Self-duality under gauging a non-invertible symmetry

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 $\mathbb{Z}_2\times \mathbb{Z}_2$ symmetry, and has $\mathsf{Rep}(H_8)$ and $\mathsf{Rep}(D_8)$ fusion category symmetries as a result. We find that gauging the entire $\mathsf{Rep}(H_8)$ fusion category symmetry maps the orbifold theory at radius $R$ to that at radius $2/R$. At $R=\sqrt{2}$, which corresponds to two decoupled Ising CFTs (Ising$^2$ in short), the theory is self-dual under gauging the $\mathsf{Rep}(H_8)$ symmetry. This implies the existence of a topological defect line in the Ising$^2$ CFT obtained from half-space gauging of the $\mathsf{Rep}(H_8)$ 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 $\mathsf{Rep}(H_8)$ symmetry to form a bigger fusion category which is a $\mathbb{Z}_2$-extension of $\mathsf{Rep}(H_8)$. We solve the pentagon equations including the additional topological line and find 8 solutions, where two of them are realized in the Ising$^2$ CFT. Finally, we show that the torus partition functions of the Monster$^2$ CFT and Ising$\times$Monster CFT are also invariant under gauging the $\mathsf{Rep}(H_8)$ symmetry.