Two dimensional sigma models describe (harmonic) maps of Riemann surfaces to Riemannian manifolds. I will present the motivations to study the complexification of this problem. I will present the novel approach, developed in my work with Igor Krichever, allowing to construct essentially all twisted complexified harmonic maps of two-torus to spheres and projective spaces. -- Part of the London Integrability Journal Club. If you are a new participant please register at integrability-london.weebly.com. The link will be emailed.

This is the live session included as part of the LonTI lecture on "From spin chain to AdS/CFT with Mathematica". Please register at https://lonti.weebly.com/registration.html to receive joining instructions for this live session which will be held via Zoom. ​In this introductory lecture we describe the XXX Heisenberg spin chain, study its spectrum, wavefunctions and discuss integrability of the system. Some examples are given with simple Mathematica code. We also discuss applications to AdS/CFT correspondence.

The release of the youtube video for the London Theory institute seminar series available at https://youtu.be/jtfmWDrC370. Gravity in asymptotic AdS spacetimes behaves in many ways quite differently to the usual asymptotically flat situation we are usually introduced to. It is very important in understanding the AdS-CFT correspondence, and many of these differences to the flat setting have important implications. In this lecture and the problem sheet I will introduce AdS spacetime, asymptotic AdS spacetimes and then focus on the physics of black holes in AdS. For the simplest static black holes we will explore their behaviour, the implications for AdS-CFT and introduce some basic calculational tools to study them.

I will present a generating function for the loop-integrands of all four-point functions of protected single-trace operators in $\mathcal{N}=4$ SYM. This function enjoys a ten-dimensional symmetry that combines spacetime and the internal R-charge symmetries. By considering a 10D light-like limit I will establish a relationship between the simplest four-point correlators (octagons) and four-particle amplitudes. --- Part of the London Integrability Journal Club. If you are a new participant, please register using the form at integrability-london.weebly.com. The link will be emailed.

The release of the youtube video for the London Theory institute seminar series https://youtu.be/jTRQTwBX9ek. In this introductory lecture we describe the XXX Heisenberg spin chain, study its spectrum, wavefunctions and discuss integrability of the system. Some examples are given with simple Mathematica code. We also discuss applications to AdS/CFT correspondence. Some derivations are formulated in the form of step-by-step exercises. They can be solved either with Mathematica or by hand. A quick introduction to Mathematica is provided in a separate video.

Two-dimensional integrable field theories are characterised by the existence of infinitely many integrals of motion. Recently, two unifying frameworks for describing such theories have emerged, based on four-dimensional Chern-Simons theory in the presence of surface defects and on Gaudin models associated with affine Kac-Moody algebras. I will explain how these formalisms can be used to construct infinite families of two-dimensional integrable field theories. The latter can all naturally be formulated as so-called E-models, a framework for describing Poisson-Lie T-duality in sigma-models. The talk will be based on the joint work [arXiv:2008.01829] with M. Benini and A. Schenkel and [2011.13809] with S. Lacroix. ---- Please register using the form at integrability-london.weebly.com if you are a new participant. The link will be emailed.

Conformal partial wave expansion provide Fourier-like decompositions of correlation functions in Conformal Field Theory. Despite their fundamental importance, conformal partial waves remain poorly understood, at least beyond the case of four local fields. In the last few years, a deep relation with integrable quantum mechanical models has emerged. It offers a wealth of powerful new algebraic methods to study and construct conformal partial waves e.g. for general supermultiplets, non-local (line-, surface-) operators and multi-point correlation functions. In my talk I will use ideas from harmonic analysis of the conformal group to embed conformal partial waves into the framework of Gaudin integrable models and then discuss several concrete ramifications as trigonometric and elliptic Calogero-Sutherland models. The latter are relevant for multi-point blocks of scalar fields. ----- Part of London Integrability Journal Club. Please register at integrability-london.weebly.com/registration.html if you are a new participant.

The behavior of dynamical correlation functions in one-dimensional quantum systems at zero temperature is now very well understood in terms of linear and non-linear Luttinger models. The "microscopic" justification of these models consists in exactly accounting for the soft-mode excitations around the vacuum state and at most a few high-energy excitations. At finite temperature, or more generically for finite entropy states, this direct approach is not strictly applicable due to the different structure of soft excitations. To address these issues we study the asymptotic behavior of correlation functions in one-dimensional free fermion models. On the one hand, we obtain exact answers in terms of Fredholm determinants. On the other hand, based on "microscopic" numerical resummations, we develop a phenomenological approach that provides results depending only on the state-dependent dressing of the scattering phase. Our main example will be the sine-kernel and correlation functions in XY model. -------------------- Part of the London Integrability Journal Club. New participants please register using the form at integrability-london.weebly.com.

Supersymmetric gauge theories in four dimensions have various interrelated connections to quantum integrable systems. I will present a new correspondence which identifies Wilson-'t Hooft lines in N=2 circular quiver theories with transfer matrices of trigonometric systems. I will explain how this correspondence is related to Costello's 4d Chern-Simons theory and other similar correspondences. This is based on my joint work with Kazunobu Maruyoshi and Toshihiro Ota. -------------------- Part of the London Integrability Journal Club. New participants please register using the form at integrability-london.weebly.com.

In the first part of this talk I will review the recent realization that a large class of two-dimensional theories of dilaton-gravity in asymptotically AdS space are holographically dual to a random matrix model. In this description the matrix represents a random boundary Hamiltonian, and its probability distribution depends on the dilaton potential in a specific way. In the second part of the talk I will explain the relation between two-dimensional dilaton-gravity and the minimal string theory. -------------------- Part of the London Integrability Journal Club. New participants please register using the form at integrability-london.weebly.com.

The $\mathfrak{psu}(2,2|4)$ integrable super spin chain underlying the AdS/CFT correspondence has integrable boundary states which describe set-ups where $k$ D3-branes get dissolved in a probe D5-brane. Overlaps between Bethe eigenstates and these boundary states encode the one-point functions of conformal operators and are expressed in terms of the superdeterminant of the Gaudin matrix that in turn depends on the Dynkin diagram of the symmetry algebra. The different possible Dynkin diagrams of super Lie algebras are related via fermionic dualities and we determine how overlap formulae transform under these dualities. As an application we show how to consistently move between overlap formulae obtained for $k=1$ from different Dynkin diagrams. -------------------- Part of the London Integrability Journal Club. New participants please register using the form at integrability-london.weebly.com.

Many integrable critical classical statistical mechanical models and the corresponding quantum spin chains possess a fractional-spin conserved current. These currents have been constructed by utilising quantum-group algebras, fermionic and parafermionic operators, and ideas from ``discrete holomorphicity''. I define them generally and naturally using a braided tensor category, a topological structure familiar from the study of knot invariants, anyons and conformal field theory. Such a current amounts to terminating a lattice topological defect, and I will touch on related work on such done with Aasen and Mong. I show how requiring a current be conserved yields simple constraints on the Boltzmann weights, and that all of the many models known to satisfy these constraints are integrable. This procedure therefore gives a linear construction for ``Baxterising'', i.e. building a solution of the Yang-Baxter equation out of topological data. -------------------- Part of the London Integrability Journal Club. New participants please register using the form at integrability-london.weebly.com.

We will have seven 10 mins talks + discussions, with: 1) Sara Bonansea (INFN Florence and NBI) - "Wilson loop correlators in defect N=4 SYM" 2) Andrea Fontanella (Humboldt) - "Lie Algebra Expansion in Coset Sigma Models and Non-relativistic String Theory" 3) Suvajit Majumder (City) - "Protected states in AdS3xS3xT4 from integrability" 4) Gerben Oling (NBI) - "Non-relativistic strings and limits of AdS/CFT" 5) Chiara Paletta (TCD) - "Integrable open quantum systems" 6) Davide Polvara (Durham) - "From tree-level perturbation theory to the S-matrix bootstrap in two dimensions" 7) Anton Pribytok (TCD) - "Deformed AdS Integrability and Free Fermion Condition" ----------------------------------- Part of the London Integrability Journal Club. New participants are required to register using the form at integrability-london.weebly.com. The link will be emailed. Abstracts on the website.

In this talk, I discuss an operator product expansion for planar form factors of local operators in N=4 SYM theory. In this expansion, a form factor is decomposed into a sequence of known pentagon transitions and a new universal object - the form factor transition. This transition is subject to a set of non-trivial bootstrap constraints, which can be used to determine it at finite coupling. I demonstrate this for MHV form factors of the chiral half of the stress tensor supermultiplet, which in particular contains the chiral Lagrangian and the 20'. ---- Part of London Integrability Journal Club. If you are a new participant, please register filling the form at integrability-london.weebly.com. The link will be emailed.

We investigate the different large N phases of a deformed Gross-Witten-Wadia U(N) matrix model. The deformation, which leads to a solvable model, corresponds to the addition of characteristic polynomial insertions and mimicks the one-loop determinant of fermion matter. In one version of the model, the GWW phase transition is smoothed out and it becomes a crossover. In another version, the phase transition occurs along a critical line in the two-dimensional parameter space spanned by the 't~Hooft coupling $\lambda $ and the Veneziano parameter $\tau $. A calculation of the $\beta $ function shows the existence of an IR stable fixed point. ---- Part of London Integrability Journal Club. If you are a new participant, please register filling the form at integrability-london.weebly.com. The link will be emailed.

This Thursday we will have six speakers presenting a 10 mins talk each. 1) Stefano Baiguera (Niels Bohr Institute), "Non-relativistic near-BPS corners of N=4 super Yang-Mills with SU(1,1) symmetry". 2) Ilija Buric (Desy), "From Integrable Gaudin Models to Multipoint Conformal Blocks", 3) Julius Julius (King's College London), "Baxter Equation for Boundary Integrability", 4) Rob Klabbers (Nordita), "How coordinate Bethe ansatz works for Inozemtsev model", 5) Yuan Miao (University of Amsterdam), "On the Q-operator and the spectrum of the XXZ model at root of unity". 6) Anne Spiering (Trinity College Dublin), "Integrability and Quantum Chaos in N=4 SYM from the Spectral Rigidity". Join us to hear the latest news in integrability! Important: Please make sure your Zoom version is updated, in order to use the break-out rooms feature for discussions after the talks. See the journal club website for abstracts of the talks. ----------------------------- Part of the London Integrability Journal Club. New participants please register using the form which can be found on the webpage integrability-london.weebly.com. The link will be emailed.

I will discuss the integrability and wall-crossing properties of Kondo line defects in rational conformal field theories. It provides a large class of interesting defect RG flow starting from topological line defects. As a surprise, I will discuss new examples of the ODE/IM correspondence and our attempts towards its physical origin using 4d Chern Simons theory. This work is part of a multi-pronged exploration of studying 4D Chern-Simons theory as an overarching structure for integrable systems. ---- Part of London Integrability Journal Club. New participants can register using the form at integrability-london.weebly.com. The link will be emailed.

AdS/CFT duality (or more general holographic principle) represents a major advance in understanding quantum gravity, and provides powerful tools for studying strongly coupled quantum field theories. This talk will give a basic introduction to AdS/CFT duality, with the focus on the computation of Witten diagrams and their Mellin transform. Witten diagrams, which play the role of Feynman diagrams, provide the means for computing correlation functions in AdS/CFT. We will show that CFT correlation functions obtained from Witten diagrams have much simpler structures after Mellin transform. Correlators in Mellin space are very analogous to the flat-space scattering amplitudes, and they are often called Mellin amplitudes. We will demonstrate the ideas by studying a few non-trivial examples.

t.b.a. -------------------- Part of the London Integrability Journal Club. New participants please register using the form at integrability-london.weebly.com.

Part of London Integrability Journal Club. New participants should register using the form at integrability-london.weebly.com. The link will be emailed.