Note on holographic torus stress tensor correlators in $AdS_3$ gravity

TL;DR

This paper computes energy-momentum tensor correlators in AdS$_3$/CFT$_2$, using perturbative, operator, and Chern-Simons methods, and explores their high- and low-temperature behaviors, including T$ar{T}$ deformations and KdV operators.

Contribution

It introduces new recursive relations for high-point stress tensor correlators and applies multiple formalisms to analyze their behavior in different temperature regimes.

Findings

Derived high-point correlators for the energy-momentum tensor.

Established recursive relations for stress tensor correlators.

Connected stress tensor correlators to KdV operator correlators.

Abstract

In the AdS$_3$/CFT$_2$ framework, the Euclidean BTZ black hole corresponds to the dominant high-temperature phase of its dual field theory. We initially employ perturbative methods to solve the Einstein equations as boundary value problems, providing correlators for the energy-momentum tensor operator at low points. Utilizing operator equations established in our previous work, we further compute arbitrary high-point correlators for the energy-momentum tensor operator in the high-temperature phase and recursive relations for these high-point functions. Concurrently, we employ the Chern-Simons formalism to derive consistent results. Further, using the cut-off AdS/$T\bar{T}$-deformed CFT duality, we calculate the energy-momentum tensor correlators, contributing to the comprehensive understanding of the system's dynamics. Finally, stress tensor correlators enable us to ascertain the corresponding KdV operator correlators at low-temperature.