Higher Gauging and Non-invertible Condensation Defects

TL;DR

This paper explores higher gauging of discrete higher-form symmetries in 2+1d quantum field theories, revealing universal fusion rules for topological defects and their implications for non-invertible symmetries.

Contribution

It provides a systematic analysis of higher gauging in 2+1d QFTs, including fusion rules and the realization of all 0-form symmetries via higher gauging.

Findings

Derived universal fusion rules for condensation defects.

Showed all 0-form symmetries in 2+1d TQFTs can arise from higher gauging.

Identified fusion coefficients as 1+1d TQFTs rather than numbers.

Abstract

We discuss invertible and non-invertible topological condensation defects arising from gauging a discrete higher-form symmetry on a higher codimensional manifold in spacetime, which we define as higher gauging. A $q$-form symmetry is called $p$-gaugeable if it can be gauged on a codimension-$p$ manifold in spacetime. We focus on 1-gaugeable 1-form symmetries in general 2+1d QFT, and gauge them on a surface in spacetime. The universal fusion rules of the resulting invertible and non-invertible condensation surfaces are determined. In the special case of 2+1d TQFT, every (invertible and non-invertible) 0-form global symmetry, including the $\mathbb{Z}_2$ electromagnetic symmetry of the $\mathbb{Z}_2$ gauge theory, is realized from higher gauging. We further compute the fusion rules between the surfaces, the bulk lines, and lines that only live on the surfaces, determining some of the most basic data for the underlying fusion 2-category. We emphasize that the fusion "coefficients" in these non-invertible fusion rules are generally not numbers, but rather 1+1d TQFTs. Finally, we discuss examples of non-invertible symmetries in non-topological 2+1d QFTs such as the free $U(1)$ Maxwell theory and QED.