Infinite Product Decomposition of Orbifold Mapping Spaces

TL;DR

This paper reveals a geometric factorization behind the infinite product formula for orbifold elliptic genera, generalizing previous results and introducing geometric Hecke operators with applications to 2D tori.

Contribution

It introduces a geometric factorization framework for orbifold mapping spaces and defines geometric Hecke operators that satisfy classical identities in specific cases.

Findings

Established a geometric interpretation of the infinite product formula.

Defined geometric Hecke operators for functors.

Verified Hecke operator identities for 2D tori.

Abstract

Physicists showed that the generating function of orbifold elliptic genera of symmetric orbifolds can be written as an infinite product. We show that there exists a geometric factorization on space level behind this infinite product formula in much more general framework, where factors in the infinite product correspond to isomorphism classes of connected finite covering spaces of manifolds involved. From this formula, a concept of geometric Hecke operators for functors emerges. We show that these Hecke operators indeed satisfy the usual identity of Hecke operators for the case of 2-dimensional tori.