Titles and Abstracts
Monica Guica, 07.04.2022, 14:45
JTbar - deformed CFTs as non-local CFTs
TTbar and JTbar - deformed CFTs provide an interesting example of non-local, yet UV-complete two-dimensional QFTs that are entirely solvable. I will start by showing that both classes of theories possess Virasoro x Virasoro or Virasoro- Kac- Moody x Virasoro - Kac- Moody symmetry. For the case of JTbar, I will discuss the classical realization of these symmetries in terms of field-dependent coordinate transformations and show how the associated generators can be used to define an analogue of "primary" operators in this non-local theory, whose correlation functions are entirely fixed in terms of those of the undeformed CFT. In particular, two and three-point functions are simply given by the corresponding momentum-space correlator in the undeformed CFT, with all dimensions replaced by particular momentum-dependent conformal dimensions. Interestingly, scattering amplitudes off the near-horizon of extremal black holes are known to take a strikingly similar form.
Shai Chester, 10.03.2022, 14:45
Bootstrapping N = 4 super-Yang-Mills on the conformal manifold
We study the N = 4 SYM stress tensor multiplet 4-point function for any value of the complexified coupling tau, and in principle any gauge group (we focus on SU(2) and SU(3) for simplicity). By combining non-perturbative constraints from the numerical bootstrap with two exact constraints from supersymmetric localization, we are able to compute upper bounds on low-lying CFT data (e.g. the Konishi) for any value of tau. These upper bounds are very close to the 4-loop weak coupling predictions in the appropriate regime. We also give preliminary evidence that these upper bounds become small islands under reasonable assumptions, in which case our method would provide a numerical solution to N = 4 SYM for any gauge group and tau.
Riccardo Borsato, 03.03.2022, 14:45
Homogeneous Yang-Baxter deformations as undeformed yet twisted models
I will review recent progress in the study of a class of integrable deformations of sigma models known as "homogeneous Yang-Baxter". These deformations can be understood as generalisations of the well known TsT transformations. In fact, rather than deformations, the homogeneous Yang-Baxter procedure too can be reinterpreted as imposing twisted worldsheet boundary conditions in the undeformed sigma model. I will explain how to construct the twist in the generic case, which generalises the twist of TsT from abelian to non-abelian. I will also use the expression for the twist to discuss the construction of the classical spectral curve in some examples. To conclude, I will mention some open questions related to the quantum integrability of these models.
Joao Caetano, 24.02.2022, 14:45
Crosscap States in Integrable Theories
In this talk, I will describe crosscap states in integrable field theories and spin chains in 1+1 dimensions. I will derive an exact formula for overlaps between the crosscap state and any excited state in integrable field theories with diagonal scattering. I will then compute the crosscap entropy, i.e. the overlap for the ground state, in some examples. In the examples analyzed, the result turns out to decrease monotonically along the renormalization group flow except in cases where the discrete symmetry is spontaneously broken in the infrared. I will discuss crosscap states in integrable spin chains, and obtain determinant expressions for the overlaps with energy eigenstates. I will comment on the realization of crosscap states in holography.
Gregory Korchemsky, 17.02.2022, 14:45
Applications of strong Szego limit theorem in AdS/CFT
I will review a recent progress in computing four-point correlation functions of infinitely heavy half-BPS operators in planar N = 4 SYM. Taking advantage of integrability of the theory, these correlation functions can be constructed in terms of fundamental building blocks - the octagon form factors. We show that the octagon form factor can be expressed as a Fredholm determinant of an integrable Bessel operator and demonstrate that this representation is very efficient in finding its dependence on the ’t Hooft coupling and two cross ratios. At weak coupling, this yields a known series representation of the octagon in terms of ladder integrals. At strong coupling, we apply strong Szego limit theorem to develop a systematic expansion of the octagon in the inverse powers of the coupling constant and calculate accompanying expansion coefficients analytically.
David Vegh, 10.02.2022, 15:00
The spectral curve of segmented strings
In this talk, I will discuss how to compute the spectral curve of ``segmented strings'' in AdS_3. The motion of a string in this target space is integrable and the worldsheet theory can be discretized while preserving integrability. The corresponding embeddings are segmented strings, which generalize piecewise linear strings in flat space. I will
present several examples. Next, I will introduce ``brane tilings'', which are doubly-periodic planar bipartite graphs. I will show that the motion of a closed segmented string can be embedded into the mutation dynamics of a certain brane tiling. This will enable us to compute the spectral curve by taking the determinant of the dressed adjacency matrix of the tiling.
Juan Miguel Nieto Garcia, 03.02.2022, 14:45
Jordan blocks and the Bethe ansatz: The eclectic spin chain as a limit
In this talk, I will present a procedure to extract the generalised eigenvectors of a non-diagonalisable matrix by considering a diagonalisable perturbation of it and computing the non-diagonalisable limit of its eigenvectors. As an example, I will show how to compute a subset of the spectrum of the eclectic spin chain by computing the appropriate limit of the Bethe states of a twisted su(3) spin chain.
Arkady Tseytlin, 27.01.2022, 14:45
Wilson loop in general representation and RG flow in 1d defect QFT
The generalized Wilson loop operator interpolating between the supersymmetric and the ordinary Wilson loop in N=4 SYM theory provides an interesting example of renormalization group flow on a line defect: the scalar coupling parameter \zeta has a non-trivial beta function and may be viewed as a running coupling constant in a 1d defect QFT.
We continue the study of this operator, generalizing previous results for the beta function and Wilson loop expectation value to the case of an arbitrary representation of the gauge group and away from the planar limit. Focusing on the scalar ladder limit where the generalized Wilson loop reduces to a purely scalar line operator in a free adjoint theory, and
specializing to the case of the rank k symmetric representation of SU(N), we also study a certain ``semiclassical'' limit where k is taken to infinity with k \zeta^2 fixed. This limit can be conveniently studied using a 1d defect QFT representation in terms of path integral over N commuting 1d bosons. Using this representation, we compute the beta function and circular loop expectation value in the large k limit, and use it to derive constraints on the structure of the beta function for general representation. We discuss the corresponding 1d RG flow and comment on the consistency of the results with the 1d defect version of the F-theorem.
Based on joint work with Matteo Beccaria and Simone Giombi.
Roland Bittleston, 09.12.2021, 15:45
Bosonic string from Beltrami Chern-Simons
It is well understood how the 2d free scalar CFT emerges from 3d Chern-Simons theory with chiral boundary conditions. Adapting a recent proposal of Costello and Stefański, I will show how bosonic string theory can be obtained from this description by coupling to a dynamical Beltrami differential in the 3d theory. In particular, I will show how this Beltrami differential restores worldsheet diffeomorphism and Weyl invariance in the 2d theory, and recover the Polyakov action explicitly. By rewriting the theory in the BV formalism, I will show how the bc ghost system arises from the 3d perspective. Finally, if there is sufficient time, I’ll provide the 3d realization of vertex operators. This talk is based on work in progress with Kevin Costello and Bogdan Stefański.
Jakub Vosmera, 02.12.2021, 14:45
D-branes in AdS3xS3xT4 at k=1 and their holographic duals
Following the recent work of Eberhardt, Gaberdiel and Gopakumar, exact comparison between various quantities living on the two sides of the AdS/CFT duality has become a possibility. The goal of this talk will be to extend the existing holographic dictionary to include some non-perturbative vacua on both sides. I will start by reviewing the original, purely closed-string setup, giving arguments that string theory on ${\rm AdS}_3\times {\rm S}^3 \times \mathbb{T}^4$ with minimal $k=1$ NS-NS flux is exactly dual to the symmetric-product orbifold CFT with the $\mathbb{T}^4$ as the seed. I will then construct various D-branes of this string theory and calculate their associated cylinder amplitudes. We will observe that these amplitudes match with the cylinder correlators of certain boundary states of the dual CFT, thus suggesting a direct correspondence between these boundary conditions. I will also show that the disk amplitudes of these D-branes localise to those points in the worldsheet moduli space where the worldsheet disk holomorphically covers the spacetime disk. This talk is based on https://arxiv.org/abs/2110.05509.
Benjamin Doyon, 18.11.2021, 15:45
Correlation functions of twist fields from hydrodynamics
The Euler-scale power-law asymptotics of space-time correlation functions in many-body systems, quantum and classical, can be obtained by projecting the observables onto the hydrodynamic modes admitted by the model and state. This is the Boltzmann-Gibbs principle; it works for integrable and non-integrable models alike. However, certain observables, such as some order parameters in thermal of generalised Gibbs ensembles, do not couple to any hydrodynamic mode: the Boltzmann-Gibbs principle gives zero. I will explain how hydrodynamics can still give the leading exponential decay of order parameter correlation functions. With the examples of the quantum XX chain and the sine-Gordon model, I will explain how large deviations of the spin and U(1) current fluctuations are related to such exponential decay. Exact predictions are given by the ballistic fluctuation theory based on generalised hydrodynamics. In the XX model, this is in agreement with results obtained previously by a more involved Fredholm determinant analysis and other techniques, and even gives a new formula for a parameter regime not hitherto studied. In the sine-Gordon model, these are new results, inaccessible by other techniques. Works in collaboration with Giuseppe Del Vecchio Del Vecchio, and Márton Kormos.
Jake Stedman, 11.11.2021, 15:45
Gauged sigma models from four-dimensional Chern-Simons
Several years ago, a new gauge theory called four-dimensional Chern-Simons was introduced by Costello in an attempt to explain the integrability of various two-dimensional models using techniques in gauge theory. My work focuses on the use of four-dimensional Chern-Simons to explain the integrability of two-dimensional sigma models. I will begin by reviewing the construction of the Wess-Zumino-Witten (WZW) model as the boundary theory of three-dimensional Chern-Simons theory as was introduced by Moore and Seiberg. This will allow me to introduce the analogous construction of Costello and Yamazaki, in which two-dimensional sigma models appear as theories on defects in four-dimensional Chern-Simons. This naturally leads to a discussion of my work in which I construct a large class of gauged sigma models by coupling together two four-dimensional Chern-Simons theories. I will argue that the structure of four-dimensional Chern-Simons suggests that these models are integrable and finish by constructing the gauged WZW model and conformal Toda theories. This talk is based on: https://arxiv.org/abs/2109.08101.
Simeon Hellerman, 04.11.2021, 10:00
Precision Correlators at Large R-Charge
Building up macroscopic physics from microscopic constituents is an idea as old as theoretical physics itself, and has been a perennial theme in quantum theory beginning with Bohr's correspondence principle and onward through the modern renormalization group. Recently there has been a semi-organized effort at applying this principle to strongly coupled conformal field theory, developing asymptotic series in inverse powers of a large quantum number as a systematic approximation scheme whose leading-order term is the familiar macroscopic limit. I will discuss this idea in the case of N=2 superconformal gauge theory in four dimensions, where it is possible to compute the large-quantum-number asymptotic series to all orders and even a hyperasymptotic series for the exponentially small corrections to it that are associated with the leading qualitative breakdown of the naive macroscopic picture. The results quite easily yield approximations accurate to 4-6 significant digits for quantum numbers of order 1, to 15 significant digits or better for quantum numbers of order 100
Mykola Dedushenko, 28.10.2021
Quantum algebras and SUSY interfaces in Bethe/gauge correspondence
I will describe my work with N.Nekrasov on supersymmetric interfaces in gauge theories in the context of the Bethe/gauge correspondence. These interfaces, viewed as operators on the Hilbert space, give linear maps between spaces of SUSY vacua, understood mathematically as generalized cohomology theories of the Higgs branch. A natural class of interfaces are SUSY Janus interfaces for masses, with the corresponding cohomological maps being either the stable envelopes or the chamber R-matrices (both due to Maulik-Okounkov and Aganagic-Okounkov). Thus, such interfaces (and their collisions) can be used to define actions of the spectrum generating algebras (such as Yangians) on the “gauge” side of the Bethe/gauge correspondence, i.e., in QFT. Further applications and possible generalizations will be mentioned as well.
Zechuan Zheng, 21.10.2021
Matrix bootstrap revisited
Matrix bootstrap is a new method for the numerical study of (multi)-matrix models in the planar limit, using loop equations for moments of distribution (Ward identities and factorization of traces at infinite N). The lack of information associated with the use of only a finite number of lower moments is supplemented by the conditions of positivity of the correlation matrix. The numerical solution of loop equations and these conditions leads to inequalities for the lowest moments, which rapidly converge to exact values with an increase in the number of used moments. In our work https://arxiv.org/pdf/2108.04830.pdf, the method was tested on the example of the standard one-matrix model, as well as on the case of an "unsolvable" 2-matrix model with the interaction tr[A, B]^2 and with quartic potentials. We propose a significant improvement of original H.Lin’s proposal for matrix bootstrap by introducing the relaxation procedure: we replace the non-convex, non-linear loop equations by convex inequalities. The results look quite convincing and matrix bootstrap seems to be an interesting alternative to the Monte Carlo method. For example, for <trA^2>, the precision reaches 6 digits (with modest computer resources). I will discuss the prospects for applying the method in other, physically interesting systems.
Elli Pomoni, 14.10.2021
Dynamical spin chains in 4D N = 2 SCFTs
In this talk we will revisit the study of spin chains capturing the spectral problem of 4d N = 2 SCFTs in the planar limit. At one loop and in the quantum plane limit, we will discover a quasi-Hopf symmetry algebra, defined by the R-matrix read off from the superpotential. This implies that when orbifolding the N = 4 symmetry algebra down to the N = 2 one and then marginaly deforming, the broken generators are not lost, but get upgraded to quantum generators. We will also demonstrate that these chains are dynamical, in the sense that their Hamiltonian depends on a parameter which is dynamically determined along the chain. At one loop we will show how to map the holomorphic SU(3) scalar sector to a dynamical 15-vertex model, which corresponds to an RSOS model, whose adjacency graph can be read off from the gauge theory quiver/brane tiling. One scalar SU(2) sub-sector is described by an alternating nearest-neighbour Hamiltonian, while another choice of SU(2) sub-sector leads to a dynamical dilute Temperley-Lieb model. These sectors have a common vacuum state, around which the magnon dispersion relations are naturally uniformised by elliptic functions. For the example of the ℤ_{2} quiver theory we study these dynamical chains by solving the one- and two-magnon problems with the coordinate Bethe ansatz approach.
Zohar Komargodski, 07.10.2021
Renormalization Group Flows on Line Defects
We will review the subject of line defects in d-dimensional Conformal Field Theories (CFTs). We discuss an exact formula governing the renormalization group flow on line defects and consider some examples involving line defects in 2,3, and 4 space-time dimensions.
Vladimir Kazakov, 10.06.2021
Dually weighted graphs and 2d quantum gravity
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
Ofer Aharony, 03.06.2021
A gravity interpretation for the Bethe ansatz expansion of the N=4 SYM superconformal index
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.
Congkao Wen, 20.05.2021
Integrated four-point correlators in N=4 super Yang-Mills
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 https://arxiv.org/abs/2102.09537 (a short version can be found at https://arxiv.org/abs/2102.08305).
Yunfeng Jiang, 13.05.2021
OPE coefficients in ABJM theory with giants
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.
Balázs Pozsgay, 06.05.2021
Current operators in integrable models (a review)
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.
Benjamin Basso, 29.04.2021
Scattering Amplitudes Near the Origins: Localization and Globalization
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.
Vladimir Korepin, 22.04.2021
Lattice nonlinear Schroedinger equation: history and open problems
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.
Sylvain Lacroix, 01.04.2021
Affine Gaudin models and integrable sigma models
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.
Igor Klebanov, 25.03.2021
Confining or Not?
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.
Takato Yoshimura, 18.03.2021
TTbar-deformed conformal field theories out of equilibrium
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.
Nikita Nekrasov, 11.03.2021
Lefschetz thimbles in sigma models
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.
Frank Coronado, 04.03.2021
Ten dimensional hidden symmetry of N=4 SYM
I will present a generating function for the loop-integrands of all four-point functions of protected single-trace operators in N=4 super Yang-Mills. 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.
Benoit Vicedo, 25.02.2021
Integrable E-models, 4d Chern-Simons theory and affine Gaudin models
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.
Volker Schomerus, 18.02.2021
Conformal Fourier Analysis and Gaudin Integrability
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.
Oleksandr Gamayun, 11.02.2021
Modeling finite-entropy states with free fermions
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.
Junya Yagi, 04.02.2021
Wilson-'t Hooft lines as transfer matrices
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.
Gustavo Joaquin Turiaci, 28.01.2021
2D dilaton-gravity, matrix models, and the minimal string
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.
Charlotte Kristjansen, 21.01.2021
Overlaps and Fermionic Dualities for Integrable Super Spin Chains
The 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.
Paul Fendley, 14.01.2021
Integrability and Braided Tensor Categories
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.
Matthias Wilhelm, 10.12.2020
An Operator Product Expansion for Form Factors
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'.
Jorge Russo, 03.12.2020
Phases of unitary matrix models and lattice QCD in two dimensions
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 λ and the Veneziano parameter τ. A calculation of the β function shows the existence of an IR stable fixed point.
Jingxiang Wu, 19.11.2020
Integrable Kondo line defect, 4D Chern Simons, and ODE/IM correspondence
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.
Nicolai Reshetikhin, 12.11.2020
Superintegrable systems on moduli spaces of flat connections
Tristan McLoughlin, 05.11.2020
Non-planar N=4 SYM: from integrability to quantum chaos
In this talk we will consider the spectrum of anomalous dimensions in N=4 super-Yang-Mills and related theories. We will first discuss the emergence of quantum chaos as one goes from infinite to finite N and how the perturbative spectrum is described by GOE random matrix theory. We will then describe how the integrability of the planar-limit can be used to rewrite the computation of the leading 1/N corrections to the one-loop anomalous dimensions in terms of scalar products of off-shell Bethe states or, alternatively, hexagon-like functions.
Roberto Tateo, 29.10.2020
The TTbar deformation and a promising 4D generalisation
Two-dimensional field theories deformed by Zamolodchikov's TTbar operator have recently attracted the attention of theoretical physicists due to the many important links with string theory and AdS/CFT.
In this talk, I will describe various classical and quantum aspects of this particular irrelevant perturbation, including its geometrical interpretation at the classical level.
I will also introduce a generalisation of this geometrical framework to 4D field theories, and discuss some of the interesting differences with the 2D case.
Lorenz Eberhardt, 22.10.2020
An exact AdS/CFT correspondence
One incarnation of the AdS/CFT correspondence is the duality between the symmetric product orbifold of T4 and superstrings on AdS3xS3xT4 with one unit of NS-NS flux. Compared to other instances of AdS/CFT, this duality is much simpler and can in principle be fully understood. I will give a broad overview of the inner workings of the duality from a worldsheet CFT point of view, physical lessons and connections to integrability.
Rouven Frassek, 15.10.2020
QQ-system construction for so(2r) spin chains
I will review the QQ-system and oscillator construction of Q-operators for su(r+1) spin chains and discuss its generalisation to so(2r) spin chains
Florian Loebbert, 08.10.2020
Integrability for Feynman Integrals
In this talk I give an overview of the Yangian symmetry of Feynman integrals. After reviewing the connection between AdS/CFT integrability and the Yangian symmetry of massless Feynman graphs, I discuss the idea to bootstrap Feynman integrals based on the Yangian constraints. Then I show that also in the massive case large classes of Feynman integrals are constrained by a Yangian extension of dual conformal symmetry. When translated to momentum space, this leads to a novel massive generalization of ordinary conformal symmetry. Finally, I argue that these features of massive Feynman integrals can be understood as the integrability of planar scattering amplitudes in a massive version of the so-called fishnet theory, which is obtained as a double-scaling limit of N=4 super Yang-Mills theory on the Coulomb branch.
Pedro Vieira, 01.10.2020
Multi-point Bootstrap and Integrability
We initiate an exploration of the conformal bootstrap for n > 4 point correlation functions. Here we bootstrap correlation functions of the lightest scalar gauge invariant operators in planar non-abelian conformal gauge theories as their locations approach the cusps of a null polygon. For that we consider consistency of the OPE in the so-called snowflake channel with respect to cyclicity transformations which leave the null configuration invariant. For general non-abelian gauge theories this allows us to strongly constrain the OPE structure constants of up to three large spin Jj operators (and large polarization quantum number lj ) to all loop orders. In N = 4 we fix them completely through the duality to null polygonal Wilson loops and the recent origin limit of the hexagon explored by Basso, Dixon and Papathanasiou.
Yifei He, 24.09.2020
Geometrical four-point functions in the 2d critical Q-state Potts model
An important example among the 2d geometrical critical phenomena is the critical Q-state Potts model, which describes the percolation in the limit Q-->1. In this talk I will consider the problem of determining the geometrical four-point functions (cluster connectivities) in this model. Connections with the minimal models are made which uncover remarkable properties of the Potts amplitudes. Such properties allow to deduce the existence of "interchiral conformal blocks" which can be constructed using the degeneracy in the Potts spectrum. Using these, I will then determine the four-point functions through numerical bootstrap. In addition, I will also discuss the logarithmic nature of the Potts CFT and hints of a full analytic solution of the model.
Davide Gaiotto, 17.09.2020
't Hooft operators and Q-functions
I will describe the role of 't Hooft operators in 4d Chern-Simons theory and applications to integrability
Carlo Meneghelli, 23.07.2020
Pre-fundamental representations for the Hubbard model and AdS/CFT
There is a class of representations of quantum groups, referred to as prefundamental representations, that plays an important role in the solution of integrable models. The first example of such representations was given by V. Bazhanov, S. Lukyanov and A. Zamolodchikov in the context of two dimensional conformal field theory in order to construct Baxter Q-operators as transfer matrices. At the same time, there is a rather exceptional quantum group that governs the integrable structure of the one dimensional Hubbard model and plays a fundamental role in the AdS/CFT correspondence. In this talk I will introduce prefundamental representations for this quantum group, explain their basic properties and discuss some of their applications.
Sergei Lukyanov, 16.07.2020
Density matrix for the 2D black hole from an integrable spin chain
Twenty years ago Maldacena, Ooguri and Son constructed a modular invariant partition function for the Euclidean black hole (cigar) NLSM. They also proposed an expression for the corresponding density matrix.
This result played a key role in the formulation of the remarkable conjecture by Ikhlef, Jacobsen and Saleur that the Euclidean black hole NLSM underlies the critical behaviour of a certain integrable spin chain.
In this talk we critically reexamine the above proposals.
The talk is based on the recent (unpublished) joint work with
V. Bazhanov and G. Kotousov.
Marius de Leeuw, 09.07.2020
Solving the Yang-Baxter equation
The Yang-Baxter equation is an important equation that appears in many different areas of physics. It signals the presence of integrable structures which appear in topics ranging from condensed matter physics to holography. In this talk I will discuss a new method to find all regular solutions of the Yang-Baxter equation by using the so-called boost automorphism. The main idea behind this method is to use the Hamiltonian rather than the R-matrix as a starting point. I will demonstrate our method by classifying all solutions of the Yang-Baxter equation of eight-vertex type. I will also consider certain 9x9 and 16x16 solutions and give new integrable models in all of these cases. As a further application, I will discuss all integrable deformations of R-matrices that appear in the lower-dimensional cases of the AdS/CFT correspondence.
Joao Caetano, 25.06.2020
Exact g-functions
The g-function is a measure of degrees of freedom associated to a boundary of two-dimensional quantum field theories. In integrable theories, it can be computed exactly in a form of the Fredholm determinant, but it is often hard to evaluate numerically. In this paper, we derive functional equations---or equivalently integral equations of the thermodynamic Bethe ansatz (TBA) type---which directly compute the g-function in the simplest integrable theory; the sinh-Gordon theory at the self-dual point. The derivation is based on the classic result by Tracy and Widom on the relation between Fredholm determinants and TBA, which was used also in the context of topological string. As a side result, we present multiple integrals of Q-functions which we conjecture to describe a universal part of the g-function, and discuss its implication to integrable spin chains.
Gwenael Ferrando, 10-06-2020
Fishnet CFT: TBA and Non-compact Spin Chain
The fishnet CFT is a non-unitary CFT of two matrix complex scalar fields interacting via a single quartic potential. The chiral nature of the interaction strongly constrains the Feynman diagrams arising at each order in perturbation theory, those that survive are of fishnet type. In this talk, I will present the TBA equations for the conformal dimensions of multi-magnon local operators in this theory. I will emphasize the need to diagonalize suitable graph-building operators in order to determine the asymptotic data, dispersion relation and S matrix, on which the TBA relies. A dual version of the TBA equations, relating D-dimensional graphs to two-dimensional sigma models, will also be examined. The last part of the talk will be devoted to the presentation of the underlying non-compact spin chain and of additional results regarding diagonalization of graph-building operators.
Olof Ohlsson Sax, 04-06-2020
Crossing equations for mixed flux AdS3/CFT2
I will give an overview of recent progress in understanding string theory in AdS3 backgrounds with a mixture of
Ramond-Ramond and Neveu-Schwarz-Neveu-Schwarz three-form flux. Such theories are integrable, but provide many features not encountered in the more familiar case of pure Ramond-Ramond flux. In this talk I will explore the analytic structure of the dispersion relation of the world-sheet excitations and how it relates to the crossing equations of the
two-particle S matrix. Determining the dressing phases of the mixed flux S matrix is the next major step in using
integrability to the AdS3/CFT2 correspondence.
Tamas Gombor, 28-05-2020
Boundary states, overlaps, nesting and bootstrapping AdS/dCFT
Recently there have been renewed interest and relevant progress in calculating overlaps between periodic multiparticle states and integrable boundary states. They appear in quite distinct parts of theoretical physics including statistical physics and the gauge/string duality.
I will give a summary of known overlap formulas and analyze the connection between selection rules and symmetries. I will introduce a nesting procedure for boundary states which provides the factorizing overlaps for higher rank algebras automatically. This method can be used for the calculation of the asymptotic all-loop 1-point functions in AdS/dCFT. In doing so I will present the solutions of the YBE for the K-matrices with centrally extended su(2|2) symmetry and the generic overlaps of the corresponding boundary states.
Based on 2004.11329
Ines Aniceto, 21-05-2020
Integrable Field Theories with an Interacting Massless Sector
Integrability techniques have played a major role in the study the AdS/CFT correspondence, providing an accurate description of different string theoretic observables beyond the weak or strong coupling perturbation theory. However, the case of string on certain AdS_3 backgrounds provided new challenges in the form of massless excitations. Difficulties in incorporating these into the integrable description have led to disagreements concerning the energy of massive physical states.
In general integrable theories, massless and massive sectors can generally be treated separately. We know this cannot be the case in AdS_3, but a full TBA description of the interaction between the sectors is yet to be found. Surprisingly, such a description can found in a family of integrable field theories — homogeneous sine-Gordon models. Here, one can take a double scaling limit of the adjustable parameters and zoom into a regime described by a TBA where the massless sector does not decouple and contributes to the energy of massive particles at the same order as for which the Bethe ansatz would suffice in a massive theory.
Shota Komatsu, 15-05-2020
Wilson Loops as Matrix Product States
In his paper in 1979, Polyakov envisaged a possibility of reformulating the gauge theory as a Principal Chiral Model defined on a space of loops and discussed "the loop-space integrability". This idea, together with a closely related idea of the loop equation, led to numerous important results in matrix models and 2d gauge theories, but its application to four-dimensional gauge theories had only limited success. Now, after 50 years, we have a concrete example of integrable four-dimensional gauge theory, N=4 SYM. However integrability in N=4 SYM is formulated mostly in terms of local operators, although important progress has been made in constructing the Yangian for the Wilson loops. In this talk, I will present a framework which would bridge these two distant notions of integrabililty. The key player in the story is a correlation function of a local operator and the Wilson loop. I reformulate the gauge-theory computation of this observable as an overlap between an energy eigenstate of a spin chain and a matrix product state (MPS). Unlike standard MPS's discussed in the literature, our MPS has infinite bond dimensions in order to accommodate infinite dimensionality of the space of loops. It provides an "intertwiner" between integrable structures of the local operators and the Wilson loops, and in particular implies the existence of a special set of deformations of the Wilson loop which satisfy the QQ-relation. I will also explain how to formulate a nonperturbative bootstrap program based on the results obtained in this framework and compute the correlator of the circular BPS Wilson loop and general non-BPS operators at finite coupling, emphasizing the relation to and the difference from other observables that were computed by a similar approach.
Dmytro Volin, 07-05-2020
Completeness of Bethe equations
We review a proof of bijection between eigenstates of the Bethe algebra and solutions of Bethe equations written as a Wronskian quantisation condition or as QQ-relations on Young diagrams. Furthermore, it is demonstrated that the Bethe algebra is maximal commutative and it has simple spectrum every time it is diagonalisable. The proof covers rational gl(m|n) spin chains in the defining representation with the famous Heisenberg spin chain being a particular subcase. The proof is rigorous (no general position arguments are used). We do not rely on the string hypothesis and moreover we conjecture a precise meaning of Bethe strings as a consequence of the proposed proof.
A short introduction with necessary facts about polynomial rings will be given at the beginning of the talk.
Based on 2004.02865
Juan Miguel Nieto Garcia, 30-04-2020
Computing scalar products in the Bethe Ansatz
Computing scalar products is a non-trivial problem in the context of the Algebraic Bethe Ansatz. In this talk I comment about the different problems one encounter and how to easily compute recurrence relations for scalar products, even beyond SU(2). I will also present a possible explanation for the existence of determinant representations of the scalar products.
Lorenzo Bianchi, 23-04-2020
Exact results with defects: an overview
I will give a summary of recent progress in the computation of exact results in the presence of superconformal defects, focusing on a special class of defect correlators. Along the way, I will comment on the possibility of using integrability and propose various future directions.
Fedor Levkovich-Maslyuk, 16-04-2020
A review of the AdS/CFT Quantum Spectral Curve
I will give an introduction to the Quantum Spectral Curve in AdS/CFT. This is an integrability-based framework which provides the exact spectrum of planar N = 4 super Yang-Mills theory (and of the dual string model) in terms of a solution of a Riemann-Hilbert problem for a finite set of functions. I review the underlying QQ-relations starting from simple spin chain examples, and describe the special features arising for AdS/CFT. I will also present some pedagogical examples to show the framework in action. Lastly I will briefly discuss its recent applications for correlation functions. Based on the review arXiv:1911.13065.
Nat Levine, 09-04-2020
Integrable sigma models and RG flow
It is often suggested that integrable 2d sigma models should be renormalizable, however this relationship has only previously been checked in the 1-loop approximation. The aim of this work is to understand what happens beyond 1-loop.
We shall pedagogically introduce the lambda- and eta-models, and non-abelian duality. Based on these examples, we confirm that classically integrable sigma models appear to be 2-loop renormalizable if supplemented with a particular choice of finite counterterms, i.e. quantum corrections to the target space geometry. The 2-loop beta-function of the lambda-model is computed, matching the known results for groups and symmetric spaces in the limit when the lambda-model becomes the corresponding non-abelian dual model. This leads to the statement that non-abelian duality commutes with the RG flow beyond 1-loop order.
Carlo Meneghelli, 02-04-2020
Bootstrapping 1/2 BPS line defects in N=4 SYM
I will present some results on the 1d Super-Conformal Field Theory (SCFT) living on a 1/2-BPS line defect in a 4d N=4 SCFT. The main realization of this setup being Wilson lines in N=4 Super-Yang-Mills (SYM). After reviewing what the modern numerical bootstrap have to say about this problem I will describe how analytic bootstrap methods can efficiently produce the perturbative expansion at strong coupling in the planar theory.
Paul Ryan, 26-03-2020
Recent advancements in Separation of Variables for higher rank
I will review recent advancements in the development of the Separation of Variables (SoV) program for rational higher rank spin chains, motivated by the recent appearance of SoV-type structures in AdS/CFT. I will discuss the main approaches for constructing a separated variable basis which include diagonalising the B-operator and the action of fused transfer matrices on a suitable vacuum state. I will explain how these approaches are linked and demonstrate how they can be unified into a
single framework governed by Yangian representation theory. The outcome is that for any finite-dimensional su(n) spin chain the wave functions (Bethe vectors) factorise into an ascending product of Slater determinants in Baxter Q-functions, allowing us to immediately link this operatorial construction of states with the recently developed functional approach of computing scalar products and form factors.
Edoardo Vescovi, 19-03-2020
TBA for the g-function
The notion of integrability can be extended to systems with boundaries. In large volume and finite temperature, the free energy of such systems – unlike those periodic – contains a non-extensive piece, called g-function, with many physical interpretations. We present a method [1906.07733] hybrid of [1003.5542, 1007.1148, 1809.05705] to calculate the g-function from the TBA partition function.
Andrea Cavaglià, 05-03-2020
A new effective theory of the Thermodynamic Bethe Ansatz
The Thermodynamic Bethe Ansatz describes the thermodynamics in 1+1 dimensional integrable quantum field theories. I will follow some recent papers and present a statistical derivation of the TBA which goes beyond the thermodynamic limit, and can be applied to more general observables beyond the partition function. This argument leads to a new effective quantum field theory of finite volume effects, which could have applications to a number of difficult problems in this field. References: https://arxiv.org/abs/1805.02591, https://arxiv.org/abs/1911.07343 .
Giuseppe Del Vecchio Del Vecchio, 20-02-2020
Thermodynamic Bethe Ansatz: a gentle introduction
The Bethe Ansatz (BA) has proved to be incredibly efficient in the study of the ground state and the low-lying excitation spectrum of a lot of one dimensional many body quantum systems. The analysis of finite temperature situations is achieved by means of a non-trivial generalization of this educated guess. The Thermodynamic Bethe Ansatz (TBA) is the framework by which such a generalization is implemented. In this seminar, I will review the solution of the paradigmatic Lieb-Liniger model, a one dimensional system of N bosons interacting via repulsive delta potentials. The choice of this model is motivated by, beyond the simplicity, its ability to capture most of the universal properties of cold atom systems. In particular, I will show how factorizability of scattering in two body processes allows, by means of the so-called coordinate BA, to fully describe the quantum states both for finite N and in the thermodynamic limit. The TBA equations for the model will be derived following the original paper by Yang and Yang. To conclude, I will try to give a general picture behind the thermodynamics of integrable systems (classical and quantum) in terms of scattering processes.
Evgeny Sobko, 13-02-2020
Double-Scaling Limit in Principal Chiral Model: a New Non-Critical String?
I will present a systematic, non-perturbative analysis of the two-dimensional SU(N) Principal Chiral Model (PCM) in the large-N limit. Starting with the known infinite-N solution for the ground state at fixed chemical potential, we devise an iterative procedure to solve the Bethe Ansatz equations order by order in 1/N. The first few orders, which are explicitly computed, reveal a systematic enhancement pattern at strong coupling calling for the near-threshold resummation of the large-N expansion. The resulting double-scaling limit bears striking similarities to the c=1 non-critical string theory and suggests that the double-scaled PCM is dual to a non-critical string with a 2+1-dimensional target space where an additional dimension emerges from the SU(N) Dynkin diagram.
Alessandro Torrielli, 30-01-2020
Massless Integrable Scattering
We will informally discuss properties of the S-matrix in massive and massless integrable systems, with a view to the interpretation of the latter in terms of massless flows.
A gravity interpretation for the Bethe ansatz expansion of the N=4 SYM superconformal index
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.
Congkao Wen, 20.05.2021
Integrated four-point correlators in N=4 super Yang-Mills
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 https://arxiv.org/abs/2102.09537 (a short version can be found at https://arxiv.org/abs/2102.08305).
Yunfeng Jiang, 13.05.2021
OPE coefficients in ABJM theory with giants
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.
Balázs Pozsgay, 06.05.2021
Current operators in integrable models (a review)
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.
Benjamin Basso, 29.04.2021
Scattering Amplitudes Near the Origins: Localization and Globalization
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.
Vladimir Korepin, 22.04.2021
Lattice nonlinear Schroedinger equation: history and open problems
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.
Sylvain Lacroix, 01.04.2021
Affine Gaudin models and integrable sigma models
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.
Igor Klebanov, 25.03.2021
Confining or Not?
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.
Takato Yoshimura, 18.03.2021
TTbar-deformed conformal field theories out of equilibrium
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.
Nikita Nekrasov, 11.03.2021
Lefschetz thimbles in sigma models
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.
Frank Coronado, 04.03.2021
Ten dimensional hidden symmetry of N=4 SYM
I will present a generating function for the loop-integrands of all four-point functions of protected single-trace operators in N=4 super Yang-Mills. 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.
Benoit Vicedo, 25.02.2021
Integrable E-models, 4d Chern-Simons theory and affine Gaudin models
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.
Volker Schomerus, 18.02.2021
Conformal Fourier Analysis and Gaudin Integrability
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.
Oleksandr Gamayun, 11.02.2021
Modeling finite-entropy states with free fermions
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.
Junya Yagi, 04.02.2021
Wilson-'t Hooft lines as transfer matrices
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.
Gustavo Joaquin Turiaci, 28.01.2021
2D dilaton-gravity, matrix models, and the minimal string
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.
Charlotte Kristjansen, 21.01.2021
Overlaps and Fermionic Dualities for Integrable Super Spin Chains
The 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.
Paul Fendley, 14.01.2021
Integrability and Braided Tensor Categories
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.
Matthias Wilhelm, 10.12.2020
An Operator Product Expansion for Form Factors
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'.
Jorge Russo, 03.12.2020
Phases of unitary matrix models and lattice QCD in two dimensions
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 λ and the Veneziano parameter τ. A calculation of the β function shows the existence of an IR stable fixed point.
Jingxiang Wu, 19.11.2020
Integrable Kondo line defect, 4D Chern Simons, and ODE/IM correspondence
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.
Nicolai Reshetikhin, 12.11.2020
Superintegrable systems on moduli spaces of flat connections
Tristan McLoughlin, 05.11.2020
Non-planar N=4 SYM: from integrability to quantum chaos
In this talk we will consider the spectrum of anomalous dimensions in N=4 super-Yang-Mills and related theories. We will first discuss the emergence of quantum chaos as one goes from infinite to finite N and how the perturbative spectrum is described by GOE random matrix theory. We will then describe how the integrability of the planar-limit can be used to rewrite the computation of the leading 1/N corrections to the one-loop anomalous dimensions in terms of scalar products of off-shell Bethe states or, alternatively, hexagon-like functions.
Roberto Tateo, 29.10.2020
The TTbar deformation and a promising 4D generalisation
Two-dimensional field theories deformed by Zamolodchikov's TTbar operator have recently attracted the attention of theoretical physicists due to the many important links with string theory and AdS/CFT.
In this talk, I will describe various classical and quantum aspects of this particular irrelevant perturbation, including its geometrical interpretation at the classical level.
I will also introduce a generalisation of this geometrical framework to 4D field theories, and discuss some of the interesting differences with the 2D case.
Lorenz Eberhardt, 22.10.2020
An exact AdS/CFT correspondence
One incarnation of the AdS/CFT correspondence is the duality between the symmetric product orbifold of T4 and superstrings on AdS3xS3xT4 with one unit of NS-NS flux. Compared to other instances of AdS/CFT, this duality is much simpler and can in principle be fully understood. I will give a broad overview of the inner workings of the duality from a worldsheet CFT point of view, physical lessons and connections to integrability.
Rouven Frassek, 15.10.2020
QQ-system construction for so(2r) spin chains
I will review the QQ-system and oscillator construction of Q-operators for su(r+1) spin chains and discuss its generalisation to so(2r) spin chains
Florian Loebbert, 08.10.2020
Integrability for Feynman Integrals
In this talk I give an overview of the Yangian symmetry of Feynman integrals. After reviewing the connection between AdS/CFT integrability and the Yangian symmetry of massless Feynman graphs, I discuss the idea to bootstrap Feynman integrals based on the Yangian constraints. Then I show that also in the massive case large classes of Feynman integrals are constrained by a Yangian extension of dual conformal symmetry. When translated to momentum space, this leads to a novel massive generalization of ordinary conformal symmetry. Finally, I argue that these features of massive Feynman integrals can be understood as the integrability of planar scattering amplitudes in a massive version of the so-called fishnet theory, which is obtained as a double-scaling limit of N=4 super Yang-Mills theory on the Coulomb branch.
Pedro Vieira, 01.10.2020
Multi-point Bootstrap and Integrability
We initiate an exploration of the conformal bootstrap for n > 4 point correlation functions. Here we bootstrap correlation functions of the lightest scalar gauge invariant operators in planar non-abelian conformal gauge theories as their locations approach the cusps of a null polygon. For that we consider consistency of the OPE in the so-called snowflake channel with respect to cyclicity transformations which leave the null configuration invariant. For general non-abelian gauge theories this allows us to strongly constrain the OPE structure constants of up to three large spin Jj operators (and large polarization quantum number lj ) to all loop orders. In N = 4 we fix them completely through the duality to null polygonal Wilson loops and the recent origin limit of the hexagon explored by Basso, Dixon and Papathanasiou.
Yifei He, 24.09.2020
Geometrical four-point functions in the 2d critical Q-state Potts model
An important example among the 2d geometrical critical phenomena is the critical Q-state Potts model, which describes the percolation in the limit Q-->1. In this talk I will consider the problem of determining the geometrical four-point functions (cluster connectivities) in this model. Connections with the minimal models are made which uncover remarkable properties of the Potts amplitudes. Such properties allow to deduce the existence of "interchiral conformal blocks" which can be constructed using the degeneracy in the Potts spectrum. Using these, I will then determine the four-point functions through numerical bootstrap. In addition, I will also discuss the logarithmic nature of the Potts CFT and hints of a full analytic solution of the model.
Davide Gaiotto, 17.09.2020
't Hooft operators and Q-functions
I will describe the role of 't Hooft operators in 4d Chern-Simons theory and applications to integrability
Carlo Meneghelli, 23.07.2020
Pre-fundamental representations for the Hubbard model and AdS/CFT
There is a class of representations of quantum groups, referred to as prefundamental representations, that plays an important role in the solution of integrable models. The first example of such representations was given by V. Bazhanov, S. Lukyanov and A. Zamolodchikov in the context of two dimensional conformal field theory in order to construct Baxter Q-operators as transfer matrices. At the same time, there is a rather exceptional quantum group that governs the integrable structure of the one dimensional Hubbard model and plays a fundamental role in the AdS/CFT correspondence. In this talk I will introduce prefundamental representations for this quantum group, explain their basic properties and discuss some of their applications.
Sergei Lukyanov, 16.07.2020
Density matrix for the 2D black hole from an integrable spin chain
Twenty years ago Maldacena, Ooguri and Son constructed a modular invariant partition function for the Euclidean black hole (cigar) NLSM. They also proposed an expression for the corresponding density matrix.
This result played a key role in the formulation of the remarkable conjecture by Ikhlef, Jacobsen and Saleur that the Euclidean black hole NLSM underlies the critical behaviour of a certain integrable spin chain.
In this talk we critically reexamine the above proposals.
The talk is based on the recent (unpublished) joint work with
V. Bazhanov and G. Kotousov.
Marius de Leeuw, 09.07.2020
Solving the Yang-Baxter equation
The Yang-Baxter equation is an important equation that appears in many different areas of physics. It signals the presence of integrable structures which appear in topics ranging from condensed matter physics to holography. In this talk I will discuss a new method to find all regular solutions of the Yang-Baxter equation by using the so-called boost automorphism. The main idea behind this method is to use the Hamiltonian rather than the R-matrix as a starting point. I will demonstrate our method by classifying all solutions of the Yang-Baxter equation of eight-vertex type. I will also consider certain 9x9 and 16x16 solutions and give new integrable models in all of these cases. As a further application, I will discuss all integrable deformations of R-matrices that appear in the lower-dimensional cases of the AdS/CFT correspondence.
Joao Caetano, 25.06.2020
Exact g-functions
The g-function is a measure of degrees of freedom associated to a boundary of two-dimensional quantum field theories. In integrable theories, it can be computed exactly in a form of the Fredholm determinant, but it is often hard to evaluate numerically. In this paper, we derive functional equations---or equivalently integral equations of the thermodynamic Bethe ansatz (TBA) type---which directly compute the g-function in the simplest integrable theory; the sinh-Gordon theory at the self-dual point. The derivation is based on the classic result by Tracy and Widom on the relation between Fredholm determinants and TBA, which was used also in the context of topological string. As a side result, we present multiple integrals of Q-functions which we conjecture to describe a universal part of the g-function, and discuss its implication to integrable spin chains.
Gwenael Ferrando, 10-06-2020
Fishnet CFT: TBA and Non-compact Spin Chain
The fishnet CFT is a non-unitary CFT of two matrix complex scalar fields interacting via a single quartic potential. The chiral nature of the interaction strongly constrains the Feynman diagrams arising at each order in perturbation theory, those that survive are of fishnet type. In this talk, I will present the TBA equations for the conformal dimensions of multi-magnon local operators in this theory. I will emphasize the need to diagonalize suitable graph-building operators in order to determine the asymptotic data, dispersion relation and S matrix, on which the TBA relies. A dual version of the TBA equations, relating D-dimensional graphs to two-dimensional sigma models, will also be examined. The last part of the talk will be devoted to the presentation of the underlying non-compact spin chain and of additional results regarding diagonalization of graph-building operators.
Olof Ohlsson Sax, 04-06-2020
Crossing equations for mixed flux AdS3/CFT2
I will give an overview of recent progress in understanding string theory in AdS3 backgrounds with a mixture of
Ramond-Ramond and Neveu-Schwarz-Neveu-Schwarz three-form flux. Such theories are integrable, but provide many features not encountered in the more familiar case of pure Ramond-Ramond flux. In this talk I will explore the analytic structure of the dispersion relation of the world-sheet excitations and how it relates to the crossing equations of the
two-particle S matrix. Determining the dressing phases of the mixed flux S matrix is the next major step in using
integrability to the AdS3/CFT2 correspondence.
Tamas Gombor, 28-05-2020
Boundary states, overlaps, nesting and bootstrapping AdS/dCFT
Recently there have been renewed interest and relevant progress in calculating overlaps between periodic multiparticle states and integrable boundary states. They appear in quite distinct parts of theoretical physics including statistical physics and the gauge/string duality.
I will give a summary of known overlap formulas and analyze the connection between selection rules and symmetries. I will introduce a nesting procedure for boundary states which provides the factorizing overlaps for higher rank algebras automatically. This method can be used for the calculation of the asymptotic all-loop 1-point functions in AdS/dCFT. In doing so I will present the solutions of the YBE for the K-matrices with centrally extended su(2|2) symmetry and the generic overlaps of the corresponding boundary states.
Based on 2004.11329
Ines Aniceto, 21-05-2020
Integrable Field Theories with an Interacting Massless Sector
Integrability techniques have played a major role in the study the AdS/CFT correspondence, providing an accurate description of different string theoretic observables beyond the weak or strong coupling perturbation theory. However, the case of string on certain AdS_3 backgrounds provided new challenges in the form of massless excitations. Difficulties in incorporating these into the integrable description have led to disagreements concerning the energy of massive physical states.
In general integrable theories, massless and massive sectors can generally be treated separately. We know this cannot be the case in AdS_3, but a full TBA description of the interaction between the sectors is yet to be found. Surprisingly, such a description can found in a family of integrable field theories — homogeneous sine-Gordon models. Here, one can take a double scaling limit of the adjustable parameters and zoom into a regime described by a TBA where the massless sector does not decouple and contributes to the energy of massive particles at the same order as for which the Bethe ansatz would suffice in a massive theory.
Shota Komatsu, 15-05-2020
Wilson Loops as Matrix Product States
In his paper in 1979, Polyakov envisaged a possibility of reformulating the gauge theory as a Principal Chiral Model defined on a space of loops and discussed "the loop-space integrability". This idea, together with a closely related idea of the loop equation, led to numerous important results in matrix models and 2d gauge theories, but its application to four-dimensional gauge theories had only limited success. Now, after 50 years, we have a concrete example of integrable four-dimensional gauge theory, N=4 SYM. However integrability in N=4 SYM is formulated mostly in terms of local operators, although important progress has been made in constructing the Yangian for the Wilson loops. In this talk, I will present a framework which would bridge these two distant notions of integrabililty. The key player in the story is a correlation function of a local operator and the Wilson loop. I reformulate the gauge-theory computation of this observable as an overlap between an energy eigenstate of a spin chain and a matrix product state (MPS). Unlike standard MPS's discussed in the literature, our MPS has infinite bond dimensions in order to accommodate infinite dimensionality of the space of loops. It provides an "intertwiner" between integrable structures of the local operators and the Wilson loops, and in particular implies the existence of a special set of deformations of the Wilson loop which satisfy the QQ-relation. I will also explain how to formulate a nonperturbative bootstrap program based on the results obtained in this framework and compute the correlator of the circular BPS Wilson loop and general non-BPS operators at finite coupling, emphasizing the relation to and the difference from other observables that were computed by a similar approach.
Dmytro Volin, 07-05-2020
Completeness of Bethe equations
We review a proof of bijection between eigenstates of the Bethe algebra and solutions of Bethe equations written as a Wronskian quantisation condition or as QQ-relations on Young diagrams. Furthermore, it is demonstrated that the Bethe algebra is maximal commutative and it has simple spectrum every time it is diagonalisable. The proof covers rational gl(m|n) spin chains in the defining representation with the famous Heisenberg spin chain being a particular subcase. The proof is rigorous (no general position arguments are used). We do not rely on the string hypothesis and moreover we conjecture a precise meaning of Bethe strings as a consequence of the proposed proof.
A short introduction with necessary facts about polynomial rings will be given at the beginning of the talk.
Based on 2004.02865
Juan Miguel Nieto Garcia, 30-04-2020
Computing scalar products in the Bethe Ansatz
Computing scalar products is a non-trivial problem in the context of the Algebraic Bethe Ansatz. In this talk I comment about the different problems one encounter and how to easily compute recurrence relations for scalar products, even beyond SU(2). I will also present a possible explanation for the existence of determinant representations of the scalar products.
Lorenzo Bianchi, 23-04-2020
Exact results with defects: an overview
I will give a summary of recent progress in the computation of exact results in the presence of superconformal defects, focusing on a special class of defect correlators. Along the way, I will comment on the possibility of using integrability and propose various future directions.
Fedor Levkovich-Maslyuk, 16-04-2020
A review of the AdS/CFT Quantum Spectral Curve
I will give an introduction to the Quantum Spectral Curve in AdS/CFT. This is an integrability-based framework which provides the exact spectrum of planar N = 4 super Yang-Mills theory (and of the dual string model) in terms of a solution of a Riemann-Hilbert problem for a finite set of functions. I review the underlying QQ-relations starting from simple spin chain examples, and describe the special features arising for AdS/CFT. I will also present some pedagogical examples to show the framework in action. Lastly I will briefly discuss its recent applications for correlation functions. Based on the review arXiv:1911.13065.
Nat Levine, 09-04-2020
Integrable sigma models and RG flow
It is often suggested that integrable 2d sigma models should be renormalizable, however this relationship has only previously been checked in the 1-loop approximation. The aim of this work is to understand what happens beyond 1-loop.
We shall pedagogically introduce the lambda- and eta-models, and non-abelian duality. Based on these examples, we confirm that classically integrable sigma models appear to be 2-loop renormalizable if supplemented with a particular choice of finite counterterms, i.e. quantum corrections to the target space geometry. The 2-loop beta-function of the lambda-model is computed, matching the known results for groups and symmetric spaces in the limit when the lambda-model becomes the corresponding non-abelian dual model. This leads to the statement that non-abelian duality commutes with the RG flow beyond 1-loop order.
Carlo Meneghelli, 02-04-2020
Bootstrapping 1/2 BPS line defects in N=4 SYM
I will present some results on the 1d Super-Conformal Field Theory (SCFT) living on a 1/2-BPS line defect in a 4d N=4 SCFT. The main realization of this setup being Wilson lines in N=4 Super-Yang-Mills (SYM). After reviewing what the modern numerical bootstrap have to say about this problem I will describe how analytic bootstrap methods can efficiently produce the perturbative expansion at strong coupling in the planar theory.
Paul Ryan, 26-03-2020
Recent advancements in Separation of Variables for higher rank
I will review recent advancements in the development of the Separation of Variables (SoV) program for rational higher rank spin chains, motivated by the recent appearance of SoV-type structures in AdS/CFT. I will discuss the main approaches for constructing a separated variable basis which include diagonalising the B-operator and the action of fused transfer matrices on a suitable vacuum state. I will explain how these approaches are linked and demonstrate how they can be unified into a
single framework governed by Yangian representation theory. The outcome is that for any finite-dimensional su(n) spin chain the wave functions (Bethe vectors) factorise into an ascending product of Slater determinants in Baxter Q-functions, allowing us to immediately link this operatorial construction of states with the recently developed functional approach of computing scalar products and form factors.
Edoardo Vescovi, 19-03-2020
TBA for the g-function
The notion of integrability can be extended to systems with boundaries. In large volume and finite temperature, the free energy of such systems – unlike those periodic – contains a non-extensive piece, called g-function, with many physical interpretations. We present a method [1906.07733] hybrid of [1003.5542, 1007.1148, 1809.05705] to calculate the g-function from the TBA partition function.
Andrea Cavaglià, 05-03-2020
A new effective theory of the Thermodynamic Bethe Ansatz
The Thermodynamic Bethe Ansatz describes the thermodynamics in 1+1 dimensional integrable quantum field theories. I will follow some recent papers and present a statistical derivation of the TBA which goes beyond the thermodynamic limit, and can be applied to more general observables beyond the partition function. This argument leads to a new effective quantum field theory of finite volume effects, which could have applications to a number of difficult problems in this field. References: https://arxiv.org/abs/1805.02591, https://arxiv.org/abs/1911.07343 .
Giuseppe Del Vecchio Del Vecchio, 20-02-2020
Thermodynamic Bethe Ansatz: a gentle introduction
The Bethe Ansatz (BA) has proved to be incredibly efficient in the study of the ground state and the low-lying excitation spectrum of a lot of one dimensional many body quantum systems. The analysis of finite temperature situations is achieved by means of a non-trivial generalization of this educated guess. The Thermodynamic Bethe Ansatz (TBA) is the framework by which such a generalization is implemented. In this seminar, I will review the solution of the paradigmatic Lieb-Liniger model, a one dimensional system of N bosons interacting via repulsive delta potentials. The choice of this model is motivated by, beyond the simplicity, its ability to capture most of the universal properties of cold atom systems. In particular, I will show how factorizability of scattering in two body processes allows, by means of the so-called coordinate BA, to fully describe the quantum states both for finite N and in the thermodynamic limit. The TBA equations for the model will be derived following the original paper by Yang and Yang. To conclude, I will try to give a general picture behind the thermodynamics of integrable systems (classical and quantum) in terms of scattering processes.
Evgeny Sobko, 13-02-2020
Double-Scaling Limit in Principal Chiral Model: a New Non-Critical String?
I will present a systematic, non-perturbative analysis of the two-dimensional SU(N) Principal Chiral Model (PCM) in the large-N limit. Starting with the known infinite-N solution for the ground state at fixed chemical potential, we devise an iterative procedure to solve the Bethe Ansatz equations order by order in 1/N. The first few orders, which are explicitly computed, reveal a systematic enhancement pattern at strong coupling calling for the near-threshold resummation of the large-N expansion. The resulting double-scaling limit bears striking similarities to the c=1 non-critical string theory and suggests that the double-scaled PCM is dual to a non-critical string with a 2+1-dimensional target space where an additional dimension emerges from the SU(N) Dynkin diagram.
Alessandro Torrielli, 30-01-2020
Massless Integrable Scattering
We will informally discuss properties of the S-matrix in massive and massless integrable systems, with a view to the interpretation of the latter in terms of massless flows.