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.

124 pages, LaTeX; v2: references added

References in corpus (9)

• Non-Invertible Duality Defects in 3+1 Dimensions
• Topological defects for the free boson CFT
• Asymptotic density of states in 2d CFTs with non-invertible symmetries
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• Twisted gauge theories in 3D Walker-Wang models
• Remarks on Boundaries, Anomalies, and Noninvertible Symmetries
• Discrete torsion defects
• Decomposition, Trivially-Acting Symmetries, and Topological Operators
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