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$T\bar{T}$ deformed CFT as a non-critical string

Nele Callebaut, Jorrit Kruthoff, Herman Verlinde

TL;DR

This work provides an exact, non-perturbative formulation of the $T\bar{T}$-deformed CFT as the worldsheet theory of a non-critical string with two light-cone directions $X^+$ and $X^-$, whose free-field OPEs are supplemented by a modified stress tensor to achieve total central charge $c=26$. By establishing an explicit dictionary with 2D dilaton gravity and constructing a BRST-consistent framework (including DDF-like operators) for arbitrary central charge $c$, the authors show that the deformed spectrum and partition function reproduce known results, and they give integral expressions for general correlation functions, yielding exact OPE-coefficient data. They extend the formalism to off-shell operators via boundary/Ishibashi constructions (including cross-caps and D-branes), derive a flow equation for local operators under the $T\bar{T}$ deformation, and provide a concrete 3-point function with holographic interpretation that matches bulk BTZ expectations. The approach offers a rigorous, non-perturbative handle on $T\bar{T}$ and suggests rich connections to holography, entanglement, and non-critical string dynamics, with several highlighted avenues for future work.

Abstract

We present a new exact treatment of $T\bar{T}$ deformed 2D CFT in terms of the worldsheet theory of a non-critical string. The transverse dimensions of the non-critical string are represented by the undeformed CFT, while the two longitudinal light-cone directions are described by two scalar fields $X^+$ and $X^-$ with free field OPE's but with a modified stress tensor, arranged so that the total central charge adds up to 26. The relation between our $X^\pm$ field variables and 2D dilaton gravity is indicated. We compute the physical spectrum and the partition function and find a match with known results. We describe how to compute general correlation functions and present an integral expression for the three point function, which can be viewed as an exact formula for the OPE coefficients of the $T\bar{T}$ deformed theory. We comment on the relationship with other proposed definitions of local operators.

$T\bar{T}$ deformed CFT as a non-critical string

TL;DR

This work provides an exact, non-perturbative formulation of the -deformed CFT as the worldsheet theory of a non-critical string with two light-cone directions and , whose free-field OPEs are supplemented by a modified stress tensor to achieve total central charge . By establishing an explicit dictionary with 2D dilaton gravity and constructing a BRST-consistent framework (including DDF-like operators) for arbitrary central charge , the authors show that the deformed spectrum and partition function reproduce known results, and they give integral expressions for general correlation functions, yielding exact OPE-coefficient data. They extend the formalism to off-shell operators via boundary/Ishibashi constructions (including cross-caps and D-branes), derive a flow equation for local operators under the deformation, and provide a concrete 3-point function with holographic interpretation that matches bulk BTZ expectations. The approach offers a rigorous, non-perturbative handle on and suggests rich connections to holography, entanglement, and non-critical string dynamics, with several highlighted avenues for future work.

Abstract

We present a new exact treatment of deformed 2D CFT in terms of the worldsheet theory of a non-critical string. The transverse dimensions of the non-critical string are represented by the undeformed CFT, while the two longitudinal light-cone directions are described by two scalar fields and with free field OPE's but with a modified stress tensor, arranged so that the total central charge adds up to 26. The relation between our field variables and 2D dilaton gravity is indicated. We compute the physical spectrum and the partition function and find a match with known results. We describe how to compute general correlation functions and present an integral expression for the three point function, which can be viewed as an exact formula for the OPE coefficients of the deformed theory. We comment on the relationship with other proposed definitions of local operators.

Paper Structure

This paper contains 20 sections, 97 equations, 3 figures.

Table of Contents

  1. Introduction
  2. A non-perturbative definition of $T \bar{T}$
  3. $T \bar{T}$ at $c=24$ as a critical string worldsheet
  4. Equivalence to 2D dilaton gravity
  5. Dilaton gravity at general central charge
  6. $T \bar{T}$ as a non-critical string worldsheet
  7. Correlation functions in the $X^{\pm}$ theory
  8. Spectrum and Partition Sum
  9. Spectrum
  10. Physical operators
  11. On-shell operators
  12. Stress tensor
  13. Off-shell operators
  14. Flow equation for local operators
  15. Three-point function
  16. ...and 5 more sections

Figures (3)

  • Figure 1: The $A$- and $B$-cycles of the $n$ punctured $z$-plane. The $A$-cycles go around the punctures, whereas the $B$-cycles go from an intersection to a reference point $P$, indicated in blue.
  • Figure 2: The local operator $\widetilde{\cal O}_h$ in the deformed theory creates a finite size cross cap. The matrix element between energy-momentum eigenstates can be computed by mapping to the Schottky double. The energy and momentum are encoded via four target space momenta given in equation \ref{['kmomenta']}.
  • Figure 3: Plot of normalized smearing function $f(\rho)$ in the limit of large $R$ and $\Delta$ for decreasing values of $R$ (decreasing shade of blue).