Random Boundary Geometry and Gravity Dual of $T\bar{T}$ Deformation

We study the random geometry approach to the $T\bar{T}$ 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 $T\bar{T}$ deformation becomes a straightforward translation of the field theory language. Namely, the dual geometry is an ensemble of AdS$_3$ 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 $T\bar{T}$ 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 $T\bar{T}$ deformation works not only for the energy spectrum and the thermal free energy but also for correlation functions.