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$T\bar{T}$ deformations as TsT transformations

Alessandro Sfondrini, Stijn J. van Tongeren

TL;DR

<3-5 sentence high-level summary> This work clarifies how $T\overline{T}$ deformations arise from gauge-fixing in the uniform light-cone formulation and shows they correspond to TsT transformations in the T-dual frame. The authors distinguish genuine TTbar deformations from simple gauge-frame changes, relate the TTbar CDD factor to a Drinfel'd-Reshetikhin twist of the worldsheet S-matrix, and illustrate with pp-wave and Lin-Lunin-Maldacena backgrounds. The results provide a geometric and S-matrix perspective on the spectral properties of deformed string backgrounds and suggest extensions to current-current deformations and AdS/CFT contexts.

Abstract

The relationship between $T\bar{T}$ deformations and the uniform light-cone gauge, first noted in arXiv:1804.01998, provides a powerful generating technique for deformed models. We recall this construction, distinguishing between changes of the gauge frame, which do not affect the theory, and genuine deformations. We investigate the geometric interpretation of the latter and argue that they affect the global features of the geometry before gauge fixing. Exploiting a formal relation between uniform light-cone gauge and static gauge in a T-dual frame, we interpret such a change as a TsT transformation involving the two light-cone coordinates. In the static-gauge picture, the $T\bar{T}$ CDD factor then has a natural interpretation as a Drinfel'd-Reshetikhin twist of the worldsheet S matrix. To illustrate these ideas, we find the geometries yielding a $T\bar{T}$ deformation of the worldsheet S matrix of pp-wave and Lin-Lunin-Maldacena backgrounds.

$T\bar{T}$ deformations as TsT transformations

TL;DR

<3-5 sentence high-level summary> This work clarifies how deformations arise from gauge-fixing in the uniform light-cone formulation and shows they correspond to TsT transformations in the T-dual frame. The authors distinguish genuine TTbar deformations from simple gauge-frame changes, relate the TTbar CDD factor to a Drinfel'd-Reshetikhin twist of the worldsheet S-matrix, and illustrate with pp-wave and Lin-Lunin-Maldacena backgrounds. The results provide a geometric and S-matrix perspective on the spectral properties of deformed string backgrounds and suggest extensions to current-current deformations and AdS/CFT contexts.

Abstract

The relationship between deformations and the uniform light-cone gauge, first noted in arXiv:1804.01998, provides a powerful generating technique for deformed models. We recall this construction, distinguishing between changes of the gauge frame, which do not affect the theory, and genuine deformations. We investigate the geometric interpretation of the latter and argue that they affect the global features of the geometry before gauge fixing. Exploiting a formal relation between uniform light-cone gauge and static gauge in a T-dual frame, we interpret such a change as a TsT transformation involving the two light-cone coordinates. In the static-gauge picture, the CDD factor then has a natural interpretation as a Drinfel'd-Reshetikhin twist of the worldsheet S matrix. To illustrate these ideas, we find the geometries yielding a deformation of the worldsheet S matrix of pp-wave and Lin-Lunin-Maldacena backgrounds.

Paper Structure

This paper contains 31 sections, 87 equations, 1 figure.

Table of Contents

  1. Introduction
  2. TTbar deformations and uniform light-cone gauge
  3. Uniform light-cone gauge
  4. Choice of the light-cone coordinates.
  5. Uniform light-cone gauge fixing.
  6. Some choices of the parameters $a,b$.
  7. Changing the gauge frame
  8. Changes of gauge frame and the Hamiltonian.
  9. Changes of gauge frame and the S matrix.
  10. TTbar deformations vs. gauge-frame choices
  11. Deformed backgrounds from TTbar
  12. TTbar deformations as a coordinate shift
  13. From uniform light-cone gauge to static gauge
  14. TTbar in the T-dual picture
  15. TsT and boundary conditions
  16. ...and 16 more sections

Figures (1)

  • Figure 1: The embedding of $(Y^+,Y^-)$ in $\mathbb{R}^{1,2}$ before and after the shift. This submanifold corresponds to the target space geometry; in the static gauge $Y^+\sim \tau$ and $Y^-\sim \sigma$ the string worldsheet has the same topology. Left: before the shift eq. \ref{['eq:embeddingnormal']} has periodic boundary conditions. Right: after the shift eq. \ref{['eq:embeddingtwist']} has twisted boundary condtions proportional to $\delta a$.