Random boundary geometry and gravity dual of $$ T\overline{T} $$ deformation

Abstract

We study the random geometry approach to the $$ T\overline{T} $$ T T ¯ deformation of 2d conformal field theory developed by Cardy and discuss its realization in a gravity dual. In this representation, the gravity dual of the $$ T\overline{T} $$ T T ¯ deformation becomes a straightforward translation of the field theory language. Namely, the dual geometry is an ensemble of AdS3 spaces or BTZ black holes, without a finite cutoff, but instead with randomly fluctuating boundary diffeomorphisms. This reflects an increase in degrees of freedom in the renormalization group flow to the UV by the irrelevant $$ T\overline{T} $$ T T ¯ operator. We streamline the method of computation and calculate the energy spectrum and the thermal free energy in a manner that can be directly translated into the gravity dual language. We further generalize this approach to correlation functions and reproduce the all-order result with universal logarithmic corrections computed by Cardy in a different method. In contrast to earlier proposals, this version of the gravity dual of the $$ T\overline{T} $$ T T ¯ deformation works not only for the energy spectrum and the thermal free energy but also for correlation functions.

Geometry and complexity of path integrals in inhomogeneous CFTs

“…It may be interesting to use this property to study more general deformations, e.g. the TT deformation [51], which has been argued to correpsond to a CFT on a random background metric [52,53] (see [24] for related progress on TT deformation and path integral optimization). We leave this as an interesting future problem.…”

Geometrizing non-relativistic bilinear deformations

“…Yet another interesting direction is to study the holographic dual of the deformed non-relativistic theories. In the relativistic case, there have been several proposals for the holographic dictionary [21,[91][92][93]. It would be interesting to see how these proposals are generalized to the non-relativistic cases.…”

$$ T\overline{T} $$ Deformation of stress-tensor correlators from random geometry

“…The results are checked against the standard conformal perturbation theory computation performed in appendix C. In section 6 we discuss the TT deformation to the stress tensor operator product expansions (OPEs) and check its consistency with the 4-point function results. In section 7, we comment on the translation of our results into the gravity dual [21] and give discussions on future works. Appendix A summarizes conventions used in this paper, and appendix B contains some details of the computations in the main text.…”

“…With these field theoretical backgrounds as our motivation as well as the aim to understand dual gravitational implications in the AdS/CFT correspondence [19][20][21], we further develop Cardy's random geometry approach to the TT deformation [22] building on our previous work on the subject [21]. In this paper, we focus on stress-tensor correlators in the TT -deformed CFTs and develop a new geometrical method to compute stress-tensor correlators.…”

“…The reference [25] uses the TT flow equation and the conformal perturbation theory, whereas the reference [26] combines the random geometry approach with the Ward-Takahashi (WT) identity for the stress tensor. Here we provide a new method that is purely based on the random geometry approach, generalizing the technique developed in our previous work [21] to the stress-tensor correlators. More specifically, we derive the TT deformation to the Polyakov-Liouville conformal anomaly action [24] and calculate three and four point correlators to…”