Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A pedagogical review on solvable irrelevant deformations of 2d quantum field theory

Yunfeng Jiang

TL;DR

The article surveys solvable irrelevant deformations of 2d QFTs, focusing on TTbar and related constructions. It develops the Lagrangian flow, exact deformed spectra via Burgers-type equations, and the modular structure of torus partition functions, then ties these results to gravity through random geometry and 2d topological gravity, and finally to holography. A key contribution is the demonstration that TTbar is uniquely characterized by modular invariance together with a universal spectral flow, with robust nonperturbative features and rich connections to holography and string theory. The review also outlines generalizations, higher-spin currents, and recent developments up to January 2021, providing a comprehensive map of the solvable deformation landscape and its physical implications.

Abstract

This is a pedagogical review on $\mathrm{T}\overline{\mathrm{T}}$ deformation of two dimensional quantum field theories. It is based on three lectures which the author gave at ITP-CAS in December 2018. This review consists of four parts. The first part is a general introduction to $\mathrm{T}\overline{\mathrm{T}}$ deformation. Special emphasises are put on the deformed classical Lagrangian and the exact solvability of the spectrum. The second part focuses on the torus partition sum of the $\mathrm{T}\overline{\mathrm{T}}$/$\mathrm{J}\overline{\mathrm{T}}$ deformed conformal field theories and modular invariance/covariance. In the third part, different perspectives of $\mathrm{T}\overline{\mathrm{T}}$ deformation are presented, including its relation to random geometry, 2d topological gravity and holography. We summarize more recent developments until January 2021 in the last part.

A pedagogical review on solvable irrelevant deformations of 2d quantum field theory

TL;DR

The article surveys solvable irrelevant deformations of 2d QFTs, focusing on TTbar and related constructions. It develops the Lagrangian flow, exact deformed spectra via Burgers-type equations, and the modular structure of torus partition functions, then ties these results to gravity through random geometry and 2d topological gravity, and finally to holography. A key contribution is the demonstration that TTbar is uniquely characterized by modular invariance together with a universal spectral flow, with robust nonperturbative features and rich connections to holography and string theory. The review also outlines generalizations, higher-spin currents, and recent developments up to January 2021, providing a comprehensive map of the solvable deformation landscape and its physical implications.

Abstract

This is a pedagogical review on deformation of two dimensional quantum field theories. It is based on three lectures which the author gave at ITP-CAS in December 2018. This review consists of four parts. The first part is a general introduction to deformation. Special emphasises are put on the deformed classical Lagrangian and the exact solvability of the spectrum. The second part focuses on the torus partition sum of the / deformed conformal field theories and modular invariance/covariance. In the third part, different perspectives of deformation are presented, including its relation to random geometry, 2d topological gravity and holography. We summarize more recent developments until January 2021 in the last part.

Paper Structure

This paper contains 61 sections, 356 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The S-matrix
  3. Deformed Lagrangian
  4. Finite volume spectrum
  5. Torus partition function
  6. Integrable deformation
  7. 2d topological gravity
  8. Holography
  9. Little string theory
  10. Definition and deformed lagrangian
  11. Definition
  12. Deformed Lagrangian
  13. Deformed spectrum
  14. The $\mathrm{T}\overline{\mathrm{T}}$ operator
  15. Factorization formula
  16. ...and 46 more sections

Figures (4)

  • Figure 2.1: The $\mathrm{T}\overline{\mathrm{T}}$ flow in the space of 2d quantum field theories. The point $t=0$ corresponds to the undeformed IR theory. As we increase $t$, we go from IR to UV.
  • Figure 5.2: The parallelogram which is identified with the torus.
  • Figure 5.3: The ordering of particles. The ordering of the rapitities are the same as the orderings of the spacetime positions.
  • Figure 5.4: The cut-off geometry. Turning on $\mathrm{T}\overline{\mathrm{T}}$ deformation for the bad sign is equivalent to putting a Dirichlet boundary condition at finite radius.