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Symmetries, operators and correlators in $J\bar{T}$ deformed CFTs

Liangyu Chen, Zhengyuan Du, Wei Song

TL;DR

This work analyzes $J\bar{T}$-deformed CFTs on the plane by constructing symmetry generators and two operator classes—dressed and physical—within a Hamiltonian-flow framework. It shows the left-moving sector retains a local Virasoro-Kac-Moody structure, while the right-moving sector becomes nonlocal, encoded by the field-dependent coordinate $\hat{v}$, and yields a nonlocal right-moving Virasoro-Kac-Moody algebra. Using conformal perturbation theory around the undeformed CFT, the authors compute two-point and $N$-point functions of physical operators; in momentum space, UV-dominated contributions resum to a CFT-like form with a momentum- and charge-dependent shift in conformal weights, $h_{\lambda}$, consistent with string-theory expectations. In position space, a Borel-resummed analysis reveals an instanton-like nonperturbative contribution, yielding a complete picture of correlation functions including a cusp/oscillation structure in certain regimes. Overall, the paper provides a concrete, symmetry-guided framework to compute observables in $J\bar{T}$-deformed CFTs and connects field-theoretic results with holographic/string-theory predictions, while identifying intriguing nonperturbative features for further exploration.

Abstract

We construct symmetry generators and operators for $J\bar{T}$-deformed conformal field theories by generalizing the framework established for $T\bar{T}$ deformations. Working in the Hamiltonian formalism on the plane, we derive the symmetry algebra of the deformed theory, which consists of a local Virasoro-Kac-Moody algebra in the left-moving sector and a non-local counterpart in the right-moving sector. This algebraic structure guides the definition of two operator classes: dressed operators, which transform as primaries under the deformed symmetries, and local physical operators. While dressed operators are local only in the left null direction, physical operators maintain locality in both directions and are constructed from dressed operators and currents. This formulation allows the powerful constraints of conformal symmetry to be leveraged for computing physical observables. Consequently, we employ conformal perturbation theory to compute the two-point and $N$-point functions of physical operators. In momentum space, we sum the most UV-sensitive contributions to all orders; the results show precise agreement with string theory predictions. Furthermore, a non-perturbative analysis of the position-space correlator reveals an instanton contribution, providing a complete characterization of the correlation functions.

Symmetries, operators and correlators in $J\bar{T}$ deformed CFTs

TL;DR

This work analyzes -deformed CFTs on the plane by constructing symmetry generators and two operator classes—dressed and physical—within a Hamiltonian-flow framework. It shows the left-moving sector retains a local Virasoro-Kac-Moody structure, while the right-moving sector becomes nonlocal, encoded by the field-dependent coordinate , and yields a nonlocal right-moving Virasoro-Kac-Moody algebra. Using conformal perturbation theory around the undeformed CFT, the authors compute two-point and -point functions of physical operators; in momentum space, UV-dominated contributions resum to a CFT-like form with a momentum- and charge-dependent shift in conformal weights, , consistent with string-theory expectations. In position space, a Borel-resummed analysis reveals an instanton-like nonperturbative contribution, yielding a complete picture of correlation functions including a cusp/oscillation structure in certain regimes. Overall, the paper provides a concrete, symmetry-guided framework to compute observables in -deformed CFTs and connects field-theoretic results with holographic/string-theory predictions, while identifying intriguing nonperturbative features for further exploration.

Abstract

We construct symmetry generators and operators for -deformed conformal field theories by generalizing the framework established for deformations. Working in the Hamiltonian formalism on the plane, we derive the symmetry algebra of the deformed theory, which consists of a local Virasoro-Kac-Moody algebra in the left-moving sector and a non-local counterpart in the right-moving sector. This algebraic structure guides the definition of two operator classes: dressed operators, which transform as primaries under the deformed symmetries, and local physical operators. While dressed operators are local only in the left null direction, physical operators maintain locality in both directions and are constructed from dressed operators and currents. This formulation allows the powerful constraints of conformal symmetry to be leveraged for computing physical observables. Consequently, we employ conformal perturbation theory to compute the two-point and -point functions of physical operators. In momentum space, we sum the most UV-sensitive contributions to all orders; the results show precise agreement with string theory predictions. Furthermore, a non-perturbative analysis of the position-space correlator reveals an instanton contribution, providing a complete characterization of the correlation functions.

Paper Structure

This paper contains 29 sections, 147 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Nonlocal symmetries in $J\bar{T}$ deformation
  3. $J\bar{T}$ deformation as a canonical transformation
  4. The $J\bar{T}$ deformed stress tensor and currents
  5. The dressed symmetry generators: stress tensor and chiral currents
  6. The left-moving sector.
  7. The right-moving sector.
  8. The dressed symmetry algebra.
  9. The physical operators
  10. The left-moving Ward identities.
  11. The right-moving Ward identities.
  12. The dressed operators
  13. Construction of the dressed operators
  14. Alternative expressions of the dressed operators
  15. Physical operators from dressed operators
  16. ...and 14 more sections

Figures (3)

  • Figure 1: The zero charge sector: we consider the case $x = \bar{x} = t_E$ with fixed $k = 1$. The UV behavior of the two-point function is controlled by \ref{['smallx']}. Figure 1(a) shows different values of $\lambda$, and Figure 2(b) is for different $h$.
  • Figure 2: The zero charge sector: we consider the case $x = -\bar{x} =iy$ with fixed $k = 1$. The UV behavior is controlled by \ref{['smallx']}. Again, the behavior of the two-point function is shown for different values of $\lambda$ and $h$.
  • Figure 3: Non-zero charge sector: the two-point function for different chiral charges $q$, with $x = \bar{x} = t_E$ and fixed parameters $\lambda = k = 1$, $h = 5/3$.