Path Integral Optimization for $T\bar{T}$ Deformation

TL;DR

The paper demonstrates that path integral optimization applied to $T\overline{T}$-deformed 2D CFTs yields optimized geometries whose time slices encode the full bulk, with positive deformation implying a finite bulk cutoff. By incorporating mixed boundary conditions via TTbar, the authors derive the generalized Liouville framework, obtain holographic entanglement entropy corrections, and define a relative path-integral complexity whose BTZ case leads to a UV-finite complexity of formation. Finite-temperature states map to BTZ-like geometries with TTbar-induced shifts in mass and temperature, consistent with the deformed energy spectrum. Overall, the work connects TTbar holography to finite-cutoff AdS, provides explicit EE and complexity data, and suggests further explorations in correlation functions and entanglement structure within TTbar-deformed holography.

Abstract

We use the path integral optimization approach of Caputa, kundu, Miyaji, Takayanagi and Watanabe to find the time slice of geometries dual to vacuum, primary and thermal states in the $T\bar{T}$ deformed two dimensional CFTs. The obtained optimized geometries actually capture the entire bulk which fits well with the integrability and expected UV-completeness of $T\bar{T}$-deformed CFTs. When deformation parameter is positive, these optimized solutions can be reinterpreted as geometries at finite bulk radius, in agreement with a previous proposal by McGough, Mezei and Verlinde. We also calculate the holographic entanglement entropy and quantum state complexity for these solutions. We show that the complexity of formation for the thermofield double state in the deformed theory is UV finite and it depends to the temperature.