Notes on gauging noninvertible symmetries, part 1: Multiplicity-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.