On orbifold elliptic genus

arXiv:math/0109005v1 [math.DG] 1 Sep 2001

On orbifold elliptic genus

Chongying DONG¹, Kefeng LIU², Xiaonan MA³

0 Introduction.

Elliptic genus was derived as the partition function in quantum field theory [26].

Mathematically it is the beautiful combination of topology of manifolds, index theory and modular forms (cf. [15], [10]). Elliptic genus for smooth manifolds has been well-studied.

Recently, Borisov and Libgober ([3], [4]) proposed some definitions of elliptic genus for certain singular spaces, especially for complex orbifolds which is a global quotient M/G, here the finite group G acts holomorphically on complex variety M. Similar definitions have been introduced by string theorists in the 80s, in the study of orbifold string theory.

One of their guiding principles is the modular invariance. More recently orbifold string theory has attracted the attention of geometers and topologists. For example Chen and Ruan (cf. [7], [22]) have defined orbifold cohomology and orbifold quantum cohomology groups.

One of most important properties for elliptic genus is its rigidity property under compact Lie group action. For smooth manifolds, the rigidity and its generalizations have been well studied. Since orbifold elliptic genus is the partition function of orbifold string theory, it is natural to expect the rigidity property for orbifold elliptic genus.

Although the global quotients from a very important class of orbifolds, many interesting orbifolds are not global quotients. For example, most of the Calabi-Yau hypersurfaces of weighted projective spaces are not global quotients. In this paper we define elliptic genus for general orbifolds which generalizes the definition of Borisov and Libgober, and prove their rigidity property. We actually introduce the more general elliptic genus involving twisted bundles and proved its rigidity. The idea of considering the weights in the definition of orbifold elliptic genus comes from [4] and our proof of the K-theory version of Witten rigidity theorems [21, §4]. The proof of the rigidity is again a combination of modular invariance and the index theory. Only more complicated combinatorics are involved in the definition and the proof.

This paper is organized as follows: In Sections 1 and 2 we review the equivariant index theorem on orbifolds. We define orbifold elliptic genus and prove its rigidity for almost complex orbifolds in Section 3. Finally in Section 4 we introduce orbifold elliptic genus for spin orbifolds, and will study its rigidity property on a later occasion.

The authors would like to than Jian Zhou for many interesting discussions regarding orbifold elliptic genus.

1Mathematics Department, University of California, Santa Cruz, CA 95064, U.S.A.

(dong@math.ucsc.edu), Partially supported by NSF grant DMS-9987656 and a research grant from the Committee on Research, UC Santa Cruz.

2Department of Mathematics, UCLA,CA 90095-1555, USA (liu@math.ucla.edu). Partially supported by the Sloan Fellowship and a NSF grant.

3Humboldt-Universit¨at zu Berlin, Institut f¨ur Mathematik, Rudower Chaussee 25, 12489 Berlin, Germany. (xiaonan@mathematik.hu-berlin.de). Partially supported by SFB 288.

1

In this section and the next section we recall the notations on orbifolds, and explain the equivariant index theorem for orbifolds (cf. [8, Chap. 14], [25]).

We first recall the definition of orbifolds, which are called V-manifolds in [11], [23].

We consider the pair (G, V), here V is a connected smooth manifold, G is a finite group acting smoothly and effectively on V. A morphism Φ : (G, V) → (G^′, V^′) is a family of open embedding ϕ :V →V^′ satisfying:

i) For eachϕ∈Φ, there is an injective group homomorphism λϕ :G→G^′ such that ϕ isλϕ-equivariant.

ii) For g ∈ G^′, ϕ ∈ Φ, we define gϕ : V → V^′ by (gϕ)(x) = gϕ(x) for x ∈ V. If (gϕ)(V)∩ϕ(V)6=φ, then g ∈λϕ(G).

iii)For ϕ∈Φ, we have Φ ={gϕ, g∈G^′}.

The morphism Φ induces a unique open embeddingiΦ :V /G→V^′/G^′of orbit spaces.

Definition 1.1 An orbifold (X,U) is a paracompact Hausdorff space X together with a covering U of X consisting of connected open subsets such that

i) For U ∈ U, V(U) = ((GU,Ue) →^τ U) is a ramified covering Ue → U giving an identification U ≃U /Ge U.

ii) For U, V ∈ U, U ⊂V, there is a morphismϕV U : (GU,Ue)→(GV,Ve) that covers the inclusion U ⊂V.

iii) ForU, V, W ∈ U, U ⊂V ⊂W, we have ϕ_{W U} =ϕ_{W V} ◦ϕ_{V U}.

In the above definition, we can replace (G, V) by a category of manifolds with an additional structure such as orientation, Riemannian metric or complex structure. We understand that the morphisms (and the groups) preserve the specified structure. So we can define oriented, Riemannian or complex orbifolds.

Remark: Let G be a compact Lie group and M a smooth manifold with a smooth G-action. We assume that the action of Gis effective and infinitesimally free. Then the quotient space M/G is an orbifold. Reciprocally, any orbifold X can be presented this way. For example, let O(X) be the total space of the associated tangential orthonormal frame bundle. We know thatO(X) is a smooth manifold and the action of the orthogonal groupO(n) (n= dimX) is infinitesimally free onO(X). TheX is identified canonically with the orbifold O(X)/O(n).

LetX be an oriented orbifold, with singular set ΣX.Forx∈X, there exists a small neighbourhood (Gx,Uex) →^τx Ux such that xe= τ_x⁻¹(x) ∈ Uex is a fixed point of Gx. Such Gx is unique up to isomorphisms for each x ∈ X [23, p468]. Let (1), (h¹_x),· · ·(h^ρ_x^x) be all the conjugacy classes in G_x. Let Z_Gx(h^j_x) be the centralizer of h^j_x in G_x. One also denotes by Ue_x^hjx the fixed points ofh^j_x in Uex. There is a natural bijection

{(y,(h^j_y))|y∈Ux, j = 1,· · ·ρy} (1.1)

≃ ∐^ρ_j=1^x Ue_x^hjx/ZGx(h^j_x).

So we can define globally [11, p77],

ΣXg ={(x,(h^j_x))|x∈X, Gx 6= 1, j = 1,· · ·ρx}.

(1.2)

Then ΣXg has a natural orbifold structure defined by n(ZGx(h^j_x)/K_x^j,Ue_x^hjx)→Ue_x^hjx/ZGx(h^j_x)o

(x,Ux,j). (1.3)

Here K_x^j is the kernel of the representation ZGx(h^j_x) → Diffeo (Ue_x^hjx). The number m=

|K_x^j| is called the multiplicity of ΣXg in X at (x, h^j_x). Since the multiplicity is locally constant on ΣX, we may assign the multiplicityg mi to each connected component Xi of ΣX. In a senseg ΣXg is a resolution of singularities of X.

Definition 1.2 A mapping π from an orbifold X to an orbifold X^′ is called smooth if forx∈X, y =π(x), there exist orbifold charts(Gx,Uex),(G^′_y,fU^′y)together with a smooth mapping φ : Uex → fU^′y and a homomorphism ρ : Gx → G^′_y such that φ is ρ-equivariant and τ_y^′ ◦φ =π◦τx.

Definition 1.3 An orbifold vector bundleξ over an orbifold(X,U)is defined as follows:

ξ is an orbifold and for U ∈ U, (G^ξ_U,peU : ξeU → Ue) is a G^ξ_U-equivariant vector bundle and (G^ξ_U,ξeU) (resp. (G^ξ_U/KU,Ue), KU = Ker(G^ξ_U → Diffeo(Ue))) is the orbifold structure of ξ (resp. X). In general, G^ξ_U does not act effectively on Ue, i.e. KU 6={1}. IfG^ξ_U acts effectively on Ue for U ∈ U, we sayξ is a proper orbifold vector bundle.

Remark: Let Gbe a compact Lie group acting effectivelty and infinitesimally freely onM. Then each G-equivariant bundle E →M defines a proper orbifold vector bundle E/G→M/G, and vice versa.

In the following, we will always denote by (Gx,Uex) (x ∈ X) the orbifold chart as above. For h ∈Gx, we have the following h-equivariant decomposition of TUex⊗RCas a real vector bundle on Ue_x^h,

TUe_x⊗RC= M

λ∈Q∩]0,1[

N_λ(h)M

TUe_x^h⊗RC.

(1.4)

Here Nλ(h) is the complex vector bundle over Ue_x^h with h acting by e^2πiλ on it. The complex conjugation provides a Canti-linear isomorphism between Nλ(h) and N(1−λ)(h). If the order of h is even, this produces a real structure on N¹

2(h), so this bundle is the complexification of a real vector bundle N^R1

2(h) onUe_x^h. Thus,TUe_x is isomorphic, as a real vector bundle, to

TUex ≃ M

λ∈Q∩]0,¹₂[

Nλ(h)⊕N^R1 2(h)

MTUe_x^h. (1.5)

Note that N_λ(h) (resp. N^R1

2(h)) extends to complex (resp. real) vector bundle on ΣX. Weg will still denote them by Nλ(h), N^R1

2(h).

Assume that a compact Lie group H acts differentially on X. For γ ∈ H, let X^γ = {x∈X, γx=x}. In the index theorem, we will use the following orbifold as fixed point set of γ which is a resolution of singularities of X^γ [8, p180]. For x∈X^γ, then on local chart (Gx,Uex),γ_Ue acts onUex as a linear map. The compatibility condition forγ_Ue means that there exists an automorphismαofG_x such that for eachg ∈G_x,γ_Ue◦g◦γ⁻¹_e

U =α(g).

For h∈G_x, let (h)_γ ={ghα(g)⁻¹;g ∈G_x} be the γ conjugacy class inG_x. Let Ub_x^(h)γ ={(y, h1)∈Uex×Gx|(h1◦γ_Ue)(y) =y, h1 ∈(h)γ}.

(1.6)

LetUex^h◦γUe be the fixed point set ofh◦γ_Ue inUe_x, thenUex^h◦γUe is connected, andx∈Uex^h◦γUe. Forg ∈Gx,g acts on Ubx^(h)γ by the transformation

(y, h)→(g(y), g◦h◦α(g)⁻¹).

Indeed, if (h◦γ_Ue)(y) =y, as α(g)⁻¹◦γ_Ue =γ_Ue ◦g⁻¹◦γ⁻¹_e

U ◦γ_Ue =γ_Ue ◦g⁻¹, we know (gh◦α(g)⁻¹)γ_Ue ◦g(y) =gh◦γ_Ue(y) = g(y).

(1.7)

LetZ_h,G^γ _x ={g ∈Gx, gh◦γ(g)⁻¹ =h}, K_h,G^γ _x = Ker{Z_h,G^γ _x →Diffeo(Uex^h◦γUe)}. Then (Z_h,G^γ _x/K_h,G^γ _x,Uex^h◦γUe)→Uex^h◦γUe/Z_h,G^γ _x =Ub_x^(h)γ/Gx

(1.8)

defines an orbifold. We denote it by Xe^γ. Clearly, m(Xe^γ) =|K_h,G^γ _x| is local constant on Xe^γ.

Definition 1.4 The oriented orbifold X is spin if there exists 2-sheeted covering of SO(X)(SO(X) is the oriented orthonormal frame bundle of T X), such that for U ∈ U, there exists a principal Spin(n) bunlde Spin(Ue) on Ue, such that Spin(X)|U → SO(X)|U

is induced by Spin(Ue)→SO(Ue), and Spin(Ue) also verifies the corresponding compatible condition.

Then spin(X) is clearly a smooth manifold.

Assume that orbifoldX is spin. Leth^{T X} be a metric onT X andS(T X) =S⁺(T X)⊕

S⁻(T X) the corresponding orbifold spinor bundle on X. Let c(·) be the Clifford action of T X on S(T X). Let ∇^{S(T X)} be the connection on S(T X) induced by the Levi-Civita connection ∇^{T X} onT X. Let W be a complex orbifold vector bundle on X. Let ∇^W be a connection on W. Then∇^{S(T X)⊗W} = ∇^{S(T X)}⊗1 + 1⊗ ∇^W is a connection onX. Let Γ(S^±(T X)⊗W) be the set of C^∞ sections ofS^±(T X)⊗W on X. Let D^X ⊗W be the Dirac operator on Γ(S⁺(T X)⊗W) to Γ(S⁻(T X)⊗W), defined by

D^X ⊗W =c(ei)∇^S_ei^{+(T X)⊗W}. (1.9)

Here {ei} is an orthonrmal basis of T X.

LetHbe a compact Lie group. Ifγ ∈H acts onXand lifts to Spin(X) andW. Then

∇^{S(T X)}isγ invariant and we can always findγ inavriant connection∇^W onW. Note that D^X ⊗W is a γ invariant elliptic operator on X. For x∈X, let K_x^W = Ker(G^W_x →^τ Gx).

On Ub^(h)γ, let N be the normal bundle of Ue^h◦γUe in Uex. Let W⁰ be the subbundle of W on Ubx which is K_x^W-invariant. Then W⁰ extends to a proper orbifold vector bundle on X. We have the following decompositions:

N =⊕0<θ<πNθ⊕Nπ, W⁰ =⊕0≤θ<2πWθ, (1.10)

where Nθ, Wθ (resp. Nπ) are complex (resp. real) vector bundle on which h◦γ_Ue acts as multiplication by e^iθ. Then ∇^{T X} induces connection ∇^Nθ on Nθ, and ∇^{T X} = ⊕∇^Nθ ⊕

∇^{T Xγ}. Let R^W, R^W0, R^Nθ, R^{T Xγ} be the curvatures of ∇^W,∇^W0,∇^Nθ,∇^{T Xγ} (∇^W0 is the connection on W⁰ induced by ∇^W).

Definition 1.5 For h∈Gx, g =h◦γ_Ue, 0< θ ≤π, we write (1.11)

chg(W,∇^W) = _|K¹W x |

P

h1∈G^W_x ,τ(h1)=hTrh

(h1◦γ_Ue) exp(^−R_2πi^W)i

= Trh

gexp(^−R_2πi^W0)i , A(Tb Ue^g,∇^TUeg) = det^{1/2 i}

4πR^TUeg sinh(_4πⁱ R^TUeg)

, Abθ(Nθ,∇^Nθ) = 1

i^12dimNθdet^1/2

1−gexp(_2πⁱ R^Nθ), Abg(N,∇^N) = Π0<θ≤πAbθ(Nθ,∇^Nθ).

If we denote by{xj,−xj}(j = 1,· · · , l) the Chern roots ofNθ,TUe^g (Here we consider Nθ as a real vector bundle) such that Πxj defines the orientation of Nθ and TUe^g, then

A(Tb Ue^g,∇^TUe) = Πj

xj

2/sinh(xj

2), Abθ(Nθ,∇^Nθ) = 2^−lΠ^l_j=1 1

sinh¹₂(xj +iθ) = Π^l_j=1 e^12(xj+iθ) e^xj+iθ −1. (1.12)

Recall that for γ ∈ H, the Lefschetz number Indγ(D^X ⊗W), which is the index of D^X ⊗W if γ = 1, is defined by

Indγ(D^X ⊗W) = Trγ|KerD^X⊗W −Trγ|CokerD^X⊗W. (1.13)

By using heat kernel, as in [8, Th. 14.1], we get

Theorem 1.1 For γ ∈H, we have the following equality:

Indγ(D^X ⊗W) = X

F∈Xe^γ

1 m(F)

Z

F

αF, (1.14)

where αF is the characteristic class

A(Tb Ue^h◦γUe,∇^TUeh◦γUe)Π0<θ≤πAbθ(Nθ,∇^Nθ)chh◦γ(W,∇^W) on Uex^h◦γUe.

LetS¹ act differentiably onX. Let X^S1 = {x∈ X, γ(x) =x,for allγ ∈S¹}. Let V be the canonical basis of Lie(S¹) =R. For x∈X, let VX be the smooth vector field on (G_x,Ue_x) corresponding to V. Then V_X isG_x-invariant [8, p181]. We still denote by V_X the corresponding smooth vector field on X. We have X^S1 ={x∈X, VX(x) = 0}.

Forx∈X, let (1),· · ·(h^j_x),· · · be the conjugacy classes ofGx. LetXe^S1 ={(x,(h^j_x))|x∈ X^S1, h^j_x∈Gx}. Then Xe^S1 has a natural orbifold structure defined by

{ZGx(h^j_x)/K_x^j,V,Ue_V^hjx=Ue_x^hjx ∩ {y∈Uex|VX(y) = 0}} →(Ue_V^hjx/ZGx(h^j_x),(h^j_x)), (1.15)

where K_x^j,V is the kernel ofZGx(h^j_x)→Diffeo{Ue_V^hjx}.

We have the following decomposition of smooth vector bundles onUe_V^h : N_λ(h) =⊕_jN_λ,j,

N^R1

2(h) =⊕j>0N¹

2,j⊕N^R1 2,0, TUe^h =⊕j>0N0,j⊕TUe_V^hjx, W⁰ =⊕_λ,jW_λ,j⁰ .

(1.16)

Note that Nλ,j, N¹

2,j, N0,j and W_λ,j⁰ extend to complex vector bundles on Xe^S1, and γ = e^2πit ∈S¹ acts on them as multiplication by e^2πijt. Also, N^R1

2,0 and TUe_V^hjx extend to real vector bundles onXe^S1, andS¹ acts on them as identity. In fact,TUe^h =TUe_V^hjx⊕v6=0N0,v,R, whereN0,v,R denotes the undeling real bundle of the complex vector bundleNv on which g ∈S¹ acts by multiplying by g^v. Since we can choose either Nv orNv as the complex vector bundle for Nv,R, in what follows, we always assume N¹

2,j, N0,j are zero ifj <0.

By (1.16), for given a∈C, the eigenspace of h◦γ_Ue with eigenvalue a is equal to the sum of the above elements Nλ,j such that

e^2πi(λ+tj) =a.

(1.17)

Let A ⊂ R consist of a ∈ R such that there exists x ∈ X^S1, more than one non-zero Nλ,j on Ue_V^hjx are in the eigenspace of h◦γ_Ue with eigenvaluee^2πia. As X is compact,A is a discrete set of R.

If γ = e^2πit, t ∈ R\A, then Xe^γ = Xe^S1 by the construction. An immediate conse- quence of Theorem 1.1 is the following.

Theorem 1.2 Under the condition of Theorem 1.1, for t∈ R\A, γ =e^2πit, we have Indγ(D^X ⊗W) = X

F∈Xe^S1

1 m(F)

Z

F

αF, (1.18)

where αF is the characteristic class

A(Tb Ue_V^h,∇^TUeVh)X

λ,j

e^2πi(λ+tj)ch(W_λ,j⁰ ,∇^W0)/Πλ,ji^12dimNλ,jdet^1/2

1−e^2πi(λ+tj)exp( i

2πR^Nλ,j) on Ue_V^h.

2

If X is an almost complex orbifold, then on the orbifold chart (Gx,Uex) for x ∈ X, we have the following h-equivariant decomposition ofTUex as complex vector bundles onUe_x^h

TUex ≃ M

λ∈Q∩[0,1[

Nλ(h). (2.1)

HereNλ(h) are complex vector bundles over Ue_x^h withh acting bye^2πiλ on it, andN0(h) is TUe_x^h. Again Nλ(h) extend to complex vector bundles on ΣX. We will still denote it byg Nλ(h).

Let F(x, h) = P

λλdimCNλ(h) be the fermionic shift, then F : X ∪ΣXg → Q is locally constant. For a connected component Xi ⊂X∪ΣXg , we define F(Xi) to be the values of F on Xi.

LetW be an orbifold complex vector bundle on X. Let D^X⊗W be the Spin^c Dirac operator on Λ(T^∗(0,1)X)⊗W [16, Appendix D].

Let H be a compact Lie group acting on X. We assume that the action H on X lifts on W, and preserves the complex structures of T X and W. Now for γ ∈ H, the decomposition (1.10) on Ubx^(h)γ also preserves the complex structure of the normal bundle N. We denote by R^N the curvature of∇^N as complex vector bundle. Then

N =⊕0<θ<2πNθ, W⁰ =⊕0≤θ<2πWθ. (2.2)

Here Nθ, Wθ are complex vector bundles on which h◦γ_Ue acts as multiplication by e^iθ. The following theorem is proved in [8, Th. 14.1].

Theorem 2.1 Let

Td(TUe^h◦γUe,∇^TUeh◦γUe) = det −R^TUeh◦γUe/2iπ 1−exp(−R^TUeh◦γUe/2iπ)

!

be the Chern-Weil Todd form of TUe^h◦γUe. Then we have Ind_γ(D^X ⊗W) = X

F∈Xe^γ

1 m(F)

Z

F

α_F. (2.3)

Here on Ue^h◦γUe, αF is the characteristic class

Td(TUe^h◦γUe,∇^TUeh◦γUe)chh◦γ(W,∇^W)/det(1−(h◦γ) exp( i

2πR^N)).

IfH =S¹, onUe_V^h as in (1.15), we have the following decomposition of complex vector bundles,

N_λ(h) =⊕_jN_λ(h),j, TUe^h =⊕N0,j⊕TUe_V^hjx. (2.4)

Here N_λ(h),j, N_0,j extend to complex vector bundles on Xe^S1, and γ =e^2πit ∈S¹ acts on them as multiplication by e^2πijt.

By Theorem 2.1, we get,

Theorem 2.2 Under the condition of Theorem 2.1, for t∈ R\A, γ =e^2πit, we have Ind_γ(D^X ⊗W) = X

F∈Xe^S1

1 m(F)

Z

F

α_F. (2.5)

Here on Ue_V^h, αF is the characteristic class Td(TUe_V^h,∇^TUeVh)X

λ,j

e^2πi(λ+tj)ch(W_λ,j⁰ ,∇^W0)/Π_λ,jdet

1−e^2πi(λ+tj)exp( i

2πR^Nλ,j) .

Note that we can also get Theorems 1.2 and 2.2 from [25, Theorem 1].

3

In this section, we define the elliptic genus for a general almost complex orbifold and prove its rigidity property. We are using the setting of Section 2.

Forτ ∈H={τ ∈C; Imτ >0}, q=e^2πiτ, t∈C, let

θ(t, τ) =c(q)q^1/82 sin(πt)Π^∞_k=1(1−q^ke^2πit)Π^∞_k=1(1−q^ke^−2πit).

(3.1)

be the classical Jacobi theta function [6], where c(q) = Π^∞_k=1(1−q^k). Set θ^′(0, τ) = ∂θ(·, τ)

∂t |t=0. (3.2)

Recall the following transformation formulas for the theta-functions [6]:

θ(t+ 1, τ) =−θ(t, τ), θ(t+τ, τ) =−q^−1/2e^−2πitθ(t, τ), θ(_τ^t,−¹_τ) = ¹_ip_τ

ie^πitτ2θ(t, τ), θ(t, τ + 1) =e^πi4θ(t, τ).

(3.3)

For a complex or real vector bundleF on a manifold X, let Sym_q(F) = 1 +qF +q²Sym²F +· · · , Λq(F) = 1 +qF +q²Λ²F +· · · , (3.4)

be the symmetric and the exterior power operations F,respectively.

Let X be an almost complex orbifold, and dimCX = l. In this Section, all vector bundles are complex vector bundles. Let W be a proper orbifold complex vector bundle on X, and dimCW = m. Then W⁰ in (2.2) is W. Now for the vector bundle W, the fermionic shift F(Xi, W) = P

λλdimWλ is well define on each connected component X_i ⊂X∪ΣX. Forg x∈X, y=e^2πiz, we use the orbifold chart (G_x,Ue_x). Forh∈G_x, by

(2.1), we define on Ue_x^h, (3.5) Θ^z_q,x,(h)(T X) = O

λ∈Q∩[0,1[

O^∞

k=1

Λ_−y⁻¹_q^k−1+λ(h)N_λ(h)^∗ ⊗Λ_−yq^k−λ(h)Nλ(h)

O

λ∈Q∩]0,1[

O^∞

k=1

Sym_q^k−1+λ(h)N_λ(h)^∗ ⊗Sym_q^k−λ(h)Nλ(h)

O∞

k=1

Sym_q^kN₀^∗⊗Sym_q^kN0

,

Θ^z_q,x,(h)(T X|W) = O

λ∈Q∩[0,1[

O^∞

k=1

Λ_−y⁻¹_q^k−1+λ(h)W_λ(h)^∗ ⊗Λ_−yq^k−λ(h)Wλ(h)

O

λ∈Q∩]0,1[

O^∞

k=1

Sym_q^k−1+λ(h)N_λ(h)^∗ ⊗Sym_q^k−λ(h)Nλ(h)

O∞

k=1

Sym_q^kN₀^∗⊗Sym_q^kN0

.

It is easy to verify that each coefficient of q^a (a ∈ Q) in Θ^z_q,x,(h)(T X), Θ^z_q,x,(h)(T X|W) defines an orbifold vector bundle onΣX. We will denote it by Θg ^z_q,Xi(T X), Θ^z_q,Xi(T X|W) on the connected component Xi of X ∪ΣX. It is the usual Witten element ong X (see [10] and [26]),

Θ^z_q(T X) = O∞

k=1

Λ_−y⁻¹_q^k−1T^∗X⊗Λ_−yq^kT XO^∞

k=1

Sym_q^kT^∗X⊗Sym_q^kT X . (3.6)

Definition 3.1 The orbifold elliptic genus of X is defined to be

F(y, q) =y^2l X

Xi⊂X∪gΣX

y^−F(Xi)Ind(D^Xi⊗Θ^z_q,Xi(T X)).

(3.7)

More generally, we define the orbifold elliptic genus associated to W as

F(y, q, W) =y^m2 X

Xi⊂X∪gΣX

y^−F(Xi,W)Ind(D^Xi ⊗Θ^z_q,Xi(T X|W)).

(3.8)

If X is a global quotient M/G where the action of finite group G on almost com- plex manifold M preserves its complex structure, the equation (3.7) coincides with [4, Definition 4.1].

We next prove that the orbifold elliptic genus is rigid forS¹ action on X.

LetS¹ act on X, preserving the complex structure on T X, and lifting on W. Natu- rally, we define the Lefschetz number for γ ∈S¹,

Fγ(y, q, W) =y^m2 X

Xi⊂X∪gΣX

y^−F(Xi,W)Indγ(D^Xi⊗Θ^z_q,Xi(T X|W)) (3.9)

LetP be a compact manifold acted infinitesimally freely by a compact Lie group G, and X = P/G the corresponding orbifold. We still denote W the corresponding vector bundle on P for W. Then KX = det(T^(1,0)X), KW = detW are naturally induced by complex line bundles on P, we still denote it by KX, KW. We may also consider KX, KW as an orbifold line bundle on X.

Recall that the equivariant cohomology group H_S^∗1(P,Z) ofP is defined by H_S^∗1(P,Z) =H^∗(P ×S¹ ES¹,Z).

(3.10)

where ES¹ is the universal S¹-principal bundle over the classifying space BS¹ of S¹. So H_S^∗1(P,Z) is a module overH^∗(BS¹,Z) induced by the projectionπ :P×S¹ES¹ →BS¹. Letp1(W)_S¹, p1(T X)_S¹ ∈H_S^∗1(P,Z) be the equivariant first Pontrjagin classes of W and T X respectively. Also recall that

H^∗(BS¹,Z) =Z[[u]]

(3.11)

with u a generator of degree 2.

Recall from [17, Theorem B] that for smooth manifoldX one needs the conditions p1(W −T X)_S¹ = 0, c1(W −T X)_S¹ = 0.

(3.12)

for the rigidity theorem.

Note that if the connected component Xi of X∪ΣXg is defined by (Ue_x^h, ZGx(h)), for g ∈Z_Gx(h), setU_V^h,g ={b ∈Ue_x|hb=gb=b, V_X(b) = 0}. Then the connected component Xik of Xg_i^S1 is defined by (U_V^h,g, ZZ_Gx(h)(g)). We have the following decomposition of complex vector bundles on U_V^h,g,

TUex= X

λ(h),λ(g)∈Q∩[0,1[,v∈Z

Nλ(h),λ(g),v, (3.13)

W = X

λ(h),λ(g)∈Q∩[0,1[,v∈Z

Wλ(h),λ(g),v.

where g, h ∈ Gx (resp. γ = e^2πit ∈ S¹) act on Nλ(h),λ(g),v, Wλ(h),λ(g),v as multiplication by e^2πiλ(g), e^2πiλ(h) (resp. e^2πivt). Let 2πix^jλ(h),λ(g),v, 2πiwλ(h),λ(g),v^j be the formal Chern roots of Nλ(h),λ(g),v, Wλ(h),λ(g),v respectively. To simplify the notation, we will omit the superscript j.

Then Nλ(h),λ(g),v, Wλ(h),λ(g),v extend to orbifold vector bundles on Xik. Now the natural generalization of (3.12) for orbifold is the following: there exists n ∈ N, such that on each connected component X_ik of Xg_i^S1,

X

λ(h),λ(g),v,j

h(w^jλ(h),λ(g),v+λ(g)−τ λ(h) +vu)² (3.14)

−(x^jλ(h),λ(g),v +λ(g)−τ λ(h) +vu)²i

=nπ^∗u² ∈H^∗(Xik,Q)[τ, t],

X

λ(h),λ(g),v,j

(w^jλ(h),λ(g),v+λ(g)−τ λ(h) +vu) (3.15)

− X

λ(h),λ(g),v,j

(x^jλ(h),λ(g),v+λ(g)−τ λ(h) +vu) = 0 ∈H^∗(Xik,Q)[τ, t].

Theorem 3.1 Assume that S¹ acts on P which induces the S¹-action on X, and lifts to W, and c1(W) = 0 modN in H^∗(P,Z)for some 1< N ∈N. Also assume that there existsn ∈Nsuch that equations (3.14) and (3.15) hold. Then for any N-th root of unity y=e^2πiz we have

i) If n= 0, then Fγ(y, q, W) is constant on γ ∈S¹. ii) If n <0, then Fγ(y, q, W) = 0.

Note that, in case W = T X, the condition (3.14), (3.15) are automatic, and as a consequence we get the rigidity and vanishing theorems for the usual orbifold elliptic genus F(y, q). In particular we know that for a Calabi-Yau almost complex manifold X, F(y, q) is rigid for any y.

P roof: Using Theorem 2.2, for γ =e^2πit, t∈R\A, y=e^2πiz, q =e^2πiτ, we get Fγ(y, q, W) =y^m2 X

Xi⊂X∪gΣX

y^−F(Xi,W) X

F⊂Xe_i^S1

1 m(F)

Z

F

αF, (3.16)

Recall that VX is the smooth vector field generated by S¹-action on X. For x ∈ X, take the orbifold chart (Gx,Uex). If Xi ⊂ X ∪ΣXg is represented by Ue_x^h/ZGx(h) on Uex

as in (1.3), the normal bundle NXi,g,V = N_Ueh

x/U_V^h,g of U_V^h,g in Ue_x^h extends to an orbifold vector bundle on Xe_i^S1. By Theorem 2.2, the contribution of the chart (Gx,Uex) for Indγ(D^Xi⊗Θ^z_q,Xi(T X|W)) is

1

|ZGx(h)|

X

gh=hg,g∈Gx

Z

U_V^h,g

Td(T U_V^h,g)chg◦γ(Θ^z_q,x,(h)(T X|W)) det(1−(g◦γ)e^{−2πi1 RNXi ,g,V}) . (3.17)

Let

Nv = X

λ(h),λ(g)∈Q∩[0,1[

Nλ(h),λ(g),v, (3.18)

Wv = X

λ(h),λ(g)∈Q∩[0,1[

Wλ(h),λ(g),v.

As the vector field VX commutes with the action of Gx, Nv and Wv extend to vector

bundles on Xe_i^S1. So the contribution of the chart (G_x,Ue_x) forF_γ(y, q, W) is (3.19)

1

|Gx|

X

gh=hg;g,h∈Gx

y^m2−F(Xi,W) Z

U_V^h,g

Td(T U_V^h,g)chg◦γΘ^z_q,x,(h)(T X|W) det(1−(g◦γ)e^{−2πi1 RNXi ,g})

= 1

|Gx|

X

gh=hg;g,h∈Gx

Z

U_V^h,g

(2πix0(h),0(g),0)y^m2−F(Xi,W) Πλ(g),v

1

1−e^2πi(x0(h),λ(g),v+λ(g)+tv)Πλ(h)>0,λ(g),v

Π^∞_k=1(1−y⁻¹q^k−1+λ(h)e^2πi(−wλ(h),λ(g),v−λ(g)−tv))(1−yq^k−λ(h)e^2πi(wλ(h),λ(g),v+λ(g)+tv)) (1−q^k−1+λ(h)e^2πi(−xλ(h),λ(g),v−λ(g)−tv))(1−q^k−λ(h)e^2πi(xλ(h),λ(g),v+λ(g)+tv))

Πλ(g),vΠ^∞_k=1(1−y⁻¹q^k−1e^2πi(−w0(h),λ(g),v−λ(g)−tv))(1−yq^ke^2πi(w0(h),λ(g),v+λ(g)+tv))

(1−q^ke^2πi(−x0(h),λ(g),v−λ(g)−tv))(1−q^ke^2πi(x0(h),λ(g),v+λ(g)+tv))

= (i⁻¹c(q)q^1/8)^l−m 1

|Gx|

X

gh=hg;g,h∈Gx

Z

U_V^h,g

(2πix0(h),0(g),0)

Πλ(h),λ(g),v,j(θ(wλ(h),λ(g),v^j +λ(g)−τ λ(h) +z+tv, τ)e^{−2πizλ(h)}) Πλ(h),λ(g),v,jθ(x^jλ(h),λ(g),v+λ(g)−τ λ(h) +tv, τ) . To get the last equality of (3.19), we use (3.1), (3.15).

We will considerFγ(y, q, W) as a function of (t, z, τ), we can extend it to a meromor- phic function on C×H×C. From now on, we denote (i⁻¹c(q)q^1/8)^{m−l θ}_θ(z,τ)^′(0,τ)m^lFγ(y, q, W) by F(t, τ, z). We also denote by Fγ(t, τ, W)_|eUx the function defined by (3.19). We set

F(t, τ, z)_|eUx = (i⁻¹c(q)q^1/8)^m−lθ^′(0, τ)^l

θ(z, τ)^mFγ(t, τ, W)_|eUx. (3.20)

Now, the equation (3.14) implies the equalities Xw²λ(h),λ(g),v−X

x²λ(h),λ(g),v = 0, X

v wλ(h),λ(g),v−X

v xλ(h),λ(g),v = 0, (3.21)

X

λ(h),λ(g)

λ(h)λ(g)(dimWλ(h),λ(g)−dimNλ(h),λ(g)) = 0, X

λ(h),λ(g),v

λ(g)v (dimWλ(h),λ(g),v −dimNλ(h),λ(g),v) = 0, Xλ(g)²(dimW_λ(g)−dimN_λ(g)) = 0, X

v²(dimW_v−dimN_v) =n.

By (3.1), for a, b∈2Z, k ∈N,

θ(x+k(t+aτ +b), τ) =e−πi(2kax+2k²at+k²a²τ)θ(x+kt, τ).

(3.22)

Asc1(KX) = 0 modN inH^∗(P,Z), by the same argument as [10,§8] or [21, Lemma 2.1, Remark 2.6], P

uvdimNv modN is constant on each connected component of X.

By (3.19), (3.21), (3.22), we know fora, b∈2Z,

F(t+aτ +b, τ, z) =e^{−2πizaPvvdimNv}F(t, τ, z).

(3.23)

For A=

a b c d

∈SL₂(Z), we define its modular transformation onC×H by

A(t, τ) = t

cτ +d,aτ +b cτ +d

. (3.24)

By (3.19), under the action A=

a b c d

∈SL2(Z), we have θ(A(t1, τ))

θ(A(t2, τ)) =e^{πic(t21−t22)/cτ+d}θ(t1, τ) θ(t2, τ), (3.25)

θ^′(A(0, τ))

θ(A(t, τ)) = (cτ +d)e^{−πict2/cτ+d}θ^′(0, τ) θ(t, τ).

For g, h ∈ G_x, by looking at the degree 2 dimCU_V^h,g part, that is the dimCU_V^h,g-th homogeneous terms of the polynomials in x, w’s, on both sides of the following equation, we get

Z

U_V^h,g

(x0(h),0(g),0)Πλ(h),λ(g),ve^2πiczwλ(h),λ(g),v(cτ+d)

θ(wλ(h),λ(g),v(cτ +d) +λ(g)(cτ +d)−(aτ +b)λ(h) +z(cτ +d) +tv, τ) θ(xλ(h),λ(g),v(cτ +d) + (cτ +d)λ(g)−(aτ +b)λ(h) +tv, τ)

= Z

U_V^h,g

(x0(h),0(g),0)Πλ(h),λ(g),ve^2πiczwλ(h),λ(g),v

θ(wλ(h),λ(g),v+λ(g)(cτ +d)−(aτ +b)λ(h) +z(cτ +d) +tv, τ) θ(xλ(h),λ(g),v + (cτ +d)λ(g)−(aτ+b)λ(h) +tv, τ) . (3.26)

By (3.3), (3.19), (3.21), (3.25) and (3.26), we easily derive the following identity:

(3.27) F(A(t, τ), z)_|eUx = 1

|Gx|(cτ +d)^le^{πicnt2/(cτ+d)} θ^′(0, τ)^l θ(z(cτ +d), τ)^m X

gh=hg;g,h∈Gx

Z

U_V^g,h

(2πix0(h),0(g),0)

×n

Πλ(h),λ(g),v(e^2πicz(wλ(h),λ(g),v+(λ(g)−^aτ+b_cτ+dλ(h))(cτ+d)+tv)e^{−2πizλ(h)})o

×Πλ(h),λ(g),vθ(wλ(h),λ(g),v+λ(g)(cτ +d)−(aτ +b)λ(h) +z(cτ +d) +tv, τ) Πλ(h),λ(g),vθ(xλ(h),λ(g),v+ (cτ +d)λ(g)−(aτ +b)λ(h) +tv, τ) .

By (3.3), (3.15), (3.21), (3.22), (3.27), we have

(cτ +d)^−le^{−πicnt2/(cτ+d)}θ(z(cτ +d), τ)^m

θ^′(0, τ)^l F(A(t, τ), z)_|Uex (3.28)

= 1

|Gx|

X

gh=hg;g,h∈Gx

Z

U_V^h,g

(2πix0(h),0(g),0)

Πλ(h),λ(g),v,j

ne^2πicz

dλ(g)−bλ(h)+τ(cλ(g)−aλ(h))

e^2πicz(wjλ(h),λ(g),v+tv)

×e−2πiz(cλ(g)−aλ(h)+λ(g^−ch^a))(cτ+d)e^{−2πizλ(h)})o

× Πλ(h),λ(g),vθ(wλ(h),λ(g),v+λ(g^dh^−b)−τ λ(g^−ch^a) +z(cτ +d) +tv, τ) Πλ(h),λ(g),vθ(xλ(h),λ(g),v+λ(g^dh^−b)−τ λ(g^−ch^a) +tv, τ)

= 1

|G_x|

X

gh=hg;g,h∈Gx

Z

U_V^h,g

(2πix0(h),0(g),0)Πλ(h),λ(g),v

e^2πicz(wλ(h),λ(g),v+tv)e^{−2πiz(cτ+d)λ(g−cha)}

Πλ(h),λ(g),vθ(wλ(h),λ(g),v+λ(g^dh^−b)−τ λ(g^−ch^a) +z(cτ +d) +tv, τ) Πλ(h),λ(g),vθ(xλ(h),λ(g),v+λ(g^dh^−b)−τ λ(g^−ch^a) +tv, τ) .

Recall that c1(KW) = 0 modN where KW = detW, this implies that the line bundle K_W^cz is well defined on P, also as an orbifold vector bundle on X. Let

(3.29) F^A(t, τ, z) = (i⁻¹c(q)q^1/8)^m−l θ^′(0, τ)^l θ(z(cτ +d), τ)^m X

Xi⊂X∪Xe

y^m2−F(Xi,W)Ind_γ

D⊗K_W^cz ⊗Θ^(cτ_q,X^+d)z_i (T X|W) .

Now, observe that as A =

a b c d

∈ SL2(Z), when g, h run through all pair of P

gh=hg,g,h∈Gx, then g^−ch^a, g^dh^−b run through all pair of P

gh=hg,g,h∈Gx. Then by (3.28), (3.29)

F(A(t, τ), z) = (cτ +d)^le^{πicnt2/(cτ+d)}F^A(t, τ, z).

(3.30)

The following lemma implies that the index theory comes in to cancel part of the poles of the functions F.

Lemma 3.1 The function F^A(t, τ, z) is holomorphic in (t, τ) for (t, τ)∈R×H.

The proof of the above Lemma is the same as the proof of [17, Lemma 1.3] or [19, Lemma 2.3].

Now, we return to the proof of Theorem 3.1. Note that the possible polar divisors of F in C×H are of the form t= ^k_j(cτ +d) withk, c, d, j integers and (c, d) = 1 or c= 1 and d = 0.

We can always find integers a, b such that ad − bc = 1, and consider the matrix A=

d −b

−c a

∈SL₂(Z).

F^A(t, τ, z) = (−cτ +a)^−le^{πicnt2/(−cτ+a)}F

A(t, τ), z (3.31)

Now, ift= ^k_j(cτ+d) is a polar divisor ofF(t, τ, z), then one polar divisor ofF^A(t, τ, z) is given by

t

−cτ +a = k j

c dτ −b

−cτ +a +d , (3.32)

which exactly gives t = k/j. This contradicts Lemma 3.1, and completes the proof of

Theorem 3.1.

4

Forτ ∈H={τ ∈C; Imτ >0}, q=e^2πiτ, let

θ3(v, τ) =c(q)Π^∞_k=1(1 +q^k−1/2e^2πiv)Π^∞_k=1(1 +q^k−1/2e^−2πiv).

(4.1)

be the other three classical Jacobi theta-functions [6], where c(q) = Π^∞_k=1(1−q^k).

LetX be a compact orbifold, dimRX = 2n. We assume that X and ΣXg are spin in the sense of Definition 1.4. For x∈ X, taking the orbifold chart (G_x,Ue_x). By (1.5), for h∈Gx, we define on Ue_x^h

Θ^′_q,x,(h)(T X) =⊗_λ∈Q∩]0,¹

2[

⊗^∞_k=1

Λ_qk−1+λ(h)N_λ(h)^∗ ⊗Λ_qk−λ(h)N_λ(h)

⊗^∞_k=1

Sym_q^k−1+λ(h)N_λ(h)^∗ ⊗Sym_q^k−λ(h)Nλ(h)

⊗^∞_k=1

Λq^k−12N^R1

2(h)⊗Sym

q^k−12N^R1 2(h)

⊗ ⊗^∞_k=1(Λ_q^k(TUe_x^h)⊗Sym_q^k(TUe_x^h)).

(4.2)

It is easy to verify that each coefficient ofq^a(a∈Q) in Θ^′_q,x,(h)(T X) defines an orbifold vector bundle on X∪ΣX. We denote as Θg ^′_q,Xi(T X) on the connected component Xi of X∪ΣX. Especially, Θg ^′_q,X(T X) is the usual Witten elements on X

Θ^′_q(T X) =N∞ k=1

Λ_q^k(T X)⊗Sym_qk(T X) . (4.3)

We propose the following definition for the elliptic genus on spin orbifold:

Definition 4.1 The orbifold elliptic genus of X is

F(q) = X

Xi⊂X∪gΣX

Ind

D^Xi⊗(S⁺(T Xi)⊕S⁻(T Xi))⊗Θ^′_q,Xi(T X) . (4.4)