McKay correspondence for elliptic genera

Abstract

We establish a correspondence between orbifold and singular elliptic genera of a global quotient. While the former is defined in terms of the fixed point set of the action, the latter is defined in terms of the resolution of singularities. As a byproduct, the second quantization formula of Dijkgraaf, Moore, Verlinde and Verlinde is extended to arbitrary Kawamata log-terminal pairs.

Equivariant elliptic genera

“…(3) The fans α are in 1 − 1 correspondence with connected components U α of For a proof, see [Borisov and Libgober 2005]. In the examples studied in this paper, it is easy to see that the proposition holds.…”

“…Proof of the change of variables formula. The method of proof used here is adapted from Borisov and Libgober's calculation of the pushforward of the orbifold elliptic genus by a toroidal morphism [Borisov and Libgober 2005].…”

“…Finally, motivated by [Borisov and Libgober 2005], we introduce the notion of the equivariant elliptic class…”

“…Following [Borisov and Libgober 2005], we define the orbifold elliptic genus of the pair (X, D) by the formula…”

“…When X possesses an action of a finite group G, there exists a notion of the orbifold elliptic genus of X which extends Batyrev's definition of the orbifold χ y genus. Recently, Borisov and Libgober [2005] proved the elliptic genus analogue of the McKay correspondence. To do this, they first define the elliptic genus and orbifold elliptic genus of a pair (X, D) for D a divisor on X and show that these definitions satisfy change of variable formulae similar to the objects e str (X, D) in Batyrev's paper.…”