Notes on gauging noninvertible symmetries. Part I. Multiplicity-free cases

Abstract

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 $$ \textrm{Rep}\left(\mathcal{H}\right) $$ for H $$ \mathcal{H} $$ 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 ∗ $$ {\mathcal{H}}^{\ast } $$ . 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 $$ \textrm{Rep}\left({\mathcal{H}}_8\right) $$ , and discuss applications in c = 1 CFTs. We also discuss decomposition in the special case that the entire noninvertible symmetry group acts trivially.

Keywords

• Discrete Symmetries
• Global Symmetries
• Topological Field Theories