A note on TT deformations and boundaries

Nicolò Brizio, Tommaso Morone, Roberto Tateo

Published Jan 1, 2026• 0 citations

Abstract

Abstract The irrelevant composite operator TT, 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 topo- logical gravity models. However, the behavior of TT-deformed theories in the pres- ence of boundaries and defects remains largely unexplored. In this note, we review analytical results obtained through various techniques. Specifically, we study the TT-deformed exact g-function within the framework of the Thermodynamic Bethe Ansatz and show that the results coincide with those obtained by solving the corre- sponding Burgers-type flow equation. Finally, we highlight some potentially signif- icant open problems.

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Abstract

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

Nicolò Brizio, Tommaso Morone, Roberto Tateo

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.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.

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arXiv:2109.13180Journal of Statistical Mechanics Theory and ExperimentApr 1, 202210 citations

A Note on

T\bar{T}

Deformations and Boundaries

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.

T\bar{T}

Deformations of Quantum Field theory

On the flow of states under

T\overline{T}

A general definition of

JT_a

-- deformed QFTs

T T ¯

\mathrm{T}\overline{\mathrm{T}}

-deformed 2D quantum field theories

It was noticed many years ago, in the framework of massless RG flows, that the irrelevant composite operator

\mathrm{T}\overline{\mathrm{T}}

, built with the components of the energy-momentum tensor, enjoys very special properties in 2D quantum field theories, and can be regarded as a peculiar kind of integrable perturbation. Novel interesting features of this operator have recently emerged from the study of effective string theory models. In this paper we study further properties of this distinguished perturbation. We discuss how it affects the energy levels and one-point functions of a general 2D QFT in finite volume through a surprising relation with a simple hydrodynamic equation. In the case of the perturbation of CFTs, adapting a result by Lüscher and Weisz we give a compact expression for the partition function on a finite-length cylinder and make a connection with the exact g-function method. We argue that, at the classical level, the deformation naturally maps the action of N massless free bosons into the Nambu-Goto action in static gauge, in N + 2 target space dimensions, and we briefly discuss a possible interpretation of this result in the context of effective string models.