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.
We study stress-tensor correlators in the $$ T\overline{T} $$ T T ¯ -deformed conformal field theories in two dimensions. Using the random geometry approach to the $$ T\overline{T} $$ T T ¯ deformation, we develop a geometrical method to compute stress-tensor correlators. More specifically, we derive the $$ T\overline{T} $$ T T ¯ deformation to the Polyakov-Liouville conformal anomaly action and calculate three and four-point correlators to the first-order in the $$ T\overline{T} $$ T T ¯ deformation from the deformed Polyakov-Liouville action. The results are checked against the standard conformal perturbation theory computation and we further check consistency with the $$ T\overline{T} $$ T T ¯ -deformed operator product expansions of the stress tensor. A salient feature of the $$ T\overline{T} $$ T T ¯ -deformed stress-tensor correlators is a logarithmic correction that is absent in two and three-point functions but starts appearing in a four-point function.
We quantize the D1-D5-P microstate geometries known as superstrata directly in supergravity. We use Rychkov's consistency condition [hep-th/0512053] which was derived for the D1-D5 system; for superstrata, this condition turns out to be strong enough to fix the symplectic form uniquely. For the (1,0,n) superstrata, we further confirm this quantization by a bona-fide explicit computation of the symplectic form using the semi-classical covariant quantization method in supergravity. We use the resulting quantizations to count the known supergravity superstrata states, finding agreement with previous countings that the number of these states grows parametrically smaller than those of the corresponding black hole.
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