Path integral optimization for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>T</mml:mi><mml:mover accent="true"><mml:mi>T</mml:mi><mml:mo accent="true" stretchy="false">¯</mml:mo></mml:mover></mml:math> deformation
Ghadir Jafari, Ali Naseh, Hamed Zolfi
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 $T\overline{T}$-deformed two-dimensional conformal field theories (CFTs). The obtained optimized geometries actually capture the entire bulk, which fits well with the integrability and expected UV completeness of $T\overline{T}$-deformed CFTs. When the 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 on the temperature.