Correlators in $T\bar{T}$ and Root-$T\bar{T}$ Deformed CFTs

TL;DR

This paper develops a geometric framework to compute correlators in 2D CFTs deformed by both $Tar T$ and root-$Tar T$, providing explicit results for two- and three-point functions.

Contribution

It introduces a geometric approach to evaluate correlators under combined $Tar T$ and root-$Tar T$ deformations, including all orders in $Tar T$ and leading order in root-$Tar T$.

Findings

Two-point function expressed as a kernel average over undeformed correlators.

Explicit kernel representation derived for pure $Tar T$ and combined deformations.

Leading correction to the three-point function computed.

Abstract

Quasi-primary correlators in two-dimensional conformal field theories deformed simultaneously by $T\bar T$ and root-$T\bar T$ are studied. A path-integral formulation motivated by the geometric realization of the combined deformation is used to develop a geometric framework for evaluating the deformed correlators. Within this framework, the two-point function is obtained to all orders in the $T\bar T$ coupling and to leading order in the root-$T\bar T$ coupling, while the leading correction to the three-point function is computed. It is further shown that the deformed two-point correlator admits a kernel representation as a weighted average of undeformed CFT correlators over conformal dimensions. This representation is derived explicitly for both the pure $T\bar T$ deformation and the combined flow. In this way, the mixed $T\bar T$/root-$T\bar T$ deformation is incorporated into the geometric description of irrelevant deformations, and the structure of local correlators beyond the pure $T\bar T$ case is characterized more explicitly.