Higher Gauging and Non-invertible Condensation Defects

TL;DR

The paper develops a systematic framework for higher gauging of discrete higher-form symmetries on codimension-p submanifolds, producing condensation defects in 2+1d QFTs. It derives universal fusion rules for condensation surfaces and their interactions with bulk and surface lines, revealing that fusion coefficients are themselves 1+1d TQFTs and that many 0-form symmetries in 2+1d TQFTs arise from higher gauging. The authors work out explicit constructions and fusion rules for Z_N and Z_{N1}×Z_{N2} 1-form symmetries, and illustrate non-invertible defects in Maxwell, U(1) CS, Z_2 and Z_p theories, among others. They also connect higher gauging to Morita theory, show how all 0-form symmetries in 2+1d TQFTs can be realized via higher gauging, and discuss RCFT interpretations through modular invariants. The work provides a cohesive, broadly applicable framework for non-invertible symmetries in low dimensions and links to both condensed matter and mathematical formalisms such as fusion 2-categories and algebra objects in UMTCs.

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.

Figures (17)

• Figure 1: The left figure shows the action of a 1-form symmetry line $a$ on a line $L$, while the right figure shows the action $S \cdot L$ of a condensation surface $S$ (e.g., the charge conjugation symmetry) on a line $L$. The condensation defect is defined by summing over insertions of topological lines. The 1-form symmetry acts on lines by a phase without changing the type of lines. In contrast, the condensation surface defect changes the types of lines.
• Figure 2: The $R$- and $F$-symbols captures the braiding and crossing relations between topological lines in 2+1d. Here we only discuss the $R$- and $F$-symbols for invertible lines (i.e. abelian anyons), where $ab$ denotes the fusion of $a$ with $b$.
• Figure 3: Topological spin $\theta(a)$ of the line $a$.
• Figure 4: Gauging a $q$-form symmetry on a codimension-$p$ manifold is equivalent to inserting a network of symmetry defects along the dual graph of a triangulation of the codimension-$p$ manifold.
• Figure 5: The fusion of two parallel surface defects $S(\Sigma)$ and $S'(\Sigma)$ on an oriented manifold $\Sigma$. The red dashed arrows denote the normal vector of $\Sigma$. The fusion "coefficients" $c_i$ are generally 1+1d TQFTs, rather than numbers.