Modular Invariance on the Torus and Abelian Chern-Simons Theory

TL;DR

This paper explores how modular invariance can be implemented at the quantum level on the torus within a group-theoretical framework, revealing restrictions on wave functions depending on the cohomology class of the symplectic form, with applications to Abelian Chern-Simons theory.

Contribution

It provides a detailed analysis of quantum modular invariance conditions on the torus, considering different cohomology classes, and applies these findings to Abelian Chern-Simons theory.

Findings

Quantum restrictions depend on the cohomology class of the symplectic form.

Wave functions are periodic or antiperiodic based on the parity of cohomology parameters.

Results have implications for the quantization of Abelian Chern-Simons theory.

Abstract

The implementation of modular invariance on the torus as a phase space at the quantum level is discussed in a group-theoretical framework. Unlike the classical case, at the quantum level some restrictions on the parameters of the theory should be imposed to ensure modular invariance. Two cases must be considered, depending on the cohomology class of the symplectic form on the torus. If it is of integer cohomology class $n$, then full modular invariance is achieved at the quantum level only for those wave functions on the torus which are periodic if $n$ is even, or antiperiodic if $n$ is odd. If the symplectic form is of rational cohomology class $\frac{n}{r}$, a similar result holds --the wave functions must be either periodic or antiperiodic on a torus $r$ times larger in both direccions, depending on the parity of $nr$. Application of these results to the Abelian Chern-Simons is discussed.