The TT̄ deformation of two-dimensional holographic conformal field theory is conjectured to be dual to quantum gravity in a finite region of three-dimensional anti-de Sitter spacetime. We study entanglement entropy in this theory and its relation to quantum fluctuations of the dual geometry. We derive the correspondence between the TT̄ flow equation and the Wheeler-DeWitt equation with a negative cosmological constant. By fixing the resulting emergent diffeomorphism symmetry, we obtain a differential equation for the sphere partition function which can be solved exactly. The solution can be expressed as a Euclidean path integral along a particular complex contour. We then apply this result to study entanglement entropy of the boundary theory for an entangling surface consisting of two antipodal points on the sphere. The entanglement entropy gives the Ryu-Takayanagi formula for a geodesic in a finite region plus quantum corrections. We suggest an interpretation of the latter as entropy of fluctuations of the bulk geodesic length, in accordance with the proposal of Faulkner, Lewkowycz, and Maldacena.
Based on work with E. LePage, Y. Li, A. Pereira and V. Shyam.
