Rigidity and Vanishing Theorems on ${\mathbb{Z}}/k$ Spin$^c$ manifolds

In this paper, we first establish an $S^1$-equivariant index theorem for Spin$^c$ Dirac operators on $\mathbb{Z}/k$ manifolds, then combining with the methods developed by Taubes \cite{MR998662} and Liu-Ma-Zhang \cite{MR1870666,MR2016198}, we extend Witten's rigidity theorem to the case of $\mathbb{Z}/k$ Spin$^c$ manifolds. Among others, our results resolve a conjecture of Devoto \cite{MR1405063}


Introduction
In [25], Witten derived a series of elliptic operators on the free loop space LM of a spin manifold M . In particular, the index of the formal signature operator on loop space turns out to be exactly the elliptic genus constructed by Landweber-Stong [13] and Ochanine [23] in a topological way. Motivated by physics, Witten conjectured that these elliptic operators should be rigid with respect 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 as well as various further generalizations. In particular, several new vanishing theorems were established in [15,16]. Furthermore, on the equivariant Chern character level, Liu and Ma ( [17,18]) generalized Witten's rigidity theorem to the family case, and also obtained several vanishing theorems for elliptic genera. In [19,20], inspired by [24], Liu, Ma and Zhang established the corresponding family rigidity and vanishing theorems on the equivariant K-theory level.
In [27], Zhang established an equivariant index theorem for circle actions on Z/k spin manifolds and pointed out that by combining with the analytic arguments developed in [20], one can prove an extension of Witten's rigidity theorem to Z/k spin manifolds. The purpose of this paper is to extend the result of [27] to Z/k Spin c manifolds and then establish Witten's rigidity theorem for Z/k Spin c manifolds. Recall that a Z/k manifold X is a smooth manifold with boundary ∂X which consists of k disjoint pieces, each of which is diffeomorphic to a given closed manifold Y (cf. [22]). It is interesting that for a Dirac operator D on a Z/k manifold, the APS-ind(D) mod kZ determines a topological invariant in Z/kZ, where APS-ind(D) is the index of D which is imposed the boundary condition of Atiyah-Patodi-Singer type [1]. Freed and Melrose [7] proved a mod k index theorem, APS-ind(D) mod kZ = t-ind(D) , (1.1) giving the APS-ind(D) mod kZ 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 circle action. Let R(S 1 ) denote the representation ring of S 1 . The equivariant topological index of D is defined by Freed and Melrose [7] as an element of Z/kZ ⊗ R(S 1 ), and we denote it by t-ind S 1 (D). Then there exist R n ∈ Z/kZ such that where by [n] (n ∈ Z) we mean the one dimensional complex vector space on which S 1 acts as multiplication by g n for a generator g ∈ S 1 . 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 kZ . The Dirac operator D on X is said to be rigid in Z/k category for the circle action if its equivariant topological index t-ind S 1 (D) verifies that for n ∈ Z, n = 0, one has R n = 0 in Z/kZ. (1.4) Furthermore, we say D has vanishing property in Z/k category if its equivariant topological index t-ind S 1 (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 spin manifolds as an S 1 -equivariant topological index in the sense of [7] of some twisted Dirac operator and conjectured that this mod k elliptic genus is rigid in Z/k category. In this paper, following the suggestion in [27, Remark 1], we present 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 (Sections 2 and 3), we extend the Z/k equivariant index theorem of Zhang [27] to the Spin c case. In step 2 (Sections 4 and 5), using the mod k localization index theorem established in step 1 and modifying the process in [19,20], we prove the main results of this paper.
This paper is organized as follows. In Section 2, we state an S 1 -equivariant index 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 1 -equivariant index theorem stated in Section 2. In Section 4, we prove our main results (cf. Theorem 4.1), the rigidity and vanishing theorems for Z/k Spin c manifolds, which generalize [16,Theorems A and B]. When applied to Z/k spin manifolds, our results resolve a conjecture of Devoto [6]. Section 5 is devoted to a proof of the recursive formula which has been used in Section 4 in the proof of our main results.
2 Spin c Dirac operators and a mod k localization formula In this section, for a Z/k manifold which admits a nontrivial Z/k circle action, we state a mod k localization formula for S 1 -equivariant Spin c Dirac operators, whose proof will be given in Section 3. As an application, we deduce the rigidity and vanishing property for several Dirac operators on a Z/k almost complex manifold. In particular, we extend Hattori's vanishing theorem [8] to the case of Z/k almost complex manifolds.
This section is organized as follows. In Section 2.1, we review the construction of Spin c Dirac operators on Z/k manifolds and the Atiyah-Patodi-Singer boundary problems. In Section 2.2, we recall the circle actions on Z/k manifolds and present a variation formula for the indices of these boundary problems. In Section 2.3, we state the mod k localization formula for Z/k circle actions. As an application, in Section 2.4, we extend Hattori's vanishing theorem [8] to the case of Z/k almost complex manifolds.

Spin c Dirac operators on Z/k manifolds
We first recall the definition of Z/k manifolds introduced by Morgan and Sullivan (cf. [22]). Definition 2.1 (cf. [27,Definition 1.1]) A compact Z/k manifold is a compact manifold X with boundary ∂X, which admits a decomposition ∂X = ∪ k i=1 (∂X) i into k disjoint manifolds and k diffeomorphisms π i : (∂X) i → Y to a closed manifold Y .
Let π : ∂X → Y be the induced map. In what follows, as in [27], we will call an object α (e.g., metrics, connections, etc.) of X a Z/k-object if there will be 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 of the k disjoint pieces of the boundary ∂X. Then X has the homotopy type of a CW complex, which implies that the first Chern class c 1 induces a 1-to-1 correspondence between the equivalence classes of the complex line bundles over X and the elements of H 2 (X; Z). As will be seen, this is essential in our proof.
We make the assumption that X is Z/k oriented and of dimension 2l. Let V be a Z/k real vector bundle over X which is of dimension 2p and Z/k oriented. Let L be a Z/k complex line bundle over X with the property that the vector bundle U = T X ⊕ V satisfies ω 2 (U ) = c 1 (L) mod (2), where ω 2 denotes the second Stiefel-Whitney class, and c 1 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 restriction on T ∂X. Let ǫ 0 > 0 be less than the injectivity radius of g T X . We use the inward geodesic flow to identify a neighborhood of the boundary with the collar [0, ǫ 0 ) × ∂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 0 such that one has the orthogonal splitting on U ǫ , 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 . We make 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 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 are of product structure near ∂X (cf. (2.2)).
By taking ǫ > 0 sufficiently small, one can always find the metrics 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 ) be the 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).
In the remaining part of this paper, we always fix an involution 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 2 -grading of S(U, L). Let ∇ S(U,L)⊗W be the Hermitian connection on S(U, L) ⊗ W obtained from the tensor product of ∇ S(U,L) and ∇ W .

Definition 2.2
The twisted Spin c Dirac operator D X on S(U, L) ⊗ W over X is defined by 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 first order differential operator D : Γ(S(U, L) ⊗ W ) −→ Γ(S(U, L) ⊗ W ) on X, if there exists ǫ > 0 sufficient small such that the following identity holds on U ǫ , with B independent of r, then we will call B the canonical boundary operator associated to D. When there is no confusion, we will also use B to denote its restriction on Γ(X, S(U, L) ⊗ W )| ∂X . We then recall the Atiyah-Patodi-Singer projection associated to a boundary operator (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 differential operator on ∂X. For any λ ∈ Spec (B), the spectrum of B, let E λ be the eigenspace corresponding to λ. For a ∈ R, let P ≥a be the orthogonal projection from the L 2 -completion of Γ(X, S(U, L) ⊗ W )| ∂X onto ⊕ λ≥a E λ . We call the particular projection P ≥0 the Atiyah-Patodi-Singer projection associated to B to emphasize its role in [1]. If we assume in addition that B preserves the Z 2 -grading of Γ(X, S(U, L) ⊗ W )| ∂X , and let B ± be the restrictions of B on Γ(X, S ± (U, L) ⊗ W )| ∂X , then we will restrict P ≥a on the L 2 -completions of Γ(X, S ± (U, L) ⊗ W )| ∂X and denote them by P ≥a,± .
Let e 1 = ∂ ∂r be the inward unit normal vector field perpendicular to ∂X. Let e 2 , · · · , e 2l be an oriented orthonormal basis of T ∂X so that e 1 , e 2 , · · · , e 2l is an oriented orthonormal basis of T X| ∂X . Then using parallel transport with respect to ∇ T X along the unit speed geodesics perpendicular to ∂X, e 1 , e 2 , · · · , e 2l forms an oriented orthonormal basis of T X over U ǫ . (2.6) By [1], B X is a formally self-adjoint first order elliptic differential operator intrinsically defined on ∂X, which is the canonical boundary operator associated to D X and preserves the natural Z 2 -grading of (S(U, L) ⊗ W )| ∂X .
We now recall the Dirac type operator [5, Definition 1.1] as well as the boundary condition of Atiyah-Patodi-Singer type [1]. Definition 2.4 By a Dirac type operator on S(U, L)⊗W , we mean a first order differential 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 canonical boundary 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 type operator.

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 it preserves ∂X and there exists a corresponding circle action on Y such that these two actions are compatible with π. The circle action is said to be nontrivial if it is not equal to identity.
In what follows we assume that X admits a nontrivial Z/k circle action preserving 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 may and 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 2 -grading.
Let E be a Z/k S 1 -equivariant vector bundle over X. Let E Y be the S 1equivariant 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 −1 x)) for g ∈ S 1 , s ∈ Γ(X, E ), x ∈ X. Similarly, the group S 1 acts on Γ(X, E )| ∂X and Γ(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 )) on which S 1 acts as multiplication by g ξ for g ∈ S 1 .
Theorem 2.6 (Compare with [5, Theorem 1.2]) The following identity holds, where sf is the notation for the spectral flow of [2]. In particular, Proof The proof is the same as that of [5, Theorem 1.2].

A mod k localization formula for Z/k circle actions
Let H be the canonical basis of Lie(S 1 ) = R, i.e., for t ∈ R, exp(tH) = e 2π √ −1t ∈ S 1 . 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 vector field 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 > 0 small enough, we know U ′ ǫ = U ǫ ∩ X H carries the metric naturally induced from g T X | Uǫ .
Let π : N → X H be the normal bundle to X H in X, which is identified to be the orthogonal complement of T X H in T X| X H . Then T X| X H admits a Z/k S 1 -equivariant decomposition (cf. [20, (1.8) 11) where N v is a Z/k complex vector bundle such that g ∈ S 1 acts on it by g v with v ∈ Z\{0}. We will regard N as a Z/k complex vector bundle and write N R for the underlying real vector bundle of N . Clearly, Similarly, let be the Z/k S 1 -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 bundles over X H on which g ∈ S 1 acts by g v , and V R 0 is the real subbundle of V such that S 1 acts as identity. For v = 0, let V v,R denote the underlying real vector bundle of V v . Denote by 2p ′ = dim V R 0 and 2l ′ = dim X H . Let us write (2.13) Recall that N v,R and V v,R (v = 0) are canonically oriented by their complex structures. The decompositions (2.11), (2.12) induce the orientations of T X H and V R 0 respectively. Let be the corresponding oriented orthonormal basis of (T X H , g T X H ) and (V R 0 , g V R 0 ). There are two canonical ways to consider S(T X H ⊕ V R 0 , L F ) as a Z 2 -graded vector bundle . Let ). Let 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 the metric g N , and let where ∧ and i denote the exterior and interior multiplications, respectively. Let τ N be the involution on Λ(N * ) given by Similarly, we can define the Clifford action of (2.16) where id denotes the trivial involution, and (2.17) Here we denote by ⊗ the Z 2 -graded tensor product (cf. [14, pp. 11]). Furthermore, isomorphisms (2.16), (2.17) give the identifications of the canonical connections on the bundles (compare with [20, (1.13)]). We still denote the involution on S(T X H ⊕ V R 0 , 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 metric and the Hermitian connection are both of product structure near ∂X H . We will denote by D X H ⊗ R the twisted Spin c Dirac operator on 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], Let S 1 act on L| X H by sending g ∈ S 1 to g lc (l c ∈ Z) on X H . Then l c is locally constant on X H . Following [20, (1.50)], we define the following elements (2.20) As explained in [20, pp. 139 ]. Clearly each R ±,ξ , R ′ ±,ξ (ξ ∈ Z) is a Z/k vector bundle over X H carrying a 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, Theorem 1.2] to the case of Z/k manifolds. It also generalizes the Z/k equivariant index theorem in [27, Theorem 2.1] to the case of Spin c -manifolds.
Theorem 2.7 For any ξ ∈ Z, the following identities hold, Proof The proof will be given in Section 3.

A Z/k extension of Hattori's vanishing theorem
In this subsection, we assume that T X has a Z/k S 1 -equivariant almost complex structure J. Then one has the canonical splitting where T (1,0) X and T (0,1) X are the eigenbundles of J corresponding to the eigenvalues We suppose that c 1 (T (1,0) X) = 0 mod (N ) (N ∈ Z, N ≥ 2). As explained in Section 2.1, the complex line bundle K 1/N X is well defined over X. After replacing the S 1 action by its N -fold action, we can always assume that S 1 acts on K 1/N X . For s ∈ Z, let D X ⊗ K s/N X be the twisted Spin c Dirac operator on Λ(T * (0,1) X) ⊗ K s/N X defined as in (2.4). Using Theorem 2.7, we can generalize the main result of Hattori [8] to the case of Z/k almost complex manifolds.
has vanishing property in Z/k category. In particular, the following identity holds, Proof Using the almost complex structure on T X H induced by the almost complex structure J on T X and by (2.11), we know 26) where N v are complex subbundles of T (1,0) X X H on which g ∈ S 1 acts by multiplication by g v .
We claim that for each ξ ∈ Z, the following identity holds, In fact, if X H = ∅, the empty set, by Theorem 2.7, (2.27) is obvious.

Remark 2.9
From the proof of Theorem 2.8, one also deduces that if X is a connected Z/k almost complex manifold with a nontrivial Z/k circle action, then D X , D X ⊗ K −1 X are rigid in Z/k category.

A proof of Theorem 2.7
In this section, following Zhang [27] and by making use of the analysis of Wu-Zhang [26] and Dai-Zhang [5] as well as Liu-Ma-Zhang [20], which in turn depend on the analytic localization techniques of Bismut-Lebeau [3], we present a 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 establish the Taylor expansions of D X and c(H) (resp. B X ) near the fixed point set X H (resp. ∂X H ). In Section 3.3, following [5, Section 3(b)], we decompose the Dirac type operators under consideration to a sum of four operators and introduce a deformation of the Dirac type operators as well as their associated boundary 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 certain operators and prove the Fredholm property of the Atiyah-Patodi-Singer type boundary problem for the deformed operators introduced in Section 3.3. In Section 3.5, we complete the proof of Theorem 2.7.

Witten deformation on flat spaces
Recall that H is the canonical basis of Lie(S 1 ) = R. In this subsection, let W be a complex vector space of dimension n with an Hermitian form. Let ρ be a unitary representation of the circle group S 1 on W such that all the weights are nonzero. Suppose W ± are the subspaces of W corresponding to the positive and 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 Hermitian structure on W takes the standard form and ρ is diagonal with weights λ i ∈ Z\{0} (1 ≤ i ≤ n), and λ i < 0 for i ≤ ν. The Lie algebra action on W is given by the vector field Let E be a finite dimensional complex vector space with an Hermitian form and suppose E carries a unitary representation of S 1 . Let ∂ be the twisted Dolbeault operator acting on Ω 0, * (W, E), the set of ) has discrete eigenvalues.

A Taylor expansion of certain operators near the fixedpoint set
Following [3, Section 8(e)], we now describe a coordinate system on X near X H . For ε > 0, set B ε = Z ∈ N |Z| < ε . Since X and X H are compact, there exists ε 0 > 0 such that for 0 < ε ≤ ε 0 , the exponential map From now on, we identify B ε with V ε and use the notation Let g T X H , g N be the corresponding metrics on T X H and N induced by g T X . Let dv X , dv X H and dv N be the corresponding volume elements on (T X, g T X ), Then k(y) = 1 and ∂k ∂Z (y) = 0 for y ∈ X H . The latter follows from the wellknown fact that X H is totally geodesic in X.
For x = (y, Z) ∈ V ε 0 , 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 1 -invariant connections ∇ S(U,L) and ∇ W respectively, along the geodesic t −→ (y, tZ). The induced identification of (S(U, L) ⊗ W ) x with (S(U, L) ⊗ W ) y preserves the metric and the Z 2 -grading, and moreover, is S 1 -equivariant. Consequently, D X can be considered as an operator acting on the sections of the bundle π * ((S(U, L) ⊗ W )| X H ) over B ε 0 commuting with the circle action.
For ε > 0, let E(ε) (resp. E) be the set of smooth sections of π * ((S(U, L) ⊗ W )| X H ) on B ε (resp. on the total space of N ). If f, g ∈ E have compact supports, we will write 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 totally geodesic, this splitting, when restricted to X H , is preserved by the connection We choose a local orthonormal basis of T X such that e 1 , · · · , e 2l ′ form a basis of T X H , and e 2l ′ +1 , · · · , e 2l , that of N R . Denote the horizontal lift of Clearly, D N acts along the fibers of N . Let ∂ N be the ∂-operator along the fibers of N , and let ∂ N * be its formal adjoint with respect to (3.6). It is easy For a first order differential operator On the boundary of X H , we choose the local orthonormal basis as in Definition 2.3. Similarly as in (2.6), we define on E ∂ (compare with (3.7)). Let J H be the representation of Lie(S 1 ) on N . Then Z → J H Z is a Killing vector field on N . We have the following analogue of [

A decomposition of Dirac type operators under consideration and the associated deformation
For p ≥ 0, let E p (resp. E p ∂ , E p , F p , F p ∂ ) be the set of sections of the bundles S(U, L) ⊗ W over X (resp. (S(U, L) ⊗ W )| ∂X over ∂X, π * ((S(U, which lie in the p-th Sobolev spaces. The group S 1 acts on all these spaces (cf. Section 2.2). For any ξ ∈ Z, let E p ξ , E p ξ,∂ E p ξ , F p ξ and F p ξ,∂ be the corresponding weight-ξ subspaces, respectively.
Let the image of I T,ξ from F p ξ be E p T,ξ = I T,ξ F p ξ ⊆ E p ξ . Denote the orthogonal complement of E 0 T,ξ in E 0 ξ by E 0,⊥ T,ξ , and let E p,⊥ T,ξ = E p ξ ∩ E 0,⊥ T,ξ . Let p T,ξ and p ⊥ T,ξ be the orthogonal projections from E 0 ξ to E 0 T,ξ and E 0,⊥ T,ξ respectively. We denote by v |v| dim N v with respect to the given circle action. Let q ξ be the orthogonal bundle projection from the vector bundle We now proceed to deduce a formula which computes p T,ξ s for s ∈ E 0 ξ explicitly under a local unitary trivialization of N .

21)
For T > 0, set   One verifies that B T,ξ (u) is the canonical boundary operator associated to D T,ξ (u) in the sense of (2.5).

Various estimates of the operators as T → +∞
We continue the discussion in the previous subsection. Corresponding to the involution τ on S(U, L), for τ = τ s (resp. τ = τ e ), let D X H ξ be the restriction of the twisted Spin c Dirac operator D X H ⊗ R + (1) (resp. D X H ⊗ R − (1)) on F 0 ξ , and let B X H ξ be the restriction of the canonical boundary operator associated to D X H ⊗ R + (1) (resp. D X H ⊗ R − (1)) on F 0 ξ,∂ . With (3.15), (3.23) and Propositions 3.1, 3.2, 3.3 at our hands, by proceeding exactly as in [3, Sections 8 and 9], we can show that the following estimates for B

28)
where O( 1 √ T ) denotes a first order differential operator whose coefficients are dominated by C √ T (C > 0).

29)
and D With Proposition 3.7 at our hands, we can complete the proof of Proposition 3.6 in the same way as in the proof of [5, Proposition 3.5] by applying the gluing argument in [3, pp. 115-117].

A proof of Theorem 2.7
Let D Y H ξ be the induced operator from B X H ξ through π X H . We first assume  Let P T,ξ,1 (resp. P T,ξ,4 ) be the Atiyah-Patodi-Singer projection associated to B (1) T,ξ (resp. B T,ξ J T,ξ,∂ . The proof of Theorem 3.9 is completed.
In general, dim ker D Y H ξ need not be zero. For any ξ ∈ Z, choose a ξ > 0 be such that To control the eigenvalues of B T,ξ near zero, we use the method in [5, Section 4(a)] to perturb the Dirac operators under consideration. Let ǫ > 0 be sufficiently small so that there exists an S 1 -invariant smooth function f : X −→ R such that f ≡ 1 on U ǫ/3 and f ≡ 0 outside of U 2ǫ/3 .
Let D X H ξ,−a ξ be the Dirac type operator defined by where for τ = τ s (resp. τ e ), D X H ξ,−a ξ is considered as a differential operator acting Let D T,ξ,−a ξ be its restriction to the weight-ξ subspace.
Let B X H ξ,−a ξ be the canonical boundary operator of D X H ξ,−a ξ in the sense of (2.5). Since D Y H ξ − a ξ , which is the induced operator from B X H ξ,−a ξ through π X H , is invertible, by the proof of Theorem 3.9, we get when T is large enough, By taking τ = τ s (resp. τ e ), we get the first equation of (2.22) (resp. (2.23)). To get the second equation of (2.22) (resp. (2.23)), we only need to apply the first equation of (2.22) (resp. (2.23)) to the case where the circle action on X defined by the inverse of the original circle action on X.
The proof of Theorem 2.7 is completed.

Rigidity and vanishing theorems on Z/k Spin c manifolds
In this section, combining the S 1 -equivariant index theorem we have established in Section 2 with the methods of [19], we prove the rigidity and vanishing theorems for Z/k Spin c manifolds, which generalize [16, Theorems A and B]. As will be pointed out in Remark 4.3, when applied to Z/k spin manifolds, our results provide a resolution to a conjecture of Devote [6]. Both the statement of the main results and their proof are inspired by the corresponding results as well as their proof for closed manifolds in [19,20]. As explained in Section 2.1, when we regard the considered Z/k manifold as a quotient space which has the homotopy type of a CW complex, by using splitting principle [11,Chapter 17], we can apply the topological arguments in [19,20] in our Z/k context with little modification. Thus we will only indicate the main steps of the proof of our results.
This section is organized as follows. In Section 4.1, we state our main results, the rigidity and vanishing theorems for Z/k Spin c manifolds. In Section 4.2, we present two recursive formulas which will be used to prove our main results stated in Section 4.1. In Section 4.3, we prove the rigidity and vanishing theorems for Z/k Spin c manifolds.

Rigidity and vanishing theorems
Let X be a 2l-dimensional Z/k manifold, which admits a nontrivial Z/k circle action. We assume that T X has a Z/k S 1 -equivariant Spin c structure. Let V be an even dimensional Z/k real vector bundle over X. We assume that V has a Z/k S 1 -equivariant spin structure. Let W be a Z/k S 1 -equivariant complex vector bundle of rank r over X. Let K W = det(W ) be the determinant line bundle of W , which is obviously a Z/k complex line bundle.
Let K X be the Z/k complex line bundle over X induced by the Spin c structure of T X. Let S(T X, K X ) be the complex spinor bundle of (T X, K X ) as in Section 2.  For N ∈ N, let y = e 2πi/N ∈ C be an N th root of unity. Set (4.2) is a module over H * (BS 1 , Z) induced by the projection π : X × S 1 ES 1 → BS 1 . Let p 1 (·) S 1 and ω 2 (·) S 1 denote the first S 1 -equivariant pontrjagin class and the second S 1 -equivariant Stiefel-Whitney class, respectively. As V × S 1 ES 1 is spin over X × S 1 ES 1 , one knows that [24, pp. 456-457]). Recall that with u a generator of degree 2.
In the following, we denote by D X ⊗ R the twisted Spin c Dirac operator acting on S(T X, K X ) ⊗ R (cf. Definition 2.2). Furthermore, for m ∈ 1 2 Z, h ∈ Z and R(q) = m∈ 1 2 Z q m R m ∈ K S 1 (X)[[q 1/2 ]], we will also denote APS-ind(D X ⊗ R m , h) (cf. (2.9)) by APS-ind(D X ⊗ R(q), m, h). Now we can state the main results of this paper as follows, which generalize [16, Theorems A and B] to the case of Z/k Spin c manifolds.
, and c 1 (W ) = 0 mod (N ). For 0 ≤ ℓ < N , i = 1, 2, 3, 4, consider the S 1 -equivariant twisted Spin c Dirac operators   (2). We note that in our case, X × S 1 ES 1 has the homotopy type of a CW complex [21]. By [9, Corollary 1.2], the circle action on X can be lifted to (K W ⊗ K −1 X ) 1/2 and is compatible with the circle action on K W ⊗ K −1 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].
Actually, as in [19], our proof of Theorem 4.1 works under the following slightly weaker hypothesis. Let us first explain some notations.
For each n > 1, consider Z n ⊂ S 1 , the cyclic subgroup of order n. We have the Z n -equivariant cohomology of X defined by H * Zn (X, Z) = H * (X × Zn ES 1 , Z), and there is a natural "forgetful" map α(S 1 , Z n ) : X × Zn ES 1 → X × S 1 ES 1 which induces a pullback α(S 1 , Z n ) * : H * S 1 (X, Z) → H * Zn (X, Z). We denote by α(S 1 , 1) the arrow which forgets the S 1 -action. Thus α(S 1 , 1) * : Finally, note that if Z n acts trivially on a space M , then there is a new arrow t * : Let Z ∞ = S 1 . For each 1 < n ≤ +∞, let i : X(n) → X be the inclusion of the fixed point set of Z n ⊂ S 1 in X, and so i induces i S 1 : 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 ≤ +∞, Remark that the relation (4.6) clearly follows from the hypothesis of Theorem 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 0 ∈ G y on W (resp. W ) by multiplication by y 0 (resp. y −1 0 ) on W (resp. W ). Set 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 , In what follows, for m ∈ 1 2 Z, 0 ≤ ℓ < N , h ∈ Z and R(q) ∈ K S 1 (X)[[q 1/2 ]], 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.
Sym q n (T X) (4.11) In particular, one has (4.12) The rest of this paper is devoted to a proof of Theorem 4.4.

Several intermediate results
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 13) where N v is the complex vector bundles on which S 1 acts by sending g to g v . We also assume that where V v , W v are complex vector bundles on which S 1 acts by sending g to g v , and V R 0 is a real vector bundle on which S 1 acts as identity. By (4.13), as in (2.16) or (2.17), there is a natural Z/k isomorphism between the Z 2 -graded C(T X)-Clifford modules over X H , For a Z/k complex vector bundle R over X H , let D X H ⊗ R, D X H,α ⊗ R be the twisted Spin c Dirac operators on S(T X H , K X ⊗ v>0 (det N v ) −1 ) ⊗ R over X H , X H,α , respectively (cf. Definition 2.2).
For i = 1, 2, 3, 4, we set Then by Theorem 2.7, we can express the global Atiyah-Patodi-Singer index via the Atiyah-Patodi-Singer indices on the fixed point set up to kZ. (4.17) To simplify the notations, we use the same convention as in [19, pp. 945].
For n 0 ∈ N * , we define a number operator P on n 0 ]], then P acts on R(q) by multiplication by n on R n . From now on, we simply denote Sym q n (T X), Λ q n (V ) and Λ 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∈ 1 n 0 we will also denote APS-ind D X ⊗ R m , h by APS-ind D X ⊗ R(q), m, h .
Recall that H is the Killing vector field on X corresponding to H, the canonical basis of Lie(S 1 ). If E is a Z/k S 1 -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 Recall that the Z 2 -grading on S(T X, K X ) ⊗ ∞ n=1 Sym(T X n ) is induced by the Z 2 -grading on S(T X, K X ). Write There are two natural Z 2 -gradings on F 1 V , F 2 V (resp. Q 1 (W )). The first grading is induced by the Z 2 -grading of S(V ) and the forms of homogeneous degrees in Λ(V n ) (resp. Q 1 (W )). We define τ e | F i± V = ±1 (i = 1, 2) (resp. τ 1 | Q 1 (W ) ± = ±1) to be the involution defined by this Z 2 -grading. The second grading is the one for which F i V (i = 1, 2) are purely even, i.e., F i+ V = F i V . We denote by τ s = id the involution defined by this Z 2 -grading. Then the coefficient of q n (n ∈ 1 2 Z) in (4.1) of R 1 (V ) or R 2 (V ) (resp. R 3 (V ), R 4 (V ) or Q(W )) is exactly the Z 2 -graded Z/k vector subbundle of (F 1 V , τ s ) or (F 1 V , τ e ) (resp. (F 2 V , τ e ), (F 2 V , τ s ) or (Q 1 (W ), τ 1 )), on which P acts by multiplication by n.
Furthermore, we denote by τ e (resp. τ s ) the Z 2 -grading on S(T X, K X ) ⊗ ⊗ ∞ n=1 Sym(T X n ) ⊗ F i V (i = 1, 2) induced by the above Z 2 -gradings. We will denote by τ e1 (resp. τ s1 ) the Z 2 -grading on S(T X, Let h Vv be the Hermitian metric on V v induced by the metric h V on V . In the following, we identity ΛV v with ΛV * v by using the Hermitian metric h Vv on V v . By (4.14), as in (4.15), there is a natural Z/k isomorphism between the Z 2 -graded C(V )-Clifford modules over X H , Let V 0 = V R 0 ⊗ R C. By using the above notations, we rewrite (4.18) on the fixed point set X H , (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 , (4.22) Furthermore, for p ∈ N, we introduce the following elements in K S 1 (X H )[[q]] (cf. [19, (3.6)]), (4.23) Note that when p = 0, F −p (X) is exactly the F 0 (X) in (4.21).
As in [19, (2.9)], we write Using the similar Z/k S 1 -equivariant isomorphism of complex vector bundles as in [20, (3.14)] and the similar Z/k G y × S 1 -equivariant isomorphism of complex vector bundles as in [19, (3.15) and (3.16)], by direct calculation, we deduce the following proposition.
For any p ∈ Z, p > 0, there is a natural Z/k G y × S 1 -equivariant isomorphism of vector bundles over X H , On X H , as in [19, (2.8)], we write (4.28) As indicated in Section 2.1, (4.28) means L is a trivial complex line bundle over each component X H,α of X H , and S 1 acts on L by sending g to g 2e , and G y acts on L by sending y to y d ′ (W ) . The following proposition is deduced from Proposition 4.6.

A proof of Theorem 4.4
Recall we assume in Theorem 4.1 that c 1 (W ) = 0 mod (N ). Then by [10,Section 8] and [19,Lemma 2.1], d ′ (W ) mod(N ) is constant on each connected component X H,α of X H . So we can extend L to a trivial complex line bundle over X, and we extend the S 1 -action on it by sending g on the canonical section 1 of L to g 2e · 1, and G y acts on L by sending y to y d ′ (W ) .
The proof of Theorem 4.4 is completed.

A proof of Theorem 4.9
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 the same refined shift operators as in [19,Section 4.2]. In Section 5.2, we construct the twisted Spin c Dirac operator on X(n j ), the fixed point set of the naturally induced Z n j -action on X. In Section 5.3, by applying the S 1 -equivariant index theorem we have established in Section 2, we prove Theorem 4.9.
For 0 ≤ j ≤ J 0 , p ∈ N * , we write Clearly, I p 0 = ∅, the empty set. We define F p,j (X) as in [19, (2.21)], which are analogous with (4.23). More specifically, we set where we use the notation that for s ∈ R, [s] denotes the greatest integer which is less than or equal to s. Then From the construction of β i , we know that for v ∈ J, there is no integer in We use the same shift operators r j * , 1 ≤ j ≤ J 0 as in [19, (4.21)], which refine the shift operator r * defined in (4.22). For p ∈ N * , set and Q W (β j ) as in [19, (4.13)].
F(β j ) = 0<n∈Z Sym (T X H,n ) ⊗ v>0,v≡0, n j 2 mod(n j ) 0<n∈Z+ Using the definition of r j * and computing directly, we get an analogue of Proposition 4.6 as follows.
There is a natural Z/k G y × S 1 -equivariant isomorphism of vector bundles over X H ,

The Spin c Dirac operators on X(n j )
Recall that there is a nontrivial Z/k circle action on X which can be lifted to the Z/k circle actions on V and W .
For n ∈ N * , let Z n ⊂ S 1 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 ) be the 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 where V (n j ) R 0 is the Z/k real vector bundle on which Z n j acts by identity, and N (n j ) R n j /2 (resp. V (n j ) R n j /2 ) is defined to be zero if n j is odd. Moreover, for 0 < v < n j /2, N (n j ) v (resp. V (n j ) v ) admits unique Z/k complex structure such that N (n j ) v (resp. V (n j ) v ) becomes a Z/k complex vector bundle on which g ∈ Z n j acts by g v . We also denote by V (n j ) 0 , V (n j ) n j /2 and N (n j ) n j /2 the corresponding complexification of V (n j ) R 0 , V (n j ) R n j /2 and N (n j ) R n j /2 . Similarly, we also have the following Z n j -equivariant decomposition of W , as Z/k complex vector bundles over X(n j ), where for 0 ≤ v < n j , g ∈ Z n j acts on W (n j ) v by sending g to g v . By [ , we know that the Z/k vector bundles T X(n j ) and V (n j ) R 0 are orientable and even dimensional. Thus N (n j ) is orientable over X(n j ). By (5.9), V (n j ) R n j /2 and N (n j ) R n j /2 are also orientable and even dimensional. In what follows, we fix the orientations of N (n j ) R n j /2 and V (n j ) R n j /2 . Then T X(n j ) and V (n j ) R 0 are naturally oriented by (5.9) and the orientations of T X, V , N (n j ) R n j /2 and V (n j ) R n j /2 . By (4.13), (4.14), (5.9) and (5.10), upon restriction to X H , we get the following identifications of Z/k complex vector bundles (cf. [19, (4.9) and (4.12)]), Also we get the following identifications of Z/k real vector bundles over X H (cf. [19, (4.11)]), T X(n j ) = T X H ⊕ v>0, v≡0 mod (n j ) N v , N (n j ) R n j /2 = v>0, v≡ n j 2 mod (n j ) Moreover, we have the identifications of Z/k complex vector bundles over X H as follows, T X(n j ) ⊗ R C = T X H ⊗ R C ⊕ v>0, v≡0 mod(n j ) As (p j , n j ) = 1, we know that, for v ∈ Z, p j v/n j ∈ Z if and only if v/n j ∈ Z. Also, p j v/n j ∈ Z + 1 2 if and only if v/n j ∈ Z + 1 2 . Remark if v ≡ −v ′ mod(n j ), then {n | 0 < n ∈ Z + p j n j v} = {n | 0 < n ∈ Z − p j n j v ′ }. Using the identifications (5.11), (5.12) and (5.13), we can rewrite F(β j ), F 1 V (β j ), F 2 V (β j ) and Q W (β j ) defined in (5.8) as follows (cf. [19, (4.7)]), Sym N (n j ) n j /2,n , (5.14) Thus F(β j ), F 1 V (β j ), F 2 V (β j ) and Q W (β j ) can be extended to Z/k vector bundles over X(n j ).
We now define the Spin c Dirac operators on X(n j ) following [19,Section 4.1].
Consider the hypothesis in (4.6). By splitting principle [11,Chapter 17] and computing as in [4, Lemmas 11.3 and 11.4], we get +r(n j ) · n j 2 · ω 2 W (N j ) n j /2 + V (n j ) n j /2 − N (n j ) n j /2 · u n j = 0 , where r(n j ) = 1 2 (1 + (−1) n j ), and u n j ∈ H 2 (BZ n j , Z) ≃ Z n j is the generator of H * (BZ n j , Z) ≃ Z[u n j ]/(n j · u n j ) . Then by (5.18), we know that 0<v< n j 2 v · c 1 V (n j ) v + W (n j ) v − W (n j ) n j −v − N (n j ) v + r(n j ) · n j 2 · ω 2 W (n j ) n j /2 + V (n j ) n j /2 − N (n j ) n j /2 is divided by n j . Therefore, we have Lemma 5.2 (cf. [19,Lemma 4.2]) Assume that (4.6) holds. Let be the complex line bundle over X(n j ). Then we have (i) L(n j ) has an n th j root over X(n j ).
Then U 1 (resp. U 2 ) has a Z/k Spin c structure defined by L 1 (resp. L 2 ).
Remark that in order to define an S 1 (resp. G y ) action on L(n j ) r(n j )/n j , we must replace the S 1 (resp. G y ) action by its n j -fold action. Here by abusing notation, we still say an S 1 (resp. G y ) action without causing any confusion.
In what follows, by D X(n j ) we mean the S 1 -equivariant Spin c Dirac operator on S(U 1 , L 1 ) or S(U 2 , L 2 ) over X(n j ) (cf. Definition 2.2).
The proof of Theorem 4.9 is completed.