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arXiv:2006.03054 · SciPost Physics

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

Jorrit Kruthoff, Onkar Parrikar

Published Nov 20, 2020• 19 citations
arXivHTMLPDF

Abstract

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.

DOI
10.21468/scipostphys.9.5.078

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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.
On the flow of states under TT‾T\overline{T}TT
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Abstract We study the evolution of correlation functions of local fields in a two-dimensional quantum field theory under the λTT‾ \lambda T\overline{T} λTT 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 TT‾ T\overline{T} TT T T ¯ deformation to a state-dependent coordinate transformation emerges in this picture.
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.
Smirnov and Zamolodchikov recently introduced a new class of two-dimensional quantum field theories, defined through a differential change of any existing theory by the determinant of the energy-momentum tensor. From this TT‾T\overline{T}TT flow equation one can find a simple expression for both the energy spectrum and the SSS-matrix of the TT‾T\overline{T}TT deformed theories. Our goal is to find the renormalized Lagrangian of the TT‾T\overline{T}TT deformed theories. In the context of the TT‾T\overline{T}TT deformation of an integrable theory, the deformed theory is also integrable and, correspondingly, the SSS-matrix factorizes into two-to-two SSS-matrices. One may thus hope to be able to extract the renormalized Lagrangian from the SSS-matrix. We do this explicitly for the TT‾T\overline{T}TT deformation of a free massive scalar, to second order in the deformation parameter. Once one has the renormalized Lagrangian one can, in principle, compute all other observables, such as correlation functions. We briefly discuss this, as well as the relation between the renormalized Lagrangian, the TT‾T\overline{T}TT flow equation, and the SSS-matrix. We also mention a more general class of integrability-preserving deformations of a free scalar field theory.
Symmetries and operators in TT‾ T\overline{T} TT deformed CFTs
A bstract TT‾ T\overline{T} TT T T ¯ -deformed CFTs are known to possess nonlocal Virasoro×Virasoro symmetries, whose currents depend on integration of the stress tensor. Unfortunately, these symmetries do not act in a tractable manner on the undeformed operators, which are defined by evolving local operators in the undeformed CFT on a constant-time slice using the deformed Hamiltonian. In this paper, we explicitly construct two distinct classes of operators: (i) dressed operators, which are primary with respect to the Virasoro×Virasoro symmetries, and (ii) physical operators, a new type of operator that depends on fundamental fields at specific points and is therefore local at the classical level. While dressed operators preserve the conformal symmetry structure, they are themselves nonlocal in the sense that they depend on integrated quantities. Physical operators, by contrast, depend only on local fields. Moreover, physical operators can be expressed in terms of dressed operators, and this relation can be promoted to a definition of such operators at the quantum level. This enables the use of Virasoro×Virasoro symmetries to systematically compute physical quantities such as correlation functions. We calculate the two-point correlation functions of these operators in momentum space and find that our results align with both string theory predictions and field theory calculations. Additionally, we explore the relationship between physical operators and alternative operator definitions proposed in the literature.
T T ¯ TT‾ \mathrm{T}\overline{\mathrm{T}} TT -deformed 2D quantum field theories
It was noticed many years ago, in the framework of massless RG flows, that the irrelevant composite operator TT‾ \mathrm{T}\overline{\mathrm{T}} TT , 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.