The Fusion Categorical Diagonal

TL;DR

This paper introduces a Frobenius algebra in fusion categories to generalize the diagonal subgroup, enabling the extension of field theories to non-invertible symmetries and providing explicit calculations for theories with symmetric group-based symmetries.

Contribution

It defines a new Frobenius algebra structure over fusion categories that generalizes the diagonal subgroup, extending field theoretical constructions to non-invertible symmetries.

Findings

Explicit calculations for theories with Rep(S_3)×Rep(S_3) symmetry.

Application of the algebra to gauging topological quantum field theories.

Discussion on Morita equivalence in symmetry categories.

Abstract

We define a Frobenius algebra over fusion categories of the form Rep$(G)\boxtimes$Rep$(G)$ which generalizes the diagonal subgroup of $G\times G$. This allows us to extend field theoretical constructions which depend on the existence of a diagonal subgroup to non-invertible symmetries. We give explicit calculations for theories with Rep$(S_3)\boxtimes$Rep$(S_3)$ symmetry, applying the results to gauging topological quantum field theories which carry this non-invertible symmetry. Along the way, we also discuss how Morita equivalence is implemented for algebras in symmetry categories.