Notes on gauging noninvertible symmetries, part 2: higher multiplicity cases

TL;DR

This work extends gauging of noninvertible symmetries in 2d to non-multiplicity-free cases, detailing Rep$(A_4)$ as a key example. It develops a comprehensive partition-function framework using symmetric special Frobenius algebras in fusion categories, and clarifies discrete torsion via lazy cohomology alongside fiber-functor data. The authors realize Rep$(A_4)$ gaugings in the exceptional $c=1$ CFT $SU(2)_1/A_4$, uncover self-duality that enlarges the symmetry to Rep$(SL(2,\,\mathbb{Z}_3))$, and analyze duality defects arising from self-dual gaugings. They further explore decomposition when a trivially-acting subcategory is present, including detailed examples with Rep$({\mathbb Z}_Q)$ inside Rep$({\mathbb Z}_{PQ})$ and Rep$(S_3)$, and discuss how these results organize into a Brauer-Picard groupoid with $D_6$ structure. Overall, the paper expands the toolkit for classifying and manipulating noninvertible symmetries in 2d QFT and reveals rich structure at special points in the $c=1$ moduli space.

Abstract

In this paper we discuss gauging noninvertible zero-form symmetries in two dimensions, extending our previous work. Specifically, in this work we discuss more general gauged noninvertible symmetries in which the noninvertible symmetry is not multiplicity free, and discuss the case of Rep$(A_4)$ in detail. We realize Rep$(A_4)$ gaugings for the $c = 1$ CFT at the exceptional point in the moduli space and find new self-duality under gauging a certain non-group algebra object, leading to a larger noninvertible symmetry Rep$(SL(2, Z_3))$. We also discuss more general examples of decomposition in two-dimensional gauge theories with trivially-acting gauged noninvertible symmetries.

Figures (5)

• Figure 1: The associator for topological lines are encoded in F-symbols. The labels on the vertices represent insertions of our fusion basis vectors, i.e. $(\lambda_{A,E}^D)_i$ and $(\lambda_{B,C}^E)_j$ on the left and $(\lambda_{A,B}^F)_k$ and $(\lambda_{F,C}^D)_\ell$ on the right.
• Figure 2: Definition of the partial trace $(Z_{A,B}^C)_{ij}$ as a $T^2$ correlation function with topological lines inserted as shown and our chosen fusion and co-fusion basis vectors at the junctions.
• Figure 3: Brauer-Picard groupoid of Rep$(A_4)$ symmetry. We denote fusion categories as objects in the groupoid with nodes. We use straight links to denote different categorical symmetries that are connected via gauging, i.e., they are Morita equivalent to each other. The link attached to a single node denotes the Brauer-Picard group $\mathfrak{Brpic}(\text{Rep}(A_4))=D_6$.
• Figure 4: (a) Gauging various subgroups $H\subset A_4$ of the $SU(2)_1$ theory and (b) various Rep$(A_4)$ gaugings of the exceptional orbifold theory $SU(2)_1/A_4$ and the respectively resulting CFTs in $c=1$ moduli space.
• Figure 5: (a) Gauging various subgroups $H\subset A_4$ of the $SU(2)_1/(\mathbb{Z}_2\times \mathbb{Z}_2)$ theory and (b) various Rep$(A_4)$ gaugings of the exceptional orbifold theory $SU(2)_1/A_4$ and the respectively resulting CFTs in $c=1$ moduli space. Note that the self-duality of $SU(2)_1/(\mathbb{Z}_2\times \mathbb{Z}_2)$ theory under gauging $(\mathbb{Z}_2\times \mathbb{Z}_2)_{\text{d.t.}}$ matches the fact that the theory enjoys at least a Rep$(D_4)$ symmetry Perez-Lona:2023djo.