Permutation Modular Invariants from Modular Functors

TL;DR

This paper constructs G-equivariant monoidal categories from modular tensor categories and G-sets, revealing how permutation actions induce modular invariants, advancing the understanding of symmetry in topological quantum field theories.

Contribution

It introduces a method to build G-equivariant monoidal categories using permutation equivariant modular functors, connecting group actions with modular invariants.

Findings

Module category structure for each g in G over the trivial component

Permutation g induces the modular invariant partition function

Formalism extends the understanding of symmetry in modular tensor categories

Abstract

For any finite group G with a finite G-set X and a modular tensor category C we construct a part of the algebraic structure of an associated G-equivariant monoidal category: For any group element g in G we exhibit the module category structure of the g-component over the trivial component. This uses the formalism of permutation equivariant modular functors that was worked out in arXiv:1004.1825. As an application we show that the corresponding modular invariant partition function is given by permutation by g.