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

TL;DR

This paper applies path integral optimization to $T\bar{T}$-deformed 2D CFTs, deriving dual geometries that reflect the deformation's effects on bulk structure, entanglement, and complexity, confirming UV finiteness and finite bulk radius interpretations.

Contribution

It introduces a method to find dual geometries for $T\bar{T}$-deformed CFTs using path integral optimization, connecting deformation parameters with bulk geometry features.

Findings

Optimized geometries capture the entire bulk structure.

Deformation parameter relates to finite bulk radius geometries.

Complexity of formation is UV finite and temperature-dependent.

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.