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

## TL;DR

This pedagogical review explains the $ ext{T}ar{ ext{T}}$ deformation in 2D quantum field theories, covering its classical, spectral, and geometric aspects, and discusses recent developments up to 2021.

## Contribution

It provides a comprehensive, accessible overview of $ ext{T}ar{ ext{T}}$ deformation, highlighting its solvability, modular properties, and connections to geometry and holography.

## Key findings

- Exact solvability of the spectrum under deformation
- Modular invariance of the deformed partition function
- Connections to random geometry and 2D gravity

## 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.