L2 dimensions of spaces of braid-invariant harmonic forms
aa r X i v : . [ m a t h . F A ] O c t L dimensions of spaces of braid-invariantharmonic forms Alexei Daletskii a, ∗ , Alexander Kalyuzhnyi b a Department of Mathematics, University of York, UK b Institute of Mathematics, NAS, Kiev, Ukraine
Abstract
Let X be a Riemannian manifold endowed with a co-compact isometric action of an in-finite discrete group. We consider L spaces of harmonic vector-valued forms on theproduct manifold X N , which are invariant with respect to an action of the braid group B N ,and compute their von Neumann dimensions ( the braided L - Betti numbers ). Keywords:
Braid group, von Neumann algebra, L -Betti number, Configuration space Primary 58A10, 20F36, 46L10; Secondary 58A12, 58J22
Subject classification:
Global Analysis, Analysis on Manifolds
1. Introduction
In recent years, there has been a growing interest in the analysis and geometry on vari-ous spaces of configurations (i.e. finite or infinite countable subsets) in a given topologicalspace X , motivated by widespread applications in various parts of mathematics, includingtopology, representation theory, statistical mechanics, quantum field theory, mathematicalbiology, etc. (see [19], [33], [21], [20], [26], [9] and an extensive bibliography in [18], [8]and [31]). The development of the general mathematical framework for suitable classes ofconfigurations is an important and challenging research theme, even in the case of finitenumber of components (particles) in each configuration. Many properties of the spaces ofinfinite configurations are in turn studied via the limit transition in the number of particles.Observe that functions on the space of N -particle configurations in X can be identifiedwith the functions on the Cartesian product X N of N copies of X , which are invariant with ∗ Corresponding author.
Email addresses: [email protected] (Alexei Daletskii ), [email protected] (Alexander Kalyuzhnyi)
Preprint submitted to Elsevier October 12, 2018 espect to the natural action of the symmetric group S N by permutations of arguments.Let us also note that certain spaces of mappings on X N (e.g. vector-valued functions anddifferential forms in the case where X possesses a manifold structure) may admit morecomplicated groups of symmetries. An important example of an extended symmetry groupis given by the braid group B N , and the spaces of B N -invariant differential forms on X N can be regarded as spaces differential forms on ” braided configuration spaces ”, which arerelated to the study of multi-particle systems obeying anyon and plekton statistics ([22],[16], [23]).An important and intriguing question in this framework is the relationship between theproperties associated with the multi-particle structure of these spaces, and the topologyand geometry of the underlying space X . In many interesting cases, X is a non-compactRiemannian manifold of infinite volume, which makes it natural to discuss the above ques-tions in the framework of the theory of L invariants. This programme has been started in[15], [13], [14], [1].Initiated by M. Atiyah, the theory of L invariants of non-compact manifolds pos-sessing infinite discrete groups of isometries serves as a brilliant example of applicationof von Neumann algebras in topology, geometry and mathematical physics. The initialframework introduced in [11] is as follows. Let X be a non-compact Riemannian mani-fold admitting an infinite discrete group G of isometries such that the quotient K = X/G is compact. Then G acts by isometries on the spaces L Ω m ( X ) of square-integrable m -forms over X . The projection P ( m ) onto the space H m ( X ) of square-integrable harmonic m -forms (the m -th L -cohomology space) commutes with the action of G and thus be-longs to the commutant M m ( X ) of this action, which is (under certain conditions) a II ∞ -factor. The corresponding von Neumann trace Tr M m P ( m ) =: β m ( X ) gives a regularizeddimension of the space H m ( X ) and is called the L -Betti number of X (or K ). Sincethe pioneering work [11], L -Betti numbers and other L invariants have been studied bymany authors (see e.g. [30] and references given there).The construction above covers in an obvious way the case of the product manifold X N endowed with the natural action of the product group G N . The commutant M m ( X N ) ofthe corresponding action in the space L Ω m ( X N ) contains the von Neumann subalgebra L k + ··· + k N = m M k ( X ) ⊗ ... ⊗ M k N ( X ) , and the K¨unneth formula shows that β m ( X N ) = X k + ··· + k N = m β k ( X ) . . . β k N ( X ) . The situation becomes much more complicated if we consider the spaces of formson X N admitting additional symmetries, for instance, invariant with respect to the ac-tion of the symmetric group S N by permutations of arguments. In this case, the pro-jection P ( m ) N onto the space of S N -invariant harmonic forms does not in general belong to2 m ( X N ) . The minimal extension n M m ( X N ) , P ( m ) N o ′′ has been studied in [15], [13], [14],[1], where the corresponding trace formulae were derived and applied to the computationof L -Betti numbers of finite and infinite configuration spaces.In the present paper, we consider the space L U Ω m ( X N , A N ) of square-integrable m -forms on X N , which take values in a finite dimensional Abelian algebra A N and are invari-ant with respect to a special representation U of the braid group B N . The representation U is constructed as the tensor product of a given unitary representation π of B N in A N andthe natural action S of B N in L Ω m ( X N ) by permutations.Let P ( m ) N be the projection onto the space H mU ( X N , A N ) of harmonic forms in L U Ω m ( X N , A N ) . The construction of the (minimal) von Neumann algebra containing P ( m ) N and computation of the trace of P ( m ) N involves the study of the von Neumann algebra (cid:8) M m ( X N ) ⊗ End( A N ) , U (cid:9) ′′ . (1)We give a description of these von Neumann algebras and study the space H mU ( X N , A N ) in the situation where A N = ( C ) ⊗ N and the representation π is generated by a × matrix C satisfying the braid equation ( C ⊗ ⊗ C )( C ⊗
1) = (1 ⊗ C )( C ⊗ ⊗ C ) . Representations of this type have been extensively studied, see [28].The structure of the paper is as follows. In Sections 2 and 3, we derive general for-mulae for the traces on von Neumann algebras of the type (1). In Sections 4 and 5, wecompute the L dimensions of the spaces H mU ( X N , A N ) . In Section 6, we define the space L N ∈ N H mU ( X N , A N ) of ” harmonic m -forms on the braided configuration space ” andcompute the supertrace of the corresponding projection (via the limit transition N → ∞ )in terms of the Euler characteristic of the underlying manifold X and certain invariant ofthe representation U .In what follows we refer to [12], [32], [27] for general notions of the theory of vonNeumann algebras. We denote by Tr K the faithful normal semifinite trace on a II ∞ - factor K .
2. Trace formula
We start with the review of some well-known facts from the theory of braid groupsthat we will need in what follows, see e.g. [10], [29]. The braid group B N is generated byelements b , . . . , b N − satisfing the commutation relations b i b j = b j b i , | i − j | ≥ ,b i b j b i = b j b i b j , | i − j | = 1 . S N is a quotient group of B N with respect to the relations b i = 1 , i = 1 , . . . N − . We denote by φ : B N → S N the corresponding canonical homomor-phism.Let us consider the pure braid group B pureN := Ker φ (or the group of color braids ).Define the forgetful homomorphism f N − : B pureN → B pureN − which acts on a pure N -braidby forgetting its last string. Observe that the kernel of f N − is isomorphic to the free group F N − . Thus one has the split exact sequence of groups → F N − → B pureN → B pureN − → . (2)Thus the pure braid group can be decomposed into the semidirect product B pureN = F N − ⋊ B pureN − of the free group F N − and the group B pureN − of pure N − braids. Iterating thedecomposition procedure one can represent B pureN as the iterated semidirect product offree groups F ≃ Z , F , . . . , F N − (the Artin normal form of B pureN ). The generators x i,j , ≤ i ≤ j , of the free group F j , j ≤ N − , can be represented in the form x i,j = b j b j − . . . b i +1 b i b − i +1 . . . b − j − b − j . Therefore the elements x i,j , ≤ i ≤ j ≤ N − , generate B pureN .In what follows, we will use the following special representation of the group B N .Let S be a unitary representation of the symmetric group S N in a separable Hilbert space H and π be a unitary representation of B N in a finite dimensional Hilbert space V . Thenthese representations generates a unitary representation U of the group B N in Hilbert space H ⊗ V by the formula U ( b i )( ξ ⊗ v ) = S ( φ ( b i )) ξ ⊗ π ( b i ) v, (3)where ξ ∈ H, v ∈ V . A direct computation shows that U ( b i ) U ( b i +1 ) U ( b i ) = U ( b i +1 ) U ( b i ) U ( b i +1 ) , which implies that formula (3) indeed defines a unitary representation of the group B N .Let us introduce the subspace ( H ⊗ V ) U = { f ∈ H ⊗ V | U ( b i ) f = f, i = 1 , . . . , N − } (4)of U -invariant elements of H ⊗ V and consider the corresponding orthogonal projection P U : H ⊗ V → ( H ⊗ V ) U . Let M be a II ∞ factor acting in H and define Q = M ⊗ End( V ) . In general, theprojection P U does not in general belong to Q . In what follows, we will extend Q to asuitable II ∞ factor containing P U . We suppose that the following condition holds.4 ondition 2.1. The operators S ( σ ) , σ ∈ S N , σ = e , do not belong to M . Condition 2.1 will be verified in the particular situations considered in the followingsections. It implies that the formula α σ ( x ) = S ( σ ) x S ∗ ( σ ) , x ∈ M (5)defines a nontrivial outer action S N ∋ σ α σ ∈ Aut( M ) of the group S N by automorphisms of the II ∞ -factor M . Consider the correspondingcross-product M ⋊ α S N . It is well known that an automorphism of a factor is free iff it isouter. Therefore (see e.g. [27], Proposition 1.4.4) M ′ ∩ ( M ⋊ α S N ) = C so that M ⋊ α S N is a ( II ∞ -) factor. Lemma 2.2.
The map i : M ⋊ α S N ∋ ( x, σ ) xU ( σ ) ∈ { M, { S ( σ ) } σ ∈ S N } ′′ (6) is an isomorphism of factors.Proof. Observe that the map (6) is a non-trivial surjective normal homomorphism. Itis well known that factors do not contain proper weakly closed two-sided ideals, whichimmediately implies that any homomorphisms between them is either identically zero orinjective, and the result follows.Let us define the space V π = { v ∈ V | π ( β ) v = v for all β ∈ B pureN } of π ( B pureN ) -invariant elements of V . Lemma 2.3. (i) The space V π is invariant under the representation π of the braidgroup B N .(ii) The restriction of the representation π of B N to the space V π defines a representa-tion ˜ π of the symmetric group S N via the formula ˜ π ( ϕ ( b i )) = π ( b i ) ↾ V π , i = 1 , . . . , N − . roof. (i). It is sufficient to show that the subspace V π is invariant under the action ofoperators π ( b i ) , i = 1 . . . , N − . Let us suppose that V π is not invariant w.r.t. π ( b N − ) .Then there exists an element w ∈ π ( b N − ) V π orthogonal to V π . Let w = π ( b N − ) v , where v ∈ V π . The equality π ( b N − ) v = v implies that π ( b N − ) w = v and π ( b N − ) w = w .Then v = π ( b N − ) π ( b N − ) π ( b − N − ) v = π ( b N − ) π ( b N − ) w, so that we have π ( b N − ) w = w . The relations b N − b k = b k b N − , k ≤ N − , and π ( x i,j ) v = v imply that π ( x i,k ) w = π ( x i,k ) π ( b N − ) v = π ( b N − ) π ( x i,k ) v = π ( b N − ) v = w for all ≤ i ≤ k ≤ N − . We have also π ( x i,N − ) w = π ( b N − ) π ( x i,N − ) π ( b − N − ) w = π ( b N − ) π ( x i,N − ) v = π ( b N − ) v = w because π ( b − N − ) w = v then for all i = 1 , . . . , N − . Finally, let us show that the vector w is invariant with respect to π ( x i,N − ) , i = 1 , . . . , N − . The equality π ( b N − ) π ( x i,N − ) π ( b − N − ) v = v implies that π ( b N − ) π ( x i,N − ) w = v , and π ( x i,N − ) w = π ( b − N − ) v = w. We see that w is invariant with respect to all operators π ( x i,j ) , ≤ i ≤ j ≤ N − .Elements x i,j generate the pure braid group, which implies that w ∈ V π . Therefore thespace V π is π ( b N − ) -invariant under the operatorBy similar arguments one can show that V π is invariant w.r.t. the action of operators π ( b k ) , k = 1 , . . . N − . (ii) follows from the equality π ( b i ) ↾ V π = id V π and (i).Consider the projection p π : V → V π . We have the following result.
Theorem 2.4.
The projection P U : H ⊗ V ( H ⊗ V ) U can be represented in the form P U = 1 ⊗ p π N ! X σ ∈ S N S ( σ ) ⊗ ˜ π ( σ ) , (7) where ˜ π is the unitary representation of the group S N constructed in Lemma 2.3. roof. Let us denote by A the right-hand side of (7). I t follows from Lemma 2.3 that therelation π ( b i ) p π = p π π ( b i ) holds for any i = 1 , . . . , N − . Therefore A ∈ End ( H ⊗ V ) is a projection.It is sufficient to show that images of P and A coincide. It is obvious that if ξ ∈ ( H ⊗ V ) U then (1 ⊗ p π ) ξ = ξ and S σ ⊗ ˜ π ( σ ) ξ = ξ . Therefore the space ( H ⊗ V ) U = P ( H ⊗ V ) is contained in the image of A . Let us prove the converse inclusion. Let anelement ξ ∈ H ⊗ V belong to the image of A . Then ξ ∈ H ⊗ V π and S ( σ ) ⊗ ˜ π ( σ ) ξ = ξ .Since π ( b i ) ↾ V π = ˜ π ( ϕ ( b i )) we have U ( b i ) ξ = S ( ϕ ( b i )) ⊗ π ( b i ) ξ = S ϕ ( b i ) ⊗ ˜ π ( ϕ ( b i )) ξ = ξ, i. e. ξ is U -invariant. Thus the image of A is contained in the space ( H ⊗ V ) U = P U ( H ⊗ V ) . Corollary 2.5.
We have the inclusion ( H ⊗ V ) U ⊂ H ⊗ V π . Let us introduce the von Neumann algebra Q U := { Q, { U ( β ) } β ∈ B N } ′′ generated by Q and the operators U ( β ) , β ∈ B N . It is obvious that P U ∈ Q U . Lemma 2.6.
Von Neumann algebras ( M ⋊ S N ) ⊗ End( V ) and Q U are isomorphic.Proof. Recall that operators S ( φ ( β )) , β ∈ B N , β = e , do not belong to M , which impliesthat operators U ( β ) = S ( φ ( β )) ⊗ π ( β ) , β ∈ B N , β / ∈ B pureN , do not belong to Q . Therefore the representation U generates a nontrivial outer action α ′ ofthe group S N on the II ∞ factor Q . Applying the arguments similar to the proof of Lemma2.2 one can see that there exists an isomorphism ( Q ⋊ α ′ S N ) ≃ { Q, { U ( β ) } β ∈ B N } ′′ . (8)On the other hand, for any β ∈ B N we have the inclusion π ( β ) ∈ End( V ) , which impliesthat the factors Q ⋊ α ′ S N and ( M ⋊ α S N ) ⊗ End( V ) are isomorphic, and the result follows. Corollary 2.7. Q U is a II ∞ -factor.Proof. Follows from the fact that ( M ⋊ S N ) ⊗ End( V ) is a II ∞ -factor. Corollary 2.8.
It follows from (2.6) that Tr Q U = Tr M ⋊ S N × tr End( V ) . Without loss of generality we assume that the trace on
End( V ) is normalized, that is, tr End( V ) (1) = 1 . 7 orollary 2.9. For any A ∈ M and B ∈ End ( V ) commuting with p π we have Tr Q U (( A ⊗ B ) P U ) = Tr M ( A ) N ! tr End( V ) ( Bp π ) . (9) Proof.
It follows the properties of the trace on a semidirect product of factors that for any σ ∈ S N the following equality holds: Tr Q U ( A S ( σ ) ⊗ Bp π ˜ π ( σ )) = Tr M ⋊ S N ( A S ( σ ))tr End( V ) ( Bp π ˜ π ( σ ))= δ e,σ Tr M ( A )tr End( V ) ( Bp π ˜ π ( σ )) , where δ e,σ is the Kronecker symbol. Formula (9) follows now from (7). Remark . Introduce the projection P s = 1 N ! X σ ∈ S N S σ . It follows from (9) that the following formula holds: Tr Q U (( A ⊗ B ) P U ) = Tr M ( P s )Tr Q U (( A ⊗ B )(1 ⊗ p π ))= 1 N ! Tr Q U (( A ⊗ B )(1 ⊗ p π )) . In order to extend formula (9) to general elements of Q U , we introduce II ∞ -factors Q := M ⊗ End ( V π ) and Q := M ⊗ End (( V π ) ⊥ ) . It is well known that any finite dimensional unitary representation of a locally compactgroup is completely reducible. Thus the representation π can be decomposed into a directsum π = π ⊕ π where π = ˜ π = π ↾ V π and π = π ↾ ( V π ) ⊥ . Observe that in the casewhen π is irreducible we have V π = { } and π is a zero representation. Therefore wecan introduce von Neumann algebras ˜ Q = { Q , { S ( σ ) ⊗ ˜ π ( σ ) } σ ∈ S N } ′′ and ˜ Q = { Q , { S ( φ ( β )) ⊗ π ( β ) } β ∈ B N } ′′ . Clearly ˜ Q and ˜ Q are isomorphic to II ∞ -factors Q ⋊ S N and Q ⋊ S N respectively, and P U ∈ ˜ Q .We define the following natural trace on the von Neumann algebra ˜ Q ⊕ ˜ Q setting Tr ˜ Q ⊕ ˜ Q ( A ⊕ A ) = dim V π dim V Tr ˜ Q ( A ) + dim ( V π ) ⊥ dim V Tr ˜ Q ( A ) , (10)8 ∈ ˜ Q , A ∈ ˜ Q .Consider an arbitrary A ∈ Q commuting with ⊗ p π . Then A ∈ Q ⊕ Q and AP U ∈ ˜ Q . Observe that dim V π dim V = tr End( V ) ( p π ) and A (1 ⊗ p π ) ∈ Q . Applying thearguments similar to the proof of Corollary 2.9 one can see that Tr ˜ Q ⊕ ˜ Q ( AP U ) = Tr Q ( A (1 ⊗ p π )) N ! tr End( V ) ( p π ) . (11) Remark . Note that formula (11) is compatible with (9). Indeed, if A ⊗ B ∈ M ⊗ End ( V ) and B ∈ End ( V ) commutes with p π , then A ⊗ B ∈ Q ⊕ Q and by (11) wehave Tr ˜ Q ⊕ ˜ Q (( A ⊗ B ) P U ) = Tr Q ( A ⊗ Bp π ) N ! tr End( V ) ( p π )= Tr M ( A ) N ! tr End( V π ) ( Bp π )tr End( V ) ( p π )= Tr M ( A ) N ! tr End( V ) ( Bp π ) because the traces tr End( V ) and tr End( V π ) are normalized.
3. Braided multi-particle spaces and the corresponding trace
In this section we apply the general construction discussed above to the case where thespace H has the form of the tensor product H ⊗ N of N copies of a separable Hilbert space H and S N ∋ σ S ( σ ) ∈ End( H ⊗ N ) (12)is the natural action of S N in H ⊗ N by permutations, so that S ( φ ( b i )) is the permutation ofthe i -th and i + 1 -th components in H ⊗ N . As in the previous section, the representation U is defined by formula (3), that is, U ( b i )( ξ ⊗ v ) = S ( φ ( b i )) ξ ⊗ π ( b i ) v, (13)where ξ ∈ H, v ∈ V , elements b , . . . , b N − are the generators of the group B N and φ : B N → S N is the canonical homomorphism. The corresponding subspace ( H ⊗ V ) U of invariant elements (cf. (4)) will be called the braided N -particle space. Let M be a II ∞ factor acting in H and define the corresponding II ∞ factors M = M ⊗ N and Q = M ⊗ End V acting in H = H ⊗ N and H ⊗ V .The following result was proved in [13] (see also [14]).9 roposition 3.1. Representation S defined by formula (12) satisfies Condition 2.1, thatis, S ( σ ) / ∈ M for all σ ∈ S N . Proposition 3.1 implies that we can apply the theory developed in the previous section.According to Lemma 2.6 we have P U ∈ Q U = ( M ⊗ N × α S N ) ⊗ End( V ) . (14)In what follows we will show that Q U is the minimal extension of M ⊗ N containing and P U .Theorem 3.2. Factors Q U and Q P := { Q, P U } ′′ are isomorphic.Proof. The inclusion Q P ⊂ ( M ⊗ N × α S N ) ⊗ End( V ) follows from (14).Let us prove the inverse inclusion. Observe that operators π ( b i ) ∈ End( V ) and hence M ⊗ N × α S N ⊗ π ( b i ) ∈ Q P . This it suffices to prove that the operators S ( φ ( b i )) ⊗ End( V ) belong to Q P . Denote P i = (1 + S ( φ ( b i )) ⊗ End( V ) ) . The II ∞ -factor M can berepresented in a form N ⊗ B ( l ) , where N is a II factor and B ( l ) is a space of allbounded operators in the Hilbert space l = l ( N ) . Therefore M contains the isometry v = 1 N ⊗ T , where T e i = e i +1 is the operator of unilateral shift in l . It is clear that ( v ∗ ) m → , m → ∞ , strongly. Therefore we have strong convergence (cid:0) (1 ⊗ ⊗ ( v ∗ ) m ⊗ · · · ⊗ ( v ∗ ) m ) ⊗ End( V ) (cid:1) · (15) · P (cid:0) (1 ⊗ ⊗ v m ⊗ · · · ⊗ v m ) ⊗ End( V ) (cid:1) → N ! P ,m → ∞ . Thus P ∈ M , which implies that P ∈ M ⊗ End( V ) ⊂ Q P . Similar argumentsshow that P i ∈ Q P for any i = 1 , . . . , N − . Remark . It follows from Theorem 3.2 that formulae (9) and (11) are valid for Tr Q P instead of Tr Q U .
4. Von Neumann dimensions of the spaces of braid-invariant harmonic forms
Let X be a smooth connected Riemannian manifold admitting an infinite discretegroup G of isometries such that the quotient K = X/G is a compact connected Rie-mannian manifold.In this section we consider the De Rahm complex of square-integrable forms on X N taking values in a finite dimensional Abelian algebra A N [24]. We apply the trace formula(9) in order to find the von Neumann dimensions of the spaces of U -invariant harmonicforms (the braided L - Betti numbers ). 10 .1. Setting: Von Neumann algebras associated with infinite coverings of compact mani-folds
Let us describe the framework introduced by M. Atiyah in his theory of L -Betti num-bers, which we will use during the rest of the paper. For a detailed exposition, see [11] ande.g. [30].We assume that there exists an infinite discrete group G acting freely on X by isome-tries and that K = X/G is a compact Riemannian manifold. That is, G → X → K (16)is a Galois (normal) cover of K .Throughout this section, we fix m = 1 , ..., d, d = dim X − , and use the followinggeneral notations: L Ω m ( X ) - the space of square-integrable m -forms on X ; L Ω m ( K ) - the space of square-integrable m -forms on K ; H ( m ) X - Hodge-de Rahm Laplacian on X (considered as a self-adjoint operator in L Ω m ( X ) ); H m ( X ) := Ker H ( m ) X - the space of square-integrable harmonic m -forms on X .The action of G in X generates in the natural way the unitary representation of G in L Ω m ( X ) , which we denote G ∋ g T g ∈ End( L Ω m ( X )) . (17)Let M m be the commutant of this representation, M m = { T g } ′ g ∈ G ⊂ End( L Ω m ( X )) . (18)It is clear that the space L Ω m ( X ) can be described in the following way: L Ω m ( X ) = L Ω m ( K ) ⊗ l ( G ) , (19)with the group action obtaining the form T g = id ⊗ L g , (20) g ∈ G, where L g , g ∈ G, are operators of the left regular representation of G . Then End( L Ω m ( X )) = End( L Ω m ( K )) ⊗ End( l ( G )) (21)and M m = End( L Ω m ( K )) ⊗ R ( G ) , (22)11here R ( G ) is the von Neumann algebra generated by the right regular representation of G . In what follows, we assume that G is an ICC group, that is, all non-trivial classes of conjugate elements are infinite. (23)This ensures that R ( G ) is a II -factor (see e.g. [32]). Thus M is a II ∞ -factor.Let us consider the orthogonal projection P ( m ) Har : L Ω m ( X ) → H m ( X ) (24)and its integral kernel k ( x, y ) ∈ T x X ⊗ T y X. (25)Then, because of the G -invariance of the Laplacian H ( m ) X , we have P Har ∈ M m . It wasshown in [11] that Tr M P ( m ) Har = Z K tr k ( x, x ) dx, (26)where tr is the usual matrix trace and dx is the Riemannian volume on K . Let us remarkthat, because of G -invariance, k ( m, m ) is a well-defined function on K . Moreover, it isknown that H ( m ) X is elliptic regular, which implies that the kernel k is smooth. Thus β m := Tr M P ( m ) Har < ∞ . (27)The numbers β m , m = 0 , , ..., d, are called the L -Betti numbers of X (or K ) associatedwith G . L -Betti numbers are homotopy invariants of K ([17]). It is known that d X m =0 ( − m β m = χ ( K ) , (28)where χ ( K ) is the Euler characteristic of K ( L index theorem [11]).Observe that all of the above can be applied to the product group G N acting on theproduct manifolds X N (instead of G acting on X ). The K¨unneth formula H m ( X N ) = M m + ··· + m N = m H m ( X ) ⊗ · · · ⊗ H m N ( X ) . (29)implies that the corresponding L -Betti numbers of X N have the form β m ( X N ) = X m + ··· + m N = m β m . . . β m N . (30)12 .2. L -dimensions of the spaces of braid-invariant harmonic forms Let A N be a finite dimensional Abelian algebra possessing a structure of a finite dimen-sional Hilbert space. Consider the Hilbert space L Ω m ( X N ) ⊗ A N of square integrable(with respect to the volume measure) A N -valued m -forms on X N .Recall that the group G acts on X by isometries. Denote by M m ( X N ) ⊂ End( L Ω m ( X N )) and Q ( m,N ) := M m ( X N ) ⊗ End( A N ) ⊂ End( L Ω m ( X N ) ⊗ A N ) the commutant of the induced natural action of G N (by isometries) on L Ω m ( X N ) and L Ω m ( X N ) ⊗ A N , respectively. Here End( A N ) is the space of endomorphisms of A N with respect to its Hilbert space structure (but not in general morphisms of the algebra A N ). Both M m ( X N ) and Q ( m,N ) are II ∞ factors.We adopt the construction of Section 2 with H = L Ω m ( X N ) and V = A N . Considera unitary representation π : B N ∋ b π ( b ) ∈ End( A N ) of the braid group B N in A N , and the corresponding representation (cf. (3)) U : B N ∋ b U ( b ) = S ( φ ( b )) ⊗ π ( b ) ∈ End( L Ω m ( X N ) ⊗ A N ) in the space L Ω m ( X N ) ⊗ A N , where S is the natural action the group S N in L Ω q ( X N ) defined in local coordinates by the formula S ( σ ) f (( x m ) m =1 ,...N ) dx i ∧ · · · ∧ dx i q = f (( x σ ( m ) ) m =1 ,...N ) dx σ ( i ) ∧ · · · ∧ dx σ ( i m ) . (31)Here σ ∈ S N is the permutation (1 , ..., N ) ( σ (1) , ..., σ ( N )) . Let L Ω mU ( X N , A N ) := (cid:0) L Ω m ( X N ) ⊗ A N (cid:1) U be the Hilbert space of U -invariant elements of L Ω m ( X N ) ⊗ A N (cf. (4)), and considerthe corresponding projection P ( m ) U : L Ω m ( X N ) ⊗ A N → L Ω mU ( X N , A N ) . Let A πN := { a ∈ A N : π ( b ) a = a, b ∈ B pureN } be the space of all π ( B pureN ) -invariantelements of A N . In what follows we will assume that A πN is a subalgebra of A N . (32)13he latter condition holds in all the examples considered in Section 5 below.Let Ω m ( X N ) be the space of smooth differential m -forms on X N , and d mX N : L Ω m ( X N ) → L Ω m +1 ( X N ) , Dom( d mX N ) = Ω m ( X N ) be the Hodge differential on X N . It can be extended to an operator d mN := d mX N ⊗ A N : L Ω m ( X N ) ⊗ A N → L Ω m +1 ( X N ) ⊗ A N , where A N is the identity operator on A N . Proposition 4.1. (i) The exterior product of two U -invariant A N -valued forms is also U -invariant.(ii) The differential d mN preserves the space of U -invariant forms, that is, d mN (cid:16) P ( m ) U (cid:0) Ω m ( X N ) ⊗ A N (cid:1)(cid:17) ⊂ P ( m +1) U (cid:0) Ω m +1 ( X N ) ⊗ A N (cid:1) . Proof. (i) The statement follows from (7) and condition (32) . (ii) The statement follows from the fact that d mX N commutes with operators S ( φ ( b i )) ⊗ id and id ⊗ π ( b i ) , i = 1 , . . . , N − .Let us consider the Hodge-de Rham Laplacian H ( m ) X N and define H ( m,N ) : = H ( m ) X N ⊗ A N . H ( m,N ) is a selfajoint operator in L Ω m ( X N ) ⊗ A N . It follows from the definition of theoperators d mN and H ( m,N ) that H ( m,N ) = d mN ( d mN ) ∗ + (cid:0) d m − N (cid:1) d m − N . Proposition 4.1 implies that H ( m,N ) preserves the space of U -invariant forms. Moreover, H ( m,N ) commutes with the operators U ( b ) , b ∈ B N .Let us consider the projection P ( m,N ) Har : L Ω m ( X N ) → H m ( X N ) . Introduce the space H mU ( X N , A N ) := (cid:0) H m ( X N ) ⊗ A N (cid:1) U , U -invariant elements of H m ( X N ) ⊗ A N . Observe that the projection P ( m,N ) Har ⊗ A N : L Ω m ( X N ) ⊗ A N → H m ( X N ) ⊗ A N commutes with the projection P ( m,N ) U : L Ω m ( X N ) ⊗ A N → L Ω mπ ( X N , A N ) . Therefore the operator P ( m,N ) := ( P ( m,N ) Har ⊗ A N ) P ( m,N ) U : L Ω m ( X N ) ⊗ A N → H mU ( X N , A N ) . is the projection, too.Our goal is to compute the L -dimension of the space H mU ( X N , A N ) , that is, the vonNeumann trace of the projection P ( m,N ) . Observe that P ( m,N ) Har ∈ M m ( X N ) , which impliesthat P ( m,N ) Har ⊗ A N ∈ Q ( m,N ) .Proposition 4.2. The factor M m ( X N ) does not contain the operators S ( φ ( b )) , b ∈ B N , b / ∈ B pureN .Proof. Note that the space L Ω m ( X N ) can be represented in the form L Ω m ( X N )= L ( X ) ⊗ N ⊗ Λ m ( C N ) , where Λ m ( C N ) is a m -th direct summand of the exterior algebra Λ( C N ) . Then the factor M m ( X N ) has the form M m ( X N ) = M ⊗ N ⊗ End(Λ m ( C N )) , where M is the commutantof the action of the group G in L ( X ) . The representation S of the group S N can bedecomposed into the product S = S ⊗ S ′ , where S and S ′ are natural actions of S N bypermutations in L ( X ) ⊗ N and Λ m ( C N ) , respectively (cf. (12) and (31)). It follows fromProposition 3.1that S ( σ ) / ∈ M ⊗ N , σ = e . Therefore S ( σ ) / ∈ M m ( X N ) , σ = e , whichimplies the result.Thus Condition 2.1 is satisfied, and we can apply the results of Section 2. Let Q ( m,N ) P := { Q ( m,N ) , P ( m,N ) U } ′′ be the minimal von Neumann algebra containing Q ( m,N ) and P ( m,N ) U . Theorem 4.3.
The von Neumann algebras Q ( m,N ) P , (cid:0) M m ( X N ) ⋊ S N (cid:1) ⊗ End ( A N ) and { Q ( m,N ) , { U ( b ) } b ∈ B N } ′′ are isomorphic. roof. The first isomorphism of can be proved by the arguments similar to the proof ofTheorem 3.2. The second isomorphism follows from Lemma 2.6.Theorem 4.3 limplies that the von Neumann algebra Q ( m,N ) P is a II ∞ factor (since (cid:0) M m ( X N ) ⋊ S N (cid:1) ⊗ End ( A N ) is so). Theorem 4.4.
The projection P ( m,N ) belongs to Q ( m,N ) P and its trace is given by the for-mula Tr Q ( m,N ) P ( P ( m,N ) ) = Tr End( A N ) ( p π ) N ! X m + ··· + m N = m β m . . . β m N (33) Here β m , m = 0 , , . . . , dim X , are the L -Betti numbers of X and p π is the projectiononto the subalgebra A πN of B pureN -invariant elements of A N .Proof. The inclusion P ( m,N ) ∈ Q ( m,N ) P follows from the definition of the von Neumannalgebra Q ( m,N ) P . The equality (33) follows from Corollary 2.9 with A ⊗ B = P ( m,N ) Har ⊗ A N and the K¨unneth formula H m ( X N ) = M m + ··· + m N = m H m ( X ) ⊗ · · · ⊗ H m N ( X ) . (34)We will use the notation b m ( X N ) = Tr Q ( m,N ) P ( P ( m,N ) ) and call b m ( X N ) the m -th braided L - Betti number of X N .Let us introduce the constant C πN = tr End( A N ) ( p π ) N ! . Observe that the trace on the algebra
End( A N ) is normalized, that is, tr End( A N ) (1 A N ) = 1 ,which implies that C πN = dim A πN N ! dim A N ≤ N ! . (35)Then formula (33) can be rewritten in the form b m ( X N ) = C πN Tr M ( m ) ( P ( m,N ) Har ) . (36) Remark . In the situation where the representation π reduces to a representation of thesymmetric group S N (i.e. π ( b i ) = 1 ) the projection p π = 1 and formula (33) can berewritten in a form Tr Q mπ ( P ( m,N ) Har ) = 1 N ! Tr M ( m ) ( P ( m,N ) Har ) In this case the braided L -Betti numbers b m do not depend on the algebra A N and coincidewith the L -Betti numbers of the space of N -point configurations Γ N ( X ) of X ([13], [14]).16 . Examples In this section we consider examples associated with particular representations of thebraid group B N . We set A N = ( C ) ⊗ N ( = C N ) equipped with the natural Abelian algebrastructure.An important class of representations π can be constructed in the following way. Let C be a complex unitary × -matrix satisfying the braid equation: ( C ⊗ ⊗ C )( C ⊗
1) = (1 ⊗ C )( C ⊗ ⊗ C ) , (37)where denotes the identity × -matrix. For all generators b i ∈ B N define an operator π ( b i ) acting in A N by the formula π ( b i ) := 1 ⊗ ⊗ · · · ⊗ C ⊗ · · · ⊗ , (38)where C acts on the i -th and i + 1 -th components of the tensor product ( C ) ⊗ N . Then π de-fines a unitary representation of B N . This representation is not reduced to a representationof S N if and only if C = 1 . Remark . Representations (38) are extensively studied. It is known that the solutionsof the equation (37) are in one-to-one correspondence with constant solutions R of theYang-Baxter equation R k k j j R l k k j R l l k k = R k k j j R k l j k R l l k k . (39)More precisely, C = R Σ , where Σ is a numerical matrix with elements Σ ijkl = δ jk δ il and δ jk denotes the Kr¨oneker symbol. All two-dimensional solutions of (39) have been classifiedin [25], which gives the complete list of unitary × -matrices satisfying the braid equation(37) .Example 1. Let C be an arbitrary unitary solution of the equation (37) and C = 1 . Forinstance, C = q ε ε q , where ε = ± and | q | = 1 . Then the representation π is reduced to a representation of S N , so that the corresponding representation ˜ π of the pure braid group B pureN is trivial( ˜ π ( b ) = id for any b ∈ B pureN ) and thus A πN = A N . Then C πN = N ! and b m ( X N ) = 1 N ! X m + ··· + m N = m β m . . . β m N . (40)17 xample 2. Let q ∈ C be such that | q | = 1 and q = 1 . Set C = q q , Then C is unitary solution of (37) and C = 1 . Proposition 5.1. dim A πN = 2 for any N ∈ N .Proof. A direct calculation using formula (38).We can apply the results of Section 4 and compute the braided L -Betti numbers b m ( X N ) of a connected cocompact Riemannian manifold X . Formula (35) implies that C πN = N − N ! . Therefore b m ( X N ) = 12 N − N ! X m + ··· + m N = m β m . . . β m N , (41)where β i , i = 0 , , , are the L -Betti numbers of X .It this example, we can give an explicit description of the braided N -particle space.A direct calculation shows that the only non-zero components of f = ( f , . . . , f N ) ∈ ( H ⊗ V ) U are f a N − and f a N , where the sequence a N is defined recursively as follows: a = 2 and a N = 2 N − a N − + 1 . Moreover f a N − and f a N are symmetric elements, sothat ( H ⊗ V ) U = H ˆ ⊗ N ⊗ C , where ˆ ⊗ denotes the symmetric tensor product. Example 3.
Let q ∈ C be such that | q | = 1 and q = 1 . Set C = q , Then C is unitary solution of (37) and C = 1 . Proposition 5.2. dim A πN = N + 1 .Proof. Let e = (1 , t , e = (0 , t be the canonical basis in C . Denote by e i ...i n = e i ⊗ · · · ⊗ e i N the corresponding basis in C N .18onsider first the case of N = 2 . We have B = { b } and π ( b ) e = qe , π ( b ) e = e , π ( b ) e = e , π ( b ) e = e . Therefore the basis of A πN consists of e , e and e ,and dim A πN = 3 .A direct check shows that e . . . | {z } N and e . . . | {z } N belong to A πN . Let us prove that forany multiindex I = i . . . i N , i k ∈ { , } , such that e I ∈ A πN the equality i N = 0 impliesthat i = . . . i N − = 1 . Indeed, let i N = 0 and k := max j ∈{ , ,...,N − } { i j = 0 } > . Thenit is easy to see that e I = π ( x k,N − ) e I = π ( b N − b N − . . . b k +1 b k b − k +1 . . . b − N − b − N − ) e I = π ( b N − b N − . . . b k +1 b k ) e i ...i k − ... = q e I , because the space A πN is invariant under the representation π and π ( b k ) e i ...i k − ... = q e i ...i k − ... (recall that π ( b i ) = 1 ⊗ · · · ⊗ ⊗ C ⊗ ⊗ · · · ⊗ ). Therefore e I = 0 .Let us prove that e I ∈ A πN iff e I ∈ A πN +1 . Indeed, let e I ∈ A πN . To prove that e I ∈ A πN +1 it suffice to check that π ( x k,N ) e I = e I for all k ≤ N . If i N = 1 wehave π ( x k,N ) e I = π ( b N x N − ,k ) e I = π ( b N ) e I = e I since π ( x k,N − ) e I = e I . If i N =0 then by previous statement we have that i = · · · = i N − = 1 and π ( x k,N ) e I = π ( b N x k,N − ) e ... = π ( b N ) e ... = e ... = e I .The equalities π ( x i,j ) e I = e I , i ≤ j ≤ N − and the fact that the operators π ( x i,j ) are acting only in first N − indexes imply that e I ∈ A πN provided e I ∈ A πN +1 . Thisstatement combined with the induction arguments and the fact that e . . . | {z } N +1 ∈ A πN +1 complete the proof.It follows from Proposition 5.2 that C πN = N +12 N N ! . Therefore the braided L -Betti num-bers b m ( X N ) of X have the following form: b m ( X N ) = N + 12 N N ! X m + ··· + m N = m β m . . . β m N , (42)where β i , i = 0 , , , are the L -Betti numbers of X .The structure of the corresponding braided N -particle space is more complicated thenis previous example. It is not hard to see that the representation ˜ π of the symmetric groupin the space A πN that corresponds to the representation π can be decomposed into a di-rect sum of two cyclic representations ˜ π and ˜ π . Representation ˜ π is trivial and acts inone-dimensional space generated by the basis element e ... while representation ˜ π per-mutetes the other N basis elements in A πN . Therefore the space H ⊗ π N can be decomposedinto the direct sum of the space L ( X ) ˆ ⊗ N of symmetric functions and the space H ⊗ ˜ π N of19lements from L ( X N ) ⊗ C N that are invariant under representation ˜ U N of the group S N given by the relation ( ˜ U N ( σ i ) f )( x , . . . , x i , x i +1 , . . . x N ) = ˜ π ( σ i ) f ( x , . . . , x i +1 , x i , . . . , x N ) , where σ , . . . , σ N − are generating elements of the group S N . Example 4.
Let q ∈ C be such that | q | = 1 and q = 1 C = q q
01 0 0 0 . Then C is unitary solution of (37) and C = 1 . Proposition 5.3. dim A πN = 0 .Proof. Similar to the Example 2 we can show that π ( b k ) e i ...i k i k +1 ...i N = q e i ...i N iff i k i k +1 = 01 or i k i k +1 = 10 . Therefore any vector v ∈ A πN should have the form v = αe ... + βe ... . On the other hand, π ( x , ) e ... = q e ... and π ( x , ) e ... = q e ... ,so that π ( x , ) v = q v , which implies that v = 0 . Therefore dim A πN = 0 .Thus, b k ( X N ) = 0 .
6. Fock space of braid-invariant harmonic forms: L -dimensions and index Let us consider the infinite disjoint union X = ∞ G N =1 X N . The space L Ω m ( X ) of square-integrable m -forms on X has the form L Ω m ( X ) = M N L Ω m ( X N ) . (43)We can define the Hodge-de Rham Laplacian H ( m ) X on L Ω m ( X ) component-wise, that is, H ( m ) X := X N H ( m ) X N . H m ( X ) := Ker H ( m ) X = M N H m ( X N ) . (44)Formulae (43) and (44) motivate us to define the space L Ω mU ( X ,A ) := M N L Ω mU ( X N , A N ) , which can be regarded as the space of ” m - forms on the braided configuration space ”.Similar to the case of the de Rham complex (43), the Hodge-de Rham Laplacian H ( m ) on L π Ω m ( X ) can be defined component-wise, H ( m ) := X N H ( m,N ) . The corresponding space of U -invariant harmonic m -forms can be decomposed into thedirect sum H mU ( X ,A ) := Ker H ( m ) = M N H mU ( X N , A N ) . Let P : = P m P ( m ) , where P ( m ) is the projection onto H mU ( X ,A ) . Moreover, we have theequality P ( m ) = P N P ( m,N ) , which implies that P ( m ) an element of the von Neumannalgebra Q ( m ) P := L N Q ( m,N ) P .We introduce a regularized dimension b m ( X ) of the space H mU ( X ,A ) by the formula b m ( X ) = ∞ X N =0 b ( N ) m = Tr Q ( m ) P P ( m ) and define the supertrace STR P = ∞ X m =0 ( − m b m ( X ) . (45)Here we use the convention b = 1 . Theorem 6.1.
The series in the right-hand side of (45) converges absolutely and
STR P = X N C πN χ ( K ) N < ∞ , where χ ( K ) is the Euler characteristic of K . roof. Formula (35) shows that C πN ≤ N ! . Then ∞ X m =0 b m ( X ) ≤ ∞ X m =0 ∞ X N =0 N ! X k + k + ... + k N = m β k . . . β k N = ∞ X N =0 N ! X k β k ! N < ∞ . Then
STR P = ∞ X m =0 ( − n b m ( X )= ∞ X m =0 ( − m X N C πN X k + k + ... + k N = m β k . . . β k N = X N C πN X k ,k ,...,k N ( − k β k ... ( − k N β k N = X N C πN X k ( − k β k ! N = X N C πN χ ( K ) N because of the equality P m ( − m β m = χ ( K ) (the L index theorem [11]).Let us consider the examples of the previous section. In all these examples we have A N = ( C ) ⊗ N so that dim A N = 2 N . Thus, we can apply Theorem 6.1 and compute STR P π . Example 1 (revisited). C πN = N ! and STR P = e − χ ( K ) . The latter expression coincideswith the formula derived in [1] for the case the de Rham complex over the configurationspace equipped with the Poisson measure. Example 2 (revisited). C πN = N − N ! . Then
STR P = X N ( − N C πN χ ( K ) N = X N ( − N N ! (cid:18) χ ( K )2 (cid:19) N = 2 e − χ ( K )2 . xample 3 (revisited). C πN = N +12 N N ! . Then
STR P = X N ( − N C πN χ ( K ) N = X N ( − N NN ! (cid:18) χ ( K )2 (cid:19) N + X N ( − N N ! (cid:18) χ ( K )2 (cid:19) N = − χ ( K )2 X N ( − N − N − (cid:18) χ ( K )2 (cid:19) N − + X N ( − N N ! (cid:18) χ ( K )2 (cid:19) N = 1 − χ ( K )2 e − χ ( K )2 + e − χ ( K )2 . Remark . Under the additional assumption of dim X = 2 , the only non-zero L -Bettinumber of X is β = β ( X ) . Then, according to formula (33), the only non-zero braided L -Betti number of X N is b N ( X N ) = C πN β N , and therefore b m ( X ) = C πN β m , m = 1 , , , .... Remark . The right-hand side of formula (45) can be understood as a regularized indexof the Dirac operator D + D ∗ , where D := P m P N d mN , see [11] and e.g. [30] for the dis-cussion of von Neumann supertraces in geometry and topology of Riemannian manifoldsand their relation to L -index theorems, and [1], [2], [3] for the extension of these notionsto the framework of infinite configuration spaces. Acknowledgments
The authors would like to thank Sergio Albeverio, Yuri Kondratiev, Eugene Lytvynovand Leonid Vainerman for helpful discussions.
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