Module categories for permutation modular invariants

TL;DR

This paper explores the structure of module categories over braided monoidal categories, linking algebraic properties to permutation invariants in conformal field theory, and demonstrating their physical relevance.

Contribution

It introduces a family of module category structures over braided monoidal categories and connects these to permutation modular invariants in conformal field theory.

Findings

Module categories over braided monoidal categories can be structured as module categories over C×C.

The internal End of the tensor unit is an Azumaya algebra if the category is modular.

Permutation modular invariants are shown to be physical in conformal field theory.

Abstract

We show that a braided monoidal category C can be endowed with the structure of a right (and left) module category over C \times C. In fact, there is a family of such module category structures, and they are mutually isomorphic if and only if C allows for a twist. For the case that C is premodular we compute the internal End of the tensor unit of C, and we show that it is an Azumaya algebra if C is modular. As an application to two-dimensional rational conformal field theory, we show that the module categories describe the permutation modular invariant for models based on the product of two identical chiral algebras. It follows in particular that all permutation modular invariants are physical.