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The $\textrm{T}\bar{\textrm{T}}$ perturbation and its geometric interpretation

Riccardo Conti, Stefano Negro, Roberto Tateo

TL;DR

The paper shows that the classical T T̄ deformation of bosonic 2D theories is equivalent to a spacetime deformation realized by a field-dependent local coordinate change that maps deformed equations of motion to their undeformed counterparts. This is established via a geometric approach rooted in the ZCR/surface formalism, linking TT̄ to JT gravity and yielding a coordinate reparametrization that preserves the intrinsic geometry of solitonic surfaces, notably for the sine-Gordon model where the Gaussian and mean curvatures remain unchanged. The authors extend the construction to N bosons with arbitrary potentials, providing explicit Jacobians that implement the map and revealing how the action acquires a TT̄ (gravitational) contribution under the transformation. Overall, the work bridges classical integrability, differential geometry of embedded surfaces, and holographic interpretations of TT̄, offering a framework to apply integrable-model techniques in TT̄/JT contexts.

Abstract

Starting from the recently-discovered $\textrm{T}\bar{\textrm{T}}$-perturbed Lagrangians, we prove that the deformed solutions to the classical EoMs for bosonic field theories are equivalent to the unperturbed ones but for a specific field-dependent local change of coordinates. This surprising geometric outcome is fully consistent with the identification of $\textrm{T}\bar{\textrm{T}}$-deformed 2D quantum field theories as topological JT gravity coupled to generic matter fields. Although our conclusion is valid for generic interacting potentials, it first emerged from a detailed study of the sine-Gordon model and in particular from the fact that solitonic pseudo-spherical surfaces embedded in $\mathbb R^3$ are left invariant by the deformation. Analytic and numerical results concerning the perturbation of specific sine-Gordon soliton solutions are presented.

The $\textrm{T}\bar{\textrm{T}}$ perturbation and its geometric interpretation

TL;DR

The paper shows that the classical T T̄ deformation of bosonic 2D theories is equivalent to a spacetime deformation realized by a field-dependent local coordinate change that maps deformed equations of motion to their undeformed counterparts. This is established via a geometric approach rooted in the ZCR/surface formalism, linking TT̄ to JT gravity and yielding a coordinate reparametrization that preserves the intrinsic geometry of solitonic surfaces, notably for the sine-Gordon model where the Gaussian and mean curvatures remain unchanged. The authors extend the construction to N bosons with arbitrary potentials, providing explicit Jacobians that implement the map and revealing how the action acquires a TT̄ (gravitational) contribution under the transformation. Overall, the work bridges classical integrability, differential geometry of embedded surfaces, and holographic interpretations of TT̄, offering a framework to apply integrable-model techniques in TT̄/JT contexts.

Abstract

Starting from the recently-discovered -perturbed Lagrangians, we prove that the deformed solutions to the classical EoMs for bosonic field theories are equivalent to the unperturbed ones but for a specific field-dependent local change of coordinates. This surprising geometric outcome is fully consistent with the identification of -deformed 2D quantum field theories as topological JT gravity coupled to generic matter fields. Although our conclusion is valid for generic interacting potentials, it first emerged from a detailed study of the sine-Gordon model and in particular from the fact that solitonic pseudo-spherical surfaces embedded in are left invariant by the deformation. Analytic and numerical results concerning the perturbation of specific sine-Gordon soliton solutions are presented.

Paper Structure

This paper contains 7 sections, 45 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Classical integrable equations and embedded surfaces
  3. Construction of the solitonic surfaces
  4. The case of sine-Gordon
  5. The $\text{T}\bar{\text{T}}$-deformed sine-Gordon model and its associated surfaces
  6. From the deformed to the undeformed model through a local change of coordinates
  7. A geometric map for N-boson fields and arbitrary potential

Figures (1)

  • Figure 1: Pseudo-spherical solitonic surfaces associated to kink and breather solutions. Figure \ref{['fig:Dini']} represents the Dini surface, corresponding to a moving kink, while in Figure \ref{['fig:Beltrami']} the famous Beltrami pseudo-sphere is represented. The latter surface is obtained from Dini's surface by taking the stationary limit of the kink solution. Figures \ref{['fig:BreatherS']} and \ref{['fig:2kinkS']} correspond to the pseudo-spherical surfaces associated to a stationary breather and to a two-kink solution, respectively.