$a\times b=c$ in $2+1$D TQFT

TL;DR

This paper explores how the fusion rules of anyons in 2+1D TQFTs reveal underlying symmetries and structural properties, including zero-form and quasi-zero-form symmetries, across various TQFT models.

Contribution

It introduces the concept of quasi-zero-form symmetries and links fusion equations to the presence of such symmetries in non-abelian anyon systems.

Findings

Fusion equations indicate the presence of zero-form or quasi-zero-form symmetries.

Fusion structures can reveal non-modular fusion subcategories.

Theoretical results apply to discrete gauge theories and Chern-Simons models.

Abstract

We study the implications of the anyon fusion equation $a\times b=c$ on global properties of $2+1$D topological quantum field theories (TQFTs). Here $a$ and $b$ are anyons that fuse together to give a unique anyon, $c$. As is well known, when at least one of $a$ and $b$ is abelian, such equations describe aspects of the one-form symmetry of the theory. When $a$ and $b$ are non-abelian, the most obvious way such fusions arise is when a TQFT can be resolved into a product of TQFTs with trivial mutual braiding, and $a$ and $b$ lie in separate factors. More generally, we argue that the appearance of such fusions for non-abelian $a$ and $b$ can also be an indication of zero-form symmetries in a TQFT, of what we term "quasi-zero-form symmetries" (as in the case of discrete gauge theories based on the largest Mathieu group, $M_{24}$), or of the existence of non-modular fusion subcategories. We study these ideas in a variety of TQFT settings from (twisted and untwisted) discrete gauge theories to Chern-Simons theories based on continuous gauge groups and related cosets. Along the way, we prove various useful theorems.