Self-duality under gauging a non-invertible symmetry

Abstract

We discuss two-dimensional conformal field theories (CFTs) which are invariant under gauging a non-invertible global symmetry. At every point on the orbifold branch of c = 1 CFTs, it is known that the theory is self-dual under gauging a ℤ2 × ℤ2 symmetry, and has Rep(H 8) and Rep(D 8) fusion category symmetries as a result. We find that gauging the entire Rep(H 8) fusion category symmetry maps the orbifold theory at radius R to that at radius 2/R. At R = 2 , which corresponds to two decoupled Ising CFTs (Ising2 in short), the theory is self-dual under gauging the Rep(H 8) symmetry. This implies the existence of a topological defect line in the Ising2 CFT obtained from half-space gauging of the Rep(H 8) symmetry, which commutes with the c = 1 Virasoro algebra but does not preserve the fully extended chiral algebra. We bootstrap its action on the c = 1 Virasoro primary operators, and find that there are no relevant or marginal operators preserving it. Mathematically, the new topological line combines with the Rep(H 8) symmetry to form a bigger fusion category which is a ℤ2-extension of Rep(H 8). We solve the pentagon equations including the additional topological line and find 8 solutions, where two of them are realized in the Ising2 CFT. Finally, we show that the torus partition functions of the Monster2 CFT and Ising×Monster CFT are also invariant under gauging the Rep(H 8) symmetry.