The $T\overline T$ deformation of quantum field theory as a stochastic process

John Cardy

Published Jan 21, 2018• 0 citations

Abstract

We revisit the results of Zamolodchikov and others on the deformation of two-dimensional quantum field theory by the determinant $\det T$ of the stress tensor, commonly referred to as $T\overline T$. Infinitesimally this is equivalent to a random coordinate transformation, with a local action which is, however, a total derivative and therefore gives a contribution only from boundaries or nontrivial topology. We discuss in detail the examples of a torus, a finite cylinder, a disk, and a truncated cone. In all cases the partition function evolves according to a linear diffusion-type equation, and the deformation may be viewed as a kind of random walk in moduli space in which domains shrink and become more symmetrical. The truncated cone allows access to information about the behavior of correlations, and of entanglement entropies, under the deformation. We also suggest a generalization of the whole formalism to higher dimensions.

Related papers

$$ T\overline{T} $$ deformation of correlation functions

John Cardy

Abstract We study the evolution of correlation functions of local fields in a two-dimensional quantum field theory under the $$ \lambda T\overline{T} $$ λT T ¯ deformation, suitably regularized. We show that this may be viewed in terms of the evolution of each field, with a Dirac-like string being attached at each infinitesimal step. The deformation then acts as a derivation on the whole operator algebra, satisfying the Leibniz rule. We derive an explicit equation which allows for the analysis of UV divergences, which may be absorbed into a non-local field renormalization to give correlation functions which are UV finite to all orders, satisfying a (deformed) operator product expansion and a Callan-Symanzik equation. We solve this in the case of a deformed CFT, showing that the Fourier-transformed renormalized two-point functions behave as k 2∆+2λk2, where ∆ is their IR conformal dimension. We discuss in detail deformed Noether currents, including the energy-momentum tensor, and show that, although they also become non-local, when suitably improved they remain finite, conserved and satisfy the expected Ward identities. Finally, we discuss how the equivalence of the $$ T\overline{T} $$ T T ¯ deformation to a state-dependent coordinate transformation emerges in this picture.

arXiv:1907.03394Journal of High Energy PhysicsDec 1, 20196 citations

$T\bar{T}$ Deformations of Quantum Field theory

Mengqi Zhu

Sep 20, 20180 citations

On the flow of states under $T\overline{T}$

Jorrit Kruthoff

Nov 1, 202020 citations

On the flow of states under $T\overline{T}$

Jorrit Kruthoff, Onkar Parrikar

arXiv:2006.03054SciPost PhysicsNov 20, 202019 citations

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

Nicolò Brizio, Tommaso Morone, Roberto Tateo

arXiv:2410.20947Oct 28, 20240 citations

T T ¯ $$ \mathrm{T}\overline{\mathrm{T}} $$ -deformed 2D quantum field theories

Andrea Cavaglià, Stefano Negro, István M. Szécsényi +1 more

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.

arXiv:1608.05534Journal of High Energy PhysicsOct 1, 2016548 citations

The $T\overline T$ deformation of quantum field theory as a stochastic processThe $T\overline T$ deformation of quantum field theory as a stochastic process

John Cardy

Published Jan 21, 2018• 0 citations

OpenAlex

Abstract

We revisit the results of Zamolodchikov and others on the deformation of two-dimensional quantum field theory by the determinant

\det T

of the stress tensor, commonly referred to as

T\overline T

. Infinitesimally this is equivalent to a random coordinate transformation, with a local action which is, however, a total derivative and therefore gives a contribution only from boundaries or nontrivial topology. We discuss in detail the examples of a torus, a finite cylinder, a disk, and a truncated cone. In all cases the partition function evolves according to a linear diffusion-type equation, and the deformation may be viewed as a kind of random walk in moduli space in which domains shrink and become more symmetrical. The truncated cone allows access to information about the behavior of correlations, and of entanglement entropies, under the deformation. We also suggest a generalization of the whole formalism to higher dimensions.

Related papers

$T\overline{T}$ deformation of correlation functions$$ T\overline{T} $$ deformation of correlation functions

John Cardy

Abstract We study the evolution of correlation functions of local fields in a two-dimensional quantum field theory under the $\lambda T\overline{T}$ λT T ¯ deformation, suitably regularized. We show that this may be viewed in terms of the evolution of each field, with a Dirac-like string being attached at each infinitesimal step. The deformation then acts as a derivation on the whole operator algebra, satisfying the Leibniz rule. We derive an explicit equation which allows for the analysis of UV divergences, which may be absorbed into a non-local field renormalization to give correlation functions which are UV finite to all orders, satisfying a (deformed) operator product expansion and a Callan-Symanzik equation. We solve this in the case of a deformed CFT, showing that the Fourier-transformed renormalized two-point functions behave as k 2∆+2λk2, where ∆ is their IR conformal dimension. We discuss in detail deformed Noether currents, including the energy-momentum tensor, and show that, although they also become non-local, when suitably improved they remain finite, conserved and satisfy the expected Ward identities. Finally, we discuss how the equivalence of the $T\overline{T}$ T T ¯ deformation to a state-dependent coordinate transformation emerges in this picture.Abstract We study the evolution of correlation functions of local fields in a two-dimensional quantum field theory under the $$ \lambda T\overline{T} $$ λT T ¯ deformation, suitably regularized. We show that this may be viewed in terms of the evolution of each field, with a Dirac-like string being attached at each infinitesimal step. The deformation then acts as a derivation on the whole operator algebra, satisfying the Leibniz rule. We derive an explicit equation which allows for the analysis of UV divergences, which may be absorbed into a non-local field renormalization to give correlation functions which are UV finite to all orders, satisfying a (deformed) operator product expansion and a Callan-Symanzik equation. We solve this in the case of a deformed CFT, showing that the Fourier-transformed renormalized two-point functions behave as k 2∆+2λk2, where ∆ is their IR conformal dimension. We discuss in detail deformed Noether currents, including the energy-momentum tensor, and show that, although they also become non-local, when suitably improved they remain finite, conserved and satisfy the expected Ward identities. Finally, we discuss how the equivalence of the $$ T\overline{T} $$ T T ¯ deformation to a state-dependent coordinate transformation emerges in this picture.

arXiv:1907.03394Journal of High Energy PhysicsDec 1, 20196 citations

$T\bar{T}$ Deformations of Quantum Field theory$T\bar{T}$ Deformations of Quantum Field theory

Mengqi Zhu

Recent work by Zamolodchikov and others has uncovered a kind of "solvable" irrelevant deformation of two dimensional QFT generated by the composite operator T T .The irrelevant composite operator T T is built with the components of the energy-momentum tensor.The T T deformation can be regarded as a peculiar kind of integrable perturbation of integrable quantum field theories (IQFT).In the holographic dual, this deformation represents a geometric cutoff of the asymptotic region of AdS.The QFT is placed on a Dirichlet boundary at finite radial distance in the bulk.In this thesis we consider the problem of exact integration of the T T deformation of two dimensional quantum field theories, as well as some higher dimensional extensions in the form of T T deformations.When the action can be shown to only depend algebraically on the background metric the solution of the deformation equation on the Lagrangian can be given in closed form in terms of solutions of the (extended) Burgers' equation.We present such examples in two and higher dimensions.Recent work by Zamolodchikov and others has uncovered a kind of "solvable" irrelevant deformation of two dimensional QFT generated by the composite operator T T .The irrelevant composite operator T T is built with the components of the energy-momentum tensor.The T T deformation can be regarded as a peculiar kind of integrable perturbation of integrable quantum field theories (IQFT).In the holographic dual, this deformation represents a geometric cutoff of the asymptotic region of AdS.The QFT is placed on a Dirichlet boundary at finite radial distance in the bulk.In this thesis we consider the problem of exact integration of the T T deformation of two dimensional quantum field theories, as well as some higher dimensional extensions in the form of T T deformations.When the action can be shown to only depend algebraically on the background metric the solution of the deformation equation on the Lagrangian can be given in closed form in terms of solutions of the (extended) Burgers' equation.We present such examples in two and higher dimensions.

Sep 20, 20180 citations

On the flow of states under $T\overline{T}$ On the flow of states under $T\overline{T}$

Jorrit Kruthoff

We study the $T\overline{T}$ deformation of two dimensional quantum field theories from a Hamiltonian point of view, focusing on aspects of the theory in Lorentzian signature. Our starting point is a simple rewriting of the spatial integral of the $T\overline{T}$ operator, which directly implies the deformed energy spectrum of the theory. Using this rewriting, we then derive flow equations for various quantities in the deformed theory, such as energy eigenstates, operators, and correlation functions. On the plane, we find that the deformation merely has the effect of implementing successive canonical/Bogoliubov transformations along the flow. This leads us to define a class of non-local, 'dressed' operators (including a dressed stress tensor) which satisfy the same commutation relations as in the undeformed theory. This further implies that on the plane, the deformed theory retains its symmetry algebra, including conformal symmetry, if the original theory is a CFT. On the cylinder the $T\overline{T}$ deformation is much more non-trivial, but even so, correlation functions of certain dressed operators are integral transforms of the original ones. Finally, we propose a tensor network interpretation of our results in the context of AdS/CFT.We study the $T\overline{T}$ deformation of two dimensional quantum field theories from a Hamiltonian point of view, focusing on aspects of the theory in Lorentzian signature. Our starting point is a simple rewriting of the spatial integral of the $T\overline{T}$ operator, which directly implies the deformed energy spectrum of the theory. Using this rewriting, we then derive flow equations for various quantities in the deformed theory, such as energy eigenstates, operators, and correlation functions. On the plane, we find that the deformation merely has the effect of implementing successive canonical/Bogoliubov transformations along the flow. This leads us to define a class of non-local, 'dressed' operators (including a dressed stress tensor) which satisfy the same commutation relations as in the undeformed theory. This further implies that on the plane, the deformed theory retains its symmetry algebra, including conformal symmetry, if the original theory is a CFT. On the cylinder the $T\overline{T}$ deformation is much more non-trivial, but even so, correlation functions of certain dressed operators are integral transforms of the original ones. Finally, we propose a tensor network interpretation of our results in the context of AdS/CFT.

Nov 1, 202020 citations

On the flow of states under $T\overline{T}$ On the flow of states under $T\overline{T}$

Jorrit Kruthoff, Onkar Parrikar

We study the TT deformation of two dimensional quantum field theories from a Hamiltonian point of view, focusing on aspects of the theory in Lorentzian signature. Our starting point is a simple rewriting of the spatial integral of the TT operator, which directly implies the deformed energy spectrum of the theory. Using this rewriting, we then derive flow equations for various quantities in the deformed theory, such as energy eigenstates, operators, and correlation functions. On the plane, we find that the deformation merely has the effect of implementing successive canonical/Bogoliubov transformations along the flow. This leads us to define a class of non-local, “dressed” operators (including a dressed stress tensor) which satisfy the same commutation relations as in the undeformed theory. This further implies that on the plane, the deformed theory retains its symmetry algebra, including conformal symmetry, if the original theory is a CFT. On the cylinder the TT deformation is much more non-trivial, but even so, correlation functions of certain dressed operators are integral transforms of the original ones. Finally, we propose a tensor network interpretation of our results in the context of AdS/CFT.We study the TT deformation of two dimensional quantum field theories from a Hamiltonian point of view, focusing on aspects of the theory in Lorentzian signature. Our starting point is a simple rewriting of the spatial integral of the TT operator, which directly implies the deformed energy spectrum of the theory. Using this rewriting, we then derive flow equations for various quantities in the deformed theory, such as energy eigenstates, operators, and correlation functions. On the plane, we find that the deformation merely has the effect of implementing successive canonical/Bogoliubov transformations along the flow. This leads us to define a class of non-local, “dressed” operators (including a dressed stress tensor) which satisfy the same commutation relations as in the undeformed theory. This further implies that on the plane, the deformed theory retains its symmetry algebra, including conformal symmetry, if the original theory is a CFT. On the cylinder the TT deformation is much more non-trivial, but even so, correlation functions of certain dressed operators are integral transforms of the original ones. Finally, we propose a tensor network interpretation of our results in the context of AdS/CFT.

arXiv:2006.03054SciPost PhysicsNov 20, 202019 citations

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.

arXiv:2410.20947Oct 28, 20240 citations

T T ¯ $\mathrm{T}\overline{\mathrm{T}}$ -deformed 2D quantum field theoriesT T ¯ $$ \mathrm{T}\overline{\mathrm{T}} $$ -deformed 2D quantum field theories

Andrea Cavaglià, Stefano Negro, István M. Szécsényi +1 more

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.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.

arXiv:1608.05534Journal of High Energy PhysicsOct 1, 2016548 citations

The

T\overline T

deformation of quantum field theory as a stochastic process

We revisit the results of Zamolodchikov and others on the deformation of two-dimensional quantum field theory by the determinant

\det T

of the stress tensor, commonly referred to as

T\overline T

T\overline{T}

deformation of correlation functions

Abstract We study the evolution of correlation functions of local fields in a two-dimensional quantum field theory under the

\lambda T\overline{T}

λT T ¯ deformation, suitably regularized. We show that this may be viewed in terms of the evolution of each field, with a Dirac-like string being attached at each infinitesimal step. The deformation then acts as a derivation on the whole operator algebra, satisfying the Leibniz rule. We derive an explicit equation which allows for the analysis of UV divergences, which may be absorbed into a non-local field renormalization to give correlation functions which are UV finite to all orders, satisfying a (deformed) operator product expansion and a Callan-Symanzik equation. We solve this in the case of a deformed CFT, showing that the Fourier-transformed renormalized two-point functions behave as k 2∆+2λk2, where ∆ is their IR conformal dimension. We discuss in detail deformed Noether currents, including the energy-momentum tensor, and show that, although they also become non-local, when suitably improved they remain finite, conserved and satisfy the expected Ward identities. Finally, we discuss how the equivalence of the

T\overline{T}

T T ¯ deformation to a state-dependent coordinate transformation emerges in this picture.

T\bar{T}

Deformations of Quantum Field theory

On the flow of states under

T\overline{T}

We study the

T\overline{T}

deformation of two dimensional quantum field theories from a Hamiltonian point of view, focusing on aspects of the theory in Lorentzian signature. Our starting point is a simple rewriting of the spatial integral of the

T\overline{T}

operator, which directly implies the deformed energy spectrum of the theory. Using this rewriting, we then derive flow equations for various quantities in the deformed theory, such as energy eigenstates, operators, and correlation functions. On the plane, we find that the deformation merely has the effect of implementing successive canonical/Bogoliubov transformations along the flow. This leads us to define a class of non-local, 'dressed' operators (including a dressed stress tensor) which satisfy the same commutation relations as in the undeformed theory. This further implies that on the plane, the deformed theory retains its symmetry algebra, including conformal symmetry, if the original theory is a CFT. On the cylinder the

T\overline{T}

deformation is much more non-trivial, but even so, correlation functions of certain dressed operators are integral transforms of the original ones. Finally, we propose a tensor network interpretation of our results in the context of AdS/CFT.

On the flow of states under

T\overline{T}

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 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.