Applications of topological recursion and matrix models
Gaëtan Borot
In diverse problems ranging from combinatorics of maps to topological strings and gauge theories, matrix models often give a direct access to the algebraic structures governing them. I will explain how a universal topological recursion organizes the all-order large N expansion in a class of matrix models with non-Coulombic repulsion between the eigenvalues. This reduces the computation of the full expansion to the construction of the spectral curve, for which we give some tools. I will describe an application to Chern-Simons theory on finite quotients of S^3, and a conjecture relating it to relativistic Toda systems of type ADE - which we justify in light of various physical dualities if time allows.
An introduction to entanglement measures in integrable quantum field theory
Olalla Castro Alvaredo
In these lectures I will introduce entanglement in many body quantum systems and various functions that are used to quantify it. I will discuss the properties of some of these functions near critical points and review an approach for their computation that is based on the use of branch point twist fields. I will briefly discuss the properties of these fields and some techniques to compute their correlation functions. I will finally describe some known results for the entanglement of integrable quantum filed theories that follow from the use of this approach.
Geometry of random Botlzmann maps
and
Geometry and spectral properties of 2D causal triangulations.
Nicolas Curien
We will survey recent results about convergence of random planar maps (and their duals) sampled according to a Botlzmann measure. In particular, in the critical generic case, these random graphs converge, in the scaling limit, towards the Brownian sphere (Le Gall). But when the weight sequence authorizes appearance of large degree vertices/faces, the geometry totally changes and displays the emergence of new universality classes for random 2D geometry.
If time permits, we will also study causal (dynamical) triangulations in 1+1 dimension. Geometrically the random graphs are obtained by adding the horizontal edges to the vertices of standard generic plane random trees (critical Galton--Watson trees with finite variance). We prove that the horizontal distances are much shorter than the vertical distances and as a result establish rigorously that the spectral dimension of these graphs is equal to 2. Based on joint work with Tom Hutchcroft and Asaf Nachmias.
Emergent hydrodynamics in integrable systems out of equilibrium
Benjamin Doyon
The hydrodynamic approximation is an extremely powerful tool to describe the behaviour of many-body systems such as gases. At the Euler scale (that is, when variations of densities and currents occur only on large space-time scales), the approximation is based on the idea of local thermodynamic equilibrium: locally, within fluid cells, the system is in a Galilean or relativistic boost of a Gibbs equilibrium state. This is expected to arise in conventional gases thanks to ergodicity and Gibbs thermalization, which in the quantum case is embodied by the eigenstate thermalization hypothesis. However, integrable systems are well known not to thermalize in the standard fashion. The presence of infinitely-many conservation laws preclude Gibbs thermalization, and instead generalized Gibbs ensembles emerge. In this lecture I will introduce the associated theory of generalized hydrodynamics (GHD), which applies the hydrodynamic ideas to systems with infinitely-many conservation laws. It describes the dynamics from inhomogeneous states and in inhomogeneous force fields, and is valid both for quantum systems such as experimentally realized one-dimensional interacting Bose gases and quantum Heisenberg chains, and classical ones such as soliton gases and classical field theory. I will first give an overview of what GHD is, how its main equations are derived, its relation to quantum integrable systems and to gases of classical solitons, and some geometry that lies at its core and that gives an exact solution to the initial value problem in the form of integral equations. I will then explain how it leads to exact results in transport problems including Drude weights and non-equilibrium currents, and exact results for large-scale space-time correlations.
This is based on various collaborations with Alvise Bastianello, Olalla Castro Alvaredo, Jean-Sébastien Caux, Jérôme Dubail, Robert Konik, Herbert Spohn, Gerard Watts and my student Takato Yoshimura, and strongly inspired by previous collaborations with Denis Bernard, M. Joe Bhaseen, Andrew Lucas and Koenraad Schalm.
CFT in 2d chiral topological phases: an introduction
Jérome Dubail
The goal of these two lectures is to give an introduction to 2d chiral topological phases to an audience who is familiar with the 2d Ising model, and with the CFT that describes its critical point. Focusing on one specific example, the p_x+i p_y superconductor, I will explain the bulk-edge correspondence (1st lecture) and the non-abelian adiabatic statistics of vortices (2nd lecture).
Trial states for the FQHE : From Conformal Field Theory to Matrix Product States
Benoit Estienne
The goal of these two lectures is to review the CFT approach to model wavefunctions in the fractional quantum Hall effect. In the first lecture I will give an introduction to the quantum Hall effect (QHE), starting with Landau levels, the integer QHE and then the fractional QHE (and in particular the Laughlin state). In the second lecture I will present the CFT approach (a.k.a. the Moore-Read construction) to model wavefunctions of the FQHE, and if time allows the more recent matrix product state description of these states.
The conformal bootstrap in Mellin space
Rajesh Gopakumar
I will describe a new approach towards analytically solving for the dynamical content of Conformal Field Theories (CFTs) using the bootstrap philosophy. This combines the original bootstrap idea of Polyakov with the modern technology of the Mellin representation of CFT amplitudes. We illustrate the power of this method in the epsilon expansion of the Wilson-Fisher fixed point by reproducing anomalous dimensions and obtaining OPE coefficients to higher orders in epsilon than currently available using other analytic techniques (including Feynman diagram calculations). Our results enable us to get a somewhat better agreement of certain observables in the 3d Ising model, with the precise numerical values that have been recently obtained.
Geometry of ``flux attachment'' in the fractional quantum Hall effect.
Deformations of Q-systems, character formulas and the completeness problem
Rinat Kedem
Motivated by the completeness problem of Bethe ansatz, one is led to the family of difference equations called Q-systems. This system can be regarded as a discrete difference equation which is integrable in itself. It can be regarded as a subalgebra of a cluster algebra, which has a canonical quantization. Integrability survives quantization. The counting problem in the quantized case becomes a character formula, associated with conformal partition functions. Further deformation of the algebra of creation operators acting on such partition functions leads naturally to spherical DAHA and associated algebras. I will explain the algebraic structures involved and the role of discrete integrability.
Application of integrability to gauge-string duality
Shota Komatsu
I will talk about the application of the integrability method to study the so-called N=4 super Yang-Mills theory (SYM), which is a four-dimensional supersymmetric gauge theory. I will first explain how the integrability, in particular integrable spin chain models, arises in the study of 4d quantum field theories and then discuss recent developments.
Liouville reflection operator and integrable structure of 2D CFT
Alexey Litvinov
In my lectures I will give basic introduction to the notion of Lioville reflection operator and its role in conformal field theory. I will disscuss its relation to quantum KdV-like integrable systems and the corresponding spectral problem.
Toward the quantization of O(3) non-linear sigma model
Sergei L. Lukyanov
In these lectures, I'll revisit the problem of quantization of the two-dimensional O(3) non-linear sigma model and its one-parameter integrable deformation -- the sausage model. The consideration is based on the so-called ODE/IQFT correspondence, a variant of the Quantum Inverse Scattering Method.
The plan of the lectures is as follows. After a general discussion of the paradigm of integrability in 2D Quantum Field Theories (QFT), I'll describe a remarkable relation of integrable Conformal QFT to certain ordinary differential equations -- the ODE/CFT correspondence. Then I'll explain how to extend the ODE/CFT correspondence to the massive Integrable QFT and discuss a general scheme of quantization within the ODE/IQFT approach. After that I am going to focus on application of the ODE/IQFT correspondence to the sausage model. Finally I'll give a brief overview of integrable structures underlying the quantum O(3)/sausage model.
Liouville quantum gravity and the Brownian map
Over the past few decades, two natural random surface models have emerged within physics and mathematics. The first is Liouville quantum gravity, which has roots in string theory and conformal field theory. The second is the Brownian map, which has roots in planar map combinatorics.
In this course, we will give an introduction to the Schramm-Loewner evolution, its relationship with Liouville quantum gravity, and ultimately build up to showing that the Brownian map is equivalent to Liouville quantum gravity with parameter $gamma=sqrt{8/3}$.
Based on joint work with Scott Sheffield.
Consequences of MBL-type and Yang-Baxter-type integrability for observable quantum dynamics
and
Successes and limitations of hydrodynamics in quantum integrable models
Joel Moore
Two of the most active areas in quantum many-particle dynamics involve systems with an unusually large number of conservation laws. Many-body-localized systems generalize ideas of Anderson localization by disorder to interacting systems. While localization with still exists with interactions and inhibits thermalization, the interactions between conserved quantities lead to some dramatic differences from the Anderson case. We discuss the emergence of slow (logarithmic) dynamics in entanglement and certain observables.
Quantum integrable models such as the XXZ spin chain or Bose gas with delta-function interactions also have infinite sets of conservation laws, again leading to modifications of conventional thermalization. A practical way to treat the hydrodynamic evolution from local equilibrium to global equilibrium in such models is reviewed. The predictions of hydrodynamics can be tested in some cases against numerics or previous exact results, and the hydrodynamical approach is found generally to have a large regime of validity.
S-matrix bootstrap
João Miguel Penedones
Exact results for KPZ universality in one dimension
Sylvain Prolhac
KPZ universality, from Kardar, Parisi and Zhang, describes several non-equilibrium phenomena exhibiting a strong interplay between randomness and non-linearity, in particular interface growth and driven particles. In the first part, we will give an overview of various systems and models with fluctuations described by KPZ universality in 1+1 dimension. The second part will be devoted to exact results for scaling functions using the one-dimensional asymmetric simple exclusion process, a stochastic integrable model described at large scales by KPZ universality.
Non-equilibrium dynamics of the Heisenberg spin chain: exact methods
Balázs Pozsgay
In this talk we will consider non-equilibrium time evolution in the XXZ spin chain and related models. After a quick overview of the subject we will concentrate on a specific class of global quenches, where the calculations can be carried out using only integrability techniques. The main example will be time evolution starting from local two-site states. The objects we will concentrate on are the Loschmidt amplitude (dynamical free energy) and the long-time limit of local correlators.
A crash course on two-dimensional CFT
Sylvain Ribault
We provide a brief but self-contained introduction to conformal field theory on the Riemann sphere. We first introduce general axioms such as local conformal invariance, and derive Ward identities and BPZ equations. We then define Liouville theory by specific axioms on its spectrum and degenerate fields. We solve the theory and study its four-point functions.
Liouville theory and log-correlated random energy models
Raoul Santachiara
Log-correlated Random Energy Models (log-REM) are a class of disordered statistical models that display a glassy behavior generated by the competition between deep metastable states and thermal excitations. Liouville field theory (LFT) is a conformal field theory that plays an important role in the study of 2D quantum gravity and that has been recently mathematically defined via free Gaussian fields.
We will discuss the LFT/log-REM connection and argue that it provides not only a statistical interpretation of subtle (and somehow unexplored) properties of the LFT, but also a route to calculate certain lower order corrections without using the replica symmetry breaking approach.
Integrability and the conformal bootstrap
Volker Schomerus
The conformal bootstrap program promises powerful new insights into the non-perturbative dynamics of conformal field theories from a combination of quantum field theoretical consistency conditions and the representation theory of conformal symmetry. In my first lecture I will interpret (spinning) conformal blocks in terms of eigenfunctions of the Laplacian on the conformal group and show that this Laplacian is closely related to the integrable Hamiltonian of a hyperbolic Calogero-Sutherland 2-particle problem. Wave functions of the latter were studied extensively through a variety of modern mathematical methods. I will describe key results from Heckman-Opdam theory of hypergeometric functions as well as the integrability based approach pioneered by Cherednik and Matsuo. The lecture concludes with a brief outline of implications for conformal blocks.
Some conjectures about duality identities associated with affine rootsystems and screened vertex operators with toroidal structure
Junichi Shiraishi
I present a construction for a certain multiple hypergeometric series based on affine root systems by using screened vertex operators with toroidal structure. Some conjectures about duality properties are found for the hypergeometric series. These conjectural identities can be regarded as affine analogues of the fundamental duality identities for the Macdonald polynomials.
By considering some particular limits, the hypergeometric series should gives us a combinatorial expression for the asymptotic eigenfunctions of the Ruijsenaars model or affine Toda system.
su(2) cosets and Liouville theory
I will talk about relations between CFT models suggested by coset constructions. I will start from the well known coset construction of Virasoro minimal models and describe a relation between minimal models and su(2) WZW model. In the second part I will discuss a similar relation between Liouville theory and an affine su(2) model with non-rational level and continuous spectrum.
