Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Shinji Hirano, Masaki Shigemori

TL;DR

This work develops Cardy’s random geometry approach to the $T\overline{T}$ deformation and provides a gravity-dual interpretation as an ensemble of AdS$_3$/BTZ spacetimes with randomly fluctuating boundary diffeomorphisms. It streamlines computations and extends the framework to correlation functions, showing that all-order energy spectra satisfy a Burgers-type flow and that thermal free energy evolves coherently under the same dynamics, with the gravity dual reproducing these results via the BTZ spectrum and on-shell action. The authors formulate precise field-theory PDEs for both energy and free energy and derive all-order correlator corrections, supporting a consistent holographic dictionary in which deformed observables are obtained by averaging undeformed gravity correlators over the diffeomorphism ensemble. This gravity-dual construction is contrasted with cutoff-AdS proposals and offers a broader, geometry-centric perspective on UV completion, non-locality, and entanglement features in $T\overline{T}$-deformed theories, with several directions for future work including rotation, other topologies, and chaos.

Abstract

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.

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

TL;DR

This work develops Cardy’s random geometry approach to the deformation and provides a gravity-dual interpretation as an ensemble of AdS/BTZ spacetimes with randomly fluctuating boundary diffeomorphisms. It streamlines computations and extends the framework to correlation functions, showing that all-order energy spectra satisfy a Burgers-type flow and that thermal free energy evolves coherently under the same dynamics, with the gravity dual reproducing these results via the BTZ spectrum and on-shell action. The authors formulate precise field-theory PDEs for both energy and free energy and derive all-order correlator corrections, supporting a consistent holographic dictionary in which deformed observables are obtained by averaging undeformed gravity correlators over the diffeomorphism ensemble. This gravity-dual construction is contrasted with cutoff-AdS proposals and offers a broader, geometry-centric perspective on UV completion, non-locality, and entanglement features in -deformed theories, with several directions for future work including rotation, other topologies, and chaos.

Abstract

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

Paper Structure

This paper contains 23 sections, 131 equations, 2 figures.

Table of Contents

  1. Introduction
  2. $T\bar{T}$ deformation as random geometry
  3. Gravity dual
  4. Energy spectrum and thermal free energy
  5. Energy spectrum
  6. Field theory
  7. $\bullet$ All orders I -- PDE (Burgers' equation)
  8. $\bullet$ All orders II -- order-by-order iteration
  9. Gravity dual
  10. Thermal free energy
  11. Field theory
  12. $\bullet$ All orders I -- PDE (Burgers' equation)
  13. $\bullet$ All orders II -- order-by-order iteration
  14. Gravity dual
  15. Micro-canonical to canonical ensemble
  16. ...and 8 more sections

Figures (2)

  • Figure 1: The gravity dual of $T\bar{T}$ deformation: AdS$_3$ and BTZ black hole with random boundary geometry with the metric $h_{ij}$ that fluctuates over the scale $\mu$. The gravity dual is an ensemble of AdS$_3$ spaces or BTZ black holes.
  • Figure 2: A finite cylinder of length $L$ and radius $R$.