On finite symmetries and their gauging in two dimensions

TL;DR

The paper generalizes finite group symmetries in 2d quantum field theories to symmetry categories, showing that gauging yields a dual category and regauging recovers the original theory. It develops a comprehensive framework using module and bimodule categories to classify gaugings, encodes anomalies within symmetry categories, and presents a 2d TFT axiomatization for theories with category symmetry, including explicit constructions of gauged theories and their Hilbert spaces. It also surveys numerous examples, such as Rep(G) versus C(G,α), RCFTs, Tambara–Yamagami categories, and gaugings of subgroups of anomalous groups, illustrating the breadth and power of the categorical approach. This work provides a unifying, rigorous language for understanding finite non-Abelian symmetries, their anomalies, and the fate of symmetries under gauging in two dimensions, with potential extensions to higher dimensions and more exotic fusion categories.

Abstract

It is well-known that if we gauge a $\mathbb{Z}_n$ symmetry in two dimensions, a dual $\mathbb{Z}_n$ symmetry appears, such that re-gauging this dual $\mathbb{Z}_n$ symmetry leads back to the original theory. We describe how this can be generalized to non-Abelian groups, by enlarging the concept of symmetries from those defined by groups to those defined by unitary fusion categories. We will see that this generalization is also useful when studying what happens when a non-anomalous subgroup of an anomalous finite group is gauged: for example, the gauged theory can have non-Abelian group symmetry even when the original symmetry is an Abelian group. We then discuss the axiomatization of two-dimensional topological quantum field theories whose symmetry is given by a category. We see explicitly that the gauged version is a topological quantum field theory with a new symmetry given by a dual category.

Figures (52)

• Figure 3.1: Two lines with the labels $a$ and $b$ can be fused to form a line with the label $a\otimes b$.
• Figure 3.2: Left: The associator relates two different orders to tensor three lines. Right: Equivalently, the local operator labeled by the associator morphism is obtained by squeezing the region between $(a\otimes b)\otimes c$ and $a\otimes(b\otimes c)$ shown above to a point.
• Figure 3.3: The pentagon identity guarantees that two distinct ways to rearrange the order of the tensoring of four lines lead to the same result.
• Figure 3.4: Folding a line and squeezing it gives rise to local operators labeled by evaluation and co-evaluation morphisms.
• Figure 3.5: Consistency condition on evaluation and co-evaluation morphisms resulting from a topological deformation.