Modularity of Trace Functions in Orbifold Theory for ℤ-Graded Vertex Operator Superalgebras

Summary

Abstract

We study the trace functions in orbifold theory for ℤ-graded vertex operator superalgebras and obtain a modular invariance result. More precisely, let V be a C₂-cofinite ℤ-graded vertex operator superalgebra and G a finite automorphism group of V. Then for any commuting pair (g, h) ∈ G, the hσ-trace function associated to a simple g-twisted V-modules is holomorphic in the upper half plane, where σ is the canonical involution on V coming from the superspace structure of V. If V is further g-rational for every g ∈ G, the trace functions afford a representation for the full modular group SL(2,ℤ).

Introduction

This work is a continuation of our study of the modular invariance for trace functions in orbifold theory. Motivated by generalized moonshine [N] and orbifold theory in physics [DVVV], the modular invariance of trace functions in orbifold theory has been studied for an vertex operator algebra [DLM3], under suitable conditions. This work has been generalized to a ½ℤ-graded vertex operator superalgebra [DZ2] (also see [H]), under suitable assumptions. In this paper we investigate the modular invariance of trace functions in orbifold theory for a ℤ-graded vertex operator superalgebra.

There is an essential difference between a ℤ-graded vertex operator superalgebra considered in this paper and a ½ℤ-graded vertex operator superalgebra studied in [DZ1]-[DZ2]. For a ½ℤ-graded vertex operator superalgebra V = ⊕_n∈½ℤV_n the even part of V is ∑_n∈ℤV_n and the odd part is ∑_n∈ℤV_n+½.