Liouville gravity at the end of the world: Deformed defects in AdS/BCFT

We study shape deformations of two-dimensional end-of-the-world (ETW) branes, such as those in bottom-up models of two-dimensional holographic boundary conformal field theories (BCFT), and derive an action for the theory of brane deformations in any bulk three-dimensional maximally symmetric spacetime. In the case of a bulk anti-de Sitter (AdS) spacetime, at leading order in the ultraviolet cutoff, the induced theory on the brane controlling its shape is Liouville gravity coupled to quantum matter. We show in certain limits the theory reduces to semi-classical AdS, dS or flat Jackiw-Teitelboim (JT) gravity, thus providing the first doubly-holographic derivation of two-dimensional models of dilaton gravity minimally coupled to a large number of conformal fields. Specializing to the AdS JT gravity limit, we discuss the dual BCFT interpretation and provide evidence that changing the boundary conditions of JT gravity on the brane is equivalent to a deformation of the dual BCFT with the displacement operator. This establishes a doubly-holographic triality between (i) brane deformations in the bulk, (ii) JT gravity in the brane description, and (iii) irrelevant deformations of the CFT boundary. Lastly, in the presence of a non-trivial dilaton profile, we prove that the Ryu-Takayanagi formula for holographic BCFTs receives a contact term whenever the minimal surface ends on the brane.