## Program (To be completed. Preliminary list of talks. Jun23)

• #### Vsevolod Adler (Landau Institute): The tangential map and associated integrable equations

Abstract: The tangential map is a map on the set of smooth planar curves. It satisfies the 3D-consistency property and is closely related to some well-known integrable equations.

• #### Herman Boos (Wuppertal University): Fermionic structure of the XXZ model and CFT.

Abstract: To be fixed

• #### Andrea Cappelli (INFN, Florence): Testing conformal field theory of quantum Hall droplets

Abstract: I dicuss applications of CFT to the quantum Hall effect, the use of the modular invariant partition functions to extract the peaks in the conductance due Coulomb blockade.

• #### Edward Corrigan (York University): Aspects of integrable defects

Abstract: Some time ago it was pointed out that some integrable models can support discontinuities (shock-like defects), with the main examples being the a_n affine Toda field theories (including the sine-Gordon model). In this talk a generalisation will be described that also encompasses the Tzitz\eica equation and possibly other integrable field theories as well.

• #### Sergei Derkachev (Steklov Inst. St. Petersburg): Yang-Baxter R operators and parameter permutations.

Abstract: We present an uniform construction of the solution to the Yang- Baxter equation with the symmetry algebra $s\\ell(2)$ and its deformations: the q-deformation and the elliptic deformation or Sklyanin algebra. The R-operator acting in the tensor product of two representations of the symmetry algebra with arbitrary spins $\\ell_1$ and $\\ell_2$ is built in terms of products of three basic operators $\\mathcal{S}_1, \\mathcal{S}_2,\\mathcal{S}_3$ which are constructed explicitly. They have the simple meaning of representing elementary permutations of the symmetric group $\\mathfrak{S}_4$, the permutation group of the four parameters entering the RLL-relation.

• #### Leonid Chekhov (Steklov Institute): Algebras of Yangian type from $D_n$ algebras of geodesics on orbifold Riemann surfaces

Abstract: We construct presumably new algebras of Yangian type arising from $D_n$ algebras of geodesic functions on the annulus with $n$ orbifold points of the second order. Different (finite-dimensional) reductions of these algebras correspond to actual algebras of geodesic functions for the annulus with $n$ $Z_2$ orbifold points and/or for the disc with $n$ $Z_2$ orbifold points and with one $Z_p$ orbifold point, $p$ indicating the reduction level. The action of the braid group and the central elements are constructed for the reductions. This is the joint paper with Marta Mazzocco (Loughborough Univ.)

• #### Vladimir Fateev (LPTA, Montpellier): Coulomb gas integrals in Conformal Field Theories

Abstract: To be fixed

• #### Rainald Flume (Bonn University): Matrix models as scalar field theories.

Abstract: To be fixed

• #### Frank Goehmann (Wuppertal University): Thermal correlation functions of quantum spin chains

Abstract: Recently the mathematical structure of the static correlation functions of the spin-1/2 Heisenberg chain has been completely understood. It has been proved by Boos, Jimbo, Miwa and Smirnov that, after appropriate regularizations, all static correlation functions become polynomials in a one-point and a special neighbour-two-point correlation function. We review this important result and show that those two functions allow for an efficient description in terms of the solutions of linear and non-linear integral equations. This opens the way for applications in quantum field theory and condensed matter physics.

• #### Ismagil Habibullin (Math. Institute, Ufa): Characteristic Lie algebras and classification of Darboux integrable chains

Abstract: We study differential-difference equation of the form $$\label{1}\frac{d}{dx}u_{n+1}=f(u_{n},u_{n+1},\frac{d}{dx}u_{n})$$ with unknown $u_n=u_n(x)$ depending on $x,n$. Equation (\ref{1}) is called Darboux integrable, if two functions of a finite number of arguments $F(x,u_n,u_{n\pm1},u_{n\pm2},...)$ and $I(x,u_n,u_{nx}, u_{nxx}, ...)$ called $x$- and $n$-integrals correspondingly exist, such that $D_xF=0$ and $DI=I$, where $D_x$ is the operator of total differentiation with respect to $x$, and $D$ is the shift operator: $Dp(n)=p(n+1)$. The Darboux integrability property can be reformulated in terms of characteristic Lie algebras that gives an effective tool for classification of integrable equations. The complete list of Darboux integrable equations (\ref{1}) is given in the case when the function $f$ is of the special form $f(x,y,z)=z+g(x,y)$.

• #### Kazuo Hosomichi (Yukawa Institute): Boundary phenomena in the dilute O(n) matrix model,

Abstract: We study the boundary critical behavior of the dilute $O(n)$ loop model with the anisotropic boundary condition introduced by Jacobsen and Saleur (JS), using the methods of 2D quantum gravity. The phase space is parametrized by two kinds of temperature couplings, which control the total loop length and the number of times the loops touch the boundary respectively. We determine the location of the critical points, identify the operators associated to the bulk / boundary thermal perturbations and explore the phase structure of the problem.

• #### Andreas Kluemper (with B. Aufgebauer) (Wuppertal University): Spectral properties of quantum spin chains of Temperley-Lieb type

Abstract: We determine the spectrum of a class of quantum spin chains of Temperley-Lieb type by utilizing the concept of Temperley-Lieb equivalence with the XXZ-model as a reference system. We consider open boundary conditions and in particular periodic boundary conditions. For both types of boundaries the identification with XXZ-spectra is performed within isomorphic representations of the underlying Temperley-Lieb algebra. For open boundaries the spectra of these models differ from the spectrum of the associated XXZ-chain only in the multiplicities of the eigenvalues. The periodic case is rather different. Here we show how the spectrum is obtained sectorwise from the spectrum of globally twisted XXZ-chains via the construction of appropriate reference states. In the latter case the determination of the multiplicities is more involved.

• #### Hitoshi Konno (Hiroshima University): The elliptic quantum group $U_{q,p}(\widehat{sl}_2)$: formulation and applications

Abstract: We discuss an $h$-Hopf algebroid structure on the elliptic algebra $U_{q,p}(\widehat{sl}_2)$ and formulate it as an elliptic quantum group. We then discuss its dynamical representations. In particular, we give an elliptic analogue of Chari-Pressley’s classification theorem of finite-dimensional irreducible representations for the quantum affine algebra $U_q(\widehat{sl}_2)$. As an application, we discuss a derivation of the elliptic hypergeometric series ${}_12V_{11}$ as an elliptic analogue of the Clebsch-Gordan coefficients and a derivation of vertex operators as intertwining operators of infinite dimensional representations of $U_{q,p}(\widehat{sl}_2)$.

• #### Igor Krichever (Landau Institute & Columbia University): Absolute extremals of integrals for solitonic equations and Whitham theory

Abstract: TBA

• #### Michael Lashkevich (Landau Institute): Form factors of descendant operators: Free field construction and reflection relations

Abstract: The free field representation for form factors in the sinh-Gordon model and the sine-Gordon model in the breather sector is modified to describe the form factors of descendant operators, which are obtained from the exponential ones, $\e^{\i\alpha\varphi}$, by means of the action of the Heisenberg algebra associated to the field~$\varphi(x)$. As a check of the validity of the construction the numbers of operators defined by the form factors at each level in each chiral sector are counted. Another check is related to the so called reflection relations, which identify in the breather sector the descendants of the exponential fields $\e^{\i\alpha\varphi}$ and $\e^{\i(2\alpha_0-\alpha)\varphi}$ for generic values of~$\alpha$. The operators defined by the obtained families of form factors are proven to satisfy such reflection relations. The construction admits some generalizations.

• #### Sergei Lukyanov (Rutgers University): On mass spectrum in 't Hooft's 2D model of mesons

Abstract: We discuss an analytic method to study 't Hooft's integral equation determining the meson masses M_n in multicolor QCD_2. The approach is based on the observation that solutions of 't Hooft's equation can be related to solutions of a certain functional equation of the type of Baxter's T-Q equation, with special analyticity. Among new results is systematic large-n expansion and exact sum rules for M_n. The results indicate rich analytic structure of M_n as functions of the t'Hooft coupling.

• #### Cyril Malyshev (with N. M. Bogoliubov) (Steklov Inst. St. Petersburg): The Correlation Functions of the XXZ Heisenberg Chain in Free Fermion and Strong Anisotropy Limits, and Random Walks of Vicious Walkers

Abstract: We are studying the thermal evolution'' of the states with no spin down on the last sites of the lattice for the two specific limits of the $XXZ$ Heisenberg chain, namely for the free fermion and strongly anisotropic limits. The correlation functions of these states are studied. The amplitude of these correlation functions in the low temperature limit for the model placed on the infinite lattice are expressed through the number of plane partitions (three-dimensional Young diagrams) in a finite box. The connection of the discussed problem with the random walks of vicious walkers is discussed.

• #### Giuseppe Mussardo (SISSA Trieste): Topological Quantum Computation: The Art of Computing with Icosahedral Group

Abstract:I will present the basic ideas behind topological computation and non-abelian anyons. Using Fibonacci anyons as an example, it will be shown how to use the discrete icosahedral group for setting up a very efficient and fast method to implement universal gates.

• #### Khazret Nirov (Wuppertal University): New solutions to loop Toda systems

Abstract: We construct solutions to non-abelian loop Toda equations using the rational dressing formalism. These include soliton-like solutions and give generalizations to the Hirota\'s constructions known for abelian systems.

• #### Sergei Oblezin (ITEP): q-difference Toda chains, Demazure modules, and Topological Field Theories.

Abstract: I my talk I present recent results (joint with A.Gerasimov and D.Lebedev) on new integral representations of q-difference gl(N)-Toda wave functions. In particular it will be shown that the constructed wave functions coincide with the characters of Demazure module of the corresponding affine Lie algebra $\hat{gl}(N)$. A special attention will be given to the discovered relations of our results to (quantum) K-theory of flag manifolds, and to topological field theories.

• #### Paul A. Pearce (Melbourne University): Logarithmic Minimal Models

Abstract: An overview of the lattice approach to logarithmic minimal models will be presented. Emphasis will be placed on the fusion rules of these logarithmic CFTs in the Virasoro and W-extended pictures.

• #### Dimitrii Polyakov (Witwatersrand University): Ground Ring of Alpha-Symmetries and Sequence of RNS String Theories

Abstract: We construct a sequence of nilpotent BRST charges in RNS superstring theory, based on new gauge symmetries on the worldsheet, found in this paper. These new local gauge symmetries originate from the global nonlinear space-time $\alpha$-symmetries, shown to form a noncommutative ground ring in this work. The important subalgebra of these symmetries is $U(3)\times{X_6}$, where $X_6$ is solvable Lie algebra consisting of 6 elements with commutators reminiscent of the Virasoro type. We argue that the new BRST charges found in this work describe the kinetic terms in string field theories around curved backgrounds of the $AdS\times{CP}_n$-type, determined by the geometries of hidden extra dimensions induced by the global $\alpha$-generators.

• #### Alexander Razumov, Yurii Stroganov (IHEP): Three-coloring statistical model with domain wall boundary conditions.

Abstract: The Baxter statistical three-coloring model is considered for the domain wall boundary conditions. In this case it is possible to prove that the partition function of the model satisfies some functional equations similar to the functional equations satisfied by the partition function of the six-vertex model for a special value of the crossing parameter.

• #### Sylvain Ribault (LPTA, Montpellier): On sl3 KZ equations and W3 null-vector equations.

Abstract: I will rewrite the sl3 Knizhnik-Zamolodchikov equations in terms of Sklyanin's separated variables for the sl3 Gaudin model. It is then possible to compare the sl3 KZ equations with certain W3 null-vector equations, in the spirit of the known relation between sl2 KZ equations and Belavin-Polyakov-Zamolodchikov equations. In the sl3 case however, a relation is found only in the critical level limit.

• #### Junichi Shiraishi (Tokyo University): Macdonald polynomials and quantum algebras

Abstract: The Ding-Iohara quantum algebra gives us a natural framework for the quantum W-algebras, the Feigin-Odesskii algebras and the Macdonald polynomials.

• #### Nikita Slavnov (Steklov Institute): Long-distance asymptotics of correlation functions of XXZ Heisenberg chain.

Abstract: We describe a method to derive the long-distance asymptotic behavior of correlation functions of XXZ Heisenberg chain by the algebraic Bethe ansatz. This method allows to reduce the asymptotic analysis to the one of series of multiple integrals of a special type. We discuss the link between this method and the form factor approach. We also discuss the relationship between critical exponents in the asymptotics and the properties of spectra of integrable models.

• #### Andrei Smirnov (ITEP): Characteristic Classes of Bundles over Elliptic Curves and Integrable Systems.

Abstract: Let G be a simply connected Lie group and $\zeta$ is a generator of its center. We find solutions $Q$, $\Lambda$ of the equation: $$\Lambda\,Q\,\Lambda^{-1}\,Q^{-1}=\zeta$$ for simple Lie groups. The solutions of this equation can be used for the construction of bundles over elliptic curves with non-trivial characteristic classes. We demonstrate that the Lax operators for some class of integrable systems arise naturally as a holomorphic sections of these bundles and suggest explicit formulae in terms of root systems of the extended Dynkin diagrams. The r-matrices for the models are also described.

• #### Vyacheslav Spriridonov (JINR) : Elliptic hypergeometry of supersymmetric dualities

Abstract:Recently some of elliptic hypergeometric integrals were interpreted as topological indices in N=1 supersymmetric field theories. Coincidence of superconformal indices in Seiberg dual theories corresponds to nontrivial identities for these integrals. Analysing such relations, from known dual field theories we conjecture many new relations for integrals, and from a number of known integral identities, we come to many new Seiberg dualities. This is a joint work with G.S. Vartanov (see, e.g., hep-th/0811.1909).

• #### Alexei Starobinsky (Landau Insitute): Infrared dynamics of light quantum fields in the 4-D de Sitter space-time.

Abstract: To be fixed

• #### Bulat Suleimanov (Math. Institute, Ufa): Quantum" linearisations of Painleve equations as one of components of their $L, A$ pairs.

Abstract: We show that the Painleve equations are compatible conditions of the corresponding nonstationar quantum'' Schrodinger equations (with Plank's constant $\hbar=-i$) and partial (generally) linear differential equations. The last equations may be seen as quantum'' analogues of linearisations of Painleve equations.

• #### Vitaliy Tarasov (Steklov Inst. St. Petersburg): On Gaudin model for vector representations

Abstract: I will discuss some properties of the algebra of integrals of motion for the Gaudin model on a tensor power of vector representations of $gl_N$. In particular, I will show an explicit formula for higher integrals of motion in terms of the standard Gaudin Hamiltonians.

• #### Valeriy Tolstoy (Moscow University) (to be confirmed): Discrete series of irreducible representations for the noncompact quantum algebra U_q(u(n-1,1)).

Abstract: To be fixed

• #### Mikhail Yurishchev (IPCP, Chernogolovka): Fluctuations of Entanglement Entropy in Quantum Systems

Abstract: It is shown that the entanglement entropy can fluctuate. This is demonstrated with examples of a two-qubit system (E.B.Fel'dman and M.A.Yurishchev, 2009) and quantum Ising chain. The fluctuations are absent in the fully ordered and fully disordered states of Ising model. It is found also that the entanglement fluctuations diverge in the vicinity of the second-order phase transition point.

• #### Anton Zabrodin (ITEP): Dyson gas simulation of Laplacian growth: Geometry and integrability

Abstract: To be fixed

• #### Alexander Zamolodchikov (Landau Insitute & Rutgers University): Confining Interaction in 1+1 dimensions

Abstract: To be fixed

• #### Paul Zinn-Justin (LPTMS): Exactly solvable models of tilings and Littlewood--Richardson coefficients

Abstract: There are various known combinatorial rules for computing Littlewood--Richardson coefficients. A particularly attractive one is the so-called puzzles of Knutson and Tao. Puzzles are related to a model of random tilings, the so-called square-triangle tiling model. We discuss the consequences of the quantum integrability of the latter. If time allows, we shall introduce a more general model, of square-triangle-rhombus tilings, which allows for equivariant'' generalizations of Littlewood--Richardson coefficients.

• #### Andrei Zotov (ITEP): Integrable Systems, Monopoles and Modifications of Bundles over Elliptic Curves

Abstract: We define a new family of integrable models associated with characteristic classes of bundles over elliptic curve. The classes correspond to the central elements of the Lie groups which are the structure groups of bundles. The interrelations between integrable models are shown to be given by the modification procedure. We also suggest some monopole solutions of the Bogomolny equations related to the modifications.

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