Quantum gravity in a finite region of spacetime is conjectured to be dual to a conformal field theory (CFT) deformed by the irrelevant operator TT. We test this conjecture with entanglement entropy, which is sensitive to ultraviolet physics on the boundary, while also probing the bulk geometry. We find that the entanglement entropy for an entangling surface consisting of two antipodal points on a sphere is finite and precisely matches the Ryu-Takayanagi formula applied to a finite region consistent with the conjecture of McGough et al. We also consider a one-parameter family of conical entropies, which are finite and verify a conjecture due to Dong. Since ultraviolet divergences are local, we conclude that the TT deformation acts as an ultraviolet cutoff on the entanglement entropy. Our results support the conjecture that the TT-deformed CFT is the holographic dual of a finite region of spacetime.

We consider sphere partition functions of T T deformed large N conformal field theories in d = 2, 3, 4, 5 and 6 dimensions, computed using the flow equation. These are shown to non-perturbatively match with bulk computations of AdS d+1 with a finite radial cut-off. We then demonstrate how the flow equation can be independently derived from a regularization procedure of defining T T operators through a local Callan-Symanzik equation. Finally, we show that the sphere partition functions, modulo bulk-counterterm contributions, can be reproduced from Wheeler-DeWitt wavefunctions.

The geometerization of the renormalization group flow triggered by the TT deformation of large c conformal field theories in two dimensions is presented. This entails the construction of the off shell Einstein-Hilbert action in three dimensions from said renormalization group flow.The crucial ingredient to this construction will be the encoding of general covariance in the emergent bulk theory in a very particular form of the Wess-Zumino consistency conditions. The utilisation of the local renormalization group, which requires putting the theory under consideration on an arbitrary background geometry, supplemented by the aforementioned covariance condition ensures that the whole construction is background independent.

We consider the $$ T\overline{T} $$ T T ¯ deformation of two dimensional Yang-Mills theory on general curved backgrounds. We compute the deformed partition function through an integral transformation over frame fields weighted by a Gaussian kernel. We show that this partition function satisfies a flow equation which has been derived previously in the literature, which now holds on general backgrounds. We connect ambiguities associated to first derivative terms in the flow equation to the normalization of the functional integral over frame fields. We then compute the entanglement entropy for a general state in the theory. The connection to the string theoretic description of the theory is also investigated.

We calculate quantum corrections to holographic entanglement entropy in the proposed duality between TT-deformed holographic 2D CFTs and gravity in AdS 3 with a finite cutoff. We first establish the dictionary between the two theories by mapping the flow equation of the deformed CFT to the bulk Wheeler-DeWitt equation. The latter reduces to an ordinary differential equation for the sphere partition function, which we solve to find the entanglement entropy for an entangling surface consisting of two antipodal points on a sphere. The entanglement entropy in the inverse central charge expansion yields the expectation value of the bulk length operator plus the entropy of length fluctuations, in accordance with the Ryu-Takayanagi formula and its generalization due to Faulkner, Lewkowycz, and Maldacena. Special attention is paid to the conformal mode problem and its resolution by a choice of contour for the gravitational path integral.