On Generalized Symmetries and Structure of Modular Categories

TL;DR

This paper explores the extension of group symmetries to category symmetries in modular categories using linear Hopf monads, aiming to classify simple modular categories akin to finite simple groups.

Contribution

It introduces a framework for generalizing symmetries in modular categories through Hopf monads and proposes a classification approach for simple modular categories.

Findings

Development of a theory of linear Hopf monads for modular categories

Proposal of an analogue of finite simple group classification for modular categories

Identification of prime modular categories as building blocks

Abstract

Pursuing a generalization of group symmetries of modular categories to category symmetries in topological phases of matter, we study linear Hopf monads. The main goal is a generalization of extension and gauging group symmetries to category symmetries of modular categories, which include also categorical Hopf algebras as special cases. As an application, we propose an analogue of the classification of finite simple groups to modular categories, where we define simple modular categories as the prime ones without any nontrivial normal algebras.