Mutual Influence of Symmetries and Topological Field Theories

TL;DR

This paper explores how the fusion 2-category symmetry in fermionic 2+1 dimensional quantum field theories can be modified through stacking with TQFTs, revealing a finite set of inequivalent symmetries influenced by group-theoretical data.

Contribution

It introduces a framework for understanding symmetry modifications via stacking with Spin( n)_1 TQFTs, connecting these to group extension maps and tangential structures.

Findings

Finite set of inequivalent symmetry modifications identified.

Relation established between symmetry modifications and group extension maps.

Connections made between symmetry changes and tangential structures on manifolds.

Abstract

We study how the fusion 2-category symmetry of a fermionic (2+1)d QFT can be affected when one allows for stacking with TQFTs to be an equivalence relation for QFTs. Focusing on a simple kind of fermionic fusion 2-category described purely by group theoretical data, our results reveal that by allowing for stacking with $\mathrm{Spin}(n)_1$ as an equivalence relation enables a finite set of inequivalent modifications to the original fusion 2-categorical-symmetry. To put our results in a broader context, we relate the order of the symmetry modifications to the image of a map between groups of minimal nondegenerate extensions, and to the tangential structure set by the initial categorical symmetry on the background manifold for the QFT.