Rigidity and Vanishing Theorems on Z/k Spin c manifolds
aa r X i v : . [ m a t h . DG ] A p r Rigidity and Vanishing Theoremson Z /k Spin c manifolds Bo LIU ∗ and Jianqing YU † Abstract
In this paper, we first establish an S -equivariant index theorem forSpin c Dirac operators on Z /k manifolds, then combining with the methodsdeveloped by Taubes [24] and Liu-Ma-Zhang [19, 20], we extend Witten’srigidity theorem to the case of Z /k Spin c manifolds. Among others, ourresults resolve a conjecture of Devoto [6]. In [25], Witten derived a series of elliptic operators on the free loop space L M of a spin manifold M . In particular, the index of the formal signatureoperator on loop space turns out to be exactly the elliptic genus constructedby Landweber-Stong [13] and Ochanine [23] in a topological way. Motivated byphysics, Witten conjectured that these elliptic operators should be rigid withrespect to the circle action.This conjecture was first proved by Taubes [24] and Bott-Taubes [4]. See also[10] and [12] for other interesting cases. By the modular invariance property,Liu ([15, 16]) presented a simple and unified proof of the above conjecture aswell as various further generalizations. In particular, several new vanishingtheorems were established in [15, 16]. Furthermore, on the equivariant Cherncharacter level, Liu and Ma ([17, 18]) generalized Witten’s rigidity theorem tothe family case, and also obtained several vanishing theorems for elliptic genera.In [19, 20], inspired by [24], Liu, Ma and Zhang established the correspondingfamily rigidity and vanishing theorems on the equivariant K -theory level.In [27], Zhang established an equivariant index theorem for circle actionson Z /k spin manifolds and pointed out that by combining with the analyticarguments developed in [20], one can prove an extension of Witten’s rigiditytheorem to Z /k spin manifolds. The purpose of this paper is to extend the re-sult of [27] to Z /k Spin c manifolds and then establish Witten’s rigidity theoremfor Z /k Spin c manifolds. Recall that a Z /k manifold X is a smooth manifoldwith boundary ∂X which consists of k disjoint pieces, each of which is diffeo-morphic to a given closed manifold Y (cf. [22]). It is interesting that for a ∗ Chern Institute of Mathematics & LPMC, Nankai University, Tianjin 300071, P. R. China.([email protected]) † Chern Institute of Mathematics & LPMC, Nankai University, Tianjin 300071, P. R. China.([email protected]) D on a Z /k manifold, the APS-ind( D ) mod k Z determines atopological invariant in Z /k Z , where APS-ind( D ) is the index of D which isimposed the boundary condition of Atiyah-Patodi-Singer type [1]. Freed andMelrose [7] proved a mod k index theorem,APS-ind( D ) mod k Z = t-ind( D ) , (1.1)giving the APS-ind( D ) mod k Z a purely topological interpretation.Assume that X is a Z /k manifold which admits a Z /k circle action (cf.Section 2.2). Let D be a Dirac operator on X which commutes with the cir-cle action. Let R ( S ) denote the representation ring of S . The equivarianttopological index of D is defined by Freed and Melrose [7] as an element of Z /k Z ⊗ R ( S ), and we denote it by t-ind S ( D ). Then there exist R n ∈ Z /k Z such that t-ind S ( D ) = X n ∈ Z R n ⊗ [ n ] , (1.2)where by [ n ] ( n ∈ Z ) we mean the one dimensional complex vector space onwhich S acts as multiplication by g n for a generator g ∈ S .On the other hand, by applying the equivariant index theorem for Z /k manifolds established by Freed and Melrose in [7], one gets for n ∈ Z , R n = APS-ind( D, n ) mod k Z . (1.3)See (2.9) for the definition of APS-ind( D, n ).The Dirac operator D on X is said to be rigid in Z /k category for the circleaction if its equivariant topological index t-ind S ( D ) verifies that for n ∈ Z , n = 0, one has R n = 0 in Z /k Z . (1.4)Furthermore, we say D has vanishing property in Z /k category if its equivarianttopological index t-ind S ( D ) is identically zero, i.e., (1.4) holds for any n ∈ Z .In [6], Devoto introduced what he called mod k elliptic genus for Z /k spinmanifolds as an S -equivariant topological index in the sense of [7] of sometwisted Dirac operator and conjectured that this mod k elliptic genus is rigidin Z /k category . In this paper, following the suggestion in [27, Remark 1], wepresent a proof of Devoto’s conjecture. Moreover, we establish our results for Z /k Spin c manifolds, thus generalizing [16, Theorems A and B] to the case of Z /k Spin c manifolds.Our proof of these rigidity results consists of two steps. In step 1 (Sections2 and 3), we extend the Z /k equivariant index theorem of Zhang [27] to theSpin c case. In step 2 (Sections 4 and 5), using the mod k localization indextheorem established in step 1 and modifying the process in [19, 20], we provethe main results of this paper.This paper is organized as follows. In Section 2, we state an S -equivariantindex theorem for Spin c Dirac operators on Z /k manifolds (cf. Theorem 2.7).As an application, we extend Hattori’s vanishing theorem [8] to the case of Z /k almost complex manifolds. In Section 3, we prove the S -equivariant indextheorem stated in Section 2. In Section 4, we prove our main results (cf. The-orem 4.1), the rigidity and vanishing theorems for Z /k Spin c manifolds, which2eneralize [16, Theorems A and B]. When applied to Z /k spin manifolds, ourresults resolve a conjecture of Devoto [6]. Section 5 is devoted to a proof of therecursive formula which has been used in Section 4 in the proof of our mainresults. c Dirac operators and a mod k localization for-mula In this section, for a Z /k manifold which admits a nontrivial Z /k circleaction, we state a mod k localization formula for S -equivariant Spin c Diracoperators, whose proof will be given in Section 3. As an application, we deducethe rigidity and vanishing property for several Dirac operators on a Z /k almostcomplex manifold. In particular, we extend Hattori’s vanishing theorem [8] tothe case of Z /k almost complex manifolds.This section is organized as follows. In Section 2.1, we review the construc-tion of Spin c Dirac operators on Z /k manifolds and the Atiyah-Patodi-Singerboundary problems. In Section 2.2, we recall the circle actions on Z /k manifoldsand present a variation formula for the indices of these boundary problems. InSection 2.3, we state the mod k localization formula for Z /k circle actions. Asan application, in Section 2.4, we extend Hattori’s vanishing theorem [8] to thecase of Z /k almost complex manifolds. c Dirac operators on Z /k manifolds We first recall the definition of Z /k manifolds introduced by Morgan andSullivan (cf. [22]). Definition 2.1 (cf. [27, Definition 1.1])
A compact Z /k manifold is a compactmanifold X with boundary ∂X , which admits a decomposition ∂X = ∪ ki =1 ( ∂X ) i into k disjoint manifolds and k diffeomorphisms π i : ( ∂X ) i → Y to a closedmanifold Y . Let π : ∂X → Y be the induced map. In what follows, as in [27], we willcall an object α (e.g., metrics, connections, etc.) of X a Z /k -object if there willbe a corresponding object β on Y such that α | ∂X = π ∗ β .We point out here that in this paper when consider the topological objects(e.g., cohomology, characteristic classes, K group, etc.) on a Z /k manifold X , we always regard X as a quotient space obtained by identifying each ofthe k disjoint pieces of the boundary ∂X . Then X has the homotopy typeof a CW complex, which implies that the first Chern class c induces a 1-to-1 correspondence between the equivalence classes of the complex line bundlesover X and the elements of H ( X ; Z ). As will be seen, this is essential in ourproof.We make the assumption that X is Z /k oriented and of dimension 2 l .Let V be a Z /k real vector bundle over X which is of dimension 2 p and Z /k oriented. Let L be a Z /k complex line bundle over X with the property thatthe vector bundle U = T X ⊕ V satisfies ω ( U ) = c ( L ) mod (2), where ω c denotes the first Chern class.Then the Z /k vector bundle U has a Z /k Spin c -structure.Let g T X be a Z /k Riemannian metric on X . Let g T ∂X be its restrictionon
T ∂X . Let ǫ > g T X . We use theinward geodesic flow to identify a neighborhood of the boundary with the collar[0 , ǫ ) × ∂X . We assume that g T X is of product structure near ∂X . That is,there is an open neighborhood U ǫ = [0 , ǫ ) × ∂X of ∂X in X with 0 < ǫ ≤ ǫ such that one has the orthogonal splitting on U ǫ , g T X | U ǫ = dr ⊕ π ∗ ǫ g T ∂X , (2.1)where π ǫ : [0 , ǫ ) × ∂X → ∂X is the obvious projection onto the second factor.Let ∇ T X be the Levi-Civita connection on (
T X, g
T X ). Then ∇ T X is a Z /k connection.Let W be a Z /k complex vector bundle over X with a Z /k Hermitian metric g W . Let ∇ W be a Z /k Hermitian connection on W with respect to g W . Wemake the assumption that g W and ∇ W are both of product structure near ∂X .That is, over the open neighborhood U ǫ of ∂X , one has g W | U ǫ = π ∗ ǫ ( g W | ∂X ) , and ∇ W | U ǫ = π ∗ ǫ ( ∇ W | ∂X ) . (2.2)Let g V (resp. g L ) be a Z /k Euclidean (resp. Hermitian) metric on V (resp. L ), and ∇ V (resp. ∇ L ) be a corresponding Z /k Euclidean (resp. Hermitian)connection on V (resp. L ). We make the assumption that g V , ∇ V , g L , ∇ L areof product structure near ∂X (cf. (2.2)).By taking ǫ > g T X , g W , g V , g L and the connections ∇ W , ∇ V , ∇ L verifying the above assumptions.The Clifford algebra bundle C ( T X ) is the bundle of Clifford algebras over X whose fibre at x ∈ X is the Clifford algebra C ( T x X ) (cf. [14]). Let C ( V ) bethe Clifford algebra bundle of ( V, g V ).Let S ( U, L ) be the fundamental complex spinor bundle for (
U, L ) (cf. [14,Appendix D]). We denote by c ( · ) the Clifford action of C ( T X ), C ( V ) on S ( U, L ).Let { e i } li =1 (resp. { f j } pj =1 ) be an oriented orthonormal basis of ( T X, g
T X ) (resp.(
V, g V )). There are two canonical ways to consider S ( U, L ) as a Z -graded vectorbundle. Let τ s = ( √− l c ( e ) · · · c ( e l ) ,τ e = ( √− l + p c ( e ) · · · c ( e l ) c ( f ) · · · c ( f p ) (2.3)be two involutions of S ( U, L ). Then τ s = τ e = 1. We decompose S ( U, L ) = S + ( U, L ) ⊕ S − ( U, L ) corresponding to τ s (resp. τ e ) such that τ s | S ± ( U,L ) = ± τ e | S ± ( U,L ) = ± τ on S ( U, L ),either τ s or τ e , without further notice.Let ∇ S ( U,L ) be the Hermitian connection on S ( U, L ) induced by ∇ T X ⊕ ∇ V and ∇ L (cf. [14, Appendix D]). Then ∇ S ( U,L ) preserves the Z -grading of S ( U, L ). Let ∇ S ( U,L ) ⊗ W be the Hermitian connection on S ( U, L ) ⊗ W obtainedfrom the tensor product of ∇ S ( U,L ) and ∇ W .4 efinition 2.2 The twisted Spin c Dirac operator D X on S ( U, L ) ⊗ W over X is defined by D X = l X i =1 c ( e i ) ∇ S ( U,L ) ⊗ We i : Γ( X, S ( U, L ) ⊗ W ) −→ Γ( X, S ( U, L ) ⊗ W ) . (2.4) Denote by D X ± the restrictions of D X on Γ( X, S ± ( U, L ) ⊗ W ) . By [14], D X is a formally self-adjoint operator. To get an elliptic operator,we impose the boundary condition of Atiyah-Patodi-Singer type [1].We first recall the canonical boundary operators (cf. [5, (1.4)]). For a firstorder differential operator D : Γ( S ( U, L ) ⊗ W ) −→ Γ( S ( U, L ) ⊗ W ) on X , ifthere exists ǫ > U ǫ , D = c (cid:18) ∂∂r (cid:19) (cid:18) ∂∂r + B (cid:19) , (2.5)with B independent of r , then we will call B the canonical boundary operatorassociated to D . When there is no confusion, we will also use B to denote itsrestriction on Γ( X, S ( U, L ) ⊗ W ) | ∂X .We then recall the Atiyah-Patodi-Singer projection associated to a boundaryoperator (cf. [1]). Assume temporarily that B : Γ( X, S ( U, L ) ⊗ W ) | ∂X −→ Γ( X, S ( U, L ) ⊗ W ) | ∂X is a first order formally self-adjoint elliptic differentialoperator on ∂X . For any λ ∈ Spec ( B ), the spectrum of B , let E λ be theeigenspace corresponding to λ . For a ∈ R , let P ≥ a be the orthogonal projectionfrom the L -completion of Γ( X, S ( U, L ) ⊗ W ) | ∂X onto ⊕ λ ≥ a E λ . We call theparticular projection P ≥ the Atiyah-Patodi-Singer projection associated to B to emphasize its role in [1]. If we assume in addition that B preserves the Z -grading of Γ( X, S ( U, L ) ⊗ W ) | ∂X , and let B ± be the restrictions of B onΓ( X, S ± ( U, L ) ⊗ W ) | ∂X , then we will restrict P ≥ a on the L -completions ofΓ( X, S ± ( U, L ) ⊗ W ) | ∂X and denote them by P ≥ a, ± .Let e = ∂∂r be the inward unit normal vector field perpendicular to ∂X . Let e , · · · , e l be an oriented orthonormal basis of T ∂X so that e , e , · · · , e l is anoriented orthonormal basis of T X | ∂X . Then using parallel transport with re-spect to ∇ T X along the unit speed geodesics perpendicular to ∂X , e , e , · · · , e l forms an oriented orthonormal basis of T X over U ǫ . Definition 2.3
Let B X : Γ( X, S ( U, L ) ⊗ W ) | ∂X −→ Γ( X, S ( U, L ) ⊗ W ) | ∂X bethe differential operator on ∂X defined by B X = − l X i =2 c (cid:18) ∂∂r (cid:19) c ( e i ) ∇ S ( U,L ) ⊗ We i . (2.6)By [1], B X is a formally self-adjoint first order elliptic differential operatorintrinsically defined on ∂X , which is the canonical boundary operator associatedto D X and preserves the natural Z -grading of ( S ( U, L ) ⊗ W ) | ∂X .We now recall the Dirac type operator [5, Definition 1.1] as well as theboundary condition of Atiyah-Patodi-Singer type [1].5 efinition 2.4 By a Dirac type operator on S ( U, L ) ⊗ W , we mean a first orderdifferential operator D : Γ( X, S ( U, L ) ⊗ W ) −→ Γ( X, S ( U, L ) ⊗ W ) such that D − D X is an odd self-adjoint element of zeroth order, and that its canonicalboundary operator B acting on Γ( X, S ( U, L ) ⊗ W ) | ∂X is formally self-adjoint.We will also call the restrictions D ± of D to Γ( X, S ± ( U, L ) ⊗ W ) a Dirac typeoperator. Let now D be a Z /k Dirac type operator with its canonical boundary oper-ator B . Obviously, B preserves the Z -grading of Γ( X, S ( U, L ) ⊗ W ) | ∂X ,Following [1], the boundary problem( D + , P ≥ , + ) : (cid:8) s (cid:12)(cid:12) s ∈ Γ( X, S + ( U, L ) ⊗ W ) , P ≥ , + ( s (cid:12)(cid:12) ∂X ) = 0 (cid:9) −→ Γ( X, S − ( U, L ) ⊗ W ) , (2.7)defines an elliptic boundary problem whose adjoint is ( D − , P > , − ). Moreover, itinduces a Fredholm operator [1]. We will call the boundary problem ( D + , P ≥ , + )the Atiyah-Patodi-Singer boundary problem associated to D + . SetAPS-ind( D ) = dim ker( D + , P ≥ , + ) − dim ker( D − , P > , − ) . (2.8) Z /k circle actions and a variation formula Definition 2.5
We will call a circle action on X a Z /k circle action if itpreserves ∂X and there exists a corresponding circle action on Y such that thesetwo actions are compatible with π . The circle action is said to be nontrivial ifit is not equal to identity. In what follows we assume that X admits a nontrivial Z /k circle actionpreserving the orientation and that the Z /k circle action on X lifts to Z /k circle actions on V , L and W , respectively. Without loss of generality, we mayand we will assume that these Z /k circle actions preserve g T X , g V , g L , g W , ∇ V , ∇ L and ∇ W , respectively. We also assume that the Z /k circle actions on T X , V and L lift to a Z /k circle action on S ( U, L ) and preserves its Z -grading.Let E be a Z /k S -equivariant vector bundle over X . Let E Y be the S -equivariant vector bundle over Y induced from E through the map π : ∂X → Y .Recall that the circle action on Γ( X, E ) is defined by ( g · s )( x ) = g ( s ( g − x )) for g ∈ S , s ∈ Γ( X, E ), x ∈ X . Similarly, the group S acts on Γ( X, E ) | ∂X andΓ( Y, E Y ). For ξ ∈ Z , by the weight- ξ subspace of Γ( X, E ) (resp. Γ( X, E ) | ∂X ,Γ( Y, E Y )), we mean the subspace of Γ( X, E ) (resp. Γ( X, E ) | ∂X , Γ( Y, E Y )) onwhich S acts as multiplication by g ξ for g ∈ S .For any ξ ∈ Z , let E ± ξ (resp. E ± ξ,∂ , E ± Y,ξ ) be the weight- ξ subspaces ofΓ( X, S ± ( U, L ) ⊗ W ) (resp. Γ( X, S ± ( U, L ) ⊗ W ) | ∂X , Γ( Y, ( S ( U, L ) ⊗ W ) Y )).Let D be a Z /k S -equivariant Dirac type operator on Γ( S ( U, L ) ⊗ W )with canonical boundary operator B acting on Γ( X, S ( U, L ) ⊗ W ) | ∂X . Let P ≥ , + be the orthogonal projection associated to B + . For ξ ∈ Z , let D ± ,ξ and P ≥ , + ,ξ (resp. P > , − ,ξ ) be the restrictions of D ± and P ≥ , + (resp. P > , − ) on thecorresponding weight- ξ subspaces E ± ξ and E + ξ,∂ (resp. E − ξ,∂ ) respectively. Then( D + ,ξ , P ≥ , + ,ξ ) forms an elliptic boundary problem. SetAPS-ind( D, ξ ) = dim ker( D + ,ξ , P ≥ , + ,ξ ) − dim ker( D − ,ξ , P > , − ,ξ ) . (2.9)6et (cid:8) D t : Γ( X, S ( U, L ) ⊗ W ) −→ Γ( X, S ( U, L ) ⊗ W ) (cid:12)(cid:12) ≤ t ≤ (cid:9) be a oneparameter family of Z /k S -equivariant Dirac type operators with the canonicalboundary operators (cid:8) B t (cid:12)(cid:12) ≤ t ≤ (cid:9) . For any t ∈ [0 , D Yt, + be the inducedoperator from B t, + through the map π : ∂X → Y , and let B t, + ,ξ (resp. D Yt, + ,ξ )be the restriction of B t, + (resp. D Yt, + ) on the weight- ξ subspace E + ξ,∂ (resp.E + Y,ξ ). We have the following variation formula.
Theorem 2.6 (Compare with [5, Theorem 1.2])
The following identity holds,
APS-ind( D , ξ ) − APS-ind( D , ξ ) = − sf (cid:8) B t, + ,ξ (cid:12)(cid:12) ≤ t ≤ (cid:9) = − k sf (cid:8) D Yt, + ,ξ (cid:12)(cid:12) ≤ t ≤ (cid:9) , (2.10) where sf is the notation for the spectral flow of [2] . In particular, APS-ind( D , ξ ) ≡ APS-ind( D , ξ ) mod k Z . Proof
The proof is the same as that of [5, Theorem 1.2]. k localization formula for Z /k circle actions Let H be the canonical basis of Lie( S ) = R , i.e., for t ∈ R , exp( t H ) = e π √− t ∈ S . Let H be the Killing vector field on X corresponding to H .Since the circle action on X is of Z /k , H | ∂X ⊂ T ∂X induces a Killing vectorfield H Y on Y . Let X H (resp. Y H ) be the zero set of H (resp. H Y ) on X (resp. Y ). Then X H is a Z /k manifold and there is a canonical map π X H : ∂X H → Y H induced by π . In general, X H is not connected. We fix a connected component X H,α of X H , and we omit the subscript α if there is no confusion.Clearly, X H intersects with ∂X transversally. Let g T X H be the metric on X H induced by g T X . Then g T X H is naturally of product structure near ∂X H .In fact, by choosing ǫ > U ′ ǫ = U ǫ ∩ X H carries themetric naturally induced from g T X | U ǫ .Let e π : N → X H be the normal bundle to X H in X , which is identified tobe the orthogonal complement of T X H in T X | X H . Then T X | X H admits a Z /kS -equivariant decomposition (cf. [20, (1.8)]) T X | X H = M v =0 N v ⊕ T X H , (2.11)where N v is a Z /k complex vector bundle such that g ∈ S acts on it by g v with v ∈ Z \{ } . We will regard N as a Z /k complex vector bundle and write N R for the underlying real vector bundle of N . Clearly, N = ⊕ v =0 N v . For v = 0,let N v, R denote the underlying real vector bundle of N v .Similarly, let W | X H = M v W v , V | X H = M v =0 V v ⊕ V R (2.12)be the Z /k S -equivariant decompositions of the restrictions of W and V over X H respectively, where W v and V v ( v ∈ Z ) are Z /k complex vector bundles7ver X H on which g ∈ S acts by g v , and V R is the real subbundle of V suchthat S acts as identity. For v = 0, let V v, R denote the underlying real vectorbundle of V v . Denote by 2 p ′ = dim V R and 2 l ′ = dim X H .Let us write L F = L ⊗ O v =0 det N v ⊗ O v =0 det V v − . (2.13)Then T X H ⊕ V R has a Z /k Spin c -structure since ω ( T X H ⊕ V R ) = c ( L F )mod (2). Let S ( T X H ⊕ V R , L F ) be the fundamental spinor bundle for ( T X H ⊕ V R , L F ) as in Section 2.1.Recall that N v, R and V v, R ( v = 0) are canonically oriented by their complexstructures. The decompositions (2.11), (2.12) induce the orientations of T X H and V R respectively. Let { e i } l ′ i =1 , { f j } p ′ j =1 be the corresponding oriented or-thonormal basis of ( T X H , g T X H ) and ( V R , g V R ). There are two canonical waysto consider S ( T X H ⊕ V R , L F ) as a Z -graded vector bundle . Let τ s = ( √− l ′ c ( e ) · · · c ( e l ′ ) ,τ e = ( √− l ′ + p ′ c ( e ) · · · c ( e l ′ ) c ( f ) · · · c ( f p ′ ) (2.14)be two involutions of S ( T X H ⊕ V R , L F ). Then τ s = τ e = 1. We decompose S ( T X H ⊕ V R , L F ) = S + ( T X H ⊕ V R , L F ) ⊕ S − ( T X H ⊕ V R , L F ) corresponding to τ s (resp. τ e ) such that τ s | S ± ( T X H ⊕ V R ,L F ) = ± τ e | S ± ( T X H ⊕ V R ,L F ) = ± C ( N R ) be the Clifford algebra bundle of ( N R , g N ). Then Λ( N ∗ ) is a C ( N R )-Clifford module. Namely, for e ∈ N , let e ′ ∈ N ∗ correspond to e by themetric g N , and let c ( e ) = √ e ′ ∧ , c ( e ) = −√ i e , (2.15)where ∧ and i denote the exterior and interior multiplications, respectively. Let τ N be the involution on Λ( N ∗ ) given by τ N | Λ even / odd ( N ∗ ) = ± C ( V v, R ) on the C ( V v, R )-Cliffordmodule Λ( V ∗ v ) with the involution τ Vv | Λ even / odd ( V ∗ v ) = ± X H , one has the following Z /k isomorphisms of Z -graded Clifford modules over X H (compare with [20, (1.49)]), (cid:0) S ( U, L ) , τ s (cid:1) | X H ≃ (cid:0) S ( T X H ⊕ V R , L F ) , τ s (cid:1) b ⊗ (cid:0) Λ N ∗ , τ N (cid:1) b ⊗ [ O v =0 (cid:0) Λ V ∗ v , id (cid:1) , (2.16)where id denotes the trivial involution, and (cid:0) S ( U, L ) , τ e (cid:1) | X H ≃ (cid:0) S ( T X H ⊕ V R , L F ) , τ e (cid:1) b ⊗ (cid:0) Λ N ∗ , τ N (cid:1) b ⊗ [ O v =0 (cid:0) Λ V ∗ v , τ Vv (cid:1) . (2.17)Here we denote by b ⊗ the Z -graded tensor product (cf. [14, pp. 11]). Fur-thermore, isomorphisms (2.16), (2.17) give the identifications of the canonical8onnections on the bundles (compare with [20, (1.13)]). We still denote theinvolution on (cid:0) S ( T X H ⊕ V R , L F ) by τ .Let R be a Z /k Hermitian vector bundle over X H endowed with a Z /k Hermitian connection. We make the assumption that the Hermitian metricand the Hermitian connection are both of product structure near ∂X H . We willdenote by D X H ⊗ R the twisted Spin c Dirac operator on S ( T X H ⊕ V R , L F ) ⊗ R and by D X H,α ⊗ R its restriction to X H,α (cf. Definition 2.2).We denote by K ( X H ) the K -group of Z /k complex vector bundles over X H (cf. [7, pp. 285]). We use the same notations as in [20, pp. 128],Sym q ( R ) = + ∞ X n =0 q n Sym n ( R ) ∈ K ( X H )[[ q ]] , Λ q ( R ) = + ∞ X n =0 q n Λ n ( R ) ∈ K ( X H )[[ q ]] , (2.18)for the symmetric and exterior power operations in K ( X H )[[ q ]], respectively.Let S act on L | X H by sending g ∈ S to g l c ( l c ∈ Z ) on X H . Then l c islocally constant on X H . Following [20, (1.50)], we define the following elementsin K ( X H )[[ q ]], R ± ( q ) = q P v | v | dim N v − P v v dim V v + 12 l c O v> (cid:0) Sym q v ( N v ) ⊗ det N v (cid:1) ⊗ O v< Sym q − v ( N v ) ⊗ O v =0 Λ ± q v ( V v ) ⊗ (cid:16) X v q v W v (cid:17) = X n R ± ,n q n , (2.19) R ′± ( q ) = q − P v | v | dim N v − P v v dim V v + 12 l c O v> Sym q − v ( N v ) ⊗ O v< (cid:0) Sym q v ( N v ) ⊗ det N v (cid:1) ⊗ O v =0 Λ ± q v ( V v ) ⊗ (cid:16) X v q v W v (cid:17) = X n R ′± ,n q n . (2.20)As explained in [20, pp. 139], since T X ⊕ V ⊕ L is spin, one gets X v v dim N v + X v v dim V v + l c ≡ . (2.21)Therefore, R ± ,ξ ( q ), R ′± ,ξ ( q ) ∈ K ( X H )[[ q ]].Clearly each R ± ,ξ , R ′± ,ξ ( ξ ∈ Z ) is a Z /k vector bundle over X H carryinga canonically induced Z /k Hermitian metric and a canonically induced Z /k Hermitian connection, which are both of product structure near ∂X H .We now state a mod k localization formula which generalizes [20, Theorem1.2] to the case of Z /k manifolds. It also generalizes the Z /k equivariant indextheorem in [27, Theorem 2.1] to the case of Spin c -manifolds.9 heorem 2.7 For any ξ ∈ Z , the following identities hold, APS-ind τ s (cid:0) D X , ξ (cid:1) ≡ X α ( − P The proof will be given in Section 3. Z /k extension of Hattori’s vanishing theorem In this subsection, we assume that T X has a Z /k S -equivariant almostcomplex structure J . Then one has the canonical splitting T X ⊗ R C = T (1 , X ⊕ T (0 , X, (2.24)where T (1 , X and T (0 , X are the eigenbundles of J corresponding to the eigen-values √− −√− 1, respectively.Let K X = det( T (1 , X ) be the determinant line bundle of T (1 , X over X .Then the complex spinor bundle S ( T X, K X ) for ( T X, K X ) is Λ( T ∗ (0 , X ) (cf.[14, Appendix D]).We suppose that c ( T (1 , X ) = 0 mod ( N ) ( N ∈ Z , N ≥ K /NX is well defined over X . Afterreplacing the S action by its N -fold action, we can always assume that S actson K /NX . For s ∈ Z , let D X ⊗ K s/NX be the twisted Spin c Dirac operator onΛ( T ∗ (0 , X ) ⊗ K s/NX defined as in (2.4).Using Theorem 2.7, we can generalize the main result of Hattori [8] to thecase of Z /k almost complex manifolds. Theorem 2.8 Assume that X is a connected Z /k almost complex manifoldwith a nontrivial Z /k circle action. If c ( T (1 , X ) = 0 mod ( N ) ( N ∈ Z , N ≥ , then for s ∈ Z , − N < s < , D X ⊗ K s/NX has vanishing property in Z /k category. In particular, the following identity holds, t-ind (cid:0) D X ⊗ K s/NX (cid:1) = 0 in Z /k Z . (2.25) Proof Using the almost complex structure on T X H induced by the almostcomplex structure J on T X and by (2.11), we know T (1 , X (cid:12)(cid:12) X H = M v =0 N v ⊕ T (1 , X H , (2.26)where N v are complex subbundles of T (1 , X (cid:12)(cid:12) X H on which g ∈ S acts bymultiplication by g v . 10e claim that for each ξ ∈ Z , the following identity holds,APS-ind (cid:0) D X ⊗ K s/NX , ξ (cid:1) ≡ k Z . (2.27)In fact, if X H = ∅ , the empty set, by Theorem 2.7, (2.27) is obvious.When X H = ∅ , we see that P v | v | dim N v > N v ’sis nonzero) on each connected component of X H . Consider R + ( q ), R ′ + ( q ) of(2.19) and (2.20) for the case that V = 0, W = K s/NX . We deduce that R + ,ξ = 0 if ξ < a = inf α (cid:16) X v | v | dim N v + (cid:16) 12 + sN (cid:17) X v v dim N v (cid:17) ,R ′ + ,ξ = 0 if ξ > a = sup α (cid:16) − X v | v | dim N v + (cid:16) 12 + sN (cid:17) X v v dim N v (cid:17) . Since − N < s < 0, we know a > a < 0. By using Theorem 2.7, we seethat (2.27) holds for any ξ ∈ Z .Now Theorem 2.8 follows easily from (1.1), (1.3) and (2.27). The proof ofTheorem 2.8 is completed. Remark 2.9 From the proof of Theorem 2.8, one also deduces that if X isa connected Z /k almost complex manifold with a nontrivial Z /k circle action,then D X , D X ⊗ K − X are rigid in Z /k category. In this section, following Zhang [27] and by making use of the analysis ofWu-Zhang [26] and Dai-Zhang [5] as well as Liu-Ma-Zhang [20], which in turndepend on the analytic localization techniques of Bismut-Lebeau [3], we presenta proof of Theorem 2.7.This section is organized as follows. In Section 3.1, we recall a result from[26] concerning the Witten deformation on flat spaces. In Section 3.2, we es-tablish the Taylor expansions of D X and c ( H ) (resp. B X ) near the fixed pointset X H (resp. ∂X H ). In Section 3.3, following [5, Section 3(b)], we decomposethe Dirac type operators under consideration to a sum of four operators andintroduce a deformation of the Dirac type operators as well as their associatedboundary operators. In Section 3.4, by using the techniques of [5, Section 3(c)],[20, Section 1.2] and [3, Section 9], we carry out various estimates for certainoperators and prove the Fredholm property of the Atiyah-Patodi-Singer typeboundary problem for the deformed operators introduced in Section 3.3. InSection 3.5, we complete the proof of Theorem 2.7. Recall that H is the canonical basis of Lie( S ) = R . In this subsection, let W be a complex vector space of dimension n with an Hermitian form. Let ρ bea unitary representation of the circle group S on W such that all the weightsare nonzero. Suppose W ± are the subspaces of W corresponding to the positive11nd negative weights respectively, with dim C W − = ν , dim C W + = n − ν . Let z = { z i } be the complex linear coordinates on W such that the Hermitianstructure on W takes the standard form and ρ is diagonal with weights λ i ∈ Z \{ } (1 ≤ i ≤ n ), and λ i < i ≤ ν . The Lie algebra action on W is givenby the vector field H = 2 π √− n X i =1 λ i (cid:18) z i ∂∂z i − ¯ z i ∂∂ ¯ z i (cid:19) . (3.1)Set K ± ( W ) = Sym(( W ± ) ∗ ) ⊗ Sym( W ∓ ) ⊗ det( W ∓ ) . (3.2)Let E be a finite dimensional complex vector space with an Hermitian formand suppose E carries a unitary representation of S .Let ∂ be the twisted Dolbeault operator acting on Ω , ∗ ( W, E ), the set ofsmooth sections of Λ( W ∗ ) ⊗ E on W . Let ∂ ∗ be the formal adjoint of ∂ . Let D = √ ∂ + ∂ ∗ ). Let c ( H ) be the Clifford action of H on Λ( W ∗ ) defined as in(2.15). Let L H be the Lie derivative along H acting on Ω , ∗ ( W, E ).The following result was proved in [26, Proposition 3.2]. Proposition 3.1 . A basis of the space of L -solutions of D + √− c ( H ) (resp. D − √− c ( H ) ) on the space of C ∞ sections of Λ( W ∗ ) is given by (cid:0) ν Y i =1 z k i i (cid:1)(cid:0) n Y i = ν +1 ¯ z k i i (cid:1) e − P ni =1 π | λ i || z i | d ¯ z ν +1 · · · d ¯ z n ( k i ∈ N ) (3.3) with weight P νi =1 k i | λ i | + P ni = ν +1 ( k i + 1) | λ i | (resp. (cid:0) ν Y i =1 ¯ z k i i (cid:1)(cid:0) n Y i = ν +1 z k i i (cid:1) e − P ni =1 π | λ i || z i | d ¯ z · · · d ¯ z ν ( k i ∈ N ) (3.4) with weight − P ni = ν +1 k i | λ i | − P νi =1 ( k i + 1) | λ i | ).So the space of L -solution of a given weight of D + √− c ( H ) (resp. D −√− c ( H ) ) on the space of C ∞ sections of Λ( W ∗ ) ⊗ E is finite dimensional. Thedirect sum of these weight spaces is isomorphic to K − ( W ) ⊗ E (resp. K + ( W ) ⊗ E ) as representations of S . . When restricted to an eigenspace of L H , the operator D + √− c ( H ) (resp. D − √− c ( H ) ) has discrete eigenvalues. Following [3, Section 8(e)], we now describe a coordinate system on X near X H . For ε > 0, set B ε = (cid:8) Z ∈ N (cid:12)(cid:12) | Z | < ε (cid:9) . Since X and X H are compact,there exists ε > < ε ≤ ε , the exponential map( y, Z ) ∈ N exp Xy ( Z ) ∈ X 12s a diffeomorphism from B ε onto a tubular neighborhood V ε of X H in X .From now on, we identify B ε with V ε and use the notation x = ( y, Z ) insteadof x = exp Xy ( Z ). Finally, we identify y ∈ X H with ( y, ∈ N .Let e π ∗ (( S ( U, L ) ⊗ W ) | X H ) be the vector bundle on N obtained by pullingback ( S ( U, L ) ⊗ W ) | X H for e π : N → X H .Let g T X H , g N be the corresponding metrics on T X H and N induced by g T X .Let d v X , d v X H and d v N be the corresponding volume elements on ( T X, g T X ),( T X H , g T X H ) and ( N, g N ). Let k ( y, Z ) (( y, Z ) ∈ B ε ) be the smooth positivefunction defined by d v X ( y, Z ) = k ( y, Z )d v X H ( y )d v N y ( Z ) . (3.5)Then k ( y ) = 1 and ∂k∂Z ( y ) = 0 for y ∈ X H . The latter follows from the well-known fact that X H is totally geodesic in X .For x = ( y, Z ) ∈ V ε , we will identify S ( U, L ) x with S ( U, L ) y and W x with W y by the parallel transport with respect to the S -invariant connections ∇ S ( U,L ) and ∇ W respectively, along the geodesic t ( y, tZ ). The inducedidentification of ( S ( U, L ) ⊗ W ) x with ( S ( U, L ) ⊗ W ) y preserves the metric andthe Z -grading, and moreover, is S -equivariant. Consequently, D X can beconsidered as an operator acting on the sections of the bundle e π ∗ (( S ( U, L ) ⊗ W ) | X H ) over B ε commuting with the circle action.For ε > 0, let E ( ε ) (resp. E ) be the set of smooth sections of e π ∗ (( S ( U, L ) ⊗ W ) | X H ) on B ε (resp. on the total space of N ). If f, g ∈ E have compactsupports, we will write h f, g i = (cid:18) π (cid:19) dim X Z X H (cid:18)Z N h f, g i ( y, Z )d v N y ( Z ) (cid:19) d v X H ( y ) . (3.6)Then k / D X k − / is a (formally) self-adjoint operator on E .The connection ∇ N on N induces a splitting T N = N ⊕ T H N , where T H N is the horizontal part of T N with respect to ∇ N . Moreover, since X H is totallygeodesic, this splitting, when restricted to X H , is preserved by the connection ∇ T X on T X | X H . Let e ∇ be the connection on ( S ( U, L ) ⊗ W ) | X H induced bythe restriction of ∇ S ( U,L ) ⊗ W to X H . We denote by e π ∗ e ∇ the pulling back of theconnection e ∇ on ( S ( U, L ) ⊗ W ) | X H to the bundle e π ∗ (( S ( U, L ) ⊗ W ) | X H ).We choose a local orthonormal basis of T X such that e , · · · , e l ′ form abasis of T X H , and e l ′ +1 , · · · , e l , that of N R . Denote the horizontal lift of e i (1 ≤ i ≤ l ′ ) to T H N by e Hi . We define D H = l ′ X i =1 c ( e i )( e π ∗ e ∇ ) e Hi , D N = l X i =2 l ′ +1 c ( e i )( e π ∗ e ∇ ) e i . (3.7)Clearly, D N acts along the fibers of N . Let ∂ N be the ∂ -operator along thefibers of N , and let ∂ N ∗ be its formal adjoint with respect to (3.6). It is easyto see that D N = √ (cid:0) ∂ N + ∂ N ∗ (cid:1) . Both D N and D H are formally self-adjointwith respect to (3.6). 13or T > 0, we define a scaling f ∈ E ( ε ) → S T f ∈ E ( ε √ T ) by S T f ( y, Z ) = f (cid:18) y, Z √ T (cid:19) , ( y, Z ) ∈ B ε √ T . (3.8)For a first order differential operator Q T = l ′ X i =1 a iT ( y, Z )( e π ∗ e ∇ ) e Hi + l X i =2 l ′ +1 b iT ( y, Z )( e π ∗ e ∇ ) e i + c T ( y, Z ) (3.9)acting on E ( ε √ T ), where a iT , b iT , and c T are endomorphisms of e π ∗ (( S ( U, L ) ⊗ W ) | X H ) which depend smoothly on ( y, Z ), we write Q T = O (cid:0) | Z | ∂ N + | Z | ∂ H + | Z | + | Z | p (cid:1) , (3.10)if there is a constant C > p ∈ N such that for any T ≥ 1, ( y, Z ) ∈ B ε √ T ,we have | a iT ( y, Z ) | ≤ C | Z | (1 ≤ i ≤ l ′ ) , | b iT ( y, Z ) | ≤ C | Z | (2 l ′ + 1 ≤ i ≤ l ) , | c T ( y, Z ) | ≤ C ( | Z | + | Z | p ) . (3.11)Let E ∂ be the set of smooth sections of e π ∗ (( S ( U, L ) ⊗ W ) | X H ) over N | ∂X H .On the boundary of X H , we choose the local orthonormal basis as in Definition2.3. Similarly as in (2.6), we define B H = − l ′ X i =2 c (cid:18) ∂∂r (cid:19) c ( e i )( e π ∗ e ∇ ) e Hi , B N = − c (cid:18) ∂∂r (cid:19) D N | ∂X H (3.12)on E ∂ (compare with (3.7)).Let J H be the representation of Lie( S ) on N . Then Z → J H Z is a Killingvector field on N . We have the following analogue of [3, Theorem 8.18], [20,Proposition 1.2] and [26, Proposition 3.3]. Proposition 3.2 As T → + ∞ , S T k / D X k − / S − T = √ T D N + D H + 1 √ T O ( | Z | ∂ N + | Z | ∂ H + | Z | ) ,S T k / c ( H ) k − / S − T = 1 √ T c ( J H Z ) + 1 √ T O ( | Z | ) ,S T k / B X k − / S − T = √ T B N + B H + 1 √ T O ( | Z | ∂ N + | Z | ∂ H + | Z | ) . For p ≥ 0, let E p (resp. E p∂ , E p , F p , F p∂ ) be the set of sections of thebundles S ( U, L ) ⊗ W over X (resp. ( S ( U, L ) ⊗ W ) | ∂X over ∂X , e π ∗ (( S ( U, L ) ⊗ W ) | X H ) over N , S ( T X H ⊕ V R , L F ) ⊗ K − ( N ) ⊗ (cid:0) b ⊗ v =0 Λ V v ⊗ W (cid:1) | X H over X H ,14 S ( T X H ⊕ V R , L F ) ⊗ K − ( N ) ⊗ b ⊗ v =0 Λ V v ⊗ W (cid:1) | ∂X H over ∂X H ) which lie inthe p -th Sobolev spaces. The group S acts on all these spaces (cf. Section2.2). For any ξ ∈ Z , let E pξ , E pξ,∂ E pξ , F pξ and F pξ,∂ be the corresponding weight- ξ subspaces, respectively.Recall that the constant ε > ε ∈ (0 , ε ]. Let ρ : R → [0 , 1] be a smooth function such that ρ ( a ) = ( a ≤ ,0 if a ≥ 1. (3.13)For Z ∈ N , set ρ ε ( Z ) = ρ ( | Z | ε ).By Proposition 3.1, the solution space of the operator D N + √− T c ( J H Z )along the fiber N y ( y ∈ X H ) is the L completion of K − ( N y ) ⊗ ( b ⊗ v =0 Λ V v ⊗ W ) y .They form an infinite dimensional Hermitian complex vector bundle K − ( N ) ⊗ ( b ⊗ v =0 Λ V v ⊗ W ) | X H over X H , with the Hermitian connection induced fromthose on N , V | X H → X H and W | X H → X H . Let θ be the isomorphism from L ( X H , K − ( N ) ⊗ ( b ⊗ v =0 Λ V v ⊗ W ) | X H ) to L ( N, e π ∗ ((Λ N ∗ ⊗ b ⊗ v =0 Λ V v ⊗ W ) | X H ))given by Proposition 3.1.Let α ∈ Γ (cid:0) X H , S ( T X H ⊕ V R , L F ) (cid:1) , φ ∈ L ( X H , K − ( N ) ⊗ ( b ⊗ v =0 Λ V v ⊗ W ) | X H ), σ = α ⊗ φ . We define a linear map I T,ξ : F pξ −→ E pξ , σ T dim N R ρ ε ( Z ) e π ∗ α ∧ S − T ( θφ ) . (3.14)In general, there exist c ( ε ) > C > c ( ε ) < k I T,ξ k < C .Let the image of I T,ξ from F pξ be E pT,ξ = I T,ξ F pξ ⊆ E pξ . Denote the orthogonalcomplement of E T,ξ in E ξ by E , ⊥ T,ξ , and let E p, ⊥ T,ξ = E pξ ∩ E , ⊥ T,ξ . Let p T,ξ and p ⊥ T,ξ be the orthogonal projections from E ξ to E T,ξ and E , ⊥ T,ξ respectively.We denote by (cid:0)(cid:0) cN v =0 Λ V v (cid:1) ⊗ (cid:0) L v W v (cid:1)(cid:1) ξ − P v | v | dim N v the subbundle of (cid:0) cN v =0 Λ V v (cid:1) ⊗ (cid:0) L v W v (cid:1) whose weight equals to ξ − P v | v | dim N v with re-spect to the given circle action. Let q ξ be the orthogonal bundle projectionfrom the vector bundle (cid:0) dO v =0 Λ N ∗ v (cid:1) b ⊗ (cid:0) dO v =0 Λ V v (cid:1) ⊗ (cid:0) M v W v (cid:1) −→ X H to its subbundle O v> det N v b ⊗ (cid:16)(cid:0) dO v =0 Λ V v (cid:1) ⊗ (cid:0) M v W v (cid:1)(cid:17) ξ − P v | v | dim N v −→ X H . We now proceed to deduce a formula which computes p T,ξ s for s ∈ E ξ explicitly under a local unitary trivialization of N .For y ∈ X H , on a small neighborhood V y ⊂ X H of y , choose a unitarytrivialization N | V y ∼ = V y × C n = { ( y, Z ) | y ∈ V y , Z = ( z , · · · , z n ) ∈ C n } suchthat for t ∈ R , exp( t H ) · ∂∂z i = e π √− λ i t ∂∂z i . λ i < i ≤ ν and λ i > ν < i ≤ n . For any T > −→ k = ( k , · · · , k n ) ∈ N n , and ( y, Z ) ∈ V y × C n , set f T, −→ k ( Z ) = (cid:0) ν Y i =1 z k i i (cid:1)(cid:0) n Y i = ν +1 ¯ z k i i (cid:1) e − T P ni =1 π | λ i || z i | ,α T, −→ k ( y ) = Z N R ,y ρ ε ( Z ) n Y i =1 (cid:16) | z i | k i e − T π | λ i || z i | (cid:17) dv N (2 π ) dim N R . Computing directly, we have for s ∈ E ξ that (compare with [3, Proposition 9.2]) p T,ξ s ( y, Z ) = X −→ k ,ξ , s.t. P ni =1 k i | λ i | + ξ = ξ α − T, −→ k ρ ε ( Z ) f T, −→ k ( Z ) · q ξ Z N R ,y ρ ε ( Z ′ ) f T, −→ k ( Z ′ ) s ( y, Z ′ ) dv N ( Z ′ )(2 π ) dim N R . (3.15)Using (3.15), we get the following analogue of [3, Proposition 9.3]. Proposition 3.3 There exists C > such that if T ≥ , σ ∈ F ξ , then k I T,ξ σ k E ξ ≤ C ( k σ k F ξ + √ T k σ k F ξ ) . (3.16) There exists C > such that for any T ≥ , any s ∈ E ξ , then k p T,ξ s k E ξ ≤ C ( k s k E ξ + √ T k s k E ξ ) . (3.17) Given γ > , there exists C ′ > such that for T ≥ , for s ∈ E ξ , then k p T,ξ | Z | γ s k E ξ ≤ C ′ T γ k s k E ξ . (3.18)Since we have the identification of the bundles( S ( U, L ) ⊗ W ) | V ε ≃ e π ∗ (cid:16) S ( T X H ⊕ V R , L F ) ⊗ Λ( N ∗ ) ⊗ (cid:0) b ⊗ v =0 Λ V v ⊗ W (cid:1) | X H (cid:17)(cid:12)(cid:12)(cid:12) B ε , we can consider k − / I T,ξ σ as an element of E pξ for σ ∈ F pξ . Set J T,ξ = k − / I T,ξ . (3.19)We denote by J T,ξ,∂ : F pξ,∂ → E pξ,∂ the restriction of J T,ξ on the boundary. LetE pT,ξ = J T,ξ F pξ (resp. E pT,ξ,∂ = J T,ξ,∂ F pξ,∂ ) be the image of J T,ξ (resp. J T,ξ,∂ ).Denote the orthogonal complement of E T,ξ (resp. E T,ξ,∂ ) in E ξ (resp. E ξ,∂ ) byE , ⊥ T,ξ (resp. E , ⊥ T,ξ,∂ ) and let E p, ⊥ T,ξ = E pξ ∩ E , ⊥ T,ξ (resp. E p, ⊥ T,ξ,∂ = E pξ,∂ ∩ E , ⊥ T,ξ,∂ ). Let¯ p T,ξ (resp. ¯ p T,ξ,∂ ) and ¯ p ⊥ T,ξ (resp. ¯ p ⊥ T,ξ,∂ ) be the orthogonal projections from E ξ ξ,∂ ) to E T,ξ (resp. E T,ξ,∂ ) and E , ⊥ T,ξ (resp. E , ⊥ T,ξ,∂ ) respectively. It isclear that ¯ p T,ξ = k − / p T,ξ k / (resp. ¯ p T,ξ,∂ = k − / p T,ξ,∂ k / ).For any (possibly unbounded) operator A (resp. B ) on E ξ (resp. E ξ,∂ ), wewrite A = (cid:18) A (1) A (2) A (3) A (4) (cid:19) ( resp. B = (cid:18) B (1) B (2) B (3) B (4) (cid:19) ) (3.20)according to the decomposition E ξ = E T,ξ L E , ⊥ T,ξ (resp. E ξ,∂ = E T,ξ,∂ L E , ⊥ T,ξ,∂ ),i.e., A (1) = ¯ p T,ξ A ¯ p T,ξ , A (2) = ¯ p T,ξ A ¯ p ⊥ T,ξ , (3.21) A (3) = ¯ p ⊥ T,ξ A ¯ p T,ξ , A (4) = ¯ p ⊥ T,ξ A ¯ p ⊥ T,ξ . ( resp. B (1) = ¯ p T,ξ,∂ B ¯ p T,ξ,∂ , B (2) = ¯ p T,ξ,∂ B ¯ p ⊥ T,ξ,∂ , (3.22) B (3) = ¯ p ⊥ T,ξ,∂ B ¯ p T,ξ,∂ , B (4) = ¯ p ⊥ T,ξ,∂ B ¯ p ⊥ T,ξ,∂ . )For T > 0, set D T = D X + √− T c ( H ) , B T = B X − √− T c (cid:18) ∂∂r (cid:19) c ( H ) . (3.23)Then D T is a Dirac type operator with its canonical boundary operator B T inthe sense of Definition 2.4. Let D T,ξ and B T,ξ be the restrictions of D T and B T on E ξ and E ξ,∂ , respectively.We now introduce a deformation of D T,ξ (resp. B T,ξ ) according to thedecomposition (3.21) (resp. (3.22)). Definition 3.4 (cf. [5, Definition 3.2], [20, (1.39)]) For any T > , u ∈ [0 , ,set D T,ξ ( u ) = D (1) T,ξ + D (4) T,ξ + u (cid:0) D (2) T,ξ + D (3) T,ξ (cid:1) ,B T,ξ ( u ) = B (1) T,ξ + B (4) T,ξ + u (cid:0) B (2) T,ξ + B (3) T,ξ (cid:1) . (3.24)One verifies that B T,ξ ( u ) is the canonical boundary operator associated to D T,ξ ( u ) in the sense of (2.5). T → + ∞ We continue the discussion in the previous subsection. Corresponding to theinvolution τ on S ( U, L ), for τ = τ s (resp. τ = τ e ), let D X H ξ be the restrictionof the twisted Spin c Dirac operator D X H ⊗ R + (1) (resp. D X H ⊗ R − (1)) on F ξ ,and let B X H ξ be the restriction of the canonical boundary operator associatedto D X H ⊗ R + (1) (resp. D X H ⊗ R − (1)) on F ξ,∂ .With (3.15), (3.23) and Propositions 3.1, 3.2, 3.3 at our hands, by proceedingexactly as in [3, Sections 8 and 9], we can show that the following estimates for B ( i ) T,ξ (1 ≤ i ≤ 4) hold. 17 roposition 3.5 (Compare with [5, Proposition 3.3]) There exists ε > suchthat (i) As T −→ + ∞ , J − T,ξ,∂ B (1) T,ξ J T,ξ,∂ = B X H ξ + O (cid:18) √ T (cid:19) , (3.25) where O ( √ T ) denotes a first order differential operator whose coefficientsare dominated by C √ T ( C > . (ii) There exist C > , C > and T > such that for any T ≥ T , any s ∈ E , ⊥ T,ξ,∂ , s ′ ∈ E T,ξ,∂ , then (cid:13)(cid:13) B (2) T,ξ s (cid:13)(cid:13) E ξ,∂ ≤ C (cid:16) √ T k s k E ξ,∂ + k s k E ξ,∂ (cid:17) , (cid:13)(cid:13) B (3) T,ξ s ′ (cid:13)(cid:13) E ξ,∂ ≤ C (cid:16) √ T k s ′ k E ξ,∂ + k s ′ k E ξ,∂ (cid:17) , (3.26) and (cid:13)(cid:13) B (4) T,ξ s (cid:13)(cid:13) E ξ,∂ ≥ C (cid:16) k s k E ξ,∂ + √ T k s k E ξ,∂ (cid:17) . (3.27)From here, by proceeding as in [5, Section 3(c)], we can deduce that thereexists T > T ≥ T , each B T,ξ ( u ), for u ∈ [0 , P T,ξ ( u ) denotethe Atiyah-Patodi-Singer projection associated to B T,ξ ( u ).For any T ≥ T and u ∈ [0 , D APS ,T,ξ ( u ) : (cid:8) s ∈ E ξ (cid:12)(cid:12) P T,ξ ( u )( s | ∂X ) = 0 (cid:9) −→ E ξ be the uniquely determined extension of D T,ξ ( u ). Proposition 3.6 (Compare with [5, Proposition 3.5]) There exists T > suchthat for any u ∈ [0 , and T ≥ T , D APS ,T,ξ ( u ) is a Fredholm operator. To prove Proposition 3.6, we modify the process in [5, Section 3(d)]. Forthe case where s is supported in X \ U ǫ ′ (0 < ǫ ′ < ǫ ), we need an analogue of [5,Lemma 3.7]. As a matter of fact, using (3.15), (3.23) as well as Propositions3.1, 3.2, 3.3 and proceeding exactly as in [3, Sections 8 and 9], we deduce thefollowing interior estimates. Proposition 3.7 There exists ε > such that (i) As T −→ + ∞ , J − T,ξ D (1) T,ξ J T,ξ = D X H ξ + O (cid:18) √ T (cid:19) , (3.28) where O ( √ T ) denotes a first order differential operator whose coefficientsare dominated by C √ T ( C > . There exist C ′ > , C ′ > and T ′ > such that for any T ≥ T ′ , any s ∈ E , ⊥ T,ξ , s ′ ∈ E T,ξ , then (cid:13)(cid:13) D (2) T,ξ s (cid:13)(cid:13) E ξ ≤ C ′ k s k E ξ √ T + k s k E ξ ! , (cid:13)(cid:13) D (3) T,ξ s ′ (cid:13)(cid:13) E ξ ≤ C ′ k s ′ k E ξ √ T + k s ′ k E ξ ! , (3.29) and (cid:13)(cid:13) D (4) T,ξ s (cid:13)(cid:13) E ξ ≥ C ′ (cid:16) k s k E ξ + √ T k s k E ξ (cid:17) . (3.30)With Proposition 3.7 at our hands, we can complete the proof of Proposition3.6 in the same way as in the proof of [5, Proposition 3.5] by applying the gluingargument in [3, pp. 115-117]. Let D Y H ξ be the induced operator from B X H ξ through π X H . We first assumethat D Y H ξ is invertible, then B X H ξ is invertible. Moreover, we have the followinganalogue of [5, Proposition 3.8]. Proposition 3.8 If D Y H ξ is invertible, then there exists T > such that forany T ≥ T , u ∈ [0 , , the boundary operator B T,ξ ( u ) is invertible. By Propositions 3.6 and 3.8, we have a continuous family of Fredholm oper-ators { D APS ,T,ξ ( u ) } ≤ u ≤ when T is large enough. Furthermore, by Proposition3.8 and Green’s formula, we know that the operators D APS ,T,ξ ( u ), 0 ≤ u ≤ h τ (cid:12)(cid:12) ker( D APS ,T,ξ (0)) i = Tr h τ (cid:12)(cid:12) ker( D APS ,T,ξ (1)) i . (3.31) Theorem 3.9 (Compare with [20, (1.43)]) If D Y H ξ is invertible, then there ex-ists T > such that for any T ≥ T , the following identity holds, APS-ind( D T,ξ ) = X α ( − P By the definitions of D APS ,T,ξ ( u ) and D T,ξ ( u ), we get thatAPS-ind( D T,ξ ) = APS-ind( D T,ξ (1)) = Tr h τ (cid:12)(cid:12) ker( D APS ,T,ξ (1)) i . (3.33)Let P T,ξ, (resp. P T,ξ, ) be the Atiyah-Patodi-Singer projection associatedto B (1) T,ξ (resp. B (4) T,ξ ) acting on E T,ξ,∂ (resp. E , ⊥ T,ξ,∂ ). Let D (1)APS ,T,ξ : (cid:8) s ∈ E T,ξ (cid:12)(cid:12) P T,ξ, ( s | ∂X ) = 0 (cid:9) −→ E T,ξ ,D (4)APS ,T,ξ : (cid:8) s ∈ E , ⊥ T,ξ (cid:12)(cid:12) P T,ξ, ( s | ∂X ) = 0 (cid:9) −→ E , ⊥ T,ξ 19e the uniquely determined extensions of D (1) T,ξ and D (4) T,ξ , respectively. UsingProposition 3.5 and proceeding as in the proof of [5, Proposition 3.5], one seesthat for T large enough, D (1)APS ,T,ξ and D (4)APS ,T,ξ are both self-adjoint Fredholmoperators. Furthermore, we deduce that for T large enough, ker( D (4)APS ,T,ξ ) = 0.Thus we get Tr h τ (cid:12)(cid:12) ker( D APS ,T,ξ (0)) i = Tr (cid:20) τ (cid:12)(cid:12) ker( D (1)APS ,T,ξ ) (cid:21) . (3.34)On the other hand, for T large enough and u ∈ [0 , D X H T,ξ ( u ) = u D X H ξ + (1 − u ) J − T,ξ D (1) T,ξ J T,ξ ,B X H T,ξ ( u ) = u B X H ξ + (1 − u ) J − T,ξ,∂ B (1) T,ξ J T,ξ,∂ . (3.35)From (3.25), one can proceed as in [5, (3.37)-(3.39)] to see that when T is largeenough, B X H T,ξ ( u ) is invertible for every u ∈ [0 , P X H T,ξ ( u ) the Atiyah-Patodi-Singer projection associated to B X H T,ξ ( u ). Using (3.25), (3.28) and applying the same gluing argument [3, pp.115-117] as in the proof of [5, Proposition 3.5], one sees that when T is largeenough and u ∈ [0 , D X H APS ,T,ξ ( u ) : (cid:8) s ∈ F ξ (cid:12)(cid:12) P X H T,ξ ( u )( s | ∂X ) = 0 (cid:9) −→ F ξ , the uniquely determined extensions of D X H T,ξ ( u ), form a continuous family ofself-adjoint Fredholm operators. Thus by the homotopy invariance of the indexof Fredholm operators, one getsTr (cid:20) τ (cid:12)(cid:12) ker( D XH APS ,T,ξ (0)) (cid:21) = Tr (cid:20) τ (cid:12)(cid:12) ker( D XH APS ,T,ξ (1)) (cid:21) = APS-ind (cid:0) D X H ξ (cid:1) . (3.36)From (2.16), (2.17), (3.14) and (3.19), one gets J − T,ξ ◦ τ ◦ J T,ξ = ( − P Assume that ω ( W ) S = ω ( T X ) S , p ( V + W − T X ) S = e · π ∗ u ( e ∈ Z ) in H ∗ S ( X, Z ) , and c ( W ) = 0 mod ( N ) . For ≤ ℓ < N , i = 1 , , , , consider the S -equivariant twisted Spin c Dirac operators D X ⊗ ( K W ⊗ K − X ) / ⊗ ∞ O n =1 Sym q n ( T X ) ⊗ R i ( V ) ⊗ Q ℓ ( W ) . (4.5)(i) If e = 0 , then these operators are rigid in Z /k category. (ii) If e < , then they have vanishing properties in Z /k category. Remark 4.2 (Compare with [19, Remark 2.1]) As ω ( W ) S = ω ( T X ) S , c ( K W ⊗ K − X ) S = 0 mod (2) . We note that in our case, X × S ES hasthe homotopy type of a CW complex [21] . By [9, Corollary 1.2] , the circle ac-tion on X can be lifted to ( K W ⊗ K − X ) / and is compatible with the circleaction on K W ⊗ K − X . Remark 4.3 If X is a Z /k spin manifold, by taking V = T X , W = 0 and i = 3 in Theorem 4.1, we resolve a conjecture of [6] . n > 1, consider Z n ⊂ S , the cyclic subgroup of order n . Wehave the Z n -equivariant cohomology of X defined by H ∗ Z n ( X, Z ) = H ∗ ( X × Z n ES , Z ), and there is a natural “forgetful” map α ( S , Z n ) : X × Z n ES → X × S ES which induces a pullback α ( S , Z n ) ∗ : H ∗ S ( X, Z ) → H ∗ Z n ( X, Z ).We denote by α ( S , 1) the arrow which forgets the S -action. Thus α ( S , ∗ : H ∗ S ( X, Z ) → H ∗ ( X, Z ) is induced by the inclusion of X into X × S ES as afiber over BS .Finally, note that if Z n acts trivially on a space M , then there is a newarrow t ∗ : H ∗ ( M, Z ) → H ∗ Z n ( M, Z ) induced by the projection t : M × Z n ES = M × B Z n → M .Let Z ∞ = S . For each 1 < n ≤ + ∞ , let i : X ( n ) → X be the inclusion ofthe fixed point set of Z n ⊂ S in X , and so i induces i S : X ( n ) × S ES → X × S ES .In the rest of this paper, we use the same assumption as in [19, (2.4)].Suppose that there exists some integer e ∈ Z such that for 1 < n ≤ + ∞ , α ( S , Z n ) ∗ ◦ i ∗ S (cid:16) p ( V + W − T X ) S − e · π ∗ u (cid:17) = t ∗ ◦ α ( S , ∗ ◦ i ∗ S (cid:16) p ( V + W − T X ) S (cid:17) . (4.6)Remark that the relation (4.6) clearly follows from the hypothesis of Theo-rem 4.1 by pulling back and forgetting. Thus it is a weaker hypothesis.Let G y be the multiplicative group generated by y . Following Witten [25],we consider the action of y ∈ G y on W (resp. W ) by multiplication by y (resp. y − ) on W (resp. W ). Set Q ( W ) = ∞ O n =0 Λ − q n ( W ) ⊗ ∞ O n =1 Λ − q n ( W ) ∈ K ( X )[[ q ]] . (4.7)Then the actions of G y on W and W naturally induce the action of G y on Q ( W ). Clearly, y · Q ( W ) = Q y ( W ). By (4.3), we know that for 0 ≤ ℓ < N , y · Q ℓ ( W ) = y ℓ Q ℓ ( W ) , where y ∈ G y . (4.8)In what follows, for m ∈ Z , 0 ≤ ℓ < N , h ∈ Z and R ( q ) ∈ K S ( X )[[ q / ]],we will denote APS-ind( D X ⊗ R ( q ) ⊗ Q ℓ ( W ) , m, h ) by APS-ind( D X ⊗ R ( q ) ⊗ Q ( W ) , m, ℓ, h ).We can now state a slightly more general version of Theorem 4.1. Theorem 4.4 Under the hypothesis (4.6) , consider the S × G y -equivarianttwisted Spin c Dirac operators D X ⊗ ( K W ⊗ K − X ) / ⊗ ∞ O n =1 Sym q n ( T X ) ⊗ R i ( V ) ⊗ Q ( W ) . (4.9)24i) If e = 0 , for m ∈ Z , h ∈ Z , h = 0 , ≤ ℓ < N , one has APS-ind (cid:16) D X ⊗ ( K W ⊗ K − X ) / ⊗ ∞ O n =1 Sym q n ( T X ) ⊗ R i ( V ) ⊗ Q ( W ) , m, ℓ, h (cid:17) ≡ k Z . (4.10)(ii) If e < , for m ∈ Z , h ∈ Z , ≤ ℓ < N , one has APS-ind (cid:16) D X ⊗ ( K W ⊗ K − X ) / ⊗ ∞ O n =1 Sym q n ( T X ) ⊗ R i ( V ) ⊗ Q ( W ) , m, ℓ, h (cid:17) ≡ k Z . (4.11) In particular, one has APS-ind (cid:16) D X ⊗ ( K W ⊗ K − X ) / ⊗ ∞ O n =1 Sym q n ( T X ) ⊗ R i ( V ) ⊗ Q ( W ) , m, ℓ (cid:17) ≡ k Z . (4.12)The rest of this paper is devoted to a proof of Theorem 4.4. Recall that X H = { X H,α } be the fixed point set of the circle action. As in[19, pp. 940], we may and we will assume that T X | X H = T X H ⊕ M v> N v ,T X | X H ⊗ R C = T X H ⊗ R C ⊕ M v> (cid:0) N v ⊕ N v (cid:1) , (4.13)where N v is the complex vector bundles on which S acts by sending g to g v .We also assume that V | X H = V R ⊕ M v> V v , W | X H = M v W v , (4.14)where V v , W v are complex vector bundles on which S acts by sending g to g v ,and V R is a real vector bundle on which S acts as identity.By (4.13), as in (2.16) or (2.17), there is a natural Z /k isomorphism betweenthe Z -graded C ( T X )-Clifford modules over X H , S ( T X, K X ) | X H ≃ S (cid:16) T X H , K X ⊗ v> (det N v ) − (cid:17) b ⊗ dO v> Λ N v . (4.15)For a Z /k complex vector bundle R over X H , let D X H ⊗ R , D X H,α ⊗ R bethe twisted Spin c Dirac operators on S ( T X H , K X ⊗ v> (det N v ) − ) ⊗ R over X H , X H,α , respectively (cf. Definition 2.2).25or i = 1, 2, 3, 4, we set R i = ( K W ⊗ K − X ) / ⊗ R i ( V ) ⊗ Q ( W ) . (4.16)Then by Theorem 2.7, we can express the global Atiyah-Patodi-Singer indexvia the Atiyah-Patodi-Singer indices on the fixed point set up to k Z . Proposition 4.5 (Compare with [19, Proposition 2.1]) For m ∈ Z , h ∈ Z , ≤ i ≤ , ≤ ℓ < N , we have APS-ind (cid:16) D X ⊗ ⊗ ∞ n =1 Sym q n ( T X ) ⊗ R i , m, ℓ, h (cid:17) ≡ X α ( − P v> dim N v APS-ind (cid:16) D X H,α ⊗ ∞ n =1 Sym q n ( T X ) ⊗ R i ⊗ Sym ( ⊕ v> N v ) ⊗ v> det N v , m, ℓ, h (cid:17) mod k Z . (4.17)To simplify the notations, we use the same convention as in [19, pp. 945].For n ∈ N ∗ , we define a number operator P on K S ( X )[[ q n ]] in the followingway: if R ( q ) = ⊕ n ∈ n Z R n q n ∈ K S ( X )[[ q n ]], then P acts on R ( q ) by multipli-cation by n on R n . From now on, we simply denote Sym q n ( T X ), Λ q n ( V ) andΛ q n ( W ) by Sym( T X n ), Λ( V n ) and Λ( W n ), respectively. In this way, P acts on T X n , V n and W n by multiplication by n , and the actions of P on Sym( T X n ),Λ( V n ) and Λ( W n ) are naturally induced by the corresponding actions of P on T X n , V n and W n . So the eigenspace of P = n is just given by the coefficient of q n of the corresponding element R ( q ). For R ( q ) = ⊕ n ∈ n Z R n q n ∈ K S ( X )[[ q n ]],we will also denote APS-ind (cid:0) D X ⊗ R m , h (cid:1) by APS-ind (cid:0) D X ⊗ R ( q ) , m, h (cid:1) .Recall that H is the Killing vector field on X corresponding to H , thecanonical basis of Lie( S ). If E is a Z /k S -equivariant vector bundle over X ,let L H denote the corresponding Lie derivative along H acting on Γ( X H , E | X H ).The weight of the circle action on Γ( X H , E | X H ) is given by the action J H = 12 π √− L H . Recall that the Z -grading on S ( T X, K X ) ⊗ ∞ n =1 Sym( T X n ) is induced by the Z -grading on S ( T X, K X ). Write F ( X ) = ∞ O n =1 Sym q n ( T X ) ⊗ Sym( ⊕ v> N v ) ⊗ v> det N v ,F V = S ( V ) ⊗ ∞ O n =1 Λ( V n ) , F V = O n ∈ N + Λ( V n ) ,Q ( W ) = ∞ O n =0 Λ( W n ) ⊗ ∞ O n =1 Λ( W n ) . (4.18)There are two natural Z -gradings on F V , F V (resp. Q ( W )). The first gradingis induced by the Z -grading of S ( V ) and the forms of homogeneous degrees in26 ∞ n =1 Λ( V n ), ⊗ n ∈ N + Λ( V n ) (resp. Q ( W )). We define τ e | F i ± V = ± i = 1, 2)(resp. τ | Q ( W ) ± = ± 1) to be the involution defined by this Z -grading. Thesecond grading is the one for which F iV ( i = 1, 2) are purely even, i.e., F i + V = F iV .We denote by τ s = id the involution defined by this Z -grading. Then thecoefficient of q n ( n ∈ Z ) in (4.1) of R ( V ) or R ( V ) (resp. R ( V ), R ( V ) or Q ( W )) is exactly the Z -graded Z /k vector subbundle of ( F V , τ s ) or ( F V , τ e )(resp. ( F V , τ e ), ( F V , τ s ) or ( Q ( W ) , τ )), on which P acts by multiplication by n . Furthermore, we denote by τ e (resp. τ s ) the Z -grading on S ( T X, K X ) ⊗⊗ ∞ n =1 Sym( T X n ) ⊗ F iV ( i = 1, 2) induced by the above Z -gradings. We willdenote by τ e (resp. τ s ) the Z -grading on S ( T X, K X ) ⊗ ⊗ ∞ n =1 Sym( T X n ) ⊗ F iV ⊗ Q ( W ) ( i = 1, 2) defined by τ e = τ e b ⊗ τ , τ s = τ s b ⊗ τ . (4.19)Let h V v be the Hermitian metric on V v induced by the metric h V on V . Inthe following, we identity Λ V v with Λ V ∗ v by using the Hermitian metric h V v on V v . By (4.14), as in (4.15), there is a natural Z /k isomorphism between the Z -graded C ( V )-Clifford modules over X H , S ( V ) | X H ≃ S (cid:16) V R , ⊗ v> (det V v ) − (cid:17) ⊗ dO v> Λ V v . (4.20)Let V = V R ⊗ R C . By using the above notations, we rewrite (4.18) on thefixed point set X H , F ( X ) = ∞ O n =1 Sym( T X H ) ⊗ ∞ O n =1 Sym (cid:16) ⊕ v> (cid:0) N v,n ⊕ N v,n (cid:1)(cid:17) ⊗ Sym( ⊕ v> N v ) ⊗ v> det N v ,F V = ∞ O n =1 Λ (cid:16) V ,n ⊕ ⊕ v> ( V v,n ⊕ V v,n ) (cid:17) ⊗ S (cid:16) V R , ⊗ v> (det V v ) − (cid:17) ⊗ v> Λ V v, ,F V = O n ∈ N + Λ (cid:16) V ,n ⊕ ⊕ v> ( V v,n ⊕ V v,n ) (cid:17) ,Q ( W ) = ∞ O n =0 Λ( ⊕ v W v,n ) ⊗ ∞ O n =1 Λ( ⊕ v W v,n ) . (4.21)We introduce the same shift operators as in [19, Section 3.2], which follow[24] in spirit. For p ∈ N , we set r ∗ : N v,n → N v,n + pv , r ∗ : N v,n → N v,n − pv ,r ∗ : V v,n → V v,n + pv , r ∗ : V v,n → V v,n − pv ,r ∗ : W v,n → W v,n + pv , r ∗ : W v,n → W v,n − pv . (4.22)27urthermore, for p ∈ N , we introduce the following elements in K S ( X H )[[ q ]](cf. [19, (3.6)]), F p ( X ) = ∞ O n =1 Sym( T X H ) ⊗ O v> (cid:16) ∞ O n =1 Sym ( N v,n ) O n>pv Sym ( N v,n ) (cid:17) , F ′ p ( X ) = O v> O ≤ n ≤ pv (cid:16) Sym ( N v, − n ) ⊗ det( N v ) (cid:17) , F − p ( X ) = F p ( X ) ⊗ F ′ p ( X ) . (4.23)Note that when p = 0, F − p ( X ) is exactly the F ( X ) in (4.21). The Z -gradingon S ( T X H , K X ⊗ v> (det N v ) − ) ⊗ F − p ( X ) is induced by the Z -grading on S ( T X H , K X ⊗ v> (det N v ) − ).As in [19, (2.9)], we write L ( N ) = ⊗ v> (det N v ) v , L ( V ) = ⊗ v> (det V v ) v ,L ( W ) = ⊗ v =0 (det W v ) v , L = L ( N ) − ⊗ L ( V ) ⊗ L ( W ) . (4.24)Using the similar Z /k S -equivariant isomorphism of complex vector bun-dles as in [20, (3.14)] and the similar Z /k G y × S -equivariant isomorphism ofcomplex vector bundles as in [19, (3.15) and (3.16)], by direct calculation, wededuce the following proposition. Proposition 4.6 (cf. [19, Proposition 3.1]) For p ∈ Z , p > , i = 1 , , thereare natural Z /k isomorphisms of vector bundles over X H , r ∗ ( F − p ( X )) ≃ F ( X ) ⊗ L ( N ) p , r ∗ ( F iV ) ≃ F iV ⊗ L ( V ) p . (4.25) For any p ∈ Z , p > , there is a natural Z /k G y × S -equivariant isomorphismof vector bundles over X H , r ∗ ( Q ( W )) ≃ Q ( W ) ⊗ L ( W ) − p . (4.26)On X H , as in [19, (2.8)], we write e ( N ) = X v> v dim N v , d ′ ( N ) = X v> v dim N v ,e ( V ) = X v> v dim V v , d ′ ( V ) = X v> v dim V v ,e ( W ) = X v v dim W v , d ′ ( W ) = X v v dim W v . (4.27)Then e ( N ), e ( V ), e ( W ), d ′ ( N ), d ′ ( V ) and d ′ ( W ) are locally constant functionson X H .Take Z ∞ = S in the hypothesis (4.6). By using splitting principle [11,Chapter 17], we get the same identities as in [19, (2.11)], c ( L ) = 0 , e ( V ) + e ( W ) − e ( N ) = 2 e . (4.28)As indicated in Section 2.1, (4.28) means L is a trivial complex line bundle overeach component X H,α of X H , and S acts on L by sending g to g e , and G y acts on L by sending y to y d ′ ( W ) .The following proposition is deduced from Proposition 4.6.28 roposition 4.7 (cf. [19, Proposition 3.2]) For p ∈ Z , p > , i = 1 , , the Z /k G y -equivariant isomorphism of vector bundles over X H induced by (4.25) , (4.26) , r ∗ : S ( T X H , K X ⊗ v> (det N v ) − ) ⊗ ( K W ⊗ K − X ) / ⊗ F − p ( X ) ⊗ F iV ⊗ Q ( W ) −→ S ( T X H , K X ⊗ v> (det N v ) − ) ⊗ ( K W ⊗ K − X ) / ⊗ F ( X ) ⊗ F iV ⊗ Q ( W ) ⊗ L − p , (4.29) verifies the following identities r − ∗ · J H · r ∗ = J H ,r − ∗ · P · r ∗ = P + p J H + p e − p e ( N ) − p d ′ ( N ) . (4.30) For the Z -gradings, we have r − ∗ τ e r ∗ = τ e , r − ∗ τ s r ∗ = τ s , r − ∗ τ r ∗ = ( − pd ′ ( W ) τ . (4.31)Then we get the following recursive formula. Theorem 4.8 (Compare with [19, Theorem 2.4]) For each α , m ∈ Z , ≤ ℓ < N , h , p ∈ Z , p > , the following identity holds, APS-ind (cid:16) D X H,α ⊗ F − p ( X ) ⊗ R i , m + 12 p e ( N ) + p d ′ ( N ) , ℓ, h (cid:17) (4.32)= ( − pd ′ ( W ) APS-ind (cid:16) D X H,α ⊗ F ( X ) ⊗ R i ⊗ L − p , m + ph + p e, ℓ, h (cid:17) . Now we state another recursive formula whose proof will be presented inSection 5. Theorem 4.9 (Compare with [19, Theorem 2.3]) For ≤ i ≤ , m ∈ Z , ≤ ℓ < N , h , p ∈ Z , p > , we have the following identity, X α ( − P v> dim N v APS-ind (cid:16) D X H,α ⊗ F ( X ) ⊗ R i , m, ℓ, h (cid:17) ≡ X α ( − pd ′ ( N )+ P v> dim N v APS-ind (cid:16) D X H,α ⊗ F − p ( X ) ⊗ R i ,m + p e ( N ) + p d ′ ( N ) , ℓ, h (cid:17) mod k Z . (4.33) Recall we assume in Theorem 4.1 that c ( W ) = 0 mod ( N ). Then by [10,Section 8] and [19, Lemma 2.1], d ′ ( W ) mod( N ) is constant on each connectedcomponent X H,α of X H . So we can extend L to a trivial complex line bundleover X , and we extend the S -action on it by sending g on the canonical section1 of L to g e · 1, and G y acts on L by sending y to y d ′ ( W ) .29s p ( T X − W ) S ∈ H ∗ S ( X, Z ) is well defined, one has the same identityas in [19, (2.27)], d ′ ( N ) + d ′ ( W ) ≡ . (4.34)From Proposition 4.5, Theorems 4.8, 4.9 and (4.34), for 1 ≤ i ≤ m ∈ Z ,1 ≤ ℓ < N , h , p ∈ Z , p > 0, we get the following identity (compare with [19,(2.28)]),APS-ind (cid:0) D X ⊗ ∞ O n =1 Sym q n ( T X ) ⊗ R i , m, ℓ, h (cid:1) (4.35) ≡ APS-ind (cid:0) D X ⊗ ∞ O n =1 Sym q n ( T X ) ⊗ R i ⊗ L − p , m ′ , ℓ, h (cid:1) mod k Z , with m ′ = m + ph + p e. (4.36)By (4.1), (4.2), if m < m ′ < 0, then either side of (4.35) is identicallyzero, which completes the proof of Theorem 4.4. In fact,(i) Assume that e = 0. Let h ∈ Z , m ∈ Z , h = 0 be fixed. If h > 0, wetake m ′ = m , then for p large enough, we get m < h < m = m , then for p large enough, we get m ′ < e < 0. For h ∈ Z , m ∈ Z , we take m = m , then for p large enough, we get m ′ < In this section, following [19, Section 4], we present a proof of Theorem 4.9.This section is organized as follows. In Section 5.1, we first introduce thesame refined shift operators as in [19, Section 4.2]. In Section 5.2, we constructthe twisted Spin c Dirac operator on X ( n j ), the fixed point set of the naturallyinduced Z n j -action on X . In Section 5.3, by applying the S -equivariant indextheorem we have established in Section 2, we prove Theorem 4.9. We first introduce a partition of [0 , 1] as in [19, pp. 942–943]. Set J = (cid:8) v ∈ N (cid:12)(cid:12) there exists α such that N v = 0 on X H,α (cid:9) andΦ = (cid:8) β ∈ (0 , (cid:12)(cid:12) there exists v ∈ J such that βv ∈ Z (cid:9) . (5.1)We order the elements in Φ so that Φ = (cid:8) β i (cid:12)(cid:12) ≤ i ≤ J , J ∈ N and β i < β i +1 (cid:9) .Then for any integer 1 ≤ i ≤ J , there exist p i , n i ∈ N , 0 < p i ≤ n i , with( p i , n i ) = 1 such that β i = p i /n i . (5.2)Clearly, β J = 1. We also set p = 0 and β = 0.30or 0 ≤ j ≤ J , p ∈ N ∗ , we write I pj = n ( v, n ) ∈ N × N (cid:12)(cid:12)(cid:12) v ∈ J, ( p − v < n ≤ pv, nv = p + 1 + p j n j o ,I pj = n ( v, n ) ∈ N × N (cid:12)(cid:12)(cid:12) v ∈ J, ( p − v < n ≤ pv, nv > p + 1 + p j n j o . (5.3)Clearly, I p = ∅ , the empty set. We define F p,j ( X ) as in [19, (2.21)], which areanalogous with (4.23). More specifically, we set F p,j ( X ) = ∞ O n =1 Sym ( T X H ) ⊗ O v> ∞ O n =1 Sym ( N v,n ) ⊗ O n> ( p − v + p j n j v Sym ( N v,n ) !O v> O ≤ n ≤ ( p − v + (cid:2) p j n j v (cid:3)(cid:16) Sym ( N v, − n ) ⊗ det N v (cid:17) (5.4)= F p ( X ) ⊗ F ′ p − ( X ) ⊗ O ( v,n ) ∈∪ ji =0 I pi (cid:16) Sym ( N v, − n ) ⊗ det N v (cid:17) O ( v,n ) ∈ I pj Sym ( N v,n ) , where we use the notation that for s ∈ R , [ s ] denotes the greatest integer whichis less than or equal to s . Then F p, ( X ) = F − p +1 ( X ) , F p,J ( X ) = F − p ( X ) . (5.5)From the construction of β i , we know that for v ∈ J , there is no integer in (cid:0) p j − n j − v, p j n j v (cid:1) . Furthermore, h p j − n j − v i = h p j n j v i − v ≡ n j ) , h p j − n j − v i = h p j n j v i if v n j ) . (5.6)We use the same shift operators r j ∗ , 1 ≤ j ≤ J as in [19, (4.21)], whichrefine the shift operator r ∗ defined in (4.22). For p ∈ N ∗ , set r j ∗ : N v,n → N v,n +( p − v + p j v/n j , r j ∗ : N v,n → N v,n − ( p − v − p j v/n j ,r j ∗ : V v,n → V v,n +( p − v + p j v/n j , r j ∗ : V v,n → V v,n − ( p − v − p j v/n j ,r j ∗ : W v,n → W v,n +( p − v + p j v/n j , r j ∗ : W v,n → W v,n − ( p − v − p j v/n j . (5.7)For 1 ≤ j ≤ J , we define F ( β j ), F V ( β j ), F V ( β j ) and Q W ( β j ) as in [19,(4.13)]. F ( β j ) = O T X H,n ) ⊗ O v> ,v ≡ , n j n j ) O There are natural Z /k isomor-phisms of vector bundles over X H , r j ∗ ( F p,j − ( X )) ≃ F ( β j ) ⊗ O v> , v ≡ n j ) Sym ( N v, ) ⊗ O v> (det N v ) (cid:2) p j n j v (cid:3) +( p − v +1 ⊗ O v> , v ≡ n j ) (det N v ) − ,r j ∗ ( F p,j ( X )) ≃ F ( β j ) ⊗ O v> , v ≡ n j ) Sym( N v, ) ⊗ O v> (det N v ) (cid:2) p j n j v (cid:3) +( p − v +1 ,r j ∗ ( F V ) ≃ S (cid:16) V R , ⊗ v> (det V v ) − (cid:17) ⊗ F V ( β j ) ⊗ O v> , v ≡ n j ) Λ( V v, ) ⊗ O v> (det V v ) (cid:2) p j n j v (cid:3) +( p − v ,r j ∗ ( F V ) ≃ F V ( β j ) ⊗ O v> , v ≡ n j n j ) Λ( V v, ) ⊗ O v> (det V v ) (cid:2) p j n j v + 12 (cid:3) +( p − v . here is a natural Z /k G y × S -equivariant isomorphism of vector bundlesover X H , r j ∗ ( Q ( W )) ≃ Q W ( β j ) ⊗ O v> (det W v ) (cid:2) p j n j v (cid:3) +( p − v +1 ⊗ O v> , v ≡ n j ) (det W v ) − ⊗ O v< (det W v ) (cid:2) − p j n j v (cid:3) − ( p − v . c Dirac operators on X ( n j ) Recall that there is a nontrivial Z /k circle action on X which can be liftedto the Z /k circle actions on V and W .For n ∈ N ∗ , let Z n ⊂ S denote the cyclic subgroup of order n . Let X ( n j )be the fixed point set of the induced Z n j action on X . Let N ( n j ) → X ( n j ) bethe normal bundle to X ( n j ) in X . As in [4, pp. 151] (see also [19, Section 4.1],[20, Section 4.1] or [24]), we see that N ( n j ) and V can be decomposed, as Z /k real vector bundles over X ( n j ), into N ( n j ) = M T X ( n j ) n (cid:1) ⊗ O Assume that (4.6) holds. Let L ( n j ) = O 2, we have the following isomorphisms ofClifford modules over X H , S ( U i , L i ) ≃ S ( U i , L i ) ′ ⊗ O v> , v ≡ n j ) Λ N v . (5.22)We define the Z -gradings on S ( U i , L i ) ′ ( i = 1 , 2) induced by the Z -gradingson S ( U i , L i ) ( i = 1 , 2) and on ⊗ v> , v ≡ n j ) Λ N v such that the isomorphisms(5.22) preserve the Z -gradings.As in [19, pp. 952], we introduce formally the following Z /k complex linebundles over X H , L ′ = (cid:16) L − ⊗ O v> , v ≡ n j ) (det N v ⊗ det V v ) O v> (det N v ⊗ det V v ) − ⊗ K X (cid:17) ,L ′ = (cid:16) L − ⊗ O v> , v ≡ n j ) det N v O v> , v ≡ nj mod ( n j ) det V v O v> (det N v ) − ⊗ K X (cid:17) . In fact, from (2.16), (2.17), Lemma 5.2 and the assumption that V is spin, oneverifies easily that c ( L ′ i ) = 0 mod (2) for i = 1 , 2, which implies that L ′ and L ′ are well defined Z /k complex line bundles over X H (cf. Section 2.1).Then by (5.20), (5.21) and the definitions of L , L , L ′ and L ′ , we get thefollowing identifications of Z /k Clifford modules over X H (cf. [19, (4.19)]), S ( U , L ) ′ ⊗ L ′ = S (cid:0) T X H , K X ⊗ v> (det N v ) − (cid:1) ⊗ S (cid:0) V R , ⊗ v> (det V v ) − (cid:1) ⊗ O v> , v ≡ n j ) Λ( V v ) , (5.23) S ( U , L ) ′ ⊗ L ′ = S (cid:0) T X H , K X ⊗ v> (det N v ) − (cid:1) ⊗ O v> , v ≡ nj mod ( n j ) Λ( V v ) . (5.24)36 emma 5.3 (cf. [19, Lemma 4.3]) Let us write L ( β j ) = L ′ ⊗ O v> (det N v ) (cid:2) p j n j v (cid:3) +( p − v +1 ⊗ O v> (det V v ) (cid:2) p j n j v (cid:3) +( p − v ⊗ O v> , v ≡ n j ) (det N v ) − ⊗ O v< (det W v ) (cid:2) − p j n j v (cid:3) − ( p − v ⊗ O v> (det W v ) (cid:2) p j n j v (cid:3) +( p − v +1 ⊗ O v> , v ≡ n j ) (det W v ) − ,L ( β j ) = L ′ ⊗ O v> (det N v ) (cid:2) p j n j v (cid:3) +( p − v +1 ⊗ O v> (det V v ) (cid:2) p j n j v + 12 (cid:3) +( p − v ⊗ O v> , v ≡ n j ) (det N v ) − ⊗ O v< (det W v ) (cid:2) − p j n j v (cid:3) − ( p − v ⊗ O v> (det W v ) (cid:2) p j n j v (cid:3) +( p − v +1 ⊗ O v> , v ≡ n j ) (det W v ) − . Then L ( β j ) and L ( β j ) can be extended naturally to Z /k G y × S -equivariantcomplex line bundles over X ( n j ) which we will still denote by L ( β j ) and L ( β j ) respectively. Now we compare the Z -gradings in (5.23). Set∆( n j , N ) = X n j For i = 1 , , the Z /k G y -equivariant somorphisms of complex vector bundles over X H , r i : S ( T X H , K X ⊗ v> (det N v ) − ) ⊗ ( K W ⊗ K − X ) / ⊗ F p,j − ( X ) ⊗ F iV ⊗ Q ( W ) −→ S ( U i , L i ) ⊗ ( K W ⊗ K − X ) / ⊗ F ( β j ) ⊗ F iV ( β j ) ⊗ Q W ( β j ) ⊗ L ( β j ) i ⊗ O v> , v ≡ n j ) Sym ( N v, ) ,r i : S ( T X H , K X ⊗ v> (det N v ) − ) ⊗ ( K W ⊗ K − X ) / ⊗ F p,j ( X ) ⊗ F iV ⊗ Q ( W ) −→ S ( U i , L i ) ⊗ ( K W ⊗ K − X ) / ⊗ F ( β j ) ⊗ F iV ( β j ) ⊗ Q W ( β j ) ⊗ L ( β j ) i ⊗ O v> , v ≡ n j ) (cid:0) Sym ( N v, ) ⊗ det N v (cid:1) have the following properties: (i) for i = 1 , , γ = 1 , , r − iγ · J H · r iγ = J H ,r − iγ · P · r iγ = P + (cid:16) p j n j + ( p − (cid:17) J H + ε iγ , (5.31) where ε i = ε i + ε ( W ) − e ( p, β j − , N ) , ε i = ε i + ε ( W ) − e ( p, β j , N ) . (ii) Recall that o (cid:0) V ( n j ) R n j (cid:1) is defined in (5.25) . Let µ = − X v> (cid:2) p j n j v (cid:3) dim V v + ∆( n j , N ) + ∆( n j , V ) mod (2) ,µ = − X v> (cid:2) p j n j v + (cid:3) dim V v + ∆( n j , N ) + o (cid:0) V ( n j ) R n j (cid:1) mod (2) ,µ = ∆( n j , N ) mod (2) ,µ = X v (cid:0)(cid:2) p j n j v (cid:3) + ( p − v (cid:1) dim W v + dim W + dim W ( n j ) mod (2) . Then for i = 1 , , γ = 1 , , we have r − iγ τ e r iγ = ( − µ i τ e , r − iγ τ s r iγ = ( − µ τ s ,r − iγ τ r iγ = ( − µ τ . (5.32) Let X ′ be a connected component of X ( n j ). By [19, Lemmas 4.4, 4.5, 4.6],we know that for i = 1 , k = 1 , , 3, the following functions are independenton the connected components of X H in X ′ , ε i + ε ( W ) mod (2) , d ′ ( p, β j , N ) + µ k + µ mod (2) , X v> (cid:2) p j n j v (cid:3) dim V v + ∆( n j , V ) mod (2) , X v> (cid:2) p j n j v + (cid:3) dim V v + o (cid:0) V ( n j ) R n j (cid:1) mod (2) , d ′ ( p, β j − , N ) + P 2) is well definedon X ( n j ). Observe that the two equalities in Theorem 2.7 are both compatiblewith the G y action. Thus, by using Proposition 5.4 and applying both thefirst and the second equalities of Theorem 2.7 to each connected component of X ( n j ) separately, we deduce that for i = 1 , 2, 1 ≤ j ≤ J , m ∈ Z , 1 ≤ ℓ < N , h ∈ Z , τ = τ e or τ s , X α ( − d ′ ( p,β j − ,N )+ P v> dim N v APS-ind τ (cid:16) D X H,α ⊗ ( K W ⊗ K − X ) / ⊗F p,j − ( X ) ⊗ F iV ⊗ Q ( W ) , m + e ( p, β j − , N ) , ℓ, h (cid:17) ≡ X β ( − d ′ ( p,β j − ,N )+ P v> dim N v + µ APS-ind τ (cid:16) D X ( n j ) ⊗ ( K W ⊗ K − X ) / ⊗ F ( β j ) ⊗ F iV ( β j ) ⊗ Q W ( β j ) ⊗ L ( β j ) i , m + ε i + ε ( W ) + ( p j n j + ( p − h, ℓ, h (cid:17) ≡ X α ( − d ′ ( p,β j ,N )+ P v> dim N v APS-ind τ (cid:16) D X H,α ⊗ ( K W ⊗ K − X ) / ⊗F p,j ( X ) ⊗ F iV ⊗ Q ( W ) , m + e ( p, β j , N ) , ℓ, h (cid:17) mod k Z , (5.33)where P β means the sum over all the connected components of X ( n j ). 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