }i=1,2,... , and let W = Cv ⊕Cu be a two dimensional vector space. Set ∂ 2i ∂si p(s) for i < 0, h(i) · p(s) := si p(s) for i > 0, ∂ c · p(s) := p(s), and d · p(s) := ( ∞ i=1 i ∂si )p(s) for any p(s) ∈ C[s1 , s2 , . . . ]. Then M := C[s1 , s2 , . . . ] ⊗ W is a ⊕i =0 Ch(i) ⊕ Cc ⊕ Cd–module with the above action on C[s1 , s2 , . . . ] and with trivial action on W . By putting h(0) · (p(s) ⊗ v) := 0 and h(0) · (p(s) ⊗ u) := −2p(s) ⊗ u, we define a k–module structure on M . Clearly, M decomposes into a direct sum of the irreducible k–modules C[s1 , s2 , . . . ] ⊗ v and C[s1 , s2 , . . . ] ⊗ u. Below we will extend the k–module structure on M to a k⊕n+ –module structure such + that (M )n = C[s1 , s2 , . . . ] ⊗ v. Then M := I (M) = U (g) ⊗U (k⊕n+ ) M will be an indecomposable g–module with composition series 0 ⊂ M1 ⊂ M with M/M1 ∼ = M2 , where M1 := I (C[s1 , s2 , . . . ] ⊗ v) and M2 := I (C[s1 , s2 , . . . ] ⊗ u), n+ acting trivially on C[s1 , s2 , . . . ]⊗v and C[s1 , s2 , . . . ]⊗u. Indeed, C[s1 , s2 , . . . ]⊗v and C[s1 , s2 , . . . ]⊗ u are irreducible k–modules on which c acts nontrivially, and hence, by Theorem 1, both + M1 and M2 are irreducible g–modules. Finally, M n = C[s1 , s2 , . . . ] ⊗ v, and hence (I ◦ R)(M) ∼ = M1 . To construct the k ⊕ n+ –structure on M we first set e(n) · (p(s) ⊗ v) := 0 and recall i that Schur’s polinomials S are given by their generating function exp( n i≥1 si x ) = n n∈Z Sn (s1 , s2 , . . . )x . (Note that Sn = 0 for n < 0.) From the definition of Sn we obtain ∂ Sn (s) = Sn−i (s). ∂si We show now that there exists a unique k ⊕ n+ –module structure on M which extends its k–module structure and such that e(0) · (1 ⊗ u) = 1 ⊗ v. The uniqueness is clear. For the existence, we first use the commutation relations [h(i), e(−n)] = 2e(i − n) with i > 0 and n > 0 to derive the equality e(−n) · (1 ⊗ u) = Sn (¯s ) ⊗ v, where s¯ stands for s1 , s2 /2, s3 /3, . . . . Furthermore, we extend the latter formula to
(−2)#L si Sn−i∈L i (¯s ) ⊗ v, (7) e(−n) · (si1 si2 . . . sil ⊗ u) = L⊂K
i∈K\L
where n ∈ Z, K is the set of all indices i1 , i2 , . . . , il with possible repetitions, and #L stands for the number of elements of L. We leave it to the reader to check that (7) together
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I. Dimitrov, V. Futorny, I. Penkov
with the equality e(n) · (p(s) ⊗ v) = 0 determines a k ⊕ n+ –module structure on the + k–module M . Finally, since M has length 2 as a k–module, (M )n is a k–submodule + of M , and e(0) · (1 ⊗ u) = 1 ⊗ v, we conclude that (M )n = C[s1 , s2 , . . . ] ⊗ v. Recall that the Borel subalgebra b of g determines a strictly decreasing sequence of toroidal Lie algebras (3). In order to be able to construct a chain of adjoint functors between ki –modules and ki+1 –modules, we need a generalization of Theorem 3 in the ◦ case of a reductive g. Theorem 2 enables us to modify the category Og (ν) and the functor I accordingly. ◦ g (ν) be the full subcategory of Og (ν) on whose Assume now that g is reductive. Let O − objects Z (ν) acts trivially, and let I˜ be the functor g (ν), I˜ : Ok (ν) → O
N˜ → U (˜g) ⊗U (k⊕n+ ) N,
where n+ acts trivially on N . Using Theorem 2 one establishes that for any object M ◦ g (ν), R(M) is an object of Ok (ν). The main result for reductive g is the following of O theorem. ◦
Theorem 4. Let g be reductive and ν(0 ) = 0. g (ν) is a left adjoint to the functor R : O g (ν) → (i) The functor I˜ : Ok (ν) → O Ok (ν), i.e. R ◦ I˜ is naturally isomorphic to the identity functor on Ok (ν). (ii) The functors R and I˜ are mutually inverse equivalences of Ok (ν) and the full k (ν) of O g (ν) whose objects are generated as g–modules by their n+ – subcategory O g invariants. The proof of Theorem 4 is a straightforward modification of the proof of Theorem 3 based on Theorem 2. Acknowledgements. The authors gratefully acknowledge MSRI’s support and excellent working conditions. We also thank the referee for several thoughtful comments. Vyacheslav Futorny is a Regular Associate of the ICTP.
References [BB] [BC] [C] [CF] [CFM] [D] [DP1] [DP2]
Berman, S., Billig, Y.: Irreducible representations for toroidal Lie algebras. J. Algebra 221, 188–231 (1999) Berman, S., Cox, B.: Enveloping algebras and representations of toroidal Lie algebras. Pacific J. Math. 165, 239–267 (1994) Cox, B.: Verma modules induced from nonstandard Borel subalgebras. Pacific J. Math. 165, 269–294 (1994) Cox, B., Futorny, V.: Borel subalgebras and categories of highest weight modules for toroidal Lie algebras. J. Algebra 236, 1–28 (2001) Cox, B., Futorny, V., Melville, D.: Categories of nonstandard highest weight modules for affine Lie algebras. Math.Z. 221, 193–209 (1996) Donaldson, S.: Anti self–dual Yang–Mills connections over complex algebraic surfaces and stable vector bundles. Proc. Lond. Math. Soc. 50(3), 1–26 (1985) Dimitrov, I., Penkov, I.: Partially and fully integrable modules over Lie superalgebras. In: Studies in Advanced Mathematics, S.-T. Yau (ed.), 4, Providence, RI: AMS and Internatl. Press, 1997, pp. 49–67 Dimitrov, I., Penkov, I.: Partially integrable highest weight modules. Transform. Groups 3, 241–253 (1998)
Reduction Theorem for Highest Weight Modules over Toroidal Lie Algebras [F1]
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Futorny, V.: Representations of Affine Lie algebras. Queen’s Papers in Pure and Applied Mathematics 106, Queen’s University, 1997 [F2] Futorny, V.: Verma type modules of level 0 for affine Lie algebras. Trans. Am. Math. Soc. 349, 2619–2661 (1997) [FK] Futorny, V., Kashuba, I.: Verma type modules for toroidal Lie algebras. Comm. Algebra 27, 3979–3991 (1999) [IKU] Inami, T., Kanno, H., Ueno, T.: Higher-dimensional WZW model on K¨ahler manifold and toroidal Lie algebra. Mod. Phys. Lett A 12, 2757–2764 (1997) [IKUX] Inami, T., Kanno, H., Ueno, T., Xiong, C.-S.: Two–toroidal Lie algebra as current algebra of four–dimensional K¨ahler WZW model. Phys. Lett B 399, 97–104 (1997) [JK] Jakobsen, H., Kac, V.: A new class of unitarizable highest weight representations of infinite dimensional Lie algebras. Lecture Notes in Physics 226, Berlin-Heidelberg-New York: Springer-Verlag, 1985, pp. 1–20 [K] Kassel, C.: K¨ahler differentials and coverings of complex simple Lie algebras extended over a commutative algebra. J. Pure Appl. Algebra 34, 265–275 (1984) [MRY] Moody, R., Rao, S., Yokonuma, T.: Toroidal Lie algebras and vertex representations. Geom. Dedicata 35, 283–307 (1990) Communicated by L. Takhtajan
Commun. Math. Phys. 250, 65–80 (2004) Digital Object Identifier (DOI) 10.1007/s00220-004-1118-3
Communications in
Mathematical Physics
Quasi-Spherical Metrics and Applications Yuguang Shi1, , Luen-Fai Tam2, 1 2
Key Laboratory of Pure and Applied Mathematics, School of Mathematics Science, Peking University, Beijing 100871, P.R. China. E-mail:
[email protected] Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong, P.R. China. E-mail:
[email protected]
Received: 7 July 2003 / Accepted: 22 October 2003 Published online: 1 June 2004 – © Springer-Verlag 2004
Abstract: In this paper, using the idea of Bartnik [B2] on quasi-spherical metrics we continue our study on the boundary behaviors of compact manifolds with nonnegative scalar curvature and nonempty boundary. Unlike the previous work [ST] of the authors and the work of Liu-Yau [LY], we only assume each boundary component has nonnegative curvature which is not identically zero. We also study the case that the boundary is embedded in the quotient of the infinity of the Euclidean space over a finite group. The regularity of the black hole boundary condition of quasi-spherical metrics is also discussed. 0. Introduction In [B2], Bartnik introduced a family of metrics with prescribed scalar curvature, which are called quasi-spherical metrics. Let 0 be a compact strictly convex surface in R3 and hence the exterior of 0 can be foliated by the level surfaces r of the distance function from 0 . One can construct a metric with zero scalar curvature on the exterior of 0 which is of the form u2 dr 2 + gr , where gr is the induced metric from R3 on r and u is a solution of
H0 ∂u = u2 r u + (u − u3 )K on0 × [0, ∞), ∂r u(x, 0) = u0 (x)
(0.1)
where H0 is the mean curvature and K is the Gauss curvature of r . If 0 is the standard sphere, then (0.1) is a special case considered in [B2]. For simplicity, we still call metrics obtained in this way quasi-spherical metrics.
Research partially supported by NSF of China, Projects 10001001 and 10231010. Research partially supported by Earmarked Grant of Hong Kong #CUHK4032/02P.
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The construction of [B2] is very useful. In [ST], by studying the properties of (0.1), the authors proved the following: Theorem 0.1. Let (3 , g) be a compact manifold of dimension three with boundary and with nonnegative scalar curvature. Suppose ∂ has finitely many components i so that each component has positive Gauss curvature and positive mean curvature with respect to the unit outward normal. Then for each boundary component i , (i) H dσ ≤ H0 dσ, (0.2) i
i
(i)
where H0 is the mean curvature of i with respect to the outward normal when it is isometrically embedded in R3 , dσ is the volume form on i induced from g. Moreover, if equality holds in (0.2) for some i , then ∂ has only one component and is a domain in R3 . This result can be interpreted as positivity of quasi-local mass defined by Brown and York [BY 1, BY 2]. Later, in [LY], Liu and Yau proposed new definitions of quasilocal energy and momentum surface energy of a spacelike 2-surface with positive intrinsic curvature in a spacetime. Using Theorem 0.1, they showed that the quasilocal energy of the boundary of a compact spacelike hypersurface which satisfies the dominant energy condition is strictly positive unless the spacetime is flat along the spacelike hypersurface. However, both papers need to assume the Gauss curvature of the boundary surface is positive. It is interesting to know whether this assumption can be removed. One purpose of this paper is to consider the case that the boundary of the manifold is only assumed to have nonnegative Gauss curvature which is positive somewhere in each boundary component. More precisely, we will show: Theorem 0.2. Let (3 , g) be a compact manifold of dimension three with smooth boundary and with nonnegative scalar curvature. Suppose ∂ has finitely many components i so that each component has nonnegative Gauss curvature which is positive somewhere and has positive mean curvature H with respect to the unit outward normal. Then for each boundary component i , (i) H dσ ≤ H0 dσ, (0.3) i
i
(i) H0
is the mean curvature of i with respect to the outward normal when it is where isometrically embedded in R3 , dσ is the volume form on i induced from g. To prove Theorem 0.1, one embeds the surface in R3 , which can be done if the Gauss curvature is strictly positive, see [N]. Then one can construct quasi-spherical metrics. If we only assume that the surface has nonnegative Gauss curvature, then there is an example by Iaia [I] of a smooth surface (S2 , g) with nonnegative Gauss curvature which cannot be C 3 embedded in R3 . However, such a surface still has a C 1,1 embedding by [GL, HZ]. In particular, the mean curvature of the embedded surface can be defined almost everywhere so that (0.3) is meaningful. If equality holds in (0.3), then we have the following: Theorem 0.3. With the same assumptions as in Theorem 0.2, if equality holds in (0.3) for some i , then ∂ has only one component and is scalar flat. If in addition each i can be C 4 embedded in R3 with positive mean curvature, then is a domain in R3 .
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An interesting boundary condition in (0.1) is the black hole boundary condition u−1 = 0. In case 0 ⊂ R3 is the sphere with radius 2M, M > 0, then the solution with this boundary data is just the Schwarzschild 3-metric. Hence it is interesting to understand what we can say about the regularity of the metric near r = 0. We are able to prove that similar to the Schwarzschild metric the constructed metric has bounded curvature, can be extended to be Lipschitz at r = 0, and the surface r = 0 is totally geodesic. This is a partial generalization of Bartnik [B2, Theorem 4.6] and is related to a question of Bartnik [B3] In certain cases the results in [ST] are still true in the asymptotically locally Euclidean (ALE) case. More precisely, if is spin and the boundary of in Theorem 0.1 is assumed to be embedded in Rn \ {0}/ , where is a finite subgroup of SO(n) which acts freely on Rn \ {0}, then under certain conditions on , results similar to those in Theorem 0.1 are true. In particular, if is a subgroup of SU(m) so that has odd order, then (0.2) is still true, where H0 is the mean curvature of the boundary surface in Rn \ {0}/ . This gives some restriction on the existence of metrics with nonnegative scalar curvature on a compact manifold with boundary whose components may not be simply connected. The paper is organized as follows: Theorems 0.2 and 0.3 will be proved in §1; in §2 we study regularity properties of solutions of (0.1) with black hole boundary condition and in §3 we discuss nonexistence of certain Riemannian metrics on compact manifolds with boundary. 1. Boundary with Nonnegative Curvature We first recall some results in [ST]. Let 0 be a strictly convex compact hypersurface in Rn . Then the exterior of 0 in Rn with the standard metric can be represented by (0 × (0, ∞), dr 2 + gr ), where gr is the induced metric on r which is the hypersurface consisting of the points outside 0 and with distance r from 0 . Consider the following initial value problem: 2H0 ∂u = 2u2 r u + (u − u3 )Rr on 0 × [0, ∞), ∂r (1.1) u(x, 0) = u0 (x) where u0 (x) > 0 is a smooth function on 0 , H0 and Rr are the mean curvature and scalar curvature of r respectively, and r is the Laplacian operator on r . By [ST, §2], we have: Lemma 1.1. For any smooth initial data u0 > 0, (1.1) has a unique solution such that r − 1 r − 1 2 2 1 + C2 exp − ξ(s)ds ≤ u(x, r) ≤ 1 − C1 exp − ϕ(s)ds , 0
where
0
Rr (x, r) > 0, x∈0 H0 (x, r) −2 C1 = 1 − max u0 + 1 ,
ϕ(r) = min
0
Rr (x, r) > 0, x∈0 H0 (x, r) −2 C2 = min u0 − 1,
ψ(r) = max
0
and ξ(r) = ϕ(r) if min0 u0 ≤ 1, ξ(r) = ψ(r) if min0 u0 > 1.
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Moreover, the metric ds 2 = u2 dr 2 + gr has zero scalar curvature and is asymptotically Euclidean in the sense that if ds 2 = ij gij dx i dx j in the standard coordinates (x 1 , . . . , x n ) on the exterior of 0 in Rn , then there exists a constant C such that |gij (x) − δij | + |x| |∇0 gij (x)| + |x|2 |∇02 gij (x)| ≤ C|x|2−n , where ∇0 , ∇02 are the gradient and Hessian operator with respect to the Euclidean metric. The uniqueness of solutions of (1.1) can be seen as follows. Suppose u, v are two solutions of (1.1) on 0 × [0, T ], T > 0 with the same initial data, and suppose u, v are bounded above and below by some positive constants. Let w = u − v. Then w satisfies: 2H0
∂w = 2u2 r w + 2(u + v)v + 1 − u2 − uv − v 2 Rr w ∂r
which is of the form 2H0
∂w = 2u2 r w + F w, ∂r
where F is a smooth function. From this it is easy to see that u ≡ v. Now suppose (, g) is a smooth compact oriented surface with smooth metric g and with nonnegative Gauss curvature which is positive somewhere. By [GL,HZ], (, g) can be C 1,1 isometrically embedded in R3 as a convex surface. Let X : (, g) → R3 be such an embedding. Define the mean curvature H0 in R3 with respect to the unit normal n by H0 = X, n , where is the intrinsic Laplacian of (, g) and X is considered as a vector valued function and the inner product is taken in R3 . In our convention, the mean curvature with respect to the outward normal of the unit sphere in R3 is 2. Note that H0 is defined almost everywhere, is bounded because X is C 1,1 and is equal to the usual mean curvature if X is C 2 . Moreover, H0 is independent of the embedding in the following sense. Suppose Y is another C 1,1 embedding of (, g) as a convex surface in R3 . Then by the result of Pogorelov [P, p.167], X() and Y() are congruent. Hence there is a C 1,1 isometry F of (, g) and a Euclidean motion A in R3 such that A ◦ X = Y ◦ F . By [Ha], F is in fact C 2 . Hence in our definition the mean curvatures at the corresponding points in X() and Y() are equal. With this definition in mind, we have the following: Theorem 1.1. Let (3 , g) be a compact manifold of dimension three with smooth boundary and with nonnegative scalar curvature. Suppose ∂ has finitely many components i so that each component has nonnegative Gauss curvature which is positive somewhere and has positive mean curvature H with respect to the unit outward normal. Then for each boundary component i ,
H dσ ≤ i (i)
i
(i)
H0 dσ,
(1.2)
where H0 is the mean curvature of i with respect to the outward normal when it is isometrically embedded in R3 , dσ is the volume form on i induced by g.
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Proof. We only prove the case that ∂ has only one boundary component . The general case can be proved similarly. As mentioned above, (, g) can be C 1,1 embedded in R3 as a convex surface by [GL, HZ]. The embedding is obtained in the following way. One can find smooth functions wi on such that (, e2wi g) has positive Gauss curvature and such that wi → 0 in C ∞ as i → ∞. In fact, since the Gauss curvature K of (, g) is nonnegative and is positive somewhere, one can find a smooth function such that g w = −1 in a neighborhood of {x ∈ | K = 0}, where g is the Laplacian of (, g). Then wi = i w will satisfy the requirements provided i is positive, small and i → 0. Let Xi : (, e2wi g) → R3 be the embedding of (, e2wi g). Passing to a subsequence and modifying Xi by rigid motions in R3 if necessary, Xi converge in C 1 to a C 1,1 isometric embedding X of (, g) in R3 . Moreover, ||∇ 2 Xi || ≤ C1 ,
(1.3)
for some constant C1 independent of i. Here ∇ is computed with respect to the metric g on . For each i, let vi be the solution of in g vi = 0 . 1 wi 2 vi = e on ∂ = Here we also denote the Laplacian of (, g) by g . By the maximum principle, vi is positive and vi converge to the constant function 1 in C ∞ . Let gi = vi4 g. Then (, gi ) = (, e2wi g) and the scalar curvature Ri of (, gi ) is given by Ri = vi−5 (−8g vi + Rvi ) ≥ 0, where R is the scalar curvature of g. Let Hi be the mean curvature of with respect to the metric gi . Since gi converge to g in C ∞ , Hi converge uniformly on to H as i → ∞. Since H > 0, Hi > 0 if i is large enough and we assume this is true for all i. By Theorem 4.2 in [ST], for each i we have
Hi dσi ≤
Hi,0 dσi ,
(1.4)
where Hi,0 is the mean curvature of (, gi ) when embedded in R3 and dσi is the volume element of (, e2wi g). By (1.4) and the fact that Hi → H as i → ∞, in order to prove the theorem, it is sufficient to prove that Hi,0 dσi = H0 dσ, (1.5) lim i→∞
where H0 is the mean curvature of (, g) when embedded in R3 which is defined after Lemma 1.1. Let and i be the Laplacians of (, g) and (, e2wi g) respectively, let ∇ and ∇i be the covariant derivatives with respect to g and e2wi g respectively, and let n and ni be the unit normals of X() and Xi () respectively. Then
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X − i Xi , n = (H − Hi,0 ) + Hi,0 n − ni , n .
(1.6)
Now n, X − i Xi dσ
= n, X dσ − n, i Xi dσi + n, i Xi e2wi − 1 dσ
=−
3
∇ns , ∇x s g dσ +
s=1
+
3 s=1
∇i ns , ∇i xis gi dσi
n, i Xi e2wi − 1 dσ,
where n = (n1 , n2 , n3 ), X = (x 1 , x 2 , x 3 ) and Xi = (xi1 , xi2 , xi3 ), , , , g , , gi are the inner products in R3 , g and gi respectively. Using the fact that wi → 0, X → Xi in C 1 as i → ∞ and using (1.3), we conclude that lim
i→∞
n, X − i Xi dσ = 0.
(1.7)
Since ni → n uniformly as i → ∞, by (1.3) we also have lim
i→∞
Hi,0 n − ni , n dσ = 0.
(1.8)
By (1.6)–(1.8), we conclude that lim
i→∞
H0 − Hi,0 dσ = 0.
Since wi → 0, (1.4) is true. This completes the proof of the theorem.
Suppose equality holds in (1.2) for a boundary component. Then we have the following: Theorem 1.2. With the same assumptions as in Theorem 1.1, if equality holds in (1.2) for some i , then ∂ is connected and is scalar flat. If in addition ∂ can be C 4 embedded in R3 with positive mean curvature, then is a domain in R3 . Before we prove the theorem, we need the following lemma. Lemma 1.2. Let (, g) as in Theorem 1.1 so that (∂, g) is connected. Suppose there is a C 4 isometric embedding X of ∂ in R3 with mean curvature H0 . Then for any > 0, there is a smooth metric g on and a C ∞ embedding X of (∂, g ) in R3 such that (i) (∂, g ) has positive Gauss curvature; (ii) ||g −g ||C 3 () ≤ ; (iii) ||X−X ||C 2 () ≤ and ( iv) g has nonnegative scalar curvature on .
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Proof. ∂ = is conformal to S2 because it has nonnegative Gauss curvature and is positive somewhere. Hence we may write (, g) = (S2 , g) so that gij = e2w λij , where λij is the standard metric of S2 . The isometric embedding can be represented by X(x) = ρ(x)x for x ∈ S2 and ρ(x) is a C 4 function on S2 . We assume that the origin is the center of the largest ball contained in the convex body bounded by 0 which is the image of ∂. Then ρ > 0. In local coordinates of S2 we have, gij = ρi ρj + ρ 2 λij .
(1.9)
ρ Let > 0, and let ρ = 1+ ρ . It is easy to see that ρ converges uniformly to ρ in C 4 as → 0. Consider the surfaces given by
X (x) = ρ (x)x. Since X has nonnegative Gauss curvature, X has positive Gauss curvature, see [LW] for example. Moreover, H0 → H0 , where H0 is the mean curvature of X (S2 ). By (1.9) and the fact that gij = e2w λij , the metric for X (x) is
1 gij = e2w + 2 ρ 3 + 2 ρ 4 λij . 4 (1 + ρ) Hence gij = e2w λij , where w is C 4 and it converges to w in C 4 as → 0. Now for each > 0, we can find smooth functions vk, on S2 such that vk, converge to w in C 4 as k → ∞. Since (, g ) has positive Gauss curvature, we can find a smooth embedding Xk of (, e2vk, λij ) such that Xk converge to X in C 2 as k → ∞, see [N]. From this we conclude that for any > 0, there exists a smooth function v on such that the metric gij = e2v λij satisfies ||g − g ||C 4 ≤ in . Moreover, there exists an embedding X of (, g ) in R3 such that ||X − X||C 2 () ≤ . Now we solve the equation in g v = 0 . 1 (v −w) v = e2 on ∂ = Consider the metric v 4 g on which is g when restricted on . We denote this metric also by g . Then as before, g is a smooth metric on with nonnegative scalar curvature. Since ||g − g ||C 4 ≤ in is small, ||(v − 1)||C 3 () ≤ C( ) such that C( ) → 0 as → 0. From this, it is easy to see the lemma is true.
Proof of Theorem 1.2. Let 1 , . . . , k be the components of ∂. Suppose equality holds in (1.2) for some i. Let us assume that i = 1 so that
(1)
1
(H0 − H ) dσ = 0.
(1.10)
We want to prove that k = 1. Suppose k > 1. For any 0 < < 1, solve the following: in , g v = 0 . v = 1 − on 2 , v = 1 on i for i = 2
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By the maximum principle, v > 0 on . Consider the metric g˜ = (v )4 g on , then it has nonnegative scalar curvature as in the proof of Theorem 1.1. It is equal to (1 − )4 g on 2 and is equal to g on i for i = 2. Moreover, the mean curvature on each i with respect to g˜ is positive if is small enough and the Gauss curvature of (i , g) ˜ is nonnegative and is positive somewhere for each i. By the strong maximum principle, v < 1 inside because v < 1 on 2 and v = 1 on i if i = 2. By the strong maximum principle again, the mean curvature of 1 with respect to g˜ is
∂v H˜ = H + 4 > H, ∂ν where H is the mean curvature of g. ˜ Since g˜ = g on 1 , we have (1) (1) ˜ H dσ ≤ H0 dσ = Hd σ < H0 dσ 1
1
1
1
by Theorem 1.1. This is a contradiction. Hence ∂ has only one component. Next we want to prove that is scalar flat. Let = ∂ and let H0 be the mean curvature of in R3 . Let R be the scalar curvature of and let u be the solution of
g u −
=0 u=1
R 8u
in . on ∂ =
By the maximal principle, we see 0 < u ≤ 1. Let H¯ be the mean curvature of with ∂u respect to the metric g¯ = u4 g, then H¯ = H + 4 ∂u ∂ν > 0, because H > 0 and ∂ν ≥ 0. Moreover,
∂u dσ ∂ν ∂u = H dσ + 4 u dσ ∂ν R = |∇u|2 + u2 H0 dσ + 4 8 R ≥ |∇u|2 + u2 , H¯ dσ + 4 8
H¯ dσ =
H dσ + 4
where we have used (1.10) and Theorem 1.1. Hence R ≡ 0. Suppose (1.10) is true and if in addition = ∂ can be C 4 embedded (with embedding X) in R3 with positive mean curvature H0 . Note that in this case, X() is convex by [Sa]. By Lemma 1.2, there exist smooth metrics gi on with nonnegative scalar curvature such that (, gi ) has positive Gauss curvature and there exists an embedding Xi of (, gi ) for each i with the following properties: lim ||Xi − X||C 2 () = lim ||g − gi ||C 3 () = 0.
i→∞
i→∞
(1.11)
By the proof of Theorem 4.1 and 4.2 in [ST], we can glue (, gi ) with the exterior of Xi () in R3 to be a smooth manifold Mi , so that we can find a metric on the exterior of Xi (), which will also be denoted by gi , then gi is a smooth metric in and
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on the exterior of Xi (), and is Lipschitz near Xi (). In fact, the metric gi is equal to u2i dr 2 + gi,r , where dr 2 + gr is the Euclidean metric on the exterior of Xi () as described at the beginning of this section and ui is the solution of (1.1) with initial condition ui = Hi,0 /Hi . Here Hi is the mean curvature of with respect to the metric gi and Hi,0 is the mean curvature of Xi () in the Euclidean space. We identify (, gi ) with Xi (). We want to define a bilipschitz map between M1 and Mi for i > 1. To begin with, let us identify as S2 as a smooth surface. We may denote the embedding Xi as a map Xi (x) = ρi (x)x and X(x) = ρ(x)x with x ∈ S2 in R3 . We may also assume that ρ(x) > 0 everywhere. Define a map Fi from R3 \ I (i ) to R3 \ I (1 ) by Y = Fi (X) =
ρ1 (X/|X|) X, ρi (X/|X|)
where I (i ) denotes the interior of Xi () in R3 . It is easy to see that Fi is bijective. Moreover, Fi (ρi (x)x) = ρ(x)x. Hence Fi induces an identity map on . Since ρ(x) and ρi (x) are bounded away from zero by a positive constant independent of i, and the C 1 norm of ρi is bounded by a constant independent of i, we conclude that C2−1 ≤ ||Fi (X1 ) − Fi (X2 )|| ≤ ||X1 − X2 || ≤ C2 ||Fi (X1 ) − Fi (X2 )||
(1.12)
for some constant C2 independent of i. On the other hand, by the construction of gi as described in the proof of Theorem 4.1 in [ST] and by Lemma 1.1, we conclude that the metric gi on R3 \ I (i ) is equivalent to the Euclidean metric, provided i is large enough. Here we have used the fact that H0 > 0 everywhere and (1.11). More precisely, there is a constant C3 > 0 independent of i so that C3−1 ge ≤ gi ≤ C3 ge , where ge is the Euclidean metric. Hence Fi is a bilipschitz map from (R3 \ I (i ), gi ) to (R3 \ I (1 ), ge ). Now we extend Fi to be a map from Mi to M1 such that Fi | is the identity map. Since gi → g in C 3 (), it is easy to see that C4−1 dg1 (Fi (p), Fi (q)) ≤ dgi (p, q) ≤ C4−1 dg1 (Fi (p), Fi (q)) for some constant C4 independent of i. Hence there is a constant C5 independent of i such that for any W01,2 (Mi ) function f we have
1 6
f dVi Mi
3
≤ C5
|∇i f |2 dVi ,
(1.13)
Mi
where dVi is the volume element of (Mi , gi ) and ∇i f is the gradient of f with respect to gi . By [ST, §3], for each i, there is a harmonic spinor i such that the norm of i satisfies i (X)||i ≤ C|X|−1 log |X| ||i (X) −
(1.14)
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i is a parallel spinor on R3 \ I (i ) for some constant C in X ∈ R3 \ I (i ), where and ||i || ≡ 1 with respect to the Euclidean metric. Here || · ||i is the norm with respect 1,2 to gi and |X| is the Euclidean norm of X. Moreover, ||i ||i is in Wloc . By (1.13) and (1.14), we conclude that
1 (||i ||i − 1)6 dVi
3
≤ C5
Mi
||∇i i ||2 dVi .
(1.15)
Mi
On the other hand, from [ST], we have
||∇i i ||2 dVi ≤ C6 Hi,0 − Hi dσi Mi
(1.16)
for some absolute constant C6 . Note that ||i ||i is uniformly bounded by 1 by construction. Using the fact that gi → g in C 3 in , we may assume that {i } converges in C 1 in to a spinor in (, g). By (1.15) and (1.16), using the fact that
lim Hi,0 − Hi dσi = (H0 − H ) dσ = 0 i→∞
we conclude that is a nontrivial parallel spinor in (, g). Hence (, g) is Ricci flat (see [Hi]), and hence it is flat because dim() = 3. Since (, g) can be C 4 embedded in R3 , we can glue (, g) along with R3 \ I (), where I () is the image of under the embedding, so that a tubular neighborhood of in is identified with a tubular neighborhood of in R3 and the outward normals are the same. The resulting manifold is a C 3 manifold M and the metric given by g on and the Euclidean metric ge on R3 \ I () is Lipschitz near . Since the Gauss curvature of is nonnegative and the curvature of is zero, as in [S, v. 5, p.284-288] one can prove that on the set {x ∈ : K(x) > 0}, the second fundamental forms with respect to g and ge are the same. Moreover, suppose x ∈ such that the Gauss curvature is zero in a neighborhood of x, then x is a parabolic point of in R3 because H0 > 0. In this case, we can still prove that the second fundamental forms are the same. By Lemma 4.1 in [ST], (M, g) is a C 2 metric on a C 3 manifold which is flat outside a compact set. Hence it is isometric to R3 by volume comparison [BC]. This completes the proof of the theorem.
2. Regularity: Black Hole Boundary Conditions In [ST], it was remarked that as in [B2] one can solve (1.1) with black hole boundary 3 data u−1 0 = 0. In case 0 ⊂ R is the sphere with radius 2M, M > 0, then the solution with this boundary data is just the Schwarzschild 3-metric. Hence it is interesting to understand what we can say about the regularity of the metric near r = 0. The following theorem answers these questions and partially generalizes Theorem 4.6 in Bartnik’s work [B2]. In our case, we only assume 0 to be strictly convex. In [B2], an equation more general than (1.1) was considered with 0 being the standard two sphere. We also prove that the quasi-spherical metric can be extended in a neighborhood of the black hole. In this section, we always assume that 0 is a compact strictly convex surface in R3 . For the sake of completeness, we will also give the details of the construction of the solution u. For any positive integer k, let uk be the solution of (1.1) with uk (x, 0) = k.
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Theorem 2.1. With above notations, uk subconverge to a solution u of the equation in (1.1) in 0 × (0, ∞) such that the metric ds 2 = u2 dr 2 + gr on 0 × (0, ∞) is asymptotically Euclidean with zero scalar curvature. Moreover, the following are true: (i) There exist positive constants C, C such that C/r ≤ u2 (x, r) ≤ C /r, for (x, r) ∈ 0 × (0, 1]. (ii) The surface 0 = 0 × {0} is totally geodesic with respect to the metric ds 2 . More precisely, let ||A|| be the norm of the second fundamental form of r = 0 × {r}, then lim sup ||A|| = 0. r→0 r
(iii) The curvature of the metric ds 2 is bounded near r = 0. (iv) The metric ds 2 is Lipschitz up to r = 0. More precisely, let F : 0 × (0, ∞) → 0 × (0, ∞) be defined by F (r, x) = (ρ, y) r ∗
with y(r, x) = x and ρ(r, x) = 0 u(x, τ ) dτ . Then the metric F −1 (ds 2 ) can be extended to be Lipschitz continuous on 0 × [0, 1]. Proof. For any a > 0, by Lemma 1.1 and using the notations in the lemma, we have 1 ≤ uk (x, r) ≤ 1 − exp −
− 1
a
2
ϕ(s)ds
0
for all x ∈ 0 and a ≤ r < ∞. From this it is easy from the results in [ST] that after passing to a subsequence if necessary, uk converges to a smooth solution of the equation in (1.1) so that the metric ds 2 = u2 dr 2 +gr is scalar flat and is asymptotically Euclidean near r = ∞. To prove (i), note that if 0 ≤ r ≤ 1, then r r ϕ(s)ds) ≥ exp(− ψ(s)ds) ≥ 1 − αr 1 − βr ≥ exp(− 0
0
for some α > 0, β > 0. Let 0 < r0 < 1 be such that 1 − αr0 > 0. Then r −2 1 − (1 − k ) exp − ϕ(s)ds ≥ 1 − (1 − k −2 )(1 − βr) 0
= k −2 + β(1 − k −2 )r ) exp −
on [0, r0 ]. Similarly we have 1 − (1 − k
−2
r
ψ(s)ds
≤ k −2 + α(1 − k −2 )r.
0
Hence by Lemma 1.1
k −2 + α(1 − k −2 )r
− 1 2
− 1 2 ≤ uk (x, r) ≤ k −2 + β(1 − k −2 )r
on 0 × (0, r0 ]. Let k → ∞; it is easy to see that (i) is true.
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To prove (ii), by the computations in [ST, §1], the second fundamental form hij of r with respect to the metric ds 2 is equal to u−1 h0ij , where h0ij is the second fundamental form with respect to the Euclidean √ metric. Hence (ii) is true. To prove (iii), if we let w = ru, then 1 1 −1 r 2 u + r 2 ur 2
1 = H0 r −1 w + r 2 2u2 r u + 2(u − u3 )K
= H0 r −1 w + r −1 2w 2 r w + (rw − w 3 )K .
∂w 2H0 = 2H0 ∂r
Hence if we let r = et , we have 2H0
∂w = 2w2 r w + H0 w + 2(et w − w 3 )K ∂t
for −∞ < t < 0. Note that C ≤ w2 ≤ C for all t by (ii). As in the proof of [ST, Lemma 2.5], for any integers k, ≥ 0, there is a constant C1 such that k ∂ w j + ||∇r w|| ≤ C1 ∂t k j =0
for all x ∈ 0 and for all −∞ < t < 0. Here, ∇r is the intrinsic covariant derivate of r . Hence on 0 × (0, 1], for any ≥ 0, there is a constant C2 such that 3 ∂u 1 j r 2 + r 2 ||∇r u|| ≤ C2 ∂r
(2.1)
j =0
for some constant C2 . To compute the curvature tensor of the metric ds 2 . Let e1 , e2 , e3 be a local ortho∂ normal frame so that e3 = u−1 ∂r which is the normal to r with respect to ds 2 . Then ∂ e1 , e2 , e˜3 = ∂r is a local orthonormal frame with respect to the Euclidean metric. By the computations in [ST, §1], on r we have R1212 = (1 − u−2 )K,
(2.2)
where K is the Gauss curvature of r and for 1 ≤ i ≤ 2 2 −R3i3i = (log u)ii + (log u)i − u−3 ur h0ii ,
(2.3)
where (log u)ii = ∇r2 log u (ei , ei ) and h0ij is the second fundamental form of r with respect to the Euclidean metric. Hence (iii) follows from (i) and (2.1)–(2.3). To prove (iv), first observe that by (i), (2.1) and the fact that u ≥ 1, it is easy to see that F is a diffeomorphism from 0 × (0, ∞) onto itself. Let x = (x 1 , x 2 ) and
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y = (y 1 , y 2 ) be the local coordinates in 0 in the domain and in the target respectively. Then the differential of F is given by ∂ρ
∂ρ ∂x 2 ∂y 1 ∂x 2 ∂y 2 ∂x 2
∂ρ ∂x 1 ∂y 1 ∂x 1 ∂y 2 ∂x 1
∂r ∂y 1 ∂r ∂y 2 ∂r
r u 0 = 0 0
∂u (τ, x) dτ ∂x 1
1 0
Here we have used (2.1) for the computation of
∂r ∂ρ ∂x 1 ∂ρ ∂x 2 ∂ρ
∂r ∂y 1 ∂x 1 ∂y 1 ∂x 2 ∂y 1
∂r ∂y 2 ∂x 1 ∂y 2 ∂x 2 ∂y 2
1
− u1
u
= 0 0
r
∂ρ . ∂x i
∂u 0 ∂x 1 (τ, x) dτ
r
∂u 0 ∂x 2 (τ, x) dτ
0 1
.
Hence − u1
r
1 0
∂u 0 ∂x 2 (τ, x) dτ
0 1
,
and the pull back metric under F −1 is (F
−1 ∗
) (ds ) = u 2
2
∂r ∂ρ
2
∂r ∂r ∂r ∂r dρ + i j dy i dy j + (dρdy i + dy i dρ) ∂y ∂y ∂ρ ∂y i
2
+gij (r(ρ, y), y)dy i dy j r r ∂u ∂u 2 = dρ + dτ dτ dy i dy j i j 0 ∂x 0 ∂x r
∂u i i − dτ dρdy + dy dρ + gij (r(ρ, y), y)dy i dy j , i 0 ∂x where gij (r, x) is smooth in r and x. By (i) and (2.1), there is a constant C3 such that near ρ = 0, i.e. near r = 0,
0
∂ ∂y j
r 0
r
∂u ≤ C3 r 21 , dτ ∂x i
r ∂ 1 ∂u ∂u dτ = j dτ ≤ C3 r 2 , i i ∂x ∂x 0 ∂x
∂ ∂ρ
r 0
∂u ∂r ∂u ≤ C3 , dτ = i ∂x i ∂x ∂ρ
∂gij ∂gij ∂r ∂gij = ≤ C3 + ∂y k ∂r ∂y k ∂x k and
∂gij ∂gij ∂r 1 = ∂ρ ∂r ∂ρ ≤ C3 r 2 .
Hence (iv) is true. This completes the proof of the theorem.
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3. Non-Simply Connected Boundary In this section, we always assume that (n , g), (n ≥ 4), is a spin compact manifold with nonnegative scalar curvature and with boundary ∂ such that (∂, g) can be isometrically embedded in Rn∗ / = Rn \ {0}/ with image , where is a nontrivial / for some compact finite subgroup of SO(n) which acts freely on Rn∗ , and = n strictly convex hypersurface in R so that τ ( ) = for all τ ∈ . We also assume . that the origin is in the interior of If we glue the exterior part of with along the boundary via the isometry of ∂ we obtain a spin smooth manifold. We denote a manifold constructed in this way by )/ , where E( ) is the exterior part M . Here the exterior part of is defined as E( in Rn . Note that the tangent cone Rn∗ / of M has a spin structure induced by the of spin structure of M . Theorem 3.1. Let (, g) and be as above. Suppose ∂ has positive mean curvature H with respect to the unit outward normal. If in addition, there is a nontrivial parallel spinor on the tangent cone of M with respect to the spin structure induced by that of M , then
H dσ ≤
∂
H 0 dσ,
(3.1)
∂
where H0 is the mean curvature of ∂ when embedded in Rn∗ / with respect to the Euclidean metric. Moreover, if equality holds then is Ricci flat. Remark 3.1. (1) If is trivial then this is the case considered in [ST]. (2) If the dimension is odd then must be trivial. It was proved in [Sh] by the first author that if n = 2m, ⊂ SU(m) such that the 2m order || of is odd, then Cm ∗ / = R∗ / has a unique spin structure. Moreover, m C∗ / has two linear independent parallel spinors. Hence we have the following: Corollary 3.1. Let (2m , g) as before. Suppose ⊂ SU(m) and || is odd. Then (3.1) is true. Moreover, if equality holds, then is Ricci flat. In order to prove Theorem 3.1, we need the following positive mass theorem for asymptotically locally Euclidean (ALE) manifold with a metric which may not be smooth. A positive mass theorem for smooth ALE manifolds under certain conditions has been proved in [D]. Let N n be a smooth manifold, such that there is a bounded domain ⊂ N with smooth boundary ∂ and such that there is a continuous Riemannian metric g on N with the following properties: (i) g is smooth on N \ and , and is Lipschitz near ∂ such that the scalar curvature on N \ is integrable. (ii) The mean curvatures at ∂ with respect to the outward normal and with respect to the metrics g|N\ and g| are the same. (iii) N has only one end, which is ALE in the following sense:
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There is a compact set K containing such that N \ K = E such that E is diffeomorphic to Rn \ BR (0)/ and in the standard coordinates in Rn , the pull back metric of g on Rn \ BR (0) satisfies: gij = δij + bij , with
bij + r∂bij + r 2 ∂∂bij = O(r 2−n ),
where r and ∂ denotes Euclidean distance and the standard gradient operator on Rn , respectively. We should remark that because of (i), the outward unit normal on ∂ is well-defined. Moreover, (i) and (iii) imply that the ADM mass of end of N is also well-defined by the proof in [B1]. Explicitly, the ADM mass at the end E is given by
C(n)mE = lim gij,j − gjj,i dSi , r→∞ S(r)
where we have already taken the universal covering of Rn \ BR (0)/ , and C(n) is a positive constant, S(r) is the Euclidean sphere in Rn and dSi . Proposition 3.1. Let (N, g) be as above with nonnegative scalar curvature. Suppose the tangent cone of N has a nontrivial parallel spinor. Then mE ≥ 0. Moreover, if mE = 0 then N is Ricci flat. Proof. The proof is similar to the proof in [ST]. Let be a nontrivial parallel spinor on the tangent cone. Then one can find a harmonic spinor 1 which is asymptotically equal to . In particular, 1 is nontrivial. This has been proved in [Sh]. Then we can apply the Lichnerowicz formula which is still true because of property (ii) (see [ST, Lemma 3.2]), and we can conclude that mE ≥ 0. If mE = 0, then we can conclude that 1 is parallel in and N \ . But 1 is continuous near ∂, see [ST, Lemma 3.3]. Hence 1 is a nontrivial parallel spinor on and is Ricci flat by [Hi]. Now we are ready to prove Theorem 3.1. ( x ) = H (π( x )) Proof of Theorem 3.1. Let π : Rn∗ → Rn∗ / be the projection. Let H 0 0 x ) = H (π( x )). By Lemma 1.1, there is a unique solution u of the following and let H ( initial value problem:
u r0 ∂ 2H ∂r u(x, 0)
= 2 u2 r u + ( u − u3 )Rr on 0 × [0, ∞) , 0 = HH
(3.2)
0 also denotes the mean curvature of the level surface r of the distance funcwhere H , and Rr is the scalar curvature of r . Since is invariant under , r is tion from also invariant under . Hence u ◦ τ is still a solution of (3.2). By the uniqueness, we conclude that u◦τ = u. Hence the metric ds 2 = u2 dr 2 + gr is well-defined, where u is the function on M such that u = u ◦ π , and gr is the induced r ) by the Euclidean metric. Moreover, if we extend the metric ds 2 metric on r = π( to such that ds 2 is equal to the given metric g, then ds 2 satisfies the conditions in
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Proposition 3.1. Since the tangent cone of M has a nontrivial parallel spinor, we can apply Proposition 3.1 to conclude that m ≥ 0, where m is the mass of the end of M . On the other hand, by Lemma 4.2 and Theorem 2.1 in [ST], we have 0 0 − H ) ≥ C(n)m || (H − H ) = (H
for some positive constant C(n) depending only on n. From this and Proposition 3.1, the theorem follows.
Acknowledgement. The authors would like to thank Robert Bartnik, Jiaxing Hong and Andrejs Treibergs for useful discussions.
References [B1] [B2] [B3] [BC] [BY 1] [BY 2] [D] [GL] [Ha] [HH] [Hi] [HZ] [I] [LW] [LY] [N] [P] [Sa] [Sh] [ST] [Sp] [Y]
Bartnik, R.: The mass of asymptotically flat manifold. Comm. Pure Appl. Math. 39, 661–693 (1986) Bartnik, R.: Quasi-spherical metrics and prescribed scalar curvature. J. Differ. Geom. 37, 31–71 (1993) Bartnik, R.: Private communication Bishop, R. L., Crittenden, R. J.: Geometry of manifolds. Landon-New York-San Diego: Academic Press, 1964 Brown, J. D., York, J. W.: Mathematical aspects of classical field theory (Seattle, WA, 1991). Providence, RI: Am. Math. Soc. 1992, pp. 129–142 Brown, J. D., York, J. W.: Quasilocal energy and conserved charges derived from the gravitational action. Phys. Rev. D (3) 47 4, 1407–1419 (1993) Dahl, M.: The positive mass theorem for ALE manifolds. Mathematics of Gravitation, Part I (Warsaw, 1996), Banach Center Publ. 41, Part I, Warsaw: Polish Acad. Sci. 1997, pp. 133–142 Guan, P., Li, Y. Y.: The Weyl problem with nonnegative Gauss curvature. J. Differ. Geom. 39, 331–342 (1994) Hartman, P.: On unsmooth two-dimensional Riemannian metrics. Am. J. Math. 74, 215–226 (1952) Hawking, S. W., Horowitz, G. T.: The gravitational Hamiltonian, action, entropy and surface terms. Classical Quantum Gravity 13, 1487–1498 (1996) Hitchin, N.: Harmonic spinors. Adv. in Math. 14, 1–55 (1974) Hong, J., Zuily, C.: Isometric embedding of the 2-sphere with nonnegative curvature in R3 . Math. Z. 219, 323–334 (1995) Iaia, J. A.: Geometric analysis and nonlinear partial differential equations (Denton, TX, 1990). In: Lecture Notes in Pure and Appl. Math. Vol 144, New York: Dekker, 1993, pp. 213–220 Li,Y., Weinstein, G.: A priori bounds for co-dimension one isometric embeddings. Am. J. Math. 121, 945–965 (1999) Liu, C.-C. M.,Yau, S.-T.: Positivity of quasilocal mass. Phys. Rev. Lett. 90, 231102-1–231102-4 (2003) Nirenberg, L.: The Weyl and Minkowski problems in differential geometry in the large. Comm. Pure Appl. Math. 6, 337–394 (1953) Pogorelov, A.V.: Extrinsic geometry of convex surfaces. Providence, RI: American Mathematical Society, 1973 Sacksteder, R.: On hypersurfaces with no negative sectional curvatures. Am. J. Math. 82, 609–630 (1960) Shi, Y.: Crepant resolution, rigidity theorem, and parallel spinors on asymptotically locally Euclidean manifolds. To appear in Pac. J. Math. Shi,Y.-G., Tam, L.-F.: Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature. J. Diff. Geom. 62, 79–125 (2002) Spivak, M.: A Comprehensive introduction to differential geometry. V.5, Berkely, CA: Publish or Perish 1970–75 Yau, S.-T.: Geometry of three manifolds and existence of black hole due to boundary effect. Adv. Theo. Math. Phys. 5, 755–767 (2001)
Communicated by G.W. Gibbons
Commun. Math. Phys. 250, 81–94 (2004) Digital Object Identifier (DOI) 10.1007/s00220-004-1085-8
Communications in
Mathematical Physics
Soliton Surfaces in the Mechanical Equilibrium of Closed Membranes Brian Smyth Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA. E-mail:
[email protected] Received: 17 July 2003 / Accepted: 18 November 2003 Published online: 28 April 2004 – © Springer-Verlag 2004
Abstract: For a closed material membrane in equilibrium in a force field we investigate whether the external observables (membrane geometry and force field) determine the internal membrane response (the stress tensor T ), when the mean stress 21 Tr T is known. For membranes with boundary the indeterminacy of the response is classical. For closed membranes the geometry decides the question. We show uniqueness for all but a class of soliton surfaces — the globally isothermic surfaces (§1, 2); the physical phenomenon exhibited by any closed globally isothermic membrane in equilibrium is that, with all observables static, there is in total a 1-parameter family of responses with the same mean stress — all canonically determined by membrane geometry (Theorem 1). There exist closed embedded globally isothermic surfaces of every genus. The recognition of the role of these soliton surfaces settles the old classification problem for the space of static shears in any closed membrane, explicitly identifying all static shears, where before only a genus-dependent dimension bound was known (Theorem 2). 1. Introduction This work treats the equilibrium of a closed thin linearly elastic material membrane under the influence of a force field, from the geometric point of view, using connexions between the geometry, the conformal structure and the stress; for exceptional membrane configurations — the surfaces in the title — there are exceptional families of stress responses. These exceptional configurations turn out to be abundant, being stable under the conformal group of R3 ∪ {∞} and, as we show, occurring in every topological genus. Given a smooth compact orientable material membrane M with smooth boundary (possibly empty) in space, in equilibrium under a given external force field F , it is of interest to consider the extent to which the internal response — that is, the stress tensor
Partially supported by NSF Grant DMS00 71729
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T — is determined. The specific question treated here concerns the extent to which the observables (membrane geometry and force field) and the mean stress function τ = 21 Tr T determine the full stress tensor T . I am grateful to Andrew Smyth who interested me in this subject and for a conversation which crystallized for me the significance of this question for closed membranes. All membrane surfaces M are assumed smooth, compact and oriented with or without smooth boundary. M is called closed when the boundary ∂M is empty. When ∂M = ∅ the boundary is given the induced orientation. The orientation and induced metric on M determine the structure of a Riemann surface on M and its complex structure is denoted J (see §2). Force and moment balance considerations between the force field F and the response T , applied in the small to M, lead to the equilibrium equation derived in §2 (see also [9]). It may be thought of as an inhomogeneous system of linear first order partial differential equations for the unknown internal response T into which the observables enter as known quantities. Our study of the uniqueness question above is the source of the identification of the distinguished class of closed soliton membrane surfaces and the extra flexibility in the stress response they allow. Our analysis will show that uniqueness is the rule but an exceptional geometric class of surfaces emerges naturally — the globally isothermic surfaces (§2). The occurrence of such a surface as a membrane configuration signals a canonical non-uniqueness in the stress tensor response — that is, there are distinct stress tensor solutions with the same mean stress (pointwise) and any two such stress tensors stand in a canonical relationship to one another determined solely by the geometry of the membrane (Theorem 1, below). The globally isothermic surfaces are defined by a condition linking the geometry with the complex structure; namely, on the complement of the umbilic set, the Hopf quadratic differential of the surface (see §2) is a pure imaginary functional multiple of a global holomorphic quadratic differential . The differential is uniquely determined to within a real constant multiple and is called the underlying holomorphic quadratic differential of the globally isothermic surface. The condition is truly geometric, being equivalent to the condition that the principal foliations of the surface coincide with the foliations determined by a global holomorphic quadratic differential — in this case i — in the manner of [20] (see p.17). In §4 we gain a better understanding of this anomalous family of surfaces. A sphere has no non-trivial holomorphic quadratic differentials and so, from the definition, the only topologically spherical examples are round spheres. The main result on the physical question posed above is: Theorem 1. Let M be a smooth closed membrane in equilibrium under an external force field F with mean stress function τ . Either the mean stress uniquely determines the stress tensor or else M is a globally isothermic surface of genus p ≥ 1. If M is a globally isothermic membrane of genus p ≥ 1, then the space of solutions of the equilibrium equation with mean stress τ is in one-to-one correspondence with R and the correspondence is canonically determined by the geometry of the membrane M. Remark. By the geometry of M is meant its configuration in R3 , that is, its first and second fundamental forms. With this result the question of the existence of globally isothermic compact embedded surfaces in R3 becomes important and is settled for every genus in §4. It also follows that there is no exceptional behaviour for topologically spherical membranes, that is, the mean stress completely determines the stress tensor on any topologically spherical membrane in equilibrium.
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Theorem 1 is proved in §3. The argument associates a complex quadratic differential to the responding stress tensor in the way of Hopf [10]. The theory of holomorphic differentials then enters naturally via divergence-trace conditions arising from the equilibrium equation (see §2). An analysis of the index of an isolated umbilic figures in the proof, in §3, but the vital algebraic identity comes from the normal component of the equilibrium equation which links the Hopf differential of the membrane surface with a quadratic differential determined by the stress tensor. This algebraic identity is the source of the appearance of soliton surfaces in the theory. Examples of globally isothermic surfaces include all constant mean curvature surfaces, their Hopf differentials being themselves holomorphic; in particular, from Kapouleas [12], there are compact examples of every genus but none are embedded, by Alexandrov [1], except for the round sphere. The class of globally isothermic surfaces is much larger since the property is stable under the conformal group of R3 ∪ {∞} (see §4). Bonnet surfaces (that is, surfaces of non-constant mean curvature admitting a non-trivial 1-parameter family of isometric deformations preserving the mean curvature) form another interesting class of globally isothermic examples but Bonnet surfaces are never compact [3]. Thus the existence of compact embedded globally isothermic surfaces in every genus is of greatest interest here (see §4). A stress tensor T is called a shear if τ ≡ 0 and a surface tension if T ≡ τ I . The equilibrium equation becomes a homogeneous system when F ≡ 0 and describing its solution space is important for the general case; these solutions are called the static or residual stress tensors. This vector space is of some physical interest. A simple model where this space presents itself is the pericardial sac at end-diastole, where F ≡ 0, and it was proposed [18] to investigate the stresses occurring in this case. For closed strictly convex membranes there are no non-trivial static stresses, by a result of Cohn-Vossen and Blaschke (see Vekua [21]). It was suspected [18] that this space might always be finite-dimensional but it can be shown that there are closed embedded surfaces of every genus for which this space is infinite-dimensional [19]. It is significant that the space of static shears for membranes related by a conformal transformation of space can be shown to be canonically isomorphic but no canonical correspondence appears between their spaces of static stresses [19]. As has been long known (see Theorem 5.2, [16]), the space of static shears is finitedimensional of real dimension bounded by 6(p − 1) if the genus p of the membrane is p > 1, 2 if p = 1 and 0 if p = 0. But, this rough bound apart, nothing more was known. Our theorem below gives a simple genus-independent explicit classification of the space of static shears on any closed membrane with the geometry deciding everything. The effectiveness of our approach originates in the recognition of the soliton surfaces in the background of all uniqueness considerations. Applying Theorem 1 with τ ≡ 0, we obtain the following result. Theorem 2. If a smooth closed membrane admits a non-trivial static shear, then it must be a globally isothermic surface of genus p ≥ 1 and this shear is uniquely determined to within a constant multiple by the membrane geometry alone. Conversely, any smooth closed globally isothermic membrane of genus p ≥ 1 has a non-trivial static shear. Any torus of revolution is globally isothermic (see §4) and so this theorem would apply. But the result was known only for circular tori of revolution and then only from explicit computation [16]. Our theorem gives the result for all globally isothermic membranes and the underlying holomorphic quadratic differential supplies all of the static shears in a canonical way (§2). If the membrane is not globally isothermic there are no non-trivial static shears.
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The fact that globally isothermic surfaces admit no umbilics of positive index (Proposition 2, §4) gives at once a simple and striking consequence. Corollary 1. If a smooth closed membrane has an isolated umbilic of positive index then it admits no non-trivial static shear. A surface in R3 whose Hopf differential is locally a real multiple of a local holomorphic quadratic differential on the complement of the umbilic set is called isothermic (§2) and such surfaces were first studied by Cayley [7] and then by Bianchi, Darboux, B¨acklund, Christoffel and others (see [4]) in the late nineteenth and early twentieth century; this property is also conformally invariant and a soliton theory was developed for these surfaces by Cie´sli´nski-Goldstein-Sym [8] in 1995; the geometric interest in these surfaces in recent years, as well as their earlier history, is chronicled in the lecture notes of Burstall [4]. Because globally isothermic surfaces emerge here as such an exceptional class among membrane configurations, it is worth contrasting them with isothermic surfaces and giving examples (see §4). 2. Surface Theory, the Soliton Surfaces and the Equilibrium Equation We first explain some of the basic differential geometry of surfaces, in particular the Codazzi equation, the Hopf differential and the simple geometric characterization of the particular soliton surfaces which dominate the uniqueness considerations in the physical problem considered later. Let M be a smooth oriented surface without boundary and x : M −→ R3 an immersion of M in R3 . The standard inner product < , > on R3 induces a Riemannian metric g on M defined by gp (X, Y ) =< x∗p (X), x∗p (Y ) > for any vectors X, Y ∈ Mp ; here Mp denotes the tangent space to M at p and x∗p denotes the differential of x at p ∈ M. The given orientation on M determines a unique unit normal vector field ξ to M along x. The second fundamental form A of the immersion is defined pointwise by Xξ = −x∗p (Ap X) for all X ∈ Mp . The operator A is symmetric with respect to g and depends to within sign on the choice of unit normal field; its characteristic functions on M are the mean curvature function H = 21 Tr A and the Gauss curvature function K = det A. The oriented area element da of the metric g is defined by da(X, Y ) = < x∗ (X) × x∗ (Y ), ξ >, where × is the cross product. If ∇ is the Riemannian connexion of the induced metric g on M, then the equality of mixed partials of the vector-valued function x on M results in the Codazzi equation for A (∇X A)Y − (∇Y A)X = 0, for all vectors X and Y tangent to M. The complex structure J on M is defined pointwise as counterclockwise rotation of the oriented inner product space (Mp , gp ) through the angle π2 . Local coordinates (u1 , u2 ) on M for which g = e2ρ (du21 + du22 ), with ρ a smooth real-valued function of these variables, are called local isothermal coordinates for the metric g and if the coordinate frame { ∂u∂ 1 , ∂u∂ 2 } gives the surface orientation the coordinates are called positive isothermal coordinates; w = u1 + iu2 is then called a local complex coordinate for the underlying Riemann surface. ∂ ∂ The complex quadratic differential defined locally by = g(A ∂w , ∂w )dw 2 , ∂ is called the Hopf differential of the immersion x; here ∂w = 21 { ∂u∂ 1 − i ∂u∂ 2 } and ∂ 1 ∂ ∂ ∂ w¯ = 2 { ∂u1 + i ∂u2 }. If
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A=
H +α β β H −α
85
with respect to the above positive local coordinates (u1 , u2 ), then the corresponding local representation of the Hopf differential is = ωdw2 = − 2i e2ρ (β + iα)dw 2 , where w = u1 + iu2 . Codazzi’s equation for A is ωw¯ = 21 e2ρ Hw and the zeros of are the umbilics of the immersion x. To say that a symmetric tensor field T of type (1,1) is divergence-free is equivalent to saying that S = j (T ) = J T J satisfies Codazzi’s equation. The associated ∂ ∂ global complex quadratic differential to T is = g(S ∂w , ∂w )dw 2 = φ(w)dw 2 = i 2ρ 2 − 2 e (b + ia)dw , where τ +a b T = , b τ −a with respect to the coordinate frame. The divergence-free condition on T is equivalent to the local equation φw¯ = − 21 e2ρ τw and so if T is also trace-free then is holomorphic. An immersion x : M −→ R3 is globally isothermic if there exists a global holomorphic quadratic differential on M such that = ik on the complement of the umbilic set U in M, k being a real-valued function on M − U . The holomorphic differential being then uniquely determined to within a real constant multiple by — and therefore by the geometry of M — will be called the underlying holomorphic quadratic differential of the globally isothermic surface. The differential then uniquely determines a smooth symmetric divergence-free trace-free tensor field T0 on M. Since on a globally isothermic surface the differential is uniquely determined to within a real constant factor, the same is true of T0 . This field ultimately accounts for the extra freedom in the response on a globally isothermic membrane. An immersion x : M −→ R3 is isothermic if there exist positive isothermal coordinates {u1 , u2 } on a neighbourhood V of each non-umbilic point and a local holomorphic quadratic differential V in the coordinate w = u1 + iu2 such that the Hopf differential satisfies = ikV V on V , kV being a real-valued function on V . Note that if A diagonalizes with respect to these isothermal coordinates then β ≡ 0 and = ikV V , where kV is a real function on V and V is the holomorphic quadratic differential idw 2 on V . Conversely, if on an isothermal patch V centred at a non-umbilic point p, = ikV V for kV and V as above, then V = f (w)dw 2 , with f a holomorphic function of w, and since p is not an umbilic, f (o) = 0. After making a holomorphic change of coordinates we may assume f ≡ i and, in these new coordinates, β ≡ 0 and so A diagonalizes. Thus, being isothermic is equivalent to the existence of isothermal coordinates on a neighbourhood of each non-umbilic point with respect to which the second fundamental form diagonalizes. It was in this form that the notion was first introduced by Cayley [7]. Examples and properties of isothermic and globally isothermic surfaces are discussed in §4. The soliton theory of these surfaces is found in [8]. An infinitesimally thin material membrane in equilibrium under the influence of an external force field F may be considered as a smooth surface M with smooth boundary all smoothly embedded in R3 . The smooth embedding x : M −→ R3 is taken to be smooth at the boundary, in the sense that it extends to a smooth embedding of a larger surface into R3 . We consider only oriented surfaces because our interest is in closed membranes and these are automatically orientable. The boundary, if non-empty, is given the canonical orientation, i.e., that for which the oriented unit normal to the boundary
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∂M points into the interior of M. The force field is a smooth vector-valued function F : M −→ R3 . The nature of the responding forces of stress within the membrane (i.e., the stress tensor T ), holding it in equilibrium in response to the applied field F, is explained below. In our discussion we also assume that these forces are C ∞ . Post facto, it will be seen that for the equilibrium equation we need no more than x to be C 3 , T to be C 1 and F to be continuous. The equilibrium of an isotropic membrane (for which T ≡ τ I , such as happens with a liquid film or bubble) acted on by a smooth force field F , was considered first byYoung [23] and then by Laplace [13] at the beginning of the nineteenth century in studies of capillary surfaces (see [17]). The more general equation was later found by Beltrami [2] and Lecornu [15]. If we imagine a small slit cut in the membrane along a geodesic emanating from p in the direction of a unit vector e ∈ Mp , then equal restorative forces must be applied in opposite directions at points on either side of the cut in order to maintain the membrane in equilibrium. At the left edge of the cut, the limiting value of this force (per unit length) as the length of the cut goes to zero, is denoted r(p, e) and called the response at p corresponding to the direction e. Clearly r is an R3 -valued function on the unit tangent bundle. The equilibrium equation of the membrane is derived in Gurtin-Murdoch [8] from force and moment balance considerations in the small, and in their notation r(p, e) = −x∗p (Tp (J e)), where T defines a smooth tensor field of type (1,1) on M which is symmetric with respect to the induced metric g. The tensor field T is called the stress tensor of the membrane in equilibrium under the field F . Let D be any compact simply-connected domain in M with oriented boundary a smooth closed curve γ (s), 0 ≤ s ≤ l, parametrized by arc length s. Thinking of this domain of the membrane in isolation, the only forces keeping it in equilibrium are F , acting over the interior, and the forces of response corresponding to the oriented unit tangent field e acting at the boundary ∂D of D. The external force field F gives rise to the R3 -valued 2-form F da, where da is the natural area element of the induced metric on M for the fixed choice of orientation. Equilibrium calls for the integral of this 2-form over D to balance the integral of the restorative forces of the response on D along its oriented boundary ∂D. Thus the equilibrium equation for D is l F da + r(γ (s), γ˙ (s))ds = 0, 0
D
or alternatively,
F da +
D
ω = 0, ∂D
where ω(X) = −x∗ (T (J X)). This defines an R3 -valued 1-form ω on M, which we call the response 1-form of the membrane, and which completely encodes the forces of stress within the membrane. By Stokes’ theorem, the equilibrium condition on D is (F da + dω) = 0, D
where dω denotes the exterior derivative of ω. The vector-valued 2-form dω may be computed from dω(X, Y ) = Xω(Y ) − Y ω(X) − ω([X, Y ]), using the fundamental equations of surface theory, and we obtain dω = [x∗ (div T ) + Tr (AT )ξ ]da. Hence
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D
(F + x∗ (div T ) + Tr (AT )ξ )da = 0
and, by continuity of F , T and div T , F + x∗ (div T ) + Tr (AT )ξ = 0 is the equilibrium equation of the membrane. Remark. If ft : M −→ R3 is a 1-parameter family of isometric deformations of an oriented surface M with oriented unit normal field ξt and associated second fundamental form At , then the Gauss equation may be written J At J At = KI , where K is the Gauss t curvature of the induced metric. Writing B = ( dA dt )t=0 we have a symmetric tensor field satisfying Codazzi’s equation, so that T0 = J BJ must have zero divergence, i.e., div T0 = 0. Differentiating Gauss’ equation above we obtain T0 A − J AT0 J = 0, from which it follows that Tr AT0 = 0. Thus, it follows that any isometric deformation gives rise to a static stress T0 (i.e., a solution of the equilibrium equation with F = 0). If H is preserved under the deformation, then Tr T0 = 0 and T0 is a static shear. Similar remarks may be made for infinitesimal isometric deformations. The connexion between static stresses and infinitesimal isometric deformations already figures in the work of Blaschke and Cohn-Vossen on the rigidity (and vanishing of static stresses) of closed strictly convex surfaces (see Vekua [19]). That there is no uniqueness in the solutions of the equilibrium equation is known classically from the infinite-dimensional space of smooth symmetric tensor fields satisfying the equilibrium equation on any planar membrane configuration in the presence of the external field F ≡ 0 — the Airy tensors. Such a tensor field is obtained by taking T = j −1 (Hφ ), where φ is any smooth function on the planar configuration M and Hφ is its Hessian operator (defined by Hφ (X) = ∇X (∇φ), where ∇ is the connexion of the induced metric) and j is the operator defined in §2. Using the fact that M is planar — in fact, vanishing Gauss curvature (flatness) is enough — it is easily checked that Hφ satisfies Codazzi’s equation. Thus, from §2, div T ≡ 0 and, since A ≡ 0 for planar surfaces, referring back to the equilibrium equation we see this is all that is needed for solutions of the equilibrium equation for planar membranes with F ≡ 0. Conversely, suppose that M is a simply-connected compact flat membrane with smooth boundary in equilibrium with F ≡ 0. If T is a solution of the equilibrium equation then, since div T ≡ 0, S = j (T ) satisfies Codazzi’s equation. Letting denote the Laplace operator of the flat induced metric on the membrane M, we solve the Poisson equation ψ = −2τ for ψ, where 2τ = Tr T = −Tr S. Now Hψ satisfies Codazzi’s equation since M is flat and so also does S. It follows that S0 = S − Hψ is a Codazzi operator with trace zero. Thus, the quadratic differential determined by S0 must be holomorphic. Now (M, g) being flat and simply-connected, it may be identified with a simply-connected region of the plane endowed with its euclidean metric. Then = − 2i (b + ia)dw 2 , where w is the complex coordinate on the plane and b + ia is a holomorphic function of w. M being simply-connected, there exists a holomorphic function f = k + il on M such that fww = b + ia. From the Cauchy-Riemann equations for f, this becomes b + ia = 2ilww and this means that S0 = Hl or S = Hψ+l is Hessian. In summary, when M is simply-connected and flat with smooth boundary, the space of solutions of div T = 0 may be identified with the space of Hessians of all smooth functions on M. This then is the space of solutions of the equilibrium equation for planar membranes when F ≡ 0, and it is infinite-dimensional. When M is flat but not planar,
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A is not identically vanishing and the normal component of the equilibrium equation gives a further restriction on the space of solutions. 3. The Proof of the Main Result In this section we prove Theorem 1 for closed membranes but, to begin with, M will be taken to be compact and oriented with smooth boundary ∂M, possibly empty. We consider the tangential and normal equilibrium equations as conditions on the quadratic differentials corresponding to the second fundamental form and the stress tensor. It is the normal equation that leads to the appearance of globally isothermic surfaces. If two distinct stress tensors T and T˜ satisfying the equilibrium equation div T + Tr (AT )ξ + F = 0 have the same mean stress, then their difference T0 = T˜ − T is non-trivial and satisfies (i) div T0 = 0, (ii) Tr (AT0 ) = 0 and (iii) Tr T0 = 0. Let be the corresponding nontrivial quadratic differential determined by T0 . As we saw in §2, (i) and (iii) imply is holomorphic. Its zero set Z() is therefore discrete in M. Let H +α β A= β H −α and T0 =
a b b −a
be the matrices of A and T0 with respect to a positive isothermal coordinate frame { ∂u∂ 1 , ∂u∂ 2 } on a neighbourhood V on which g = e2ρ (du21 + du22 ). Then the corresponding local representations of the associated quadratic differentials, encountered in §2, are = ωdw2 and = φdw 2 , where ω = − 2i e2ρ (β + iα) and φ = − 2i e2ρ (b + ia). Since Tr T0 = 0, condition (ii) above translates into ωφ¯ being pure imaginary and so ω = ikφ on V − Z(), k being a real function on the latter set. Hence = ik on M − Z(), k being a smooth real function on the latter set. There are just two possibilities. The first is that k ≡ 0 on M − Z() and in this case we show that M must be globally isothermic. If we show that Z() consists of umbilics then we will have = ik on the complement of the umbilic set, as required. Our analysis gives somewhat more: all points of Z() are umbilics of negative index and all umbilic points outside of Z() are of index zero, and are even removable singularities of the principal foliations (see Proposition 2, §4). Assume p ∈ Z() is not an umbilic. We may take p to be the origin o of the local isothermal coordinates. Let m > 0 be the order of vanishing of at p. Since p is assumed not an umbilic, ω(o) = 0 and we may assume that ω does not vanish on V . From the identity ω = ikφ on V − {o} it follows that k and φ do not vanish on V − {o}. Any smooth non-vanishing complex-valued function v = r + is on V − {o} may be thought of as a vector field on this punctured neighbourhood with an isolated singularity at o and the index jv (o) of v at o is defined to be the integer rds − sdr 1 jv (o) = , 2π C r 2 + s 2
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the value of which is independent of the choice of simple closed curve C circulating within V once around the singularity o, counterclockwise. From the definition, j gives a homomorphism from the ring of germs of non-vanishing complex functions on punctured neighbourhoods of o onto the additive group of integers Z. Note that jv (o) = 0 if v is purely real, purely imaginary or if v extends to a non-vanishing continuous complex function on a full neighbourhood of o. Moreover jv (o) = m > 0 if v is holomorphic with a zero of order m > 0 at o, as follows from the previous remark on representing v(w) = wm u(w) with u holomorphic and non-vanishing on V −{o}. The earlier identity ω = ikφ on V − {o} therefore gives jω (o) = ji (o) + jk (o) + jφ (o) = m > 0. In particular, ω must vanish at o, a contradiction. Hence each p ∈ Z() must be an umbilic and so we certainly have = ik on the complement M − U of the umbilic set U . This completes the proof that M is globally isothermic when k ≡ 0. In the event that k ≡ 0 on M − Z(), then ≡ 0 on M − Z() and therefore on M. Hence M is composed of umbilics and, if closed, must be the round sphere. This completes the proof of Theorem 1. If k ≡ 0 but M is not closed, then it must be a proper subdomain of a round sphere or plane in R3 . We may further note that in the two exceptional cases of surfaces with boundary just mentioned, M is conformally equivalent to a domain D in C and may be represented globally as = φ(z)dz2 , where φ(z) is a holomorphic function of the coordinate z on C. Such differentials form an infinite-dimensional vector space over C and, conversely, each such determines a T0 satisfying (i) and (iii) above — and (ii) holds trivially when M is planar and, in consequence of (iii), also when M is spherical. Thus, for proper subdomains of the round sphere or plane, any solution T of the equilibrium equation determines an infinite family of solutions T + cT0 , where c is any real constant and T0 is a tensor field corresponding to any holomorphic quadratic differential via the correspondence used in §2; these solutions all have the same mean stress as T . We summarize these results in the following theorem which contains Theorem 1 and of which Theorem 2 is a simple consequence. Theorem 3. Let M be a smooth compact orientable membrane with smooth boundary ∂M (possibly empty) in equilibrium under an external force field F , and let T be the responding stress tensor on M and τ = 21 Tr T the mean stress of T . Either T is the unique solution of the equilibrium equation with mean stress τ, or M is a globally isothermic surface but not the round sphere S 2 . Conversely, if M is globally isothermic but not the round sphere, then either i) M is a proper subdomain of the round sphere S 2 or the plane R2 and the space of solutions of the equilibrium equation with mean stress τ is infinite-dimensional, or ii) M is closed and the space of solutions of the equilibrium equation with mean stress τ is one-dimensional and canonically determined by the geometry. Note that when ∂M = ∅ the proof does not require any information on the behaviour of the stress tensor T at the boundary. If two solutions of the equilibrium equation with the same mean stress also coincide in directions tangential to the boundary, then their difference T0 is zero in these directions. Since Tr T0 ≡ 0 on account of the mean stress condition, it follows that T0 is zero in all directions at each point of the boundary. Hence the holomorphic differential , associated to T0 , vanishes on ∂M and so is identically
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zero. Thus if ∂M = ∅ and two solutions T and T˜ of the equilibrium equation with the same mean stress on M coincide in tangential (or normal) directions along the boundary, then they coincide everywhere on M. Corollary 2. If the boundary of M is nonempty, then T is uniquely determined by τ and the value of T in directions tangent (or normal) along the boundary.
4. Remarks on Globally Isothermic Surfaces In this study of the equilibrium of material membranes, globally isothermic surfaces came to the fore with the phenomenon described in Theorem 1, and for that reason existence of such compact embedded surfaces in every genus is important. We also point out some properties of globally isothermic surfaces which serve to better distinguish them within the larger family of isothermic surfaces with examples of both types. If is a complex quadratic differential on the Riemann surface M, then at each point of the complement M − Z() of its zero set Z() there is a unique linear subspace of the tangent space, defined by φ(w)dw2 > 0, where = φ(w)dw 2 is the representation of in any local complex coordinate about this point; this then defines a foliation of M − Z() (see [20], p.17). The pair of orthogonal foliations on M − Z() given locally by Im (φ(w)dw 2 ) = 0 we call the foliations of ; we will be interested in the case where is holomorphic and non-trivial so that Z() is discrete. The foliations of the differential i intersect those of the differential at an angle π4 . Now if is the Hopf quadratic differential of an immersion x of M, then the complement M − U of the umbilic set U is foliated by the principal foliations and these are described by Im (ω(w)dw2 ) = 0, where = ω(w)dw 2 in any local complex coordinate on M − U . From §2, if x is globally isothermic then = ik on M − U , where is a holomorphic quadratic differential on M and k is a real function on M − U ; it follows as in the proof in §3, that Z() is contained in U and, by the definition above, that the principal foliations of x coincide with the foliations of i on M − U ; moreover the foliations of i extend the principal foliations of x onto the set M − Z(). Conversely, if the principal foliations of x (defined only on the set M − U ) coincide with the foliations of a global holomorphic quadratic differential i, then Z() is contained in U and, from the definition above, = ik on M − Z() with k some real function on M − Z() and therefore the relation holds on M − U and x is globally isothermic. Hence Proposition 1. An immersion x is globally isothermic if and only if its principal foliations coincide with the foliations of some global holomorphic quadratic differential. When M is globally isothermic and is the underlying holomorphic quadratic differential then i determines the principal foliations. A distinctive feature of any globally isothermic surface is the character of its isolated umbilics. Proposition 2. On a globally isothermic surface every isolated umbilic has index ≤ 0. On the complement of the set of umbilics of negative index, the foliations of the underlying holomorphic quadratic differential extend the principal foliations.
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Proof. On M −U we have = ik with a global holomorphic quadratic differential, and since does not vanish there neither does . Hence Z() is in U . Any isolated umbilic outside Z() has index zero since the foliations of i extend the principal foliations on M − Z(). We now give a formula for the index of any isolated umbilic p0 . Obviously k is defined and nowhere zero on a punctured isothermal neighbourhood V − {p0 } of p0 . Writing = ω(w)dw 2 and = φ(w)dw 2 in the local complex coordinate, φ(w) is holomorphic with a zero of order m ≥ 0 at w = 0. Interpreting the complex-valued functions ω(w), ik(w) and φ(w) as vector fields with isolated singularities at w = 0, the identity = ik on V − {p0 } and the usual index property gives the relation jω = jik + jφ between their indices. Now since ik is pure imaginary its index is zero. Since φ is holomorphic its index is −m, the order of vanishing of at p0 . Hence jω = −m. The umbilic index of p0 is easily determined by to be 21 jω . Hence the umbilic index of p0 is − m2 ≤ 0. Surfaces of constant mean curvature are globally isothermic since for them Codazzi’s equation amounts to the statement that the Hopf differential is itself holomorphic. Thus, from Kapouleas [12] we know that there exist compact globally isothermic surfaces in R3 of every genus; but none of these, except the round sphere, is embedded [1]. The Bonnet surfaces — surfaces not of constant mean curvature but admitting a one-parameter family of isometric deformations preserving the mean curvature function — give another class of globally isothermic surfaces; but none of these can be compact [3]. Example. Let γ (t) = (γ1 (t), γ2 (t)) be a smooth curve in the upper-half xy-plane (i.e., γ2 (t) > 0), and f (t, θ ) = (γ1 (t), γ2 (t) cosθ, γ2 (t) sin θ) the surface M obtained by T γ1 2 +γ2 2 rotating γ about the x-axis. Let s(T ) = dt be the arc length parameter of γ γ2 with respect to the Poincar´e metric on the upper half plane. Then (s, θ ) are isothermal coordinates for the reparametrization F (s, θ ) = f (t (s), θ ) of the given surface. It is easily checked that = k(ds + idθ )2 with k real. Since ds + idθ is a globally defined holomorphic 1-form on M, the surface M is globally isothermic. If γ is a smooth simple closed curve in the upper half plane then we get an embedded globally isothermic torus. It is worth noting that unlike surfaces of constant mean curvature, globally isothermic surfaces are not necessarily analytic. To see this one need only take a torus of revolution generated by a smooth non-analytic simple closed curve in the upper half plane. Further embedded examples which are not of revolution follow from the next remark. Remark. The property of being a globally isothermic surface is preserved under the conformal group. Indeed let x : M −→ R3 be globally isothermic with Hopf differential and underlying holomorphic differential . Then for any conformal transformation ψ, x˜ = ψ ◦ x has the same induced conformal structure and second fundamental forms related by A˜ = pA + qI with p and q functions on M and p vanishing nowhere. Thus the umbilic ˜ is a non-zero functional sets of both immersions coincide and the Hopf differential multiple of . So x˜ is globally isothermic also with as underlying holomorphic differential. Proposition 3. In R3 there exist closed embedded globally isothermic surfaces of every genus.
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Proof. Let x : M −→ S 3 be a compact oriented surface minimally embedded in the unit sphere S 3 with oriented unit normal field ξ in S 3 and corresponding second fundamental form A and Hopf differential . Such surfaces exist in every genus, by Lawson [14]. Stereographic projection into R3 gives an embedding x˜ with the same induced conformal structure and A˜ = pA + qI with p and q functions on M and p vanishing nowhere. ˜ is a non-vanishing multiple of — which is holomorphic by the minimality Hence of x — so that x˜ is globally isothermic. Finally a few remarks on isothermic surfaces, for which there is now the fundamental result of Cie´sli´nski-Goldstein-Sym [8] on the integrability of the Gauss-Codazzi equation and the resultant soliton theory (see also [4]). The soliton theory for such surfaces has as its setting simply-connected isothermal coordinate neighbourhoods; the implications of the soliton theory for globally isothermic surfaces is not yet understood. It is easy to see that the definition of an isothermic surface given in §2 is equivalent to the requirement that, locally on the non-umbilic set, the principal foliations coincide with the foliations of some local holomorphic quadratic differential. It was seen in §2 that being isothermic is equivalent to the requirement that there exist local isothermal coordinates on a neighbourhood of each non-umbilic point with respect to which the second fundamental form diagonalizes. This notion of considering the local simultaneous diagonalization of the first and second fundamental forms was Cayley’s [7] point of departure in his study. The property of being isothermic is also invariant under conformal transformations of R3 ∪ {∞}. Another characterization is due to Calapso [6], who showed that any non-trivial solution of the fourth-order p.d.e. (
σuv σuv )uu + ( )vv + (σ 2 )uv = 0 σ σ
determines a four-dimensional family of isothermic surfaces. Globally isothermic surfaces are isothermic, but Cayley had already noted that any ellipsoid is isothermic and among topologically spherical surfaces only round spheres are globally isothermic. Every smooth surface of revolution M intersecting its axis of revolution is isothermic but it can be easily checked that it is not globally isothermic — unless it be a piece of a round sphere or of a plane. Proposition 4. There exist closed embedded isothermic surfaces of every genus in R3 which are not globally isothermic. Proof. In genus p = 0 the ellipsoids are isothermic but not globally isothermic as remarked above. For any genus p ≥ 2, examples can be constructed as follows: in the example above first choose the generating curve to be convex and containing two congruent straight line segments orthogonal to the axis of revolution and at the same distance from it. The original curve may be chosen so that no umbilics of the surface of revolution occur outside the planar part. The resulting torus of revolution contains two congruent planar annuli. Mark off a small contractible circle in one of these annuli as well as the congruent circle in the congruent annulus. Take p ≥ 2 copies of this surface (together with its pair of marked circles) stacked along the axis of revolution. We may smoothly join the neighbouring planar annuli of revolution by gluing a cylinder of revolution of negative curvature between neighbouring annuli along the circular markings. Then no new umbilics are introduced so that all umbilics of the new genus p surface are in the interior or boundary of the planar regions and therefore none are isolated. The resulting
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surface of genus p ≥ 2 is isothermic since it is locally of revolution in a neighbourhood of any non-umbilic point. If M were globally isothermic then applying Riemann-Roch to the underlying holomorphic quadratic differential of Propositions 1 and 2, we have 0 > χ (M) = 2(1 − p) = j, where the latter summation of umbilic indices (all negative, by Proposition 2) is taken over Z(); in particular, since p ≥ 2, M must have an isolated umbilic of negative index. As the construction gives no isolated umbilics whatsoever, we have a contradiction to Proposition 2 and M is not globally isothermic. Finally we construct an example in genus p = 1; in the initial torus of revolution in the previous construction remove the planar disk bounded by the small contractible circle and smoothly attach a rotationally symmetric disk having a single umbilic (which must be of index 1); the resulting torus is isothermic since it is locally a surface of revolution in a neighbourhood of each non-umbilic point but it is not globally isothermic, by Proposition 2, since it contains an umbilic of positive index. Remark. A simply connected isothermic surface without umbilics is globally isothermic. To see this fix an orientation of the surface M and cover it with positive isothermal coordinate patches {Vα }α∈I with = ikα α for some real function kα and some holomorphic quadratic differential α on each patch Vα . By holomorphicity kαβ = kkβα is a non-zero real constant on Vα ∩ Vβ and satisfies the cocycle condition kαβ kβγ = kαγ on each non-empty Vα ∩ Vβ ∩ Vγ . Thus there is a 0-cocycle assigning a non-zero constant cα to each Vα such that kαβ = ccβα for each non-empty Vα ∩ Vβ (see, for example, Weil [22], p. 85, Lemma 1). Then k = kcαα defines a global smooth function on M and = cα α defines a global holomorphic quadratic differential on M with = ik on M, as required. Acknowledgement. I am grateful to the referee for pointing out that Theorem 1 resonates with a recent characterization of isothermic surfaces [5], namely that they are precisely those surfaces which are not determined up to conformal diffeomorphism by the trace-free part of the second fundamental form (see also [11]). This work was begun at the Max Planck Institute for Mathematics in the Sciences in Leipzig where I was a visitor in 2000 and continued at the Max-Planck-Institut f¨ur Mathematik in Bonn in 2001. It is a pleasure to record my thanks to both institutes. I thank Robert Finn for conversations on this work when we were both visitors in Leipzig and Chuu-Lian Terng for pointing me toward the lecture notes of Burstall [4] on isothermic surfaces.
References 1. Alexandrov, A.D.: Uniqueness theorems for surfaces in the large I, II. Am. Math. Soc. Transl. 21(2), 341–388 (1962) 2. Beltrami, E.: Sull’equilibrio delle superficie flessibili ed inestendibili. Memorie dell’Academia delle Scienze dell’Instituto di Bologna 3, 217–265 (1882) 3. Bobenko, A.I., Eitner, U.: Painlev´e Equations in the Differential Geometry of Surfaces. Lecture Notes in Mathematics, Vol. 1753, Berlin-Heidelberg-New York: Springer Verlag, 2000 4. Burstall, F.E.: Isothermic surfaces: conformal geometry, Clifford algebras and integrable systems. Lecture Notes, Tsing Hua University, Taipei, 1999 5. Burstall, F.E., Pedit, F., Pinkall, U.: Schwarzian Derivatives and Flows on Surfaces. In: Differential Geometry and Integral Systems M.A. Guest, R. Miyaoka, Y. Ohnita, eds., Contemp. Math. 308, Providence, RI: Am. Math. Soc., 2002, pp. 39–61
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6. Calapso, P.: Sulla superficie a linee di curvatura isoterme. Rendiconti Circ. Mat. Palermo 17, 275–286 (1903) 7. Cayley, A.: On surfaces divisible into squares by their curves of curvature. Proc. London Math. Soc. 4, 8–9, 120–121 (1872) 8. Cie´sli´nski, J., Goldstein, P., Sym, A.: Isothermic surfaces in E3 as soliton surfaces. Phys. Lett. A 205, 37–43 (1995) 9. Gurtin, M., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Rational Mech. Anal. 57, 293–323 (1975) 10. Hopf, H.: Differential geometry in the large. Lecture Notes in Mathematics, Vol. 1000, BerlinHeidelberg-New York: Springer Verlag, 1990 11. Kamberov, G., Pedit, F., Pinkall, U.: Bonnet pairs and isothermic surfaces. Duke Math. J. 92, 637–644 (1998) 12. Kapouleas, N.: Compact constant mean curvature surfaces in Euclidean three-space. J. Differ. Geom. 33, 683–715 (1990) 13. Laplace, P.S.: Trait´e de M´echanique C´eleste. Suppl´ement au livre X (1806), Vol. IV. Paris: GauthierVillars, 1806. Annotated English translation by Nathaniel Bowditch (1839); reprinted by New York: Chelsea Publishing Company, 1996 14. Lawson, B.: Complete minimal surfaces in S 3 . Ann. of Math. 92, 335–374 (1970) 15. Lecornu, L.: Sur l’´equilibre des surfaces flexibles et inextensibles. J. de l’Ecole Polytechnique (Paris) 29, 1–109 (1880) 16. Molzon, R., Man, C-S.: Residual stress in membranes. J. Elasticity 20, 181–202 (1988) 17. Nitsche, J.C.C.: Lectures on minimal surfaces. Vol. 1. Cambridge: Cambridge University Press, 1989 18. Shivakumar, P.N., Man, C-S., Rabkin, S.W.: Modelling of the heart and pericardium at end-diastole. J. Biomech. 22, 201–209 (1989) 19. Smyth, B.: The space of residual stress tensors of a closed membrane. In preparation 20. Strebel, H.: Quadratic Differentials. Ergebnisse der Mathematik und ihrer Grenzgebiete 5, BerlinHeidelberg-New York: Springer Verlag, 1984 21. Vekua, I.N.: Generalized Analytic Functions. New York: John Wiley, 1965 22. Weil, A.: Vari´et´es kaehleriennes. Paris: Hermann, 1956 23. Young, T.: An essay on the cohesion of fluids. Phil. Trans. Roy. Soc. (London) 1, 65–87 (1805) Communicated by G.W. Gibbons
Commun. Math. Phys. 250, 95–117 (2004) Digital Object Identifier (DOI) 10.1007/s00220-004-1120-9
Communications in
Mathematical Physics
Hausdorff Measure of the Singular Set of Landau-Lifshitz Equations with a Nonlocal Term Shijin Ding1, , Boling Guo2 1 2
Department of Mathematics, South China Normal University, Guangzhou, Guangdong 510631, P.R. China. E-mail:
[email protected] Center for Nonlinear Studies, Institute of Applied Physics and Computational Mathematics, P.O.Box 8009, Beijing 100088, P.R. China. E-mail:
[email protected]
Received: 22 July 2003 / Accepted: 9 December 2003 Published online: 11 June 2004 – © Springer-Verlag 2004
Abstract: This paper is concerned with the partial regularity of the stationary solutions to the Landau-Lifshitz system of ferromagnetic spin chain with Gilbert damping and a nonlocal term in 3-dimensions. The Hausdorff measure of the singular set is discussed. 1. Introduction Steady state soft ferromagnetic problem is to find minimizers of the energy E=
1 |∇u|2 + Aϕ(u) − Bu · H0 − u · H, 2
where u = (u1 , u2 , u3 ) is the spin vector, |u| = 1, A is the anisotropy constant, n is the symmetric axis of the uniaxial crystal. For example, n = (0, 0, 1), B is a constant, H0 is the applied fields, H is the induced field. The magnetic moment u links the magnetic field and the magnetic induction B by the relation B = H˜ + 4πu,
divB = 0,
curlB = j
satisfying the decomposition H˜ = H0 +H . The applied field H0 arises from the external current j : divH0 = 0,
curlH0 = j,
and the demagnetizing field satisfying div(H + 4πu) = 0,
curlH = 0.
The author is partially supported by the National Natural Science Foundation of China (Grant No.19971030) and by Guangdong Provincial Natural Science Foundation (Grant No.000671 and No.031495).
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The density ϕ(u) of the anisotropy energy A ϕ(u) favors some special direction of the magnetization such that Aϕ(u) attains its minimum. In this paper, without loss of generality, we ignore the applied energy and the anisotropy energy. That is, we only consider the following energy in = B 3 (0): 1 E= |∇u|2 − u · H. 2 B3 B3 The Euler-Lagrange equation for the energy is −u − u|∇u|2 = H − < H, u > u. If the right-hand side is identically 0, then this is the equation of typical harmonic maps. The dynamical equation of the ferromagnetic spin chain is the following Landau-Lifshitz flow: δE(u) δE(u) ut = −λ1 u × u × + λ2 u × , δu δu where λ1 > 0 is the Gilbert damping constant. This equation can be rewritten as (for λ1 = λ2 = 1) 1 1 ut − (u × ut ) = u + u|∇u|2 + H (u) − H (u)u, in B 3 × R+ , 2 2
(1.1)
where H (u) is the nonlocal term which satisfies the following quasi-steady state Maxwell equations: curlH (u) = 0, in D (R3 ),
(1.2)
div(H (u) + u) = 0, in D (R3 ).
(1.3)
We impose on Eqs (1.1)–(1.3) the initial condition u(x, 0) = u0 (x)
(1.4)
∂u | 3 =0 ∂ν ∂B
(1.5)
and boundary condition
in (1.4), |u0 (x)| ≡ 1; in (1.5), ν is the unit outer normal to the boundary of B 3 , u is the zero extension of u from B 3 to R3 , u = (u1 , u2 , u3 ) is the spin vector, “×” denotes the vector cross product in R 3 . We should notice that this extension guarantees u ∈ L∞ (R3 × R+ ) ∩ L∞ (0, ∞; W −1,∞ (R3 )). For the existence of weak solutions to Eq. (1.1) (without nonlocal terms), Zhou, Guo et al have done much work years ago (see [22 – 25]). We also refer to F. Alouges 1 (R3 ), and A. Soyeur [1]; they proved that if λ2 = 1, and the initial data u0 ∈ Hloc 2 3 ∇u0 ∈ L (R ), |u0 | = 1, a.e., then there exists a weak solution. If u0 ∈ H 1 (), λ1 > 0, then the Neumann Boundary problem admits infinitely many weak solutions. If λ1 → 0, then the equation tends to ut = u × u; but if λ1 → ∞, the equation tends to ut = u + u|∇u|2 , the harmonic map heat flow.
Hausdorff Measure of Landau-Lifshitz Equations
97
For the equation with a nonlocal term ((1.1)–(1.3)), Carbou and Fabrie in [2] obtained the global smooth solution for small initial data and Neumann boundary condition, and the local existence of smooth solutions. Carbou [3] discussed the partial regularity of stationary solution of the steady state equations with a nonlocal term and got some results similar to those of harmonic maps. Another important problem for Eq. (1.1)–(1.3)) is the regularity problem. Up to now, we have understood that the discussions on the regularity problem of Landau-Lifshitz equations is similar to that for the harmonic map heat flow although it is much more difficult. So let us first recall some results on the regularity of the harmonic map heat flow. First of all, Chen and Struwe in [7] established the existence and the partial regularity results for some weak solutions of harmonic map heat flow. Later, Chen and Lin [5] generalized the main results of [7] to the flow with a Dirichlet condition. However, Coron [10] observed that there are infinitely many weak solutions to the flow different from those constructed in [7] or [5]. On the other hand, Riviere’s example [20] showed that a weakly harmonic map from B 3 into S 2 may be everywhere singular on B 3 . Therefore, one suitable admissible space is needed to guarantee the partial regularity. Feldman [14] introduced, motivated by the studies on stationary harmonic maps by Evans [13], a notion of “stationary weak solution” for the flow and proved that a stationary solution must be partially regular. The Hausdorff measure of the singular set was given. He also pointed out that the solution constructed by Chen and Struwe is stationary. Since the conditions of “stationary solution” seem unnatural, Chen, Li and Lin in [6] studied the partial regularity for a class of more general weak solutions than those in [14], i.e., the weak solutions satisfying the monotonicity inequality and energy inequality which include the solutions in [14]. They proved the same results for such solutions as in [14] and [7]. The uniqueness was obtained by Freire in [15] for the two dimensional problem. Chen and Wang [8] extended the conclusions of [6] to the case that the target is a homogeneous Riemannian manifold. However, the study on the partial regularity of the Landau-Lifshitz system is much more difficult than that of the harmonic map heat flow. The essential difficulty lies in the derivation of the monotonicity inequality because we have a bad term u × u or u × ut in Landau-Lifshitz systems. In this aspect, the first progress was made by Guo and Hong in 1993 [17] in which they revealed the links between system (1.1) (n = 2 and without a nonlocal term and applied field) and the harmonic maps heat flow and established the same partial regularity results as those for the harmonic map heat flow [7]. The uniqueness of weak solution with finite energy for the 2-dimensional problem was obtained by Chen, Ding and Guo in 1998 [4]. Recently, Liu [18] has got the partial regularity for the stationary solutions of the high dimensional equation (1.1) (without a nonlocal term again and the domain has no boundary) using a similar method to [14]. However, this method is not applicable to the Dirichlet problem. Later, the authors proved in [11] and [12], by somewhat different arguments, similar partial regularity theorems for the weak solutions of the initial value problem and initial-boundary value problem satisfying the energy inequality and generalized monotonicity inequality in which the singular set was characterized in a different way from [18]. In [18] and [11], the authors found that they have to express the possible singular set by two sets both of which have some dimensional Hausdorff measure zero. In 2002, R. Moser [19] studied the partial regularity of stationary solutions to Landau-Lifshitz equations without nonlocal terms in 3 or 4 dimensions. He characterized the singular set only by one set due to the smallness of the dimensions. His method
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is different from that of [18] and [11] but can not be applied to higher dimensional problems. As we know, the key point to get the partial regularity for the harmonic map heat flow is the monotonicity inequalities. However, they are no longer true for Landau-Lifshitz equations as pointed out in [18] and [11]. We can only get the generalized monotonicity inequality to give the energy decay. This is why we have the two-set characterization of the singular set in higher dimensions in [18] and [11]. However, as in [19] for small dimensional problems, the singular set can be characterized only by one set, as usual as for harmonic map heat flow. It is worth noticing that his method is also different from and much more simple than the one usually used for the harmonic map heat flow although it can not be applied to high dimensional problems. In this paper, we adopt some ideas of Moser’s since n = 3. However, to define the stationary solutions, in view of the nonlocal term, we must have detailed studies on the quasi-steady state Maxwell equations although there were some discussions by Carbou in [3]. Secondly, the nonlocal term also leads to some difficulties in the derivations of the generalized monotonicity inequality and energy decay when we intend to improve the generalized monotonicity on almost every time layer t to the cylinders and when we prove the compactness claim for the energy decay. Unlike the usual method for the harmonic map heat flow, we do not need the theories of Hodge decompositions, Hardy space, BMO space and Hardy maximal functions. Only the H´elein technique is used. The proof is direct and concise. It is obvious that in physical settings, in view of the Maxwell equations, it is reasonable to restrict ourselves to three dimensional problems. Our main results are Theorem 5.1 and Theorem 5.2 about the partial regularity of stationary solutions to (1.1)–(1.5). 2. Quasi-Static Maxwell Equations In this section we discuss Maxwell equations (1.2)–(1.3). We recall that Carbou [3] proved the following lemma Lemma 2.1. Let u ∈ H 1 (B 3 , S 2 ). Let H = ∇ ∈ L2 (R3 , R3 ) be the solution of curlH = 0, div(H + u) ˜ =0
(2.1)
in D (R3 ) where u˜ is equal to u in B 3 (|u| = 1 a.e in B 3 ) and zero outside B 3 . Then H ∈ ∩1≤p<∞ Lp (R3 )
(2.2)
and for all p ∈ (1, ∞) there exists a constant Kp > 0 such that
H Lp (R3 ) ≤ Kp u Lp (B 3 ) .
(2.3)
Proof. This is because |u| ˜ = |u| = 1 a.e in B 3 and u˜ = 0 outside of B 3 , then u˜ ∈ ∞ 3 −1,∞ L (R ) and divu ∈ W . Consider ∈ H 1 (R3 ) such that H = ∇ and = −divu in R3 . We have that ∈ ∩1≤p<∞ W 1,p and ∀1 < p < ∞,
∇
Lp (R3 ) = H Lp (R3 ) ≤ Kp u Lp (B 3 ) . The lemma is proved.
(2.4)
Hausdorff Measure of Landau-Lifshitz Equations
99
Remark 2.1. For n = 3, taking p = 3∗ = 6, we get from (2.4)
∇
L6 (R3 ) = H L6 (R3 ) ≤ K u L6 (B 3 ) ≤ C u H 1 = C ∇u L2
(2.5)
1
Therefore, ∈ W 1,6 (R3 ) with embedding W 1,6 (R3 ) ⊂ C 0, 2 (R3 ) (note that n = 3 < 6)
L∞ ≤ C ∇u L2
(2.6)
Lp ≤ C ∇u L2 , ∀1 ≤ p < ∞.
(2.7)
and
Remark 2.2. We have
u·H =− u · ∇ =
divu 3 R3 R =−
= |∇ |2 = |H |2 .
−
B3
R3
R3
R3
(2.8)
Remark 2.3. Next we recall some results concerning the singular integral transforms, the Riesz transform. Let G(r) = Cr be the Laplace kernel in R3 , u as above. Then the solution of the Poisson equation with parameter t (x, t) = −divu, in D (R3 ) can be expressed by G(|x − y|)divu(y, t)dy +
(x, t) = − B3
∂B 3
G(|x − y|) < u(y, t), n(y) > dσ (y).
On the other hand, since divu ∈ L∞ (0, T ; L2 (B 3 )), from the Lp theory of Riesz transforms (see [21]) we have ∂ 2
= −Ri Rj (−divu), i, j = 1, 2, 3, ∂xi ∂xj where Ri are the Riesz transforms Ri (−divu) =
((3 + 1)/2) (xi − yi )(−divu(y, t)) dy (3+2)/2 π |x − y|(3+1) R2
and ϕ ∈ W 2,2 (R3 ) with
W 2,2 (R3 ) (t) ≤ C divu L2 (R3 ) (t). This result can also be deduced from Coifman-Fefferman [9] for general singular integral operators. We should also note that (2.4)–(2.7) hold for H = ∇ and .
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Remark 2.4. If
Lε (x, t) =
y∈B 3 \B(x,ε)
x−y G (|x − y|)divu(y, t)dy |x − y|
(2.9)
and L(x, t) = lim Lε (x, t) = −H + v, then we have from the theory of Riesz transε→0 dσ (y) ≤ form (see [21]) that H (x, t) ∈ L∞ (0, ∞; Lp (B 3 )), |v(x, t)| ≤ K ∂B 3 |x − y|2 K ln(1 − |x|) ∈ L∞ (0, ∞; Lp (B 3 )) for any p > 1 and
L(x, t) L2 ≤ C ∇u L2 .
(2.10) L∞ (0, T ; H 1 (B 3 ))
with Remark 2.5. (Definition of weak solution). A function u ∈ ut ∈ L2 (0, T ; L2 (B 3 )) is called a weak solution if (1.1)–(1.3) hold in the sense of distribution. 3. Definition of Stationary Solution Definition 3.1. A weak solution u of (1.1) is called a stationary solution if for any ξ(x, t) ∈ C01 (B 3 × R+ , R3 ), θ (x, t) ∈ C01 (B 3 × R+ , R) with ξ(x, t), θ (x, t), ∇(x,t) ξ, ∇(x,t) θ bounded on B 3 × R+ and ξ, θ ≡ 0 for t = 0 and t ≥ t ∗ > 0 such that x + τ ξ |∂B 3 = I d, t + τ θ |∂B 3 = I d, there holds +∞ +∞ 1 1 ∂uτ )τ =0 + ∂τ+ ( ut − u × ut )( e(uτ ) + |H (uτ )|2 dxdt ≤ 0, 3 3 2 2 ∂τ B B 0 0 (3.1) where uτ (x, t) = u(x + τ ξ(x, t), t + τ θ (x, t)), e(u) = 21 |∇u(x, t)|2 . We want to make the definition applicable in the following. To this aim we first compute the right derivative in the definition. The computation can be done as in [14], for simplicity, we simply compute the derivative at τ = 0 and without loss of generality, we simply assume that u is smooth. It follows from (1.2) and (1.3) that Hτ = H (uτ ) satisfies curlHτ = 0, in D (R3 ),
(3.2)
div(Hτ + uτ ) = 0, in D (R3 ).
(3.3)
Let the potential ατ (x, t) be such that Hτ = ∇ατ , then (3.2) and (3.3) becomes ατ = −divuτ , in D (R3 ). Therefore
ατ = −
B3
(3.4)
G(|x − y|)divuτ dy +
∂B 3
G(|x − y|) < uτ , n(y) > dσ (y), (3.5)
where G is the fundamental solution in three dimensions. Denote (xτ , tτ ) =2 (x + +∞ τ (x, t)|2 dxdt, B = − +∞ |∇u τ ξ(x, t), t+τ θ (x, t)), Aτ = 21 0 τ 0 B3 B 3 |Hτ | dxdt. It is simple to compute j
divuτ (x, t) = divu(xτ , tτ ) + τ uixj (xτ , tτ )ξxi + τ uit (xτ , tτ )θxi .
(3.6)
Hausdorff Measure of Landau-Lifshitz Equations
101
According to [14], we have +∞ 1 +∞ dAτ i i k i i = (uxj uxk ξxj + ut uxj θxj ) − |∇u|2 (divξ + θt ).(3.7) 3 3 dτ τ =0 2 B B 0 0 On the other hand
+∞
Bτ = 0
B3 +∞
=−
0
+∞
|Hτ |2 dxdt
B3
Hτ · uτ dxdt
∇ατ · uτ dxdt =− B3 0 +∞ (divuτ ) · ατ dxdt − = B3
0
+∞ ∂B 3
0
(uτ · n)ατ dσ.
(3.8)
It follows from (3.5), (3.6) and (3.8) that dBτ dB1 dB2 dB3 dB4 = + + + , dτ dτ dτ dτ dτ where
+∞
−B1 =
dt 0
B3 B3
(3.9)
j
[divu(xτ , tτ ) + τ uixj (xτ , tτ )ξxi + τ uit (xτ , tτ )θxi ] · j
·[divu(yτ , tτ ) + τ uiyj (yτ , tτ )ξyi + τ uit (yτ , tτ )θyi ]G(|x − y|)dxdy. (3.10) Hence we have dB1 − dτ τ =0 +∞ j = dt [uixi xk ξ k + uixi t θ + uixj ξxi + uit θxi ][divu(y, t)]G(|x − y|)dxdy B3 B3 0 +∞ j k i dt [uiyi yk ξ+ uyi t θ + uiyj ξyi + uit θyi ][divu(x, t)]G(|x − y|)dxdy + 0
B3 B3
(3.11) It follows from [3] that j [uixi xk ξ k + uixj ξxi ][divu(y, t)]G(|x − y|)dxdy 3 3 B B j + [uiyi yk ξ k + uiyj ξyi ][divu(x, t)]G(|x − y|)dxdy B3 B3 j = −2 [divudivξ − uixj ξxi ][divu(y, t)]G(|x − y|)dxdy 3 3 B B x−y + , ξ(x, t) − ξ(y, t) > divu(x, t)divu(y, t)dxdy. G (|x − y|) < 3 3 |x − y| B B (3.12)
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S. Ding, B. Guo
Therefore +∞ dB1 j − = −2 dt [divudivξ − uixj ξxi ][divu(y, t)]G(|x − y|)dxdy 3 3 dτ τ =0 B B 0 +∞ x−y , ξ(x, t) − ξ(y, t) dt G (|x − y|) < + 3 3 |x − y| B B 0 > divu(x, t)divu(y, t)dxdy +∞ dt (uixi t θ + uit θxi )divu(y, t)G(|x − y|)dxdy + B3 B3 0 +∞ dt (uiyi t θ + uit θyi )divu(x, t)G(|x − y|)dxdy. (3.13) + B3 B3
0
The last two terms can be estimated as follows: (uixi t θ + uit θxi )divu(y, t)G(|x − y|)dxdy B3
=−
B3
uit (x, t)θ (x, t)divu(y, t)
xi − yi G (|x − y|)dxdy, |x − y|
(3.14)
B3
(uiyi t θ + uit θyi )divu(x, t)G(|x − y|)dxdy
=−
B3
uit (y, t)θ (y, t)divu(x, t)
xi − yi G (|x − y|)dxdy. |x − y|
(3.15)
So we have from (3.13)–(3.15) that +∞ dB1 j =2 dt [divudivξ − uixj ξxi ][divu(y, t)]G(|x − y|)dxdy 3 3 dτ τ =0 B B 0 +∞ x−y , ξ(x, t) − ξ(y, t) dt G (|x − y|) < − |x − y| B3 B3 0 > divu(x, t)divu(y, t)dxdy +∞ xi − yi +2 dt uit (x, t)θ (x, t)divu(y, t) G (|x − y|)dxdy. |x − y| B3 0 (3.16) Moreover, since B2 , B3 in (3.9) are +∞ B2 = dt x∈B 3
0
y∈∂B 3
j
[divu(xτ , tτ ) + τ uixj ξxi + τ uit θxi ]
(u(y, t), n(y))G(|x − y|)dxdσ (y),
+∞
B3 =
dt 0
x∈∂B 3
y∈B 3
(3.17) j
[divu(yτ , tτ ) + τ uiyj ξyi + τ uit θyi ]
(u(x, t), n(x))G(|x − y|)dydσ (x),
(3.18)
Hausdorff Measure of Landau-Lifshitz Equations
103
we have +∞ dB2 j = dt [uixi xk ξ k + uixi t θ + uixj ξxi + uit θxi ](u(y, t), n(y)) 3 3 dτ τ =0 x∈B y∈∂B 0 G(|x − y|)dxdσ (y), (3.19)
+∞ dB3 j = dt [uiyi yk ξ k + uiyi t θ + uiyj ξyi + uit θyi ](u(x, t), n(x)) dτ τ =0 x∈∂B 3 y∈B 3 0 G(|x − y|)dydσ (x), (3.20) and similarly to the above we have dB2 dB3 + dτ τ =0 dτ τ =0 +∞ =2 dt x∈B 3
0
j
y∈∂B 3
[−divu(x, t)divξ(x, t) + uixj ξxi + ui xi t +uit θxi ](u(y, t), n(y))G(|x − y|)dxdσ (y). (3.21)
We note that the term B4 is determined only by the boundary data which is indepen 4 dent of τ then dB = 0. We finally get dτ τ =0
+∞ dBτ = −2 dt divu(x, t)divξ(x, t) (x, t)dxdt dτ τ =0 B3 0 +∞ j dt
(x, t)uixj (x, t)ξxi (x, t) +2 3 B 0 +∞ x−y , ξ(x, t) − ξ(y, t) dt < + 3 3 |x − y| B B 0 > G (|x − y|)divu(x, t)divu(y, t)dxdy +∞ x−y , ut (x, t) dt < −2 |x − y| B3 B3 0 > θ (x, t)divu(y, t)G (|x − y|)dxdy,
(3.22)
where
(x, t) = −
B3
G(|x − y|)divu(y, t)dy +
B3
G(|x − y|) < u(y, t), n(y) > dσ (y),
which satisfies (2.7). Combining (3.7) and (3.22), we may now simplify the definition of a stationary weak solution for our problem (1.1)–(1.5) as follows:
104
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Remark 3.1. A stationary solution u of (1.1) satisfies for any ξ(x, t) ∈ C01 (B 3 ×R+ , R3 ) and θ (x, t) ∈ C01 (B 3 × R+ , R) as above, the following inequality: +∞ +∞ 1 1 ( ut − u × ut )(∇u · ξ + ut θ ) + dt (uixj uixk ξxkj + uit uixj θxj )dx 2 B3 2 B3 0 0 1 +∞ dt |∇u|2 (divξ + θt )dx − 2 0 B3 +∞ dt divu(x, t)divξ(x, t) (x, t)dxdt −2 B3 0 +∞ j dt
(x, t)uixj (x, t)ξxi (x, t) +2 3 B 0 +∞ x−y dt < , ξ(x, t) − ξ(y, t) + 3 3 |x − y| B B 0 > G (|x − y|)divu(x, t)divu(y, t)dxdy +∞ x−y dt < , ut (x, t) > θ (x, t)divu(y, t)G (|x − y|)dxdy ≤ 0. −2 |x − y| B3 B3 0 (3.23) In what follows we want to derive some more applicable inequalities and equalities from (3.23) to be used in the future. Now let Mε = B 3 × B 3 \ {|x − y| < ε} for ε > 0 small enough. Since ξ(·, t) is ξ(x, t) − ξ(y, t) smooth, then in (3.22) is uniformly bounded. |x − y| +∞ x−y dt < , ξ(x, t) − ξ(y, t) > G (|x − y|)divu(x, t)divu(y, t)dxdy |x − y| B3 B3 0 +∞ x−y , ξ(x, t) − ξ(y, t) dt < = lim ε→0 0 |x − y| Mε
> G (|x − y|)divu(x, t)divu(y, t)dxdy +∞ x−y dt ξ(x, t) G (|x − y|)divu(x, t)divu(y, t)dxdy = lim 2 ε→0 |x − y| Mε 0 +∞ x−y dt ξ(x, t)divu(x, t) G (|x − y|)divu(y, t)dy dx. = lim 2 ε→0 x∈B 3 y∈B 3 \B(x,ε) |x − y| 0 (3.24)
Let Lε (x, t) = 2
x−y G (|x − y|)divu(y, t)dy. y∈B 3 \B(x,ε) |x − y|
We have lim Lε (x, t) = L(x, t) = −H (x, t) + v(x, t)
ε→0
with H (x, t) ∈ L∞ (0, ∞; Lp (B 3 )), |v(x, t)| ≤ K
∂B 3
L∞ (0, ∞; Lp (B 3 )) for any p > 1 and there holds
L(x, t) L2 ≤ C ∇u L2 .
(3.25)
dσ (y) ≤ K ln(1 − |x|) ∈ |x − y|2
Hausdorff Measure of Landau-Lifshitz Equations
105
Then (3.24) can be rewritten as +∞ x−y , ξ(x, t) − ξ(y, t) > G (|x − y|)divu(x, t)divu(y, t)dxdy dt < |x − y| B3 B3 0 ∞ ξ(x, t) · L(x, t)divu(x, t)dxdt. (3.26) = B3
0
Similarly we deduce that −
+∞
dt 0 ∞ =−
< B3
B3
0
B3
x−y , ut (x, t) > θ (x, t)divu(y, t)G (|x − y|)dxdy |x − y|
ut · L(x, t)θ (x, t)dxdt.
(3.27)
Finally, inequality (3.23) can be rewritten as Assumption (S).
+∞
+∞ 1 1 ( ut − u × ut )(∇u · ξ + ut θ ) + dt (uixj uixk ξxkj + uit uixj θxj )dx 2 B3 2 B3 0 0 1 +∞ dt |∇u|2 (divξ + θt )dx − 2 0 B3 +∞ dt divu(x, t)divξ(x, t) (x, t)dxdt −2 B3 0 +∞ j dt
(x, t)uixj (x, t)ξxi (x, t) +2 3 B 0 ∞ L(x, t)[ξ divu(x, t) − ut θ ]dxdt ≤ 0. (3.28) + B3
0
Formula (3.28), Assumption (S), is just the starting point of all the following discussions. From Assumption (S), one easily derive the following lemma as in [14] which will be used to get the generalized monotonicity inequality in the following section. Lemma 3.1. Let u be a stationary weak solution of (1.1)–(1.5) and ξ , θ as before. Then we have (ut − u × ut )∇u · ξ − |∇u|2 divξ + 2uxj uxk ξxkj B 3 ×{t}
j
−2divu(x, t)divξ(x, t) (x, t) + 2 (x, t)uixj ξxi + 2divu(x, t)(ξ · L(x, t))dxdt = 0 (3.29) and B 3 ×{t2 }
−
|∇u| θ dx ≤ 2
B 3 ×{t1 }
t2
t1
B3
|∇u|2 θt − |ut |2 θ − 2(∇u · ut )∇θ
+ 2(ut · L(x, t))θ dxdt.
(3.30)
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4. Estimates for Local Energy In this section, we use inequalities (3.29) and (3.30) to derive the generalized monotonicity inequality which will be used to deduce the energy decay in the next section. We apply a different method from [6] by using the approaches of deriving the monotonicity inequality for the harmonic map rather than the method for the harmonic map heat flow so that we may avoid a term involving the integration of |ut |2 on the right-hand side of the inequality as stated in [18]. Such an idea was first introduced by Moser [19]. Now we use this idea to derive a generalized monotonicity for the L-L equation with a nonlocal term. We think that such a proof of energy decay, though it is only applicable to low dimensional problems, is much more simple than that usually used for the harmonic map heat flow. Denote Bρ = Bρ (0), Pρ (z) = Bρ (x) × (t − ρ 2 , t + ρ 2 ) for z = (x, t). In the following, we always denote by u the stationary solution of (1.1)–(1.5): Lemma 4.1. For any Bs (x0 ) ⊂ Br/2 (0) we have for the stationary solution u that 1 1 2 2 |∇u| ≤ |∇u| + Kr |ut |2 . (4.1) s Bs (x0 ) r Br Br Proof. Taking ξ(x) = φ(|x|)x in inequality (3.29) where for any fixed 0 < r, r + h < 1, h > 0, 1, if s ≤ r , if r ≤ s ≤ r + h φ(s) = 1 + r−s 0, ifh r + h ≤ s < 1 and noting that ξxkj (x) = δj k φ(|x|) + φ (|x|)
x j xk , divξ = 3φ(|x|) + φ (|x|)|x|, |x|
(4.2)
where φ (s) = 0 if s ≤ r, φ (s) = − h1 if r < s < r + h and φ (s) = 0 if r + h < s < 1, we obtain from (3.29) that
(ut − u × ut )(∇u · x)φ(|x|) − |∇u|2 (3φ(|x|) + φ (|x|)|x|)
Br+h
x j xk ) − 2 (x, t)divu(x, t)(3φ(|x|) + φ (|x|)|x|) |x| xj xk j +2 (x, t)uxk (δj k φ(|x|) + φ (|x|) ) + 2divu(x, t)φ(|x|)(x · L(x, t))dx = 0. |x| (4.3)
+2uxj uxk (δj k φ(|x|) + φ (|x|)
Sending h → 0 we may estimate every term in (4.3) as follows: (ut − u × ut )(∇u · x)φ(|x|) = (ut − u × ut )(∇u · x), lim h→0 Br+h
− lim
h→0 Br+h
(4.4)
Br
|∇u| (3φ(|x|) + φ (|x|)|x|) = −3 2
|∇u| + r
|∇u|2 dσ, (4.5)
2
Br
∂Br
Hausdorff Measure of Landau-Lifshitz Equations
lim
h→0 Br+h
2uxj uxk (δj k φ(|x|) + φ (|x|)
107
x j xk )=2 |x|
|∇u|2 − Br
2 r
|x · ∇u|2 dσ, ∂Br
(4.6) − lim
h→0 Br+h
= −6
2 (x, t)divu(x, t)(3φ(|x|) + φ (|x|)|x|)
(x, t)divu(x, t) + 2r
divu(x, t)dσ,
Br
− lim
h→0 Br+h
= −2 Br
x j xk j 2 (x, t)uxk (δj k φ(|x|) + φ (|x|) ) |x| 2
(x, t)divu(x, t) −
(x · ∇u) · xdσ, r ∂Br
lim
h→0 Br+h
(4.7)
∂Br
(4.8)
2divu(x, t)φ(|x|)(x · L(x, t)) = 2
(x · L(x, t))divu(x, t).
(4.9)
Br
Substituting (4.4)–(4.9) into (4.3), we get
(ut − u × ut )(∇u · x) − |∇u| + r |∇u|2 dσ Br ∂Br 2 2 − |x · ∇u| dσ − 8
(x, t)divu(x, t) + 2r
divu(x, t)dσ r ∂Br Br ∂Br 2 −
(x · ∇u) · xdσ + 2 (x · L(x, t))divu(x, t) = 0. (4.10) r ∂Br Br 2
Br
On the other hand, since 1 |∇u|2 − (ut − u × ut )(∇u · x) + 8 (x, t)divu(x, t) r Br −2(x · L(x, t))divu(x, t)]} 1 =− 2 |∇u|2 − (ut − u × ut )(∇u · x) + 8 (x, t)divu(x, t) r Br −2(x · L(x, t))divu(x, t)] 1 + |∇u|2 − (ut − u × ut )(∇u · x) + 8 (x, t)divu(x, t) r ∂Br −2(x · L(x, t))divu(x, t)] , d dr
(4.11)
108
S. Ding, B. Guo
we get from (4.10) and (4.11) that d 1 |∇u|2 − (ut − u × ut )(∇u · x) + 8 (x, t)divu(x, t) dr r Br −2(x · L(x, t))divu(x, t)]} 1 = [−(ut − u × ut )(∇u · x) + 8 (x, t)divu(x, t) − 2(x · L(x, t))divu(x, t)] r ∂Br 2 2 2 2 + |x · ∇u| − divu(x, t) + 3 (x · ∇u) · x 3 r r ∂B 3 r 2 |x · ∇u| (x · ∇u)(ut − u × ut ) 3 divu (x · L)divu =2 + − − 3 |x| 2|x| |x| |x| ∂B 3
(x · ∇u) · x + . (4.12) |x|3 Denote ψ(ρ, t) =
1 ρ
Bρ
|∇u| − (ut − u × ut )(∇u · x) + 8 (x, t)divu(x, t) − 2(x · L(x, t))divu(x, t) . 2
(4.13) Then by integrating (4.12) from s to ρ (s < ρ), we obtain
|x · ∇u|2 (x · ∇u)(ut − u × ut ) − 3 |x| 2|x| Bρ \Bs 3 divu (x · L)divu (x · ∇u) · x + − + |x| |x| |x|3
ψ(ρ, t) − ψ(s, t) = 2
(4.14)
or
|x · ∇u|2 (x · ∇u)(ut − u × ut ) − 3 |x| 2|x| Bρ \Bs 3 divu (x · L)divu (x · ∇u) · x , + − + |x| |x| |x|3
ψ(s, t) = ψ(ρ, t) − 2
(4.15)
that is 1 s
|∇u|2 = Bs
1 [(ut − u × ut )(∇u · x) − 8 (x, t)divu(x, t) s Bs +2(x · L(x, t))divu(x, t)] |x · ∇u|2 (x · ∇u)(ut − u × ut ) +ψ(ρ, t) − 2 − 3 |x| 2|x| Bρ \Bs 3 divu (x · L)divu (x · ∇u) · x + − . (4.16) + |x| |x| |x|3
Hausdorff Measure of Landau-Lifshitz Equations
109
By filling the hole and note that |x| ≤ s if x ∈ Bs , we have |(x · ∇u)(ut − u × ut )| 4| divu| 1 2 |∇u| ≤ ψ(ρ, t) + 2 [ + s Bs 2|x| |x| Bρ +
|(x · L)divu| | ||x · ∇u| ]. + |x| |x|2
(4.17)
Now we can estimate all the terms on the right-hand side of (4.17) and get the generalized monotonicity inequality. It follows from (4.17) and H¨older inequality that 1 1 |∇u|2 ≤ |∇u|2 + Kρ [|ut |2 + | |2 + | |8 + |L|2 ]. (4.18) s Bs ρ Bρ Bρ In this inequality, we should note that (x, t) ∈ L∞ (0, T ; W 1,p ()), L(x, t) ∈ L∞ (0, T ; Lp ()) for any p > 1. Then it follows from Lemma 2.1, (2.7) and (2.10) that 1 1 |∇u|2 ≤ K |∇u|2 + Kρ |ut |2 (4.19) s Bs ρ Bρ Bρ with K independent of t. It is not difficult to get, for any Bs (x0 ) ⊂ Br/2 (0), 1 1 |∇u|2 ≤ |∇u|2 + Kr |ut |2 . s Bs (x0 ) r Br Br
(4.20)
Lemma 4.1 is proved.
In the following we want to prove Lemma 4.2 (Generalized monotonicity inequality). There exists a constant C > 0 such that for any θ ∈ (0, 1/16] there is ε0 > 0, if u ∈ H 1 (Pr (z0 ), S 2 ) is a stationary solution, we have |∇u|2 dz ≤ δε2 + C1 r −5 |u − (u)Pr (0) |2 dz r −3 Pr/8 (0)
Pr (0)
under the condition r
−3
Pr (0)
|∇u|2 dz ≤ ε2 ≤ ε02 .
Proof. It follows from (3.30) that |∇u|2 θ dx − ≤
B 3 ×{t2 } t2
t1
B3
Therefore we have − B 3 ×{t2 }
B 3 ×{t1 }
|∇u|2 θt − |ut |2 θ − 2(∇u · ut )∇θ + 2(ut · L(x, t))θ dxdt. (4.21)
B 3 ×{t1 }
|∇u|2 θ dx ≤ C
t2
t1
B3
|∇u|2 (θt + |∇θ |2 ) + |L(x, t)|2 θ.
110
S. Ding, B. Guo
Hence if taking θ (x, t) such that |∇θ |2 , |θt | ≤ rC2 , θ(x, t1 ) = 0 and θ ≡ 1 for x ∈ Br/2 , t ∈ ((− 2r )2 , ( 2r )2 ), we obtain from Lemma 2.1 for almost every t = t2 ∈ (−( 2r )2 , ( 2r )2 ), 1 1 2 2 |∇u| ≤ K1 3 |∇u| dxdt . (4.22) r Br/2 (0)×{t} r Pr (0) Now we prove the following claim. Claim. There exists a constant C > 0 such that for any given 0 < λ < 1, there exists a set ⊂ (−r 2 /2, r 2 /2) with || ≤ λ satisfying 1 Cε2 |ut |2 dx ≤ (4.23) r B r (0)×{t} λ 2
for almost every t ∈ , where 13 Pr |∇u|2 dz ≤ ε2 ≤ ε02 . r Proof. Taking the smooth test function θ (x, t) such that θ ≡ 1 in Pr/2 (0) but θ ≡ 0 outside Pr (0), t1 = −r 2 , t2 = r 2 , and noticing that L(x, t) ∈ L∞ (0, ∞; Lp (B 3 )) for any p ≥ 1, we get from Lemma 2.1 that |ut |2 dz ≤ C0 |∇u|2 (θt + |∇θ |2 ) + θ |L(x, t)|2 dz Pr/2 (0)
Pr (0)
≤ C0 r
1 r3
|∇u|
2
≤ C0 rε 2 .
(4.24)
Pr (0)
If the Claim is false, then for any C > 0, there exists 0 < λ < 1 such that for some set ⊂ (−r 2 /2, r 2 /2) with || > λ there holds: if t ∈ then 1 1 |ut |2 dz ≥ |ut |2 dz ≥ Cε2 . r Pr/2 (0) r Br/2 (0) This contradicts (4.24) by taking C > C0 . The Claim follows. We can even prove by (4.20) (Lemma 4.1) that if t ∈ then 1 Cε 2 sup . ( |∇u|2 dx) ≤ λ Bs (x0 )⊂Br/4 (x0 ) s Bs (x0 )×{t}
(4.25)
In the sequence, we mean t ∈ . In order to prove the lemma, we take ξ ∈ C0∞ (Br/4 (0)) such that 0 ≤ ξ ≤ 1, ξ ≡ 1 in Br/8 (0), |∇ξ | ≤ 16 r to compute ξ(x)|∇u|2 dx = ξ(x)∇u · ∇(u − (u)Pr (0) )dx Br (0)×{t} Br (0)×{t} =− (u − (u)Pr (0) )∇ξ · ∇u Br (0)×{t} − ξ(x)(u − (u)Pr (0) )u. Br (0)×{t}
Hausdorff Measure of Landau-Lifshitz Equations
111
Using the equation, we have ξ(x)|∇u|2 dx = − (u − (u)Pr (0) )∇ξ · ∇u Br (0)×{t} Br (0)×{t} 1 1 − ξ(x)(u − (u)Pr (0) )( ut − u × ut ) 2 2 B (0)×{t} r + ξ(x)(u − (u)Pr (0) )u|∇u|2 Br (0)×{t} − ξ(x)(u − (u)Pr (0) )uH (u). (4.26) Br (0)×{t}
In the following we estimate every term on the right hand side of (4.26). Estimate of the first term. Br (0)×{t}
16
∇u L2 (Br ) u − (u)Pr (0) L2 (Br ) r 1/2 16 1 ≤ √ |∇u|2 dx
u − (u)Pr (0) L2 (Br ) r r Br (0) 1 ≤ C0 ε √ u − (u)Pr (0) L2 (Br ) (by (4.22)) r C 1 δλr ≤ Cε2 +
u − (u)Pr (0) 2L2 (B ) . (4.27) r 2 δλ r 2
(u − (u)Pr (0) )∇ξ · ∇u ≤
Estimate of the second term. Since t ∈ , we get
1 1 ξ(x)(u − (u)Pr (0) )( ut − u × ut ) ≤ C ut L2 (Br ) u − (u)Pr (0) L2 (Br ) 2 2 Br (0)×{t} 1/2 √ 1 2 ≤C r |ut | dx r Br (0)
u − (u)Pr (0) L2 (Br ) √ Cε r ≤ √ u − (u)Pr (0) L2 (Br ) λ (by (4.23)) δr C ≤ Cε2 + u − (u)Pr (0) 2L2 (B ) . r 2 δλ (4.28)
Estimate of the third term To estimate this term, we use the H´elein method to decompose it. Since |u| = 1 a.e, we have ui |∇u|2 =
3 j =1
∇uj (ui ∇uj − uj ∇ui ),
112
S. Ding, B. Guo
and then Br (0)×{t}
=
ξ(x)(u − (u)Pr (0) )u|∇u|2
3 3 i=1 j =1 Br (0)×{t}
(ui − (ui )Pr (0) )∇uj (ξ(x)(ui ∇uj − uj ∇ui )).
On the other hand div(ui ∇uj − uj ∇ui ) = ui uj − uj ui = ui w j − uj w i , where w = 21 ut + 21 u × ut ; we obtain
div(ξ(x)(ui ∇uj − uj ∇ui )) L2 (Br ) ≤ ∇ξ(x) · (ui ∇uj − uj ∇ui ) L2 (Br ) + ξ(x)(ui w j − uj w i ) L2 (Br ) 32
∇u L2 (Br/4 (0)) + 2 ut L2 (Br/4 (0)) ≤ r Cε ≤ √ (by (4.22) and (4.23)). (4.29) λr To continue the proof, we recall a lemma by Feldman [14]: Lemma 4.3 ([14]). Let f, h ∈ H 1 (Rn ) and g ∈ L2 (Rn , Rn ) with divg ∈ L2 (Rn ) in the distribution sense, and r 2−n |∇h|2 dx = A2 < ∞. sup x0 ∈Rn , r>0
Then
R
Br (x0 )
fg · ∇h ≤ CA( ∇f L2 g L2 + f L2 divg L2 ) n
for some universal constant C. Now we apply Lemma 4.3 to f = ui − (ui )Pr (0) , h = uj and g = ξ(x)(ui ∇uj − by extending them properly to the whole space R3 . By (4.22) and (4.29), 2 ξ(x)(u − (u)Pr (0) )u|∇u| = (ui − (ui )Pr (0) )∇uj
uj ∇ui )
Br (0)×{t}
Br (0)×{t} i
(ξ(x)(u ∇uj − uj ∇ui )) Cε ≤ √ ∇f L2 g L2 + f L2 divg L2 λ Cε Cε ≤ √ Cε2 r + √ u − (u)Pr (0) L2 λ λr Cε3 r Cε 2 ≤ (4.30) + 2 u − (u)Pr (0) 2L2 . λ λr
Hausdorff Measure of Landau-Lifshitz Equations
113
Finally we estimate the last term on the right hand side of (4.26) as follows. Since H (u) = ∇ and = −divu, divu ∈ L∞ (0, ∞; H −1 ), we know H (u) 2L2 (B ) ≤ r
C0 ∇u 2L2 (B ) ≤ Crε 2 . Hence we derive r
Br (0)×{t}
√ ξ(x)(u − (u)Pr (0) )uH (u) ≤ C rε u − (u)Pr (0) L2 (Br (x0 )) ≤ Cε 2
C δλr + u − (u)Pr (0) 2L2 (B (x )) . r 0 2 δλ (4.31)
Combining (4.26)–(4.31), we have obtained for t ∈ (|| < λ), Cε C δ |∇u|2 dx ≤ (
u − (u)Pr (0) 2L2 (B (0)) . + )ε 2 r + r λ 2 δλr 2 Br/8 (0)×{t}
(4.32)
Integrating over (−r 2 /2, r 2 /2) and using (4.22), we get
r 2 /2
−r 2 /2 Br/8 (0)×{t}
|∇u|2 dx ≤ (
δ Cε C + Cλ + )ε 2 r 3 +
u − (u)Pr (0) 2L2 (P (0)) . r λ 2 δλr 2 (4.33)
Hence, choosing C, λ and ε0 properly we obtain r −3 |∇u|2 dz ≤ δε2 + C1 r −5 Pr/8 (0)
Pr (0)
under the condition r −3
|u − (u)Pr (0) |2 dz
(4.34)
Pr (0)
|∇u|2 dz ≤ ε2 ≤ ε02 .
Lemma 4.2 is proved.
5. Energy Decay and Partial Regularity In this section we derive the energy decay which will be used to prove the regularity. Lemma 5.1. There exists a constant C > 0 such that for any θ ∈ (0, 21 ], there is a number ε0 > 0 such that for any stationary solution u ∈ H 1 (Pr (z0 ); S 2 ) of (1.1)–(1.5) satisfying −3 |∇u|2 dz ≤ ε2 ≤ ε02 , r Pr (z0 )
we have −5
(θ r)
Pθ r (z0 )
|u − (u)Pθ r (z0 ) |2 dz ≤ Cθ 2 ε 2 .
(5.1)
114
S. Ding, B. Guo
Proof. Since the integrals are invariant under the transformation (x, t) → (rx+x0 , r 2 t + t0 ), we may assume Pr (z0 ) = P1 (0). We argue by contradiction: If the conclusion is untrue, then for any C > 0 we may find θ ∈ (0, 1/2] and the stationary weak solution uk ∈ H 1 (P1 (0); S 2 ) of the considered problem such that |∇uk |2 dz = εk2 → 0, k → ∞ (5.2) P1 (0)
but
Pθ (0)
|uk − (uk )Pθ (0) |2 > Cθ 3+4 εk2 .
(5.3)
It follows from these assumptions and the lemmas in the above sections that the sequence {vk } = { ε1k (uk − (uk )Pθ (0) )} is bounded in H 1 (P1/2 (0)) which allows us to assume that there is a map v ∈ H 1 (P1/2 (0); R3 ) such that vk → v, weakly in H 1 (P1/2 (0); R3 );
with
vk → v, strongly in L2 (P1/2 (0); R3 )
vdz = 0;
Pθ (0)
|∇v|2 dz ≤ 1. P1/2 (0)
It is obvious that we can assume that uk → p strongly in L2 (P1/2 (0)) for some constant map p ∈ S 2 . Note that uk solves the following equation: 1 1 ukt − (uk × ukt ) = uk + uk |∇uk |2 + H (uk ) − H (uk )uk , in B 3 × R+ , 2 2 (5.4) where curlH (uk ) = 0, in D (R3 ),
(5.5)
div(H (uk ) + uk ) = 0, in D (R3 ).
(5.6)
Let H (uk ) = ∇ k . Then k = −divuk . Note that uk = εk vk + (uk )Pθ (0) . Then for any φ ∈ C0∞ (P1/2 (0), R3 ) we have by multiplying (5.4) by φ, 1 1 ( ukt − (uk × ukt ))φ 2 2 P1/2 (0) = (uk + uk |∇uk |2 + H (uk ) − H (uk )uk )φ P1/2 (0)
=
(−∇uk · ∇φ + uk |∇uk |2 φ + φ∇ k − uk φ∇ k ).
(5.7)
Note that uk = εk vk + (uk )Pθ (0) . We have from (5.7) that 1 1 ε ( vkt − (uk × vkt ))φ 2 2 P1/2 (0) = (−εk ∇vk · ∇φ + uk |∇uk |2 φ + φ∇ k − uk φ∇ k )
(5.8)
P1/2 (0)
P1/2 (0)
Hausdorff Measure of Landau-Lifshitz Equations
115
with k = −εk divvk . Divide both sides of (5.8) by εk and send k → ∞ to give 1 1 ( vt − (p × vt ))φ 2 P1/2 (0) 2 = (−∇v · ∇φ + φ∇ ∞ − pφ∇ ∞ ), (5.9) 1 k→∞ εk
since lim
P1/2 (0)
2 P1/2 (0) |∇uk | uk φ
= 0 from (5.2), where ∞ = −divv. Since v ∈
H 1 (P1/2 (0)) we know divv ∈ L2 (P1/2 (0)) then ∞ ∈ W22,2 . Denote H∞ = ∇ ∞ . It follows from (5.9) that v satisfies 1 1 vt − (p × vt ) = v + H∞ − pH∞ . (5.10) 2 2 By a standard estimate and boot-strapping method, we know that v is smooth and there holds |v|2 dz ≤ |v − vPθ (0) |2 dz ≤ C0 θ 3+4 (5.11) Pθ (0)
Pθ (0)
by the Poincar´e inequality. Inequality (5.11) contradicts (5.3) from the strong L2 -convergence of vk to v if one chooses C > C0 .
Combining these lemmas we get by the iteration method: Proposition 5.1. There is a constant C > 0 such that for any θ ∈ (0, 1/16] there exists a number ε0 > 0 such that if u ∈ H 1 (Pr (z0 ), S 2 ) is a stationary solution of (1.1)–(1.5) satisfying the small energy condition r −3 |∇u|2 dz ≤ ε2 ≤ ε02 , (5.12) Pr (z0 )
then −3
|∇u|2 dz ≤ Cθ 2 ε 2 .
(θ r)
(5.13)
Pθ r (z0 ) 1 1 Proof. Given 0 < θ < 1/16, taking k such that 8k+2 ≤ θ ≤ 8k+1 . Denote (r) = −3 2 −5 2 r Pr (z0 ) |∇u| dz, (r) = r Pr (z0 ) |u − (u)Pr (z0 ) | . It follows from (5.12) and Lemma 4.2 that for any δ > 0, r
( ) ≤ δε2 + C1 (r). (5.14) 8 Similarly we have r r r r
( 2 ) ≤ δ ( ) + C1 ( ) ≤ δ(δε2 + C1 (r)) + C1 ( ). (5.15) 8 8 8 8 Iterating, applying Lemma 5.1 and choosing δ properly, we get
(
r
8
) ≤ δ k+1 ε 2 + C1 k+1
k j =1
δ j (
r r 1 ) + ( k ) ≤ Cδ k+1 ε 2 + C( k )2 ε 2 ≤ Cθ 2 ε 2 , j 8 8 8 (5.16)
116
S. Ding, B. Guo
where we have used Lemma 5.1 again which shows ( 8rj ) ≤ C( 81j )2 ε 2 . Then we have −3
|∇u| dz ≤ θ
(θr) = (θ r)
2
Pθ r
≤ (8k+1 θ )−3 (
r
8
)−3 k+1
−3
(
r 8k+1
)
−3
(8
k+1 −3
|∇u|2 dz
)
P
|∇u|2 dz ≤ 83 ( P
r 8k+1
r 8k+1
r 8k+1
) ≤ Cθ 2 ε 2 (by (5.16)).
Remark 5.1. By virtue of (4.25), Proposition 5.1 also holds if replacing Pθr (z0 ) by Ps (z1 ) for any z1 ∈ P15r/16 (z0 ) and s ∈ (0, r/16). Theorem 5.1 (Main Theorem 1). There exist constants ε0 > 0 and Ckl > 0 such that any stationary solution u ∈ H 1 (Pr (z0 )) of (1.1)–(1.4) satisfying the small energy condition (5.10) is smooth in Pr/2 (z0 ) and
∂tl ∇ k u L∞ (Pr/2 (z0 )) ≤ Ckl r −k−2l ε, k, l = 0, 1, 2, · · · .
(5.17)
Proof. For any given Pr (z0 ) ⊂ B 3 × (0, T ), Proposition 5.1 shows that for any λ ∈ (0, 1), if ε0 is small enough, we have (|∇u|2 + s 2 |ut |2 )dz ≤ C1 s n+2λ (5.18) Ps (z1 )
for any z1 ∈ P15r/16 (z0 ) and s ∈ (0, r/16) with C1 only depending on λ. In fact, let θr = s then θ = rs . Substituting this θ into (5.13), we have s
−3
s |∇u|2 dz ≤ ( )2 ε02 . r Ps (z1 )
(5.19)
On the other hand we may control Ps (z1 ) s 2 |ut |2 dz as in Feldman [14]. Equation (5.18) is proved. By a standard method we deduce that u is smooth in P15r/16 (z0 ) by Morrey’s Lemma which was done by Feldman [14] . The estimates (5.17) can be obtained by a scaling argument.
By a standard method as in Giaquinta’s book [16], it is easy to conclude Theorem 5.2 (Main Theorem 2). Let u ∈ H 1 ( × (0, T ); S 2 ) be a stationary solution of (1.1)–(1.4). There is an open set Q ⊂ × (0, T ) such that u is smooth in Q and H3 ( × (0, T ) \ Q) = 0,
(5.20)
where × (0, T ) \ Q = {z = (x, t)| lim inf r r→0
−3
|∇u|2 dz ≥ ε0 }. Pr (z)
(5.21)
Hausdorff Measure of Landau-Lifshitz Equations
117
References 1. Alouges, F., Soyeur, A.: On global weak solutions for Landau-Lifshitz equations: existence and nonuniqueness. Nonlinear Analysis, TMA 18(11), 1084 (1992) 2. Carbou, G., Fabrie, P.: Regular solutions for Landau-Lifshitz equation in a bounded domain. Diff. Int. Eqns. 14, 213–229 (2001) 3. Carbou, G.: Regularity for critical points of a nonlocal energy. Calc. Var. and PDEs 5, 409–433 (1997) 4. Chen, Y., Ding, S., Guo B.: Partial regularity for two-dimensional Landau-Lifshitz equations. Acta Math. Sinica, 14(3), 423–432 (1998) 5. Chen , Y., Lin, F. H.: Evolution of harmonic maps with Dirichlet boundary conditions. Comm. Anal. Geom. 1, 327–346 (1993) 6. Chen , Y., Li, J., Lin, F. H.: Partial Regularity for Weak Heat Flows into Spheres. Comm. Pure Appl. Math. XLVIII, 429–448 (1995) 7. Chen, Y., Struwe, M.: Existence and partial regularity results for the heat flow of harmonic maps. Math. Z. 201, 83–103 (1989) 8. Chen , Y., Wang, C.: Partial regularity for weak flows into Riemannian homogeneous space. Comm. P. D. E. 21(5–6), 735–761 (1996) 9. Coifman, R. R., Fefferman, C.L.: Weighted norm inequality for maximal functions and singular integrals. Studia Math. 51, 241–250 (1974) 10. Coron, J.: Nonuniqueness for the heat flow of harmonic maps. Ann. Inst. H. Poincar´e, Non-Lin´eair´e 7, 335–344 (1990) 11. Ding, S., Guo , B.: Partial Regularity for Higher Dimensional Landau-Lifshitz Systems. Preprint 2003 12. Ding, S., Guo , B.: Initial-Boundary Value Problem for Higher Dimensional Landau-Lifshitz Systems. Appl. Anal., to appear 13. Evans, L. C.: Partial regularity for harmonic maps into spheres. Arch. Rat. Mech. Anal. 116, 101–113 (1991) 14. Feldman, M.: Partial Regularity for Harmonic Maps of Evolution into Spheres. Comm. PDE. 19, 761–790 (1994) 15. Freire, A.: Uniqueness for the harmonic heat flow in two dimensions. Calc. Var. PDE. 4, 761–790 (1996) 16. Giaquinta, M.: Introduction to Regularity theory for nonlinear elliptic systems. Basel-Boston-Berlin: Birkh¨auser Verlag, 1993 17. Guo, B., Hong, M.: The Landau-Lifshitz equations of the ferromagnetic spin chain and harmonic maps. Calc. Var. PDE. 1, 311–334 (1993) 18. Liu, X.: Partial regularity for the Landau-Lifshitz systems. Calc. Var. 20(2), 153–173 (2003) 19. Moser, R.: Partial regularity for the Landau-Lifshitz equation in small dimensions, MPI Preprint 26 2002, www.mis.mpg.de/preprints/2002 20. Rivi´ere, T.: Everywhere discontinuous harmonic maps from B 3 into S 2 . C. R. Acad. Sci. Paris 314, 719–723 (1992) 21. Stein, E. M.: Singular integrals and differentiability properties of functions. Princeton, NJ: Princeton Univ. Press, 1970 22. Zhou, Y., Guo, B.: Existence of weak solution for boundary problems of systems of ferromagnetic chain . Science in China, 27A, 799–811 (1984) 23. Zhou, Y., Guo , B.: Weak solution systems of ferromagnetic chain with several variables. Science in China, 30A, 1251–1266 (1987) 24. Zhou, Y., Guo, B., Tan, S.: Existence and uniqueness of smooth solution of system of ferromagnetic chain. Science in China 34A, 257–266 (1991) 25. Zhou, Y., Sun, H., Guo, B.: Multidimensional system of ferromagnetic chain type. Science in China 36A, 1422–1434 (1993) Communicated by P. Constantin
Commun. Math. Phys. 250, 119–131 (2004) Digital Object Identifier (DOI) 10.1007/s00220-004-1121-8
Communications in
Mathematical Physics
Singularity Dominated Strong Fluctuations for Some Random Matrix Averages P. J. Forrester1 , J. P. Keating2 1 2
Department of Mathematics and Statistics, University of Melbourne, Victoria 3010, Australia School of Mathematics, University of Bristol, Bristol, BS8 1TW, UK
Received: 24 July 2003 / Accepted: 26 January 2004 Published online: 11 June 2004 – © Springer-Verlag 2004
Abstract: The circular and Jacobi ensembles of random matrices have their eigenvalue support on the unit circle of the complex plane and the interval (0, 1) of the real line respectively. The averaged value of the modulus of the corresponding characteristic polynomial raised to the power 2µ diverges, for 2µ ≤ −1, at points approaching the eigenvalue support. Using the theory of generalized hypergeometric functions based on Jack polynomials, the functional form of the leading asymptotic behaviour is established rigorously. In the circular ensemble case this confirms a conjecture of Berry and Keating.
1. Introduction Random matrices from the classical groups — N × N unitary matrices U (N ), N × N real orthogonal matrices O(N ), and N × N unitary matrices with real quaternion elements embedded as 2N × 2N complex unitary matrices Sp(N ) — play a special role in the application of random matrix theory to number theory (see e.g. the recent review [21]). Of particular interest in such applications are the random matrix averages of the modulus of the characteristic polynomial raised to some power 2µ say. Thus in [20] it was shown how knowledge of
| det(zI − U )|2µ
U ∈U (N)
(1.1)
for |z| = 1 (in this case (1.1) is in fact independent of z) allows the mean value of the 2µth power of the modulus of the Riemann zeta function on the critical line to be predicted. Knowledge of the analogue of (1.1) for the classical groups O(N ) and Sp(N ) allows for similar predictions in the case of families of L-functions [5, 19, 6].
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P.J. Forrester, J.P. Keating
Our interest is in the asymptotic behaviour of (1.1) and its analogues as |z| approaches unity. Consider in particular the generalization of (1.1) N
|z − eiθl |2µ
CβEN
l=1
,
(1.2)
where CβEN (circular β-ensemble) refers to the eigenvalue probability density function proportional to |eiθk − eiθj |β , −π ≤ θl < π. (1.3) 1≤j
The cases β = 1 and β = 4 are well known in random matrix theory as the COE (circular orthogonal ensemble) and CSE (circular symplectic ensemble) respectively, while the case β = 2 is just (1.1). For 2µ ≤ −1 it has been argued by Berry and Keating [3] that (1.2) diverges as |1 − |z||−δ where, provided N |µ|, δ as a function of µ is the non-smooth function, (k − 1) β β (k − 1) δ = int + 1 k − 1 + − int[ + 1] , k = 2|µ|, (1.4) β 2 2 β with int[·] denoting the integer part. It was also suggested that for k values such that (k − 1)/β is an integer, there is a logarithmic correction to the behaviour of (1.2) which then diverges as |1 − |z||−δ log |1 − |z||. The intricate behaviour exhibited by (1.4) is due to a phenomenon termed singularity dominated strong fluctuations. Thus for 2µ ≤ −1 and |z| close to unity, degeneracies of the spectrum of varying order are being probed [3]. A similar mechanism has been identified in the analysis of twinkling starlight [1], van Hove-type singularities [2], and the influence of classical periodic orbit bifurcations on quantum energy level [4] and wavefunction statistics [18]. The arguments presented in [3] leading to (1.4) were heuristic. Subsequently Fyodorov and Keating [11] gave a rigorous derivation for the exponent (1.4) in the case β = 1, 2µ ∈ Z<0 , for the Gaussian orthogonal ensemble analogue of (1.2), | det(xI − X)|2µ . (1.5) X∈GOEN
Thus they proved that for N µ and k = 2|µ| ∈ Z>0 , 1 | det(iI − X)|2µ ∼ CN,k −k(k−1)/2 log X∈GOEN →0
(1.6)
(the explicit evaluation of CN,k was also given). In this paper we will give a rigorous derivation of the exponent (1.4), together with the logarithmic corrections when present, for the average (1.2) in the case of general β > 0 and general 2µ ≤ −1. Our analysis relies on identifying (1.2) as a special generalized hypergeometric function based on Jack polynomials [25, 17]. We are then able to use known asymptotic properties of the latter to deduce the sought asymptotic behaviour of (1.2). Also studied will be the x → 1+ asymptotic behaviour of N l=1
|x − xl |2µ
JβEN
,
(1.7)
Singularity Dominated Strong Fluctuations for Random Matrix Averages
121
where JβEN (Jacobi β-ensemble) refers to the eigenvalue probability density function proportional to N
xla (1 − xl )b
|xk − xj |β ,
0 < xl < 1.
(1.8)
1≤j
l=1
In the special case β = 2, (1.7) includes the analogue of (1.1) in the cases of the classical groups O(2N +1), O(2N ) and Sp(N ). Again our analysis proceeds via identifying (1.7) as a generalized hypergeometric function based on Jack polynomials. As a final issue, it is pointed out in [12] that for fixed |z| = 1 and N → ∞, Szeg¨o’s theorem [24] on the asymptotic behaviour of Toeplitz determinants with smooth generating functions implies (1.1) is a simple Gaussian in µ, 2 2 lim | det(zI − U )|2µ = e−µ log |1−|z| | , |z| < 1. (1.9) U ∈U (N)
N→∞
This gives for the exponent characterizing the divergence as |z| → 1− , δ = µ2 , and so does not reproduce (1.4). Therefore the limit N → ∞ with |z| = 1 fixed has the effect of smoothing out the singularity dominated strong fluctuations. Note that in this limit the point z is a macroscopic distance away from the eigenvalue support, as measured in units of the inter-eigenvalue spacing. A theorem of Johansson [13, 15] will be used to establish the analogue of (1.9) for the average (1.2). 2. Generalized Hypergeometric Functions Based on Jack Polynomials 2.1. Definitions. The Gauss hypergeometric function 2 F1 is, for |x| < 1, defined by the series 2 F1 (a, b; c; x)
:=
∞ (a)k (b)k k=0
(c)k k!
xk .
(2.1)
A multivariable generalization of (2.1) can be defined in which the sum over the non-negative integers k is replaced by a sum over partitions κ := (κ1 , . . . , κN ), κi ≥ κj (i < j ) and κi ∈ Z ≥ 0 ; the function (u)k := u(u + 1) · · · (u + k − 1) =
(u + k) (u)
is replaced by [u](α) κ := N
N (u − (j − 1)/α + κj ) ; (u − (j − 1)/α)
j =1
k! is replaced by |κ|! := (
k j =1 κj )!; and x is replaced by the homogeneous symmet(α) Cκ (x1 , . . . , xN ) which in turn is proportional to the Jack polynomial
ric polynomial (α) Pκ (x1 , . . . , xN ). Thus for |x1 | < 1, . . . , |xN | < 1 one defines [25, 17] (α)
2 F1 (a, b; c; x1 , . . . , xN ) :=
(α) [a](α) κ [b]κ (α)
κ
[c]κ |κ|!
Cκ(α) (x1 , . . . , xN ).
(2.2)
122
P.J. Forrester, J.P. Keating (α)
Let us say a little on the definition of the Jack polynomials Pκ (x1 , . . . , xN ) =: (α) (α) and their renormalized version Cκ (x1 , . . . , xN ) =: Cκ (x). The former are the unique homogeneous polynomials of degree |κ| with the structure Pκ(α) (z) = mκ + aκµ mµ (2.3) (α) Pκ (x)
µ<κ
(the aκµ are some coefficients in Q(α)) and which satisfy the orthogonality Pκ(α) , Pρ(α) (α) ∝ δκ,ρ , where
f, g (α) :=
1/2 −1/2
dθ1 · · ·
×
1/2 −1/2
dθN f (z1 , . . . , zN )g(z1 , . . . , zN )
|zk − zj |2/α ,
zj := e2πiθj .
1≤j
In (2.3) mκ denotes the monomial symmetric function p p(polynomial) in x1 , . . . , xN corresponding to κ, and µ < κ means j =1 κj ≥ j =1 µj for each p = 1, . . . , N. (α)
Regarding the definition of Cκ (x1 , . . . , xN ), let dκ = α(a(i, j ) + 1) + l(i, j ) ,
(2.4)
(i,j )∈κ
where the notation (i, j ) ∈ κ refers to the diagram of κ, in which each part κi becomes the nodes (i, j ), 1 ≤ j ≤ κi on a square lattice labelled as is conventional for a matrix. The quantity a(i, j ) is the so called arm length (the number of nodes in row i to the right of column j ), while l(i, j ) is the leg length (number of nodes in column j below row i). In terms of dκ we have Cκ(α) (z) :=
α |κ| |κ|! (α) Pκ (z). dκ
(2.5)
2.2. Integration formulas. Introduce the Selberg integral
1
SN (λ1 , λ2 , λ) :=
1
dt1 · · ·
0
dtN 0
N
tlλ1 (1 − tl )λ2
|tk − tj |2λ
1≤j
l=1
and the Morris integral
MN (a, b, λ) :=
1/2 −1/2
×
dθ1 · · ·
|e
1/2 −1/2
2πiθk
dθN
−e
N
eπiθl (a−b) |1 + e2πiθl |a+b
l=1
|
2πiθj 2λ
1≤j
(both these integrals have gamma function evaluations — see e.g. [7] — although these will not be needed here). Fundamental to our ability to express (1.7) and (1.2) in terms
Singularity Dominated Strong Fluctuations for Random Matrix Averages
123
of the generalized hypergeometric function (2.2) are the generalized Selberg integral evaluation [22, 16, 17], 1 SN (λ1 , λ2 , 1/α) ×
1
1
dt1 · · ·
0
dtN 0
N
tlλ1 (1 − tl )λ2 Pκ(α) (t1 , . . . , tN )
l=1 (α)
|tk − tj |2/α
= Pκ(α) (1N )
1≤j
[λ1 + (N − 1)/α + 1]κ
(α)
[λ1 + λ2 + 2(N − 1)/α + 2]κ
, (2.6)
and the generalized Morris integral evaluation [9], 1 MN (a, b, 1/α) ×
N
1/2 −1/2
dθ1 · · ·
1/2 −1/2
dθN
eπ iθl (a−b) |1 + e2πiθl |a+b Pκ(α) (−e2πiθ1 , . . . , −e2πiθN )
l=1
×
(α)
|e2πiθk − e2πiθj |2/α = Pκ(α) (1N )
1≤j
[−b]κ
(α)
[1 + a + (N − 1)/α]κ
(2.7)
((2.6) and (2.7) are in fact equivalent [16]), where
Pκ(α) (1N ) := Pκ(α) (x1 , . . . , xN )
x1 =···=xN =1
.
Equally important is the generalized binomial summation formula [25, 23] (α) 1 F0 (a; x1 , . . . , xN )
:=
[a](α) κ κ
|κ|!
Cκ(α) (x1 , . . . , xN )
=
N
(1 − xj )−a .
(2.8)
j =1
Thus we see by multiplying both sides of (2.6) and (2.7) by (xα)|κ| [r]κ dκ
(α)
and summing over κ, using (2.8) on the left-hand sides and using the definition (2.2) on (α) the right-hand sides together with the fact that Cκ is homogeneous of degree |κ|, that [8] 1 SN (λ1 , λ2 , 1/α) =
(α) 2 F1 (r,
1 0
1
dt1 · · ·
dtN 0
N l=1
tlλ1 (1 − tl )λ2 (1 − xtl )−r
1≤j
|tk − tj |2/α
1 2
(N − 1) + λ1 + 1; (N − 1) + λ1 + λ2 + 2; t1 , . . . , tN ) t1 =···=tN =x α α (2.9)
124
P.J. Forrester, J.P. Keating
and
1/2
1/2 N 1 dθ1 · · · dθN eπiθl (a−b) |1 + e2πiθl |a+b (1 + xe2πiθl )−r MN (a, b, 1/α) −1/2 −1/2 l=1 2πiθk 2πiθj 2/α × |e −e | 1≤j
= 2 F1 (r, −b;
1
. (N − 1) + a + 1; t1 , . . . , tN ) t1 =···=tN =x α
(2.10)
3. Random Matrix Averages as Generalized Hypergeometric Functions (α)
Here we will express (1.2) and (1.7) in terms of 2 F1 . Consider first (1.2). By definition and simple manipulation N
|z − eiθl |2µ
l=1
=
CβEN
1/2
1/2 N 1 dθ1 · · · dθN |1 + ze2πiθl |2µ MN (0, 0, β/2) −1/2 −1/2 l=1 2πiθk 2πiθj β × |e −e | . 1≤j
Now
|1 + ze2πiθl |2µ = (1 + ze2πiθl )µ (1 + z¯ e−2πiθl )µ .
Regarding the integrals over θl ∈ [−1/2, 1/2] as parametrizing the unit circles wl = e2πiθl in the complex plane, then deforming each wl by wl → z¯ e2πiθl (this is immediate for 2µ, β/2 ∈ Z ≥ 0 by Cauchy’s theorem; it remains valid for general complex µ and β by Carlson’s theorem) we see that we can write N
|z − eiθl |2µ
l=1
CβEN
1/2
1/2 N 1 dθ1 · · · dθN |1 + |z|2 e2πiθl |µ (1 + e−2πiθl )µ MN (0, 0, β/2) −1/2 −1/2 l=1 2π iθk 2πiθj β × |e −e |
=
1≤j
=
1/2
1/2 N 1 dθ1 · · · dθN |1 + |z|2 e2πiθl |µ e−πiθl µ |1 + e−2πiθl |µ MN (0, 0, β/2) −1/2 −1/2 l=1 2πiθk 2πiθj β × |e −e | 1≤j
= 2 F1
(−µ, −µ;
β
(N − 1) + 1; t1 , . . . , tN ) , t1 =···=tN =|z|2 2
(3.1)
Singularity Dominated Strong Fluctuations for Random Matrix Averages
125
where the third equality follows from (2.10), and is valid for |z|2 ≤ 1. For |z|2 ≥ 1 one notes the simple symmetry N
|z − eiθl |2µ
l=1
= |z|2µN
CβEN
N 1 | − eiθl |2µ , CβEN z l=1
thus relating this case back to the case |z|2 ≤ 1. We remark that in the case β = 2 the same generalized hypergeometric function evaluation (3.1) has recently been given in [10]. Consider next (1.7). By definition and simple manipulation, for x real and greater than or equal to unity N l=1
=
|x − xl |2µ
JβEN
x 2µN SN (a, b, β/2)
1
1
dt1 · · ·
0
dtN 0
N
tla (1 − tl )b (1 − tl /x)2µ
|tk − tj |β
1≤j
l=1
β (2/β) = x 2µN 2 F1 (−2µ, (N − 1) + a; β(N − 1) 2
+a + b + 2; t1 , . . . , tN ) ,
(3.2)
t1 =···=tN =1/x
where the second equality follows from (2.9). The task now is to analyze the generalized hypergeometric functions in (3.1) and (3.2) as |z| → 1 and x → 1+ respectively. Fortunately, as will be presented in the next section, the required asymptotic formulas are available in the literature. 4. Asymptotic Forms Using the series definition (2.2), Yan [25] has analyzed the behaviour of the function (α) as x1 , . . . , xN → 1− . Thus according to Proposition 4.4 of [25] the following 2 F1 result holds, where the notation A(x) ≈ B(x) means that there exist two positive numbers C1 and C2 independent of x such that C1 ≤ Proposition 1. Let
A(x) ≤ C2 . B(x)
γ := a + b − c,
and suppose for all k (α)
(α)
[a]κ [b]κ (α)
> 0.
[c]κ We have for −1 < xi < 1 (i = 1, . . . , N ) (i) If γ > (N − 1)β/2, then (α)
2 F1 (a, b; c; x1 , . . . , xN ) ≈
N i=1
(1 − xi )−γ .
(4.3)
126
P.J. Forrester, J.P. Keating
(ii) If γ < −(N − 1)β/2, then there exists a constant C such that (α) 2 F1 (a, b; c; x1 , . . . , xN )
≤ C.
(4.4)
(iii) If γ = β(− N−1 2 + j − 1), j = 1, . . . , N, then for x1 = · · · = xN = t,
(α) 2 F1 (a, b; c; x1 , . . . , xN )
x1 =···=xN =t
≈ (1 − t)−(j −1)jβ/2 log
1 . 1−t
(4.5)
N−1 (iv) If β(− N−1 2 + j − 1) < γ < β(− 2 + j ), j = 1, . . . , N − 1, then for x1 = · · · = xN = t,
(α) F (a, b; c; x , . . . , x ) ≈ (1 − t)−j (γ +(N−j )β/2) . (4.6)
1 N 2 1 x1 =···=xN =t (α)
Application of Proposition 1 to the 2 F1 evaluations (3.1) and (3.2) gives the desired functional form of the singular behaviour of (1.2) and (1.7) in the case of a negative exponent 2µ. Corollary 1. Consider the circular ensemble average (1.2). Suppose µ < 0 and |z| < 1. For 2|µ| = β(j − 1) + 1 (j = 1, . . . , N ) we have N
|z − eiθl |2µ
l=1
CβEN
≈ (1 − |z|)−(j −1)jβ/2 log
1 1 − |z|
(4.7)
while for β(j − 1) + 1 < 2|µ| < βj + 1
(j = 1, . . . , N ),
and thus j = int[(2|µ| − 1)/β + 1],
(2|µ| − 1)/β + 1 ∈ / Z,
we have N
|z − eiθl |2µ
l=1
CβEN
≈ (1 − |z|)−j (2|µ|−1−(j −1)β/2) .
(4.8)
Consider the Jacobi ensemble average (1.7). Suppose µ < 0 and x > 1. For 2|µ| = β(j − 1) + 2 + b, we have x −2µN
N l=1
|x − xl |2µ
JβEN
≈ (1 − 1/x)−(j −1)jβ/2 log
1 1 − 1/x
while for β(j − 1) + b + 2 < 2|µ| < βj + b + 2
(j = 1, . . . , N ),
(4.9)
Singularity Dominated Strong Fluctuations for Random Matrix Averages
127
and thus j = int[(2|µ| − b − 2)/β + 1],
(2|µ| − b − 2)/β + 1 ∈ / Z,
we have x −2µN
N l=1
|x − xl |2µ
JβEN
≈ (1 − 1/x)−j (2|µ|−b−2−(j −1)β/2) .
(4.10)
The asymptotic behaviour (4.8) indeed exhibits the exponent (1.4) predicted by Berry and Keating [3], and the existence of a logarithmic correction in the cases that (2|µ| − 1)/β is an integer is confirmed. Finally, let us apply the results (4.9) and (4.10) to deduce the asymptotic z → 1+ behaviours of | det(zI − U )|2µ , µ<0 U ∈G
for G the classical groups Sp(N ), O + (2N ) and O − (2N ) (we leave the cases of O + (2N − 1) and O − (2N − 1) as an exercise for the interested reader). Consider first Sp(N ). The eigenvalues come in complex conjugate pairs e±iθj (j = 1, . . . , N ) so we have N 2 z +1
det(zI − U ) = (2z)N
2z
j =1
− cos θj .
(4.11)
Furthermore, we know (see e.g. [7]) that in the variable cos θj =: xj the eigenvalue probability density function is of the form (1.8) with a = b = 1/2, β = 2. Thus setting z = 1 + , 0 < 1 and noting (z2 + 1)/2z ∼ 1 + 2 /2 it follows from (4.9) that for 2|µ| = 2j + 1/2
(j = 1, . . . , N )
we have | det((1 + )I − U )|2µ
U ∈Sp(N)
while for we have
≈ −2(j −1)j log
1
(4.12)
2j + 1/2 < 2|µ| < 2(j + 1) + 1/2
| det((1 + )I − U )|2µ
U ∈Sp(N)
≈ −2j (2|µ|−3/2−j ) .
(4.13)
For the classical group O + (2N ) the eigenvalues again come in complex conjugate pairs e±iθj (j = 1, . . . , N ), so (4.11) remains valid. In the variable cos θj =: xj the eigenvalue probability density function is proportional to (1.8) with a = b = −1/2, β = 2 (see e.g. [7]). Thus setting z = 1 + , 0 < 1 it follows from (4.9) that for 2|µ| = 2j − 1/2
(j = 1, . . . , N )
we have | det((1 + )I − U )|2µ
U ∈O + (2N)
≈ −2(j −1)j log
1
(4.14)
128
P.J. Forrester, J.P. Keating
while for we have
2j − 1/2 < 2|µ| < 2(j + 1) − 1/2
| det((1 + )I − U )|2µ
U ∈O + (2N)
≈ −2j (2|µ|−1/2−j ) .
(4.15)
In the case of the classical group O − (2N ), there is a pair of fixed eigenvalues at ±1, with the remaining eigenvalues coming in complex conjugate pairs e±iθj (j = 1, . . . , N − 1). Hence in this case det(zI − U ) = (z − 1)(2z) 2
N−1
N−1 z2
+1 − cos θj . 2z
j =1
(4.16)
In the variable cos θj =: xj the eigenvalue probability density function is proportional to (1.8) with N → N − 1 and a = 1/2, b = −1/2, β = 2. As only the value of b enters in (4.9) and (4.10), the sole modification of (4.14) and (4.15) is multiplication by to account for the factor of z2 − 1 in (4.16). Thus for 2|µ| = 2j − 1/2
(j = 1, . . . , N − 1)
we have | det((1 + )I − U )|2µ
U ∈O − (2N)
≈ −2(j −1)j +1 log
1
(4.17)
while for 2j − 1/2 < 2|µ| < 2(j + 1) − 1/2 we have
| det((1 + )I − U )|2µ
U ∈O − (2N)
(j = 1, . . . , N − 1)
≈ −2j (2|µ|−1/2−j )+1 .
(4.18)
5. The Macroscopic Limit Consider the general Toeplitz determinant 1 2π DN [ea(θ) ] := det ea(θ) ei(j −k)θ dθ . j,k=1,...,N 2π 0
(5.1)
Such determinants are related to averages over U (N ) by the simple to establish formula DN [ea(θ) ] =
N l=1
With a(θ ) =
∞
p=−∞ ap e
ipθ
ea(θl )
U (N)
.
(5.2)
and the Fourier coefficients falling off fast enough that ∞ p = −∞
|p||ap |2 < ∞,
(5.3)
Singularity Dominated Strong Fluctuations for Random Matrix Averages
129
an asymptotic formula of Szeg¨o [24] gives ∞ pap a−p + o(1) . DN [ea(θ) ] ∼ exp N a0 + N→∞
(5.4)
p=1
Now, in the case a(θ ) = 2µ log |z − eiθ |,
|z| < 1,
(5.5)
the Fourier coefficients have the explicit form ap = −
µ¯zp (p > 0), p
ap =
µz−p (p < 0), p
a0 = 0.
(5.6)
Since we are assuming |z| < 1 these coefficients satisfy (5.3). Substituting (5.6) in (5.4) gives (1.9). In this section, as our final issue, we will make use of a generalization of (5.4) due to Johansson to establish the generalization of (1.9) when the average over U (N ) is replaced by an average over CβEN . But before doing so, following [12], we make note of the interpretation of (1.9) as specifying the distribution of the linear statistic A(z) =
N
log |z − eiθj |2 ,
|z| < 1.
(5.7)
j =1
Thus let Pz (t) denote the probability density that A(z) takes on the value t after averaging over the eigenvalue distribution of U (N ). Then we have Pz (t) = δ(t − A(z)) (5.8) U (N)
and consequently P˜z (k) :=
∞ −∞
Pz (t)eitk dt =
N
e2ik log |z−e
l=1
iθl |
U (N)
.
(5.9)
Thus P˜z (k) is precisely (5.2) with the substitution (5.5), and µ therein set equal to ik. Consequently lim Pz (t) = e−k
N→∞
2 σ 2 /2
,
σ 2 = − log |1 − |z|2 |2 ,
(5.10)
telling us that the limiting distribution of (5.7) is a Gaussian with an O(1) variance taking the explicit value − log |1 − |z|2 |2 . To study (5.8) with the average over U (N ) replaced by the probability density function CβEN , we make note of the following generalization of the Szeg¨o theorem (5.4) due to Johansson [13, 15]. ipθ and the Fourier coefficients fall off fast Proposition 2. Suppose a(θ ) = ∞ p=−∞ ap e enough that the inequality (5.3) holds. We have N l=1
e
a(θl )
CβEN
∞ 2 ∼ exp N a0 + pap a−p + o(1) . N→∞ β p=1
(5.11)
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P.J. Forrester, J.P. Keating
As an immediate corollary, by substituting (5.6) for the Fourier coefficients the sought generalization of (1.9) can be deduced. Corollary 2. For |z| < 1 we have lim
N→∞
N l=1
|z − eiθl |2µ
CβEN
= e−(2µ
2 /β) log |1−|z|2 |
.
(5.12)
Of course the result (5.12) affords an interpretation as a fluctuation formula as in (5.10), the only difference being that σ 2 = −(2/β) log |1 − |z|2 |. An obvious question is to seek the analogue of (5.12) for the average (1.7) with x∈ / [0, 1]. Johansson [14] has derived the analogue of (5.11), but only in the case β = 2 and further with the restriction that a(θ ) is a polynomial in θ , which is not the case for (5.5). Acknowledgements. The work of PJF was supported by the Australian Research Council. We thank Nalini Joshi for inviting us both to lecture at Sydney University during July 2003 thus facilitating the present collaboration.
References 1. Berry, M.V.: Focusing and twinkling: critical exponents from catastrophes in non-Gaussian random short waves. J. Phys. A 10, 2061–2081 (1977) 2. Berry, M.V.: Universal power-law tails for singularity-dominated strong fluctuations. J. Phys. A 15, 2735–2749 (1977) 3. Berry, M.V., Keating, J.P.: Clusters of near degenerate levels dominate negative moments of spectral determinants. J. Phys. A 35, L1–L6 (2002) 4. Berry, M.V., Keating, J.P., Schomerus, H.: Universal twinkling exponents for spectral fluctuations associated with mixed chaology. Proc. R. Soc. A 456, 1659–1668 (2000) 5. Conrey, J.B., Farmer, D.W.: Mean values of L-functions and symmetry. Int. Math. Res. Notices 17, 883–908 (2000) 6. Conrey, J.B., Farmer, D.W., Keating, J.P., Rubinstein, M.O., Snaith, N.C.: Integral moments of zetaand L-functions. Submitted to Proc. of the London Math.Soc. 7. Forrester, P.J.: Log-gases and Random Matrices. http:// www.ms.unimelb.edu.au/ ˜matpjf/matpjf.html, 2003 8. Forrester, P.J.: Addendum to Selberg correlation integrals and the 1/r 2 quantum many body system. Nucl. Phys. B 416, 377–385 (1994) 9. Forrester, P.J.: Integration formulas and exact calculations in the Calogero-Sutherland model. Mod. Phys. Lett B 9, 359–371 (1995) 10. Forrester, P.J., Witte, N.S.: Discrete Painlev´e equations, orthogonal polynomials on the unit circle and n-recurrences for averages over U (N) – PVI τ -functions. http:// arXiv.org/abs/math-ph/0308036, 2003 11. Fyodorov, Y.V., Keating, J.P.: Negative moments of characteristic polynomials of random GOE matrices and singularity dominated strong fluctuations. J. Phys. A 36, 4035–4046 (2003) 12. Hughes, C.P., Keating, J.P., O’Connell, N.: On the characteristic polynomial of a random unitary matrix. Commun. Math. Phys. 220, 429–451 (2001) 13. Johansson, K.: On Szeg¨o’s formula for Toeplitz determinants and generalizations. Bull. Sc. Math., 2e s´erie 112, 257–304 (1988) 14. Johansson, K.: On random matrices from the compact classical groups. Ann. of Math. 145, 519–545 (1997) 15. Johansson, K.: On fluctuation of eigenvalues of random Hermitian matrices. Duke Math. J. 91, 151–204 (1998) 16. Kadell, K.W.J.: The Selberg-Jack symmetric functions. Adv. Math. 130, 33–102 (1997) 17. Kaneko, J.: Selberg integrals and hypergeometric functions associated with Jack polynomials. SIAM J. Math Anal. 24, 1086–1110 (1993) 18. Keating, J.P., Prado, S.D.: Orbit bifurcations and the scarring of wave functions. Proc. R. Soc. A 457, 1855–1872 (2001)
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19. Keating, J.P., Snaith, N.C.: Random matrix theory and L-functions at s = 1/2. Commun. Math. Phys. 214, 91–110 (2000) 20. Keating, J.P., Snaith, N.C.: Random matrix theory and ζ (1/2 + it). Commun. Math. Phys. 214, 57–89 (2001) 21. Keating, J.P., Snaith, N.C.: Random matrices and L-functions. J. Phys. A 36, 2859–2881 (2003) 22. Macdonald, I.G.: Commuting differential operators and zonal spherical functions. In A.M. Cohen et al., editor, Algebraic Groups, Utrecht 1986, Vol. 1271 of Lecture Notes in Math., Heidelberg: Springer-Verlag, 1987, pp. 189–200 23. Macdonald, I.G.: Hall polynomials and symmetric functions. 2nd edition, Oxford: Oxford University Press, 1995 24. Szeg¨o, G.: On certain Hermitian forms associated with the Fourier series of a positive function. Comm. Seminaire Math. de l’Univ. de Lund, tome suppl´ementaire, d´edi´e a` Marcel Riesz, 1952, pp. 228–237 25. Yan, Z.: A class of generalized hypergeometric functions in several variables. Can J. Math. 44, 1317–1338 (1992) Communicated by P. Sarnak
Commun. Math. Phys. 250, 133–160 (2004) Digital Object Identifier (DOI) 10.1007/s00220-004-1123-6
Communications in
Mathematical Physics
Long Term Stability of Proper Rotations of the Perturbed Euler Rigid Body Giancarlo Benettin1,2 , Francesco Fass`o1 , Massimiliano Guzzo1 1
Universit`a di Padova, Dipartimento di Matematica Pura e Applicata, and GNFM (INDAM), Via G. Belzoni 7, 35131 Padova, Italy. E-mail:
[email protected];
[email protected];
[email protected] 2 INFM, Universit`a di Padova, 35131, Padova, Italy Received: 3 August 2003 / Accepted: 2 February 2004 Published online: 18 June 2004 – © Springer-Verlag 2004
Abstract: We study the long term stability of the proper rotations of the perturbed Euler rigid body, in the framework of Nekhoroshev theory. For simplicity we treat here in detail only the kinetically symmetric case (the potential needs not to be symmetric), but we indicate how to extend the results to the triaxial case. We show that the proper rotations around the symmetry axis are Nekhoroshev stable: more precisely, if the initial datum is sufficiently close to a proper rotation, then for a very long time it remains such, and the tip of the unit vector µ parallel to the angular momentum precesses, up to a small noise, along the level curves of a regular function on the unit sphere. If the proper rotations are resonant, chaotic motions with positive Lyapunov exponents are possible, but chaos (unlike the case of ordinary motions, that is motions not close to proper rotations) is always localized, and does not affect in an essential way the motion of the angular momentum in space. Preliminary numerical results indicate that the theory is, in many aspects, optimal, although in some points it can still be improved. 1. Introduction and Statement of the Result 1.1. The purpose of the paper. In this paper we are concerned with a classical problem, namely the stability of the proper rotations (rotations about a principal axis of inertia) of the perturbed Euler–Poinsot rigid body (the rigid body with a fixed point in a small positional potential), and prove a result of long term stability in the sense of Nekhoroshev [24]. The Hamiltonian has the form H = K + εV ,
(1.1)
where K is the kinetic energy; the potential V is assumed to depend only on the configuration of the body and to be regular, in fact analytic. The problem is equivalent1 to 1 Precisely, the motions of Hamiltonian (1.1), for given initial data and small ε, are identical, up to a rescaling, to the motions of the same Hamiltonian with ε = 1 and angular velocity rescaled by a factor ε −1/2 .
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G. Benettin, F. Fass`o, M. Guzzo
the problem of the fast rotating rigid body, with angular velocity ε −1/2 , in a (non-small) potential V. We consider for simplicity a kinetically symmetric body, with moments of inertia I1 = I2 = I3 = 0 (the potential needs not to be symmetric) and show that, for small ε, the proper rotations around the symmetry axis are stable for a time scale O(exp 1/ε a ), a > 0. Stable means here that: (i) Motions initially close to proper rotations remain close to proper rotations, the modulus of the angular momentum staying almost constant. (ii) The unit vector µ parallel to the angular momentum precesses in space, up to a small noise, along the level lines of a (known) regular function on the unit sphere. As indicated in Sect. 4, such a stability result extends to the triaxial case, too (of course concerning proper rotations about the axes of inertia with minimal or maximal moment of inertia). This paper is based on [4] and [5], where the perturbed Euler rigid body was studied in the framework of Nekhoroshev theory respectively for ordinary motions and for motions close to proper rotations about the symmetry axis; [5], in particular, provides some partial results which are the starting point of this paper, see below Proposition 0. Familiarity with [4] and [5], however, is useful but not necessary. 1.2. The framework. Let (O, ex , ey , ez ) be a frame fixed in space, with origin in the fixed point, and (O, e1 , e2 , e3 ) be a body frame with the same origin, e3 being the symmetry axis. The configuration of the body is represented by a matrix R ∈ SO(3), say the matrix such that e1 = Rex and so on. Let m = (m1 , m2 , m3 ) and ms = (mx , my , mz ) = Rm denote the angular momentum of the body respectively in the proper and in the spatial frame. The phase space of the system T ∗ SO(3) is known to be diffeomorphic to R3 × SO(3), and a dynamical state of the body is identified by a pair (m, R). As is well known, the unperturbed system with Hamiltonian K is “properly degenerate” (or “superintegrable”), namely it has three degrees of freedom but four constants of motion: the three components of ms and either m3 or equivalently K. Correspondingly the phase space is foliated in low dimensional invariant tori T2 , on which motions are quasi–periodic with only two independent frequencies. (See [14] and references therein for comments on the geometric structure of the phase space.) This is evident if one recalls the Euler–Poinsot description of the unperturbed motions, according to which ms stays fixed in space, while e3 regularly precesses about it with a frequency 1 depending on G = m and L = m3 , and at the same time the body rotates about e3 , or equivalently m precesses in the body about e3 , with a second independent frequency 2 (G, L). The kinetic energy depends only on G and L, K(G, L) =
1 2 β 2 G + L , 2I1 2I3
β=
I 1 − I3 , I1
K K and 1 = ∂∂G , 2 = ∂∂L . Since 0 < I3 ≤ 2I1 , it is −1 ≤ β < 1. The value β = −1 corresponds to a flat body, and will require a special treatment. Besides β = 1, i.e. I3 = 0, we also exclude the spherically symmetric case β = 0; quantitatively we assume a condition of the form
|β|(1 − β) ≥ β∗ > 0 .
(1.2)
Understanding the perturbed system means understanding what happens, for small ε = 0, to the constants of motion, in particular how m and ms do move respectively in the body and in space.
Stability of Proper Rotations
135
For motions which are not close to proper rotations, more precisely for m⊥ ≥ const ε1/4 ,
def
m2⊥ = m2 − m23 ,
the situation was clarified in [4]. Assuming V to be analytic, it is there proved that: (a) For all initial data, up to a time scale |t| ≤ const e(1/ε)
1/4
,
(1.3)
in the worst conditions (resonance between 1 and 2 ) it is |G(t) − G(0)| , |L(t) − L(0)| ≤ const ε 1/2 and the body performs an almost regular Euler–Poinsot precession, with almost constant 1 and 2 , about the instantaneous direction of the angular momentum ms , which in turn moves slowly with speed proportional to ε. So, for all initial data the behavior of m in the body is regular up to a small noise. (b) If the initial datum is nonresonant, more precisely if |ν · | ≥ const ε 1/4
∀ν ∈ Z2 :
0 < |ν| ≤ const ε−1/4 ,
where = (1 , 2 ) and as usual |ν| = |ν1 | + |ν2 | , then for the same time scale (1.3) the unit vector µ parallel to ms precesses slowly, up to a small noise, along the level lines of a regular function F : S 2 → R (F is constructed and turns out to be close to a suitable average of the potential V). So, in the absence of resonances, ms moves regularly in space, too, and the only effect of the perturbation is to remove the degeneracy by adding besides 1 and 2 a third small frequency. The theory provided in [4] does not say anything on the long term behavior of µ for resonant initial conditions, but there is some theoretical evidence (based on the theory of adiabatic invariants [12, 21, 22, 11, 23]), also supported by numerical computations [3], that (for no matter how small ε) chaotic motions, in which µ fills unpredictably a wide region of S 2 in times larger than 1/ε but much shorter than (1.3), do exist. 1.3. The main result. In this paper (going beyond some partial results already proved in [5]) we prove that motions of the perturbed Euler rigid body which are close to proper rotations are substantially more regular than all other motions: even in presence of a resonance, the possibility of large chaotic movements of the angular momentum vector in space, discussed at the end of the previous subsection, is inhibited. In particular, the proper rotations about e3 are long term stable. The stability properties turn out to depend on the presence of resonances, and these in turn depend on the arithmetic properties of β. Following [5] we say that β is resonant, for a given ε and with reference to some constant ε0 , if there exists ν ∈ Z2 , 0 < |ν| ≤ N ,
def
N = (ε0 /ε)1/4 ,
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G. Benettin, F. Fass`o, M. Guzzo
such that
ν1 1 ; β − < ν2 2N|ν|
(1.4)
β is instead said to be nonresonant if (1.4) is not satisfied by any ν ∈ Z2 , 0 < |ν| ≤ N . One easily checks that (1.4) is satisfied by at most one independent ν with |ν| ≤ N . According to (1.2) it is 1 ≤ |ν1 | ≤ |ν2 | , (1.5) so the lowest order resonance, characteristic of flat bodies, is ν = (−1, 1), while for all other resonances it is |ν2 | ≥ 2; for all of them, as will be crucial, ν2 = 0. Proposition 1. Let V be analytic, assume G(0) ≥ G∗ > 0 and m⊥ (0) < M∗ ε 1/4 with some M∗ > 0; let β satisfy (1.2). There exist positive constants ε0 , ε∗ , A, B, B⊥ , C, a, b, b⊥ , c and a regular function F : S 2 → R (constructed in the proof, and close to the average of the potential V along the proper rotations), such that for the time scale ε a ∗ |t| ≤ A exp (1.6) ε it is
m(t) − m(0) < B εb ,
(1.7)
|m⊥ (t) − m⊥ (0)| < B⊥ ε b⊥ , |F(µ(t)) − F(µ(0))| < C ε c ,
(1.8) (1.9)
where as before µ = ms /m. Possible choices of the stability exponents are as follows: (i) In the nonresonant case, a = 41 , b = 43 , b⊥ = 21 , c = 1. (ii) In the resonant case, for all resonances ν = (−1, 1), 0
1 , 4
b=
1 , 2
b⊥ =
1 (1 − a) , 2
c=
1 −a 2
c=
1 (1 − 4a) . 3
(1.10)
(c improves up to c = 1 − a, if β gets closer to ν1 /ν2 ). (iii) For the resonance ν = (−1, 1), 0
1 , 4
b=
1 , 2
b⊥ =
1 (1 − a) , 3
The nonresonant case is included here for completeness, but in fact is already proven in [5], see Sect. 3.1A. The notion of “average of the potential V along the proper rotations” is made precise in Sect. 2.1. As far as we know, this is the first proof of long term stability of proper rotations of the perturbed Euler rigid body in a non-symmetric potential. Such a result can be considered as a rigorous version, extended to a long time scale, of a known heuristic procedure, see for example [17, 28], in which one assumes a priori that m and e3 stay parallel, and replaces the instantaneous torque by its average on a rotation, so deducing an approximate equation for m; ˙ see for example [25–27]. Clearly, due to the possible presence of chaotic motion, the approximate equation turns out to be valid, at most, for
Stability of Proper Rotations
137
short times. In fact, an essential point of our approach is that we include chaotic resonant motions, where averaging is not allowed (see Sect. 3.1C,D), and in spite of the presence of chaos it turns out that there exists an approximate constant of motion like F, which persists for exponentially long times. Proposition 1 applies to any potential V, provided it is analytic as a function of the configuration of the body (say analytic as a function of the entries of the matrix R ∈ SO(3) determining the orientation of the body in space); this is the case, in particular, of the gravitational potential. Some comments on the relevance of this paper for Celestial Mechanics are deferred to the Conclusions. Remark. In presence of resonances, motions near proper rotations can be chaotic (the maximal Lyapunov exponent χ can be positive, χ ∼ ε, and is actually found to be positive in numerical computations [6]), but chaos is localized. In particular, at variance with the case of ordinary motions far from proper rotations, even in presence of resonances µ remains almost confined to a (predictable) curve on S 2 . The technical element which is responsible for such extra stability in the vicinity of proper rotations is the presence of a second small quantity, besides ε, in the problem, namely the distance of the initial datum from the proper rotation. Such a smallness was not used in [5], where the stability of resonant motions was missed. Its use in our perturbation construction was suggested in [3].
1.4. Comments. For a better understanding of Proposition 1, it is convenient to comment here on the notion of resonance in the vicinity of proper rotations. For motions close to proper rotations it is natural to introduce, in place of L, the small quantity =G−L, related to m⊥ by
m2⊥ = (2G − ) ,
(1.11)
and correspondingly to replace = (1 , 2 ) by the linear combination ω = (1 + 2 , −2 ). Denoting K(G, ) = K(G, G − ), one easily checks that 1 (G2 + β2 − 2βG) 2I3
(1.12)
∂K ∂K 1 , = (G − β, −β(G − )) . ∂G ∂ I3
(1.13)
K(G, ) = and ω(G, ) = On the proper rotation it is
G (1, −β) , (1.14) I3 and the two components of ω are easily identified as the frequency of the proper rotation itself and the frequency of the small oscillations of m about e3 . If (1.4) holds, then ν · ω is small in the vicinity of the proper rotation (for β = ν1 /ν2 it is ν · ω(G, 0) = 0, i.e. the proper rotation is exactly resonant). This explains the choice of (1.4) as definition of resonance. Accurate numerical results suggest that the exponents b⊥ and c as estimated in this theory are optimal, with the exception of the special resonance ν = (−1, 1) for which ω(G, 0) =
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G. Benettin, F. Fass`o, M. Guzzo
Fig. 1. The motion of m in the body (Poincar´e section mz = 0) and of µ on S 2 (projection). Parameters: (c) (c) ε = 6.4 × 10−4 , β = 0.504, I1 = 1. Initial data: G = 1, γ = 0, mz = 0, j = 0.4π ; m1 , m2 (“xcorot” and “ycorot” in the plot) graphically chosen in the section
Fig. 2. χ (x, ξ, t) as function of t, for some tangent vector ξ and three initial data x in the chaotic region, two initial data respectively in the inner and in the outer regular region. Parameters and data as in Fig. 1
Stability of Proper Rotations
139
stability seems to be a little stronger than what is predicted here (see Sect. 3.1.D for a comment on this point). The stability properties can be made evident numerically by following the motion of µ on the unit sphere and of m in the body. To represent µ, we use on S 2 the cylindrical coordinates (mz /G, j ), j = arctan mx /my ; concerning m, it is convenient to introduce, for each resonance ν, the “corotating” vector m(c) = (c) (c) (m1 , m2 ) related to (m1 , m2 ) by cos ν1 γ − sin ν1 γ m(c) m1 ν2 ν2 1 , = (c) m2 sin νν21 γ cos νν21 γ m2 where γ is a suitable angle associated to the proper rotation, γ˙ ω1 (see the next section for the exact definition of γ ). Figure 1, left, shows a typical Poincar`e section of m(c) , corresponding to the section mz = 0; the right part of the figure represents instead the projection of µ on the unit sphere for the big chaotic orbit appearing in the section. The choice of the potential, as a function of the usual Euler angles ϑ, ϕ, ψ, is a simple one, namely V = const sin ϑ(sin ϑ sin ϕ sin ψ + cos ϕ + sin ϑ cos2 ψ) . The resonance is ν = (1, 2), with nearly resonant β = 0.504 and ε = 6.4 × 10−4 . Figure 2 shows the behavior in time of χˆ (x, ξ, t) =
Dtx ξ 1 log −→ χ (x) t ξ
for different initial data x and some tangent vector ξ (the choice of ξ as usual is irrelevant), t denoting the Hamiltonian flow. The three upper curves refer to initial data in the chaotic region, the two lower ones to initial data in the inner and in the outer regular regions. By varying ε, it turns out that both images in Fig. 1 scale exactly as theoretically predicted; in turn χ (in the chaotic region) decreases, as expected, proportionally to ε. Such figures illustrate the main content of Proposition 1: inside resonances, which are met for nearly rational β, motions are indeed chaotic, but chaos is localized and, up to a small noise, µ simply precesses along a line on the sphere. Such a regular behavior is substantially different from the case of resonant motions far from proper rotations, where chaotic motions of µ on S 2 are wide for no matter how small ε, and regularity is lost [4, 3]. A more complete numerical analysis of the problem will be published separately [6]. 2. Preliminaries for the Proof 2.1. Action–angle and Birkhoff coordinates. The action–angle coordinates (G, L, J, g, l, j ) of the symmetric Euler rigid body, known as Andoyer–Deprit coordinates [1, 10], are recalled in Fig. 3. Three of them are intrinsic (i.e. independent of the choice of the spatial frame), namely the already introduced quantities G = m and L = m3 entering the expression of K and the angle l, defined as the angle from the nodal line n = m × e3 to e1 in the plane orthogonal to e3 . The remaining coordinates make instead reference to the spatial frame, and are J = mz , j = angle from ex to the nodal line n = ez × m in the plane ex ey (J /G and j are the already used coordinates of µ on S 2 ) and g = angle from n to n in the plane orthogonal to the angular momentum. (The angle g depends on the choice of the spatial frame very marginally, for the choice of
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Fig. 3. The action–angle coordinates G, L, J, g, l, j for the symmetric Euler rigid body
the origin.) The domain of a chart of such coordinates excludes the intrinsic singularity e3 × m = 0 and the chart singularity ez × m = 0; two charts referring to spatial frames with different z–axis cover the subset M0 = (m, R) ∈ T ∗ SO(3) : e3 × m = 0 , that is the whole phase space but the proper rotations, providing an analytic symplectic atlas on it. In fact, the proper rotations are singular leaves of the foliation of T ∗ SO(3) in tori T2 : a family of invariant tori there shrinks to a circle, see Fig. 4. Both g and l cease to be defined on the proper rotation, but their sum γ = g + l is, there coinciding with the familiar Euler angle ψ = angle from n = ez × e3 to e1 . “Less singular” canonical action–angle coordinates, more adapted to work around the proper rotation, are (G, γ , , λ, J, j ), where =G−L,
λ = −l
(see [5]). Indeed the singularity, now occurring at = 0, is still present, but it affects only the pair of conjugated variables (, λ), see Fig. 4. The singularity is removed [18] (see also [5]) by passing from polar–like to Cartesian coordinates √ √ x1 = 2 cos λ , x2 = 2 sin λ . In fact, as is proven in [18], both the rotation matrix R and the angular momentum m are analytic functions of (G, γ , x1 , x2 , J, j ) also for x1 = x2 = 0; conversely, x1 , x2 are analytic functions of m, precisely √ √ 2m2 2m1 , x2 = − √ . x1 = √ m + m3 m + m3 Two charts of coordinates (G, γ , x1 , x2 , J, j ), relative to different spatial frames (for example, frames obtained by a cyclic permutation of the axes), cover the subset M+ = (m, R) ∈ T ∗ SO(3) : m3 > 0 . In each chart it is possible to assure, for example, J /G < 3/4, as we shall later assume.
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For technical reasons, in the proof of Proposition 1 it is convenient to use both the adapted action–angle coordinates (G, γ , , λ, J, j ), outside a small neighborhood of the proper rotation so as to avoid the singularity, and the Cartesian coordinates (G, γ , x1 , x2 , J, j ) inside that neighborhood. In the practical work, the real Cartesian coordinates x1 , x2 are replaced as usual by their complex linear combinations (Birkhoff coordinates) x1 − ix2 x1 + ix2 z1 = , z2 = √ √ i 2 2 and it is (z1 , z2 ) = iz1 z2 . Concerning the Hamiltonian, in the adapted action–angle variables it has the form H (G, γ , , λ, J, j ) = K(G, ) + εV (G, γ , , λ, J, j )
(2.1)
with K as in (1.12); in the Birkhoff coordinates it assumes instead the form h(G, γ , z1 , z2 , J, j ) = K(G, (z1 , z2 )) + εv(G, γ , z1 , z2 , J, j ) .
(2.2)
Analytic perturbation theory deals with real analytic functions, namely functions which are analytic and bounded in suitable complex neighborhoods of their real domain and are real for real variables. With reference to the Birkhoff coordinates, we shall call real any domain in which G, γ , J, j are real and x1 , x2 are real, i.e. z2 , iz1 are complex conjugate (correspondingly, we shall call real analytic any function which is analytic and bounded in some neighborhood of such a real domain, and is real in the real domain). As is well known, perturbation theory often makes use of Fourier series expansions. It is worthwhile to recall that in the Birkhoff coordinates the Fourier series of any function f (G, γ , z1 , z2 , J, j ) is written fν (G, γ , (z1 , z2 ), J, j )Eν (γ , z1 , z2 ) f (G, γ , z1 , z2 , J, j ) = ν∈Z2
with
Eν (γ , z1 , z2 ) =
eiν1 γ z1ν2 if ν2 ≥ 0 |ν |
eiν1 γ z2 2 if ν2 < 0 (see [5] for comments). In particular, v0 (G, γ , , J, j ) is the average of the potential v in the Birkhoff variables, and (out of the proper rotation, when both functions are defined) v0 (G, z1 , z2 , J, j ) = V0 (G, iz1 z2 , J, j ) , V0 denoting the usual average of V on γ , λ,
2π 2π V0 (G, , J, j ) = (2π)−2 V (G, γ , , λ, J, j ) dγ dλ . 0
0
The Fourier expansion allows us to define, in any set of variables, the “Fourier norm” fν ∞ , f = ν∈Z2
which is well known to be useful in perturbation constructions. (The Fourier norm . reduces to the sup–norm . ∞ for functions independent of the angles.) Whenever we say that a function is bounded in some domain, we refer to the Fourier norm. Functions with zero average are necessarily small for small , and this will play a crucial role in the proof of Proposition 1.
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Fig. 4. The change of coordinates from (G, L, g, l) to (G, , γ , λ)
Remark. As remarked above, the singularity m×e3 = 0 is removed by using, near proper rotations, the Cartesian–like variables x1 , x2 (or z1 , z2 ) in place of the polar ones (, λ). However, even restricting to a neighborhood of proper rotations (with m and e3 almost parallel, but possibly pointing in any direction of the space), we are still confronted with the chart singularities ez × m = 0, i.e. J = ±G. Our aim is to prove regularity of all motions, so our perturbation treatment must cover the whole sphere m = G, and this requires at least two sets of coordinates. A simple choice [4, 5] is to use two sets of coordinates (J, j ) relative to two spatial frames (ex , ey , ez ) and (ex , ey , ez ), the latter frame being a permutation of the former so as the two z–axes are orthogonal; in this way we can define the domain of each chart by requiring, for example, |J | < 43 G, and nevertheless cover the whole phase space we are interested in, actually cover M+ . If, as we assume in Proposition 1, the potential V is analytic as a function of R ∈ SO(3), then such charts provide an analytic atlas for M+ . Other choices of coordinates are possible: for example, for potentials for which the z-axis plays a special role, instead of changing the axis it is probably convenient to use, around the poles, Cartesian–like coordinates (defined similarly to x1 , x2 , the components mx , my , mz of ms replacing m1 , m2 , m3 ; see for example [16]). All choices are in principle equivalent. In general, however, it is not possible to guarantee a priori that motions stay inside a single chart,2 so the problem arises of how to develop perturbation theory, and discuss about long term stability, in presence of motions which possibly pass from one chart domain to the other. Such a question is raised and solved in [13]: provided some elementary care is taken (like keeping the same ultraviolet cut–off for all charts), the perturbation constructions in the different chart domains glue smoothly. In particular the local normal forms, as constructed in the different chart domains, are proven to be the local representatives of a global normal form defined in the union of the chart domains. So, in spite of the presence of chart singularities, local normal forms can be used to provide long term confinement. This idea, after [13], has been succesfully used in connection with the rigid body in [4] and [5]. Conceptually, the prescription is elementary: (i) make any choice of the charts (possibly adapted, if useful, to the potential at hand); write the perturbing potential in each chart, and determine the relevant constants describing it (its norm, its analyticity radii), taking the worst among charts. (ii) Construct the normal form with exponentially small remainder in each chart, and do not pose the question whether motions stay inside 2 This is evidently the case of the “wide chaotic motions” studied in [4] and [3], where µ = ms /m wanders unpredictably on the unit sphere. But even for regular motions as those found in this paper, requiring that a motion stays inside a single chart domain would require making a choice of the chart depending on the initial datum; this introduces annoying complications in the perturbation work, with no real advantage.
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the domain of a chart (as might happen with particular potentials, or for a good selection of initial data) or instead wander from one chart to the other: the question is indeed irrelevant; the worst of the local estimates will hold globally. For details and further comments we refer to [13], or to [4, 5]. In the rest of the paper, the problem of chart singularities will be ignored.
2.2. Previous results. Here we recall in a proposition some results proved in [5], which are the starting point of the proof of Proposition 1. We take the liberty to slightly reformulate some statements so as to adapt them to our needs, as well as to include in the proposition some useful estimates which are present only in the proofs, or can be trivially deduced from estimates contained in the proofs. (The main change is the passage from the framework of large angular velocity at ε = 1, which was chosen in [5], to the framework of bounded angular velocity and small ε, that we found convenient to use here.) The statement below concerns a Hamiltonian of the form (2.2), in a convenient complex neighborhood of a real domain D of the form G ≥ G∗ > 0 ,
(z1 , z2 ) ≤ ∗ ε 1/2 ,
|J /G| ≤ 3/4 ,
γ , j ∈ S1 ;
such a neighborhood, denoted Dρ , depends on an “extension vector” with positive entries 1/2 ρ = (ρG , ε1/2 ρz , ρj ), ρz ≤ ∗ , and is defined by |G|, |J | ≤ ρG ,
|z1 |, |z2 | ≤ ρz ε 1/4 ,
|γ |, |j | ≤ ρj ≤ 1 .
The notion of resonance entering the statement is the one introduced in Sect. 3.1; in the resonant case, to label a resonance, we use (here and in the following) ν = (ν1 , ν2 ) such that ν1 , ν2 have no common divisors and, on account of (1.5), ν2 > 0. Proposition 0. Consider the Hamiltonian (2.2) and assume v is real analytic in Dρ defined as above. Let ε be small. There exist ε0 , ε∗ > 0 such that, with reference to the notion (1.4) of resonance (where ε0 enters): (i) In the nonresonant case, there exists an ε3/4 –near to the identity canonical transformation (G, γ , z1 , z2 , J, j ) = C(G , γ , z1 , z2 , J , j ) ,
C : Dρ/2 → Dρ ,
(2.3)
such that the new Hamiltonian h = h ◦ C is in nonresonant normal form up to an exponentially small remainder, namely h (G , γ , z1 , z2 , J , j ) = K(G , ) + εu(G , , J , j ) +ε e−(˜ε/ε) R(G , γ , z1 , z2 , J , j ) , 1/4
where stands for (z1 , z2 ) = iz1 z2 and u, R are bounded by quantities independent of ε; u is ε1/4 –close to the average v0 of v.
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(ii) In the resonant case, there exists an ε 3/4 –near to the identity canonical transformation of the form (2.3), which gives the new Hamiltonian h the ν–resonant normal form h (G , γ , z1 , z2 , J , j ) = K(G , ) + ε u(G , , J , j ) + f (G , γ , z1 , z2 , J , j ) + ε e−(˜ε/ε) R(G , γ , z1 , z2 , J , j ) , 1/4
where f (G, z1 , z2 , J, j ) =
(2.4)
fn (G, , J, j )Enν (z1 , z2 , γ ) ;
(2.5)
0=n∈Z
u, f and R are bounded by quantities independent of ε, u is ε 1/4 –close to v0 , f is as small as max(|z1 |ν2 , |z2 |ν2 ). The normalized Hamiltonian (2.4) can be written in action–angle coordinates, giving H (G , γ , , λ , J , j ) = K(G , ) + ε U (G , γ , , J , j ) + F (G , γ , , λ , J , j ) +ε e−(˜ε/ε) R(G , γ , , λ , J , j ), 1/4
with
F (G, γ , , λ, J, j ) =
(2.6)
Fn (G, , J, j ) ein(ν1 γ +ν2 λ) ,
0=n∈Z 1 2 |nν2 |
each Fn being divisible by . All terms U , F and R are bounded, U is ε 1/4 –close to the average V0 of the perturbation and F is divisible by ν2 /2 . As stressed in Sect. 1.3, it is always ν2 = 0, so F as well as f in (2.5) are small. 2.3. Corotating coordinates. In the resonant case it is convenient to pass from the coordinates (G , γ , z1 , z2 , J , j ) to the “corotating coordinates” (, γ , w1 , w2 , J , j ), where ν1 w1 = eiγ ν1 /ν2 z1 , w2 = e−iγ ν1 /ν2 z2 , = G − . (2.7) ν2 In such coordinates the Hamiltonian (2.4) takes the form (dropping all primes) h(c) (, γ , w1 , w2 , J, j ) = K (c) (, ) + ε u(c) (, , J, j ) + f (c) (, w1 , w2 , J, j ) +εe−(˜ε/ε)
1/4
R(, γ , w1 , w2 , J, j ),
with = iw1 w2 and K (c) (, ) = having denoted δ=β−
ν1 , ν2
1 ( 2 + M 2 − 2δ) , 2I3 M=
ν
1
ν2
+δ
1−
ν1 ; ν2
(2.8)
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δ is a small quantity and thanks to (1.2) M is bounded away from zero. The perturbing term f (c) is analytic in a neighborhood of the proper rotation and is small proportionally to ν2 /2 . The advantage of this change of variables is that γ disappears from the normal part of the Hamiltonian and, as far as the remainder can be neglected, the Hamiltonian reduces to two degrees of freedom, being a parameter. The slow component of the frequency is now ∂K (c) ω(c) = = M − δ ∂ and vanishes for res () =
δ . M
(2.9)
(If res is negative, ω(c) never vanishes, nevertheless if res is close to zero ω(c) is small near = 0, and this must be taken into account.) Using res the corotating kinetic energy assumes the useful expression K (c) =
1 (1 − δ 2 /M) 2 + M( − res ())2 . 2I3
(2.10)
The action–angle coordinates corresponding to the corotating coordinates (, γ , w1 , w2 , J, j ) are (, γ , , σ, J, j ), with σ =
ν1 γ +λ; ν2
in such coordinates the Hamiltonian gets the form H (c) (, γ , , σ, J, j ) = K (c) (, ) + ε U (c) (, , J, j ) + F (c) (, , σ, J, j ) +εe−(˜ε/ε)
1/4
R(, γ , w1 , w2 , J, j ).
(2.11)
Now F (c) depends only on σ , and is small proportionally to ν2 /2 . Remark. Let us better explain the change of variables (γ , λ) → (γ , σ ), which might surprise since rational rather than integer coefficients are used. (Equivalent considerations hold for the corresponding change in Cartesian variables.) The procedure, in more detail, is as follows: (i) We preliminarily pass, for the angle γ , from the standard circle S 1 = R/(2π ) to its covering Sν12 = R/(2ν2 π). If for clarity we use γ for the variable on the covering, the ν2 –to–one correspondence is γ = γ mod 2π . The domain of all other variables does not change, the Hamiltonian is the same. Motions in the covering project trivially on motions in the original domain. (ii) In the nonstandard torus Sν12 × S 1 we introduce the smooth one–to–one change of variables (γ , λ) → (γ , σ ) with σ = νν21 γ + λ; the extension to the actions is (G, ) → (, ) with = G − νν21 . The resulting Hamiltonian H (c) is periodic of period 2π in σ and of period 2ν2 π in γ (the overline has been there dropped). As is relevant, only σ appears in the normal part of H (c) , on which we still need to work, while γ enters only the already exponentially small remainder.
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(iii) In the perturbative construction that we are going to start, we average on σ , never on γ , so the uncommon periodicity of γ is never exploited.3 As usual, we introduce near–the–identity canonical transformations to produce normal forms. Using normal forms we infer confinement of the system in the covering, and confinement goes back to the original domain. Practically, the presence of noninteger coefficients, and the fact that the periodicity of H (c) on γ is 2ν2 π , can be safely ignored. (For a similar procedure, see for example [19]; for more general considerations on this subject see [2].) (c)
(c)
Remark. It is natural to introduce the real counterpart x1 , x2 of the Birkhoff corotating coordinates, (c) cos νν21 γ − sin νν21 γ x1 x1 = ; (c) x2 sin νν1 γ cos νν1 γ x 2
2
2
(c)
(c)
such coordinates are indeed close to the coordinates (m1 , m2 ) appearing in Fig. 1, (c) (c) (c) (c) more precisely it is (m1 , m2 ) = (G − 21 )1/2 (x2 , −x1 ). 3. Proof of Proposition 1 3.1. The proof, up to a technical point. A. The nonresonant case. In the nonresonant case, part (ii) of Proposition 0 already implies Proposition 1: indeed, G and stay obviously almost constant (oscillations are much smaller than ε) for an exponentially long time scale as in (1.6), while G, remain ε–close to them. On the other hand, thanks to the overall energy conservation and to the almost constancy of K(G , ), the normal term u(G , , J , j ) also stays almost constant. So, (J , j ) stays almost exactly on a level line of F(J , j ) = u(G (0), (0), J , j ) , and in turn (J, j ) stays ε–close to (J , j ). The motion is regular, the Lyapunov exponent is as small as exp −(ε∗ /ε)(1/4) with ε∗ = ε˜ ; the values of the stability exponents appearing in Proposition 1 are immediately recovered. B. The resonant case ν = (−1, 1): regular motions. So, to prove Proposition 1 it suffices to consider the (much less trivial) resonant case. Here we restrict ourselves to ν = (−1, 1); below in paragraph D we consider that special resonance. The proof proceeds as follows: we make any choice of the exponent a in the range specified in (1.6), and consider the resonant value res given by (2.9); we take care of a (possibly empty) “resonant zone” Z surrounding res , of thickness depending on res : precisely, we define ζ and α by ∗ ε ζ = max(∗ ε 1−a , res ) , and then Z by
α=
| − res | ≤ ∗ ε α .
1 (ζ + 1 − a) 2
(3.1)
(3.2)
3 More precisely, the canonical transformations we shall construct have the following form: stays constant, the new coordinates , σ , J , j are near–the–identity functions of , σ, J, j and (independent of γ ), γ changes a little so the transformation is canonical.
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(c)
(c)
Fig. 5. The resonant zone Z (left) and the corresponding region in the corotating plane m1 m2 (right)
(Z is empty if res < −∗ ε 1−a .) Clearly ζ ≤α ≤1−a . Figure 5 provides a sketch of the situation for the case res > ∗ ε 1−a . In the complement of Z, more precisely in the slightly larger set Z¯ such that | − res | ≥
3 ∗ ε α , 4
we are able to perform on our Hamiltonians (2.4) or (2.6) a further normalization procedure4 which eliminates all harmonics nν = 0 entering the perturbation up to an exponentially small remainder. More precisely, we prove that ¯ which is close to : There exist a constant ε∗ and a function defined in Z, | − | < const ε(1+ζ +a)/2 , and stays almost constant: a
˙ | < const ε e−(ε∗ /ε) . |
(3.3)
This is the only part of the proof requiring perturbation techniques, and is deferred to the next subsection (as is easily imagined, is the new coordinate of the near to the identity canonical transformation which gives the Hamiltonian the appropriate normal form). The consequence of (3.3) is that for initial data out of Z, and up to the time scale (1.6), the motion is as regular as in the nonresonant case: and stay almost constant, while (J, j ) closely follows a level line of U (c) ; the Lyapunov exponent is negligibly small. In particular, during this time scale, motions starting out of Z cannot significantly enter it, and conversely, motions starting in Z cannot significantly escape. C. The resonant case ν = (−1, 1): bounded chaotic motions. Once such a work is done, the proof of Proposition 1 is practically over: indeed, to prove the stability of 4 In principle it would be possible, and simpler, to work only on Hamiltonian (2.4), in Cartesian–like variables. But as it will be clear in the proof (see in particular the remark at the end of Sect. 3), in this way the results worsen a little. As a matter of fact, the best values of the stability exponents are found by using Cartesian coordinates in a specially small neighborhood of the proper rotations, and polar coordinates in a larger neighborhood, namely when ≥ const εα . A similar situation occurred in [15].
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proper rotations as claimed in the proposition, it is not necessary to do any work inside Z. Motions inside Z can be chaotic (and are found numerically to be chaotic, see Fig. 2), but chaos is well localized, and both (J, j ) and m⊥ are confined as specified in Proposition 1. Concerning (J, j ), the confinement follows from the energy conservation applied to Hamiltonian (2.11). Indeed, since stays almost constant, the oscillations of K (c) , see (2.10), are bounded by const ε 2α , while the oscillations of F (c) , which is as small as ν2 /2 , for near res ≤ ∗ ε ζ are bounded at least by const ε ζ . As a consequence, the oscillations of U (c) are bounded by const εc ,
c = min(ζ, 2α − 1) = 2α − 1 = ζ − a .
This gives c = for res in the bulk of the domain 0 ≤ ≤ ∗ ε 1/2 , and c = 1−2a for res very near the origin, i.e. when δ is small and the proper rotation is almost exactly resonant. This is in agreement with Proposition 1, the stability exponent c being as in (1.8). Finally, to determine the exponent b⊥ appearing in (1.8), we must compute the maximal oscillation m⊥ of m⊥ , which according to (1.11) is bounded by
m⊥ ≤ const res + ∗ ε α − res + ∗ ε α . 1 2 −a
If res ≤ ∗ ε 1−a , then α = ζ = 1 − a, and so
m⊥ ≤ const ∗ ε (1−a)/2 ,
(3.4)
in agreement with (1.8). In the opposite case it is res = ∗ ε ζ , and so
√ m⊥ = const ∗ ε ζ /2 1 + ε α−ζ − 1 − ε α−ζ ; from this expression (3.4) follows as well. D. The resonance ν = (−1, 1). As remarked above, the resonance ν = (−1, 1) is somewhat special. A first reason is that the perturbation is larger, F (c) being small only as 1/2 . A second reason is that now, recalling β ≥ −1, it is δ =β +1≥0,
res = −
δ ≤0, 2(1 − δ)
so the resonance either is exactly on the proper rotation (δ = 0) or is out of the phase space (δ > 0). We proceed as follows: to take care of the worst situation δ = 0, we exclude a resonant zone Z defined now by ≤ ∗ ε α ,
α=
2 (1 − a) , 3
0
1 . 4
(3.5)
In the complement of Z we perform as above a normalization procedure, see the next subsection, leading to (3.3); in this way we prove that the motion outside Z is regular. For motions inside Z, we use as above the energy conservation: since inside Z the kinetic part oscillates at most by const ε 2α , while the oscillations of F (c) are bounded by const ε α/2 , the oscillations of U are necessarily bounded by const εc , with α 1 c = min , 2α − 1 = 2α − 1 = (1 − 4a) . 2 3
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The corresponding exponent b⊥ , determining the maximal oscillations of m⊥ , is obviously 1 1 b⊥ = α = (1 − a) . 2 3 This concludes the proof of Proposition 1, up to the proof of (3.3). Remark. The stability of the proper rotation about e3 in the resonant case ν = (−1, 1) is in fact slightly better than what we proved here. Indeed, for this special resonance the perturbation has a quite peculiar form which should be taken into consideration. In essence, it turns out that it is possible to reduce the size of the resonant zone by taking α as close as one wishes to 4/5, in place of 2/3, the exponent a remaining positive. However, the perturbation scheme leading to (3.3) in the larger domain needs relevant changes (and becomes essentially more complicated). Work is in progress on this point in collaboration with A.I. Neishtadt. Remark. The necessity of improving b⊥ and c in the resonance ν = (−1, 1) is evident in the numerical computations mentioned in Sect. 1.4. On the contrary, for ν = (−1, 1) the exponents seem to be optimal.
3.2. The proof of (3.3). This is a rather standard perturbation proof, with the only peculiarity that most small quantities — norms of functions, small divisors — are small with : so we must work locally in this variable in a convenient neighborhood of some value 0 ≤ ε 1/2 ∗ . We start from the Hamiltonian (2.4) or (2.6), but provisionally forget the small remainder, so as to deal with a system with two degrees of freedom in which is a parameter; at the end of the proof it will be evident how to extend the result to the complete system. We begin with the case of the resonance ν = (−1, 1), which is easier because the domain has a simple form and moreover, a neighborhood of = 0 being already excluded, we can safely use the action–angle variables, with no need of the Birkhoff ones. A. The case ν = (−1, 1). So, we consider the Hamiltonian (2.6), we cut off the remainder and (with little abuse) denote the resulting function by H (, σ, J, j ) = K() + ε[U (, J, j ) + F (, σ, J, j )] , with K() =
2 , 2I
and F (, σ, J, j ) =
def
I =
(3.6)
I3 I3 =− M 2(1 − δ)
Fn (, J, j ) einσ ;
(3.7)
0=n∈Z
F is divisible by 1/2 . We consider any 0 such that ε α ∗ ≤ 0 ≤ ε 1/2 ∗ , and introduce the real domain D: = 0 ,
σ ∈ S1 ,
We denote r = 0 /∗ ,
|J | ≤
3 , 4
εα ≤ r ≤ ε1/2 ,
j ∈ S1 .
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and consider an extension vector (with positive entries) ρ = (rρ , ρσ , ρJ , ρj ) ,
ρ =
1 ∗ . 2
The extended domain Dρ is defined (as usual) as the union of the polydisks of radii rρ , ρσ , ρJ , ρj , centered in all points of D. We shall assume for simplicity ρσ ≤ 1. With no possibility of confusion, we indicate by ρ also the positive number defined by ρ 2 = max(ρ ρσ , ρJ ρj ) , and denote, as it turns out to be practical, = ρ 2 /I . For any extension ρ ≤ ρ (the inequality is intended to work on all components), we use the Fourier norm in Dρ , denoted . ρ . Norms of derivatives of functions are useful as well, and we find it convenient to use the compact notation ∂. . ∂ρ = max . ρ , rρ , ρσ ∂ ρ
∂ . , ρJ ∂σ ρ
∂ . , ρj ∂J ρ
∂ . . ∂j ρ
(3.8)
Averages on σ will be denoted . . We plan to perform a number l of normalization steps, which will later depend on ε, in progressively reduced domains Dρk , with ρk = ρ(1 − 2kη) ,
η=
1 , 4l
so that ρl = ρ/2. We assume recursively that after k steps, 0 ≤ k ≤ l−1, the Hamiltonian has the form H (k) (, σ, J, j ) = K() + ε[U (k) (, J, j ) + F (k) (, σ, J, j )] and is real analytic in Dρk , F (k) has zero average on σ , and U (k) , F (k) satisfy the estimates U (k) ∂ρk ≤ Uk , with
F (k) ∂ρk ≤ r 1/2 k ,
k Uk = 1 + e−k U < 2U ,
(3.9)
k = e−k U ;
(3.10)
i=1
the derivative
∂U (k) ∂
is however better,5 namely ∂U (k) Uk ≤ 1/2 ∂ ρk r ρ
5
(3.11)
This peculiarity might surprise, but it is easily understood if one makes the decomposition (k)
(k)
(k)
U (k) (, J, j ) = U0 (J, j ) + U1 (, J, j ) ,
U1 (0.J, j ) = 0 . (k)
(k)
Indeed the assumption we are making on Uk is equivalent to U0 ∂ρk ≤ Uk , U1 ∂ρk ≤ r 1/2 Uk , and this is probably the best way to understand it. In practice, however, using the above decomposition is a little cumbersome.
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(r 1/2 in place of r in the denominator). As is immediately recognized, Proposition 0 ensures that for k = 0 there exist an extension vector ρ and a constant U such that these inequalities do hold; the estimate on ∂U ∂ is in fact better, but this is useless. We perform a near to the identity canonical transformation C : Dρk+1 → C(Dρk+1 ) ⊂ Dρk using the Lie method: that is, we introduce an auxiliary (small) Hamiltonian χ , denote by sχ its flow, and look for χ such that C = 1χ behaves as we wish. The new Hamiltonian H (k+1) = H (k) ◦ C has the form H (k+1) = K + εU (k) + εF (k) + {K, χ} + εR1 [χ , U (k) + F (k) ] + εR2 [χ , K] , the remainders R1 , R2 being defined as
1
s R1 [χ , . ] = { . , χ} ◦ χ ds , R2 [χ , . ] = 0
χ is taken to be a solution of
{K, χ } + εF (k)
1
s
ds 0
0
{{ . , χ}, χ } ◦ sχ ds ;
= 0, i.e.
∂χ = εF (k) , ρ 2 ∂σ
(3.12)
and thus U (k+1) , F (k+1) are given by U (k+1) = U (k) + R1 + R2 ,
F (k+1) = R1 + R2 − R1 + R2 .
(3.13)
We must check that χ is sufficiently small so as 1χ (Dρk+1 ) ⊂ Dρk , and of course verify that U (k+1) , F (k+1) satisfy the recurrent assumptions (3.9), (3.11). The solution of (3.12) can be written in form of Fourier series, namely χ=
(k) ερ Fn inσ . e in 0=n∈Z
By writing one by one the derivatives of χ , using the recurrent assumption (3.9) on F (k) , it is not difficult to deduce that χ ∂ρk ≤ The smallness condition
2ε k . r ρ
8 ε l k ≤1 r 3/2 ρ 2
(3.14)
thus ensures that for (, σ, J, j ) ∈ Dρk −ηρ it is sχ (, σ, J, j ) ∈ Dρk for any |s| ≤ 1; this implies, in particular, that the canonical transformation C from Dρk+1 ⊂ Dρk −ηρ to Dρk is well defined. The next estimates, in view of the estimates of the remainders R1 , R2 , concern the Poisson brackets {U (k) , χ } and {F (k) , χ }. By naively using the definition of Poisson bracket it is not difficult to work out the inequalities {U (k) , χ }ρk ≤
8ε Uk k , r ρ 2
{F (k) , χ }ρk ≤
8ε k2 . r ρ 2
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Good estimates on the remainders are then easily worked out: concerning R1 , from its very definition it follows (also recalling Uk < 2U) R1 ρk −ηρ ≤ ε{U (k) + F (k) , χ }ρk ≤
24ε U k , r ρ 2
while concerning R2 , using {K, χ } = −εF (k) and k ≤ Uk , it follows similarly R2 ρk −ηρ ≤
1 4ε U ε{F (k) , χ }ρk ≤ k . 2 r ρ 2
Cauchy estimates, with domain reduction from Dρk −ηρ to Dρk+1 , allow us to estimate the derivatives of R1 , R2 , and the result is R1 + R2 ∂ρk+1 ≤
26 ε l U k . r ρ 2
From (3.13) it then follows that U (k+1) , F (k+1) satisfy the recurrent assumptions (3.9), (3.11) as long as the inequality 26 e ε l U ≤1, r 3/2 ρ 2
(3.15)
stronger than (3.14), is satisfied. If (3.15) is satisfied, then the non–normal part εF (l) of the final Hamiltonian H (l) has norm as small as ε e−l U. Considering the worst case r = ε α , and choosing l such that ε a ∗ l≤
ε∗a =
1 ρ 2 . 26 e U
The conclusion is that F (l) is such that a
εF (l) ∂ρ/2 < ε e−(ε∗ /ε) U ,
(3.17)
and so, if we forget the remainder in (2.6), the inequality (3.3) is satisfied with the appropriate choice (3.5) of α, as claimed in Sect. 3.1D. The overall canonical transformation Ctot , produced after l steps, is small: namely denoting (, σ, J, j ) = Ctot ( , σ , J , j ), it is not difficult to deduce (just by summing the maximal possible displacements ∂χ ∂σ ρk at each step) | − | ≤
l−1 U ε 1−α/2 k ≤ 2ε (2+a)/3 ρ ρ 2 k=0
and similarly |σ − σ | ≤ 16ε a
U ρσ , ρ 2
|x − x| ≤ 8ε (2+a)/3
U ρx ρ 2
for
x = J, j .
The estimate for σ is somehow poor, but for our purposes this is totally irrelevant.
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Up to now we considered the truncated Hamiltonian with two degrees of freedom. However, the extension to the complete Hamiltonian is trivial: namely at each perturbation step we maintain the definition C = 1χ with the same χ , but now we interpret the auxiliary Hamiltonian χ as having three degrees of freedom, the variable being no more a parameter but a canonical coordinate. Since χ does not depend on γ , at each step remains unchanged. The angle γ , entering the remainder of the complete Hamiltonian, of course changes a little, but this is obviously irrelevant, and the remainder itself, after composition with C, remains as small as it was. It should be observed that, since a < 41 , the remainder in (2.6) is (much) smaller than the new remainder εF (l) estimated in (3.17); the conclusion is that, after a minor correction of the constant in (3.3), the inequality holds for the complete Hamiltonian, too. B. The case ν = (−1, 1), for not too small. Here we work in action–angle variables and exclude, in addition to the resonant zone Z, a neighborhood of the proper rotation given by < εα ∗ . The scheme of the proof is very similar to the previous case, with a very minor complication due to the (possible) presence of the resonant zone Z around res = εζ ∗ ≥ εα ∗ . Now the two–degrees of freedom Hamiltonian we start with has the form (3.6), with 1 K() = (3.18) ( − res )2 2I and F as in (3.7); F however is smaller, being now divisible (at least) by . For any 0 such that 0 , |0 − res | ≥ εα ∗ , we introduce as above r = 0 /∗ , but we need a second variable r˜ = min(r, |0 − res |/∗ ) to be used to bound the denominators and the analyticity domains of functions depending on (previously it was r˜ = r). In particular, r˜ enters the extension vector, which is now ρ = (˜r ρ , ρσ , J, j ) , and also replaces r in the definition (3.8) of the norm . ∂ρ . The recurrent assumption now is U (k) ∂ρk ≤ Uk , F (k) ∂ρk ≤ rk , with Uk , k as before,
∂U (k) ∂
satisfying however the better inequality ∂U (k) r Uk . ≤ ∂ ρk r˜ ρ
The perturbation scheme is then identical, and the main estimates found in the way are χ∂ρk ≤ 2
r k , r˜
{U (k) , χ }ρk ≤ 6ε
r2 U k , r˜ 2 ρ 2
{F (k) , χ }ρk ≤ 8ε
leading to R1 + R2 ∂ρk+1 ≤ 72εl
r2 U k . r˜ 2 ρ 2
r2 U k , r˜ 2 ρ 2
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The smallness condition replacing (3.15) is 72 ε l
r U ≤ e−1 , r˜ 2 ρ 2
(3.19)
and the worst situation is when r˜ = εα , r = εζ + ε α < 2εζ ; making the same choice of l as in (3.16), condition (3.19) is satisfied if 2α − ζ + a = 1 ,
def
ε∗ ≤ ε∗∗ =
1 ρ 2 1/a . 18 e U
(3.20)
The former equality explains the choice (3.1) of α. The extension of the canonical transformation to the pair of conjugated variables (, γ ) is clearly identical as before. However, at variance with the previous case, now the value a = 41 is possible and correspondingly the remainder in (2.6) can be relevant. To take care of this situation, it is enough to take ε∗ = min(˜ε , ε∗∗ ) and to choose appropriately the constant in (3.3). An overall estimate on the canonical transformation is provided by | − | ≤ 4ε (1+ζ +a)/2
U ρ ρ 2
and |σ − σ | ≤ 4ε a
U ρσ , ρ 2
|x − x| ≤ 4ε (1+ζ +a)/2
U ρx ρ 2
for
x = J, j .
C. The case ν = (−1, 1), for small . It remains to consider the case ν = (−1, 1) in a neighborhood < εα ∗ of the proper rotation, where we cannot enter with the action– angle coordinates due to their singularity at = 0. This work is useful when such a neighborhood is not included in Z, that is when res > ε α ∗ , to show that the motion is there regular and well confined as regular motions are, with vanishing Lyapunov exponent. (Strictly speaking, only for the sake of proving long term stability of proper rotations, regardless of whether motions are regular or not, such work is unnecessary.) We use the Birkhoff (corotating) coordinates and consider the two–degrees of freedom Hamiltonian h(w1 , w2 , J, j ) = K() + ε[u(, J, j ) + f (w1 , w2 , σ, J, j )] , with = iw1 w2 , K() as in (3.18) and f (w1 , w2 , σ, J, j ) =
fnν2 (w1 , w2 , σ, J, j ) ,
0=n∈Z
having denoted, for m ∈ Z, fm (w1 , w2 , J, j ) = fˆm (, σ, J, j )w|m| ,
w=
w1 w2
for m ≥ 0 . for m < 0
The real domain to be extended is given by w1 = w2 = 0 ,
|J | ≤
3 , 4
j ∈ S1 ;
(3.21)
Stability of Proper Rotations
155
the extension vector, denoting r = ε α , is now 1
ρ = (r 2 ρw , ρJ , ρj ) ,
ρw =
∗ /2 ,
and the complex extended domain is defined as the union of the polydiscs |w1 |, |w2 | ≤ r 1/2 ρw ,
|J − J0 | ≤ ρJ ,
|j − j0 | ≤ ρj
for real J0 , j0 satisfying (3.21). Similarly to the previous case, we denote ρ 2 = min(ρw2 , ρJ ρj ) and ∂. ∂. ∂ . ∂ . . ∂ρ = max . ρ , rρw , rρw , ρJ , ρj . ∂w1 ρ ∂w2 ρ ∂J ρ ∂j ρ Once more, the proof proceeds along the same scheme. The recurrent hypothesis is that after k ≤ l − 1 perturbation steps we deal with h(k) (w1 , w2 , J, j ) = K() + ε[u(k) (λ, J, j ) + f (k) (w1 , w2 , J, j )] , f (k) containing only the harmonics nν2 with 0 = n ∈ Z; u(k) and f (k) are real analytic in Dρk and satisfy the estimates u(k) ∂ρk ≤ Uk ,
f (k) ∂ρk ≤ rk
as well as
∂f ≤ r 1/2 Uk , i = 1, 2 , ∂wi ρk with Uk , k as in (3.10). The equation for χ reads now ∂χ ∂χ i 2 ( − res ) w1 − w2 = ε fk ρ ∂w1 ∂w2
and is solved by χ=
fnν ερ 2 2 . i( − res ) nν2 0=n∈Z
The procedure then goes on exactly as in the previous cases, the relevant estimates being χ ∂ρk ≤
3 ε k , 2 r 1/2 ρ 2
{uk , χ }ρk ≤ 8 ε
U k , ρ 2
and finally
{uk , χ }ρk ≤ 4 ε
U k ρ 2
U k . ρ 2 The appropriate smallness condition which allows to verify the recurrent assumption on uk , fk , also ensuring C(Dk+1 ) ⊂ Dk , is R1 + R2 ∂ρk+1 ≤ 48 ε l
48
εl U ≤ e−1 ; r ρ 2
using r = εα and l as in (3.16) the condition turns into 48 e ε1−α−a
U ≤1 ρ 2
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and is guaranteed by (3.20). The overall canonical transformation is estimated by | − | ≤ 8 ε
U ∗ . ρ 2
Remark. One might wonder why this perturbation scheme in Birkhoff variables has been used only for < εα ∗ and not for larger (the obvious advantage would be avoiding the double proof, in two different sets of variables). The answer is that the scheme works, but some results (including the crucial relation (3.20) between α, ζ and a) get worse. The reason is easy to understand: confining in this way requires a correspondingly good confinement of w1 , w2 , but if is large, this produces an unnecessary and “expensive” good confinement of the angle σ , too. The use of the polar–like coordinates , σ allows instead to treat separately the action and the angle, with poorer control on the angle. A similar motivation is at the basis of the analogous choice in [15]. We do not know if it is possible to unify the proofs through a more elaborated construction. 4. The Triaxial Case As remarked in the Introduction, we considered the case of a kinetically symmetric body mainly for simplicity: indeed in this case the Andoyer–Deprit variables (G, L, J, g, l, j ) are the action–angle variables of the system, and everything can be written explicitly; the same choice was made in [4] and [5]. The results however are more general and naturally extend to the triaxial case, too. In particular the proper rotations about e1 and e3 , if I2 is intermediate between I1 and I3 , are Nekhoroshev stable. Here we produce a sketch of the work to be done to perform such an extension, which includes only minor changes with respect to the symmetric case. It is clearly sufficient to consider motions near the proper rotation about e3 , assuming either I1 < I2 < I3 or I3 < I2 < I1 . The main question is not the extension of Proposition 1, but the extension of Proposition 0 proved in [5], which is at the basis of Proposition 1. To this purpose the starting point is showing that in the triaxial case, too, there exist Birkhoff coordinates, analytic in a neighborhood of the proper rotation, which integrate the unperturbed system. The existence of such variables is a byproduct of [18]. Indeed, the kinetic energy of the triaxial rigid body in the Andoyer–Deprit coordinates has the expression sin2 l
K(G, L, J, g, l, j ) = (G2 − L2 )
2I1
+
cos2 l L2 ; + 2I2 2I3
the system is integrable, but the coordinates are not the action–angle ones. We can nevertheless introduce the coordinates (G, γ , , λ) as in Sect. 2.1, and then the Cartesian coordinates x1 , x2 ; the kinetic energy assumes the form 1 2 1 k= G − G(β2 x12 + β1 x22 ) + (x12 + x22 )(β2 x12 + β1 x22 ) , 2I3 4 with βi = 1 −
I3 , Ii
i = 1, 2
(both constants are positive for I3 < I2 < I2 , negative if the opposite inequality holds). The rescaling x1 = (β2 /β1 )1/4 x˜1 , x2 = (β1 /β2 )1/4 x˜2
Stability of Proper Rotations
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gives then 1 2 1 G ∓ β1 β2 G (x˜12 + x˜22 ) + (β1 x˜12 + β2 x˜22 )(x˜12 + x˜22 ) k˜ = 2I3 4 (upper sign if β1 , β2 are positive). Such a Hamiltonian can be put in normal form by means of a near to the identity analytic transformation6 x˜1 = xˆ1 + X1 (xˆ1 , xˆ2 , G) ,
x˜2 = xˆ2 + X2 (xˆ1 , xˆ2 , G) ,
ˆ = with X1 , X2 at least quadratic in xˆ1 , xˆ2 ; denoting ˆ becomes a function of G and , and has the form
1 2 2 (xˆ 1
+ xˆ22 ), the Hamiltonian
1 1 2 ˆ ˆ + (β1 + β2 ) ˆ 2 + kˆ (G, ) ˆ , ˆ = G ∓ 2 β1 β2 G k(G, ) 2I3 2 ˆ 3 . This expression is the generalization of (1.12) to the triaxial k being divisible by case. The problem thus becomes essentially identical to the problem studied in [5] and in the present paper. In particular: (i) Near the proper rotation the frequency has still the form ˆ = ω(G, )
G ˆ , (1, −β) + O() I3
√ with β = ± β1 β2 , and one easily checks that −1 ≤ β < 1. The assumption corresponding to (1.2), ensuring the crucial inequality (1.5), turns out to be |β2 |(1 − β1 ) ≥ β∗ > 0 . (ii) The Hessian matrix on the proper rotation is 1 √1 I3 ∓ β 1 β 2
√ ∓ β1 β2 , 1 2 (β1 + β2 )
so kˆ is there convex if β1 , β2 > 0 (I3 < I2 < I1 ) and is immediately seen to be quasi–convex in the opposite case; this is essential to start the perturbation work of [5]. (iii) The perturbation, being analytic in the Birkhoff coordinates, is appropriately small with the distance from the proper rotation exactly as in the symmetric case, the ˆ |ν2 |/2 . norm of the Fourier component ν = (ν1 , ν2 ) being divisible by Once these basic properties are satisfied, all of the perturbation work done in [5] and in the present paper proceeds smoothly, with no important changes, and results are the same up to the value of the constants. 6 This is evident, see for example [20], if k˜ is thought of as a system with one degree of freedom (a nonlinear oscillator) parametrized by G; the extension of the transformation to the pair (G, γ ) is immediate.
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5. Conclusions and Open Problems A. A final comment. Let us here shortly summarize what we did in this paper. As remarked in the Introduction, the Euler rigid body has two fast degrees of freedom and a slow degenerate one, which does not appear in the expression of the kinetic energy. In presence of a resonance between the fast frequencies, a new slow degree of freedom is produced, and there is a chance that such degree of freedom interacts with the degenerate one, thus producing chaotic motions. A crucial fact is that degeneracy (complete lack of convexity or steepness, in the language of Nekhoroshev theory) makes useless the standard mechanisms of confinement entering Nekhoroshev theory, and wide chaotic motions become possible. As remarked in the Introduction, chaotic motions in which µ = ms /m wanders unpredictably on wide portions of the unit sphere for no matter how small ε have been observed numerically [3], precisely in connection with resonances between the two fast frequencies. The novelty of this paper is that an important class of motions of the rigid body, namely motions close to proper rotations, are more regular than all the other ones. The key observation is that the distance to the proper rotation can be successfully used as a second small parameter in the system, in addition to ε. Profiting by this fact, we are able to show that for a long time scale all motions close to proper rotations either are regular (all Lyapunov exponents vanish) or are possibly chaotic (the maximal Lyapunov exponent is positive), but nevertheless they are confined to a small neighborhood of a predictable regular curve, essentially as in nondegenerate systems (Figs. 1 and 2). In a sense, the vicinity to a proper rotation removes the degeneracy. Best results are obtained if the relation between the two small parameters is appropriately tuned, i.e., if the diameter of the neighborhoods of the proper rotation entering the game are appropriately chosen as functions of ε. We made some efforts to be careful on this point, and the result are the powers of ε, i.e. the stability exponents, entering Proposition 1. The main reason of our interest in the value of the exponents is that at least some of them can be computed numerically, and so the theory admits a critical test. The question, important in our opinion, is whether the mechanism producing stability of motions close to proper rotations is now essentially understood, or other elements producing stability are still neglected. Such a test is in progress; preliminary results are encouraging. B. Applications. Our Proposition 1 is very general — it holds for any analytic potential — so it is natural to discuss about its possible applications. To this purpose, it is worthwhile to stress that the proposition is interesting only if the potential does depend on the proper rotation (on the angle l, in the Andoyer–Deprit coordinates). Otherwise, if both the kinetic and the potential energy have rotational symmetry about e3 , the system reduces to two degrees of freedom, and everything gets substantially easier. In particular, the mechanism leading to slow chaos, shortly recalled above in paragraph A, is excluded a priori. Rigid bodies which are kinetically symmetric, but the potential is not, do exist: for example, restricting to gravitational forces, prismatic bodies, or cylinders with two or more symmetrically disposed legs, as is frequent among artificial satellites.7 The main interest however, in our opinion, is the case of triaxial bodies, to which Sect. 4 is devoted.Although in Celestial Mechanics the main objects are symmetric, non-symmetric satellites or asteroids are well known to exist (non-symmetric artificial satellites exist as well). 7 Adding other forces, for example elastic ones, is an easier way to break the rotational symmetry, but it looks very artificial.
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As it was suggested to us by the referee, to whom we express our sincere gratitude, there are other problems of Celestial Mechanics which do not fit the assumptions of Proposition 1, but likely can be studied by the same techniques used in this paper and in the same spirit. An important case is the spin–orbit problem for a symmetric rigid body in an elliptic orbit. Indeed the symmetry of the body eliminates the fast angle l, but the mean anomaly of the orbital motion can play the same role. The question then assumes the same form: in presence of a resonance between the spin motion (the angle g) and the orbital motion, a slow degree of freedom is produced, and there is a chance it effectively interacts with (J, j ) producing chaotic motions. For models of this kind, see for example [9] and [8]. A rough preliminary analysis makes us confident that the perturbation techniques used in this paper, in principle, extend to those models too. We plan to come back to this problem in the near future. Acknowledgement. we are indebted to A.I. Neishtadt for very helpful discussions and to G. Gallavotti, editor of CMP, as well as to the referees, for valuable suggestions.
References 1. Andoyer, H.: Cours de M´echanique Celeste. Paris: Gautier-Villars, 1923 2. Arnol’d, V.I.: Geometrical methods in the theory of ordinary differential equations. Berlin: Springer– Verlag, 1982 3. Benettin, G., Cherubini, A.M., Fass`o, F.: Regular and Chaotic motions of the fast rotating rigid body: a numerical study. Discr. Cont. Dyn. Sys. 4, 521–540 (2002) 4. Benettin, G., Fass`o, F.: Fast rotations of the rigid body: A study by Hamiltonian perturbation theory. Part I. Nonlinearity 9, 137–186 (1996) 5. Benettin, G., Fass`o, F., Guzzo, M.: Fast rotations of the rigid body: A study by Hamiltonian perturbation theory. Part II: Gyroscopic rotations. Nonlinearity 10, 1695–1717 (1997) 6. Benettin, G., Fass`o, F., Guzzo, M.: Paper in preparation 7. Benettin, G., Hjorth, P., Sempio, P.: Exponentially long equilibrium times in a one dimensional collisional model of a classical gas. J. Stat. Phys. 94, 871–892 (1999) 8. Biasco, L., Chierchia, L.: Effective Hamiltonian for the d’Alembert planetary model near a spin/orbit resonance. Cel. Mech. & Dyn. Astr 83, 223–237 (2002) 9. Chierchia, L., Gallavotti, G.: Drift and diffusion in phase space. Ann. Inst. H. Poincar´e Phys. Th´eor. 60, 1–144 (1994) 10. Deprit, A.: Free rotation of a rigid body studied in phase plane. Am. J. Phys. 55, 424 (1967) 11. Elskens, Y., Escande, D.F.: Slowly pulsating separatrices sweep homoclinic tangles where islands must be small: an extension of classical adiabatic theory. Nonlinearity 4, 615–667 (1991) 12. Escande, D.F.: Change of adiabatic invariant at separatrix crossing; application to slow Hamiltonian Chaos. In: Advances in Nonlinear Dynamics and Stochastic Processes, R. Livi, A. Politi, Eds., Singapore: World Scientific, 1985 13. Fass`o, F.: Hamiltonian perturbation theory on a manifold. Cel. Mech. & Dyn. Astr. 62, 43–69 (1995) 14. Fass`o, F.: The Euler–Poinsot top: a non-commutatively integrable system without global action– angle coordinates. J. Appl. Math. Phys. 47, 953–976 (1996) 15. Guzzo, M., Fass`o, F., Benettin, G.: On the stability of elliptic equilibria. MPEJ - Mathematical Physics Electronic Journal 4 (1), 1–16 (1998) 16. Henrard, J.: Virtual singularities in the artificial satellite theory. Cel. Mech. & Dyn. Astr. 10, 437–447 (1973) 17. Levi–Civita, T., Amaldi, U.: Lezioni di Meccanica Razionale, Vol. 2 – Part 2, Bologna: Zanichelli, 1927 18. Lidov, M.L., Neishtadt, A.I.: The method of canonical transformations in problems of the rotation of celestial bodies and Cassini Laws (in Russian). In: Determination of the motion of a spacecraft (in Russian), P.E. Eliasberg, Ed., Moscow: Nauka 1975, pp. 75–107 19. Morbidelli, A.: Modern Celestial Mechanics. London: Taylor and Francis, 2002 20. Moser, J.: Lectures in Celestial Mechanics. In: Mem. American Math. Soc. No 81, 1 (1968) 21. Neishtadt, A.I.: Change in adiabatic invariant at a separatrix. Sov. J. Plasma Phys. 12, 568–573 (1986) 22. Neishtadt, A.I.: On the change in the adiabatic invariant on crossing a separatrix in systems with two degrees of freedom. Prikl. Matem. Mekan. 51, 750–757 (1987)
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23. Neishtadt, A.I., Sidorenko, V.V., Treshev, D.V.: Stable periodic motions in the problem on passage through a separatrix. Chaos 3, 2–11 (1997) 24. Nekhoroshev, N.N.: An exponential estimate of the time of stability of nearly integrable Hamiltonian systems. Usp. Mat. Nauk 32:6, 5–66 (1977) [Russ. Math. Surv. 32:6, 1–65 (1977)] 25. Signorini, A.: Complementi alla dinamica dei giroscopi e equazioni del problema completo della balistica esterna. Atti Accad. Naz. Lincei, Serie 8, Vol. 1, Memorie, 1–41 (1946) 26. Stoppelli, F.: Sul principio dell’effetto giroscopico. Giornale di Matem. di Battaglini, serie 4, 80 (1950–51) 27. Stoppelli, F.: Un fenomeno di risonanza nei solidi in rapida rotazione. Ricerche di Matem. 9, 213–241 (1960) 28. Synge, J.L.: Classical Dynamics. In: Handbuch der Physik Vol. III/I, Principles of classical mechanics and field theory. Berlin: Springer–Verlag, 1960 Communicated by G. Gallavotti
Commun. Math. Phys. 250, 161–193 (2004) Digital Object Identifier (DOI) 10.1007/s00220-004-1084-9
Communications in
Mathematical Physics
Virasoro Symmetries of the Extended Toda Hierarchy Boris Dubrovin1 , Youjin Zhang2 1 2
SISSA, Via Beirut 2–4, 34014 Trieste, Italy Department of Mathematical Sciences, Tsinghua University, Beijing 100084, P.R. China
Received: 19 August 2003 / Accepted: 6 November 2003 Published online: 28 April 2004 – © Springer-Verlag 2004
Abstract: We prove that the extended Toda hierarchy of [1] admits a nonabelian Lie algebra of infinitesimal symmetries isomorphic to half of the Virasoro algebra. The generators Lm , m ≥ −1 of the Lie algebra act by linear differential operators onto the tau function of the hierarchy. We also prove that the tau function of a generic solution to the extended Toda hierarchy is annihilated by a combination of the Virasoro operators and the flows of the hierarchy. As an application we show that the validity of the Virasoro constraints for the CP 1 Gromov-Witten invariants and their descendents implies that their generating function is the logarithm of a particular tau function of the extended Toda hierarchy. 1. Introduction The extended Toda hierarchy was introduced in [30, 16, 1] in an attempt to encode the recursion relations among the CP 1 Gromov-Witten invariants into that of a hierarchy of integrable systems. As it was shown in [1], this hierarchy can be represented in a Lax pair formalism through the Lax operator L = + v(x) + eu(x) −1 .
(1.1)
The functions v, u serve as the dependent variables for the hierarchy with spatial variable x and = e∂x is the shift operator, is a small parameter. We will also introduce a two-component vector w = (w1 , w2 ),
w1 := v, w 2 := u
to use in the formulae where many summations enter. The flows of the hierarchy are defined via Lax representation
∂L = [Aβ,q , L] := Aβ,q L − LAβ,q , ∂t β,q
β = 1, 2; q ≥ 0.
(1.2)
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Here the operators Aβ,q have the expression A1,q =
2 q L (log L − cq ) + , q!
A2,q =
c0 = 0, cq = 1 +
1 Lq+1 , + (q + 1)!
1 1 + ··· + 2 q
(1.3)
(1.4)
with part B+ of the difference operator B = Bk k given by B+ = the positive k k≥0 Bk . The logarithm of the operator L is defined as follows [1]. Let us first introduce the dressing operators P and Q of the form P = pk −k , Q = qk k , p0 = 1 (1.5) k≥0
k≥0
such that L = P P −1 = Q−1 Q−1 .
(1.6)
Then we define log L :=
1 P ∂x P −1 − Q∂x Q−1 . 2
(1.7)
Equations (1.2) for β = 2 coincide with the standard flows of the Toda hierarchy. In particular for q = 0 one obtains the equations of Toda lattice k ∂v 1 u(x+) u(x) = − e = e ∂ k+1 eu , ∂t 2,0 (k + 1)! x k≥0
∂u k 1 k = (−1) − v(x − )) = ∂ k+1 v, (v(x) ∂t 2,0 (k + 1)! x
(1.8)
k≥0
written in the interpolated form. To return to the original discrete setup of [26] one introduces the lattice variables un := u(n), vn := v(n). The parameter plays the role of the mesh of the lattice. Another part, for β = 1 is a new one. For q = 0 one obtains just the spatial translations ∂v ∂v , = ∂t 1,0 ∂x
∂u ∂u . = ∂t 1,0 ∂x
(1.9)
The flow for β = 1 and q = 1 is less trivial: ∂v 1 u(x+) u(x) e = vv + u(x + ) − 2) − e u(x − ) − 2) , (B (B x − − ∂t 1,1 1 ∂u = v(x) (B− u(x) − 2) − v(x − ) (B− u(x − ) − 2) 1,1 ∂t + B+ v(x + ) − B+ v(x − ) , (1.10)
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where the operators B± are defined by B+ := ( − 1)−1 ∂x =
Bk k≥0
B− := (1 − −1 )−1 ∂x =
k!
(∂x )k ,
Bk k≥0
k!
(−∂x )k .
(1.11)
Here Bk are the Bernoulli numbers. The flows of the extended Toda hierarchy can be represented as Hamiltonian systems ∂w α αγ δHβ,q = {w α (x), Hβ,q }1 ≡ U1 β,q ∂t δw γ (x)
(1.12)
with the Hamiltonian operators U111 = U122 = 0, and the Hamiltonians
U112 =
1 ( − 1),
U121 =
1 (1 − −1 ),
(1.13)
Hβ,q =
hβ,q dx,
β = 1, 2; q ≥ −1,
with the densities hβ,q defined by 2 res Lq+1 (log L − cq+1 ) = res A1,q+1 , (q + 1)! 1 res Lq+2 = res A2,q+1 . = (q + 2)!
h1,q = h2,q
(1.14)
By definition the residue of a difference operator a k k A= k∈Z
is given by res A := a0 . Note that h1,−1 = B− u(x),
h2,−1 = v(x)
(1.15)
are densities of Casimirs of the Poisson bracket, i.e. { . , H1,−1 }1 = { . , H2,−1 }1 ≡ 0. Denote A the graded ring of formal power series of the form
k fk (w, wx , . . . ),
k≥0
where fk are polynomials of v, u, e±u , v (m) , u(m) , m ≥ 1. The gradation is defined by deg v (m) = 1 − m, deg u(m) = −m, for m ≥ 0, deg eu = 2, deg = 1.
(1.16)
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As it was shown in [1], all the Hamiltonian densities (1.14) as well as the right hand sides of the flows of the extended Toda hierarchy are homogeneous elements of the ring A. Introduce functions α,p;β,q ∈ A by ∂hα,p−1 1 = ( − 1)α,p;β,q := ∂t β,q
1 2 p! res 1 1 (p+1)!
Aβ,q , Lp (log L − cp ) , α = 1, α = 2. res Aβ,q , Lp+1 , (1.17)
Existence of such functions and an important property of τ -symmetry β,q;α,p = α,p;β,q was established in [1]. These elements of the ring A are uniquely determined by the above formulae and by the homogeneity condition deg α,p;β,q = p + q + 1 + µα + µβ . Here 1 µ1 = − , 2
µ2 =
1 . 2
In this paper we will consider the solutions to the extended Toda hierarchy in the class of formal series in k wkα (x, t), α = 1, 2, w α (x, t; ) = k≥0
t = (t 1,0 , t 2,0 , t 1,1 , t 2,1 , . . . ).
(1.18)
As it follows from the definition (1.3), A1,0 = ∂x modulo terms commuting with L. So ∂ ∂ , = ∂t 1,0 ∂x i.e., the solution depends on x, t 1,0 only via the combination x + t 1,0 . We will therefore often suppress the explicit dependence on x in the formulae. As it follows from Proposition 4.1 of [30], after the transformation of dependent variables u → u, ˆ v → v, ˆ v(x) ˆ = B+ v(x), u(x) ˆ = B+ B− u(x),
(1.19)
only even powers of remain in the -expansions of the right hand sides of equations of the hierarchy (1.2).
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Definition ([1]). For any solution v(x, t; ), u(x, t; ) of the extended Toda hierarchy there exists a function τ = τ (x, t; ) = e
g≥0
2g−2 Fg (x,t)
such that the functions α,p;β,q evaluated on this solution can be represented in the form α,p;β,q = 2
∂ 2 log τ ∂t α,p ∂t β,q
(1.20)
for any α, β = 1, 2, p, q ≥ 0. It is called the tau function of the solution (1.18) of the extended Toda hierarchy. In particular, we have hα,p = ( − 1)
∂ log τ , ∂t α,p+1
α = 1, 2, p ≥ −1,
(1.21)
∂ log τ ∂ τ (x + , t; ) , = 2,0 log 2,0 ∂t τ (x, t; ) ∂t τ (x + , t; )τ (x − , t; ) u = ( − 1)(1 − −1 ) log τ = log . τ 2 (x, t; )
v = ( − 1)
(1.22)
Another important property of this hierarchy is that, apart from its Hamiltonian structure described above, it also possesses a second Hamiltonian structure which is compatible with the first one (see (4.15) below). The bihamilonian structure and the tau symmetry property of the extended Toda hierarchy imply, due to a general theorem of [9], quasi-triviality of the extended Toda hierarchy. The precise formulation of the quasitriviality property in the case of interest is given by the following theorem (cf. Corollary 3.10.23 of [9]): Theorem 1.1. Any solution v, u of the extended Toda hierarchy is obtained from a solution v0 , u0 of the dispersionless extended Toda hierarchy through the quasi-Miura transformation of the form v = v0 +
g≥1
u = u0 +
(3g−2)
2g−1 ( − 1)
∂Fg (w0 , . . . , w0 ∂t 2,0
)
,
2g−2 ( − 1)(1 − −1 )Fg (w0 , . . . , w0
(3g−2)
),
(1.23)
g≥1
(cf. (1.19)) and the corresponding tau function of the solution admits the following genus expansion (3g−2) 2g−2 Fg (w0 , . . . , w0 ). (1.24) log τ = −2 log τ [0] + g≥1
In these formulae the functions Fg (w, wx , . . . , w(3g−2) ) do not depend on the choice of the solution.
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Here τ [0] = τ [0] (x, t) is the tau function for the solution v0 , u0 of the dispersionless extended Toda hierarchy, i.e., it is related to the leading term of the solution (1.18) w0 = (v0 , u0 ) via
∂ 2 log τ [0] (x, t) , ∂x∂t 2,0 ∂ 2 log τ [0] (x, t) u0 (x, t) = . ∂x 2
v0 (x, t) =
(1.25)
Recall that the dispersionless extended Toda hierarchy is obtained from the extended Toda hierarchy by setting = 0. All the flows of the dispersionless extended Toda hierarchy are systems of hydrodynamic type, i.e. systems of two first order quasilinear PDEs. For β = 1, q = 0 the dispersionless flow still coincides with the spatial translations (1.9). For β = 2, q = 0 one obtains ∂v ∂ u = e , 2,0 ∂t ∂x ∂u ∂v = . ∂t 2,0 ∂x Eliminating v yields the so-called long wave limit of the Toda lattice equations utt = eu xx , where t = t 2,0 . The dispersionless limit of (1.10) reads
∂v ∂ 1 2 u = v + (u − 1)e , ∂t 1,1 ∂x 2 ∂u ∂ (u v). = ∂t 1,1 ∂x Changing the sign of time t = −t 1,1 one identifies these with the equations of motion of the one-dimensional polytropic gas with the speed v and density u and the equation of state of the form p = (u2 − 2u + 2)eu − 2. It is time to remind the reader that the theory of dispersionless (extended) Toda hierarchy can be nicely encoded [5, 4] in terms of a particular two-dimensional Frobenius manifold MToda . The latter can be identified with the quantum cohomology of complex projective line MToda = QH ∗ (CP 1 ). Alternatively, the Frobenius manifold in question is isomorphic to the orbit space of the simplest extended affine Weyl group [6] MToda = C2 /W˜ (A1 ). Denote v, u the coordinates on MToda . The potential of the Frobenius manifold reads F =
1 2 v u + eu . 2
(1.26)
The third derivatives of the potential define the multiplication law of tangent vectors at each point of M such that ∂/∂v is the unity and ∂ ∂ ∂ · = eu . ∂u ∂u ∂v
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The flat metric on M reads < , >= 2du dv.
(1.27)
The Euler vector field is
∂ ∂ +2 . ∂v ∂u We will not recall here the universal construction, due to [3] of a “dispersionless” integrable hierarchy valid for an arbitrary Frobenius manifold M. The hierarchy can be considered as an infinite family of pairwise commuting flows on the loop space L(M). All these flows are represented by first order quasilinear PDEs; for this reason this hierarchy is called dispersionless. The word “hierarchy” means that the systems of integrable PDEs are organized by means of the action of a bihamiltonian recursion operator. In [9] we addressed the problem of extending the correspondence E=v
Frobenius manifolds → hierarchies of integrable PDEs to an arbitrary Frobenius manifold. We proved that, indeed such a universal correspondence exists for an arbitrary semisimple M provided a suitable completion of the loop space L(M) is made allowing to work with infinite order PDEs. By definition semisimplicity of a Frobenius manifold means that, for a generic point w ∈ M the algebra on Tw M is semisimple. We leave as an exercise to the reader to verify that MToda is a semisimple Frobenius manifold. So, according to the main result of [9], there exists an integrable hierarchy associated with the Frobenius manifold MToda . The main aim of the present paper is to identify this integrable hierarchy with the extended Toda hierarchy (1.2). The crucial role in such identification, apart from the Lax representation and tau structure obtained in [1], play Virasoro symmetries. According to our paper [8] for an arbitrary Frobenius manifold there exists a universal construction of second order linear differential operators ∂ Lm = Lm ( −1 t, ), m ≥ −1 ∂t satisfying the Virasoro commutation relations [Li , Lj ] = (i − j )Li+j ,
i, j ≥ −1.
(1.28)
For MToda these Virasoro operators are given by the following explicit expressions (cf. [12]): L−1 =
k≥1
L0 =
∂
t α,k
∂t α,k−1
k
k≥1
t 1,k
+
1 1,0 2,0 t t , 2
∂ ∂ + t 2,k−1 2,k−1 ∂t 1,k ∂t
+2
k≥1
t 1,k
∂ ∂t 2,k−1
+
1 1,0 2 t , 2
(m + k)! ∂ ∂2 Lm = t 1,k 1,m+k k! (m − k)! 2,k−1 2,m−k−1 + ∂t ∂t (k − 1)! ∂t k≥1 k=1 ∂ ∂ +t 2,k−1 2,m+k−1 + 2 αm (k)t 1,k 2,m+k−1 , m ≥ 1, (1.29) ∂t ∂t 2
m−1
k≥0
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where αm (0) = m!, αm (k) =
m+k (m + k)! 1 , k > 0. (k − 1)! j j =k
Again, we will not reproduce here the universal construction of the Virasoro operators; however, we will give below the free field realization of these operator for the MToda case obtained in Sect. 3.10.2 of our paper [9]. It is the Virasoro invariance property of a hierarchy of integrable PDEs that allows to reconstruct it uniquely starting from a given semisimple Frobenius manifold, along with bihamiltonian structure and existence of a tau function. That means that the linear action of the Virasoro operators onto tau functions defines symmetries of the hierarchy1 . The central result of this paper is the following Main Theorem. 1. The transformations τ → τ + δ Lm τ,
δ→0
(1.30)
for any m ≥ −1 are infinitesimal symmetries of the extended Toda hierarchy, i.e., given a tau function τ = τ (t; ) of a solution v(t; ), u(t; ) of the form (1.18), the functions ∂ Lm τ + O(δ 2 ), ∂t 2,0 τ Lm τ u(t; ˜ ) = u(t; ) + δ( − 1)(1 − −1 ) + O(δ 2 ) τ v(t; ˜ ) = v(t; ) + δ( − 1)
satisfy equations of the extended Toda hierarchy modulo terms of order O(δ 2 ). 2. For a generic solution (1.18) of the extended Toda hierarchy, the corresponding tau function satisfies the Virasoro constraints ∂ Lm −1 (t − c()), τ = 0, m ≥ −1 (1.31) ∂t for some c(). Here c() = (cα,p ()) is a collection of formal power series in . We will describe the class of generic solutions in Sect. 4. Note that in the above formulae we omit writing explicitly the x-dependence since x enters only through the combination x + t 1,0 . The above two theorems and the uniqueness result of [9] (see Theorem 3.10.31) imply Corollary 1.2. The hierarchy of PDEs associated, according to [9], with the Frobenius manifold MToda coincides with (1.2). (3g−2)
According to [9], the functions Fg = Fg (w0 ; . . . , w0 ) in the genus expansion (1.24) are uniquely determined by the loop equation that was introduced in Sect. 3.10.7 1 In Sect. 3.10.4 of [9] we called this property linearization of the Virasoro symmetries. The reason for this name was the following. The dispersionless hierarchy on L(M) constructed in [3] is always invariant with respect to an action of half of the Virasoro algebra [8]. However, the generators of the Virasoro action do not act linearly onto the dispersionless tau function τ [0] (t). The full hierarchy constructed in [9] is a deformation of the dispersionless one that transforms the nonlinear action of the Virasoro symmetries to the linear one.
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of [9] for any semisimple Frobenius manifold. For MToda the loop equation will be discussed in Sect. 5. It will also be explained how one can compute Gromov - Witten invariants of CP 1 and their descendents of any genus using the loop equation. Due to the uniqueness of solution of the loop equation and the validity of the Virasoro constraints for CP 1 Gromov-Witten invariants [20], we have the following Corollary 1.3. The generating function of the CP 1 Gromov-Witten invariants and their descendents is uniquely determined by the system of Virasoro constraints. It coincides with the logarithm of the tau function of a particular solution of the extended Toda hierarchy. This particular solution for the CP 1 Gromov-Witten invariants will be described in Sect. 5. Summarizing, we can say that, in the theory of Gromov - Witten invariants of CP 1 and their descendents the extended Toda hierarchy (1.2) plays the role similar to one played by the KdV hierarchy in the Kontsevich - Witten formulation [28, 21, 29] of the intersection theory of the tautological classes on the Deligne - Mumford moduli spaces of stable punctured curves. 2. Some Formulae for the Functions Ωα,p;β,q We present in this section some important identities for the functions α,p;β,q defined in (1.17) and for the Hamiltonians Hα,p . Let us first recall the definitions of [1] of the logarithmic x-derivatives of the dressing operators P and Q. Let us look for these logarithmic derivatives in the form Px P −1 = bj −j , j ≥1
Qx Q−1 =
cj j .
(2.1)
j ≥0
Lemma 2.1. There exist unique elements bj , cj ∈ A homogeneous of the degrees deg bj = j,
deg cj = −j
such that the operators (2.1) satisfy the following system of equations: res [Px P −1 , Lm ] − ∂x Lm = res [Qx Q−1 , Lm ] − ∂x Lm = 0,
m ≥ 1.
Proof. See [1]. For example, b1 = −B+ v,
c0 = B− u,
c1 = e−u(x+) B− v.
Lemma 2.2. The following identities hold true ∂α,p;β,q = α,p−1;β,q + α,p;β,q−1 + (δα,1 δβ,2 + δα,2 δβ,1 )δp,0 δq,0 ; ∂v
m≥0
v (m)
(2.2)
∂ ∂ + 2 α,p;β,q ∂u ∂v (m) γ
= (p + q + 1 + µα + µβ )α,p;β,q + Rαγ γ ,p−1;β,q + Rβ α,p;γ ,q−1 +2 δα,1 δβ,1 δp,0 δq,0 .
(2.3)
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γ Here the numbers µα and a 2 × 2 matrix R = Rβ are defined by 1 µ1 = −µ2 = − , 2
γ
γ
Rβ = 2δ2 δβ,1 ,
(2.4)
and we assume that α,p−1;β,q = β,q;α,p−1 = 0 when p = 0. Proof. Let us consider the difference operators B :=
∂bj ∂ P ∂x P −1 = − −j , ∂v ∂v j ≥1
∂cj ∂ C := Q∂x Q−1 = − j . ∂v ∂v j ≥0
The coefficients bj ∈ A, cj ∈ A were defined in (2.1). We want to show that LB = 1,
LC = −1.
(2.5)
Indeed, differentiating the identity [P ∂x P −1 , L] = 0 with respect to v we derive that the operators B and L commute. Therefore [B − P −1 P −1 , Lm ] = 0, Since b1 = −B+ v,
m ≥ 1.
∂b1 = −1. ∂v
So the expansion of the difference operator B − P −1 P −1 begins with −2 . From the equations res[B − P −1 P −1 , Lm ] = 0 for any m ≥ 1, we prove that the coefficient of −2 of the operator B − P −1 P −1 is a constant. Since deg
∂bj = j − 1, ∂v
the degree of this coefficient, as an element of A, is equal to 1. So, the coefficient must be equal to zero. Continuing this process we prove that B = P −1 P −1 = −1 + O(−2 ). In a similar way we can prove that C = −QQ−1 = −e−u(x+) + O(2 ). Now, using the definition (1.7) of log L we obtain 1 1 ∂ log L = (B − C) = (P −1 P −1 + QQ−1 ). ∂v 2 2
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Therefore ∂ log L = Lq−1 . ∂v From the last equation it readily follows that
2 ∂ 2 p L (log L − cp ) = Lp−1 (log L − cp−1 ), ∂v p! (p − 1)!
∂ 1 1 p+1 = Lp , p ≥ 0. L ∂v (p + 1)! p! Lq
(2.6)
p ≥ 1,
These two equations yield, together with h1,−1 = (1 − −1 )−1 ∂x u,
h2,−1 = v
and the definition (1.17), the formula (2.2). To prove the second formula of the lemma, let us introduce the operators H and E that act on the space of difference operators of the form bk k with coefficients in A as follows: H bk k = kbk k , (2.7) ∂bk ∂bk k E bk k = . v (m) (m) + 2 (2.8) ∂u ∂v m≥0
It is easy to see that these operators satisfy the Leibnitz rule bl l ) = P ak k ak k P bl l + b l l . P ( ak k )( (2.9) Here P = H or P = E. By our definition it follows that (H + E) L = L.
(2.10)
(H + E) log L = 1.
(2.11)
Due to Lemma 2.1
Now by using (2.10) and (2.11) we obtain 1 ( − 1)E2,p;1,q = Eres A1,q , Lp+1 (p + 1)! 1 = (H + E) res A1,q , Lp+1 (p + 1)! 1 1 = res qA1,q + 2A2,q−1 , Lp+1 + res A1,q , Lp+1 (p + 1)! p! = (p + q + 1)( − 1)2,p;1,q + 2( − 1)2,p;2,q−1 . (2.12) So, from the homogeneity condition for α,p;β,q we arrive at E2,p;1,q = (p + q + 1)2,p;1,q + 2 2,p;2,q−1 . Other cases of the formula (2.3) can be proved in a similar way. The lemma is proved.
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Lemma 2.3. The variational derivatives of the Hamiltonians Hβ,q are given by the following formulae δHβ,q ∂ log τ = hβ,q−1 = ( − 1) β,q , δv(x) ∂t 2 log τ δHβ,q ∂ = 2,0;β,q = 2 2,0 β,q . δu(x) ∂t ∂t
(2.13) (2.14)
Proof. From the Hamiltonian representation (1.12) of the extended Toda hierarchy we have δHβ,q ∂v 1 1 ∂u −1 δHβ,q = = − 1) , 1 − . (2.15) ( ∂t β,q δu ∂t β,q δv On the other hand, the formulae (1.21) and (1.22) imply that ∂v 1 = ( − 1) 2,0;β,q , ∂t β,q
∂u 1 −1 hβ,q−1 . 1 − = ∂t β,q
(2.16)
So, due to the homogeneity property of the densities of the Hamiltonians we arrive at the expressions (2.13) and (2.14). The lemma is proved. 3. Virasoro Operators for the Frobenius Manifold MToda Recall that, according to the general algorithm of [9] the Virasoro operators associated with a given Frobenius manifold are obtained by the following free field realization2 that we now present for the example of MToda . Let a1,p , a2,p , p ∈ Z be free bosonic operators satisfying the following commutation relation3 [a1,p , a2,q ] = (−1)p δp+q+1,0 .
(3.1)
Introduce vectors of operators p ∈ Z.
ap := (a1,p , a2,p ),
(3.2)
Consider the generating function f(z) :=
p∈Z
ap z
z =
p+µ R
p∈Z
ap
1
zp− 2 0 p+ 21 p+ 21 2z log z z
.
(3.3)
Here the diagonal matrix µ and nilpotent matrix R correspond to the spectrum at the origin of the Frobenius manifold MToda , 1 −2 0 , R = 0 0 . µ= (3.4) 20 0 21 2 3
In [8] we used a different free field realization of the Virasoro operators inspired by [10]. Note change of notations: in [9] we used half integer labels.
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The current φ (ν) (λ) for any non-integer ν is defined by a (suitably defined) Laplace-type transform ∞ dz −λ z φ (ν) (λ) = e f(z) 1 0 z 2 +ν ap
(p − ν) 0 = . 2 1 λp−ν λ (p − ν + 1) − log λ (p − ν + 1) λ (p − ν + 1) p∈Z
(3.5) (ν)
The generating function of the regularized Virasoro operators Lm is defined by the following quadratic combination of the derivatives of the currents T (ν) (λ) L(ν) m = m+2 λ m∈Z
T 1 1 1 (−ν) (ν) 2 −µ = : ∂λ φ G(ν) ∂λ φ : + 2 tr 2 4λ 4 1 = : ap Mpq (ν, λ) aqT :, where λp+q+3
(3.6)
p,q∈Z
Mpq (ν, λ)
sin πν 2π (p
0
+ ν + 1) (q − ν + 2)
= . − sin2ππν (q − ν + 1) (p + ν + 2) λ1 ∂ν sinππν (p + ν + 2) (q − ν + 2) In this formula the Gram matrix G(ν) reads 1 π i R π i(µ+ν) e + e−π i R e−π i(µ+ν) η−1 e G(ν) = 2π 1 0 sin πν , = π − sin πν 2π cos πν
(3.7)
η is the Gram matrix of the metric (1.27), i.e., in our case 01 −1 ; η=η = 10 the term with the trace in (3.6) does not contribute since µ2 = 41 . The normal ordering in (3.6) is defined in such a way that all the operators a1,p and a2,p with nonnegative p are to be written on the right. So
sin πν = : a a : ∂
(p + ν + 2) (q − ν + 2) L(ν) 2,p 2,q ν m π p+q=m−2
+
p+q=m−1
: a1,p a2,q :
sin πν [ (p + ν + 1) (q − ν + 2) 2π
− (p − ν + 1) (q + ν + 2)] .
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Using the standard identities of the theory of gamma-functions
(x + 1) = x (x),
(x) (1 − x) =
π , sin π x
we finally obtain the following expression for the regularized Virasoro operators
(−ν − p + m + 1) 1 (ν) p+1 (ν − p + m + 1) Lm = + : a1,p a2,m−p−1 : (−1) 2 p
(ν − p)
(−ν − p)
(m − ν − p) p + (−1) ∂ν (3.8) : a2,p a2,m−p−2 : .
(−ν − p − 1) p The operators (ν)
L−1 =
(−1)p+1 : a1,p a2,−p−2 : p
and
(ν)
L0 =
(−1)p p : a1,p a2,−p−1 : + (−1)p+1 : a2,p a2,−p−2 : p
p
do not depend on ν. The operators (3.8) with m > 0 depend polynomially on ν; those with m < −1 depend rationally on ν. Therefore there exist limits Lm := lim L(ν) m , ν→0
m ≥ −1.
To arrive at the Virasoro operators given above in the Introduction one use the following realization of the bosonic operators aα,p : p≥0 ∂t∂α,p , . (3.9) aα,p = −1 (−1)p+1 tα,−p−1 , p < 0 Here we use the matrix η for lowering the indices tα,p := ηαβ t β,p . In the next section we will prove that the linear action of the Virasoro operators Lm with m ≥ −1 defines infinitesimal symmetries of the extended Toda hierarchy. 4. Proof of the Main Theorem We first consider the following system of Euler-Lagrange equations:
t˜α,p
δHα,p−1 = 0, δv(x)
t˜α,p
δHα,p−1 =0 δu(x)
p≥0
p≥0
(4.1)
with t˜α,p = t α,p − cα,p () + δ1α δp,0 x
(4.2)
Virasoro Symmetries of the Extended Toda Hierarchy
175
for some formal power series cα,p (). Here and henceforth summations with respect to the repeated Greek indices are assumed. We assume that only finitely many of them are nonzero. The series must satisfy the condition of genericity that we shall now formulate. Let us expand the Hamiltonian densities (1.14) in powers of , hα,p = θα,p+1 (v, u) + O().
(4.3)
The explicit formulae for the functions θα,p (v, u) can be found in [9] (see also the formulae (A.12), (A.13) below). Let us impose the following assumptions for the leading terms of the series cα,p (). 1. There exist values v, ¯ u¯ such that
∂θα,p (v, u) c (0) = 0, ∂v v=v,u= ¯ u¯ p≥0 ∂θα,p (v, u) α,p c (0) = 0, ∂u v=v,u= ¯ u¯
α,p
p≥0
(4.4) and 2. The operator of multiplication by the vector cα,p (0)θα,p−1 (v, ¯ u) ¯ ∇
(4.5)
p≥1
is invertible element of the Frobenius algebra Tv, ¯ u¯ MToda . Under these assumptions the following lemma holds true (cf. Lemma 3.8.2 of [9]). Lemma 4.1. There exists a unique solution to the Euler - Lagrange equations (4.1) in the class of formal series v = v(x, t, ) = a0 () + aα1 ,p1 ;...;αk ,pk ()t α1 ,p1 . . . t αk ,pk |t 1,0 →t 1,0 +x , k>0
u = u(x, t, ) = b0 () +
bα1 ,p1 ;...;αk ,pk ()t α1 ,p1 . . . t αk ,pk |t 1,0 →t 1,0 +x , (4.6)
k>0
where a0 (0) = v, ¯
b0 (0) = u. ¯
(4.7)
Proof. In the leading order in the Euler - Lagrange equations (4.1) become just equations for the critical points of the function p (t α,p − cα,p (0) + xδ1α δ0 )θα,p (v, u). p≥0
The above two assumptions imply that there exists a unique critical point v0 = v0 (x, t, ), u0 = u0 (x, t, )
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of the function
p
(t α,p − cα,p () + xδ1α δ0 )θα,p (v, u)
p≥0
in the class of formal power series of the structure similar to (4.6) with v0 (0, 0, 0) = v, ¯ u0 (0, 0, 0) = u. ¯ It is easy to see that these functions can be uniquely extended to a solution to the full Euler - Lagrange equations (4.1). The lemma is proved. Lemma 4.2. The space of solutions of the Euler-Lagrange equation (4.1) is invariant with respect to the flows of the extended Toda hierarchy. Proof. Let us represent the difference operators Aβ,q defined in (1.3) by aβ,q;k k , β = 1, 2, q ≥ 0. Aβ,q =
(4.8)
k≥0
Then by using Lemma 2.3 we obtain δHα,p = aα,p;0 (x), δv(x)
δHα,p = aα,p;1 (x − ) eu(x) . δu(x)
(4.9)
The Lax pair representation (1.2) of the extended Toda hierarchy yields ∂Aα,p ∂Aβ,q − α,p = [Aβ,q , Aα,p ], β,q ∂t ∂t ∂eu = aβ,q;0 (x) − aβ,q;0 (x − ) eu(x) . β,q ∂t These equations together with (4.8) imply δHα,p δHβ,q ∂ ∂ = , α, β, γ = 1, 2; p, q ≥ 0. ∂t β,q δw γ (x) ∂t α,p δw γ (x)
(4.10)
So, under the flows of the extended Toda hierarchy we have ξ δHβ,q ∂ α,p δHα,p−1 δHβ,q−1 ∂ ∂w t˜ t˜α,p α,p−1. = + ∂m ∂t β,q δw γ (x) δw γ (x) ∂w ξ,m δw γ (x) x ∂t p≥0
m≥0
p≥1
(4.11) Here and below we use the following notations for the x-derivatives of the functions u and v ∂ m wξ wξ,m := , ξ = 1, 2, m ≥ 0. ∂x m So w 1,m = v (m) , w2,m = u(m) . By using (4.12) that we will prove in Lemma 4.4 below we know that the r.h.s. of (4.11) can be rewritten as δHβ,q−1 δHβ,q ∂ − ∂v δw γ (x) δw γ (x) which equals zero due to (4.9), (2.6) and the identity
∂L ∂v
= 1. The lemma is proved.
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Due to the uniqueness of solutions of the initial value problem for the Euler-Lagrange equation (4.1) and the above theorem, we have Theorem 4.3. Any solution of Eq. (4.1) gives a solution to the extended Toda hierarchy. Using quasitriviality it can be shown that the class of solutions of the extended Toda hierarchy that is given by the above theorem forms a dense subset of the class of its analytic solutions w α (x, t, ), α = 1, 2 (see Theorem 3.6.15 and 3.10.31 in [9]). We call this class of solutions the generic class of solutions of the extended Toda hierarchy, and we will restrict ourselves to it henceforth. Lemma 4.4. Any solution (v, u) of the Euler-Lagrange equation (4.1) satisfies the equations ∂v t˜α,p α,p−1 + 1 = 0, ∂t p≥1
p≥1
q
q≥1
q
q≥1
∂u = 0, ∂t α,p−1 ∂v ∂v 2,q−1 ∂v 1,q ˜t 1,q ˜ ˜ + 2t + v = 0, +t ∂t 1,q ∂t 2,q−1 ∂t 2,q−1 ∂u ∂u ∂u t˜1,q 1,q + t˜2,q−1 2,q−1 + 2t˜1,q 2,q−1 + 2 = 0. ∂t ∂t ∂t
t˜α,p
(4.12)
(4.13)
Proof. Equation (4.12) is the result of the application of the operators ∂t∂2,0 and 1 (−1) on the Euler-Lagrange equation (4.1). To prove Equation (4.13), we need to use the following bihamiltonian recursion relation of the extended Toda hierarchy [1]: αγ
U2
δHβ,q−1 1 αγ δHβ,q γ αξ δHγ ,q−1 = (q + µβ + )U1 + R β U1 . δw γ δw γ δw ξ 2
(4.14)
αβ
Here the first Hamiltonian structure U1 is defined in (1.13) and the second one is given by 1 ∂x u(x) 1 U211 = − eu(x) e− ∂x , U212 = v(x) e ∂x − 1 , e e 1 1 ∂x 21 − ∂x 22 U2 = − e− ∂x . v(x), U2 = (4.15) 1−e e The matrices R and µ are defined by (2.4) (see also (3.4)). Then Eq. (4.13) is obtained αγ by applying the operator U2 to both sides of the Euler-Lagrange equation (4.1) and by using the bihamiltonian recursion relation (4.14). The lemma is proved. Let v, u be any solution of the Euler-Lagrange equation (4.1) specified by a choice of the series cα,p () and of the leading term v, ¯ u¯ in (4.4). Due to Theorem 1.1 and Theorem 4.3 this solution can be obtained from a solution v0 , u0 of the dispersionless Toda hierarchy. Denote by τ [0] and τ the corresponding tau functions with the relation (3g−2) 2g−2 Fg (w0 , . . . , w0 ). (4.16) log τ = −2 log τ [0] + g≥1
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Note that the genus zero tau function τ [0] is defined up to multiplication by a function [0] α,p [0] of the form e cα,p t with constants cα,p . We now fix this ambiguity by taking log τ [0] =
1 [0] α,p;β,q (v0 , u0 )t˜α,p t˜β,q , 2
(4.17)
where [0] α,p;β,q = α,p;β,q =0 .
(4.18)
The validity of this definition for the tau function of the solution v0 , u0 of the dispersionless extended Toda hierarchy is based on the fact that v0 , u0 satisfy the genus zero Euler-Lagrange equation
t˜α,p
∂h[0] α,p−1 (v0 , u0 ) ∂v0
= 0,
t˜α,p
∂h[0] α,p−1 (v0 , u0 ) ∂u0
=0
(4.19)
with h[0] α,p−1 = hα,p−1 =0 . From this equation it readily follows that [0] α,p;β,q =
∂ 2 log τ [0] , ∂t α,p ∂t β,q
(4.20)
and that the genus zero tau function satisfies the string equation p≥1
t˜α,p
∂ log τ [0] + t˜1,0 t˜2,0 = 0. ∂t α,p−1
(4.21)
The proof of the above statement can be found in [5]. It was proved in [8] that such a tau function also satisfies the genus zero Virasoro constraints given by the Virasoro operators (1.29). The action of these operators on tau functions of the form (4.16) can be expressed as ∂ Lm ( −1 t˜, )τ = 2g−2 Zg τ, ∂t
m ≥ −1.
(4.22)
g≥0
The genus zero Virasoro constraints are given by Z0 = 0. We are to prove below that the tau function of a generic solution to the extended Toda hierarchy satisfies the full genera Virasoro constraints Zg = 0, g ≥ 0. Let us begin with the L−1 and L0 constraints. Lemma 4.5. The tau function (4.16) satisfies the constraints L−1 ( −1 t˜,
∂ ∂ [g] )τ = 0. L0 ( −1 t˜, )τ = c0 () = 2g−2 c0 ∂t ∂t g≥1
with a certain constant c0 ().
(4.23)
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179
2
Proof. Let us apply the operator ∂t σ,k∂ ∂t ρ,l to the l.h.s. of the first equation of (4.23). Using the definition (1.20) for the tau function and Eq. (4.12) we get 2 ∂ ∂ log τ 1 2 σ,k ρ,l t˜α,p α,p−1 + 2 t˜1,0 t˜2,0 ∂t ∂t ∂t p≥1
∂σ,k;ρ,l + δσ,1 δρ,2 + δσ,2 δρ,1 δk,0 δl,0 ∂t α,p−1 p≥1 ∂σ,k;ρ,l m ∂wξ t˜α,p = σ,k−1;ρ,l + σ,k;ρ,l−1 + ∂ ∂w ξ,m x ∂t α,p−1 p≥1 + δσ,1 δρ,2 + δσ,2 δρ,1 δk,0 δl,0 ∂σ,k;ρ,l = σ,k−1;ρ,l + σ,k;ρ,l−1 − + δσ,1 δρ,2 + δσ,2 δρ,1 δk,0 δl,0 ∂v = 0. (4.24) = σ,k−1;ρ,l + σ,k;ρ,l−1 +
t˜α,p
Here the last equality is due to (2.2). On the other hand, the Euler-Lagrange equation (4.1) implies that the l.h.s. of the first formula of (4.23) does not depend on t˜1,0 and t˜2,0 , [g] [g] so there exist constants c−1 , cα,p , α = 1, 2, p ≥ 0, g ≥ 1 such that p≥1
t˜α,p
∂ log τ 1 + 2 t˜1,0 t˜2,0 = α,p−1 ∂t
[g]
2g−2 cα,p−1 t˜α,p +
p≥1,g≥1
[g]
2g−2 c−1 . (4.25)
g≥1
Here the vanishing of the −2 term in the r.h.s. of the above identity is due to (4.21). Thus if we modify the tau function by τ → τ˜ = τ e−
p≥0,g≥1
2g−2 c[g] t˜α,p α,p
,
(4.26)
then we obtain L−1 ( −1 t˜,
∂ [g] 2g−2 c−1 )τ˜ . ) τ˜ = c−1 () τ˜ = ( ∂t
(4.27)
g≥1
[g]
We will prove the vanishing of the constants cα,p , c−1 () in a moment. By using the formula (2.3) and a similar argument as that given in the proof of (4.24), we can prove the validity of the following identity: ∂2 L0 τ˜ = 0. (4.28) ∂t σ,k ∂t ρ,l τ˜ Here
∂ ). ∂t So there exist constants c0 () and bα,p () such that L0 = L0 ( −1 t˜,
L0 τ˜ = bα,p ()t˜α,p + c0 (). τ˜ α,p
(4.29)
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By using the commutation relation [L−1 , L0 ] = −L−1 we obtain L−1
bα,p ()t˜α,p + c0 () τ˜ − L0 (c−1 () τ˜ ) = −c−1 ()τ˜ .
(4.30)
The l.h.s. of the above equality reads
bα,p−1 t˜α,p τ˜ +
p≥1
=
bα,p t˜α,p + c0 () L−1 τ˜ − c−1 ()L0 τ˜
bα,p−1 t˜α,p τ˜ .
(4.31)
p≥1
So from (4.30) it follows that bα,p−1 t˜α,p τ˜ = −c−1 () τ˜ ,
(4.32)
p≥1
from which we obtain c−1 () = bα,p () = 0. [g] Now we proceed to proving the vanishing of the constants cα,p . From the above argument we already have the identity L−1 ( −1 t˜,
2g−2 c[g] t˜α,p ∂ α,p = 0. ) τ e− g≥1 ∂t
At the genus one level we have
t˜α,p
p≥1
∂F1 (w, wx ) α,p [1] − t˜ cα,p−1 = 0. ∂t α,p−1
(4.33)
p≥1
Starting from this formula till the end of the proof of the lemma we will redenote for the sake of brevity the arguments w0 = (v0 , u0 ) and w0x = (v0x , u0x ) of the function F1 (w0 , w0x ) by w = (v, u) and wx = (vx , ux ). Since τ [0] satisfies the genus zero Virasoro constraints, we can use the vanishing of the genus zero Virasoro symmetries to obtain, as we did in [8, 9], the following formula:
t˜α,p
p≥1
∂F1 (w, wx ) ∂F1 =− . ∂t α,p−1 ∂v
(4.34)
Thus the identity (4.33) can be rewritten as −
[1] = t˜α,p cα,p−1
p≥1
By applying the operator
p ∂ p≥0 z ∂t α,p
∂F1 . ∂v
to the above identity we get
(4.35)
Virasoro Symmetries of the Extended Toda Hierarchy
−
181
[1] cα,p−1 zp
p≥1
∂ 2 F1 ∂θα,p+1 ∂ 2 F1 2 ∂θα,p+1 + zp ∂ ∂ x γ x ∂v∂w γ ∂wγ ∂wγ ∂v∂w x p≥0
2 ∂ F1 γ σ ρ ∂ 2 F1 ∂ 2 F1 γ σ1 ρ1 σ γσ ρ ρ2 ∂θα (z) = c w + . γ ∂x (cρ wx ) + z γ cρ wx cσ1 ρ2 wx ∂v∂w γ ρ x ∂w σ ∂v∂wx ∂v∂wx 1 (4.36) =
Here the functions θα,p = θα,p (w), cαβγ = cαβγ (w) are given by θα,p zp , θα,p = hα,p−1 |=0 , θα (z) = p≥0
cαβγ =
∂3
1 ( v 2 u + eu ), ∂w α ∂w β ∂w γ 2
(4.37)
and the raising of indices in cαβγ is done by the metric (1.27), i.e. η11 = η22 = 0, η12 = η21 = 1. In the above computation we used the horizontality of the differentials of the functions θα (w; z) w.r.t. the deformed flat connection on MToda , i.e. the equations ∂ 2 θα (z) ξ ∂θα (z) = zcβγ . ∂w β ∂w γ ∂w ξ
(4.38)
So from (4.36) we get ∂ 2 F1 γ σ ρ ∂ 2 F1 γσ ρ c w + γ ∂x (cρ wx ) = 0, ρ x ∂v∂w γ ∂v∂wx 2 ∂ F1 γ σ1 ρ1 σ [1] ρ2 ησ α = −cα,0 , γ cρ wx cσ1 ρ2 wx ∂v∂wx 1 and together with (4.36) these formulae in turn yield [1] p cα,p z = cγ[1],0 ∂ γ θα (z).
(4.39)
(4.40)
p≥0
By differentiating both sides of the above equation w.r.t. x we get γ
0 = cγ[1],0 cξ σ wxσ ∂ ξ θα (z) which implies cγ[1],0 = 0. So from (4.40) we obtain cγ[1],p = 0,
p ≥ 1.
In a completely similar way we can prove that [g]
cγ ,p = 0, The lemma is proved.
p ≥ 0, g ≥ 2.
(4.41)
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The following lemma represents the bihamiltonian recursion relation for the extended Toda hierarchy in terms of its tau functions: Lemma 4.6. The following recursion relations hold true for any q ≥ 1 for the tau functions of generic solutions to the extended Toda hierarchy: ∂ log τ ∂ log τ ∂ log τ = R 1,q−1 − 2 ( − 1) 2,q−1 , ∂t 1,q ∂t ∂t ∂ log τ ∂ log τ (q + 1) ( − 1) = R 2,q−1 , 2,q ∂t ∂t q ( − 1)
(4.42) (4.43)
where the operator R is defined by R = v(x)( − 1) + ( + 1)
∂ . ∂t 2,0
(4.44)
Proof. Denote Wβ,q := R
∂ log τ 1 ∂ log τ γ ∂ log τ − (q + µβ + )( − 1) β,q − ( − 1)Rβ γ ,q−1 . (4.45) ∂t β,q−1 2 ∂t ∂t
We are to prove that Wβ,q = 0 for β = 1, 2, q ≥ 1. From Lemma 2.3 and from the bihamiltonian recursion relation (4.14) with α = 2 we obtain by a direct calculation that ( − 1)Wβ,q = 0.
(4.46)
We note that Wβ,q can be expressed as homogeneous differential polynomials in w i,m , e±u , i = 1, 2, m ≥ 0 of degree q + 21 + µβ . Recall that the degree of such differential polynomials is defined in (1.16). So the lemma follows from the above equation (4.46). The lemma is proved. Proof of the Main Theorem. Let us first prove that the following recursion relation holds true: R
Lm τ Lm+1 τ = ( − 1) . τ τ
Here and below Lm = Lm ( −1 t˜,
∂ ). ∂t
From the definition of the operator R we have R
m−1
k!(m − k)!
k=1 m−1
=
k=1
+
1 ∂ 2τ τ ∂t 2,k−1 ∂t 2,m−1−k
k!(m − k)!
∂
∂t
R 2,k−1
∂ log τ ∂ log τ , R ∂t 2,k−1 ∂t 2,m−1−k
∂ log τ ∂ log τ ∂ log τ 2,m−1−k + R 2,m−1−k 2,k−1 ∂t ∂t ∂t m ≥ 1.
(4.47)
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So by using the recursion relations (4.42), (4.43) we can deduce the relation (4.47) for m ≥ 1 as follows: Lm τ 1 ∂ 2τ k!(m + 1 − k)! = 2 ( − 1) τ τ ∂t 2,k−1 ∂t 2,m−k k=1
∂ log τ ∂ 2 log τ ∂ log τ 2 − m!( − 1) 2,m−1 2,0 − m! ( − 1) 2,0 ( + 1) 2,m−1 ∂t ∂t ∂t ∂t
(m + k)! ∂ log τ ∂ log τ 1,k 2,k−1 + + t˜ ( − 1) (m + k + 1) t˜ (k − 1)! ∂t 1,m+k+1 ∂t 2,m+k k≥1 ∂ log τ ∂ log τ +2t˜1,k 2,m+k + (m + 1)!( + 1) 2,m ∂t ∂t ∂ log τ (m + k + 1)αm (k)t˜1,k ( − 1) 2,m+k +2 ∂t m
R
k≥0
∂ log τ ∂ 2 log τ + 2 2 αm (0) 2,m−1 2,0 2,m−1 ∂t ∂t ∂t 2 log τ Lm+1 τ ∂ = ( − 1) − 2 m!( − 1) 2,m−1 2,0 τ ∂t ∂t
∂ log τ ∂ log τ −m! ( − 1) 2,0 ( + 1) 2,m−1 ∂t ∂t ∂ log τ ∂ log τ +(m + 1)!( + 1) 2,m − 2(m + 1)! 2,m ∂t ∂t ∂ log τ ∂ 2 log τ +2αm (0)v 2,m−1 + 2αm (0) 2,m−1 2,0 ∂t ∂t ∂t ∂ log τ Lm+1 τ ∂ log τ + m! R 2,m−1 − (m + 1)!( − 1) 2,m = ( − 1) τ ∂t ∂t Lm+1 τ . (4.48) = ( − 1) τ It can be easily checked that the recursion relation (4.47) is also true for m = −1, 0. From (4.23) and the recursion relation (4.47) we know that L1 τ ( − 1) = 0. (4.49) τ +2αm (0)v
Since the function τ [0] satisfies the genus zero Virasoro constraints, it follows from (4.16) that L1 τ (m ) = 2g−2 Wg (w0 , w0 , . . . , w0 g ). (4.50) τ g≥1
Thus from (4.49) we arrive at the formula L1 τ [g] 2g−2 c1 = c1 () = τ g≥1
[g]
for some constants c1 .
(4.51)
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On the other hand, by using the the commutation relation (1.28) and Eqs. (4.23) we obtain L−1 (Lm τ ) = −(m + 1)Lm−1 τ,
L0 (Lm τ ) = (c0 () − m)Lm τ.
These formulae can be rewritten as Lm τ Lm−1 τ ˆ L−1 = −(m + 1) , τ τ
Lˆ 0
(4.52)
Lm τ Lm τ = −m . (4.53) τ τ 1,0 2 . By putting m = 1 into the t˜
Here Lˆ −1 = L−1 − 12 t˜1,0 t˜2,0 and Lˆ 0 = L0 − 12 above two relations we get L1 τ L1 τ L1 τ ˆ ˆ L−1 = −2 c0 (), L0 =− . τ τ τ
(4.54)
So from (4.51) we have c0 () = c1 () = 0, and we proved the vanishing of L0 τ and L1 τ . By using the recursion relation (4.47) with m = 1 we obtain L2 τ ( − 1) = 0. (4.55) τ Due to the same reason as the one we used to derive (4.51) we have L2 τ [g] 2g−2 c2 = c2 () = τ
(4.56)
g≥1
[g]
for some constants c2 . So by using the second formula in (4.53) we get L2 τ = 0. Now, the Virasoro commutation relation (1.28) implies the validity of all the Virasoro constraints ∂ τ = 0, m ≥ −1. Lm −1 t˜, ∂t It remains to prove that linear action of the Virasoro operators (1.29) defines infinitesimal symmetries of the extended Toda hierarchy. To this end we observe that ∂ ∂ ∂ Lm −1 (t − c()), = Lm −1 t, − a α,p α,p − bα,p t α,p − c, ∂t ∂t ∂t where a α,p , bα,p and c are some series in that may also depend on m. Note that the a series contains only nonnegative powers of . From the already proven Virasoro constraints it follows that, for any generic solution to the extended Toda hierarchy the action of the Virasoro operators on the tau function can be recast as ∂ −1 α,p ∂ α,p Lm t, τ= a + bα,p t + c τ. ∂t ∂t α,p So, for a small parameter δ α,p ∂ ∂ α,p −1 τ + δ Lm t, τ = eδ(bα,p t +c) eδ a ∂t α,p τ + O(δ 2 ). ∂t
(4.57)
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185
The operator eδ a
α,p
∂ ∂t α,p
is nothing but the shift along trajectories of the hierarchy. Note that such a shift leaves invariant the class of generic solutions. Multiplication by the exponential of a linear function in the times for obvious reasons maps a tau function to another one for the same solution to the hierarchy. This proves that (4.57) is again a tau function of some solution of the extended Toda hierarchy. The theorem is proved. 5. The Topological Solution of the Extended Toda Hierarchy Let us briefly recall the definition of Gromov - Witten invariants and their descendents and the construction of Witten’s generating function (physicists call it the free energy of the two-dimensional CP 1 topological sigma model). Denote φ1 = 1 ∈ H 0 (CP 1 ), φ2 = ω ∈ H 2 (CP 1 ) the basis in the cohomology space H ∗ (CP 1 ). The 2-form ω is assumed to be normalized by the condition ω = 1. CP 1
The free energy of the CP 1 topological sigma-model is a function of infinite number of coupling parameters t = (t 1,0 , t 2,0 , t 1,1 , t 2,1 , . . . ) and of defined by the following genus expansion form: 2g−2 Fg (t). F(t; ) =
(5.1)
g≥0
The parameter is called here the string coupling constant, and the function Fg = Fg (t) is called the genus g free energy which is given by Fg =
1 t α1 ,p1 . . . t αm ,pm τp1 (φα1 ) . . . τpm (φαm ) g , m!
(5.2)
where τp (φα ) are the gravitational descendents of the primary fields φα , t α,p is the corresponding coupling constants, and the rational numbers τp1 (φα1 ) . . . τpm (φαm ) g are given by the following intersection numbers on the moduli spaces of CP 1 -valued stable curves of genus g: τp1 (φα1 ) . . . τpm (φαm ) g = qβ β
[M¯ g,m (CP 1 ,β)]virt
ev∗1 φα1 ∧ ψ1 1 ∧ · · · ∧ ev∗m φαm ∧ ψmm . p
p
(5.3)
Here M¯ g,m (CP 1 , β) is the moduli space of stable curves of genus g with m markings of the given degree β ∈ H2 (CP 1 ; Z), evi is the evaluation map evi : M¯ g,m (CP 1 , β) → CP 1 corresponding to the i th marking, ψi is the first Chern class of the tautological line bundle over the moduli space corresponding to the i th marking. According to the divisor axiom
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[22] the dependence of the Gromov - Witten potential on the indeterminate q appears 2,0 only through the combination q et . We will therefore omit the dependence on q in the formulae. Let us clarify the relationship between our theory of Virasoro symmetries of the extended Toda hierarchy and the Virasoro conjecture of T.Eguchi, K.Hori, and C.-S.Xiong [12, 13] extended by S.Katz. Denote ZCP 1 (t; ) := eF (t;) the partition function of the CP 1 topological sigma-model. Here F(t; ) is the generating function of the CP 1 Gromov - Witten invariants and their descendents defined above. According to the results of A.Givental [19, 20] this partition function satisfies the following infinite sequence of linear Virasoro constraints: L−1 ZCP 1 =
∂ Z 1, ∂t 1,0 CP
Lm ZCP 1 = (m + 1)!
∂
∂t 1,m+1
where κm =
+ 2κm
m+1 j =1
∂ ∂t 2,m
ZCP 1 ,
m ≥ 0,
(5.4)
1 . j
Here Lm are just the Virasoro operators defined in (1.29). For the particular case of CP 1 (5.4) coincide with the Virasoro constraints conjectured in [12]. However, in their papers Eguchi, Hori and Xiong formulated a somewhat bolder conjecture that says that all g ≥ 1 Gromov - Witten invariants and their descendents of a smooth projective variety can be uniquely determined by solving recursively the linear system of Virasoro constraints. Although this conjecture seems to be too nice to be true in general (Calabi - Yau manifolds give counterexamples to uniqueness, see [2]), in certain cases it can be justified. Let us give our version of the Eguchi - Hori - Xiong Virasoro constraints programme adapted to computing Gromov - Witten invariants of CP 1 . Step 1. Computation of the genus zero Gromov - Witten potential F0 (t). This can be done in terms of the Frobenius manifold MToda as in [3, 4]. For the reader’s convenience we recall the algorithm of computation of F0 (t) in the Appendix below. Introduce functions ∂ 2 F0 (t) , ∂t 1,0 ∂t 2,0 ∂ 2 F0 (t) u0 = u0 (t) := 1,0 1,0 . ∂t ∂t
v0 = v0 (t) :=
We will denote u 0 , v0 , u
0 , v0
etc. the derivatives of these functions along t 1,0 . Step 2. Eguchi - Hori - Xiong (3g − 2)-ansatz for the higher genus corrections. Look for the g ≥ 1 terms in the genus expansion (5.1) in the form Fg (t) = Fg (v0 (t), u0 (t), v0 (t), u 0 (t), . . . , v0
(3g−2)
(3g−2)
(t), u0
(t)),
g ≥ 1.
(5.5)
The ansatz (5.5) was proved by E. Getzler in [18]. In the setup of our theory [9] of integrable systems the (3g − 2)-ansatz is a consequence of a deep result about quasitriviality of tau-symmetric deformations of Poisson pencils (see Theorem 3.9.5 in [9]).
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Step 3. Virasoro Conjecture. Part 1. The series (5.1), where F0 (t) is the genus zero Gromov - Witten potential and the terms of positive genera have the form (5.5), satisfies the Virasoro constraints (5.4). (Clearly the (3g − 2)-ansatz is of no importance so far.) Part 2. The degree 2g − 2 homogeneous functions Fg on the (3g − 2) jet space of MToda for all g ≥ 1 are uniquely determined from the Virasoro constraints (5.4) by solving recursively systems of linear equations. Part 1 of the Virasoro Conjecture was proved by A. Givental [19, 20]. Part 2 was proved in the much more general framework of an arbitrary semisimple Frobenius manifold in Theorem 3.10.20 of [9]. Combining these results we arrive at Theorem 5.1. 1. The partition function ZCP 1 (t; ) of the CP 1 topological sigma-model is uniquely determined by the Virasoro Conjecture equations. 2. It coincides with the tau function τCP 1 of a particular solution to the extended Toda hierarchy (1.2) ZCP 1 (t; ) = τCP 1 (t; )
(5.6)
specified by the following choice of the shift parameters cα,p () and the initial point v, ¯ u: ¯ p
cα,p () = δ1α δ1 ,
v¯ = u¯ = 0.
(5.7)
The choice (5.7) selects the solution satisfying the string equation
t α,p
p≥1
∂F 1 ∂F + 2 t 1,0 t 2,0 = 1,0 . ∂t α,p−1 ∂t
(5.8)
Proof. As it was shown in Sect. 3.10.7 of [9], from validity of the Virasoro constraints for the sum F of all g ≥ 1 corrections to the Gromov - Witten potential represented via the (3g − 2)-ansatz, F :=
2 g Fg (v, u; vx , ux , . . . , v (3g−2) , u(3g−2) ),
g≥1
the loop equation below (cf. Example 3.10.27 in [9]) follows: ∂F
−λ ∂F r 1 − 2 (r) ∂x D D ∂u r≥0 r ∂F r−k+1 v − λ ∂F r−k+1 1 r k−1 1 ∂x √ ∂ ∂ + − 2 √ √ x k ∂u(r) x D ∂v (r) D D r≥1 k=1 = D −3 eu 4 eu + (v − λ)2 v ∂r ∂v (r) x
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2 ∂ 2 F v−λ v−λ ∂F ∂F ∂xk+1 √ ∂xl+1 √ + + (k) − 4 ∂v (k) ∂v (l) ∂v ∂v (l) D D k,l 2 ∂F ∂F v−λ 1 ∂ F ∂xk+1 √ ∂xl+1 √ + (k) +4 (k) (l) (l) ∂v ∂u ∂v ∂u D D 2 ∂F ∂F ∂ F k+1 1 l+1 1 ∂ + −4 ∂ √ √ x x ∂u(k) ∂u(l) ∂u(k) ∂u(l) D D 2 u
2 ∂F k+1 u 4 e (v − λ)u − [(v − λ) + 4 eu ] v
∂ e − 2 D3 ∂v (k) x k ∂F k+1 u 4 (v − λ) v − [(v − λ)2 + 4 eu ] u
, + (k) ∂x e D3 ∂u
(5.9)
where D = (v − λ)2 − 4 eu . Here λ is an arbitrary complex parameter. Expanding the loop equation near λ = ∞ reproduces the Virasoro constraints for F. The proof of existence and uniqueness of the solution to this equation is based on expanding the loop equation near zeroes u± = v ± 2eu/2 of D (these are the canonical coordinates on the Frobenius manifold MToda ). The uniqueness of the solution to the loop equation proves the first part of the theorem. To prove the second part we use the following arguments. From [4, 15, 11, 14] we already know that τ [0] (t) := F0 (t) is the tau function of the dispersionless extended Toda hierarchy. This solution is specified by the shift parameters and the leading term (5.7). The transformation log τ [0] → log τ [0] + F =: 2 log τ
(5.10)
maps dispersionless tau functions to tau functions of the full hierarchy associated with the semisimple Frobenius manifold MToda . The full hierarchy is uniquely determined for the given semisimple Frobenius manifold by the following properties: – bihamiltonian structure satisfying certain nondegeneracy conditions; – tau symmetry that provides existence of a tau function for a generic solution; – invariance with respect to the linear action of the Virasoro operators Lm , m ≥ −1 onto the tau functions. As we explained in the Introduction, the first two properties are met by the extended Toda hierarchy due to results of [1]. The last property of Virasoro invariance is established in the present paper. This implies that the full hierarchy associated with the Frobenius manifold MToda coincides with the extended Toda hierarchy. Therefore the transformation (5.10) maps the tau function τ [0] of an arbitrary solution of the dispersionless hierarchy to the tau function τ of a solution of the full extended Toda hierarchy. Taking τ [0] = F0 one obtains τ = ZCP 1 . The theorem is proved.
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Clearly the theorem covers Corollaries 1.2 and 1.3 formulated in the Introduction. To illustrate the algorithm of computation of the genus expansion (5.1) for CP 1 let us write down the first two terms of the expansion. The formulae become simpler when written in the canonical coordinates u0
u± = v0 ± 2e 2 . Genus 1: F1 =
1 1 u+ − u− log u + u − − log . 24 12 4
Genus 2: 242 F2 =
4 u
+ 3 (u+ − u− ) 5 u + 4
3 u
+ 1 + 4 u + 3 2
3 u
− 1 + 4 u − 3 2 +
1
−
4 u
− 3 (u+ − u− )
u
+ u − − u
− u +
−
5 u − 4
u
+ u
− 4 u + u −
7
u (u+ − u− ) 5 +
7 + u
(u+ − u− ) 5 −
33
2 9
1
u − u u + u u + uI+V (u+ − u− ) 10 + 10 + − 10 + −
4 u + 2
33
2 9
1
1 IV − u + u − u (u − u ) + u u u + − − − 10 − + 10 + − 4 u − 2 10 1 17
1
17
1
1 −
u+ + u− − u + u
4 u+ 5 2 4 u− 5 − 2 + 3 u + u − 3 1 1 11
2
2 − − u+ − u u + u− +
10 (u+ − u− )2 u − u+ (u+ − u− )2 5 + − u
+ − u
− u − u + + + +1 . (5.11) u+ − u− 5 u + 5 u −
Remark 1. In [16], Getzler proved that, under the assumption of the recursion relation (4.43), validity of the Virasoro constraints for τCP 1 is equivalent to (4.42). In his proof a recursion relation of the form (4.47) was used. The recursion (4.43) for τCP 1 was proved in [24] on the subspace {t 1,k = 0, k > 1} of the large phase space of all couplings. Using this result Getzler also proved (4.42) and (4.43) under the assumption of the Virasoro constraints for τCP 1 . He did not consider connections between recursion relations and Virasoro constraints for other solutions to the extended Toda hierarchy. Our Corollary 1.3 shows that the recursion relations (4.42), (4.43) for τCP 1 follow directly from validity of the Virasoro constraints. Remark 2. In [25] A. Okounkov and R. Pandharipande proved that the Gromov - Witten potential of the equivariant GW invariants of CP 1 and their descendents is the logarithm of the tau function of the 2D Toda hierarchy of K. Ueno and K. Takasaki [27]. The tau function of [25] depends on an additional small parameter t. The non-equivariant limit corresponds to t → 0. It would be interesting to derive the Lax representation of the extended Toda lattice by applying a suitable limiting procedure to that of the 2D
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Toda hierarchy of [27]. An interesting construction, due to Getzler [17] of a nontrivial reduction of the 2D Toda hierarchy depending on the parameter t (it was called the equivariant Toda lattice) could give a clue to such a limiting procedure. We plan to study the relationships between 2D and extended Toda hierarchies in a subsequent publication. Appendix. Genus Zero Gromov-Witten Potential of CP 1 To compute the genus zero Gromov-Witten potential F0 (t) according to the general scheme of [3, 4] one is to perform the following computations (cf. Sects. 3.6.2, 3.6.10 of [9]). 1. Compute the functions θα,p (v, u) as the coefficients of expansion of the following series: θ1 (v, u; z) = θ1,p (v, u)zp p≥0
1 1 = −2 ezv K0 (2ze 2 u ) + (log z + γ )I0 (2ze 2 u ) = −2ez v θ2 (v, u; z) =
m≥0
z2 m 1 (γ − u + ψ(m + 1))em u , 2 (m!)2
(A.12)
θ2,p (v, u)zp
p≥0
= z−1 ezv I0 (2ze 2 u ) − z−1 = z−1 1
m≥0
em u+z v
z2 m (m!)2
− 1 .
(A.13)
Here γ denotes Euler’s constant, ψ(z) stands for the digamma function, K0 (x) and I0 (x) are modified Bessel functions. 2. Compute the functions [0] α,p;β,q (v, u) as the coefficients of the following generating series [0] α,p;β,q (v, u)zp w q p,q≥0
=
∂θα (v, u; z) ∂θβ (v, u; w) ∂θα (v, u; z) ∂θβ (v, u; w) 1 + − ηαβ . z+w ∂v ∂u ∂u ∂v (A.14)
3. Define the functions v(t), u(t) as the unique solution of the system
∂θβ,q , ∂u ∂θβ,q u= t β,q ∂v v=
t β,q
having the expansion v(t) = t 1,0 + o(t), u(t) = t 2,0 + o(t).
(A.15)
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4. The genus zero Gromov - Witten potential of CP 1 is given by F0 (t) = Here
1 α,p β,q [0] t˜ t˜ α,p;β,q (v(t), u(t)). 2
1,1 t − 1, t˜α,p = t α,p ,
α = 1, p = 1 otherwise.
(A.16)
(A.17)
Let us write the first few terms of the expansion of the resulting genus zero Gromov Witten potential. For simplicity we denote tp := t 1,p ,
sp := t 2,p ,
p ≥ 0.
The potential is expanded in powers of tp , sp and in es0 . The powers of the exponential separate the intersection numbers on the moduli spaces M¯ 0,m (CP 1 , β) for different β. Thus, the terms without exponential correspond to β = 0 etc. In our expansion we collect the intersection numbers up to degree β = 3 and up to m ≤ 4 punctures. We also restrict the potential onto the subspace of couplings t α,p with p ≤ 3. The maximal number of descendents is restricted to three. F0 =
t02 s0 t 2 t 1 s0 t 3 s1 t 3 t 2 s0 t 4 s2 t 2 t 2 s0 + 0 + 0 + 0 + 0 + 0 1 2! 2! 3! 3! 4! 2! t04 t3 s0 t05 s3 t03 t1 t2 s0 t04 t2 s1 t 4 t1 s2 t03 t1 s1 + + +3 +3 +3 0 +2 4! 5! 3! 4! 4! 3! 2 2 t t0 t3 +es0 1 − 2 t1 + 2 1 − 2 t0 t2 + t1 s0 + t0 s1 − 2 2! 2! t1 2 s0 t 0 2 t1 t3 t 0 2 s2 t 0 2 t 3 s0 + t0 t2 s0 + −2 − t 0 t 1 t 2 s0 + 2! 2! 2! 2! t1 2 s0 2 t0 2 t1 s2 t0 3 s3 t 0 2 t 2 s1 t 0 2 s1 2 +2 + t + + − t s s + 0 1 0 1 (2!)2 2! 2! 2! 3!
2 3 3 1 5 t2 1 3 1 s1 2 +e2s0 − t3 + s2 + + t1 t3 + t3 s0 − t2 s1 + 4 4 4 2! 4 4 4 2 2!
−2
1 1 3 t2 2 s0 3 − t1 s2 + t0 s3 + 2 t0 t2 t3 − − t1 t3 s0 − 2 t0 t3 s1 4 4 2 2! 2 1 t 0 2 t3 2 1 + t2 s0 s1 − t0 t2 s2 + t1 s0 s2 + t0 s1 s2 + 2 t0 t1 t2 t3 + 4 2 (2!)2 2 2 2 2 2 2 t 1 t 2 s0 t0 t2 s1 t 2 s0 t1 t3 s0 − 3 t0 t2 t3 s0 + + + + 2! (2!)2 2! 2! 2 t 0 t 2 s1 t1 s0 s1 2 −2 t0 t1 t3 s1 − t1 t2 s0 s1 + t0 t3 s0 s1 − 2 + 2! 2! t0 s1 3 t0 2 t3 s2 +3 − t 0 t 1 t 2 s2 − 3 + t0 t2 s0 s2 + t0 t1 s1 s2 3! 2! t 0 2 t2 s 3 t0 t 1 s 0 s3 3 t0 2 s 1 s3 t 0 2 s2 2 − + + +2 (2!)2 2! 2 2 2!
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+e
3s0
50 t3 2 1 s2 2 1 t 2 2 t3 7 2 14 t3 2 s0 − t3 s2 + − t2 s3 + s1 s3 − 2 − 27 2! 9 3 2! 9 6 2! 9 2!
t 3 s1 2 s 1 2 s2 t 2 2 s2 1 + + t 3 s0 s2 − t 2 s 1 s2 + − t0 t3 s3 2! 2! 3 2! 1 1 t 0 t2 t3 2 t 2 2 t 3 s0 t1 t3 2 s 0 2 t 3 2 s0 2 + t 2 s 0 s3 + t 0 s2 s3 − 4 +5 +4 + 6 2 2! 2! 2! 3 (2!)2 2 2 2 t0 t3 s1 t 3 s0 s 1 t 2 s0 s2 − 3 t2 t3 s0 s1 + +8 + 3 t0 t2 t3 s2 − 2 2! 2! 2! 2 t0 t2 s2 t1 s 0 s 2 2 −2 t1 t3 s0 s2 − 5 t0 t3 s1 s2 + t2 s0 s1 s2 − 2 + 2! 2! t0 s1 s2 2 1 1 t0 t2 2 s3 +3 + − t0 t1 t3 s3 − t1 t2 s0 s3 + t0 t3 s0 s3 2! 2! 2 2 3 1 t 0 s1 2 s 3 t 0 2 s3 2 1 . − t 0 t 2 s1 s3 + t 1 s0 s 1 s3 + 2 + t0 t1 s2 s3 + 2 2 2! 2 (2!)2 +2 t2 t3 s1 − 2
Acknowledgements. The research of B.D. was partially supported by Italian Ministry of Education research grant Cofin2001 “Geometry of Integrable Systems”. The research of Y.Z. was partially supported by the Chinese National Science Fund for Distinguished Young Scholars grant No.10025101 and the Special Funds of Chinese Major Basic Research Project “Nonlinear Sciences”. Y.Z. thanks Abdus Salam International Centre for Theoretical Physics and SISSA where part of the work was done for their hospitality. The authors are grateful to the referee for the suggested improvements of the paper.
References 1. Carlet, G., Dubrovin, B., Zhang, Y.: The extended Toda hierarchy. nlin-SI/0306060, Moscow Math. J., to appear. 2. Cox, D., Katz, S.: Mirror symmetry and algebraic geometry. Mathematical Surveys and Monographs, 68. Providence, RI: American Mathematical Society, 1999 3. Dubrovin, B.: Integrable systems in topological field theory. Nucl. Phys. B 379, 627–689 (1992) 4. Dubrovin, B.: Integrable systems and classification of 2-dimensional topological field theories. In: Integrable systems, Proceedings of Luminy 1991 conference dedicated to the memory of J.-L. Verdier, O. Babelon, P. Cartier, Y. Kosmann-Schwarzbach, (eds.), Basel-Boston: Birkh¨auser, 1993 5. Dubrovin, B.: Geometry of 2D topological field theories. In: Integrable Systems and Quantum Groups, Montecatini Terme, 1993. M.Francaviglia, S. Greco, (eds.), Springer Lecture Notes in Math. 1620, Berlin-Heidelberg-New York: Springer-Verlag, 1996, pp. 120–348 6. Dubrovin, B., Zhang, Y.: Extended affine Weyl groups and Frobenius manifolds. Compositio Math. 111, 167–219 (1998) 7. Dubrovin, B., Zhang,Y.: Bihamiltonian hierarchies in 2d topological field theory at one loop approximation. Commun. Math. Phys. 198, 311–361 (1998) 8. Dubrovin, B., Zhang, Y.: Frobenius manifolds and Virasoro constraints, Selecta Mathematica. New series 5, 423–466 (1999) 9. Dubrovin, B., Zhang,Y: Normal forms of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants. math/0108160 10. Eguchi, T., Jinzenji, M., Xiong, C.-S.: Quantum cohomology and free field representation. Nucl. Phys. B 510, 608–622 (1998) 11. Eguchi, T., Hori, K., Yang, S.-K.: Topological σ models and large-N matrix integral. Internat. J. Modern Phys. A 10, 4203–4224 (1995) 12. Eguchi, T., Hori, K., Xiong, C.-S.: Quantum cohomology and Virasoro algebra. Phys. Lett. B 402, 71–80 (1997) 13. Eguchi, T., Xiong, C.-S.: Quantum cohomology at higher genus: topological recursion relations and Virasoro conditions. Adv. Theor. Math. Phys. 2, 219–229 (1998) 14. Eguchi, T., Yamada, Y., Yang, S.-K.: On the Genus Expansion in the Topological String Theory. Rev. Math. Phys. 7, 279 (1995)
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15. Eguchi, T., Yang, S.-K.: The Topological CP 1 Model and the Large-N Matrix Integral. Mod. Phys. Lett. A 9, 2893–2902 (1994) 16. Getzler, E.: The Toda conjecture, math.AG/0108108. In: Symplectic geometry and mirror symmetry (Seoul, 2000), River Edge, NJ: World Sci. Publishing, 2001, pp. 51–79 17. Getzler, E.: The equivariant Toda lattice. I, math.AG/0207025; II, math.AG/0209110 18. Getzler, E.: The jet-space of a Frobenius manifold and higher-genus Gromov - Witten invariants. math.AG/0211338 19. Givental, A.: Semisimple Frobenius structures at higher genus. Intern. Math. J. 48, 295–304 (2000) math.AG/0008067 20. Givental, A.: Gromov - Witten invariants and quantization of quadratic Hamiltonians. Moscow Math. J. 1, 1–23 (2001). math.AG/0108100 21. Kontsevich, M.: Intersection theory on the moduli space of curves and the matrix Airy function. Commun. Math. Phys. 147, 1–23 (1992) 22. Kontsevich, M., Yu.Manin: Gromov-Witten classes, quantum cohomology and enumerative geometry. Commun. Math. Phys. 164, 525–562 (1994) 23. Okounkov, A.: Generating functions for intersection numbers on moduli spaces of curves. math.AG/0101201 24. Okounkov, A., Pandharipande, R.: Gromov-Witten theory, Hurwitz numbers, and matrix models I. math.AG/0101147 25. Okounkov, A., Pandharipande, R.: The equivariant Gromov-Witten theory of P 1 . math.AG/0207233 26. Toda, M.: Wave propagation in anharmonic lattices. J. Phys. Soc. Japan 23, 501–506 (1967) 27. Ueno, K., Takasaki, K.: Toda lattice hierarchy. In: Group representations and systems of differential equations (Tokyo, 1982), Adv. Stud. Pure Math. 4, Amsterdam: North Holland, 1984, pp. 1–95 28. Witten, E.: Two-dimensional gravity and intersection theory on moduli space. Surv. in Diff. Geom. 1, 243–310 (1991) 29. Witten, E.: On the Kontsevich model and other models of two-dimensional gravity. Preprint IASSNS-HEP-91/24 30. Zhang, Y.: On the CP 1 topological sigma model and the Toda lattice hierarchy. J. Geom. Phys. 40, 215–232 (2002) Communicated by L. Takhtajan
Commun. Math. Phys. 250, 195–213 (2004) Digital Object Identifier (DOI) 10.1007/s00220-004-1149-9
Communications in
Mathematical Physics
Hochschild- and Cyclic-Homology of LCNT-Spaces Christian-Oliver Ewald University of Kaiserslautern, Department of Mathematics, 67653 Kaiserslautern, Germany Received: 19 August 2003 / Accepted: 24 February 2004 Published online: 5 August 2004 – © Springer-Verlag 2004
Abstract: We define a class of topological spaces ( LCNT spaces ) which come together with a nuclear Fr´echet algebra. Like the algebra of smooth functions on a manifold, this algebra carries the differential structure of the object. We compute the Hochschild homology of this algebra and show that it is isomorphic to the space of differential forms. This is a generalization of a result obtained by Alain Connes in the framework of smooth manifolds. 1. Introduction In this paper we present a result about the Hochschild- and cyclic homology groups of certain algebras corresponding to certain singular spaces, which we will call locally coned topological spaces or in short “lcnt-spaces”. These algebras represent in some sense the “smooth” functions on these spaces. The results are a slightly modified version of similar results about locally coned stratifolds ( see [Ew, Kr1 and Kr2] ) which were part of my PhD thesis. Some of the results have already been included in a sectional talk which I gave at the DMV-Jahrestagung 2000 in Dresden. The remaining results were obtained within the years 2000 and 2001 when I was a member of the Topology group at the University of Heidelberg. Theorem 1. Let (X, C∞ X ) be a finite dimensional lcnt-space, then the antisymmetrization map : ∗C∞ → H H∗ (C∞ X) X
is a topological isomorphism from the space of differential forms over the algebra C∞ X to its Hochschild homology, where differential forms are defined by Definition 3.4. Theorem 2. If in addition to the conditions above X has finite dimensional homology groups, then for all n ∈ N there is a natural topological isomorphism n−2 H Cn (C∞ ) ∼ (X, C) ⊕ H n−4 (X, C) ⊕ H n−6 (X, C).... = n ∞ /dn−1 ∞ ⊕H X
CX
CX
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These results are generalizations of results obtained byAlain Connes about the Hochschild- and cyclic-homology of the algebra of smooth functions on a compact manifold (see [Co]). Related results in the context of Whitney-stratified spaces have appeared in [Bra/Pf]. The result in this paper is in a quite general form, but can for example be applied on spaces which are locally coned stratifolds (see [Ew]). It is however not clear in what sense lcnt-spaces are stratified spaces and if Whitney stratified spaces provide examples for lcnt-spaces. The latter should be true under some mild extra conditions for the class of so called pseudo-manifolds, but we won’t give a proof for this. Besides being a generalization of a famous result (Connes’) the result was intended to be a first step toward a generalization of the concept of non-commutative geometry to non-smooth (“singular”) spaces which may or may not have applications in theoretical physics. I thank Prof. Dr. Matthias Kreck and Dr. Anna Grinberg who have both actively contributed to this work. 2. LCNT-Spaces Given a topological space X, we consider the algebra C(X) of continuous complex valued functions on X together with the topology of uniform convergence on compact subsets of X. Definition 2.1. A sub-algebra A of C(X) is called ample, if the following two conditions hold: 1. A is dense in C(X). 2. For any x ∈ X and any open neighborhood V of x, there exists an element a ∈ A such that supp(a) ⊂ V and a(x) = 0. In case the space X is paracompact it is well known that the second condition above implies that for any open covering of X there exists a subordinated partition of unity. Any topological space considered in this paper is assumed to be paracompact and has a countable base of topology. Also, since we work in the categories of topological vector-spaces and algebras, homomorphism always means continuous homomorphism and isomorphism is a continuous homomorphism which has a continuous inverse. Definition 2.2. A pair (X, C∞ X ) consisting of a topological space and a unital, associasuch that tive algebra C∞ X echet1 , 1. C∞ X is a topological algebra and is nuclear Fr´ 2. C∞ is an ample sub-algebra of C(X) and the inclusion i : C∞ X X → C(X) is continuous, is called a nuclear topological space or in short “nt-space”. Now let (X, C∞ X ) be an nt-space, U ⊂ X an open subset, x ∈ X and f ∈ C(U ). Then we say that f is of C∞ -type in x if there exists f˜ ∈ C∞ X and a neighborhood Ux of X in U such that f|Ux = f˜|Ux . We say f is of C∞ -type on U if f is of C∞ -type in x for all x ∈ U . Definition 2.3. Let (X, C∞ X ) be a nuclear topological space and U ⊂ X an open subset. Then we define C∞ (U ) := {f ∈ C(U ) | f is of C∞ -type on U }. 1
See [Tr] p. 85 for the definition of “Fr´echet space”, p. 510 for the definition of “nuclear”.
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∞ ∞ The existence of partitions of unity in C∞ X implies that CX = C (X) and for any ∞ open U ⊂ X, the topology on CX induces a locally convex topology on C∞ (U ) in the canonical way.
We will use the algebras C∞ (U ) corresponding to the nt-space (X, C∞ X ) to prescribe the local structure of (X, C∞ ). The local structure within a locally coned nuclear topological X space is that of a cone. The construction is as follows: ∞ ˆ ∞ Let cX = X×[0,1) X×{0} denote the open cone over X. We consider C (X)⊗C ([0, 1)) as a sub-algebra of C(X × [0, 1)). We define the jet-map:
ˆ ∞ ([0, 1)) → C∞ (X)[[z]], J : C∞ (X)⊗C ∞ 1 ∂i f (x, t) → f (x, 0)zi . i! ∂t i
(1) (2)
i=0
In Eq. (1) C∞ (X)[[z]] denotes the power series algebra with coefficients in C∞ (X). We consider this object as a locally convex topological algebra. The topology is given by the product topology. It follows from the Leibniz rule, that J is a continuous algebra homomorphism. Clearly C[[z]] is a closed subspace of C∞ (X)[[z]]. We define −1 C∞ (C[[z]]). cX := J
(3)
ˆ ∞ ([0, 1)) and hence a nuclear This object is a closed sub-algebra of C∞ (X)⊗C Fr´echet algebra. We consider it as a subspace of C(cX) = {f : X × [0, 1) → C|f (·, 0) = const.}. It is not hard to show that C∞ cX is an ample sub-algebra of C(cX) so that we are able to define the cone over the nt-space (X, C∞ X ) as ∞ c(X, C∞ X ) := (cX, CcX ),
(4)
and a locally coned nuclear topological space ( in short lcnt-space ) as an nt-space which has locally the structure of a product of an open ball and a cone over another lcnt-space. More precisely: Definition 2.4. A nuclear topological space (X, C∞ X ) is called a locally coned nuclear topological space (lcnt-space) if for any x ∈ X there exists an open neighborhood Ux ⊂ X of x, k ∈ N and a homeomorphism φ : Ux → B k × cLx , where B k ⊂ Rk denotes the open unit ball and cLx the cone over another lcnt-space Lx , such that the induced map ∞ ˆ ∞ φ ∗ : C∞ (B k )⊗C cLx → C (Ux )
(5)
is an isomorphism. ∞ Clearly, if (X, C∞ X ) is an lcnt-space, then c(X, CX ) is also an lcnt-space. Hence the class of lcnt-spaces is closed under the cone construction. Also any smooth manifold M together with the algebra of smooth functions on M defines an lcnt-space (M, C∞ (M)). So the class of lcnt-spaces contains at least all spaces which are locally iterated cones
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over smooth manifolds. The concept of lcnt-spaces is related to that of pseudo-manifolds2 , but emphasizes the analytic structure of the underlying space. To use inductive methods, we have to define some kind of (local) dimension. For this let x ∈ X be an arbitrary point. We consider C as a module over C∞ X via the evaluation map evx : C∞ → C. We define the tangent space at x as X TxC X := Der(C∞ X , C). Definition 2.5. Let at x as
(X, C∞ X)
(6)
be an lcnt-space and x ∈ X. We define the dimension of X
C dimx (X, C∞ X )) := dimC (Tx X) − 1.
(7)
From the definition it is not clear that dimx (X) is a finite number. Nevertheless we have the following proposition: Proposition 2.1. Let (X, C∞ X ) be an lcnt-space, x ∈ X and Ux a neighborhood of x ˆ ∞ (cLx ) ∼ such that according to Definition 2.4 we have C∞ (B k )⊗C = C∞ (Ux ). Then ∞ dimx (X, CX ) = k. Proof. From the second condition in Definition 2.1 it follows immediately that any derivation on the algebra C∞ X is a local operator. From this it follows that ∞ TxC X = Der(C∞ X , C) = Der(C (Ux ), C)
ˆ ∞ = Der(C∞ (B k )⊗C cLx , C) = Der(C∞ (B k ), C) ⊕ Der(C∞ cLx , C). The statement follows, if we show that dimC Der(C∞ cLx , C) = 1. We have C ˆ C = Der(C∞ Der(C∞ cLx )⊗C∞ cLx , C) = Tx (cLx ), cL x
where x is now considered as the vertex of the cone. Let D ∈ Der(C∞ cLx ) and let ∞ ⊂ C∞ ⊗C ∞ ([0, 1)).3 Then we have ˆ ˆ λ f (y) ⊗g (t) be an element in C i i i i cLx Lx ˆ i (t)) = ˆ i (t) + ˆ λi fi (y)⊗g λi (Dfi )(y)⊗g λi fi (y)⊗(Dg D( i )(t). i
i
i
ˆ C∞ C gives Tensoring this with ⊗ cL x
ˆ C∞ (D ⊗ cL
x
ˆ i (t)) = ˆ i (0) + ˆ 1)( λi fi (y)⊗g λi (Dfi )(y)⊗g λi fi (y)⊗(Dg i )(0) . i
i
I
i
II
ˆ By definition of C∞ i λi fi (y)⊗gi (0) is constant, hence cLx the function y → ˆ i (0)) = 0. λi fi (y)⊗g I = D( i 2 3
See [Bo] for a definition of “pseudo-manifold”. See [Tr] p. 459 about the representation of elements in the projective tensor-product.
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Hence any derivation D ∈ TxC (cLx ) is uniquely determined by its effect on the factor and we have
C∞ ([0, 1))
TxC (cLx ) = Der(C∞ ([0, 1)), C), where the right hand side is clearly of dimension 1 over C.
The last proposition clearly justifies the somehow artificial term −1 in Definition 2.5. There is still one awkward thing, the dimension of an isolated point according to this definition is −1. However, if the point is non-isolated, then thinking of stratified spaces, the dimension is precisely what it should be. In fact one can decompose X with respect to dimx (X, C∞ X ) and gets something like a stratified (pre-stratified) space. Definition 2.6. Let (X, C∞ X ) be an lcnt-space. We define the (global) dimension of (X, C∞ X ) as ∞ dim(X, C∞ X ) = sup dimx (X, CX ).
(8)
x∈X
We conclude this section with a lemma, which reflects some metric aspects of nt-spaces and will later be used to localize the Hochschild complex of C∞ X. ˆ ∞ Lemma 2.1. Let (X, C∞ nt-space. Then there exists a function δ ∈ C∞ X ) be an√ X ⊗CX which has values in R+ such that δ(x, y) is a metric on X which generates the topology of X. Proof. To show the existence of such a δ, we will modify the proof of the Urysohn metrization theorem, which states that every regular T1 space with countable base of topology is metrizable. Using that C∞ X is ample one can easily prove that for any two ∞ and disjoint, closed subsets A and B of X there is a function fA,B ∈ CX A,B > 0 −1 −1 such that A ⊂ fA,B (0) and B ⊂ fA,B (A,B ). Let us now consider a complete bounded family F of such function, i.e. choose for any two disjoint, closed subsets A and B of X a function fA,B ∈ F as above such that {fA,B } is bounded.4 We can assume that F is countable and given by the family {fn |n ∈ N}. We define δ as follows: δ : X × X → R+ ∞ (x, y) → λi µi (fi (x) − fi (Y ))2 , i=1
where λi , µi > 0 are chosen such that lim µi fi = 0 and (see [Tr], p. 459). The map
i
ˆ ∞ λi < 1. Then δ ∈ C∞ X ⊗CX
ψ : X → [0, 1]F x → (f (x))f ∈F is a topological embedding, where [0, 1]F is considered with the product topology (see [Ke], p. 125). It is well known that this topology can be realized via the metric d((xi ), (yi )) =
∞ 1 |xi − yi |, 2i i=1
4
To achieve this we have to allow the constants A,B to be arbitrary small.
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and hence also via the map ˜ i ), (yi )) = δ((x
∞
λi µi (xi − yi )2 .
i=1
We have δ = δ˜ ◦ (ψ × ψ) which shows that δ satisfies the conditions in the lemma.
In a forthcoming paper we will study the geometric aspects of lcnt-spaces. It is possible to generalize most of the concepts of Riemannian geometry and also stochastic analysis to this class of spaces. 3. Hochschild Homology and Differential Forms In this section we will give a short introduction into Hochschild homology and differential forms. We work in the category of locally convex spaces and use the projective tensor product. A reference for this section is the original work [Co] as well as the articles [Wo] and [Bro/Lyk] about excision in Hochschild homology. From now on we assume that A is a locally convex topological algebra over C. For any natural number n ∈ N let ˆ
Cn (A) = A⊗(n+1)
(9)
be the (n + 1)-fold completed tensor product over C. We define operators bn : Cn (A) → Cn−1 (A), ˆ ⊗a ˆ n) bn (a0 ⊗...
=
n−1
ˆ ⊗a ˆ i ai+1 ⊗... ˆ ⊗a ˆ n, (−1)i a0 ⊗...
i=0
bn : Cn (A) → Cn−1 (A), ˆ ⊗a ˆ n ) = bn (a0 ⊗... ˆ ⊗a ˆ n ) + (−1)n an a0 ⊗a ˆ 1 ⊗... ˆ ⊗a ˆ n−1 . bn (a0 ⊗... An easy computation shows bn−1 ◦ bn = 0 = bn−1 ◦ bn = 0, hence we get two chain complexes
C∗ (A) = (Cn (A), bn ), C∗bar (A) = (Cn (A), bn ).
(10) (11)
The first complex is called the Hochschild complex, the second complex is called the bar-complex. Both complexes give rise to homology groups. The homology groups of the bar complex are only considered for technical reasons. In fact, the bar complex is acyclic, if the algebra A in case is unital. In general, the acyclicity of the bar-complex is related to the question of excision. Definition 3.1. Let A be a unital algebra. For n ∈ N we define the nth Hochschild homology group of A as H Hn (A) =
ker(bn : Cn (A) → Cn−1 (A)) . im(bn+1 : Cn+1 (A) → Cn (A))
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We consider these homology groups as topological vector-spaces but keep in mind that unless im(b) is not closed, they are not Hausdorff. In general this can cause some problems. For the problem addressed in this paper though, they will turn out to be Hausdorff and everything is fine. Clearly the construction above is functorial, and any continuous algebra homomorphism induces a continuous homomorphism of the Hochschild homology groups. For computational reasons, it is important that Hochschild homology can be represented as a derived functor, in fact as a special kind of T or-product. Definition 3.2. A locally convex topological vector-space M which is also a topological module over A is called topological projective if it is a topological direct summand of ˆ a module of the form N = A⊗E, where E is a locally convex vector-space. Definition 3.3. Let M be a locally convex topological module over A. A topological projective resolution of M is an exact sequence of topological projective A-modules and A-linear maps b2
...M2
/ M1
b1
/ M0
b0
/M,
which admits a C-linear continuous contraction, i.e. for all i ∈ N maps si satisfying si : Mi → Mi+1 , id = bi+1 si + si−1 bi . ˆ op . Now let Aop denote the algebra A with the opposite multiplication and B := A⊗A The algebra A itself becomes a topological B-module by setting ˆ · c = acb. (a ⊗b) The following proposition gives an answer on how to compute Hochschild homology groups using projective resolutions. Proposition 3.1. Let (Mn , bn ) be a topological projective resolution of A over B. Then the Hochschild homology groups of A coincide with the homology groups of the complex ˆ BA ...M3 ⊗ Proof. (see [Co]).
b3
/M ⊗ 2 ˆ BA
b2
/M ⊗ 1 ˆ BA
b1
/M ⊗ 0 ˆ BA .
We will use this proposition to give two examples. Example 3.1. 1. Consider C together with its natural topology. Then ˆ ∼ Cop ⊗C =C via the multiplication map and 0
/C
id
/C
/0
ˆ Hence we have is in fact a topological projective resolution of C over Cop ⊗C. C if n = 0 H Hn (C) = . 0 if n = 0
(12)
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2. Consider the algebra C[[x]] of formal power series with product topology. Then we can identify ∼ ˆ C[[x]]op ⊗C[[x]] = C[[x, y]] ˆ and a topological projective resolution of C[[x]] over C[[x]]op ⊗C[[x]] is given by 0
/ C[[x, y]]
·(x−y)
/ C[[x, y]]
x=y
/ C[[x]]
/0.
The C-linear contraction can easily be found by induction on the coefficients of the ˆ C[[x,y]] C[[x]] leads to corresponding power series. Tensoring this resolution with ⊗ the sequence 0
/ C[[x]]
0
/ C[[x]] .
Passing to homology, we get
C[[x]] if n = 0 C[[x]] if n = 1 . H Hn (C[[x]]) = 0 if n = 0, 1
The concept of Hochschild homology is strongly related to the concept of differential forms. In fact it is a refinement. We will show later, that for lcnt-spaces the two concepts coincide. The next proposition is a first step in this direction. Proposition 3.2. For any k ∈ N we have H Hk (C∞ (B n )) ∼ = k (B n ),
(13)
where B n denotes the open unit ball and k the complex-valued differential k-forms. Proof. We use Proposition 3.1 with A = C∞ (B n ). We construct an explicit projective resolution of A over C∞ (B n × B n ) by defining, for each k ∈ N, the C∞ (B n × B n )modules Mk := C∞ (B n × B n , k (Cn∗ )). Here Cn∗ denotes the space of linear forms on Cn . Clearly ˆ k (Cn∗ ), Mk ∼ = C∞ (B n × B n )⊗ where completion is actually unnecessary, since the vector-space k (Cn∗ ) is finite dimensional. Nevertheless, it follows that each Mk is free, hence projective. Furthermore let γ denote the difference function γ : B n × B n → Rn ⊂ Cn , γ (a, b) = b − a. This map induces maps iγ : Mk+1 → Mk , iγ ω(a, b)(v1 , ..., vk ) = ω(a, b)(γ (a, b), v1 , ...vk ) = ω(a, b)(b − a, v1 , ...vk ).
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Here ω ∈ Mk+1 denotes a form, a, b are points in B n ⊂ Rn ⊂ Cn and v1 , ...vk are elements of Cn . In other words, iγ is contraction with the vector-field γ . Let us now consider the following sequence: ...
iγ
/ M2
iγ
/ M1
iγ
/ M0 = C∞ (B n × B n )
∗
/0,
/ C∞ (B n )
where : B n → B n × B n denotes the diagonal map. Clearly iγ ◦ iγ = 0. To show that the chain complex above is a topological projective resolution of C∞ (B n ) over C∞ (B n × B n ), we have to give a continuous C-linear contraction. For this let sk : Mk → Mk+1 be defined as follows. Let e1∗ , ..., en∗ denote the dual basis of the standard canonical basis of Cn , and let ω ∈ Mk be given as ω(x, y) = f (x, y)ei∗1 ∧ ... ∧ ei∗k , where f ∈ C∞ (B n × B n ) is a smooth function on B n × B n and i1 , ..., ik ∈ {1, ...n}. In this case we define n 1 ∂f sk ω(a, b) := (a, a + t (b − a))t k ej∗ ∧ ei∗1 ∧ ... ∧ ei∗k dt. ∂y j 0 j =1
In the following we suppress the subscript k and simply write sω. We have
n 1 ∂f (iγ sω)(a, b) = (a, a + t (b − a)) t k ∂yj 0 j =1 k l+1 ∗ ∗ ∗ ∗ · (−1) (b − a)i e ∧ e ∧ ... ∧ e ∧ ... ∧ e l
j
i1
il
ik
l=1
+(b − a)j (ei∗1 ∧ ... ∧ ei∗k ). From this we get the expression n 1 ∂f (a, a + t (b − a))t k (b − a)j ei∗1 ∧ ... ∧ ei∗k dt (iγ sω)(a, b) = 0 ∂yj j =1
n k
1
∂f (a, a + t (b − a))t k ∂yj 0 l=1 j =1 ×(b − a)il ej ∧ ei∗1 ∧ ... ∧ ei∗l ∧ ... ∧ ei∗k dt . +
(−1)l+1
The chain-rule of differential calculus applied to the first sum gives 1 d f (a, a + t (b − a))t k ei∗1 ∧ ... ∧ ei∗k dt iγ sω(a, b) = dt 0 n 1 k ∂f + (−1)l+1 (a, a + t (b − a))t k ∂y j l=1 j =1 0 ∗ ∗ ∗ ×(b − a)il ej ∧ ei1 ∧ ... ∧ eil ∧ ... ∧ eik dt .
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Now we perform partial integration with the first integral on the right side. This yields the following expression: iγ sω(a, b) = f (a, a + t (b − a))t k |10 ei∗1 ∧ ... ∧ ei∗k =ω(a,b)
1
− 0
f (a, a + t (b − a))kt k−1 ei∗1 ∧ ... ∧ ei∗k dt
n k
1
∂f (a, a + t (b − a))t k ∂yj 0 l=1 j =1 ∗ ∗ ∗ ×(b − a)il ej ∧ ei1 ∧ ... ∧ eil ∧ ... ∧ eik dt . +
(−1)l+1
The first term on the right side clearly coincides with ω(a, b). In total we get iγ sω(a, b) = ω(a, b) + R(a, b), where R(a, b) denotes the rest, i.e. the integral and the double sum on the right side. Now we compute siγ ω(a, b). The definition give us siγ ω(a, b) =
k
(−1)l {
n
1
j =1 0
l=1
∂f (a, a + t (b − a))(a + t (b − a) − a)il ∂yj
−f (a, a + t (b − a))δj,il t k−1 dt ej∗ ∧ ei∗1 ∧ ...ei∗l ∧ ... ∧ ei∗k }. Reordering the terms and considering the Kronecker symbol δj,il yields 1 f (a, a + t (b − a))t k−1 dt ei∗l ∧ ei∗1 ∧ ... ∧ ei∗l ∧ ... ∧ ei∗k siγ ω(a, b) = (−1)l+1 0
k
n
1
∂f (a, a + t (b − a))(b − a)il t k dt ∂yj 0 l=1 j =1 ∗ ∗ ∗ ∗ ej ∧ ei1 ∧ ...eil ∧ ... ∧ eik .
+
(−1)l
Shuffling ei∗l from the first to the ilth position in ei∗l ∧ ei∗1 ∧ ... ∧ ei∗l ∧ ... ∧ ei∗k changes the sign by the factor (−1)l−1 . This cancels with the factor (−1)l+1 in front of the first term on the right side, and we see that this term is actually independent of the summation index l. Hence for this term summation over l is just multiplication with k. Taking a close look at the summands we can recognize that we end up with −R(a, b), where R(a, b) was defined on the previous page. So we get siγ ω(a, b) + iγ sω(a, b) = −R(a, b) + ω(a, b) + R(a, b) = ω(a, b). This proves siγ + iγ s = id.
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It is not hard to see that s is continuous and C-linear. So far we have constructed a topological projective resolution of C∞ (B n ) over C∞ (B n × B n ). We do now compute the Hochschild homology of C∞ (B n ) by tensoring the topological projective resolution from above over C∞ (B n × B n ) with C∞ (B n ). For any k ∈ N we have ˆ k (Cn∗ ))⊗ ˆ C∞ (B n ×B n ) C∞ (B n ) ˆ C∞ (B n ×B n ) C∞ (B n ) = (C∞ (B n × B n )⊗ Mk ⊗ ˆ ∞ (B n ) = k (B n ). = k (Cn∗ )⊗C Since γ restricted to the diagonal is zero, we have ˆ C∞ (B n ×B n ) idC∞ (B n ) = 0. iγ ⊗ Therefore the tensored complex has zero differentials and we get k (B n ) for the k th homology group of this complex. To study the relationship between Hochschild homology and differential forms in general, we have to say what we mean by differential forms for a general Fr´echet-algebra. There are various concepts of this, we will use the following one:5 The first Hochschild homology group of a locally convex algebra A is given as H H1 (A) =
ˆ → A) ker(b1 : A⊗2 ˆ ˆ im(b2 : A⊗3 → A⊗2 )
.
The boundary maps b1 and b2 are explicitly given as ˆ = ab − ba, b1 (a ⊗b) ˆ ˆ = ab⊗c ˆ − a ⊗bc ˆ + ac⊗b. ˆ b2 (a ⊗b⊗c)
(14) (15)
We assume from now on that A is commutative. Then in Eq. (14) b1 = 0. We denote ˆ the class of a ⊗b ˆ in H H1 (A). We have with [a ⊗b] ˆ = [a(1⊗b ˆ − b⊗1) ˆ + ab⊗1] ˆ [a ⊗b] ˆ − b⊗1] ˆ + [ab⊗1] ˆ = a[1⊗b =∗
ˆ − b⊗1]. ˆ = a[1⊗b That ∗ = 0 follows by taking c = 1 in (15). We define the formal differential of b as ˆ − b⊗1]. ˆ db := [1⊗b An elementary computation shows that d(ab) = adb + bda.
(16)
From above, we also see that H H1 (A) = span{adb|a, b ∈ A}. This leads us to the following definition. 5 This version of differential forms works very well in the case of commutative algebras. In the case of non-commutative algebras this definition of differential forms has to be slightly modified.
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Definition 3.4. Let A be a locally convex topological algebra and k ∈ N. We define the locally convex topological vector-space of differential k-forms on A as kA := kA H H1 (A), where kA denotes the k th exterior product over A.6 By definition we have 0A = A and 1A := H H1 (A). In general we have the antisymmetrization map : kA → H Hk (A), a0 da1 ∧ ... ∧ dak → sign(σ )(a0 ⊗ aσ −1 (1) ... ⊗ aσ −1 (k) ). σ ∈ k
This map is well defined, continuous and has a continuous left inverse ( see [Lo], p. 27ff). Hence it is always injective and kA is even a topological direct summand in H Hk (A). The question is now, when is it also surjective or what can Hochschild homology see, which differential forms do not see. In the case of lcnt-spaces we will answer this question in the next section. To use inductive methods it is very useful to have at hand certain long exact sequences of Hochschild homology groups associated to short exact sequences of certain algebras. Those long exact sequences do not always exist. The question of their existence is referred to as the Excision-problem in Hochschild homology, because it somehow resembles the situation of the Mayer-Vietoris sequence in ordinary homology. In general the obstruction is a property called H -unitality ([Lo], p. 32). By definition A is H -unital, if C∗bar (A) is acyclic. For a non-unital algebra I such as for example an ideal in a bigger unital algebra, the Hochschild homology groups are defined as coker(i∗ ) : H Hn (C) → H Hn (I + ), where I + denotes the unitalization of I and i the canonical inclusion. By definition I is called H unital, if C∗bar (I ) is acyclic. The following proposition can be found in [Bro/Lyk]. Proposition 3.3. Let 0 → I → A → A/I → 0 be an exact sequence of nuclear Fr´echet algebras such that A is unital and I is H-unital. Then there is a long exact sequence of continuous Hochschild homology groups ..
/ H Hn (I )
/ H Hn (A)
/ H Hn (A/I )
δ
/ H Hn−1 (I )
/ .. .
The task to determine whether an algebra is H -unital or not can turn out to be quite difficult. We will later make use of the following proposition. Proposition 3.4. Let A be a unital nuclear Fr´echet algebra and let ∞ ∞ C∞ 0 ([0, 1)) = ker(res : C ((−1, 1)) → C (−1, 0))
(17)
be the algebra of smooth function on [0, 1) with derivatives of any order vanishing at ˆ zero. Then the non-unital nuclear Fr´echet algebra C∞ 0 ([0, 1))⊗A is H -unital. 6
In the category of locally convex topological vector-spaces [He, Ew] p. 64
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Proof. We have to show that the corresponding bar-complex (11) is acyclic. Let ˆ 0i )⊗... ˆ ni ) ∈ Cn (C∞ ˆ ⊗(f ˆ ni ⊗b ˆ α= λi (f0i ⊗b 0 ([0, 1))⊗A) i
be an element in the bar complex. Here λi is a sequence of complex numbers such that ∞ i i i=0 |λi | < 1 and fj respectively bj converge to zero as i goes to infinity. A factori ization theorem (see [Vo], Thm. 3.4) applied to C∞ 0 ([0, 1)) and the sequence (f0 ) says ∞ ∞ there are functions g i ∈ C0 ([0, 1)) for all i ∈ N and h ∈ C0 ([0, 1)) with the following properties : 1. f0i = h · g i for all i ∈ N,
i 2. g i ∈ C∞ 0 ([0, 1)) · (f0 |i ∈ N).
The expression in Condition 2 denotes the closure of the ideal in C∞ 0 ([0, 1)) which i ∞ ˆ is generated by the functions f0 . Let us define β ∈ Cn (C0 ([0, 1))⊗A) as β=
∞
ˆ ⊗(f ˆ ni ⊗b ˆ 0i )⊗... ˆ ni ). λi (g i ⊗b
i=0
From Condition 2 above it follows that ∞ ˆ ˆ β ∈ C∞ 0 ([0, 1))⊗A) · α ⊂ Cn (C0 ([0, 1))⊗A).
(18)
Here the term in the middle denotes the closure of the ideal generated by α. A simple calculation shows that ˆ A )⊗β) ˆ + (h⊗1 ˆ A )⊗b ˆ (β). α = b ((h⊗1
(19)
Let us now assume that α is a cycle in the bar complex. Then b (α) = 0. Hence by ˆ continuity and C∞ 0 ([0, 1))⊗A linearity of b it follows from (18) that b (β) = 0. Hence by (19) we have that ˆ A )⊗β) ˆ α = b ((h⊗1 is a boundary in the continuous bar complex and the bar complex is acyclic.
4. Hochschild Homology of lcnt-Spaces The following theorem is the main result of this work. It generalizes a result of A. Connes7 about Hochschild homology in the case where the underlying space is a smooth manifold to the case where the underlying space is an lcnt-space. We will not use Connes’s result in the proof, but rather Proposition 3.2 and a localization method introduced in Teleman [Te]. Theorem 3. Let (X, C∞ X ) be a finite dimensional lcnt-space, then the antisymmetrization map : ∗C∞ → H H∗ (C∞ X) X
is an isomorphism. 7
See [Co] Lemma 45 and Theorem 46 (Connes works in the cohomological setting).
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The proof of the theorem will follow after a couple of lemmas. The first step is: Lemma 4.1. The antisymmetrization map : kC[[x]] → H Hk (C[[x]]) is an isomorphism for all k ∈ N. Proof. By definition for k = 0, 1 there is nothing to show. On the other side it follows from (16) that d(x n ) = n · x n−1 dx. Since obviously dx ∧ dx = 0 we have kC[[x]] = 0 for all k ≥ 2 which together with Example 3.1 (2) shows the rest. Lemma 4.2. Let (X, C∞ X ) be an lcnt-space such that the antisymmetrization map : ∗C∞ → H H∗ (C∞ X) X
is an isomorphism and furthermore H H∗ (C∞ X ) is Hausdorff, then the antisymmetrization map : ∗C∞ → H H∗ (C∞ cX ) cX
is also an isomorphism and
H H∗ (C∞ cX )
is also Hausdorff.
Proof. Let us first consider the following short exact sequence of algebras 0
/ C∞ ⊗C ˆ ∞ ([0, 1))
incl.
0
X
/ C∞ ⊗C ˆ ∞ ([0, 1)) X
J
/ C∞ [[z]] X
/0,
(20)
∞ˆ ∞ where C∞ 0 ([0, 1)) is given by (17). It follows from Proposition 3.4 that CX ⊗C0 ([0, 1)) is H -unital. Hence we have the following commutative diagram with an exact upper row
ˆ ∞ ([0, 1))) H Hk (C∞ X ⊗C O 0
incl.∗
J∗
/ H Hk (C∞ [[z]]) OX
∼ =
incl.∗
kC∞ ⊗ˆ C∞ ([0,1)) X
/ H Hk (C∞ ⊗C ˆ ∞ ([0, 1))) X O
0
/ k ∞ ˆ ∞ CX ⊗C ([0,1))
∼ = J∗
/ k ∞ CX [[z]]
The two antisymmetrization maps on the right side are in fact isomorphisms. This fol∼ ∞ˆ lows from C∞ X [[z]] = CX ⊗C[[z]], Lemma 4.1 and Proposition 3.2 (slightly modified, to cover half open intervals) once there is a K¨unneth-Theorem for Hochschild homology at hand. Such a theorem indeed exists, if the corresponding Hochschild homology groups of the involved algebras are Hausdorff (see [Ka]). This is the case.8 It is not hard to see that the second map in the lower sequence is always surjective. Therefore by commutativity and our assumption, the second map in the upper sequence is also surjective. Since the upper sequence is actually a part of a long exact sequence, the first map in the upper sequence is in addition injective and the upper sequence is in fact a short exact sequence. It follows from this information and that is in a general split injective via a small diagram chase, that ˆ ∞ : ∗C∞ ⊗ˆ C∞ ([0,1)) → H H∗ (C∞ X ⊗C0 ([0, 1))) X
8
0
For more details about the K¨unneth-Theorem in Hochschild homology see [Ew], p. 86.
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is also an isomorphism. Let us now consider the short exact sequence of algebras / C∞ ⊗C ˆ ∞ ([0, 1))
0
incl.
0
X
/ C∞ cX
/ C[[z]]
J
/0.
(21)
As before this sequence induces a commutative diagram ˆ ∞ H Hk (C∞ ([0, 1))) X ⊗C O 0
incl.∗
∼ =
proj.∗
incl.∗
/ kC∞
proj.∗
cX
0
/ H Hk (C[[z]]) O ∼ =
kC∞ ⊗ˆ C∞ ([0,1)) X
/ H Hk (C∞ ) O cX
/ k C[[z]]
where the upper row is exact. Furthermore the right and the left antisymmetrization maps are isomorphisms. This follows from Lemma 4.1 and the first part. Also again, it is not hard to see that the second map in the lower sequence is always surjective. It follows from this and split injectivity, via a small diagram chase that the map in the middle : kC∞ → H Hk (C∞ cX )) cX
is also an isomorphism.
Lemma 4.3. Let (X, C∞ X ) be an nt-space, such that there exists a locally finite open covering (Ui |i ∈ I ) of X such that any of the antisymmetrization maps : ∗C∞ (Ui ) → H H∗ (C∞ (Ui )) is an isomorphism, then the antisymmetrization map : ∗C∞ → H H∗ (C∞ X) X
is also an isomorphism. ˆ ∞ ⊗(k+1) in the Hochschild complex of Proof. We consider the spaces Ck (C∞ X ) = (CX ) C∞ X as subspaces of the space of continuous complex valued functions on the (k +1)-fold Cartesian product of X and use a method first presented in [Te] to localize the Hochschild complex around the diagonal. First let λ : [0, ∞) → [0, 1] be a smooth function, such that supp(λ) ⊂ [0, 1] and λ|[0,1/2] ≡ 1. For t > 0 we define
λt : [0, ∞) → [0, 1] λt (s) := λ(s/t). These functions have the following properties: 1. supp(λt ) ⊂ [0, t], 2. λt|[0,t/2] ≡ 1. Now for any k ∈ N let us define functions ρk as ρk : Xk+1 → [0, ∞), ρk (x0 , x1 , ..., xk ) = δ(x0 , x1 ) + δ(x1 , x2 ) + ... + δ(xk , x0 ).
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Here δ denotes the function on X × X of Lemma 2.1. ρk measures the distance of a point in Xk+1 from the diagonal with respect to the “distance-function” δ. Clearly ˆ ⊗(k+1) . Let ρk ∈ (C∞ X) Ut,k := {(x0 , x1 , ..., xk )|ρk (x0 , ..., xk ) < t} be the t-neighborhood of the diagonal k+1 ⊂ X k+1 . Let C∗t (C∞ X ) be the sub-complex of ˆ t ∞ ∞ ⊗(k+1) the Hochschild complex C∗ (CX ), where Ck (CX ) contains the elements of (C∞ X) vanishing on Ut,k . Let C∗0 (C∞ C t (C∞ ), X ) = lim − → ∗ X where the limit is for t to zero. The complex C∗0 (C∞ X ) consists of the chains vanishing in an arbitrary neighborhood of the diagonal. We will now show that this complex is acyclic. We define an operator ∞ Et : Ck (C∞ X ) → Ck+1 (CX ),
Et (F )(x0 , ..., xk+1 ) = λt (δ(x0 , x1 )2 ) · F (x1 , ..., xk+1 ), ∀F ∈ Ck (X). ∞ This operator maps Cks (C∞ X ) into Ck+1 (CX ) which can be verified by computation. Another computation shows that s/4
b ◦ E t + Et ◦ b = 1 − N t , where Nt is defined as Nt (F )(x0 , ..., xk ) = (−1)k λt (δ(x0 , x1 )) · {F (x1 , x2 , ...xk , x0 ) − F (x1 , x2 , ...xk , x1 )}, th ∀F ∈ Ck (C∞ X ). Again by computation b ◦ Nt = Nt ◦ b. Let’s consider the k power of ∞ Nt . For F ∈ Ck (CX ) we get
(Nt )k F (x0 , ..., xk ) =
k−1
λt (δ(xi , xi+1 )) · G(x0 , ..., xk ),
i=0
where G(x0 , ..., xk ) is a linear combination of functions built out of F by restricting to certain diagonals and permutation of some arguments. For the product in front of G not to be zero, we must have δ(xi , xi+1 ) √ < t for each 0 ≤ i ≤ k − 1. The triangle equation shows that in this case we also have δ(x0 , xk ) < kt 1/2 . Thus we have ρk (x0 , ..., xk ) =
k−1
δ(xi , xi+1 ) + δ(xk , x0 ) < kt + k 2 t.
i=0 (k+k 2 )t
Hence for F ∈ Ck operator
Kt := Et ·
k−1 r=0
k (C∞ X ) we have that (Nt ) (F ) = 0. Let’s define another
(k+k 2 )t
(Nt )r : Ck
(k+k 2 )4−(k+1) t
(C∞ X ) → Ck+1
(C∞ X ).
Hochschild- and Cyclic-Homology of LCNT-Spaces
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By construction this operator satisfies b ◦ Kt + Kt ◦ b = 1, which proves the acyclicity of C∗0 (C∞ X ) by taking the direct limit where t goes to zero. This in fact shows that any Hochschild-homology class can be represented by a Hochschild-cycle F ∈ Ck (C∞ X ) which has support in an arbitrary small neighborhood of the diagonal X ≈ ⊂ Xk+1 . Now let (Ui |i ∈ I ) be an open covering of X which satisfies the requirements of the proposition. Let F ∈ Ck (C∞ X ) be an arbitrary Hochschild-cycle, then by the discussion above we can assume that supp(F ) ⊂ Uik+1 . i∈I ∞ ∞ Now we can use the C∞ X -module structures on Ck (CX ) and H Hk (CX ) as well as a partition of unity in C∞ subordinated to the covering (U |i ∈ I ) to see that i X
[F ] =
[Fi ], i∈I
where supp(Fi ) ⊂ Uik+1 . It does now follow from the surjectivity assumption on the antisymmetrization map corresponding to each Ui that each of the [Fi ] is in the image of the antisymmetrization map corresponding to X. Hence [F ] is also in the image of the antisymmetrization map. This shows that the antisymmetrization map is surjective. Since it always has a continuous left inverse it is in fact an isomorphism. With these tools we are able to proof Theorem 4.1. Proof. Let n ∈ N. We have to show that the antisymmetrization map is surjective and equally that any Hochschild cycle in Cn (C∞ X ) is homologous to an antisymmetric Hochschild cycle. From Lemma 4.3. this follows if we show that this is locally the case. ˆ ∞ Hence we can assume that our lcnt-space is given as in (5) by (B k ×cL, C∞ (B k )⊗C cL ), k where B denotes the open unit ball of dimension k and cL denotes the cone over an lcnt-space of dimension less than the dimension of X (see (3),(4),(7) and Proposition 2.1). Using induction on the dimension, we can assume that all antisymmetrization maps corresponding to L are isomorphisms and all Hochschild homology groups of L are Hausdorff. Therefore by Lemma 4.2 the same holds for (cL, C∞ cL ). Furthermore by Proposition 3.2 the antisymmetrization maps corresponding to B k are isomorphisms. Hence the corresponding Hochschild-homology groups are also Hausdorff and Theorem 4.1 follows as in the proof of Lemma 4.2 from the K¨unneth Theorem in Hochschildhomology (see [Ka]). 5. Cyclic Homology of lcnt-Spaces In this section we will say something about the cyclic homology groups corresponding to an lcnt-space (X, C∞ X ). The result we will present is a straight forward consequence of the general relationship between Hochschild homology and cyclic homology and Theorem 4.1. Using Theorem 4.1 the proof is identical to the proof in [Co]. To shorten this article, we keep this section very informal.
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If we divide out a cyclic action from the Hochschild complex (10), in equal identifying cycles, which arise from another by cyclic permutation, we get another complex, which is sometimes called Connes’ complex. The homology groups of this complex are called cyclic homology groups and will be denoted by H C∗ (A). These groups are related to the Hochschild homology groups by the so-called Connes’ exact sequence ...H Hn (A)
/ H Cn (A)
I
S
/ H Cn−2 (A)
B
/ H Hn−1 (A)
I
/ ... .
The operator S is the so-called Connes periodicity operator and corresponds via some identifications to the Bott periodicity operator in K-theory. In the commutative case it is not hard to show that via the antisymmetrization map, up to a factor the operator B ◦ I : H Hn (A) → H Hn+1 (A) exactly corresponds to the operator d : nA → n+1 A , a0 da1 ∧ ... ∧ dan → da0 ∧ da1 ∧ ... ∧ dan . Theorem 5.1. Let (X, C∞ X ) be an lcnt-space with finite dimensional homology groups. Then for all n ∈ N there is a natural topological isomorphism n−1 n−2 ∼ n (X, C) ⊕ H n−4 (X, C) ⊕ H n−6 (X, C).... H Cn (C∞ X ) = C∞ /dC∞ ⊕ H X
X
Here H k (X, C) stands for singular cohomology with coefficients in C. There is also a de Rham theorem for lcnt-spaces9 and these singular cohomology groups can be identified with the corresponding de Rham cohomology groups. References [Bo]
Borel, A.: Intersection cohomology. (Notes of a Seminar on Intersection Homology at the University of Bern, Switzerland, Spring 1983). (English) Progress in Mathematics, Vol. 50, Swiss Seminars. Boston-Basel-Stuttgart: Birkh¨auser, 1984 [Bra/Pf] Brasselet, J-P., Pflaum, M.: On the homology of algebras of Whitney functions over sub-analytic sets. Institut de Math´ematiques de Luminy, Preprint 2002-02, 2002 [Bre] Bredon, G.: Topology and Geometry. Graduate Texts in Math 139, Berlin-Heidelberg-New York: Springer, 1991 [Bro/Lyk] Brodzki,J., Lykova,Z.: Excision in Cyclic Type Homology of Fr´echet Algebras. Bull. Lond. Math. Soc. 33(3), 283–291 (2001) [Co] Connes, A.: Non-commutative Differential Geometry. Publications Mathematiques IHES 62, 1985 [Ew] Ewald, C.: Hochschild homology and de Rham cohomology of stratifolds. Dissertation Universit¨at Heidelberg 2002, Heidelberg Dokumenten Server [He] Helemskii, A.Ya.: The Homology of Banach and Topological Algebras. Dordrecht: Kluwer Academic Publishers, 1989 [Ka] Karoubi, M.: Formule de K¨unneth en homologie cyclique I et II. C.R. Acad. Sci. Paris S´erie A-B 303, 507–510 (1986) [Ke] Kelley, J.L.: General Topology. New York: Van Nostrand, 1955 [Kr1] Kreck, M.: (Co)Homology via Differential topological Varieties. Preprint Mainz, 1999 [Kr2] Kreck, M.: Lecture Notes in Differential Algebraic Topology. Mainz/Heidelberg, 2000 [Lo] Loday, J.L.: Cyclic homology. Grundlehren der Mathematik, Berlin-Heidelberg-New York: Springer, 1991 9
Compare [Ew] Theorem 4.6.1, p. 49
Hochschild- and Cyclic-Homology of LCNT-Spaces [Pf] [Te] [Tr] [Vo] [We] [Wo]
213
Pflaum, M.: Ein Beitrag zur Geometrie und Analysis auf stratifizierten Räumen. Habilitationsschrift, Humboldt Universit¨at Berlin, 2000 Teleman, N.: Microlocalisation de l’homologie de Hochschild. C.R. Acad. Sci. Paris S´erie 1, 326, 1261–1264 (1998) Treves, J.F.: Topological Vectorspaces, Distributions and Kernels, London-New-York: Academic Press, 1967 Voigt, J.: Factorization in some Fr´echetalgebras of Differentiable Functions, Juergen Voigt, Studia Mathematica, 77, 333–348 (1983) Weibel, C.: Homological Algebra. Oxford: Oxford University Press, 1995 Wodzicki, M.: Excision in Cyclic Homology, Annals of Math, 129, 591–639 (1989)
Communicated by A. Connes
Commun. Math. Phys. 250, 215–239 (2004) Digital Object Identifier (DOI) 10.1007/s00220-004-1141-4
Communications in
Mathematical Physics
A Generalization of the Bargmann’s Theory of Ray Representations Jarosław Wawrzycki Institute of Nuclear Physics PAS, ul. Radzikowskiego 152, 31-342 Kraków, Poland. E-mail: [email protected] Received: 12 June 2003 / Accepted: 2 March 2004 Published online: 27 July 2004 – © Springer-Verlag 2004
Abstract: The paper contains a complete theory of factors for ray representations acting in a Hilbert bundle, which is a generalization of the known Bargmann’s theory. With its help, we have reformulated the standard quantum theory so that the gauge freedom emerges naturally from the very nature of quantum laws. The theory is of primary importance in the investigations of covariance (in contradistinction to symmetry) of a quantum theory which possesses a nontrivial gauge freedom. In that case the group in question is not any symmetry group but a covariance group only – the case not yet investigated in depth. It is shown that the factor of a covariance group representation depends on space and time when the system in question possesses gauge freedom. In nonrelativistic theories, the factor depends on time only. In relativistic theory, the Hilbert bundle is built over spacetime while in the nonrelativistic case-over time. We explain two applications of this generalization: in the theory of a quantum particle in gravitational field in the nonrelativistic limit, and in quantum electrodynamics. 1. Introduction In standard Quantum Mechanics (QM) and in Quantum Field Theory (QFT), the spacetime coordinates are essentially classical variables. Therefore, the question about the general covariance of QM and QFT emerges naturally just like in the classical theory, namely: What is the effect of a change in spacetime coordinates in QM and QFT when this change does not form any symmetry transformation? It is a commonly accepted belief that there are no substantial difficulties if we refer the question to the wave equation. We simply treat the wave equation, without saying why, in such a manner as if it was a classical equation. The only problem arising is to find the transformation rule ψ → Tr ψ for the wave function ψ. This procedure, which on the other hand can be seriously objected, does not solve the above-stated problem. The
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heart of the matter, as well as that of QM and QFT, lies in the Hilbert space of states, and specifically in finding the representation Tr of the covariance group in question. The trouble originates from the fact that the covariance transformation changes the form of the wave equation so that ψ and Tr ψ do not belong to the same Hilbert space, which means that Tr does not act in the ordinary Hilbert space. This is not compatible with the paradigm worked out while dealing with symmetry groups. We show that the covariance group acts in a Hilbert bundle RH over time in nonrelativistic theory, and in a Hilbert bundle MH over spacetime M in the relativistic case. The waves are the respective cross-sections of the bundle in question. The exponent ξ(r, s, p) in the formula Tr Ts = eiξ(r,s,p) Trs , depends on point p of the base of the bundle in question. That is, ξ depends on time t in the nonrelativistic theory, and on spacetime point p in the relativistic theory if there exists nontrivial gauge freedom. Moreover, we argue that the bundle MH is more appropriate for treating the covariance as well as the symmetry groups than the Hilbert space itself. Namely, we show that from a more general assumption that the representation Tr of the Galilean group acts in RH and has an exponent ξ(r, s, t) depending on time t one can reconstruct nonrelativistic Quantum Mechanics. Even more, in the more complex case of theory with nontrivial time-dependent gauge describing a spinless quantum particle in Newtonian gravity, we are able to infer the wave equation and prove the equality of the inertial and gravitational masses, cf. [17]. In doing so, we apply extensively the classification theory for exponents ξ(r, s, t) of Tr acting in RH and depending on time. The main task of this paper is to construct the general classification theory of spacetime-dependent exponents ξ(r, s, p) of representations acting in MH. At the same time, the theory presented here can be viewed as a generalization of Bargmann’s [1] classification theory of exponents ξ(r, s) of representations acting in ordinary Hilbert spaces, with the exponents independent of p ∈ M. In the presented theory which is slightly more general than the standard one, gauge freedom emerges from the very nature of the fundamental laws of Quantum Mechanics. We hope this reformulation will be useful in dealing with the problems in QFT caused by gauge freedom. Section 2 presents physical motivation in detail. In Sect. 3 we discuss a generalization of the ordinary state vector ray and operator ray introduced by Weyl. In Sect. 4 we analyze the local exponents of Lie groups and introduce algebras which are important tools for the classification theory of local exponents presented in that section. In Sect. 5 we investigate globally defined exponents, classifying them in some special cases. Sect. 6 includes two examples. The first example is the Galilean group. We analyze the group from the point of view of the generalized theory. By way of the other example, the exponents of the Milne group, the covariance group relevant in the theory of a nonrelativistic particle in the gravitational field, are analyzed. The proof of differentiability of the (generalized) exponent and of the theorem about exponents defined on one-parameter groups proceeds in a quite analogous way to that presented in Bargmann’s work [1]. However, it is not trivial that the theorems are also true in this generalized situation. We present their proof explicitly for the reader’s convenience. The remainder of our reasoning is not a simple analogue of [1] and proceeds differently.
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2. Setting the Motivation In this section we carry out a general analysis of the representation Tr of a covariance group, and compare it with the representation of a symmetry group. We also describe the correspondence between the space of wave functions ψ( x , t) and the Hilbert space. The analysis is performed in the nonrelativistic case, but it can be derived in the relativistic quantum field theory as well. Before we give the general description, it will be instructive to investigate the problem for a free particle in the flat Galilean spacetime. The set of solutions ψ of the Schrödinger equation, which are admissible in Quantum Mechanics, is precisely given by t kx 3 −i 2m kk+i ψ( x , t) = (2π)−3/2 ϕ(k)e d k, is any square integrable function. The where p = k is the linear momentum and ϕ(k) functions ϕ (wave functions in the “Heisenberg picture”) form a Hilbert space H with the inner product 2 (k) d3 k. (ϕ1 , ϕ2 ) = ϕ1∗ (k)ϕ The correspondence between ψ and ϕ is one-to-one. In general, however, the construction fails if the Schrödinger equation possesses nontrivial gauge freedom. Let us explain it. For example, the above construction fails for the nonrelativistic quantum particle in the curved Newton-Cartan spacetime. Besides, in this spacetime we do not have any plane wave, see [15]. Thus, there does not exist any natural counterpart for the Fourier transform. However, we do not need to use the Fourier transform. What is the role of the Schrödinger equation in the above construction of H? Please note that in general ψ2 ≡ ψ ∗ ( x , 0)ψ( x , 0) d3 x = (ϕ, ϕ) = ψ ∗ ( x , t)ψ( x , t) d3 x. This is in accordance with the Born interpretation of ψ. Namely, if ψ ∗ ψ( x , t) is the probability density, then ψ ∗ ψ d3 x has to be preserved over time. In the above construction, the Hilbert space H is isomorphic to the space of square integrable functions ϕ( x ) ≡ ψ( x , 0), namely the set of square integrable initial data for the Schrödinger equation, cf. e.g. [5]. The connection between ψ and ϕ is given by the time evolution operator U (0, t) (equivalently by the Schrödinger equation): U (0, t)ϕ = ψ. The correspondence between ϕ and ψ has all formal properties, such as in the Fourier construction above. Of course, the initial data for the Schrödinger equation do not cover the whole Hilbert space H of square integrable functions, but the time evolution given
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by the Schrödinger equation can be uniquely extended over the whole Hilbert space H by the unitary evolution operator U . The construction can be applied to the particle in the Newton-Cartan spacetime. As we implicitly assumed, the wave equation is such that the set of its admissible initial data is dense in the space of square integrable functions (we need this for the uniqueness of the extension). Because of the Born interpretation, the integral ψ ∗ ψ d3 x has to be preserved over time. Let us denote the space of the square-integrable initial data ϕ on the simultaneous hyperplane t ( x , t) = t by Ht . Then, the evolution is an isometry between H0 and Ht . But such an isometry has to be a unitary operator, and the construction is well defined, i.e. the inner product of two states corresponding to the wave functions ψ1 and ψ2 does not depend on the choice of Ht . Let us mention that the wave equation has to be linear in accordance with the Born interpretation of ψ (since any unitary operator is linear the time evolution operator is linear as well). The space of wave functions ψ( x , t) = U (0, t)ϕ( x ) isomorphic to the Hilbert space H0 of ϕ’s is commonly called the “Schrödinger picture”. However in general, the connection between ϕ( x ) and ψ( x , t) is not unique if the wave equation possesses a gauge freedom. Namely, let us consider two states ϕ1 and ϕ2 and ask when these two states are equivalent, and indistinguishable. The answer is that they are equivalent if |(ϕ1 , ϕ)| ≡ ψ1∗ ( x , t)ψ( x , t) d3 x = |(ϕ2 , ϕ)| x , t)ψ( x , t) d3 x , (1) ≡ ψ2∗ ( for any state ϕ from H, or for all ψ = U ϕ (ψi are defined to be = U (0, t)ϕi ). By substituting ϕ1 and then ϕ2 for ϕ and making use of the Schwarz’s inequality, one gets: ϕ2 = eiα ϕ1 , where α is any constant1 . The situation for ψ1 and ψ2 is however different. In general, condition (1) is fulfilled if ψ2 = eiξ(t) ψ1 and the phase factor can depend on time. Of course, this has to be consistent with the wave equation, that is, together with a solution ψ of the wave equation, the wave function eiξ(t) ψ is also a solution of the appropriately gauged wave equation. A priori one cannot exclude the existence of such a consistent time evolution. This is not a new observation, as it was noticed by John von Neumann2 , but it seems that it has never been deeply investigated (probably because the ordinary nonrelativistic Schrödinger equation has gauge symmetry with constant ξ ). The space of waves ψ describing the system cannot be reduced in the above way to any fixed Hilbert space Ht with a fixed t. So, the existence of the nontrivial gauge freedom leads to the following 1 This gives the conception of the ray, introduced to Quantum Mechanics by Hermann Weyl [H. Weyl, Gruppentheorie und Quantenmechanik, Verlag von S. Hirzel in Leipzig (1928)]: a physical state does not correspond uniquely to a normed state ϕ ∈ H, but it is uniquely described by a ray; two states belong to the same ray if they differ by a constant phase factor. 2 J. v. Neumann, Mathematical Principles of Quantum Mechanics, University Press, Princeton (1955). He did not mention gauge freedom on that occasion. However, gauge freedom is necessary for the equivalence of ψ1 and ψ2 = eiξ(t) ψ1 .
A Generalization of the Bargmann’s Theory of Ray Representations
219
Hypothesis. The two waves ψ and eiξ(t) ψ are quantum-mechanically in-distinguishable. Moreover, we are obliged to use the whole Hilbert bundle RH : t → Ht over time instead of a fixed Hilbert space Ht , with the appropriate cross-sections ψ as the waves (see the next section for details). Let us consider now an action Tr of a group G in the space of waves ψ. Before we infer some consequences of the assumption that G is a symmetry group, we need to state a: Classical-like Postulate. Group G is a symmetry group if and only if the wave equation is invariant under the transformation x = rx, r ∈ G of independent variables and the transformation ψ = Tr ψ of the wave function. The above postulate is indeed commonly accepted in Quantum Mechanics even when the gauge freedom is not excluded. But it is a mere application of the symmetry definition for a classical field equation applied to the wave equation without any change. The wave ψ is not a classical quantity, such as e.g. electromagnetic intensity. The above Hypothesis is not true for classical fields, and we have to be careful in forming the appropriate postulate for the wave equation compatible with the Hypothesis. Namely, the two wave equations differing by a mere gauge are indistinguishable. We call them gauge-equivalent. It is therefore natural to assume the Quantum Postulate. Group G is a symmetry group if and only if the transformation x = rx, r ∈ G of independent variables and the transformation ψ = Tr ψ of the wave function transform the wave equation into a gauge-equivalent one. Please note that not all possibilities admitted by the Hypothesis are included in the Classical-like postulate. From the Classical-like Postulate it follows that ψ as well as Tr ψ are solutions of exactly the same wave equation, in view of the invariance of the equation. Therefore, ψ and Tr ψ belong to the same “Schrödinger picture”, so that Tr Ts ψ = eiξ(r,s) Trs ψ, with ξ = ξ(r, s) independent of time t! This is in accordance with the known theorem that Theorem 1. If G is a symmetry group, then the phase factor ξ should be time-independent. But if we start from the Quantum Postulate, we obtain instead Tr Ts ψ = eiξ(r,s,t) Trs ψ
(2)
and get Theorem 1’. If G is a symmetry group, then the phase factor ξ = ξ(r, s, t) is timedependent in general. In this paper we propose to accept the Quantum Postulate, which is compatible with the Hypothesis, and is more in spirit of Quantum Mechanics than the Classical-like Postulate. It should be noted that in the special case when gauge freedom degenerates to the case of constant phase, the Quantum Postulate is equivalent to to the Classical-like Postulate.
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Acceptance of the Quantum Postulate throws some light on the two very difficult problems: (a) generally covariant formulation of Quantum Mechanics, (b) the problems in Quantum Field Theory caused by gauge freedom. Moreover, with the help of the Hypothesis we can see that both (a) and (b) are deeply interconnected. Let us consider the standard treatment in which the Hypothesis is not taken into account and H0 is assumed to be the Hilbert space of all states and the Classical-like Postulate is accepted. Then a problem arises if we intend to formulate a representation theory of a covariance group in contradistinction to a symmetry group. The troubles have their source in the fact that the covariance group transforms the solution of the wave equation into the solution of the transformed wave equation, but the transformed equation is different in form in comparison to the initial one. That is Tr ψ does not belong to the same “Schrödinger picture” as ψ, and Tr does not act in the Hilbert space H0 of states. In view of the paradigm that any reasonable treatment of action of any group in Quantum Mechanics reduces to a unitary representation of the group in the Hilbert space of states, there was no natural way of treating the covariance group. The difficulty disappears if we start from the Hypothesis and the Quantum Postulate. Now the states are the respective cross-sections in the bundle RH, and Tr unitarily transforms fibre Ht onto a fibre Hr −1 t , acting in the same space RH as the symmetry group. Please note that the space of states degenerates to a fixed fibre H0 over t = 0 if gauge freedom degenerates to a constant phase. Before we explain the connection to problem (b), let us make a general comment concerning relation (2). There is a physical motivation to investigate representations Tr fulfilling (2) with ξ depending on spacetime point p: Tr Ts = eiξ(r,s,p) Trs .
(3)
Namely, in the Quantum Field Theory the spacetime coordinates of p ∈ M play the role of parameters in the same way as time does in the nonrelativistic theory (recall that, for example, the wave functions ψ of the Fock space of the quantum electromagnetic field are functions of the Fourier components of the field, with the spacetime coordinates playing the role of parameters like time t in nonrelativistic Quantum Mechanics). By this token, the two wave functions ψ and ψ = eiξ(p) ψ are indistinguishable in the sense that they give the same transition probabilities: |(ψ(p), φ(p))| = |(ψ (p), φ(p))| for any φ. In an analogous way, we get the Hilbert bundle MH over spacetime M and respective cross-sections as the wave functions ψ (see the next section for details). Now, let us return to problem (b). It should be mentioned at this place that the problems in QFT generated by gauge freedom are of a general character, and are well known. For example, there are no zero mass vector particles with helicity = 1, which is a consequence of the theory of unitary representations of the Poincaré group, as shown by Łopusza´nski [9]. This is apparently in contradiction to the experiment, because the photon is a vector particle with helicity = 1. What is the solution of this paradox? First, let us describe the solution on the grounds of the existing theory, which constitutes at the same time an orthodox view. We observe that we can build a zero mass vector state with h = 1 but we must admit finite-dimensional irreducible and thus non-unitary representations of the small group, that is the two-dimensional non-compact Euclidean group. Next, please note that the representation of G induced by the non-unitary representation of the small group remains “unitary” if we admit the inner product in the “Hilbert” space to be not positively defined, cf. [6 or 18]. However, even with the most favorable attitude towards
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the orthodox view, this solution is rather obscure. We propose to proceed in another way. First, let us observe that the case of the free quantum vector field3 Aµ (x) with zero mass is exactly the same. As long as the inner product in the Hilbert space is positively defined, we are not able to introduce any vector potential which transforms as a vector field. However, we can introduce a local real electromagnetic field Fµν (x) = −Fνµ (x) which is a linear combination of a self-dual and an antiself-dual field with helicity +1 and −1 respectively. If we introduce a vector potential Aµ in some Lorentz frame such that Fµν = ∂µ Aν − ∂ν Aµ then in another Lorentz frame we will have ( νµ is the Lorentz transformation matrix corresponding to the Poincaré transformation r) ν
Aµ (x) → Ur Aµ (x)Ur −1 = ( −1 )µ Aν (r −1 x) + ∂µ R(r, x), ∂µ R = 0,
(4)
while Fµν transforms as a tensor field. We infer that gauge transformation of the second kind has to accompany the Poincaré transformation, or that gauge freedom is indispensable in the construction of the vector potential in the quantum field theory, cf. [9 or 18, vol. I]. Let denote the vacuum state. According to QFT, we should define a photon state ψµ (x) in the following way ψµ (x) = Aµ (x) . It immediately follows from Eq. (4) that ν
Ur ψµ (x) = ( −1 )µ ψν (r −1 x) + ∂µ (x),
(5)
where ∂µ (x) denotes the vector-valued distribution ∂µ R(x) . The above representation spanned by the generalized vectors ψµ (x) induces a representation in the appropriate Hilbert space. Indeed, let us write ϕµ (x) and θ (x) for test functions which “smear” the distributions ψµ (x) and (x) respectively. From formula (5) we get the transformation law for ϕµ , ν
Tr ϕµ (x) = ( −1 )µ ϕν (r −1 x) + ∂µ θ(x).
(6)
By construction, the space of test functions is dense in the corresponding Hilbert space and the above representation Tr can be uniquely extended. As we are dealing with a gauge-invariant theory, the two quantum vector potentials Aµ (x) and Aµ + ∂µ (x) are unitarily equivalent. Accordingly, two photon states differing by a gradient, as well as their two corresponding vectors ϕµ (x) and ϕµ (x) + ∂µ φ(x) should be unitarily equivalent. This means that it is more adequate to consider all ϕµ (x) + ∂µ φ(x) instead of the respective ϕµ (x) alone. We write ϕµ (x) + ∂µ φ(x) as a pair {φ(x), ϕµ (x)}. The action of our representation Tr in the space of pairs {φ(x), ϕµ (x)} is as follows: Tr {φ(x), ϕµ (x)} = {φ(r −1 x) + θ(r, x), Ur ϕµ (x)}, where Ur acts as an ordinary vector transformation: ν
Ur ϕµ (x) = ( −1 )µ ϕν (r −1 x). 3
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Moreover, we have Tr Ts {φ(x), ϕµ (x)} = Trs {φ(x), ϕµ (x)} + {ξ(r, s, x), 0},
(7)
where ξ(r, s, x) = θ (r, x) + θ (s, r −1 x) − θ(rs, x). But this is nothing else but a (generalized) ray representation of the Poincaré group G fulfilling (3) with the spacetime-dependent exponent ξ(r, s, x)! To sum up the discussion concerning problem (b), we have just seen that the generalized representation in the sense of Eq. (3) seems to be indispensable if we wish to work with ordinary Hilbert spaces with positive norms while having a theory which describes photons. We shall resolve the following paradox. Namely, a natural question arises why the phase factor eiξ in (2) is time-independent for the Galilean group (even when the Galilean group is considered as a covariance group). The explanation of the paradox is as follows. The Galilean covariance group G induces the representation Tr in the space RH and fulfills (2). But, as we will show later on, the structure of G is such that there always exists a function ζ (r, t) continuous in r and differentiable in t, with the help of which one can define a new equivalent representation Tr = eiζ (r,t) Tr fulfilling Tr Ts = eiξ(r,s) Trs with a time-independent ξ . The representations Tr and Tr are equivalent because Tr ψ and Tr ψ are equivalent for all r and ψ. However, this is not the case in general, when the exponent ξ depends on time, and this time dependence cannot be eliminated in the same way as for the Galilean group. We have a similar situation when we try to find the most general wave equation for a nonrelativistic quantum particle in the Newton-Cartan spacetime. The relevant covariance group in this case is the Milne group which possesses representations with time-dependent ξ not equivalent to any representations with a time-independent ξ . Moreover, the only physical representations of the Milne group are those with time-dependent ξ . The difference between the relativistic and nonrelativistic theories, where we consider the bundle over time and over spacetime, is artificial and depends on the structure of the corresponding group. Namely, we could consider the spacetime-dependent ξ with Tr acting in a Hilbert bundle over the spacetime M for the Galilean group as well. However, the relevant group structure is such that Tr is equivalent to a representation Tr with time-dependent ξ . We have a similar situation with the Milne group. Let us close this section with more speculative considerations, trying to answer the following question. We have already mentioned that spacetime coordinates are classical variables in QFT. What does this mean? We have interpreted the term ‘classical’ coordinate as indicating that spacetime coordinates are numerical parameters. In other words, that they do not form any operators. But could we interpret this word in a somewhat deeper sense? Already in the classic works of Dirac [3] and Heisenberg [8], spacetime coordinates are interpreted as representing a special kind of operators, called c-numbers. The spacetime coordinate operators x µ 1 commute with each other and all other operators of our quantum algebra R, which we do not specify here. Here we propose to treat this interpretation seriously. Let us consider an algebra a of functions f (p) on spacetime M which encodes the geometry of M. At the moment, the exact structure of a is not important for us. However, it will be a commutative algebra with point-wise operations. For simplicity, we confine ourself to the topological structure of M (i.e. the
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algebra a of continuous functions), assume M to be compact and R ⊂ B(H). Let us form an operator Df = f (p)1 corresponding to the function p → f (p) in our algebra a. It is therefore natural to assume that the set of all Df , f ∈ a in a natural manner composes an algebra A isomorphic to a, i.e. the point-wise function multiplication in a corresponds to the operator composition in A. This is precisely what we mean when assuming the spacetime coordinates to be classical. Moreover, it is also natural to assume that A is closed with respect to norm and dense in the center of R with respect to the strong operator topology. But this is possible if the operators in R are decomposable and act in a direct integral of Hilbert spaces Hp , p ∈ M, or in a Hilbert bundle MH over spacetime. This is the peculiar property of the Galilean group structure that the whole construction degenerates as if we started from an ordinary ray representation acting in the ordinary Hilbert space. Theorem 1 is then true in this case, but only incidentally. The generalization to the relativistic case is natural. We postulate the spacetime coordinates to be classical commutative variables, which leads to the Hilbert bundle MH over the space-time manifold M. The factor of the representation of the Poincaré group acting in the bundle MH does not have to be constant with respect to space-time coordinates even when it is a symmetry group. In order to implement the above program consistently, we are forced to generalize Bargmann’s theory of factors to embrace the spacetime-dependent factors of representations acting in a Hilbert bundle over space-time (time). Similar ideas was already suggested in the early seventies by Piron, cf. [12, 13]. 3. Generalized Wave Rays and Operator Rays In this section we give strict mathematical definitions of the notions of the preceding section, and formulate the problem stated there in an exact way. From the pure mathematical point of view, the analysis of spacetime-dependent ξ(r, s, p) is more general, so at the outset we confine ourselves to this case 4 . Let us recall some definitions, cf. e.g. [10]. Let M be a set endowed with an analytic Borel structure. By a Hilbert bundle over M or a Hilbert bundle with base M we shall mean an assignment H : p → Hp of a Hilbert space Hp to each p ∈ M. The set of all pairs (p, ψ) with ψ ∈ Hp will be denoted by MH and called the space of the bundle. By a cross section of our bundle we shall mean an assignment ψ : p → ψp of a member of Hp to each p ∈ M. If ψ is a cross section and (p0 , φ0 ) a point of MH, we may form a scalar product (φ0 , ψp0 ). In this way, every cross-section ψ defines a complex-valued function fψ on MH. By a Borel Hilbert bundle we shall mean a Hilbert bundle together with an analytic Borel structure in MH such that the following conditions are fulfilled: (1) Let π(p, ψ) = p. Then E ⊆ M is a Borel set if and only if π −1 (E) is a Borel set in MH. (2) There exist countably many cross-sections ψ 1 , ψ 2 , . . . such that (a) the corresponding complex-valued functions on MH are Borel functions, (b) these Borel functions separate points in the sense that no two distinct points (pi , φi ) of MH assign the same values to all ψ j unless φ1 = φ2 = 0, and (c) p → (ψ i (p), ψ j (p)) is a Borel function for all i and j . 4 It becomes clear in further analysis that the group G in question has to fulfill the consistency condition requiring that for any r ∈ G, rt is a function of time only in the case of nonrelativistic theory with (2).
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A cross-section is said to be a Borel cross-section if the function on MH defined by the cross-section is a Borel function. All Borel cross-sections compose a linear space under the obvious operations, cf. [10]. Now let µ be a measure on M. The cross-section p → ϕp is said to be square summable with respect to µ if (ϕp , ϕp ) dµ(p) < ∞. M
L2 (M, µ, H)
of all equivalence classes of square-summable cross-sections, The space where two cross-sections ϕ and ϕ are in the same equivalence class if ϕp = ϕp for almost all p ∈ M, forms a separable Hilbert space with the inner product given by (ϕp , θp ) dµ(p), (ϕ, θ ) = M
cf. [10]. It is called the direct integral of the Hp with respect to µ and is denoted by M Hp dµ(p). Identification with the previous section is partially suggested by the notation itself. We shall make this identification more explicit. The set M plays the role of spacetime or real line R of time t respectively. The wave functions ψ of the preceding section are the Borel cross-sections of MH but they do not belong to the subset L2 (M, µ, H) of cross-sections which are square integrable. Rather, the separate Hilbert spaces Hp with their inner products play a role in experiments than the inner product in their direct integral product. We have also used ψ(p) and ψp as well as (ψp , θp ) and (ψ, θ )p interchangeably. The physical interpretation ascribed to the cross-section ψ is as follows. Each experiment is, by its very nature, a spatiotemporal event. To each act of measurement carried out at the spacetime point p0 we ascribe a self-adjoint operator Qp0 acting in the Hilbert space Hp0 assigning standard interpretation to the spectral theorem for Qp0 . Hence, for simplicity assuming that Qp0 possess discrete spectrum, if φ0 ∈ Hp0 and λ0 = λo (p0 ) is a characteristic vector and its corresponding characteristic value of Qp0 respectively, then the following statement holds true. If the experiment corresponding to Qp was performed at the spatiotemporal event p0 on a system in the state described by the crosssection ψ, then the probability of the measurement value to be λ0 (p0 ) and the system to be found in the state described by φ such that φ(po ) = φ0 after the experiment is given by the square of the absolute value of the Borel function |fψ (p0 , φ0 )|2 = |(φ0 , ψp0 )|2 induced by the cross-section ψ. In the nonrelativistic case, the above statement is a mere rephrasing of well-established knowledge. By an isomorphism of the Hilbert bundle MH with the Hilbert bundle M H we shall mean a Borel isomorphism T of MH on M H such that for each p ∈ M the restriction of T to p × Hp has some q × Hq for its range and is unitary when regarded as a map of Hp on Hq . The induced map carrying p into q is clearly a Borel isomorphism of M with M and we denote it by T π . The above-defined T is said to be an automorphism if MH = M H . Please note that for any automorphism T we have (T ψ, T φ)T π p = (ψ, φ)p , but in general (T ψ, T φ)p = (ψ, φ)p . By this token, any automorphism T is what is frequently called bundle isometry. The function r → Tr from group G into the set of automorphisms (bundle isometry) of MH is said to be a general factor representation of G associated to the action G × M r, p → r −1 p ∈ M of G on M if Trπ (p) ≡ r −1 p for all r ∈ G, and Tr satisfy condition (3).
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Of course, Tr is to be identified with that of the preceding section. Our further specializing assumptions partly following from the above interpretation are as follows. We assume M to be endowed with the manifold structure inducing a topology associated with the above-assumed Borel structure. We confine ourselves to a finite dimensional Lie group G which acts smoothly and transitively on spacetime M, such that a G-invariant measure µ exists on M. By a factor representation of a Lie group we mean a general factor representation with the exponent ξ(r, s, p) differentiable in p ∈ M. Now we define the operator ray T corresponding to a given bundle isometry operator T to be the set of operators T = {τ T , p → τ (p) ∈ D and |τ | = 1}, where D denotes the set of all differentiable real functions on M. Any T ∈ T will be called a representative of the ray T . The product T V is defined as the set of all point-wise products T V such that T ∈ T and V ∈ V . Please note that not all Borel sections are physically realizable. By interpreting the discussion of the preceding section in the Hilbert bundle language, we see that the role of the Schrödinger equation is essentially to establish all the physical sections. Any two sections ψ(p) and ψ (p) = eiζ (p) ψ(p) are indistinguishable giving the same probabilities |fψ |2 = |fψ |2 . After this, any group G acting in M induces a ray representation of G, i.e. a mapping r → T r of G into the space of rays of bundle automorphisms (bundle isometrics) of MH, fulfilling the condition T r T s = T rs . For any cross-section ψ we define its corresponding ray ψ = {eiζ (p) ψ(p), ζ ∈ D}. If ψ is a physical cross-section, then we get the physical ray of the preceding section. Selecting a representative Tr for each T r , we get a factor representation fulfilling (3). Please note that Tr transforms rays into rays, and we have Tr (eiξ(p) ψ) = eiξr (p) Tr ψ. Further on we assume that that operators Tr are such that ξr (p) = ξ(r −1 p), where r −1 p denotes the action of r −1 ∈ G on the spacetime point p ∈ M. This is a natural assumption which does actually take place in practice. Now we shall make the last assumption, namely that all transition probabilities vary continuously with the continuous variation of the coordinate transformation s ∈ G: For any element r in G, any ray ψ and any positive , there exists a neighborhood N of r on G such that dp (T s ψ, T r ψ) < if s ∈ N and p ∈ M, where dp (ψ1 , ψ2 ) = inf ψ1 − ψ2 p = 2|1 − |(ψ1 , ψ2 )p | |. ψi ∈ψ i
Basing on the continuity assumption, one can prove the following Theorem 2. Let T r be a continuous ray representation of a group G. For all r in a suitably chosen neighborhood N0 of the unit element e of G one may select a strongly continuous set of representatives Tr ∈ T r . That is, for any compact set C ⊂ M, any wave function ψ, any r ∈ N0 and any positive there exists a neighborhood N of r such that Ts ψ − Tr ψp < if s ∈ N and p ∈ C. There are numerous possible selections of such factor representations. But many among them merely differ by a differentiable phase factor and are physically indistinguishable. We call them equivalent. Our task then is to classify all possible factor representations with respect to this equivalence.
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4. Local Exponents The representatives Tr ∈ T r selected as in Theorem 2 will be called admissible, with the representation Tr obtained in this way referred to as an admissible representation. There are infinitely many possibilities of such a selection of admissible representations Tr . We confine ourselves to the local admissible representations defined on a fixed neighborhood No of e ∈ G, as in Theorem 2. Let Tr be an admissible representation. With the help of the phase eiζ (r,p) with a real function ζ (r, p) differentiable in p and continuous in r, we can define Tr = eiζ (r,p) Tr ,
(8)
which is a new admissible representation. This is trivial if one defines the continuity of ζ (r, p) in r appropriately. Namely, from Theorem 2 it follows that the continuity has to be defined in the following way. The function ζ (r, p) will be called strongly continuous in r at r0 if and only if for any compact set C ⊂ M and any positive there exists a neighborhood N0 of r0 such that |ζ (r0 , p) − ζ (r, p)| < , for all r ∈ N0 and for all p ∈ C. But the converse is also true. Indeed, if Tr is also an admissible representation, then (8) has to be fulfilled for a real function ζ (r, p) differentiable in p because Tr and Tr belong to the same ray. Moreover, because both Tr ψ and Tr ψ are strongly continuous (in r for any ψ), then ζ (r, p) has to be strongly continuous (in r). Let Tr be an admissible representation, and thus continuous in the sense indicated in Theorem 2. One can always choose the above ζ in such a way that Te = 1 as will be assumed from now on. Because Tr Ts and Trs belong to the same ray, one has Tr Ts = eiξ(r,s,p) Trs
(9)
with a real function ξ(r, s, p) differentiable in p. From the fact that Te = 1, we have ξ(e, e, p) = 0.
(10)
From the associative law (Tr Ts )Tg = Tr (Ts Tg ) one gets ξ(r, s, p) + ξ(rs, g, p) = ξ(s, g, r −1 p) + ξ(r, sg, p).
(11)
Formula (11) is very important and our analysis largely rests on this relation. From the fact that the representation Tr is admissible it follows that the exponent ξ(r, s, p) is continuous in r and s. Indeed, let us take a ψ belonging to a unit ray ψ. Then, making use of (9), we get eiξ(r,s,p) (Trs − Tr s )ψ + (Tr (Ts − Ts )ψ + (Tr − Tr )Ts ψ
= (eiξ(r ,s ,p) − eiξ(r,s,p) )Tr s ψ.
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Taking norms p of both sides, we get
|eiξ(r ,s ,p) − eiξ(r,s,p) | ≤ (Tr s − Trs )ψp + +Tr (Ts − Ts )ψp + (Tr − Tr )Ts ψp . From this inequality and the continuity of Tr ψ, the continuity of ξ(r, s, p) in r and s follows. Moreover, from Theorem 2 and the above inequality follows the strong continuity of ξ(r, s, p) in r and s. Formula (8) suggests the following definition. Two admissible representations Tr and Tr are called equivalent if and only if Tr = eiζ (r,p) Tr for some real function ζ (r, p) differentiable in p and strongly continuous in r. Thus, making use of (9), we get Tr Ts = eiξ (r,s,p) Trs , where ξ (r, s, p) = ξ(r, s, p) + ζ (r, p) + ζ (s, r −1 p) − ζ (rs, p).
(12)
Then the two exponents ξ and ξ are equivalent if and only if (12) is fulfilled with ζ (r, p) strongly continuous in r and differentiable in p. From (10) and (11) it immediately follows that ξ(r, e, p) = 0 and ξ(e, g, p) = 0,
(13)
ξ(r, r −1 , p) = ξ(r −1 , r, r −1 p).
(14)
Relation (12) between ξ and ξ will be written in short by ξ = ξ + [ζ ].
(15)
The relation (12) between exponents ξ and ξ defines an equivalence relation, which preserves the linear structure. We introduce now group H , a very important notion for our further investigations. It is evident that all operators Tr contained in all rays T r form a group under multiplication. Indeed, let us consider an admissible representation Tr with a well-defined ξ(r, s, p) in formula (9). Because any Tr ∈ T r has the form eiθ(p) Tr (with a real and differentiable θ), one has −1 (16) eiθ(p) Tr eiθ (p) Ts = ei{θ(p)+θ (r p)+ξ(r,s,p)} Trs . This important relation suggests the following definition of the local group H connected with the admissible representation or with the exponent ξ(r, s, p). Namely, H consists of the pairs {θ (p), r}, where θ (p) is a differentiable real function and r ∈ G. The multiplication rule, suggested by the above relation, is defined as follows: {θ (p), r} {θ (p), r } = {θ (p) + θ (r −1 p) + ξ(r, r , p), rr }.
(17)
The associative law for this multiplication rule is equivalent to (11) (in complete analogy with the classical Bargmann theory). The pair eˇ = {0, e} plays the role of the unit element in H . For any element {θ (p), r} ∈ H there exists the inverse {θ(p), r}−1 = {−θ (rp) − ξ(r, r −1 , rp), r −1 }. Indeed, from (14) it follows that {θ, r}−1 {θ, r} = {θ, r} {θ, r}−1 = e. ˇ The elements {θ (p), e} form an Abelian subgroup N of H . Any {θ, r} ∈ H can be uniquely written as {θ (p), r} = {θ(p), e} {0, r}. The same element
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can be also uniquely expressed in the form {θ (p), r} = {0, r} {θ (rp), e}. Thus, we have H = N G = G N . The Abelian subgroup N is a normal factor subgroup of H . But this time, G does not form any normal factor subgroup of H (contrary to the classical case investigated by Bargmann, when the exponents do not depend on p). So, this time H is not a direct product N ⊗ G, but a semidirect product N G, cf. e.g. [11]. In this case, however, the theorem that G is locally isomorphic to the factor group H /N is still valid, cf. [11]. Then group H composes a semicentral extension of G, and not a central extension of G as in Bargmann’s theory. The rest of this paper is based on the following reasoning (the author being largely inspired by Bargmann’s work [1]). If the two exponents ξ and ξ are equivalent, that is ξ = ξ + [ζ ], then the semicentral extensions H and H connected with ξ and ξ are homomorphic. The homomorphism h : {θ, r} → {θ , r } is given by θ (p) = θ (p) − ζ (r, p), r = r.
(18)
Using an Iwasawa-type construction we show that any exponent ξ(r, s, p) is equivalent to a differentiable one (in r and s). We can then confine ourselves to the differentiable ξ and ξ . We show that ζ (r, p) is also a differentiable function of (r, p). Moreover, we show that any ξ is equivalent to the canonical one, that is such ξ which is differentiable and for which ξ(r, s, p) = 0 whenever r and s belong to the same one-parameter subgroup. Then we can restrict our investigation to the canonical ξ considering the subgroup of all elements {θ (p), r} ∈ H with differentiable θ (p). For simplicity let us denote the subgroup by the same symbol H . We embed the subgroup in an infinite dimensional Lie group D with manifold structure modeled on a Banach space. Then we consider the subgroup H which is the closure of H in D. After this, H turns into a Lie group and the homomorphism (18) becomes an isomorphism of the two Lie groups. Thus, the group H has the Banach Lie algebra H. We apply the general theory of analytic groups developed in [2 and 4]. From this theory it follows that the correspondence between the local H and H is bi-unique and one can construct uniquely the local group H from the algebra H as well. As we will see, the algebra defines a spacetime-dependent antilinear form on the Lie algebra G of G, the so-called infinitesimal exponent . By this we reduce the classification of local ξ ’s which define H ’s to the classification of ’s which define H’s. So, we will simplify the problem of the classification of local ξ ’s to a largely linear problem. Here are the details. Iwasawa construction. Let us denote by dr and d∗ r the left and right invariant Haar measure on G. Let ν(r) and ν ∗ (r) be two infinitely differentiable functions on G with compact supports contained in the fixed neighborhood N0 of e. Bymultiplying them by the appropriate constants, we can always obtain: G ν(r) dr = G ν ∗ (r) d∗ r = 1. Let ξ(r, s, p) be any admissible local exponent defined on N0 . We will construct a differentiable (in r and s) exponent ξ (r, s, p) equivalent to ξ(r, s, p) and defined on N0 , in the following two steps: ξ = ξ + [ζ ] and ξ = ξ + [ζ ], where ζ (r, p) is the left invariant integral of l → −ξ(r, l, p)ν(l), while ζ (r, p) is the right invariant integral of u → −ξ (u, r, up)ν ∗ (u). A rather simple computation in which we use (12) and (11) and the invariance property of the Haar measures shows that ξ (r, s, p) is a differentiable (up to any order) exponent in all variables. Next we shall show that if two differentiable exponents ξ and ξ are equivalent, that is, if ξ = ξ + [ζ ], then ζ (r, p) is differentiable in r. Clearly, the difference ξ − ξ is differentiable. Similarly, both (ξ − ξ )ν ( with ν defined as above), as well as its left invariant integral η are differentiable. It is easy to show that ζ = η − ζ is differentiable. In this way we arrive at
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differentiability of ζ = η − ζ . A slightly more complicated argument shows that every (local) exponent of one-parameter group is equivalent to zero. However, the argument is quite analogous to that of Bargmann. We can treat such a group as the additive group of real numbers, so that the first two arguments of ξ are real numbers. Let us set ϑ(τ, σ, p) as the derivative of ξ with respect to the second argument. It is not hard to show that ξ + [ζ ] = 0, where ζ (τ, p) is the ordinary Riemann integral of µ → τ ϑ(µτ, 0, p) over the unit interval [0, 1]. But it means that ξ is equivalent to zero. Let us recall that the continuous curve r(τ ) in a Lie group G is a one-parameter subgroup if and only if r(τ1 )r(τ2 ) = r(τ1 + τ2 ), i.e. r(τ ) = (r0 )τ for some element r0 ∈ G. (Please note that the real power r τ is well defined on a Lie group, at least on some neighborhood of e.) The coordinates ρ k in G are canonical if and only if any curve of the form r(τ ) = τρ k (where the coordinates ρ k are fixed) is a one-parameter subgroup (the curve r(τ ) = τρ k will be denoted in short by τ a, with the coordinates of a equal to ρ k ). The “vector” a is called by physicists the generator of the one-parameter subgroup τ a. A local exponent ξ of a Lie group G is called canonical if ξ(r, s, p) is differentiable in all variables, and ξ(r, s, p) = 0 if r and s are elements of the same one-parameter subgroup. Almost the same argument used to show that every ξ on a one-parameter group is equivalent to zero also shows that every local exponent ξ of a Lie group is equivalent to a canonical local exponent. In order to prove this, we shall apply the argument to the exponent ξ0 (τ, σ, p) := ξ(τ a, σ a, p), cf. [16]. Up to now, the argumentation has been more or less analogous to that of Bargmann. From now on, the argumentation becomes entirely different. Let ξ and ξ be two differentiable and equivalent local exponents of a Lie group G, assuming ξ to be canonical. Then ξ is canonical if and only if ξ = ξ +[ ], where (r, p) is a linear form in the canonical coordinates of r fulfilling the condition that (a, (τ a)p) is constant as a function of τ , i.e. it follows that 5 a (a, p) =
(a, (a)p) − (a, p) d (a, (τ a)p) = lim = 0. →0 dτ
(19)
While sufficiency of the condition in the above statement is almost evident, proving its necessity is quite nontrivial. Hereafter we outline the argumentation. Because the exponents are equivalent we have ξ (r, s, p) = ξ(r, s, p) + [ζ ]. Since both ξ and ξ are differentiable then ζ (r, p) is also a differentiable function, which follows from what has been said above. Let us suppose that r = τ a and s = τ a. Because both ξ and ξ are canonical, we have ξ(τ a, τ a, p) = ξ (τ a, τ a, p) = 0, so that [ζ ](τ a, τ a, p) = 0. Applying the last formula recurrently one gets ζ (τ a, p) =
n−1 k=0
τ k ζ ( a, (− τ a)p). n n
5
The limit in the expression can be understood in the ordinary point-wise sense with respect to p, but also in any linear topology in the function linear space (with obvious addition) of θ (p), providing that p → (a, p) is differentiable in the sense of this linear topology. Further on, the simple notation af (p) = will be used.
df ((τ a)p) f ((a)p) − f (p) , = lim τ =0 dτ →0
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Then we apply the Taylor Theorem to each summand in the above expression, and pass to the limit n → +∞. In this way, we obtain τ ζ (τ a, p) = ς(a, (−σ a)p) dσ, (20) 0
where ς = ς (r, p) is a differentiable function, cf. [16]. If we differentiate now expression (20) with respect to τ at τ = 0, we will immediately see that the function ς (a, p) is linear with respect to a. Let us suppose that the spacetime coordinates are chosen in such a way that the integral curves p(x) = (xa)p0 are coordinate lines, which is possible for appropriately small x. There are of course three remaining families of coordinate lines besides p(x), which can be chosen in an arbitrary way, with their parameters denoted by yi . After this, 1 τ 1 x ζ (a, x, yi ) = ς(a, x − σ, yi ) dσ = ς (a, z, yi ) dz, τ 0 τ x−τ for any τ (of course, with appropriately small |τ |, in our case |τ | ≤ 1) and for any (appropriately small) x. But this is possible for the function ς (a, x, yk ) continuous in x (in our case, differentiable in x) if and only if ς(a, x, yk ) does not depend on x. This means that ζ (a, x, yk ) does not depend on x and the condition of the statement is hereby proved. Infinitesimal exponents and embedding of H in a Lie group D. According to what has been shown already, we can assume that the exponent is canonical. We also confine ourselves to the subgroup of {θ (p), r} ∈ H with differentiable θ , and denote this subgroup by the same letter H . We embed this subgroup H in an infinite dimensional Lie group with the manifold structure modeled on a Banach space. We will extensively use the theory developed by Birkhoff [2] and Dynkin [4]. For the systematic treatment of manifolds modeled on Banach spaces, see e.g. [7]. By this embedding we ascribe bi-uniquely a Lie algebra to the group H with the convergent Baker-Hausdorff series. Please note first that the formula H × L2 (M, µ, H) ({θ (p), r}, φ) → eiθ(p) Tr φ (together with(16)) can be viewed as a rule giving the action of H in the direct integral Hilbert space M Hp dµ(p) defined in Sect. 3. Moreover, this is a unitary action, provided µ is G-invariant. In accordance to [2], the group D of all unitary operators of a Hilbert space is an infinite dimensional Lie group. Hence, H = N G can be viewed as a subgroup of a Lie group. We consider now the closure H of H in the sense of the topology in D. It is remarkable that the subgroup H has locally the structure of the semi-direct product N G as well. This is a consequence of the following four facts. (1) N is a normal subgroup of H = N G. (2) G is finite dimensional, so G = G. (3) Locally (in a neighborhood O), the multiplication in D is given by the Baker-Hausdorff formula in the Banach algebra of D. Because N is normal in H , then the above exponential mapping converts locally the multiplication N S of N by any subset S of H into the sum N + S. Because G is finite-dimensional, and hence locally compact, the neighborhood O can be chosen in such a way that locally (in the closure of O + O) the following holds: N + G = N + G = H.
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(4) The local N (intersected with O) has a finite co-dimension in local N + G (intersected with O + O) and thus it splits locally N + G. So, we have locally i.e. in O + O: N + G = H = N ⊕ G , where G = G and ⊕ stands for a direct sum. From this it follows that G = G locally. This shows that H = N G. Because H = N G, every h ∈ H is uniquely representable in the form ng, where n ∈ N and g ∈ G. Please note now that (n1 g1 )(n2 g2 ) = n1 g1 n2 g1−1 g1 g2 = [n1 (g1 n2 g1−1 )](g1 g2 ) and that g1 n2 g1−1 ∈ N because N is normal in H . Let us denote the automorphism n → gng −1 of N by Rg . The group H can be locally viewed as a topological product of Banach spaces N × G, one of which (namely G) is finite-dimensional and isomorphic to the Lie algebra of G. The multiplication in H can be written as (n1 , g1 )(n2 , g2 ) = (n1 Rg1 (n2 ), g1 g2 ). Moreover, N can be viewed locally as the Banach space N with the multiplication law given by the vector addition in N. Our task now is to reconstruct the Lie algebra H corresponding to the subgroup H . Let λ → λa be a one-parameter subgroup of G. The mapping (λ, n) → (Rλa n, λa) of the Banach space R × N into the Banach space N × G is continuous. In consequence, R λ → Rλa n ∈ N as well as N n → Rλa n are continuous. Therefore, the function λ → Rλa n can be integrated over any compact interval and τ τ → (nτ a , τ a) := Rσ a n dσ, τ a 0
is a one-parameter subgroup of H with generator6 (n, a), cf. [2, 4]. Having obtained this, we are able to reconstruct the algebra. The elements of H ⊂ H are representable in the ordinary form {α, r} with differentiable α = α(p), p ∈ M, and r ∈ G. Let us consider the above-defined operator Rλa . Its restriction to H ⊂ H is given by (please remember that ξ is canonical) α(p) → (Rλa α)(p) = α((λa)−1 p). We can now compute explicitly the Lie bracket and the Jacobi identity for all the elements {α(p), a} of the subalgebra H ⊂ H corresponding to the subgroup H . The result is as follows7 ˇ = {aβ − bα + (a, b, p), [a, b]}, [a, ˇ b]
(22)
6 The limit process with the help of which the generator is computed refers to the topology in D, of course. 7 Let us stress once more that
aθ (p) = lim
→0
θ ((a)p) − θ (p) ,
and the limit is in the sense of topology induced from the Lie group D.
(21)
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(a, b, p) = lim τ −2 {ξ((τ a)(τ b), (τ a)−1 (τ b)−1 , p)+ τ →0
+ξ(τ a, τ b, p) + ξ((τ a)−1 , (τ b)−1 , (τ b)−1 (τ a)−1 p)}.
(23)
From the associative law in H one gets ([a, a ], a , p) + ([a , a ], a, p) + ([a , a], a , p) = = a(a , a , p) + a (a , a, p) + a (a, a , p),
(24)
which can be shown to be equivalent to the Jacobi identity [[a, ˇ aˇ ], aˇ ] + [[aˇ , aˇ ], a] ˇ + [[aˇ , a], ˇ aˇ ] = 0.
(25)
ˇ for all Thus, in this way we have reconstructed the Lie algebra H giving explicitly [a, ˇ b] a, ˇ bˇ ∈ H ⊆ H. Because H is dense in H, the local exponent determines the algebra H uniquely. But from the theory of Lie groups the correspondence between the algebras H and local Lie groups H is bi-unique, at least locally, cf. e.g. [2 and 4]. Therefore, the correspondence H → H between the local group H and the algebra H is one-to-one. Because the exponent ξ determines the multiplication rule in H and vice-versa, then it follows that the correspondence ξ → between the local ξ and the infinitesimal exponent is one-to-one. Please note that the term ’local ξ = ξ(r, s, p)’ means that ξ(r, s, p) is defined for r and s belonging to a fixed neighborhood N0 ⊂ G of e ∈ G, but in our case it is defined globally as a function of the spacetime variable p ∈ M. Infinitesimal exponents and local exponents. Now, let us move on to describing the relation between the infinitesimal exponents and the local exponents ξ . First, let us compute the infinitesimal exponents and given by (23), which correspond to the two equivalent canonical local exponents ξ and ξ = ξ +[ ]. Inserting ξ = ξ +[ ] into formula (23), one gets (a, b, p) = (a, b, p) + a (b, p) − b (a, p) − ([a, b], p).
(26)
According to what has been said, we can confine ourselves to the canonical exponents. Then, as one of our previous statements said, = (a, (τ b)p) is a constant function of τ if a = b, and (a, p) is linear with respect to a (we use the canonical coordinates on G). Hence (a, b, p) is antisymmetric in a and b and fulfills (24) only if (a, b, p) is antisymmetric in a and b and fulfills (24). This suggests the definition: two infinitesimal exponents and will be called equivalent if and only if relation (26) holds. For brevity, we write relation (26) as follows: = + d[ ]. Finally, we maintain that two canonical local exponents ξ and ξ are equivalent if and only if the corresponding infinitesimal exponents and are equivalent. Indeed. (1) Assume ξ and ξ to be equivalent. Then, by the definition of equivalence of infinitesimal exponents: = + d[ ]. (2) Assume and to be equivalent: = + d[ ] for some linear form (a, t) such that (a, (τ a)p) does not depend on τ . Then ξ +[ ] → , and by the uniqueness of the correspondence ξ → we have ξ = ξ + [ ], i.e. ξ and ξ are equivalent. In this way, we arrive at the following:
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Theorem 3. (1) On a Lie group G, every local exponent ξ(r, s, p) is equivalent to a canonical local exponent ξ (r, s, p) which, on some canonical neighborhood N0 , is analytic in canonical coordinates of r and s, and vanishes if r and s belong to the same one-parameter subgroup. Two canonical local exponents ξ, ξ are equivalent if and only if ξ = ξ + [ ] on some canonical neighborhood, where (r, p) is a linear form in the canonical coordinates of r such that (r, sp) does not depend on s if s belongs to the same one-parameter subgroup as r. (2) To every canonical local exponent of G there corresponds uniquely an infinitesimal exponent (a, b, p) on the Lie algebra G of G, i.e. a bilinear antisymmetric form which satisfies the identity ([a, a ], a , p) + ([a , a ], a, p) + (a , a], a , p) = a(a , a , p) + a (a , a, p) + a (a, a , p). The correspondence is linear. (3) Two canonical local exponents ξ, ξ are equivalent if and only if the corresponding , are equivalent, i.e. (a, b, p) = (a, b, p) + a (b, p) − b (a, p) − ([a, b], p), where (a, p) is a linear form in a on G such that τ → (a, (τ b)p) is constant if a = b. (4) There exist a one-to-one correspondence between the equivalence classes of local exponents ξ (global in p ∈ M) of G and the equivalence classes of infinitesimal exponents of G. 5. Global Extensions of Local Exponents Theorem 3 provides full classification of exponents ξ(r, s, p) local in r and s, defined for all p ∈ M. But if G is both connected and simply connected, then we have the following theorems. (1) If an extension ξ of a given local (in r and s) exponent ξ does exist, then it is uniquely determined (up to the equivalence transformation (12)) (Theorem 4). (2) There exists such an extension ξ (Theorem 5), proved for G, which possess the finite-dimensional extension H only. We are not able to prove that the (global) homomorphism (18) is continuous when ξ is not canonical. Please note that any ξ is equivalent to its canonical counterpart, but only locally! This is why the topology of H induced from D is not applicable in the global analysis. We introduce another topology. Because of the semidirect structure of H = N G, it is sufficient to introduce it into N and G separately in such a manner that G acts continuously on N , cf. e.g. [10]. From the discussion of Sect. 4 it is sufficient to introduce the Fréchet topology of almost uniform convergence in the function space N . Indeed, from the strong continuity of ξ and ζ in (18) it follows that the multiplication rule as well as the homomorphism (18) are continuous. Theorem 4. Let ξ and ξ be two equivalent local exponents of a connected and simply connected group G, so that ξ = ξ + [ζ ] on some neighborhood, assuming the exponents ξ1 and ξ1 of G to be extensions of ξ and ξ respectively. Then, for all r, s ∈ G: ξ1 (r, s, p) = ξ1 (r, s, p) + [ζ1 ], where ζ1 (r, p) is strongly continuous in r and differentiable in p, and ζ1 (r, p) = ζ (r, p), for all p ∈ M and for all r belonging to some neighborhood of e ∈ G. Here is the proof outline. The two exponents ξ1 and ξ1 being strongly continuous (by assumption) define two semicentral extensions H1 = N1 G and H1 = N1 G, which are continuous groups. Please note that the linear groups N1 , N1 are connected and simply connected. Because both H1 and H1 are semi-direct products of two connected and simply connected groups they are both connected and simply connected. Equation (18) defines a local isomorphism mapping h : rˇ → rˇ = h(ˇr ) of H1 into H1 . Because H1 and H1 are connected and simply connected, the isomorphism h given by (18) can be uniquely extended to an isomorphism h1 of the entire groups H1 and
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H1 such that h1 (ˇr ) = h(ˇr ) on some neighborhood of H1 , cf. [14], Theorem 80. The isomorphism h1 defines an isomorphism of the two Abelian subgroups N1 and h1 (N1 ). By (18), h1 (θ, e) = {θ, e} locally in H1 , that is for θ lying appropriately close to 0 (in the metric sense defined previously). Both N1 and h1 (N1 ) are connected, and N1 is in addition simply connected, so applying once again Theorem 80 of [14], one can see that h1 (θ, e) = {θ, e} for all θ . A rather simple computation shows that ζ1 defined by the equality h1 (0, r) = {−ζ1 (p), g(r)} fulfills the conditions of our theorem. The following theorem is proved for the group G with a finite-dimensional extended algebra H . Theorem 5. Let G be a connected and simply connected Lie group. Then to every exponent ξ(r, s, X) of G defined locally in (r, s) there exists an exponent ξ0 of G defined on the whole group G which is an extension of ξ . If ξ is differentiable, ξ0 may be chosen differentiable. Because the proof of Theorem 5 is almost identical to that of Theorem 5.1 in [1], we do not present it explicitly8 . Please note that the proof rests largely on the global theory of classical (finite-dimensional) Lie groups. Namely, it rests on the theorem that to any finite dimensional Lie group there always exists a universal covering group . We can use those methods because of the existence of a finite-dimensional extension H of G. We have obtained the full classification of time-dependent ξ defined on the whole group G for Lie groups G which are connected and simply connected in nonrelativistic theory. But for any Lie group G there exists a universal covering group G∗ which is connected and simply connected. Thus, for G∗ the correspondence ξ → is one-toone, that is, to every ξ there exists a unique and vice versa, to every corresponds a unique ξ defined on the whole group G∗ , and the correspondence preserves the equivalence relation. Because G and G∗ are locally isomorphic, the infinitesimal exponents ’s are exactly the same for G and for G∗ . Since to every there does exist exactly one ξ on G∗ , so, if to a given there exists the corresponding ξ on the whole G, then such a ξ is unique. In this way, we have obtained the full classification of ξ defined on a whole Lie group G for any Lie group G, in the sense that no ξ can be omitted in the classification. The set of equivalence classes of ξ is considerably smaller than that for ; it may happen that to some local ξ there does not exist any global extension.
6. Examples Example 1: The Galilean Group. According to the conclusions of Sect. 2 one should a priori investigate such representations of the Galilean group G which fulfill Eq. (2), with ξ depending on time. Then, the following paradox arises. Why has the transformation law Tr under the Galilean group a time-independent ξ in (2), regardless of whether it is a covariance group or a symmetry group? We will solve the paradox in this subsection. Namely, we will show that any representation of the Galilean group fulfilling (2) is equivalent to a representation fulfilling (2) with time-independent ξ . This is a rather 8 In this proof we consider the finite-dimensional extension H of G instead of the Lie group H in the proof presented in [1]. The remaining replacements are rather trivial, but we mark them here explicitly to simplify the reading. (1) Instead of the formula r¯ = t¯(θ )¯r = r¯ t¯(θ ) of (5.3) in [1], we have rˇ = tˇ(θ(r −1 p))ˇr = rˇ tˇ(θ (p)). Thus, from the formula (hˇ 1 (r)hˇ 1 (s))hˇ 1 (g) = hˇ 1 (r)(hˇ 1 (s)hˇ 1 (g)) (see [1]) it follows that ξ(r, s, p)+ξ(rs, g, p) = ξ(s, g, r −1 p)+ξ(r, sg, p) instead of (5.8) in [1]. (2) Instead of (4.9), (4.10) and (4.11) we use the Iwasawa-type construction presented in this paper.
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peculiar property of the Galilean group, not valid in general. For example, this is not true for the group of Milne transformations. In nonrelativistic theory ξ = ξ(r, s, t) depends on the time. In this case, according to our assumption about G, any r ∈ G transforms simultaneous hyperplanes into simultaneous hyperplanes. Thus, there are two possibilities for any r ∈ G. First, when r does not change time: t (rp) = t (p), and the second in which time is changed, but in such a way that t (rp) − t (p) = f (t). We assume in addition that the base generators ak ∈ G can be chosen in such a way that only one acts on time as translation and the remaining ones do not act on time. We can assume that the operators a are ordinary differential operators. Hence, the Jacobi identity (24) reads ([a, a ], a ) + ([a , a ], a) + ([a , a], a ) = ∂t (a , a ),
(27)
if one and only one among a, a , a is the time-translation generator, namely a, and ([a, a ], a ) + ([a , a ], a) + ([a , a], a ) = 0,
(28)
in all of the remaining cases. The Jacobi identities (27) and (28) can be treated as a system of ordinary differential linear equations for the finite set of unknown functions ij (t) = (ai , aj , t), where ai is the base in the Lie algebra of G. According to Sect. 3, in order to classify all ξ of G we shall determine all equivalence classes of infinitesimal exponents of the Lie algebra G of G. The commutation relations for the Galilean group are as follows [aij , akl ] = δj k ail − δik aj l + δil aj k − δj l aik ,
(29)
[aij , bk ] = δj k bi − δik bj , [bi , bj ] = 0,
(30)
[aij.dk ] = δj k di − δik dj , [di , dj ] = 0, [bi , dj ] = 0,
(31)
[aij , τ ] = 0, [bk , τ ] = 0, [dk , τ ] = bk ,
(32)
where bi , di and τ stand for the generators of space translations, with the proper Galilean transformations and time translation respectively and aij = −aj i being rotation generators. Please note that the Jacobi identity (28) is identical to that in the ordinary Bargmann theory of time-independent exponents (see [1], Eqs (4.24) and (4.24a)). Thus, using (29) – (31) we can proceed exactly after Bargmann (see [1], pp. 39, 40) and show that any infinitesimal exponent defined on the subgroup generated by bi , di , aij is equivalent to an exponent equal to zero, with the possible exception of (bi , dk , t) = γ δik , where γ = γ (t). Hence, the only components of defined on the whole algebra G which can be a priori not equal to zero are: (bi , dk , t) = γ δik , (aij , τ, t), (bi , τ, t) and (dk , τ, t). First, we compute the function γ (t). Substituting a = τ, a = bi , a = dk to (27), we get dγ /dt = 0, so that γ is a constant, denoting the constant value of γ by m. By inserting a = τ, a = ais , a = asj to (27) and summing up with respect to s, we get (aij , τ, t) = 0. In the same way, but with the substitution a = τ, a = ais , a = bs , one shows that (bi , τ, t) = 0. At last, the substitution a = τ, a = ais , a = ds to (27) and summation with respect to s gives (di , τ, t) = 0. In this way, we have proved that any time-dependent on G is equivalent to a time-independent one. In other words, we get a one-parameter family of possible , with the parameter equal to the inertial mass m of the system in question. Any infinitesimal time-dependent exponent of the
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Galilean group is equivalent to the above time-independent exponent with some value of the parameter m; and any two infinitesimal exponents with different values of m are nonequivalent. As was argued in Sect. 3 (Theorems 3 ÷ 5), the classification of gives a full classification of ξ . Moreover, it can be shown that the classification of ξ is equivalent to the classification of possible θ -s in the transformation law Tr ψ(p) = eiθ(r,p) ψ(r −1 p)
(33)
for the spinless nonrelativistic particle. On the other hand, the exponent ξ(r, s, t) of the representation Tr given by (33) can be easily computed to be equal to θ(rs, p) − θ(r, p) − θ (s, r −1 p), and the infinitesimal exponent belonging to θ defined as θ(r, p) = −m v x + m2 v2 t, covers the whole one-parameter family of the classification (its infinitesimal exponent is equal to that infinitesimal exponent , which has been found above). Thus, the standard θ (r, p) = −m v x + m2 v2 t, covers the full classification of possible θ -s in (33) for the Galilean group. Inserting the standard form for θ we see that ξ does not depend on time but only on r and s. By this, any time-dependent ξ on G is equivalent to its time-independent counterpart. In this way, we have reconstructed the standard result. Using now the formula9 (Tσ ai ) − 1)ψ(p) , σ →0 σ
Ai ψ(p) = lim
for the generator Ai corresponding to ai , we get the standard commutation relations for the ray representation Tr of the Galilean group [Aij , Akl ] = δj k Ail − δik Aj l − δj l Aik , [Aij , Bk ] = δj k Bi − δik Bj , [Bi , Bj ] = 0, [Aij , Dk ] = δj k Di − δik Dj , [Di , Dj ] = 0, [Bi , Dj ] = mδij , [Aij , T ] = 0, [Bk , T ] = 0, [Dk , T ] = Bk . Please note that to any ξ (or ) there exists a corresponding θ (and such a θ is unique up to a trivial equivalence relation). As we will see, this is not the case for the Milne group, where some ’s do exist which cannot be realized by any θ. Example 2: Milne group as a covariance group. In this subsection we apply the theory of Sect. 3 to the Milne transformations group. We proceed like with the Galilean group in the preceding section. The Milne group G does not form any Lie group, but in the physical application it is sufficient for us to consider some Lie subgroups G(m) of the Milne group. We will go on according to the following plan. First, we compute the 9 The transformation T does not act in the ordinary Hilbert space but in the Hilbert bundle space r RH, hence we cannot immediately appeal to the Stone and Gårding Theorems. Nonetheless, Tr induces a unique unitary representation acting in the Hilbert space R Ht dµ(t) and it can be shown that it is meaningful to talk about the generators A of Tr .
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infinitesimal exponents and exponents for each G(m), m = 1, 2, . . . , and then the θ in (33) for G(m). Please compare [16], where the result is extended on the whole group. The Milne transformation is defined as follows: ( x , t) → (R x + A(t), t + b),
(34)
where R is an orthogonal matrix, and b is constant. The extent of arbitrariness of the in (34) will be left undetermined for now. It is convenient to rewrite the function A(t) Milne transformations (34) in the following form: x = R x + A(t) v , t = t + b, where v is a constant vector which does not depend on time t. We define the subgroup G(m) of G as the group of the following transformations t2 tm x = R x + v(0) + t v(1) + v(2) + . . . + v(m) , t = t + b, 2! m! k are the group parameters; in particular, the group G(m) has where R = (Rab ), v(n) dimension equal to 3m + 7. Now let us investigate the group G(m), that is, classify its infinitesimal exponents. The commutation relations of G(m) are as follows
[aij , akl ] = δj k ail − δik aj l + δil aj k − δil aik , (n)
(n)
[aij , dk ] = δj k di
(n)
(n)
(k)
− δik dj , [di , dj ] = 0,
(0)
(n)
(n−1)
[aij , τ ] = 0, [di , τ ] = 0, [di , τ ] = di (n)
where di
,
(35) (36) (37)
i ): is the generator of the transformation r(v(n)
x = xi + i
tn i v , n! (n)
which will be called the n-acceleration, and 0-acceleration in the particular case of the ordinary space translation. All relations (35) and (36) are identical to (29) ÷ (31) with the n-acceleration instead of the Galilean transformation. Thus, the same argu(n) mentation as that used for the Galilean group gives: (aij , akl ) = 0, (aij , dk ) = 0, (n) (n) and (di , dj ) = 0. Substituting aih , ahi , τ for a, a , a into Eq. (27), making use of the commutation relations and summing up with respect to h, we get (aij , τ ) = (l) (n) 0. In an analogous way, substituting aih , dh , dk for a, a , a into Eq. (28), we get (l) (n) (n) (n) (di , dk ) = 13 (d (l)h , dh ) δik . Substituting aih , dh , τ for a, a , a into Eq. (27), making use of the commutation relations, and summing up with respect to h, we get (n) (n) (0) (di , τ ) = 0. Now, we substitute dk , di , τ for a, a , a in (27), and proceed recur(0) (n) rently with respect to n, obtaining in this way (di , dk ) = P (0,n) (t)δik , where (0,n) P (t) is a polynomial of degree n − 1, and the time derivation of P (0,n) (t) has to be (n) (l) equal to P (0,n−1) (t), and P (0,0) (t) = 0. Substituting dk , di , τ into (27), in the same (l) (n) way we get (dk , di ) = P (l,n) (t)δki , where dtd P (l,n) = P (l−1,n) + P (l,n−1) . This
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allows us to determine all P (l,n) by the recurrent integration process. Please note that P (0,0) = 0, and P (l,n) = −P (n,l) , so given the P (0,n) we can compute all P (1,n) . Indeed, we have P (1,0) = −P (0,1) , P (1,1) = 0, dP (1,2) /dt = P (0,2) + P (1,1) , dP (1,3) /dt = P (0,3) +P (1,2) , . . . , and after m−1 integrations we compute all P (1,n) . Each elementary integration introduces a new independent parameter (the arbitrary additive integration constant). Exactly in the same way, given all P (1,n) we can compute all P (2,n) after m−2 elementary integration processes. In general, the P (l−1,n) allows us to compute all P (l,n) after m−l integrations. Thus, P (l,n) (t) are l+n−1-degree polynomial functions of t, and (n) (l) all are determined by m(m + 1)/2 integration constants. Because d[ ](di , dk ) = 0, the exponents defined by different polynomials P (l,n) are inequivalent. Therefore, the space of nonequivalent classes of is m(m + 1)/2-dimensional. However, not all can be realized by the transformation Tr of the form (33). It can seen that any integration constant γ(l,q) of the polynomial P (l,q) (t) has to be equal to zero if l, q = 0, provided the exponent belongs to the representation Tr of the form (33). By this, all exponents of G(m) which can be realized by the transformations Tr of the form (33) are determined by the polynomial P (0,m) , that is, by m constants. We omit the proof of this fact, and refer the reader to [16], Example 2. In the proof we compute the exponent directly for the transformation Tr of the form (33) and compare it with the classification results above. Consider the θ , given by the formula θ (r, p) = γ1
dA d2 A dm A θ (t), + γ2 2 + . . . + γ m m + dt dt dt
(38)
for r ∈ G(m), where γi are the integration constants which define the polynomial t m−1 t (m−2) + γ2 (m−2)! + . . . + γm , and θ (t) is any function of time t, and evenP (0,m) = γ1 (m−1)! tually of the group parameters. A rather simple computation shows that this θ covers all possible which can be realized by (33). That is, the infinitesimal exponents corresponding to the θ given by (38) yield all possible with all integration constants γ(k,n) = 0, for k, n = 0. Thus, the most general θ (r, p) defined for r ∈ G(m) is given by (38). At this point we make use of the assumption that the wave equation is local. It can be shown then (we leave this without proof) that the θ (r, p) can be a function of a finite order of derivatives of A(t), say k th at most, while the higher derivatives cannot enter into θ. Therefore, the most general θ (r, p) defined for r ∈ G(m) has the following form: θ (r, X) = γ1
dA dk A + . . . + γk k + θ (t). dt dt
(39)
Having obtained this we can infer the most general Schrödinger equation for a spinless particle in Newton-Cartan spacetime, cf. [17]. The inertial and the gravitational masses are always equal in this equation. Acknowledgements. The author is indebted to A. Staruszkiewicz and A. Herdegen for helpful discussions. The author also wishes to thank the Referee who suggested the explicit use of the Hilbert bundle formalism. The paper was financially supported by the KBN grant no. 5 P03B 09320.
References 1. Bargmann, V.: Ann. Math. 59, 1 (1954)
A Generalization of the Bargmann’s Theory of Ray Representations
239
2. Birkhoff, G.: Continuous Groups and Linear Spaces. Recueil Mathématique (Moscow) 1(5), 635 (1935); Analytical Groups, Trans. Am. Math. Soc. 43, 61 (1938) 3. Dirac, P.A.M.: Lectures on Quantum Field Theory. New York: Academic Press, 1966 4. Dynkin, E.: Uspekhi Mat. Nauk 5, 135 (1950); Am. Math. Soc. Transl. 9(1), 470 (1950) 5. Giulini, D.: States, Symmetries and Superselection. In: Decoherence: Theoretical, Experimental and Conceptual Problems, Lecture Notes in Physics, Berlin-Heidelberg-New York: Springer Verlag, 2000, p. 87 6. Gupta, S.N.: Proc. Phys. Soc. 63, 681 (1950); Bleuler, K.: Helv. Phys. Acta 23, 567 (1950) 7. Lang, S.: Differential Manifolds. Berlin-Heidelberg-New York: Springer-Verlag, 1985 8. Heisenberg, W.: The Physical Principles of the Quantum Theory. New York: Dover, 1949 9. Łopusza´nski, J.: Forts. der Phys. 26, 261, (1978); Rachunek spinorów, Warszawa: PWN, 1985 (in Polish) 10. Mackey, G.W.: Unitary Group Representations in Physics, Probability, and Number Theory. New York, Amsterdam, Wokingham-UK: Addison-Wesley Publishing Company, Inc., 1989 11. Nachbin, L.: The Haar Integral. Princeton-New Jersey-Toronto-New York-London: D. Van Nostrad Company Inc., 1965 12. Piron, C.: Field Theory Revisited. In: Trends in Quantum mechanics, H.D. Doebnen (ed.), Singapore: World Scientific, 2000, pp. 270–273 13. Piron, C.: A Unified Concept of Evolution in Quantum Mechanics. In: Interpretation and Foundations of Quantum Theory, Holger Neumann (ed.), Mannheim: Bibliographisches Institut, 1981 14. Pontrjagin, L.: Topological groups. Moscow, 1984 (in Russian) 15. Wawrzycki, J.: Int. J. Theor. Phys. 40, 1595 (2001) 16. Wawrzycki, J.: A Generalization of the Bargmann’s Theory of Ray Representations. http://arxiv.org/abs/ 17. Wawrzycki, J.: Acta Phys. Polon. B 35, 613 (2004) 18. Weinberg, S.: The Quantum Theory of Fields. volume II, Cambridge: Univ. Press, 1996 Communicated by G.W. Gibbons
Commun. Math. Phys. 250, 241–257 (2004) Digital Object Identifier (DOI) 10.1007/s00220-004-1143-2
Communications in
Mathematical Physics
Bounds on the Unstable Eigenvalue for the Asymmetric Renormalization Operator for Period Doubling B.D. Mestel1 , A.H. Osbaldestin2 , A.V. Tsygvintsev3 1
Department of Computing Science and Mathematics, University of Stirling, Stirling, FK9 4LA, UK. E-mail: [email protected] 2 Department of Mathematics, University of Portsmouth, Portsmouth, PO1 3HE, UK. E-mail: [email protected] 3 Unite de Math´ematiques Pures et Appliqu´ees, Ecole Normale Sup´erieure de Lyon, 46 All´ee d’Italie, 69364 Lyon Cedex 07, France. E-mail: [email protected] Received: 8 July 2003 / Accepted: 1 February 2004 Published online: 5 August 2004 – © Springer-Verlag 2004
Abstract: We establish rigorous bounds for the unstable eigenvalue of the period-doubling renormalization operator for asymmetric unimodal maps. Herglotz-function techniques and cone invariance ideas are used. Our result generalizes an established result for conventional period doubling. 1. Introduction The remarkable universality of the scalings witnessed in the period-doubling route to chaos now has a well-established, mathematically rigorous basis. Soon after discovery by Feigenbaum [8] and Coullet-Tresser [3], the first (computer-assisted) proof was given by Lanford [11], closely followed by the analytic proofs of Epstein [5] and coworkers. More recently the rigorous analysis has reached new levels of sophistication in the works of Sullivan [17] and McMullen [12]. Contemporaneously, Arneodo et al [2] initiated the investigation of asymmetric unimodal maps. In a recent series of articles [13–15], we have given a rigorous renormalization analysis of period doubling in degree-d asymmetric unimodal maps. These are unimodal maps possessing a degree-d maximum, but with differing left and right d th derivatives. The maps we have in mind take the form fL (x) = 1 − a1 |x|d if x ≤ 0 ; f (x) = (1.1) fR (x) = 1 − a2 |x|d if x ≥ 0 . (The case of differing left- and right-hand degrees appears to be somewhat different in nature. See, e.g., [9].) In brief, for each d > 1, the standard Feigenbaum period-doubling renormalization operator has been shown to possess a family of period-two orbits, parametrized by an invariant asymmetry modulus, µ, measuring the ratio of the left and
This research is supported by The Leverhulme Trust (grant number F/00144/W).
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right d th derivatives at the maximum. The period-two orbit is then given by a quartet of functions (fL , fR , f˜L , f˜R ) satisfying the functional equations f˜L (x) = −λ−1 fR fR (−λx) , f˜R (x) = −λ−1 fR fL (−λx) , ˜ , fL (x) = −λ f˜R f˜R (−λx) −1 ˜ , fR (x) = −λ˜ f˜R f˜L (−λx) ˜ −1
(1.2a) (1.2b) (1.2c) (1.2d)
with the normalizations fL (0) = fR (0) = f˜L (0) = f˜R (0) = 1 so that λ = −fR (1) > 0 and λ˜ = −f˜R (1) > 0. The solutions of (1.2) depend on two parameters, viz., the degree d of the critical point and the modulus µ, which (for the case when d is an even integer) is the ratio (d)
µ=
fL (0−) (d)
(1.3)
.
fR (0+)
The case µ = 1 is the standard Feigenbaum scenario in which case the period-two orbit is in fact a fixed point. Let us denote by R the period-doubling renormalization operator acting on a unimodal map f with f (0) = 1, so that R(f )(x) = −λ−1 f (f (−λx)),
λ = −f (1) .
(1.4)
Then R acts on both symmetric and asymmetric unimodal maps, preserving the degree d and inverting the asymmetry modulus µ. The scaling of the parameters in (1.1) undergoing a period-doubling cascade is determined by the expanding eigenvalue of the derivative of R 2 at the period-2 point f . This derivative dR 2 (f ) is compact on a suitable Banach space of tangent functions δf and numerical results suggest that it is hyperbolic with a single expanding eigenvalue δ 2 . It is this expanding eigenvalue which we investigate in this paper. More precisely, we study an associated operator T (defined below), which has a positive expanding eigenvalue δ. We give a brief description of the relationship between T and dR 2 (f ) in Sect. 4. The operator T is defined on a pair of functions (v, v) ˜ and is given by: −1 ˜ t˜x))L˜ (t˜x)−1 ) v(x) ˜ t˜x) + v( ˜ L( t˜ (v( , (1.5) T = −1 v(x) ˜ t (v(tx) + v(L(tx))L (tx)−1 ) ˜ where t = µλd , t˜ = µ−1 λ˜ d , and L(x) = F (x)d , L(x) = F˜ (x)d . In this article we analyze the positive unstable eigenvalue of T , and, in particular, we shall establish the following theorem. Our work mirrors closely the analysis of Eckmann and Epstein [4] on the expanding eigenvalue of the symmetric Feigenbaum fixed-point. We shall establish the following result: Theorem 1. There exists a Banach space of function pairs on which the operator T is well defined, compact and has an eigenvalue δ > 0 satisfying 1<
1 ˜ (d−1)/2 (1 + (λλ)
˜ λλ)
<δ<
1 . ˜ d/2 (λλ)
(1.6)
Bounds on the Unstable Eigenvalue for the Asymmetric Renormalization Operator
243
Several remarks are appropriate for this theorem. Firstly, the theorem establishes the existence of an expanding eigenvalue but does not prove the hyperbolicity of the operator dRf2 . Secondly, the lower bound for δ, whilst greater than 1, is suboptimal and, indeed, is worse than the bound 1/λd − 1/λ obtained in [4] for the symmetric period-doubling case. Unfortunately, some of the estimates in that paper do not readily generalize to the asymmetric case and our results are accordingly weaker, although they do apply to all degree d and modulus µ. 2. Notation and Background Material In this section we establish our notation and give a brief summary of previous results from [13–15] that we shall use in this paper. The Herglotz function approach [5] has been an extremely fruitful technique in the analysis of the accumulation of period-doublings. It was used in [14] to prove the existence of a solution of Eqs. (1.2) for all real µ > 0 and d > 1. We recall here how Eqs. (1.2) may be recast as an anti-Herglotz function problem. Firstly we build the singularity into our functions by defining fR (x) = FR (|x|d ) ,
f˜R (x) = F˜R (|x|d ) .
(2.1)
The left-hand functions are given in terms of the right-hand ones by fL (x) = FR (µ|x|d ) ,
f˜L (x) = F˜R (µ−1 |x|d ) .
(2.2)
We then consider the inverses of these functions by defining FR (x) = U −1 (x) ,
F˜R (x) = U˜ −1 (x) .
(2.3)
The functions U and U˜ satisfy the conditions U (1) = 0, U (−λ) = 1, U˜ (1) = 0, ˜ = 1. We may further normalize by setting U (x) = kψ(x), U˜ (x) = k˜ ψ(x), ˜ U˜ (−λ) where k = U (0), k˜ = U˜ (0), so that the functions ψ and ψ˜ satisfy ψ(1) = 0, ψ(0) = 1, ˜ ˜ ˜ ψ(1) = 0, ψ(0) = 1. We then have U (x) = z1 d ψ(x), U˜ (x) = z˜ 1d ψ(x), where z1 = −1/d −1/d ˜ ˜ ψ(−λ) , z˜ 1 = ψ(−λ) . In this new setting our equations become ˜ φ(x)) ˜ ψ(x) = τ˜ −1 ψ( ,
˜ ψ(x) = τ −1 ψ(φ(x)) ,
(2.4)
˜ λx) ˜ 1/d = U˜ (−λx) ˜ ˜ 1/d , and where φ(x) = z1 ψ(−λx)1/d = U (−λx)1/d , φ(x) = z˜ 1 ψ(− ˜ z1 ) satisfy τ = ψ(z1 ), τ˜ = ψ(˜ λd =
τ z1d µ˜z1d
,
λ˜ d =
µτ˜ z˜ 1d z1d
.
(2.5)
˜ d . In terms of the functions U , U˜ , Eqs. (2.4) become Note that τ τ˜ = (λλ) U (x) = µλ˜ −d U˜ (U˜ (−λ˜ x)1/d ) ,
U˜ (x) = µ−1 λ−d U (U (−λx)1/d ) .
(2.6)
The method of the existence proof is now to show that (2.4) has a solution in a space of anti-Herglotz functions.
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Let C+ , C− denote the upper and lower half planes in C. Recall that a complex ana¯+ lytic function on C+ ∪ C− is said to be Herglotz (resp. anti-Herglotz) if f (C+ ) ⊂ C ¯ ¯ ¯ and f (C− ) ⊂ C− (resp. f (C+ ) ⊂ C− and f (C− ) ⊂ C+ ). For A < B ∈ R, we let (A, B) denote C+ ∪ C− ∪ (A, B). We denote by H(A, B) and AH(A, B) (respectively) the space of Herglotz and anti-Herglotz functions (respectively) analytic on the interval (A, B). Furthermore, if [0, 1] ⊂ (A, B), let E(A, B) denote the space of anti-Herglotz functions ψ ∈ AH(A, B) which satisfy the normalizations ψ(0) = 1, ψ(1) = 0. As is normal, we equip H(A, B), AH(A, B) and E(A, B) with the topology of uniform convergence on compact subsets of (A, B). In [14] we prove the following existence theorem: ˜ Theorem. For each µ > 0 and for each d > 1, there exists a solution pair (ψ, ψ) −1 −1 −1 −1 ˜ ˜ ˜ ˜ for (2.4) with ψ ∈ E(−λ , (λλ) ) and ψ ∈ E(−λ , (λλ) ). From this it is straightforward to reverse the transformation above to show that (1.2) has a solution. See [14]. One crucial feature of the Herglotz and anti-Herglotz functions is that they satisfy ˜ these bounds the so-called a priori bounds. (See [5, 6, 14].) For the solution pair (ψ, ψ) are, for x < 0 and x > 1: 1−x ˜ 1 − λλx
≤ ψ(x) ≤
1−x ˜ 1 + λx
,
and for 0 < x < 1: 1−x 1−x ≤ ψ(x) ≤ , ˜ ˜ 1 + λx 1 − λλx
1−x
1−x ; 1 + λx
(2.7)
1−x 1−x ˜ . ≤ ψ(x) ≤ ˜ 1 + λx 1 − λλx
(2.8)
˜ 1 − λλx
˜ ≤ ψ(x) ≤
In addition, as in [5], it is straightforward to derive a priori bounds on the first and second derivatives: −2λ˜ ψ (x) U (x) 2λλ˜ ≤ = ≤ , ˜ ˜ ψ (x) U (x) (1 + λx) (1 − λλx) −2λ ψ˜ (x) 2λλ˜ U˜ (x) ≤ ≤ = , ˜ (1 + λx) ψ˜ (x) (1 − λλx) U˜ (x)
˜ −1 ) , x ∈ (−λ˜ −1 , (λλ)
(2.9a)
˜ −1 ) . x ∈ (−λ−1 , (λλ)
(2.9b)
Let us define t = µλd , t˜ = µ−1 λ˜ d . Then we have the following properties which are a consequence of the definitions and the results of [14]. 1. t, t˜, z1 , z˜ 1 , τ , τ˜ ∈ (0, 1) ; 2. t < z1d , and t˜ < z˜ 1d . ˜ z1 x 1/d ). Following [5], we define the functions V (x) = τ −1 ψ(z1 x 1/d ), V˜ (x) = τ˜ −1 ψ(˜ −d ˜ λ). ˜ Then V ∈ AH(0, α(λλ) ˜ Let us further define α = ψ(−λ), α˜ = ψ(− ) and V˜ ∈ ˜ −d ) and, in view of Eqs. (2.4), we have AH(0, α(λ ˜ λ) ˜ λx)) ˜ ψ(x) = V˜ (ψ(− ,
˜ ψ(x) = V (ψ(−λx) .
(2.10)
Note that V (1) = V˜ (1) = 1 and V (α) = V˜ (α) ˜ = 0. Differentiating (2.10), and evaluating at 0, gives V (1) =
−ψ˜ (0) , λψ (0)
−ψ (0) V˜ (1) = , λ˜ ψ˜ (0)
1 V (1)V˜ (1) = . λλ˜
(2.11)
Bounds on the Unstable Eigenvalue for the Asymmetric Renormalization Operator
245
Lemma 1. The functions U (x) and U˜ (x) are injective respectively in domains = −1 , (λλ) ˜ −1 ) and ˜ = (−λ ˜ ˜ −1 ). (−λ˜ −1 , (λλ) Proof. Our proof is based on the ideas of Epstein given in [6] and [7]. It is sufficient to show the injectivity of U , for the injectivity of U˜ then easily follows from Eqs. (2.6). From the same equations we have ˜ −d U ( 3 (x)), U (x) = (λλ)
x ∈ ,
(2.12)
˜ where 3 (x) = u(−µ−1/d u(u(λλx))) with u(x) = U (x)1/d . To elucidate the proof we −1/d ˜ ˜ ˜ u(u(λλx)), 1 (x) = u(λλx), 0 (x) = λλx. further define 2 (x) = −µ The functions 0 , 3 are Herglotz in and 1 , 2 are anti-Herglotz in the same domain. Indeed, it is clear that they are Herglotz and anti-Herglotz respectively on C+ ∪ C− and any interval on which they are well-defined. Indeed they are Herglotz ˜ −1 ) as follows from the inclusions and anti-Herglotz respectively on J0 = (−λ˜ −1 , (λλ) −1 ˜ proved below. Let s ∈ (1, (λλ) ) be arbitrarily chosen and let J = (−λ˜ −1 , s). Our first aim is to show that i ((J )) (J ) for 0 ≤ i ≤ 3, where we adopt the notation that A B means that A is strictly inside of B, i.e., A ⊂ B. Taking into account the Herglotz (resp. anti-Herglotz) character of the functions i it is sufficient to prove the inclusions i (J ) J . We consider each function in turn. (a) 0 (J ) J is seen from the inequality 0 < λλ˜ < 1. ˜ ∈ (z1 ψ((λλ)(λ ˜ λ) ˜ −1 )1/d , z1 ψ((λλ)(− ˜ ˜ −1 )1/d ) = (b) 1 (J ) J . We have u(λλx) λ) 1/d (0, z1 ψ(−λ) ) = (0, 1) for all x ∈ J . The result follows from this and the fact that s > 1. (c) 2 (J ) J . We have −µ−1/d < 2 (x) < 0 for all x ∈ J ⊂ J0 since 0 < ˜ u(u(λλx)) < 1 for arbitrary x ∈ J0 . The inequality µ−1/d λ˜ < 1 implies −µ−1/d > −1 −λ˜ and we obtain 2 (J ) = ( 2 (s), 2 (−λ˜ −1 )) (−λ˜ −1 , 0) as required. (d) 3 (J ) J . We have 3 (J ) = ( 3 (−λ˜ −1 ), 3 (s)), 0 < 3 (−λ˜ −1 ) < 3 (s). But 3 (s) < s. Indeed, by the Schwarz Lemma, x = 1 is the unique attracting fixed point of 3 in the interval J0 . Hence, 3 (1) = 1, 3 (1) < 1 and the graph of ˜ −1 ). y = 3 (x) lies below the line y = x for all x ∈ (1, (λλ) Following [6] we define z−a D(a, b, θ) = z ∈ C : 0 < arg <θ , b−z
(2.13)
which is the domain between the real segment (a, b), a < b and the circular arc in C+ b, θ ) will denote with ends a and b whose tangent at a has argument θ ∈ (0, π ). D(a, the domain between the same arc and its reflection in the real axis. We also use the b, θ )), where c denotes ˆ θ, r)) for D(a, b, θ) (resp. D(a, notation δ(c, θ, r) (resp. δ(c, the mid-point of (a, b) and r = (b − a)/2 denotes its radius. Let C and R be the corresponding mid-point and radius of the interval J . Then, according to the conditions i (J ) J , 0 ≤ i ≤ 3 and Lemma 4 from [6], there exists k ∈ (0, 1) such that each
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B.D. Mestel, A.H. Osbaldestin, A.V. Tsygvintsev
ˆ ˆ i maps δ(C, θ, R) into δ(C, θ, kR) for an arbitrary θ ∈ (0, π ). In addition, as follows ˆ ˆ from Lemma 5 of [6], each i maps δ(C, θ, R) into δ(C, θ , R), with tan
θ θ = k tan . 2 2
(2.14)
Suppose that z0 and w0 are distinct points in (J ) such that U (z0 ) = U (w0 ). There ˆ θ0 , R). For n = 1, 2, ... we exists a θ0 ∈ (0, π ) such that z0 and w0 are both in δ(C, ˆ define inductively a pair (zn , wn ) of distinct points, both lying in δ(C, θ0 , kR) such that U (zn ) = U (wn ). This we do as follows. Assume (zn , wn ), n ≥ 0 have been constructed. Then we set ( 3 (zn ), 3 (wn )) if 3 (zn ) = 3 (wn ) 3 (zn ) = 3 (wn ) ( 2 (zn ), 2 (wn )) if (z ) = (w ) 2 n 2 n (z ) = 1 (wn ) 1 n (zn+1 , wn+1 ) = ( 1 (zn ), 1 (wn )) if 2 (zn ) = 2 (wn ) 3 (zn ) = 3 (wn ) 1 (zn ) = 1 (wn ) ( 0 (zn ), 0 (wn )) if 2 (zn ) = 2 (wn ) 3 (zn ) = 3 (wn ) ˆ θn , R), with According to (2.14) applied to i , 0 ≤ i ≤ 3, zn and wn are both in δ(C, tan
θn θ0 = k n tan . 2 2
(2.15)
ˆ Moreover, for n > 1 one has (zn , wn ) ∈ δ(C, θ0 , kR). We note that the restriction of ˆ U to the real interval I = δ(C, θ0 , kR) ∩ R is injective. This implies that there exists a small complex neighborhood UI ⊂ C of I , where U will be also injective. As n → ∞ we have θn → 0 and both points zn and wn will eventually enter UI for n sufficiently large. This contradiction proves that U is injective in . 3. Lower Bound for d In this section we shall prove the inequality d > (1 + λλ˜ )/(1 − λλ˜ ) which is important ˜ V and V˜ . The inequality is analogous to in the proof of convexity of the functions L, L, Epstein’s result for the symmetric Feigenbaum function, viz., d > (1 + λ2 )/(1 − λ2 ). However, our proof differs somewhat from that in [5]. Lemma 2. ˜ ˜ . d > (1 + λλ)/(1 − λλ)
(3.1)
Bounds on the Unstable Eigenvalue for the Asymmetric Renormalization Operator
247
Proof. We start from the estimates contained in the paper [15] 1−x , ˜ ˜ ˜ (τ˜ + λτ˜ − λ)(1 − x) + (1 − τ˜ )(1 + λ) 1−x ˜ ψ(x) ≤ , (τ + λτ − λ)(1 − x) + (1 − τ )(1 + λ)
ψ(x) ≤
x ∈ (−λ˜ −1 , 0) ,
(3.2a)
x ∈ (−λ−1 , 0) .
(3.2b)
The first inequality gives ψ(−λ) ≤ so that
z1 = ψ(−λ)
−1/d
≥
1+λ , ˜ (λ + λλ)τ˜ + 1 − λλ˜
˜ τ˜ + 1 − λλ˜ (λ + λλ) 1+λ
(3.3)
1/d
= (ρ τ˜ + (1 − ρ))1/d ≥ ρ τ˜ 1/d + 1 − ρ = 1 − ρ(1 − τ˜ 1/d ) ,
(3.4)
˜ where we have set ρ = (λ + λλ)/(1 + λ) ∈ (0, 1) and have used the property (ρx + ˜ − τ 1/d ), where (1 − ρ)y)1/d ≥ ρx 1/d + (1 − ρ)y 1/d . Similarly, we have z˜ 1 ≥ 1 − ρ(1 ˜ ˜ ∈ (0, 1).