A KK-theoretic perspective of quantization commutes with reduction
aa r X i v : . [ m a t h . K T ] N ov A KK-THEORETIC PERSPECTIVE ON QUANTIZATION COMMUTES WITH REDUCTION
RUDY RODSPHONA
BSTRACT . We reframe Paradan–Vergne’s approach to quantization commutes with reduction in KK-theory through arecent formalism introduced by Kasparov, focusing more especially the index theoretic parts that lead to their "Wittennon-abelian localization formula". While our method uses the same ingredients as their’s in spirit, interesting conceptualsimplifications occur, and the relationship to the Ma–Tian–Zhang analytic approach becomes quite transparent. I NTRODUCTION
The quantization commutes with reduction problem introduced by Guillemin–Sternberg [ ] has generated a greatdeal of interest since its inception, particularly because of the wide range of mathematics that have been involvedin its study. The most recent approaches use localization techniques and are deeply rooted in index theory. Theymostly split into two schools of thought: • Analytic techniques (inspired by Witten’s deformation of the de Rham complex) introduced in this context byTian–Zhang [ ] , subsequently extended by Ma–Zhang [ ] with the optic of solving the Vergne conjecture,and systematized by Braverman [ ] . • K-theoretic and transverse index techniques (in the sense of Atiyah [ ] ) introduced in this context by Paradan [ ] , subsequently extended by Paradan–Vergne [
10, 11 ] also in view of tackling the Vergne conjecture, inthe slightly more general setting of spin c manifolds.Note also the work of Hochs–Song [ ] , which consists in implementing the analytic approach within theaforementioned framework of Paradan and Vergne.The present article proposes a reformulation of the index theoretic constructions and results used by Paradanand Vergne [ ] within a recent formalism developed by Kasparov [ ] . The latter reframes and generalizesextensively Atiyah’s work on transversally elliptic operator in view of deriving a powerful index theorem, whichrelates Atiyah’s transverse analytic index to a topological index. To be more precise, we prove that Paradan–Vergne’s non-abelian localization theorem, which is a key step in their solution to the quantization commuteswith reduction problem, is a direct consequence of Kasparov’s index theorem together with natural functorialarguments. The context is as follows.Let ( M , g ) be a (complete Riemannian) manifold without boundary equipped with the action of a compact Liegroup G with Lie algebra g , let E be a G -equivariant Clifford module bundle over M , let ν : M → g ∗ be a momentmap (in a weak sense, see Section 2.1), and assume that 0 is a regular value of ν . The zero set of the vector field ν associated to ν admits a decomposition Z ν = G β ∈B G · (cid:0) M β ∩ ν − ( β ) (cid:1) where B ⊂ g ∗ is a finite set consisting of representatives of the coadjoint orbits within ν ( Z ν ) . The zero set Z ν can besingular, but the idea to overcome this issue is to work, for each of its components in the above decomposition, onappropriate open neighborhoods that are diffeomorphic to "slices". With this in mind, Paradan–Vergne’s theoremcalculates the transverse index of the Dirac operator by localization around the zero set Z ν , and can be stated asfollows: Date : October 31, 2020.
Theorem.
The transverse index of the symbol σ ν ( x , ξ ) = i c ( ξ + ν ) ∈ C ∞ ( T ∗ M , π ∗ T ∗ M End ( E )) is given by the fixedpoint formula: Index G [ σ ν ] = X ( V π , π ) ∈ b G (cid:0) [ E ⊗ ( ν − ( ) × G V π )] ⊗ C ( M ) [ D ] (cid:1) V π + X β = β ∈B Index G β [ Λ • C ( g / g β )] ⊠ [ σ ν | Y β ] P k ( − ) k [ Λ k N β ] ∈ b R ( G ) where E is the Clifford module bundle over the reduced space M = ν − ( ) / G induced by E ; [ E ⊗ ( ν − ( ) × G V π )] isthe K-theory class of the vector bundle E ⊗ ( ν − ( ) × G V π ) → M ; G β ⊂ G denotes the stabilizer subgroup of β ∈ B relative to the coadjoint action of G on g ∗ ; g β = Lie ( G β ) ; Y β is a small open neighborhood of M β ∩ ν − ( β ) insidethe fixed point set M β ; [ σ ν ] ∈ K G ( T ∗ G M ) ; [ σ ν | Y β ] ∈ K G β ( T G β Y β ) denote respectively the symbol classes of σ ν and ofits restriction to T ∗ Y β ; and N β is the normal bundle of M β in M. In the above, K ( T ∗ G M ) is a receptacle for symbol classes of transversally elliptic operators introduced byAtiyah, which involves the transverse (co)tangent bundle T ∗ G M . In the paper, it will subsequently be replacedby Kasparov’s K-group K ( Cl Γ ( T M )) , which offers the possibility to define an intrinsic topological tranvserse index.Some elements of our proof are close to Paradan–Vergne’s in spirit. Nevertheless, it should be noted thatthey are not a mere translation of their work in KK-theory. Instead, Kasparov’s formalism enables substantialconceptual simplifications which are mostly due to the existence of the KK-product and of a topological (trans-verse) index. As a consequence, the process ends up being rather orthogonal to Paradan–Vergne’s approach,and provides a direct derivation of the aforementioned formula from a simple systematic use of appropriateThom isomorphisms. In Paradan–Vergne’s work, the unavailability of these powerful functorial tools hampersthe construction of these natural Thom isomorphisms, perfomed via a rather technical and cumbersome procedure.In the course of constructing these Thom isomorphisms, it is also worth mentioning that we avoid the use ofa quite non-trivial transverse index calculation of Atiyah for toral actions on C n . Rather, the latter becomes adirect consequence of the above formula, whereas it is utilized as a crucial intermediary tool in Paradan–Vergne’sapproach.Lastly, let us note that gathering the material of the present article, our previous work [ ] and the work of Hochs–Song, one sees that the analytic and topological approaches to the quantization commutes with reduction are thesame up to Poincaré duality (in the sense of KK-theory). In sum, the present paper advocates KK-theory as an idealframework to synthetize and unify the different array of techniques developed in view of solving the quantizationcommutes with reduction problem. Plan of the article.
Section 1 gives a short account of Kasparov’s work [ ] on transverse index theory. Section2 begins with an exposition of Paradan–Vergne’s framework [ ] and ends on a KK-theoretical proof of theirnon-abelian localization formula. Acknowledgement.
I want to warmly thank Gennadi Kasparov for many stimulating discussions over the years.The present paper would probably have remained a manuscript note without his impulse.
Framework.
We consider thoughout the paper a complete Riemannian manifold ( M , g ) (without boundary)equipped with an (isometric) action of a compact Lie group G (the results of Section 1 also hold if G is non-compact and acts properly, isometrically). Let τ = T M and g = Lie ( G ) denote respectively the tangent bundle of M and the Lie algebra of G . We will not make much distinctions between T M and the cotangent bundle T ∗ M ,and will most of the time identify them tacitely via the metric. Similarly, g and its dual g ∗ are most of the timeidentified via a G -invariant product on g . If A , B are C ∗ -algebras, KK G • ( A , B ) , K G • ( A ) and K • G ( B ) the associated G -equivariant KK-theory, K-theory, K-homology groups. If moreover A , B are C ( X ) -algberas, R KK G • ( X ; A , B ) denotesthe representable KK-theory group of A and B . KK-THEORETIC PERSPECTIVE ON QUANTIZATION COMMUTES WITH REDUCTION 3
1. K
ASPAROV ’ S APPROACH OF TRANSVERSE INDEX THEORY
In this section, we review Kasparov’s KK-theorerical approach of transverse index theory for Lie group actions [ ] and the associated index theorem.1.1. Orbital Clifford algebra.
Let ( M , g ) be a complete Riemannian manifold (without boundary) equipped withan (isometric) action of a compact Lie group G . For every x ∈ M , the derivative of the action of G on x reads: ρ x : v ∈ g ddt (cid:12)(cid:12)(cid:12)(cid:12) t = exp ( t v ) · x ∈ T x M and yields in an obvious way a smooth bundle map ρ : g M : = M × g → T M . There is a natural extension of the G -action on M to g M given by g · ( x , v ) = ( g · x , Ad ( g ) v ) , for which ρ is G -equivariant. This action being proper,one can equip g M with a G -invariant Riemannian metric on the bundle g M . Definition 1.1.
The orbital tangent field Γ ⊂ T M is the continuous field of subspaces defined as the image Γ = Im ( ρ ) of ρ .Recall that Γ being a continuous field means that it admits a set of sections { x v x ∈ Γ x } that spans Γ fiberwise,with the requirement that k v x k be continuous in x . If we denote C ∞ c ( M , g M ) the set of smooth sections of thetrivial bundle g M = g × M , we define the set of smooth compactly supported sections of Γ as the set ρ ( C ∞ c ( M , g M )) ,which makes Γ into a continuous field. As sets, the fibers of Γ are the tangent spaces to the orbits which mayexhibit dimension jumps, but equipping Γ with the continuous field above gives a way to handle this issue.After choosing a G -invariant Riemannian metric on g M , consider the transpose operator ρ ⊤ : T M → g M .and define a smooth bundle map ϕ : T M → T M to be the composition ϕ = ρ ◦ ρ ⊤ . The relevance of the map ϕ lies in the fact that a vector ξ ∈ T x M is orthogonal to Γ x if and only if ϕ x ( ξ ) = C ∗ -algebra Cl τ ( M ) , consisting of C -sections of the Clifford algebrabundle Cliff ( T M ) over M . The initial step is to attach a Clifford algebra Cliff ( Γ x ) to each fiber Γ x of the orbitalfield Γ , and the associated family of such spaces Cliff ( Γ ) = F x ∈ M Cliff ( Γ x ) ⊂ Cliff ( T M ) inherits a continuous fieldstructure from Γ . Definition 1.2 (Kasparov [ ] ) . The orbital Clifford algebra Cl Γ ( M ) is the C ∗ -algebra generated by sections of thecontinuous field Cliff ( Γ ) over M vanishing at infinity (equipped with the sup-norm).Up to isomorphism, this definition if independent of the choice of the metric on M .1.2. Transverse index and Dirac elements.
Let A be a G -equivariant pseudodifferential operator with symbol σ A acting on sections of a G -equivariant Hermitian vector bundle E . Recall that the operator A is said to be transversally elliptic if supp ( σ A ) ∩ T ∗ G M is compact, where T ∗ G M = { ( x , ξ ) ∈ T ∗ M ; 〈 ξ , Γ x 〉 = } , and supp ( σ A ) isthe support of the symbol σ A (i.e the subset of T ∗ M where σ A fails to be invertible). Whence it defines naturallya K-theory class [ σ A ] ∈ K G ( T ∗ G M ) = K G ( C ( T ∗ G M )) .When M is compact, Atiyah proved [ ] that the restriction A π ( π ∈ b G ) of A to each isotypical component isFredholm, hence A has a well-defined equivariant indexIndex G ( A ) = X π ∈ b G Index ( A π ) π ∈ b R ( G ) = Z b G .taking values in the generalized character ring b R ( G ) . in the usual Hörmander ( ρ , δ ) = (
1, 0 ) class RUDY RODSPHON
When M is not compact, Atiyah defines the index of A by a reduction to the compact case, via an embedding ofa relatively compact neighborhood U of supp ( σ A ) ∩ T ∗ G M into a compact manifold, cf. Section 1.5 for more details.Suppose now that A has order 0; we can reformulate the above discussion from a K-homological perspective asfollows. Proposition 1.3. (cf. for example [
6, Proposition 6.4 ] ) The pair ( L ( M , E ) , A ) induces a K-homology class [ A ] ∈ K ( G ⋉ C ( M )) that we call the transverse index of A. If M is compact, then one can also view [ A ] as a class in the K-homology group K ( C ∗ ( G )) by crushing M to apoint, and the Peter-Weyl theorem shows that [ A ] coincides with Atiyah’s index.Suppose now that E is a G -equivariant ( Z -graded) Clifford module bundle over M , and let D be the associatedDirac-type operator acting on sections of E . As an elliptic operator, the Dirac operator induces a canonical K -homology class [ D ] ∈ K ( C ( M )) represented by the K -cycle (cid:0) L ( M , E ) , F : = D ( + D ) − / (cid:1) , and the same K-cyclealso induces a K-homology class [ D ] ∈ K ( G ⋉ C ( M )) , which describes the transverse index of D . The ingredientsintroduced in previous subsection offer an alternative choice of transverse index class: Definition 1.4. (cf. for example [
6, 7 ] )(i) The pair ( L ( M , E ) , F ) determines a K -homology class [ D M , Γ ] ∈ K ( G ⋉ Cl Γ ( M )) ,where the crossed product G ⋉ Cl Γ ( M ) acts on L ( M , E ) by multiplication (and convolution in G ). This classwill be referred to as the transverse Dirac element .(ii) When E is the (complexified) exterior algebra bundle Λ • C T M : = Λ • T M ⊗ C , with D being the canonicalDirac-type operator associated to the de Rham differential d , the class [ D M , Γ ] promotes to a K -homologyclass [ d M , Γ ] ∈ K ( G ⋉ Cl τ ⊕ Γ ( M )) ,where Cl τ ⊕ Γ ( M ) : = Cl τ ( M ) ⊗ C ( M ) Cl Γ ( M ) . This class will be referred to as the transverse de Rham-Diracelement .The action of Cl τ ⊕ Γ ( M ) on L ( M , Λ • C T M ) is given, on real (co)vectors, by ξ ⊕ ξ ext ( ξ ) + int ( ξ ) + i ( ext ( ξ ) − int ( ξ )) where ξ , ξ are respectively sections of T M and Γ .The fact that one indeed gets K-homology classes this way is not absolutely obvious. Details can be found in [ ] for(ii), and [ ] for (i). The general idea is that the G -action by convolution allows to make up for the unboundednessof the elements in [ D , Cl Γ ( M )] .1.3. Transversally elliptic symbols and the symbol algebra S Γ ( M ) . The precise relationship between theclasses [ σ A ] ∈ K G ( T ∗ G M ) and [ A ] ∈ K ( G ⋉ C ( M )) is the object of a highly non-trivial index theorem of Kas-parov that we describe in next subsection. Before that, a replacement of the algebra C ( T ∗ G M ) is needed. Definition 1.5. ( [ ] , Definition-Lemma 6.2) The symbol algebra S Γ ( M ) is the norm-closure in C b ( T ∗ M ) (thealgebra of continuous bounded functions on T ∗ M ) of the set of all smooth, bounded functions b ( x , ξ ) on T ∗ M ,which are compactly supported in the x -variable, and satisfy (a) together with either (b) or (c) (which are equiv-alent assuming (a) is satisfied): KK-THEORETIC PERSPECTIVE ON QUANTIZATION COMMUTES WITH REDUCTION 5 (a) The exterior derivative d x b ( x , ξ ) in x is norm-bounded uniformly in ξ , and there is an estimate | d ξ b ( x , ξ ) | ≤ C ( + k ξ k ) − for a constant C which depends only on b but not on ( x , ξ ) .(b) The restriction of b to T ∗ G M belongs to C ( T ∗ G M ) .(c) For any ǫ > c ǫ > | b ( x , ξ ) | ≤ c ǫ + k ϕ x ( ξ ) k + k ξ k + ǫ , ∀ x ∈ M , ξ ∈ T x M .Loosely speaking, item (c) says that S Γ consists of symbols with negative order in the transverse directions.Let π T ∗ M : T ∗ M → M denote the canonical projection. Given a G -equivariant Z -graded Hermitian vector bundle E , we can similarly define a Hilbert S Γ ( M ) -module, denoted S Γ ( E ) , as the norm-closure in the space of boundedsections of the pull-back bundle π ∗ T ∗ M E satisfying similar conditions to those in Definition 1.5 (using the norm onthe fibres of π ∗ T ∗ M E induced by the Hermitian structure).From now on, we refer to transversally elliptic operators (or symbols) according to the following definition. Definition 1.6.
Let A be a properly supported, odd, self-adjoint G -invariant pseudodifferential operator of order0 acting on sections of a G -equivariant Z -graded Hermitian vector bundle E . We will say that A (or its symbol σ A ) is transversally elliptic if for every a ∈ C ( M ) , a · ( − σ A ) ∈ S Γ ( M ) .Therefore, a transversally elliptic symbol naturally determines a class [ σ A ] = [( S Γ ( E ) , σ A )] ∈ R KK G ( M ; C ( M ) , S Γ ( M )) .By construction there is a ∗ -homomorphism S Γ ( M ) → C ( T ∗ G M ) , hence a map R KK G ( M ; C ( M ) , S Γ ( M )) → R KK G ( M ; C ( M ) , C ( T ∗ G M )) .In this sense the element [ σ A ] ∈ R KK G ( M ; C ( M ) , S Γ ( M )) can be viewed as a ‘refinement’ of the ‘naive’ class in R KK G ( M ; C ( M ) , C ( T ∗ G M )) defined by the symbol.1.4. Kasparov’s index theorem.
To state Kasparov’s index theorem relating the classes [ A ] ∈ K ( G ⋉ C ( M )) and [ σ A ] ∈ R KK G ( M ; C ( M ) , S Γ ( M )) , it will be convenient to introduce the C ∗ -algebraCl Γ ( T M ) : = C ( T M ) ⊗ C ( M ) Cl Γ ( M ) .which is KK-equivalent to the symbol algebra S ( M ) through the KK-class described in the definition below. Definition 1.7. [
6, p. 1344 ] The element [ f M , Γ ] ∈ R KK G ( M ; S Γ ( M ) , Cl Γ ( T M )) is the class represented bythe pair ( Cl Γ ( T M ) , f M , Γ ) where at a point ( x , ξ ) ∈ T x M , the operator f M , Γ ( x , ξ ) is left Clifford multiplication by − i ϕ x ( ξ )( + k ϕ x ( ξ ) k ) − / .This definition is in fact one of the main reasons to introduce the symbol algebra S Γ ( M ) . Being the symbol classof an orbital Dirac element, the class [ f Γ ] should be thought of as an orbital Bott element. For convenience, let usdefine an alternative receptacle for symbol classes. Definition 1.8.
Let A be a transversally elliptic operator, and let [ σ A ] ∈ R KK ( M ; C ( M ) , S Γ ( M )) be its standardsymbol class. The tangent Clifford symbol class of A is the element [ σ tcl A ] obtained as the KK-product: [ σ tcl A ] : = [ σ A ] ⊗ S Γ ( M ) [ f Γ ] ∈ R KK ( M , C ( M ) , Cl Γ ( T M )) Heuristically, the meaning of this definition is that one completes σ A into an elliptic symbol, which is paired withthe Dolbeault element to recover the transverse index of A from the classical index theorem. To convert this vagueidea into a theorem, we will need to define the appropriate Dolbeault element. RUDY RODSPHON
Notice first that the C ∗ -algebras Cl Γ ( T M ) and Cl τ ⊕ Γ ( M ) are KK-equivalent. Indeed, recall that the C ∗ -algebras C ( T M ) are KK-equivalent via an element [ d ξ ] ∈ R KK G ( M ; C ( T M ) , Cl τ ( M )) referred to as the fiber-wise Dirac element . It is represented by a family of Dirac operators acting on the fibres T M , i.e the pair (cid:0) ( L ( T x M ) ⊗ Cl τ x ) x ∈ M , ( F x = D x ( + D x ) − / ) x ∈ M (cid:1) , where for each x ∈ M , D x is a Dirac operator acting on thefiber T x M . Its inverse can be described explicitely via a family of fiberwise Bott elements. The element [ d ξ ] ⊗ C ( M ) Cl Γ ( M ) ∈ R KK G ( M ; Cl Γ ( T M ) , Cl τ ⊕ Γ ( M ))) then implements a KK-equivalence between Cl Γ ( T M ) and Cl τ ⊕ Γ ( M ) . Motivated by the fact that the Dirac elementon T M induced by the Dolbeault operator splits as a product between [ d ξ ] and the Dirac element induced by thede Rham operator, Kasparov makes the following definiton. Definition 1.9.
The transverse Dolbeault element is the class [ ∂ cl T M , Γ ] obtained as the KK-product [ ∂ cl T M , Γ ] = j G ([ d ξ ] b ⊗ M Cl Γ ( M ) ) b ⊗ G ⋉ Cl τ ⊕ Γ ( M ) [ d M , Γ ] ∈ KK ( G ⋉ Cl Γ ( T M ) , C ) .We can now state Kasparov’s index theorem. Theorem 1.10. (Kasparov, [
6, Theorem 8.18 ] ) Let ( M , g ) be a complete Riemannian manifold (without boundary) equipped with a proper and isometric action of aLie group G. Let A be a transversally elliptic operator of order on M, with symbol σ A . Then [ A ] = j G [ σ tcl A ] ⊗ G ⋉ Cl Γ ( T M ) [ ∂ cl T M , Γ ] ∈ K ( G ⋉ C ( M )) , where j G denotes the descent map. It is not hard to show that in the setup of the classical index theorem (i.e G is the trivial group), the right-hand-sideis exactly the topological index of Atiyah–Singer [ ] . In general, we can therefore refer to it as a topologicaltransverse index .When σ A is transversally elliptic in Atiyah’s sense, note that [ σ tcl A ] can also be seen as a class in K ( Cl Γ ( T M )) ;the next section discusses this point further together with the compatibility between Atiyah’s and Kasparov’s ap-proaches.1.5. Relationship to Atiyah’s index and reduction to compact manifolds.
Atiyah [ ] defines the transverseindex more generally for any element α M ∈ K G ( T G M ) where M is a not-necessarily compact G -manifold. Theconstruction proceeds as follows. Atiyah proves [
1, Lemma 3.6 ] that one can find a Z -graded Hermitian vectorbundle E = E ⊕ E on M and σ M ∈ C b ( T M , π ∗ T M
End ( E )) an odd, self-adjoint bundle endomorphism whoserestriction to T G M represents the class α , and such that one has σ M = π − T M ( K ) for a G -invariant compactsubset K of M . Choose a Hermitian vector bundle F → M such that e E = E ⊕ F is trivial, and fix a trivialization.Let e E = E ⊕ F and e σ M = σ M ⊕ id F . Via e σ M we obtain a trivialization of ( E ⊕ F ) | M \ K . Choose a relatively compact G -invariant open neighborhood U of K , and let ι U , M : U , → M be the inclusion; we will use the same symbol for theinduced open inclusion T G U , → T G M . The pair ( e E | U , e σ M | U ) represents a class α U ∈ K G ( T G U ) and α M = ( ι U , M ) ∗ α U by construction. Choose a G -equivariant open embedding ι U , X of U into a compact G -manifold X ; again we usethe same symbol for the induced open inclusion T G U , → T G X . Using the trivializations over U \ K , the bundle e E | U and endomorphism e σ M | U can be extended trivially to X (denoted e E X , e σ X respectively) and represent the class α X = ( ι U , X ) ∗ α U ∈ K G ( T G X ) . Atiyah definesIndex G ( α M ) = Index G ( A X ) ∈ b R ( G ) where A X is any transversally elliptic operator on X such that the (naive) K-theory class of its symbol is α X . Atiyahproves an excision property [
1, Theorem 3.7 ] showing that the index can be determined just from data on U , andhence the construction is independent of the various choices. KK-THEORETIC PERSPECTIVE ON QUANTIZATION COMMUTES WITH REDUCTION 7
In Kasparov’s framework, this construction can be reformulated as follows. Suppose that one manages tochoose σ M such that, in addition to the conditions above, one has ( − σ M ) ∈ S Γ ( M ) . Then σ M determinesa class [ σ M ,c ] = [( S Γ ( E ) , σ M )] ∈ KK G ( C , S Γ ( M )) refining the class α M . The subscript ‘c’ is to emphasizethat this is a K-theory class whose support is compact over M , in contrast with the symbols defining elementsof the group R KK G ( M ; C ( M ) , S Γ ( M )) that were considered in Section 1.3. One then obtains similar classes [ e σ U ,c ] = [( S Γ ( e E | U ) , e σ | U )] ∈ KK G ( C , S Γ ( U )) refining α U , [ e σ X ,c ] ∈ [( S Γ ( e E X ) , e σ X )] ∈ KK G ( C , S Γ ( X )) refining α X ,and moreover [ e σ X ,c ] = ( ι U , X ) ∗ [ e σ U ,c ] , [ σ M ,c ] = ( ι U , M ) ∗ [ e σ U ,c ] . (1)Let [ e σ tcl X ,c ] , [ e σ tcl U ,c ] , [ σ tcl M ,c ] be the corresponding tangential Clifford symbols obtained by KK-product with f X , Γ , f U , Γ , f M , Γ respectively. Functoriality of the classes f − , Γ under open embeddings implies the tangential Clifford symbolssatisfy analogous formulas.Let p : X → pt be the collapse map, and [ σ A , X ] ∈ R KK G ( X ; C ( X ) , S Γ ( X )) the class defined by the symbol of A X , sothat p ∗ [ σ A , X ] = [ e σ X ,c ] . By Theorem 1.10,Index ( A X ) = p ∗ [ A X ] = j G ([ e σ tcl X ,c ]) b ⊗ G ⋉ Cl Γ ( T X ) [ ∂ cl T X , Γ ] .Equation (1), as well as the functoriality of the KK-product and of the transverse Dolbeault class, give the equivalentformulaIndex ( A X ) = j G ([ e σ tcl U ,c ]) b ⊗ G ⋉ Cl Γ ( T U ) [ ∂ T U , Γ ] = j G ([ σ tcl M ,c ]) b ⊗ G ⋉ Cl Γ ( T M ) [ ∂ cl T M , Γ ] .We thus obtain the following formula for the index in Atiyah’s sense of α M = ι ∗ T G M [ σ M ,c ] ∈ K G ( T G M ) :Index ( ι ∗ T G M [ σ M ,c ]) = j G ([ σ tcl M ,c ]) b ⊗ G ⋉ Cl Γ ( T M ) [ ∂ cl T M , Γ ] .2. L OCALIZATION
We begin with a summary of Paradan–Vergne’s setup, following closely their work [ ] apart from its index the-oretic content, which is treated via Kasparov’s formalism and KK-theory. The section ends with a KK-theoreticalproof of their non-abelian localization formula and a discussion explaining why this point of view unifies differentapproaches to the quantization commutes with reduction problem.2.1. Moment map and excision.
As in previous section, we work with a complete Riemannian manifold ( M , g ) equipped with an action of a compact Lie group G . Moreover, we shall identify g and its dual g ∗ via a G -invariantproduct on g when appropriate.Consider in addition that we have a smooth G -equivariant map ν : M → g ≃ g ∗ such that the zero set Z ν of the (orbital) vector field: ν ( x ) = ρ x ( ν ( x )) = dd t (cid:12)(cid:12)(cid:12)(cid:12) t = exp ( t ν ( x )) · x is compact . As is customary when dealing with the quantization commutes with reduction problem, we supposethat 0 is a regular value of ν .We now use the vector field ν to perturb the symbol of the Dirac operator. Definition 2.1.
Let σ ν ∈ C ∞ ( T ∗ M , π ∗ T ∗ M End ( E )) be the symbol defined by σ ν ( x , ξ ) = i c ( ξ + ν ) ; ∀ ( x , ξ ) ∈ T ∗ M RUDY RODSPHON
This is a transversally elliptic symbol, as the set on which σ ν fails to be invertible is precisely the compact set Z ν .Choose a relatively compact neighborhood U of Z ν . Without loss of generality, we can suppose that | ν | = U . Then, the vector field ν induces a K-theory class [ ν ] : = [( Cl Γ ( M ) , c ( ν )] ∈ K ( Cl Γ ( M )) In [ ] , it is proved that the transverse indexIndex G [ σ ν ] ∈ b R ( G ) of [ σ ν ] ∈ K ( Cl Γ ( T M )) is provided by the KK-product:Index G [ σ ν ] = j G [ ν ] ⊗ Cl G ⋉ Γ ( M ) [ D M , Γ ] = j G [ σ ν ] ⊗ G ⋉ Cl Γ ( T M ) [ ∂ cl T M , Γ ] ∈ b R ( G ) Denoting ι U , M : Cl Γ ( U ) → Cl Γ ( M ) the natural extension-by-0 homomorphism, one sees without difficulty that ( ι U , M ) ∗ [ ν | U ] = [ ν ] , where [ ν | U ] ∈ K ( Cl Γ ( U )) is the class defined similarly to [ ν ] from the restriction ν | U of ν to U . Associativity of the KK-product together with the formula above applied to the manifold U therefore yield thefollowing excision / localization result: Proposition 2.2.
Let U be a relatively compact open neighborhood of the zero set Z ν . Then, Index G [ σ ν ] = [ ν | U ] ⊗ Cl G ⋉ Γ ( U ) [ D U , Γ ] = j G [ σ tcl ν (cid:12)(cid:12) U ] ⊗ G ⋉ Cl Γ ( T U ) [ ∂ cl T U , Γ ] ∈ b R ( G ) (2)(The rightmost side is Kasparov’s formula applied to the manifold U ). The middle part of the formula is set merelyin view of the final discussion of the paper, but is not necessary in the argument above: the relationship betweenthe leftmost and rightmost sides can in fact be derived directly as in Section 1.5.2.2. Coadjoint orbits and slices.
Next, we describe, following Paradan–Vergne, a neighborhood of the zero set Z ν in which the Equation (2) becomes a fixed-point formula. For that purpose, we need an additional piece ofstructure on the smooth map ν . It is a weakening of the notion of Hamiltonian action which requires neither the2-form of M to be symplectic, nor non-degenerate. Definition 2.3.
We say that the smooth G -equivariant map ν is a moment map if there exists a G -invariant closed2-form ω on M such thatint ( β ) ω + d 〈 ν , β 〉 = ∀ β ∈ g where β ( x ) = ddt (cid:12)(cid:12) t = e − t β · x is the vector field on M generated by β ∈ g , and int ( β ) ω denotes the contraction ofthe differential form ω by the vector field β .Here is an example of moment map to have in mind, because of its relevance to the quantization commutes withreduction problem. Example 2.4.
Let L be a G -equivariant line bundle endowed with a connection ∇ , and let ω = − i R , where R isthe curvature 2-form of ∇ . The Lie derivative L ν acting on smooth sections of L is defined by ( L β ϕ )( x ) = dd t (cid:12)(cid:12)(cid:12)(cid:12) t = e t β · φ ( e − t β x ) .Then, the smooth map ν : M → g ∗ defined by the equation 〈 ν , β 〉 = ∇ β − L β , ∀ β ∈ g is a moment map.Under the additional assumption that ν is a moment map, one observes that the compact zero set ν ( Z ν ) ⊂ g ∗ is afinite union of coadjoint orbits. Then, let B be a finite set consisting of representatives of those coadjoint orbitscovering ν ( Z ν ) . This leads to the following description of Z ν : Z ν = G β ∈B G · (cid:0) M β ∩ ν − ( β ) (cid:1) KK-THEORETIC PERSPECTIVE ON QUANTIZATION COMMUTES WITH REDUCTION 9 where M β = { x ∈ M ; β ( x ) = } is the stabilizer of the infinitesimal action of β ∈ B ⊂ g ∗ . (Recall that β is thevector field β ( x ) = ddt (cid:12)(cid:12) t = e − t β · x ).We now describe, for each component G · (cid:0) M β ∩ ν − ( β ) (cid:1) in the decomposition above, a neighborhood diffeomorphicto a ‘slice’. In what follows, let g β = Lie ( G β ) , where G β ⊂ G is the stabilizer subgroup of β with respect to thecoadjoint action. Proposition 2.5. [
11, Proposition 8.4 ] Let β ∈ B r { } . Then, for any sufficiently small neighborhood W β ⊂ g ∗ β of β in g ∗ β , (i) V β : = ν − ( W β ) is a G β -invariant submanifold of M. (ii) G · V β = ν − ( G · W β ) is diffeomorphic to the slice U β : = G × G β V β .Moreover, if β = is a regular value of ν , then G acts locally freely on the submanifold ν − ( ) . Whence the reduced space M : = ν − ( ) / G is an orbifold, and inherits a natural G -equivariant Clifford modulestructure E . To keep the exposition simple, let us suppose that the above action is free , so that M is a manifold.In the noncommutative geometric viewpoint adopted, this is a very minor technical point and does not makemuch difference.At this point, Equation (2) of Proposition 2.2 readsIndex G [ σ ν ] = Index G [ σ ν | U ] + X β = β ∈B j G [ σ tcl ν (cid:12)(cid:12) U β ] ⊗ G ⋉ Cl Γ ( T U β ) [ ∂ cl T U β , Γ ] ∈ b R ( G ) (3)where U is a neighborhood of ν − ( ) that we choose small enough to be acted on freely by G .The first term Index G [ σ ν | U ] : = j G [ σ tcl ν (cid:12)(cid:12) U ] ⊗ G ⋉ Cl Γ ( T U ) [ ∂ cl T U , Γ ] ∈ b R ( G ) represents the transverse index of thesymbol σ ν | U , but since G acts freely on U , the algebras G ⋉ C ( U ) and C ( U / G ) are Morita equivalent, so σ ν | U can be seen as an elliptic symbol on the manifold U / G . As the latter is a principal bundle over M : = ν − ( ) / G ,the usual Thom isomorphism shows that Index G [ σ ν | U ] is equal to the equivariant index of the Dirac operator D on the reduced space M = ν − ( ) / G . In other words: Proposition 2.6.
Let [ D ] ∈ K ( M ) denote the K-homology class of the Dirac operator on the reduced space M .Denote E = E | M the Clifford module bundle on M induced by E. Then, Index G [ σ ν | U ] = X π ∈ b G (cid:0) [ E ⊗ ( ν − ( ) × G V π )] ⊗ C ( M ) [ D ] (cid:1) V π where [ E ⊗ ( ν − ( ) × G V π )] ∈ K ( M ) is the K-theory class of the bundle E ⊗ ( ν − ( ) × G V π ) over M , and ( . ⊗ C ( M ) . ) is the KK-product over C ( M ) ( which is in this situation the index of D twisted by the vector bundle in question ) . In particular, its G -invariant part is the term associated to the trivial representation.2.3. Normal bundles and complex structures.
To deal with the terms of (3) depending on non-zero β ∈ B ⊂ g ∗ , we shall descend to the fixed point sets M β by a slightly more elaborated Thom isomorphism.We will now describe complex structures on the normal bundles N β → M β in U β = G × G β V β allowing us to doso. Note that by the tubular neighborhood theorem, we can identify U β to the total space of the normal bundle N β .Consider a non-zero element β ∈ B ⊂ g ∗ . Because of the decomposition T M | M β = T M β ⊕ N β ,the infinitesimal action L β on vector fields on M induces a fiberwise linear isomorphism L β ∈ End ( N β ) . It isskew-symmetric (with respect to the metric g ), hence diagonalizable with "imaginary positive" eigenvalues. Wecan therefore make the following definition: Definition 2.7.
We define the complex structure J β on N β as the operator J β = L β |L β | − , where |L β | − =( −L β ) / .A similar construction applies to Y β : = ( V β ) β = M β ∩ V β in M β (which is open in M β ), so the normal bundle η β relative to the embedding ι Y β , V β : Y β , → V β comes equipped with a complex structure.We will also need another construction of the same type the quotient space g / g β . Recall that g and g ∗ are identifiedwith a G -invariant inner product. Via this identification, the endomorphism of g induced by the adjoint action of β yields a linear skew-symmetric isomorphism on g / g β . Hence we can make the following definition. Definition 2.8.
The linear map J β = β | β | − provides a complex structure on the vector space g / g β .2.4. KK-theory and the non-abelian localization formula.
We are now ready to re-interpret Paradan–Vergne’scalculation of Index G [ σ ν ] in KK-theory. We shall decompose the terms j G [ σ tcl ν (cid:12)(cid:12) U β ] ⊗ G ⋉ Cl Γ ( T U β ) [ ∂ cl T U β , Γ ] with non-zero elements β ∈ B ⊂ g ∗ in Equation (3), appealing to appropriate Thom isomorphisms.Recall that U β : = G · V β is the slice diffeomorphic to G × G β V β , with V β being the G β -invariant submanifold ν − ( W β ) in M obtained from a small neighborhood W β of β in the infinitesimal stabilizer g β . In addition, consider the fixedpoint sets M β = { x ∈ M ; β ( x ) = } and Y β = M β ∩ V β , which is a open neighborhood of M β ∩ ν − ( β ) in M β . Proposition 2.9.
For a non-zero β ∈ B ⊂ g ∗ , let G β ⊂ G be the stabilizer subgroup of β for the coadjoint action ofG on g ∗ . Then, j G [ σ tcl ν (cid:12)(cid:12) U β ] ⊗ G ⋉ Cl Γ ( T U β ) [ ∂ cl T U β , Γ ] is the transverse index of the symbol Ind GG β [ Λ • C ( g / g β )] ⊠ [ σ tcl ν (cid:12)(cid:12) Y β ] P k ( − ) k [ Λ k N β ] ∈ K (cid:0) Cl Γ ( T ( G × G β Y β )) (cid:1) = K ( Cl Γ ( T U β )) , where ⊠ denotes the external product ; [ σ tcl ν (cid:12)(cid:12) Y β ] ∈ K ( Cl Γ ( T Y β )) is the symbol class associated to the restriction of σ ν to Y β ; Ind GG β denotes Atiyah’s induction functor, and N β is the normal bundle of M β in M.Proof. Let Γ β denote the orbital field in V β generated by the action of G β . By Bott periodicity, we have an isomor-phism of K-theory groups: K G β ( Cl Γ β ( T V β )) −→ K G β (cid:0) Cl Γ β ( T V β ) ⊗ C ( g / g β ) ⊗ Cliff ( g / g β ) (cid:1) = K G β (cid:0) C ( T V β × g / g β ) ⊗ C ( V β ) Cl Γ β ⊕ g / g β ( V β ) (cid:1) provided as usual by external multiplication with [ Λ • ( g / g β )] ∈ K G β (cid:0) C ( g / g β ) ⊗ Cliff ( g / g β ) (cid:1) , where Λ • ( g / g β ) isthe exterior algebra of the quotient space g / g β , which has a natural complex structure (cf. Definition 2.8).Next, consider Kasparov’s induction functor Ind GG β : K G β (cid:0) C ( T V β × g / g β ) ⊗ C ( V β ) Cl Γ β ⊕ g / g β ( V β ) (cid:1) −→ K K G (cid:0) C ( G / G β ) , Cl Γ ( T U β ) (cid:1) where the second entry in the right-hand side is obtained from the fact that the tangent bundle of the slice U β is given by T ( G × G β V β ) = G × G β ( T V β × g / g β ) . Then, observe that G / G β = T G ( G / G β ) , and recall that the zerooperator on G / G β is G -transversally elliptic. The composition of the two homomorphisms above with the (right)KK-product by [ ] ∈ K ( T ∗ G ( G / G β )) (that we shall absorb in the induction morphism to avoid heavier notations)therefore yields a homomorphism: K G β ( Cl Γ β ( T V β )) −→ K G ( Cl Γ ( T U β )) KK-THEORETIC PERSPECTIVE ON QUANTIZATION COMMUTES WITH REDUCTION 11
This construction was already considered by Atiyah, [
1, Section 4 ] , who also proves that it is an isomorphism(this can be seen directly via the construction and properties of Kasparov’s induction functor). Following Atiyah,we still denote this homomorphism Ind GG β (this is the one considered in the statement of the proposition).On the other hand, [ σ tcl ν (cid:12)(cid:12) U β ] ∈ K G ( Cl Γ ( T U β )) is the image of [ σ tcl ν (cid:12)(cid:12) V β ] ∈ K G β ( Cl Γ ( T V β )) by this isomorphism, i.e. [ σ tcl ν (cid:12)(cid:12) U β ] = Ind GG β (cid:0) [ Λ • ( g / g β )] ⊠ [ σ tcl ν (cid:12)(cid:12) V β ] (cid:1) In addition, the symbol class [ σ tcl ν (cid:12)(cid:12) V β ] ∈ K G β ( Cl Γ ( T V β )) can be decomposed further by descending to the fixedpoint set M β . To this end, let Y β : = ( V β ) β = { x ∈ V β ; β ( x ) = } = M β ∩ V β which is open in M β . Let η β be the normal bundle relative to the embedding ι Y β , V β : Y β , → V β and identify V β withits total space. We endow η β with the complex structure described in Section 2.3. Lemma 2.10.
One has a KK-equivalence Cl Γ β ( T Y β ) ∼ Cl Γ β ( T V β ) provided by a "Thom" element: [ T β ] : = [ T β ] ⊗ Cl Γ β ( Y β ) ∈ KK G β (cid:0) Cl Γ β ( T Y β ) , Cl Γ β ( T V β ) (cid:1) where [ T β ] ∈ KK G β (cid:0) C ( T Y β ) , C ( T V β ) (cid:1) is the standard Thom element that implements the KK-equivalence C ( T Y β ) ∼ C ( T V β ) .Proof. (of Lemma 2.10) This is simply a consequence of the local decompositionCl Γ β ( V β ) = C ( V β ) ⊗ C ( Y β ) Cl Γ β ( Y β ) combined with the usual Thom isomorphism C ( V β ) ∼ C ( Y β ) . (cid:3) Tensoring both sides of the KK-equivalence in the lemma by C ( g / g β ) ⊗ Cliff ( g / g β ) and combining with the afor-mentioned Bott periodicity isomorphism yields another a KK-equivalence C ( T Y β × g / g β ) ⊗ C ( Y β ) Cl Γ β ⊕ g / g β ( Y β ) ∼ C ( T V β × g / g β ) ⊗ C ( V β ) Cl Γ β ⊕ g / g β ( V β ) .implemented by a Thom element [ T β ] ∈ KK G β (cid:0) C ( T Y β × g / g β ) ⊗ C ( Y β ) Cl Γ β ⊕ g / g β ( Y β ) , C ( T V β × g / g β ) ⊗ C ( V β ) Cl Γ β ⊕ g / g β ( V β ) (cid:1) .Since Kasparov’s induction functor preserves KK-equivalences, we haveCl Γ ( T ( G × G β Y β )) ∼ Cl Γ ( T U β ) induced by Ind GG β [ T β ] ∈ KK G ( Cl Γ ( T ( G × G β Y β ) , Cl Γ ( T U β )) , which is also a Thom element (this can be seen via theproperties of the induction functor).After these preparations, we may now write (we remove the algebra in subscript of the KK-products to alleviatethe notations): j G [ σ tcl ν (cid:12)(cid:12) U β ] ⊗ [ ∂ cl T U β , Γ ] = j G Ind GG β (cid:0) [ Λ • ( g / g β )] ⊠ ([ σ tcl ν (cid:12)(cid:12) V β ]) ⊗ [ T β ] − ) (cid:1) ⊗ j G Ind GG β [ T β ] ⊗ [ ∂ cl T U β , Γ ] | {z } Dolbeault element [ ∂ cl T ( G × G β Y β ) , Γ ] To finish, the embedding ι Y β , V β induces the homomorphism ι ∗ Y β , V β : K ( Cl Γ ( T V β )) → K ( Cl Γ ( T Y β )) , and a straight-forward adaptation of a classical formula of Atiyah–Singer gives [ T β ] ⊗ Cl Γ ( T V β ) ι ∗ Y β , V β = P k ( − ) k [ Λ k η β ] Hence, j G [ σ tcl ν (cid:12)(cid:12) U β ] ⊗ [ ∂ cl T U β , Γ ] is the transverse index of the symbolInd GG β [ Λ • ( g / g β )] ⊠ [ σ | Y β ] P k ( − ) k [ Λ k η β ] ∈ K (cid:0) Cl Γ ( T ( G × G β Y β ) (cid:1) .Noticing that the normal bundle N β relative to the embedding Y β , → U β decomposes as N β = η β ⊕ ( g / g β × Y β ) finishes the proof. (cid:3) As a conclusion, we obtain the non-abelian localization formula of Paradan and Vergne:
Theorem 2.11.
Let ( M , g ) be a complete Riemannian manifold equipped with the action of a compact Lie group G, letE be a G-equivariant Clifford module bundle over M and let ν : M → g ∗ be a moment map in the sense of Definition2.3 . According to the decomposition of the zero setZ ν = G β ∈B G · (cid:0) M β ∩ ν − ( β ) (cid:1) of the vector field ν associated to ν , where B ⊂ g ∗ is a finite set consisting of representatives of the coadjoint orbitswithin ν ( Z ν ) , the transverse index of the symbol σ ν ( x , ξ ) = i c ( ξ + ν ) ∈ C ∞ ( T ∗ M , π ∗ T ∗ M End ( E )) is given by the fixedpoint formula: Index G [ σ ν ] = X π ∈ b G (cid:0) [ E ⊗ ( ν − ( ) × G V π )] ⊗ C ( M ) [ D ] (cid:1) V π + X β = β ∈B Index G β [ Λ • C ( g / g β )] ⊠ [ σ ν | Y β ] P k ( − ) k [ Λ k N β ] where E is the Clifford module bundle over the reduced space M = ν − ( ) / G induced by E ; [ E ⊗ ( ν − ( ) × G V π )] isthe K-theory class of the vector bundle E ⊗ ( ν − ( ) × G V π ) → M ; G β ⊂ G denotes the stabilizer subgroup of β ∈ B relative to the coadjoint action of G on g ∗ ; g β = Lie ( G β ) ; Y β is a small open neighborhood of M β ∩ ν − ( β ) insidethe fixed point set M β ; [ σ ν ] ∈ K G ( T ∗ G M ) ; [ σ ν | Y β ] ∈ K G β ( T G β Y β ) denote respectively the symbol classes of σ ν and ofits restriction to T ∗ Y β ; and N β is the normal bundle of M β in M. The quantization commutes with reduction is subsequently proved without much difficulty with a separate cal-culation by hand, showing that only the term associated to the trivial representation in the first sum contributeswhen extracting the G -invariant part, cf. [ ] .3. C ONCLUDING COMMENTS
Comparison to Paradan–Vergne’s work.
Paradan–Vergne’s proof of Theorem 2.11 is actually quite KK-theoretical in spirit, but relies the K-theoretical machinery of Atiyah–Singer [ ] . Let us just point out mentionsome improvements our approach brings.Applying this theorem to the case of a toral action of M = C n with moment map ν ( z ) = ( | z | /
2, . . . , | z n | / ) ,one recovers directly the results of Atiyah [
1, Theorem 6.6 ] , in contrast to Paradan–Vergne’s work which uses thisresult (which is quite non-trivial!) as an intermediary step in the construction of their Thom isomorphism (thatthey refer to as push-forward). As a byproduct, we can avoid the technical constructions from [
11, Sections 4 and6.3 ] utilized to this end, which are meant to overcome the non-availibility of the KK-product in Atiyah–Singer’sframework.By way of comparison, a similar phenomenon already appears when analyzing back to back the original proof ofthe classical theorem [ ] via the ‘index axioms’ (and notably the one on multiplicativity) and the KK-theoreticalproof: the former suffers from complications which are subsequently removed after the introduction of the KK-product. KK-THEORETIC PERSPECTIVE ON QUANTIZATION COMMUTES WITH REDUCTION 13
Comparison with the analytic approach. In [ ] , it is shown that the class j G [ ν ] ⊗ [ D ] mentioned briefly inSection 2.1 corresponds to the K-homology class [ D f ν ] ∈ K ( C ∗ ( G )) of a deformed Dirac operator of the form D f ν = D + i f c ( ν ) where f is a smooth G -invariant positive function satisfying a certain growth at infinity.The crux of Ma–Tian–Zhang’s solutions [
12, 8 ] to quantization commutes with reduction is that the localizationthe index of D f ν around the zero set ν − ( ) (and not only Z ν as in Proposition 2.2) can be performed directly atthe analytic level, at the cost of very technical estimates on the square of D f ν .Hochs–Song’s methods [ ] blend the analytic approach within Paradan–Vergne’s framework: the idea is also tostart from the decomposition of the zero set Z ν = F G · ( M β ∩ ν − ( β )) . Then, one analyzes the quantities j G [ σ tcl ν (cid:12)(cid:12) U β ] ⊗ G ⋉ Cl Γ ( T U β ) [ ∂ cl T U β , Γ ] = j G [ ν | U β ] ⊗ [ D | U β ] directly from the defomed Dirac operator by establishing the Ma–Tian–Zhang estimates in slices around thedifferent components of Z ν .The present article combined with [ ] hopefully highlights that these approaches are similar up to a KK-theoreticalPoincaré duality. To make this analogy more exact, it might be interesting (or not) to see whether working out theanalytic approach within the K-homological side j G [ ν ] ⊗ [ D ] leads to simplifications parallel to the calculationsmade in previous subsection. R EFERENCES [ ] M.F. Atiyah,
Elliptic operators and compact groups , Lecture Notes in Mathematics, Springer, 1974. [ ] M.F. Atiyah, I. Singer, The index of elliptic operators on compact manifolds.
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Invent. Math. , 67(3):515–538, 1982. [ ] P. Hochs, Y. Song, Equivariant indices of Spin c -Dirac operators for proper moment maps. Duke Math. J. (2017), no. 6, 1125–1178. [ ] G. Kasparov. Elliptic and transversally elliptic index theory from the viewpoint of KK-theory.
J. Noncommut. Geom. , 10(4):1303–1378,2016. [ ] Y. Loizides, R. Rodsphon, and Y. Song. A KK-theoretic perspective on deformed dirac operators. arXiv:1907.06150 , 2019. [ ] X. Ma and W. Zhang. Geometric quantization for proper moment maps: the Vergne conjecture.
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