Gauging Non-Invertible Symmetries: Topological Interfaces and Generalized Orbifold Groupoid in 2d QFT

TL;DR

This work develops a comprehensive framework for gauging discrete generalized symmetries in two-dimensional QFTs through topological defect lines and interfaces, recasting generalized gauging in terms of algebra objects and module categories over fusion categories. It shows that many familiar properties of ordinary group gauging extend to non-invertible symmetries, with the machinery of Morita theory and the Brauer-Picard groupoid organizing all consistent gaugings, sequential gaugings, and dualities. The authors provide bootstrap-type classification techniques, explicit constructions in ${ m Rep}(H_8)$ and ${ m Rep}(D_8)$, and a suite of CFT realizations including Ising$^2$, irrational $c=1$ orbifold branches, and $SU(2)_{10}$, revealing extensive self-dualities and generalized orbifold groupoid structure. Collectively, the results widen the landscape of dualities and RG-flow relations in 2d QFTs, offering concrete algebraic handles to navigate generalized gauging and its consequences in conformal and irrational CFTs.

Abstract

Gauging is a powerful operation on symmetries in quantum field theory (QFT), as it connects distinct theories and also reveals hidden structures in a given theory. We initiate a systematic investigation of gauging discrete generalized symmetries in two-dimensional QFT. Such symmetries are described by topological defect lines (TDLs) which obey fusion rules that are non-invertible in general. Despite this seemingly exotic feature, all well-known properties in gauging invertible symmetries carry over to this general setting, which greatly enhances both the scope and the power of gauging. This is established by formulating generalized gauging in terms of topological interfaces between QFTs, which explains the physical picture for the mathematical concept of algebra objects and associated module categories over fusion categories that encapsulate the algebraic properties of generalized symmetries and their gaugings. This perspective also provides simple physical derivations of well-known mathematical theorems in category theory from basic axiomatic properties of QFT in the presence of such interfaces. We discuss a bootstrap-type analysis to classify such topological interfaces and thus the possible generalized gaugings and demonstrate the procedure in concrete examples of fusion categories. Moreover we present a number of examples to illustrate generalized gauging and its properties in concrete conformal field theories (CFTs). In particular, we identify the generalized orbifold groupoid that captures the structure of fusion between topological interfaces (equivalently sequential gaugings) as well as a plethora of new self-dualities in CFTs under generalized gaugings.

Figures (14)

• Figure 1: Positive configurations of TDLs in the positive gauge ($\theta_{ijk}>0$) for the F-matrices.
• Figure 2: Discrete gauging on the torus.
• Figure 3: Trivalent topological junction $m$ for the algebra object $A$ which specifies the multiplication morphism and $m^\dagger$ for the comultiplication morphism.
• Figure 4: The separability, associativity, Frobenius, and unit conditions on the algebra object $(A,m,m^\dagger,u,u^\dagger)$. Here we have written them in the positive gauge. In the general gauge, the comultiplication morphism $m^\dagger$ and the counit $u^\dagger$ should be replaced by $m^\vee$ and $u^\vee$ to avoid confusions.
• Figure 5: Topological gauging interface from half-gauging algebra object $A\in {\mathcal{C}}$.

Theorems & Definitions (2)

• Theorem 1
• Theorem 2