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

A bstract 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) Rep H for H 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 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(H₈) 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.