Notes on gauging noninvertible symmetries, part 1: Multiplicity-free cases

TL;DR

The paper develops a concrete framework for gauging noninvertible 0-form symmetries in 2D by restricting to multiplicity-free fusion categories Rep(ℋ) and constructing modular-invariant partition functions Z from special symmetric Frobenius algebras on ℋ*. Ordinary G-orbifolds arise as Vec(G) within Rep(ℋ). It provides explicit calculations of crossing kernels, associators, and genus-one (and higher-genus) partition functions for Rep(S3), Rep(D4), Rep(Q8), and Rep(𝓗8), establishing modular invariance and illustrating discrete-torsion-like phases. Applications include generating duality defects in c=1 CFTs via partial gauging of Frobenius subalgebras, and exploring decomposition when the whole noninvertible symmetry acts trivially. The work lays a concrete, first-principles pathway from categorical symmetry data to gauged theories and their partition functions, with several explicit, checkable examples and a roadmap for extending to nonmultiplicity-free cases.

Abstract

In this paper we discuss gauging noninvertible zero-form symmetries in two dimensions. We specialize to certain gaugeable cases, specifically, fusion categories of the form Rep(H) for H a suitable Hopf algebra (which includes the special case Rep(G) for G a finite group). We also specialize to the case that the fusion category is multiplicity-free. We discuss how to construct a modular-invariant partition function from a choice of Frobenius algebra structure on H^*. We discuss how ordinary G orbifolds for finite groups G are a special case of the construction, corresponding to the fusion category Vec(G) = Rep( C[G]^* ). For the cases Rep(S_3), Rep(D_4), and Rep(Q_8), we construct the crossing kernels for general intertwiner maps. We explicitly compute partition functions in the examples of Rep(S_3), Rep(D_4), Rep(Q_8), and Rep(H_8), and discuss applications in c=1 CFTs. We also discuss decomposition in the special case that the entire noninvertible symmetry group acts trivially.

Figures (23)

• Figure 1: Defining property of the crossing kernel $\tilde{K}$, taken from Chang:2018iay. The sum is over simple objects.
• Figure 2: Diagrammatic definition of the partial trace $Z^{L_3}_{L_1,L_2}$, the noninvertible analogue of the partial trace $Z_{g,h}$ appearing in genus one orbifold partition functions $[X/G]$ for $G$ an ordinary group. Our conventions for partial traces essentially follow Robbins:2019zdb.
• Figure 3: The effect of the modular $T$ transformation. Diagram (a) displays the original $Z_{L_1, L_2}^{L_3}(\tau)$; (b) displays $Z_{L_1, L_2}^{L_3}(\tau+1)$, which is equivalent to diagram (c). The left- and right-hand sides of diagram (c) are related using crossing as in figure \ref{['fig:crossing-kernel']}, with $L_3$ as the internal line before crossing.
• Figure 4: Three successive views of $Z_{L_1,L_2}^{L_3}(-1/\tau)$. Part (a) shows the original modular transformation. Part (b) rewrites the result of the modular transformation. The right-hand side of (c) is obtained from the left-hand side of (b) using the cyclic identity (\ref{['eq:fig:cyclic']}). Part (c) uses the crossing identity (figure \ref{['fig:crossing-kernel']}) to write the result in terms of other $Z_{L_1,L_2}^{L_3}(\tau)$, where we use the abbreviation $* = \sum_{L_4} \tilde{K}^{\overline{L}_1, L_2}_{\overline{L}_2, L_1}(\overline{L}_3, \overline{L}_4) \tilde{K}^{\overline{L}_1, 1}_{\overline{L}_4, L_2} (\overline{L}_2,L_1)$. Each part is an equivalent expression for $Z_{L_1,L_2}^{L_3}(-1/\tau)$.
• Figure 5: Resolution of four-point vertex in the Frobenius algebra ${\cal A}$ into a pair of three-point vertices.