Modular invariance on the torus and Abelian Chern-Simons theory

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.