Dually weighted graphs (DWG) are planar Feynman graphs bearing two sets of couplings: one set of usual couplings $t_n$ attached to the vertices of valence $n$, and another set of dual couplings $t_n^*$ attached to the faces (dual vertices) of valence $n$. Such couplings allow a deep control on possible shapes of planar graphs. For example, if one turns on only the couplings $t_4$ and $t_4^*$ the graph takes a ''fishnet form'' of a regular square lattice. The problem of counting of such graphs can be formulated as a modified hermitian one matrix model with an extra constant matrix. The partition function can be then represented in terms of the ''character expansion'' over Young tableaux, solvable by the saddle point approximation. I will review old results on DWG from my papers with M.Staudacher and Th.Wynter, including the techniques of computing Schur characters of a large Young tableau and deriving the elliptic algebraic curve for counting of planar quadrangulations. Then I will present new results from our ongoing work with F.Levkovich-Maslyuk where we count the disc quadrangulations with large, macroscopic area and boundary. This allows to extract interesting continuous limit of fluctuating 2d geometry, interpolating between the ''almost'' flat disc with a few dynamical conical defects and the disc partition function for pure 2d quantum gravity, generalizing old results for the spherical topology. ---- Part of the London Integrability Journal Club. Please register at integrability-london.weebly.com if you are a new participant. The link will be emailed on Tuesday.

This (blackboard) talk is based on 2104.13932 and on work in progress with Francesco Benini, Ohad Mamroud and Paolo Milan. I will begin by briefly reviewing the superconformal index of the d=4 N=4 SU(N) supersymmetric Yang-Mills theory, how it is related (in the large N limit) to counting black hole microstates, and how it can be computed. I will then review a specific way to compute the index called the Bethe ansatz expansion, and describe the known solutions to the Bethe ansatz equations, and what they contribute to the index in the large N limit, including both perturbative and non-perturbative terms in 1/N. The index is related to the partition function of N=4 SYM on S^3xS^1, and in the large N limit this should be related by the AdS/CFT correspondence to a sum over Euclidean gravity solutions with appropriate asymptotic behavior. I will show that each known Bethe ansatz contribution arises from a specific supersymmetric (complex) black hole solution, which reproduces both its perturbative and its non-perturbative behavior (the latter comes from wrapped Euclidean D3-branes). A priori there are many more gravitational solutions than Bethe ansatz contributions, but we show that by considering the non-perturbative effects, the extra solutions are ruled out, leading to a precise match between the solutions on both sides.---- Part of the London Integrability Journal Club. Please register at integrability-london.weebly.com if you are a new participant. The link will be emailed on Tuesday.

We will study the correlation function of four superconformal primaries in N=4 super Yang-Mills (SYM) with SU(N) gauge group. Recently, very powerful methods have been developed to compute this correlator at finite coupling based on a new concept of integrated correlators, which are defined by integrating the correlator over spacetime coordinates with suitable integration measures. The integrated correlators can be computed using supersymmetric localisation. We will mostly focus on one of the integrated correlators. An exact expression was found for this integrated correlator for arbitrary values of coupling and N. The integrated correlator can be expressed as a two-dimensional lattice sum, which manifests the SL(2, Z) modular invariance of N=4 SYM. Furthermore, the result obeys an elegant Laplace-difference equation that relates the correlator of SU(N) theory with those of SU(N-1) and SU(N+1) theories. In perturbation, the formula is checked to be consistent with known results from more standard methods. Finally, one can reconstruct the unintegrated correlator with finite coupling for first few orders in large-N expansion, the results are shown to agree with known type IIB superstring amplitudes due to AdS/CFT duality. This talk will be mainly based on arxiv.org/abs/2102.09537 (a short version can be found at arxiv.org/abs/2102.08305). --- Part of the London Integrability Journal Club. Please register at integrability-london.weebly.com if you are a new participant. The link will be emailed on Tuesday.

In this talk, I will discuss a family of three-point functions in ABJM theory, both at weak and strong coupling. This family of three-point functions involve two BPS sub-determinant operators called giant gravitons and one single trace operator, which can be BPS or non-BPS. In the first part of the talk, I will explain how to compute this type of three-point function at weak coupling using a large N effective field theory. The structure constant is given by the overlap of an integrable matrix product state and a Bethe state. In the second part, I will first clarify the prescription of computation at strong coupling. I will show that it is important to perform an average over the moduli space and also take into account the contributions from wave functions. The prescription is tested in N=4 SYM theory and then applied to ABJM theory.--------- Part of the London Integrability Journal Club. Please register at integrability-london.weebly.com if you are a new participant. The link will be emailed on Tuesday.

Current operators describe the flow of the conserved charges in integrable models. Whereas lots of information was known about the charges, surprisingly the current operators remained unexplored for a very long time. I review recent results in this topic, which include an exact finite volume formula for the mean values of the current operators, their embedding into the Quantum Inverse Scattering Approach (Algebraic Bethe Ansatz), and connections with long range deformations and TTbar deformations. --- 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 on Tuesday.

I will talk about the behaviour of gluon scattering amplitudes in planar N=4 SYM near kinematical corners coined Origins where maximally-helicity-violating amplitudes are expected to be exactly Gaussian in the logs of the cross ratios and exactly predictable. In part 1, I will recall how much is known about this behaviour for the 6-gluon amplitude and interpret the result as a sort of quantum area convoluting the minimal surface TBA data with an effective coupling constant, known as the tilted cusp anomalous dimension. In part 2, I will present a generalization to any number of gluons and explore (some bit of) the emerging parterre of Origins. If time permits, I will discuss applications to scattering amplitudes in the null limit where all two-particle Mandelstam vanish simultaneously. The talk is based on a work in progress with Lance Dixon, Yu-Ting Liu and Georgios Papathanasiou. ---- 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.

The model has many names: Lieb-Liniger, Bose gas with delta interaction and nonlinear Schroedinger. A limiting case is called Tonks-Girardeau. It is solvable by algebraic Bethe ansatz. We shall use notations of quantum inverse scattering method. Applications will be mentioned. ----- Part of the London Integrability Journal Club. Please register at integrability-london.weebly.com if you are a new participant. The link will be emailed.

In this talk, I will discuss how the formalism of affine Gaudin models can be used to construct new integrable sigma-models. I will start by reviewing the construction of affine Gaudin models and their interpretation as integrable two-dimensional field theories. I will then explain how well-chosen realisations of these models give integrable coupled sigma model on an arbitrary number of copies of a Lie group G^N as well as integrable coset models on the quotient of G^N by a diagonal subgroup. In particular, I will discuss application of the latter construction to T^11 manifolds. --- 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 the Superconformal Index. Please register at https://lonti.weebly.com/registration.html to receive joining instructions for this live session which will be held via Zoom. The superconformal index is a very useful tool that allows us to probe the protected spectrum of a superconformal field theory (SCFT). After a brief review of the construction of unitary irreducible representations of the superconformal algebra, I will then motivate and give its definition for generic SCFTs before specialising to 4D N=2. Special limits of the index with additional supersymmetry, as well as applications will also be discussed. No prior knowledge of the topic is required.

The problem of Color Confinement in Yang-Mills theory is one of the deepest problems in theoretical physics. There is convincing numerical evidence from Lattice Gauge Theory, yet the proof of Confinement in Asymptotically Free theories has not been found. I will briefly introduce the Confinement problem and review some results on large N theories using the gauge/gravity duality. I will then discuss two-dimensional SU(N) theory coupled to an adjoint Majorana fermion. I will show that, when the adjoint mass is sent to zero, the spectrum retains a mass gap but the confinement disappears. Using the Discretized Light-Cone Quantization, I will discuss the spectrum of color singlet states and exhibit certain threshold states. Similar threshold states are also present in a model with a massless adjoint and a massive fundamental fermion. They provide new evidence for the lack of confinement. When the adjoint mass is turned on, the theory becomes confining, and the spectrum of bound states becomes discrete. ---- Part of the London Integrability Journal Club. If you are a new participant, please register at integrability-london.weebly.com/registration.html. The link will be emailed.

This is the live session included as part of the LonTI lecture on Entanglement in 1+1D Quantum Field Theory. 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 short course I will introduce branch point twist fields and explain how they emerge in the context of computing entanglement measures in 1+1D. I will focus on massive 1+1D integrable quantum field theory (IQFT) and also comment on some well-known results in conformal field theory (CFT). The talk will be structured into three main parts: First, I will introduce entanglement measures, focussing on the entanglement entropy, explain how these measures relate to partition functions in multi-sheeted Riemann surfaces and how these, in turn, may be expressed as correlators of branch point twist fields. Second, I will show how several well-known results in CFT and IQFT are very easily derived in this branch point twist field picture and how they can also be recovered numerically in a quantum spin chain. Finally, I will explain how more involved computations with branch point twist fields may be performed by exploiting form factor technology and will end the talk by showing an example of one such calculation.

This recorded tutorial is available via youtube at https://youtu.be/L-C6Xx0uI3c. The superconformal index is a very useful tool that allows us to probe the protected spectrum of a superconformal field theory (SCFT). After a brief review of the construction of unitary irreducible representations of the superconformal algebra, I will then motivate and give its definition for generic SCFTs before specialising to 4D N=2. Special limits of the index with additional supersymmetry, as well as applications will also be discussed. No prior knowledge of the topic is required.

In this talk I will discuss the universal properties of transport phenomena in TTbar-deformed conformal field theories. TTbar-deformed CFTs are exactly solvable and admit a number of approaches, each of which is seemingly unrelated. Amongst them, for our purpose, which is to study transport phenomena in TTbar-deformed CFTs, we make use of the following: integrability and holography. I will apply these two approaches to study non-equilibrium steady states and Drude weights, finding perfect agreement. I will also discuss a curious connection between TTbar-deformed CFTs and an integrable cellular automaton model called the Rule 54 chain. ------- 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 Gravity and black holes in AdS. Please register at https://lonti.weebly.com/registration.html to receive joining instructions for this live session which will be held via Zoom.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.

This tutorial is available via youtube at https://youtu.be/zU-BRF6xLik. In this short course I will introduce branch point twist fields and explain how they emerge in the context of computing entanglement measures in 1+1D. I will focus on massive 1+1D integrable quantum field theory (IQFT) and also comment on some well-known results in conformal field theory (CFT). The talk will be structured into three main parts: First, I will introduce entanglement measures, focussing on the entanglement entropy, explain how these measures relate to partition functions in multi-sheeted Riemann surfaces and how these, in turn, may be expressed as correlators of branch point twist fields. Second, I will show how several well-known results in CFT and IQFT are very easily derived in this branch point twist field picture and how they can also be recovered numerically in a quantum spin chain. Finally, I will explain how more involved computations with branch point twist fields may be performed by exploiting form factor technology and will end the talk by showing an example of one such calculation.

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.