Commutative exact algebras and modular tensor categories

TL;DR

This paper investigates conditions under which the category of local modules over a commutative exact algebra in a braided finite tensor category forms a non-semisimple modular tensor category, with applications to vertex operator algebras.

Contribution

It provides new criteria for the modularity, rigidity, and ribbon structure of local module categories, and introduces methods to construct such algebras.

Findings

Criteria for local module categories to be modular tensor categories

Construction methods via simple current algebras and right adjoints

Connections between Witt equivalence and VOA extensions

Abstract

Inspired by the study of vertex operator algebra extensions, we answer the question of when the category of local modules over a commutative exact algebra in a braided finite tensor category is a (non-semisimple) modular tensor category. Along the way we provide sufficient conditions for the category of local modules to be rigid, pivotal and ribbon. We also discuss two ways to construct such commutative exact algebras. The first is the class of simple current algebras and the second is using right adjoints of central tensor functors. Furthermore, we discuss Witt equivalence and its relation with extensions of VOAs.