Modular Invariance for Twisted Modules over a Vertex Operator Superalgebra

TL;DR

This paper extends Zhu's theorem to vertex operator superalgebras with rational weights, demonstrating SL_2(Z)-invariance of supertrace functions on twisted modules and providing examples.

Contribution

It generalizes Zhu's theorem to superalgebras with rational weights, incorporating twisted modules and odd traces to establish SL_2(Z)-invariance.

Findings

Supertrace functions span a finite-dimensional SL_2(Z)-invariant space.

Inclusion of twisted modules and odd traces is necessary for invariance.

Theoretical framework is supported by several examples.

Abstract

The purpose of this paper is to generalize Zhu's theorem about characters of modules over a vertex operator algebra graded by integer conformal weights, to the setting of a vertex operator superalgebra graded by rational conformal weights. To recover SL_2(Z)-invariance of the characters it turns out to be necessary to consider twisted modules alongside ordinary ones. It also turns out to be necessary, in describing the space of conformal blocks in the supersymmetric case, to include certain `odd traces' on modules alongside traces and supertraces. We prove that the set of supertrace functions, thus supplemented, spans a finite dimensional SL_2(Z)-invariant space. We close the paper with several examples.