We introduce and study a class of two-dimensional integrable quantum field theories that carry an internal ℤ structure. These models extend factorised scattering beyond the conventional framework, featuring both the usual hierarchy of integer-spin conserved charges and an additional tower of fractional-spin ones. Our construction relies on a reparametrisation of rapidity space that lifts standard scattering amplitudes to a multiplet related by an internal cyclic symmetry. This construction is naturally embedded within a generalised Gibbs ensemble, which provides the natural framework for a consistent graded Thermodynamic Bethe Ansatz. This leads to new Y-systems encoding the graded spectrum. In a special case, these functional relations match those obtained via the ODE/IM correspondence from the monodromy analysis of the quantum cubic oscillator. Even in the simplest models, for one sign of the auxiliary temperature, the finite-volume ground-state energy spectrum undergoes an infinite sequence of level crossings as the coupling strength increases. A preliminary analysis also suggests that these theories exhibit structural connections with cyclic orbifolds. Within this setup, one can consistently include extra CDD factors that realise fractional-spin analogues of the $$ T\overline{T} $$ deformation. In analytically tractable cases, a Hagedorn-like behaviour is observed for a sign of the flow parameter, and the deformed spectrum develops a finite limiting temperature.

Italy

Nicolò Brizio

Tommaso Morone

Nicolò Primi

Roberto Tateo

Graded S-matrices, generalised Gibbs ensembles and fractional-spin CDD deformations

The four-dimensional Chern-Simons (CS) theory provides a systematic procedure for realizing two-dimensional integrable field theories. It is therefore a natural question to ask whether integrable deformations of the theories can be realized in the four-dimensional CS theory. In this work, we study $$T\overline{T}$$ and root- $$T\overline{T}$$ deformations of two-dimensional integrable field theories, formulated in terms of dynamical coordinate transformations, within the framework of four-dimensional CS theory coupled to disorder defects. We illustrate our procedure in detail for the degenerate $$\mathcal{E}$$ -model, a specific construction that captures and unifies a broad range of integrable systems, including the principal chiral model.

Japan

Jun-ichi Sakamoto

Masahito Yamazaki

$$T\overline{T}$$ and root- $$T\overline{T}$$ deformations in four-dimensional Chern-Simons theory

We study the analytic continuation in the spin of the planar spectrum of ABJM theory using the integrability-based Quantum Spectral Curve (QSC) method. Under some minimal assumptions, we classify the analytic properties of the Q-functions appearing in the QSC compatible with the spin being non-integer. In this way, we find not one but $\textit{two}$ distinct possibilities. While one is related to standard Regge trajectories, we show that the second choice can be used to build bridges which connect leading and subleading Regge trajectories, thus giving a shortcut to reach infinitely many sheets of the spin Riemann surface without going explicitly around the branch points in the complex spin plane. Moreover, the bridges are exact spin reflections of standard Regge trajectories. Together, these results reveal the existence of a hidden symmetry which we call “twist/co-twist symmetry”: every non-BPS local operator has an exact image — living below unitarity on a different Regge trajectory — with the same ∆ and the spin flipped by a Weyl reflection. We discuss how an analogous phenomenon, based on the same mechanism at the level of the QSC, also occurs at non-perturbative level in $$ \mathcal{N} $$ =4 SYM. This provides a framework to understand recent independent observations of the symmetry in this model at weak and strong coupling. We present numerical results for Regge trajectories in planar ABJM theory, in particular we compute exactly the coupling dependence of the position of the leading Regge pole in the correlator of four stress tensors. The shape of this leading trajectory shows a behaviour at weak coupling that strongly resembles the BFKL limit in $$ \mathcal{N} $$ =4 SYM.

Andrea Cavaglià

Valerio Tripodi

Regge trajectories and bridges between them in integrable AdS/CFT

<p>We study <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>T</mi><mover accent="true"><mi>T</mi><mo stretchy="false">¯</mo></mover></math>-like deformations of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>d</mi><mo>&gt;</mo><mn>2</mn></math> Yang-Mills theories. The standard <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>T</mi><mover accent="true"><mi>T</mi><mo stretchy="false">¯</mo></mover></math> flows lead to multitrace Lagrangians, and the non-Abelian gauge structures make it challenging to find Lagrangians in a closed form. However, within the geometric approach to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>T</mi><mover accent="true"><mi>T</mi><mo stretchy="false">¯</mo></mover></math>, we obtain the closed-form solution to the metric flow and stress-energy tensor, and show that instanton solutions are undeformed. We also introduce new symmetrized single-trace <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>T</mi><mover accent="true"><mi>T</mi><mo stretchy="false">¯</mo></mover></math>-like deformations, whose solutions in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>d</mi><mo>=</mo><mn>4</mn></math> include the non-Abelian Born-Infeld Lagrangian proposed by Tseytlin in 1997.</p>

United States

Christian Ferko

China

Jue Hou

Australia

Gabriele Tartaglino-Mazzucchelli

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>T</mi><mover accent="true"><mrow><mi>T</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow></math>-like Flows of Yang-Mills Theories

Two dimensional conformal field theories with the extended $$ {\mathcal{W}}_3 $$ symmetry algebra have an infinite number of mutually commuting conserved charges, which are referred to as the quantum Boussinesq charges. In this work we construct local operators whose zero modes are precisely these conserved charges. For this purpose we study the higher spin conformal field theory on the torus and compute thermal correlators involving the stress tensor and the spin-3 current in a higher spin module of the $$ {\mathcal{W}}_3 $$ algebra. In addition we independently obtain the excited state eigenvalues of the quantum Boussinesq charges within the higher spin module via the ODE/IM correspondence. A judicious combination of these data allows us to derive the local operators, whose integrals are the conserved charges of the integrable hierarchy.

India

Sujay Ashok

Sanhita Parihar

Tanmoy Sengupta

Adarsh Sudhakar

Thermal correlators and currents of the $$ {\mathcal{W}}_3 $$ algebra

We develop a generic geometric formalism that incorporates both <math><mi>T</mi><mover><mi>T</mi><mo>¯</mo></mover></math>-like and root-<math><mi>T</mi><mover><mi>T</mi><mo>¯</mo></mover></math>-like deformations in arbitrary dimensions. This framework applies to a wide family of stress-energy tensor perturbations and encompasses various well-known field theories. Building upon the recently proposed correspondence between Ricci-based gravity and <math><mi>T</mi><mover><mi>T</mi><mo>¯</mo></mover></math>-like deformations, we further extend this duality to include root-<math><mi>T</mi><mover><mi>T</mi><mo>¯</mo></mover></math>-like perturbations. This refinement extends the potential applications of our approach and contributes to a deeper exploration of the interplay between stress tensor perturbations and gravitational dynamics. Among the various original outcomes detailed in this Letter, we have also obtained a deformation of the flat Jackiw-Teitelboim gravity action.

Iran

H. Babaei-Aghbolagh

Germany

Song He

Hao Ouyang

Geometric Formulation of Generalized Root-<math><mi>T</mi><mover><mi>T</mi><mo>¯</mo></mover></math> Deformations

We study two dimensional systems with extended conformal symmetry generated by the   <math> <mi>W</mi> </math>  $$ \mathcal{W} $$  3 algebra. These are expected to have an infinite number of commuting conserved charges, which we refer to as the quantum Boussinesq charges. We compute the eigenvalues of the quantum Boussinesq charges in both the vacuum and first excited states of the higher spin module through the ODE/IM correspondence. By studying the higher spin conformal field theory on the torus, we also calculate thermal correlators involving the energy-momentum tensor and the spin-3 current by making use of the Zhu recursion relations. By combining these results, we show that it is possible to derive the current densities, whose integrals are the quantum Boussinesq charges. We also evaluate the thermal expectation values of the conserved charges, and show that these are quasi-modular differential operators acting on the character of the higher spin module.

Integrable structure of higher spin CFT and the ODE/IM correspondence

We study systems in arbitrary space-time dimensions where matter, deformed by <math><mi>T</mi><mover><mrow><mi>T</mi></mrow><mo>‾</mo></mover></math>-like irrelevant operators, is coupled to gravity in the Palatini formalism. The dynamically equivalent perspective is investigated, wherein the deformation transitions from the matter action to the gravitational one or vice versa. This alternative viewpoint leads to the emergence of Ricci-based gravity theories, thus providing a high-dimensional generalisation of the well-known equivalence between two-dimensional <math><mi>T</mi><mover><mrow><mi>T</mi></mrow><mo>‾</mo></mover></math> deformations and coupling to Jackiw-Teitelboim gravity. This dynamical equivalence is examined within the framework of the recently introduced Lagrangian flow equation, which notably led to the discovery of a direct link between Nambu-Goto theory and <math><mi>T</mi><mover><mrow><mi>T</mi></mrow><mo>‾</mo></mover></math> in <math><mi>d</mi><mo>=</mo><mn>2</mn></math>, as well as significant insights into nonlinear electrodynamics models in <math><mi>d</mi><mo>=</mo><mn>4</mn></math>. The investigation involves explicit examples in <math><mi>d</mi><mo>=</mo><mn>4</mn></math> dimensions; it builds upon earlier research concerning the metric interpretation of <math><mi>T</mi><mover><mrow><mi>T</mi></mrow><mo>‾</mo></mover></math>-like perturbations, incorporates and extends recent findings in the cosmology-related literature associated to the concept of reframing. We focus on scenarios where the resulting modified gravity theories manifest as Born-Infeld and Starobinsky types.

United Kingdom

Stefano Negro

Gravity and Flows in Higher Dimensions

In this study, we introduce and investigate a family of quantum mechanical models in 0 + 1 dimensions, known as generalized Born quantum oscillators. These models represent a one-parameter deformation of a specific system obtained by reducing the Nambu-Goto theory to 0 + 1 dimensions. Despite these systems showing significant similarities with $$ \textrm{T}\overline{\textrm{T}} $$ -type perturbations of two-dimensional relativistic models, our analysis reveals their potential as interesting regularizations of the Berry-Keating theory. We quantize these models using the Weyl quantization scheme up to very high orders in $\textit{ħ}$. By examining a specific scaling limit, we observe an intriguing connection between the generalized Born quantum oscillators and the Riemann-Siegel $\textit{θ}$ function.

Francesco Giordano

The generalized Born oscillator and the Berry-Keating Hamiltonian

The ODE/IM correspondence is an exact link between classical and quantum integrable models. The primary purpose of this work is to show that it remains valid after   <math> <mi>T</mi> <mover> <mi>T</mi> <mo>¯</mo> </mover> </math>  $$ \textrm{T}\overline{\textrm{T}} $$  perturbation on both sides of the correspondence. In particular, we prove that the deformed Lax pair of the sinh-Gordon model, obtained from the unperturbed one through a dynamical change of coordinates, leads to the same Burgers-type equation governing the quantum spectral flow induced by   <math> <mi>T</mi> <mover> <mi>T</mi> <mo>¯</mo> </mover> </math>  $$ \textrm{T}\overline{\textrm{T}} $$ . Our main conclusions have general validity, as the analysis may be easily adapted to all the known ODE/IM examples involving integrable quantum field theories.

Fabrizio Aramini

Deforming the ODE/IM correspondence with   <math> <mi>T</mi> <mover> <mi>T</mi> <mo>¯</mo> </mover> </math>  $$ \textrm{T}\overline{\textrm{T}} $$