Notes on gauging noninvertible symmetries, part 2: higher multiplicity cases

TL;DR

This paper extends the study of gauging noninvertible symmetries in two-dimensional theories, exploring non-multiplicity-free cases, detailed examples like Rep(A_4), and discovering new self-dualities and larger symmetry structures.

Contribution

It introduces the gauging of noninvertible symmetries with multiplicities, provides explicit examples including Rep(A_4), and uncovers new self-duality and symmetry enhancements in 2D conformal field theories.

Findings

Rep(A_4) gaugings realized at c=1 CFT point

Discovery of self-duality under non-group algebra gauging

Identification of larger noninvertible symmetry Rep(SL(2,Z_3))

Abstract

In this paper we discuss gauging noninvertible zero-form symmetries in two dimensions, extending our previous work. Specifically, in this work we discuss more general gauged noninvertible symmetries in which the noninvertible symmetry is not multiplicity free, and discuss the case of Rep$(A_4)$ in detail. We realize Rep$(A_4)$ gaugings for the $c = 1$ CFT at the exceptional point in the moduli space and find new self-duality under gauging a certain non-group algebra object, leading to a larger noninvertible symmetry Rep$(SL(2, Z_3))$. We also discuss more general examples of decomposition in two-dimensional gauge theories with trivially-acting gauged noninvertible symmetries.