Thermal conformal field theories (CFTs) describe quantum systems at finite temperature, relevant both for real-world experiments and for the holographic description of black holes. While the thermal background breaks global conformal symmetry, key local features of the zero-temperature theory—such as the spectrum and the operator product expansion—remain intact. Thermal correlators are further constrained by the Kubo-Martin-Schwinger (KMS) condition, which imposes periodicity on the thermal circle and plays a role analogous to crossing symmetry. In this talk, I will present a new analytic bootstrap framework that relies on dispersion relations to solve the KMS condition for thermal two-point functions. This universal method reconstructs correlators from minimal input and applies broadly to both weakly and strongly-coupled theories. I will illustrate it with examples from the critical O(N) model and holographic CFTs, and briefly report on ongoing work in related setups.

I will discuss the behavior of open correlation functions in planar N = 4 SYM in the so-called Origin limits, where many cross ratios simultaneously approach zero — corresponding to corners of the positive kinematic region. Focusing on two-dimensional kinematics, I’ll explain how these limits can be explored using tools from cluster algebras and the hexagon formalism. I will also present a conjecture for the all-order behavior of correlation functions in this regime, based on a tilted version of the cusp anomalous dimension.

In this talk I will discuss some aspects of AdS/CFT worldsheet magnon kinematics and scattering based on algebraic reduction from (quantum) affine 3D Poincaré (super)algebra. In particular, I will consider a simple algebra origin of the magnon S-matrix, its deviation from a difference form and a deformed Lorentz boost.

We construct orthogonality relations in the Separation of Variables framework for the SL(2) sector of planar N = 4 supersymmetric Yang–Mills theory. Specifically, we find simple universal measures that make Q-functions of operators with different spins vanish at all orders in perturbation theory, prior to wrapping corrections. To analyze this rank-one sector, we relax some of the assumptions thus far considered in the Separation of Variables framework. Our findings may serve as guidelines for extending this formalism to other sectors of the theory, as well as other integrable models.

In holographic dualities, non-perturbative coefficients at large ’t Hooft coupling play an important role, as they encode different saddle point corrections for the string worldsheet. However, calculating these corrections from the string theory side turns out to be extremely difficult. In my talk, I present a systematic method to determine these corrections for a special class of observables in four-dimensional N=2 and N=4 supersymmetric Yang-Mills theories to arbitrary order. This method is based on the fact that, for arbitrary ’t Hooft coupling, these observables can be expressed in the form of a Fredholm determinant. From this representation, a large number of constraints can be derived in the form of differential and integral equations. These exact relations, together with certain analytical properties, completely determine the non-perturbative corrections of these observables. With this method, it is possible to effectively determine the entire transseries of several observables in supersymmetric gauge theories and it also gives the opportunity to investigate their resurgence properties.

The Virasoro-Shapiro (VS) amplitude describes closed string scattering at tree level in string theory, and is the most basic quantity in string theory. String theory is best understood in AdS background, where AdS/CFT gives a non-perturbative definition. But surprisingly the corresponding amplitude in AdS was not computed until recently, as worldsheet calculations are hard in the presence of Ramond-Ramond flux. We review the calculation of the AdS VS for AdS_5 x S^5 in an expansion around flat space, which follows from an ansatz for the amplitude in terms of a worldsheet integral of a special class of functions, combined with the block expansion of the flat space limit of the correlator in the dual CFT. We then generalize this calculation to AdS_4 x CP^3 and AdS_3 x S^3 x K3 and AdS_3 x S^3 x T^4. Our results give predictions for CFT data of heavy single trace operators, e.g. the Konishi, that can be compared to integrability.

I will present a numerical study of four-point correlation functions of the stress-tensor multiplet in N=4 super Yang-Mills (sYM) theory by leveraging integrability and localization techniques. We combine dispersive sum rules and spectral information from integrability with integrated constraints from supersymmetric localization. This results in two-sided bounds on the OPE coefficient of the Konishi operator in the planar limit at any value of the ‘t Hooft coupling ranging from weak to strong coupling. Lastly, in the limit of large ‘t Hooft coupling, I will connect this analysis with that of an analogous flat space problem involving the Virasoro-Shapiro amplitude.

I will present recent work on integrable twist deformations of the AdS_5 x S^5 superstring and its proposed spin chain formulation, focusing on non-constant, non-abelian Drinfel’d twists of the Jordanian type. On the string side, these deformations lead to particle production, raising important questions about the structure of its S-matrix integrability.
The Jordanian-twisted spin chain, however, suggests that the appropriate eigenstates for integrability-based studies are organised according to a non-Cartan residual symmetry, unrelated to particle number. Using a reformulation with twisted boundary conditions on algebra operators—that we explicitly propose for general Drinfel’d twists—I will support this argument by an explicit match between a classical string energy and the spin chain ground state in a thermodynamic limit.
We also analyse the short chain (length two) spectrum and the asymptotics of its proposed Q-functions at general length, which all indicate that a non-zero residual charge is necessary for the deformation to have non-trivial effects.

This talk presents some work in progress with Z. Kong and P. Kravchuk on the application of broken symmetries to defects, including integral identities and varied anomalies that arise in them.

I will explain a general technique that evaluates one-loop string amplitudes exactly in terms of an infinite series originating from certain Lorentzian singularities in moduli space. For the open string, this amounts to a rather direct application of the classical Rademacher procedure developed in the context of analytic number theory. For the closed string I will explain a non-holomorphic extension of the Rademacher procedure that applies to any integral of a modular function (with certain boundedness conditions). I will then discuss the concrete example of the one-loop four-graviton scattering amplitude for type II superstrings and the physics we can extract from these methods. This talk is based on joint work with M.M. Baccianti, J. Chandra, T. Hartman and S. Mizera.

Conformal line defects can be described as one dimensional CFTs. However, in that framework the locality and conformal symmetry associated with the ambient space are not manifest. I will present integrated constraints that are a direct consequence of the conformal symmetry of the full space-time (broken by the presence of the defect). Finally I will will rewrite these identities as OPE sum rules and test them in a few theories where we know the form of the 4 point function of the displacement operator.

In this talk I will present a integrability-based conjecture for the three-point functions of single-trace operators in planar N=4 super-Yang-Mills theory at finite coupling, focusing on the case where two operators are protected. This proposal is based on the hexagon representation for structure constants of long operators, which we completed to incorporate operators of any length using data from the TBA/QSC formalism. I will then, using the small spin and classical limit, be able to verifying recent two-loop results for structure constants in string theory and generalising them to operators with arbitrary R-charges.

The Froissart bound predates QCD, yet it remains shrouded in mystery. In this talk, I will present recent quantitative advances enabled by the interplay between analytic techniques and the numerical S-matrix Bootstrap. As a byproduct, I will highlight intriguing properties of the amplitude we conjecture saturates the maximal growth of the total cross section in four dimensions. Remarkably, this amplitude exhibits striking similarities to experimental features of proton-proton scattering.

Three point-correlation functions of half-BPS operators in 𝒩=4 SYM are deceivingly complex observables. Non-renormalization theorems tell us that they can be exactly computed at the free point of the theory making them a perfect target for understanding non-perturbative gravitational effects in the large N expansion in regimes where the planar expansion is inadequate. I will review techniques to efficiently compute such correlators in cases where the dimensions of the participating operators scale with powers of N. These methods allow us to systematically reproduce one-point functions of light single trace in general half-BPS backgrounds of type IIB supergravity. For the inverse reconstruction problem, I will explain how to build explicit operators dual to arbitrary LLM geometries. Then I will demonstrate how to extend these results to compute novel one-point functions of heavy probes in the large N limit via a D-instanton type formula. This gives sharp predictions for exact holographic vev’s of giant graviton branes in LLM backgrounds. I will then present some results for correlation function of three heavy (Δ~N^2) operators. I will finally comment on extensions of these techniques beyond the half-BPS sector, and the possible relation of the correlation functions of 1/4 and 1/8 BPS to lattice reductions of gauge theories.

I will discuss integrability of N=4 SYM on the Coulomb branch and use it to compute vacuum condensate at finite ‘t Hooft coupling.

I will give an overview of the integrability properties of ½ BPS Nahm pole defects in N=4 SYM paying special attention to the case of Gukov-Witten surface defects which have only very recently been studied in the integrability context. Furthermore, I will discuss how localization results for one-point functions in these set-ups imply an intriguing structure of perturbation theory that might help us get new insight on wrapping interactions.

In this talk, we review the semiclassical quantisation of M2-branes in the AdS4xS7/Zk background. As shown previously, the one-loop partition function of the M2-brane dual to the 1/2-BPS Wilson loop reproduces the 1/N correction obtained from localisation in ABJM theory. Extending this approach to non-supersymmetric configurations, we compute one-loop energy corrections for M2-branes with angular momenta in AdS4xS7/Zk. These yield predictions for 1/N corrections to the anomalous dimensions of dual operators in ABJM theory.

I will discuss a recent proposal of a precise duality between pure (2+1)-dimensional de Sitter quantum gravity and a double-scaled matrix integral. There are two main aspects of this correspondence. First, by discussing the canonical quantization of the gravitational phase space, I will arrive at a novel proposal for the quantum state of the universe at future infinity, which differs from the usual no-boundary proposal. I will then discuss the computation of cosmological correlators of massive particles in the universe specified by this wavefunction. Remarkably, these integrated cosmological correlators are precisely computed by the string amplitudes of the recently-introduced complex Liouville string (CLS), thereby establishing a direct connection between the cosmological correlators and resolvents of the matrix integral dual of CLS. The second aspect of the duality involves the Gibbons-Hawking entropy of the cosmological horizon of the de Sitter static patch. I will show that the de Sitter entropy can be reproduced exactly by counting degrees of freedom in the matrix model dual.

In recent years there have been radical developments in our understanding of symmetries, with the discovery of generalization to the usual notion of global symmetries, such as higher-form, non-invertible, or, more generally, of higher-categorical symmetries. These extensions replace the familiar group structures with more general algebraic or categorical structures and, just like the standard symmetries we all know, they can be used to constrain the dynamics of physical systems. Here I will focus on a particular class of examples: 2-dimensional Conformal Field Theories (CFTs). I will show how the categorical structures formed by the many non-invertible symmetries in 2D CFT imposes severe constraints on the allowed Renormalisation Group flows between them. Finally I will also show how, quite shockingly, even though most of these flows appear to be not integrable, they still admit an integrability-like exact description of their ground-state spectrum, obtained as a deformation of the familiar integrable ones for the {3,1}, {5,1} and {1,2} flows.

String scattering in flat space is well understood via the worldsheet, but generalising to curved backgrounds remains a major challenge. In this talk, I will review how AdS string amplitudes emerge through AdS/CFT by combining bootstrap techniques, flat space-inspired number theory, and integrability. Starting from the high-energy and Regge limits, I will present exact results for the AdS Virasoro–Shapiro amplitude, providing a direct window onto the putative AdS worldsheet. I will conclude by proposing the building blocks of AdS string amplitudes, with a natural extension of key flat space structures—most notably the KLT relations—with an explicit AdS kernel.

It is known that the double-scaled SYK model (DSSYK) reduces to JT gravity with a negative cosmological constant by zooming in on the lower edge E=-E_0 of the spectrum. We find that the de Sitter JT gravity (i.e. JT gravity with a positive cosmological constant) is reproduced from DSSYK by taking a scaling limit around the upper edge E=E_0 of the spectrum. This talk is based on arXiv:2505.08116.

It has been known for some years that strings on AdS3 × S3 × T4 can be described by an integrable family of non-linear sigma models with two free parameters: the string tension and the ratio between Ramond-Ramond and Neveu-Schwarz-Neveu-Schwarz fluxes. After fixing a light cone gauge it is possible to describe the worldsheet fluctuations of these strings in terms of particles whose S-matrix can be obtained through the so-called bootstrap approach. This S-matrix is almost entirely fixed through symmetry considerations up to certain scalar functions called dressing factors. These functions are fundamental to understanding the particle content of the theory, comprising both fundamental particles and bound states, and ultimately finding the energy levels of the string through the (Thermodynamic) Bethe Ansatz equations. In this talk I will present recent proposals for both massive and massless dressing factors of this model, and I will discuss their analytic properties.

This talk presents recent progress on spin chains in 4D N=2 SCFTs, emerging in the planar one-loop spectral problem. The first part, based on [2408.03365] and [2507.08934], provides explicit three- and four-magnon eigenstates with long-range interactions. These solutions are compatible with both twisted and untwisted periodic boundary conditions and align with exact diagonalization. We uncover a recursive structure between three- and four-magnon states. What is more, the coefficients of these solutions satisfy an infinite tower of Yang–Baxter-like equations. The second part, based on [2411.11612], shifts to a symmetry viewpoint: the Hilbert space of spin chains (= set of single-trace operators) consists of paths on the quiver diagram of the SCFT, forming a path groupoid with an Z_2 SU(4) Lie algebroid symmetry. This structure survives marginal deformations away from the orbifold point.

In this talk, I will present ongoing work on the study of generalised cusped Wilson loops with the insertion of a local operator carrying large R-charge L at the cusp’s tip. We investigate this system from the perspective of the dual string theory, both at the classical level and at one-loop order in the strong coupling regime. The focus will be on certain regions of the string phase space displaying an interesting dynamics for the flux tube. In particular, I will analyse the limit of small separation between the Wilson lines, where we observe a discontinuous behaviour at a critical value of L, signalling a “transition’’ from a regime dominated by the familiar Coulomb-like singularity to an “unbound” configuration. At the transition the energy vanishes, and the energy versus the R-charge exhibits a different “critical exponent’’ in the two regions. I will also discuss connections to earlier work in the literature and highlight how our findings extend those results.

The statistics of fluctuations on large regions of space encodes universal properties of many-body systems. At equilibrium, it is described by thermodynamics. However, away from equilibrium such as after quantum quenches, the fundamental principles are more nebulous. In particular, although exact results have been conjectured in integrable models, a correct understanding of the physics is largely missing. In this talk, I will discuss these principles, taking the example of the number of particles within a large interval in one-dimensional interacting systems. These are based on simple hydrodynamic arguments from the theory of ballistically transported fluctuations, and in particular the Euler-scale transport of long-range correlations. This allows to obtain a formula for the full counting statistics in terms of thermodynamic and hydrodynamic quantities, whose validity though depends on the structure of hydrodynamic modes. In fermionic-statistics interacting integrable models with a continuum of hydrodynamic modes, such as the Lieb-Liniger model for cold atomic gases, the formula reproduces previous conjectures, but is in fact not exact: more specifically, it gives the correct cumulants up to, including, order 5, while long-range correlations modify higher cumulants.

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