Orbifold Elliptic Genera and Rigidity

Let M be a closed almost complex manifold on which a compact connected Lie group G acts non-trivially. If the first Chern class of M is divisible by an integer N greater than 1, then its equivariant elliptic genus φ(M) of level N is rigid, i.e., it is constant as a virtual character of G. This result was predicted by Witten [W] and proved by Taubes [T], Bott-Taubes [BT] and Hirzebruch [H]. Elliptic genus can be defined even for almost complex orbifolds. Moreover another genus called orbifold elliptic genus is defined for orbifolds. A natural question is whether the rigidity property holds for these genera on orbifolds or not. It turns out that the answer is no in general. In [HM] we were concerned with related topics. In this note a modified orbifold elliptic genus of level N will be defined for closed almost complex orbifolds such that N is relatively prime to the orders of all isotropy groups. One of main results, Theorem 3.1, states that the modified orbifold elliptic genus φ̆(X) of level N of an almost complex orbifold X of dimension 2n such that ΛTX = L for some orbifold line bundle L is rigid for non-trivial G action. As to the orbifold elliptic genus itself Theorem 3.3 states that the orbifold elliptic genus φ̂(X) of level N of X is rigid for non-trivial G action if ΛTX = L for some genuine G line bundle L. Furthermore the orbifold elliptic genus φ̂(X) is rigid for non-trivial G action if (ΛTX) is trivial as an orbifold line bundle for some positive integer d (Theorem 3.4). When ΛTX is trivial the rigidity was proved by Dong-Liu-Ma [DLM]. Liu [L] gave a proof of rigidity by using modular property of elliptic genera for manifolds. Our proof of the rigidity for the genera φ̆(X) and φ̂(X) also uses Liu’s method. The organization of the paper is as follows. In Section 2 we review basic materials concerning orbifolds in general. The notion of sectors is particularly relevant for later use. In Section 3 we give the definitions of orbifold elliptic genus and modified orbifold ellitic genus and the main theorems are stated here. Section 4 is devoted to exhibiting fixed point formulae for the above genera. The proof of the main results will be given in Section 5 and Section 6. In Section 6 some additional results related to vanishing property are given. Main results in this section are Propositions 6.6, 6.8 and 6.9. Section 7 concerns the orbifold Ty genus and its modified one. They are always rigid for nontrivial actions of campact connected Lie groups and take special forms when the orbifold elliptic genera vanish. In Section 8 the generalization to the case of stably almost complex orbifolds are discussed and it will be shown that main results in Section 3, Section 6 and Section 7 also hold for stably almost complex orbifolds. The author is grateful to M. Furuta, M. Futaki, A. Kato, Y. Mitsumatsu and other members of Furuta’s Seminar who attended his talks and gave him several useful comments. The presentation of this paper was largely improved by their help. He is also grateful to M. Masuda , coauthor of the related paper [HM], for his collaboration which initiated the work of present paper.