Anyon Condensation in Virasoro TQFT: Wormhole Factorization

Anyon condensation in wormhole geometries is investigated in the Virasoro TQFT (VTQFT) formulation, a proposed reformulation of 3d AdS quantum gravity. We first review some elementary techniques of VTQFT and summarize a gauging scheme for non-invertible symmetries referred to as anyon condensation. We then exhibit that anyon condensation is applicable to VTQFT even though the category of Wilson lines associated with it is not strictly a modular tensor category (MTC) due to the continuously infinite label $p\in {\mathbb{R}}_{+}$. More specifically, it is shown that the partition function of the wormhole factorizes upon condensing the so-called diagonal condensable anyon $\mathcal{A}={\int }_0^{\mathrm{\infty }}dp{L}_p⊠{\bar{L}}_p$ in VTQFT. The resulting $2$d boundary theory is Liouville CFT by symmetry TFT construction, and to our knowledge, this is among the very few explicit computational examples of gauging $\mathit{\text{continuous}}\mathit{\text{non-invertible}}$ symmetries in the literature.