A Note on $T\bar{T}$ Deformations and Boundaries

TL;DR

This paper reviews analytical results on $T\bar{T}$ deformations in 2D quantum field theories, focusing on boundary effects, and demonstrates the consistency of different computational approaches.

Contribution

It provides the first detailed analysis of $T\bar{T}$ deformations with boundaries, connecting the TBA and Burgers flow approaches.

Findings

The $T\bar{T}$-deformed g-function matches TBA and Burgers equation results.

Boundary effects in $T\bar{T}$ deformations are systematically analyzed.

Highlights open problems in boundary $T\bar{T}$ theories.

Abstract

The irrelevant composite operator $T\bar{T}$, constructed from components of the stress-energy tensor, exhibits unique properties in two-dimensional quantum field theories and represents a distinctive form of integrable deformation. Significant progress has been made in understanding the bulk aspects of the theory, including its interpretation in terms of coordinate transformations and its connection to topological gravity models. However, the behavior of $T\bar{T}$-deformed theories in the presence of boundaries and defects remains largely unexplored. In this note, we review analytical results obtained through various techniques. Specifically, we study the $T\bar{T}$-deformed exact g-function within the framework of the Thermodynamic Bethe Ansatz and show that the results coincide with those obtained by solving the corresponding Burgers-type flow equation. Finally, we highlight some potentially significant open problems.