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Summer School on Quantum Groups and Integrability –
Algebraic, Analytic and Geometric Aspects

July 21-25, 2014
Hamburg

Titles and abstracts of lectures

Bazhanov: From Fuchsian differential equations to integrable Quantum Field Theory

In my lectures will talk about some new developments in integrable QFT, following my recent joint work with Sergei Lukyanov (arxiv:1310.4390). A tentative plan is as follows:

1. An overview of connections between classical and quantum integrable systems. Introduction to the Fateev model and its sigma model representation (two-parameter deformation of the O(4) sigma model).

2. Generalised hypergeometric equation. Fundamental solutions, monodromy group and connection coefficients. Connection with the Liouville equation.

3. Hidden algebraic structures behind the hypergeometric equation and its "perturbations". Monodromy free singular points. Functional equations for the connection coefficients and their relation to the Bethe Ansatz equations for the exceptional affine superalgebra U_q(D(2,1|α)). Quasi-classical expansion of the the Wilson loop and local integrals of motion related to the Fateev model.

4. Modified sinh-Gordon equation on the punctured Riemann sphere and stationary states in the (massive) Fateev model. Non-linear integral equations. Calculation of the vacuum energies in the deformed O(4) sigma model with twisted boundary conditions.

Hernandez: Baxter's relations and spectra of quantum integrable models

The aim of this series of lectures is to explain the recent proof of the conjecture of Frenkel-Reshetikhin describing the spectrum of quantum integrable systems (associated to quantum affine algebras) in terms of (generalized) Baxter's polynomial and q-characters.

First we introduce and study a "category O" of representations of the Borel algebra, associated with a quantum affine algebra of non-twisted type. We construct (pre)fundamental representations for this category as a limit of the Kirillov-Reshetikhin modules over the quantum loop algebra. We prove that general simple modules in this category are classified by n-tuples of rational functions in one variable, which are regular and non-zero at the origin but may have a zero or a pole at infinity. In the sl_2-case, some of these representations had been constructed by Bazhanov-Lukyanov-Zamolodchikov.

Then we generalize Baxter's relations (also known as Baxter's TQ relations) to an arbitrary untwisted quantum affine algebra. We interpret them as relations in the Grothendieck ring of the category O involving infinite-dimensional prefundamental representations. We then construct the (twisted) transfer-matrices associated to these prefundamental representations and prove that their eigenvalues on any finite-dimensional representation are polynomials up to a universal factor. These are the generalized Baxter polynomials. We derive from these results the conjecture of Frenkel-Reshetikhin.

The talks are based on a joint paper with M. Jimbo and a joint paper with E. Frenkel.

Ip: Positive representations and higher quantum Teichmüller Theory

I will introduce the notion of the positive representations of split real quantum groups introduced in a joint work with I. Frenkel, which generalize earlier work by Ponsot-Teschner in the case of Uq(sl(2,R)) that is closely related to Liouville's conformal field theory and quantum Teichmüller theory. I will then talk about the notion of multiplier Hopf algberas and how the C*-algebraic approach helps to define a braiding structure for the category of positive representations, as well as a candidate for the higher quantum Teichmüller theory via the representations restricted to the Borel part, generalizing the construction by Frenkel-Kim using the quantum plane.

Nakanishi: Cluster algebras and Y-systems

Cluster algebras were introduced by Fomin and Zelevinsky around 2000 as an underlying algebraic/combinatorial structure in Lie theory. Beyond the original expectation they are now recognized as a common structure in several branches of mathematics, for example, besides representation theory, hyperbolic geometry and Teichmüuller theory, Poisson geometry, discrete dynamical systems, exact WKB analysis, etc. Such ubiquity reminds us of root systems, a basic combinatorial structure underlying in mathematics. Indeed the theory of cluster algebras may be regarded as a certain extended theory of root systems.

In this series of lectures we focus on the application of the theory of cluster algebras to Y-systems, which appeared in 90’s through the thermodynamic Bethe ansatz (TBA) approach to study the integrable deformation of conformal field theory. The plan of the lectures is as follows:

Lecture 1. Cluster algebras.
Lecture 2. Dilogarithm identities.
Lecture 3. Y-systems.

Smirnov: Fermionic structure for XXZ spin chain (six vertex model)

In these lectures I shall concentrate on the algebraic construction of the fermionic basis in the space of local operators for XXZ spin chain (equivalently six vertex model). I shall explain that the determinant formulae for the expectation values emphasising relation to the q-deformation of hyperelliptic curve.

Plan:

1. Expectation values for six vertex model on a cylinder.
2. Quantum loop algebra.
3. Construction of annihilation fermionic opearoes.
4. Construction of creation fermionic operators.
5. Computation of expectation values.

Tateo: Novel approaches to finite-size effects in integrable models

The discovery of integrability in the planar limit of AdS/CFT has triggered a renewed interest toward the study of finite-size corrections of exactly solvable systems. In this series of lectures I will discuss a couple of recent developments. They are both, to some extent, directly relevant to the study of integrability in the supersymmetric gauge theory context. I will start by discussing a link between simple differential equations and the Bethe Ansatz for CFTs in two dimensions and extend it to a correspondence between Lax operators for particular classical integrable field equations and functional equations associated to 2D massive quantum field theories. In the second part of the course, I will present a very recent idea which has made possible the reduction of complicated AdS/CFT-related Thermodynamic Bethe Ansatz equations to much simpler nonlinear matrix Riemann-Hilbert problems.
 
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