**Alex Arvanitakis** - *Non-invertible (duality) symmetries, topological defects and hamiltonian mechanics* - Slides

Non-invertible symmetry is a fairly new concept. It is a generalization of the concept of symmetry between two "phases" A and B which are separated by a "wall" with specific properties, called a topological defect. These concepts appear in many problems of condensed matter and high energy physics. I will talk about my recent research where I explained the physics of topological defects and the corresponding non-invertible symmetries using familiar ideas of hamiltonian mechanics (lagrangian correspondences, in fact), and give applications to string theory and integrable structures in Chern-Simons theories. **Zoltan Bajnok** - *Integrable sigma models in a magnetic field: the full analytic trans-series* - Slides

We analyse the energy densities of integrable quantum field theories in the presence of an external field coupled to a conserved charge. By using the Wiener-Hopf technique we solve the linear Thermodynamic Bethe Ansatz equations in terms of a trans-series, which incorporates both perturbative and non-perturbative corrections. We demonstrate the intricate interplay between these corrections in the principle chiral and O(N) models. **Riccardo Borsato** - *Integrability of Jordanian deformations* - Slides

In recent years various constructions have appeared, giving rise to deformations of 2-dimensional sigma-models that preserve the (classical) integrability of the seed model. An interesting direction is the study of deformations of the AdS_5 x S^5 superstring background, with the motivation of constructing generalisations of the AdS/CFT correspondence where one can still apply the powerful methods of integrability. I will focus on a family of integrable deformations called "Yang-Baxter", and in particular on the "Jordanian" sub-class. During the first part of my talk, I will review various aspects of Yang-Baxter deformations, including their relation to non-abelian T-duality and their reformulation in the language of Double Field Theory. In the second part of the talk, to address some of the open questions on integrability, I will discuss some of the unexpected features that one finds when looking at the perturbative worldsheet S-matrix of a Jordanian deformation of AdS_5 x S^5, focusing in particular on the undeformed limit of this story. **Alejandra Castro** - *Designing Gravitational Theories via Symmetric Product Orbifolds* - Slides

I will discuss the large-N limit of two-dimensional symmetric product orbifolds. The goal is to single out which symmetric product orbifold theory could lead to a strongly coupled point in their moduli space, whose dual could be a semi-classical theory of AdS_3 gravity. To this end, we consider the symmetric product orbifold of N=(2,2) SCFT_2, and classify them according to two criteria. The first criterion is the existence of a single-trace twisted exactly marginal operator in their moduli space. The second criterion is a sparseness condition on the growth of light states in the elliptic genera. In this context, we encounter a strange variety: theories that obey the first criterion but the second criterion falls into a Hagedorn-like growth. I will explain why this may be counter-intuitive and discuss how it might be accounted for in conformal perturbation theory. I will also present a new infinite class of theories that obey both criteria, which are necessary conditions for their moduli spaces to contain a supergravity points. **Lucía Córdova** - *O(N) monolith reloaded: sum rules and form factor bootstrap* - Slides

In this talk I will discuss the space of two-dimensional quantum field theories with a global O(N) symmetry. Previous works using S-matrix bootstrap revealed a rich space in which integrable theories appear at special points and an abundance of unknown models hinting at a non conventional UV behaviour. In order to gain more information about the unknown models, we extend the S-matrix set-up by including into the bootstrap form factors and spectral functions for operators like stress tensor and symmetry currents. I will explain how this extended set-up works and show how the associated sum rules allow us to put bounds on quantities like the central charge of the underlying conformal theories in the UV. Based on work with M. Correia, A. Georgoudis and A. Vuignier. **Paul Fendley** - *“Dualities” from non-invertible defects* - Slides

Recent work has revealed a host of “dualities” between strongly interacting models. As apparent from the canonical example of Kramers and Wannier, such dualities are much subtler than a one-to-one mapping of energy levels, but rather are non-invertible. I describe an algebraic structure in the XXZ spin chain and three other Hamiltonian that yields non-invertible maps between them and also guarantees all are integrable. Several of these models also possess useful non-invertible symmetries, with the spontaneous breaking of one resulting in an unusual ground-state degeneracy. The maps are developed using topological defects coming from fusion categories and the lattice analog of the orbifold construction. **Falk Hassler** - *Supergeneralized geometry, dualities and integrable deformations* - Slides

Generalized geometry has become a powerful tool to study integrable sigma-models, their deformations and transformation under duality symmetries of string theory. Its success originates from unifying all bosonic symmetries, consisting of diffeomorphisms and form-field gauge transformations. This idea can be pushed further by including supersymmetry to obtain supergeneralized geometry. After reviewing its salient features, I show how it captures fermionic T-dualities and allows to describe integrable deformations without the ambiguities on the R-R sector one faces in generalized geometry. As an example, I present the eta- and lambda-deformation in this new framework. **Sylvain Lacroix -*** Elliptic integrable sigma-models and the geometry of the spectral parameter* - Slides

The first part of this talk will concern the construction of new integrable sigma-models with Lax connections and R-matrices which are elliptic functions of the spectral parameter. We will focus on an elliptic deformation of the Principal Chiral Model based on the Lie group SL(N). For N=2 and up to reality conditions, this theory coincides with a model previously found by Cherednik, while the results are new for N>2. In a second part of the talk, we will discuss some aspects of the geometry of the spectral parameter of integrable sigma-models. In particular, we will explain the role of their so-called twist function in the formulation of their renormalisation group flow, supported by various results and conjectures. This talk is based on work in progress with Anders Heide Wallberg.

**Eric Lescano**-

*Statistical matter in Double Field Theory*- Slides

In this talk I will explain how to introduce hydrodynamical variables in Double Field Theory (DFT) to include statistical matter, particularly, the perfect fluid. A generalization of the scalar field-perfect fluid correspondence will be explained in detail and with it we will prove, on the one hand, that the perfect fluid dynamics do not receive the alpha'-corrections generated by the generalized Bergshoeff-de Roo identification and, on the other, that a family of O(D,D) invariant string cosmologies can be easily recovered from this model. Since DFT has proved to be useful to deal with both Riemannian and non-Riermannian backgrounds in the last part of the talk I will show the first steps into a non-commutative DFT which can be potentially used to explored non-commutative corrections in these geometries.

**Nat Levine**-

*Universal 1-loop divergences for integrable sigma-models*- Slides

I will show that the 1-loop divergences of a broad class of integrable sigma-models take a universal form. "Universal" means the result only depends on the positions of the poles of the theory's Lax connection in the spectral plane. I will show that these universal formulae have roots in the 4d Chern-Simons theory, and that the 2d and 4d RG flows coincide at 1-loop order.

**Stefano Negro -**

*Solving the Form Factor bootstrap for Solvable Irrelevant Deformations*- Slides

Solvable Irrelevant Deformations – also known as “generalised TTbar deformations” – are a large class of perturbations of Integrable Quantum Field Theories (IQFTs). From the perspective of the factorised scattering theory, they can be defined as deformations of the two-body S-matrix by a CDD factor. While still being integrable, the resulting theories display unusual properties in their high-energy regime. In particular, the original UV fixed point is lost and it is replaced by a Hagedorn behaviour, reminiscent of the string-theoretic one. This is expected, due to the deformations being irrelevant in nature. However, to the contrary of a generic irrelevant perturbation, these theories offer an enormous amount of control, allowing us to probe their deep UV regime. In a sense, they constitute a robust extension of the standard Wilsonian paradigm for Quantum Field Theories.

In this talk I will present some recent developments in the study of the Solvable Irrelevant Deformations: the determination, in full generality, of their Form Factors. The latter are matrix elements of operators between a vacuum an an n-particle state and constitute a set of building blocks that can be used to compute correlation functions. In IQFTs, these objects satisfy a set of equations that allow us to bootstrap their exact expressions. Carrying on this procedure for Solvable Irrelevant Deformations one finds that the Form Factors take a factorised form as products of the unperturbed objects with a factor containing the effects of the perturbation. With this result, it is then possible to analyse the effect of the perturbation on correlation functions. We will see that, depending on the sign of the deformation parameters, the form factor expansion of correlation functions can be divergent or “hyper-convergent” and that these behaviours possess an intuitive interpretation in terms of particles acquiring a positive or negative size, as was recently proposed by Cardy and Doyon.

**Chiara Paletta**-

*An XYZ-type deformation of the Hubbard model*- Slides

The Hubbard model is an example of an integrable system, often used as a toy model to describe the motion of electrons in the conduction band of a solid. Its integrability properties make it an excellent playground for different fields of research, including Condensed Matter and High-Energy Physics. In this talk, I will present a new integrable range 3 elliptic model in both the Hermitian and the Lindbladian formulation: it can be employed to describe the dynamics of a physical system in contact with a Markovian environment. I will analyze three different aspects:

1. Integrability: I will demonstrate that the range 3 model is integrable and can be embedded into the standard framework of the Yang-Baxter equation.

2. Duality: I will show its relation to a Nearest Neighbour model through the use of the bond-site transformation.

3. Deformation: I will illustrate how this new model is a deformation of the Hubbard model.

The talk is based on the works 2301.01612 and 2305.01922 with M. de Leeuw, B. Pozsgay and E. Vernier. **Davide Polvara** - *Relativist mixed-flux worldsheet scattering in AdS3/CFT2* - Slides

It has been known for some years that strings on AdS3 x S3 x T4 with mixed Ramond-Ramond (RR) and Neveu-Schwarz-Neveu-Schwarz (NSNS) flux are amenable to integrability techniques. This allows for the non-perturbative study of the dynamics of strings propagating in this background, which otherwise would result incredibly difficult. Despite this fact, due to the complicated dispersion relations for the fundamental particles of this model, the worldsheet S matrix in the lightcone gauge has not been fully computed yet.

In this talk, I will consider a relativistic limit of the S matrix whereby the bootstrap program can be completed, including the dressing factors for fundamental particles and bound states. In this limit, the model features k-1 (being k the quantised NSNS flux) massive particles all connected by fusion and 2 types of massless particles, distinguished by their highest-weight states. While this relativistic model is interesting in its own right, it also constitutes a test for future proposals of the dressing factors of the full theory. **Anton Pribytok** - *Supersymmetric r-deformed CP ^{n-1} and chiral duals* - Slides

In this work we investigate the supersymmetric deformation of the

*CP*, its properties and various conformal limits, with special attention devoted to n=2. We begin by introducing fermionic coupling to bosonic model with

^{n-1}*CP*symmetric target space on 2-dim Euclidean worldsheet, which can be shown to be dual to the (gauged) super-Gross-Neveu model. The latter is a superinvariant chiral integrable model with interaction. It is then possible to construct a superdeformation by implementing r-matrix action on moment maps, which in the

^{n-1}*CP*case appears to be consistent with supersymmetric Fateev-Onofri-Zamolodchikov model. In this regard we address the novel relation with undeformed model, the geometric interpretation and beta-function of this "supersausage" model. We also study its superconformal limits, including "supercigar" and other important supersymmetric reductions. In addition a relation with 2-dim Liouville theory and mirror dual integrable hierarchies is proposed.

^{1}**Paul Ryan**-

*Exploring the AdS3/CFT2 Quantum Spectral Curve*- Slides

The spectrum of single-trace local operators in N=4 Super Yang Mills, dual to Type IIB string theory on AdS5 x S5, in the planar limit can be encoded in a set of Quantum Spectral Curve (QSC) equations, a handful of Riemann-Hilbert relations for a set of Q-functions. The QSC has allowed for extensive precision spectroscopy of N=4 SYM perturbatively, numerically at finite coupling and analytically in some regimes. QSC equations have recently been proposed for string theory on AdS3 x S3 x T4 with pure RR flux. In this talk I will review this construction and solve the resulting set of equations numerically at finite coupling and perturbatively at weak coupling. New tools are developed to deal with a salient feature of string theory on this background - massless modes. We obtain predictions for non-protected string excitations on this background which should help shed light on the mysterious CFT2 dual.

**Peter Schupp**-

*Interactions via deformation: From noncommutative gauge theory to generalized geometry*- Slides

In this talk, we will provide an overview of various interconnected topics spanning from gauge theory on noncommutative spaces to gravity in a graded geometry framework. Our guiding principle revolves around deformation methods, wherein deformations of symplectic structures or algebras of observable offer a slightly more general alternative to the traditional gauge principle for describing fundamental interactions. This approach is known to be useful in the description of charged particles in magnetic monopole backgrounds, where it leads to intriguing nonassociative structures. Similar deformation methods have been instrumental in discovering exact solutions for Seiberg-Witten maps that relate ordinary and noncommutative gauge theory in a construction that has later found an interesting reinterpretation within the realm of generalized geometry. The deformation approach generalizes the minimal coupling principles of gauge theory and in a sense unifies it with the geometric description of gravity as free fall in curved spacetime. Applied to gravity the approach requires a graded geometry setting and suggests a novel somewhat more algebraic interpretation of some key notions of general relativity. Further examples of applications of these methods include tensor gauge theories and non-local interactions.

**Konstantinos Sfetsos -**

*Classical and quantum aspects of a constrained FT*- Slides

The classical and quantum properties of systems maybe drastically affected by imposing constraints in their phase space. Desirable properties such as unitarity and renormalizability may not be retained.

In this general context we consider a specific model which by construction is also classically integrable. After imposing a constraint we show that at tree level integrability is preserved and particle production or transmutation are not-allowed. In addition, the constrained model remains one-loop renormalizable. We compute its beta-function and argue consistency with the expected reduction of the degrees of freedom due to the constraint. **Richard Szabo -** *Homotopy double copy of noncommutative gauge theories* - Slides

This talk will summarise recent work attempting to understand how standard noncommutative gauge theories, such as those which arise naturally from string theory, fit into the paradigm of colour-kinematics duality and the double copy of gauge theory to gravity. The treatment will focus on the elegant formulation of the double copy prescription using homotopy algebras. Along the way we shall encounter some novel noncommutative scalar field theories with rigid colour symmetry that have no commutative counterparts, whose double copy prescriptions are deformations of some integrable theories related to self-dual Yang-Mills theory and gravity in four dimensions.