A Geometric Construction for Permutation Equivariant Categories from Modular Functors

TL;DR

This paper introduces a geometric method to construct permutation equivariant categories from modular functors, expanding the understanding of G-equivariant fusion categories with explicit structures, especially for Z/2-permutation cases.

Contribution

It provides a new geometric construction of G-equivariant fusion categories using modular functors, completing previous work with explicit structure morphisms for specific cases.

Findings

Constructed a weak G-equivariant fusion category from modular functors.

Explicitly detailed structure morphisms for Z/2-permutation equivariant categories.

Advanced the program of understanding permutation equivariant categories in topological quantum field theory.

Abstract

Let G be a finite group. Given a finite G-set X and a modular tensor category C, we construct a weak G-equivariant fusion category, called the permutation equivariant tensor category. The construction is geometric and uses the formalism of modular functors. As an application, we concretely work out a complete set of structure morphisms for Z/2-permutation equivariant categories, finishing thereby a program we initiated in an earlier paper arXiv:0812.0986 [math.CT].