We consider a one-parameter family of composite fields — bi-linear in the components of the stress-energy tensor — which generalise the   <math> <mi>T</mi> <mover> <mi>T</mi> <mo>¯</mo> </mover> </math>  $$ \mathrm{T}\overline{\mathrm{T}} $$  operator to arbitrary space-time dimension d ≥ 2. We show that they induce a deformation of the classical action which is equivalent — at the level of the dynamics — to a field-dependent modification of the background metric tensor according to a specific flow equation. Even though the starting point is the flat space, the deformed metric is generally curved for any d &gt; 2, thus implying that the corresponding deformation can not be interpreted as a coordinate transformation. The central part of the paper is devoted to the development of a recursive algorithm to compute the coefficients of the power series expansion of the solution to the metric flow equation. We show that, under some quite restrictive assumptions on the stress-energy tensor, the power series yields an exact solution. Finally, we consider a class of theories in d = 4 whose stress-energy tensor fulfils the assumptions above mentioned, namely the family of abelian gauge theories in d = 4. For such theories, we obtain the exact expression of the deformed metric and the vierbein. In particular, the latter result implies that ModMax theory in a specific curved space is dynamically equivalent to its Born-Infeld-like extension in flat space. We also discuss a dimensional reduction of the latter theories from d = 4 to d = 2 in which an interesting marginal deformation of d = 2 field theories emerges.

Portugal

Riccardo Conti

Germany

Jacopo Romano

Italy

Roberto Tateo

Metric approach to a   <math> <mi>T</mi> <mover> <mi>T</mi> <mo>¯</mo> </mover> </math>  $$ \mathrm{T}\overline{\mathrm{T}} $$ -like deformation in arbitrary dimensions

The   <math> <mi>T</mi> <mover> <mi>T</mi> <mo>¯</mo> </mover> </math>  $$ \mathrm{T}\overline{\mathrm{T}} $$ -deformed classical Lagrangian of a 2D Lorentz invariant theory can be derived from the original one, perturbed only at first order by the bare   <math> <mi>T</mi> <mover> <mi>T</mi> <mo>¯</mo> </mover> </math>  $$ \mathrm{T}\overline{\mathrm{T}} $$  composite field, through a field-dependent change of coordinates. Considering, as an example, the nonlinear Schrödinger (NLS) model with generic potential, we apply this idea to non-relativistic models. The form of the deformed Lagrangian contains a square-root and is similar but different from that for relativistic bosons. We study the deformed bright, grey and Peregrine’s soliton solutions. Contrary to naive expectations, the   <math> <mi>T</mi> <mover> <mi>T</mi> <mo>¯</mo> </mover> </math>  $$ \mathrm{T}\overline{\mathrm{T}} $$ -perturbation of nonlinear Schrödinger NLS with quartic potential does not trivially emerge from a standard non-relativistic limit of the deformed sinh-Gordon field theory. The c → ∞ outcome corresponds to a different type of irrelevant deformation. We derive the corresponding Poisson bracket structure, the equations of motion and discuss various interesting aspects of this alternative type of perturbation, including links with the recent literature.

Paolo Ceschin

<math> <mi>T</mi> <mover> <mi>T</mi> <mo>¯</mo> </mover> </math>  $$ \mathrm{T}\overline{\mathrm{T}} $$ -deformed nonlinear Schrödinger

We study the large momentum limit of the monster potentials of Bazhanov-Lukyanov-Zamolodchikov, which — according to the ODE/IM correspondence — should correspond to excited states of the Quantum KdV model.

Davide Masoero

Counting monster potentials

The T  <math> <mover> <mi>T</mi> <mo>¯</mo> </mover> </math>  $$ \overline{\mathrm{T}} $$  deformation of 2-dimensional QFTs is closely-related to Jackiw- Teitelboim gravity. It has been shown that, at the classical level, this perturbation induces an interaction between the stress-energy tensor and space-time and the equations of motion of the deformed theory map onto the original ones through a field-dependent coordinate transformation. At the quantum level, instead, the perturbation is induced by a modification of the original S-matrix by a specific CDD factor and, correspondingly, the quantised energy levels evolve according to a Burgers-type equation. In this paper, we point out that, in the framework of integrable field theories, there exist infinite families of perturbations characterised by a coupling between space-time and local conserved currents, labelled by the Lorentz spin. Similarly to the T  <math> <mover> <mi>T</mi> <mo>¯</mo> </mover> </math>  $$ \overline{\mathrm{T}} $$  case, the deformed models emerge through a field-dependent coordinate transformation involving conserved currents with higher Lorentz spin. Furthermore, using a geometric construction, we present a general method to derive the integrable hierarchy of the corresponding deformed models. The resulting expressions of the conserved currents turn out to be essential for the identification of the scattering phase factors which generate the deformations of the S-matrix, at the quantum level. Finally, the effect of the perturbations on the finite-volume spectrum is investigated using a non-linear integral equation. Exact spectral flow equations are derived, and links with previous literature, in particular on the J  <math> <mover> <mi>T</mi> <mo>¯</mo> </mover> </math>  $$ \overline{\mathrm{T}} $$  model, are discussed. While the classical setup is very general, the sine-Gordon model and its CFT limit are used as illustrative quantum examples. Most of the final equations and considerations are, however, of broader validity, or easily generalisable to more complicated systems.

United States

Stefano Negro

Conserved currents and T  <math> <mover> <mi>T</mi> <mo>¯</mo> </mover> </math>  $$ \overline{\mathrm{T}} $$   s  irrelevant deformations of 2D integrable field theories

Starting from the recently-discovered   <math> <mi>T</mi> <mover> <mi>T</mi> <mo>¯</mo> </mover> </math>  $$ \mathrm{T}\overline{\mathrm{T}} $$ -perturbed Lagrangians, we prove that the deformed solutions to the classical EoMs for bosonic field theories are equivalent to the unperturbed ones but for a specific field-dependent local change of coordinates. This surprising geometric outcome is fully consistent with the identification of   <math> <mi>T</mi> <mover> <mi>T</mi> <mo>¯</mo> </mover> </math>  $$ \mathrm{T}\overline{\mathrm{T}} $$ -deformed 2D quantum field theories as topological JT gravity coupled to generic matter fields. Although our conclusion is valid for generic interacting potentials, it first emerged from a detailed study of the sine-Gordon model and in particular from the fact that solitonic pseudo-spherical surfaces embedded in ℝ3 are left invariant by the deformation. Analytic and numerical results concerning the perturbation of specific sine-Gordon soliton solutions are presented.

The   <math> <mi>T</mi> <mover> <mi>T</mi> <mo>¯</mo> </mover> </math>  $$ \mathrm{T}\overline{\mathrm{T}} $$  perturbation and its geometric interpretation

Surprising links between the deformation of 2D quantum field theories induced by the composite   <math> <mi>T</mi> <mover> <mi>T</mi> <mo>¯</mo> </mover> </math>  $$ \mathrm{T}\overline{\mathrm{T}} $$  operator, effective string models and the AdS/CFT correspondence, have recently emerged. The purpose of this article is to discuss various classical aspects related to the deformation of 2D interacting field theories. Special attention is given to the sin(h)-Gordon model, for which we were able to construct the   <math> <mi>T</mi> <mover> <mi>T</mi> <mo>¯</mo> </mover> </math>  $$ \mathrm{T}\overline{\mathrm{T}} $$ -deformed Lax pair. We consider the Lax pair formulation to be the first essential step toward a more satisfactory geometrical interpretation of this deformation within the integrable model framework.

Leonardo Iannella

Generalised Born-Infeld models, Lax operators and the   <math> <mi>T</mi> <mover> <mi>T</mi> <mo>¯</mo> </mover> </math>  $$ \mathrm{T}\overline{\mathrm{T}} $$  perturbation

The Quantum Spectral Curve (QSC) equations for planar  <math><mi>N</mi><mo>=</mo><mn>6</mn></math> $$ \mathcal{N}=6 $$ super-conformal Chern-Simons (SCS) are solved numerically at finite values of the coupling constant for states in the  <math><mi>s</mi><mi>l</mi><mfenced><mrow><mn>2</mn><mo>|</mo><mn>1</mn></mrow></mfenced></math> $$ \mathfrak{s}\mathfrak{l}\left(2\Big|1\right) $$ sector.