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

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 $\mathrm{Rep}\left(\mathcal{H}\right)$ \( \textrm{Rep}\left(\mathcal{H}\right) \) for $\mathcal{H}$ \( \mathcal{H} \) 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 ${\mathcal{H}}^{\ast }$ \( {\mathcal{H}}^{\ast } \) . 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(S₃), Rep(D₄), and Rep(Q₈), we construct the crossing kernels for general intertwiner maps. We explicitly compute partition functions in the examples of Rep(S₃), Rep(D₄), Rep(Q₈), and $\mathrm{Rep}\left({\mathcal{H}}_8\right)$ \( \textrm{Rep}\left({\mathcal{H}}_8\right) \) , and discuss applications in c = 1 CFTs. We also discuss decomposition in the special case that the entire noninvertible symmetry group acts trivially.