**Tuesday, August 16th, 2016**

**AVAN Jean **
*Elliptic deformations of quantum Virasoro and W algebras *

We propose a construction of quantum and classical deformed Virasoro algebras starting from the elliptic algebra $A_{qp}(\widehat{gl}(N)_c)$ and its vertex-operator modification (stemming from the construction
of ZF algebra by Awata et al.)
The generators are constructed from bilinears of the quantum Lax matrices. The procedure is two-step: a closure relation between p,q,c ensures existence of a strict subalgebra generated by the
bilinears. A criticality relation then ensures abelianity of the subalgebra and related Poisson structures on it. Extended centers are identified at central charge $\pm N$ . The original DVA is recovered from a suitable bilinear of
an elliptic algebra.

pdf file of the presentation

**CANTINI Luigi **
* Asymmetric simple exclusion process with open boundaries and Koornwinder polynomials*

In this talk I shall present a new approach to the study of the steady state of the Asymmetric Simple Exclusion Process (ASEP) on a finite strip with two particle reservoirs at the two ends. Our approach consists in exploiting the integrability of the model in order to introduce a set of "extra" parameters, usually called spectral parameters. The (unnormalized) probabilities of the particle configurations get promoted to Laurent polynomials in the spectral parameters and are constructed in terms of non-symmetric Koornwinder polynomials. In particular we show that the normalization coincides with a symmetric Macdonald-Koornwinder polynomial. As an outcome we compute the steady current and the average density of first particles.

pdf file of the presentation

**CAO Junpeng **
* Bethe states of quantum integrable models solved via the off-diagonal Bethe Ansatz *

Based on the inhomogeneous T-Q relation constructed via the off-diagonal
Bethe Ansatz, a systematic method for retrieving the Bethe-type
eigenstates of integrable models without obvious reference state is
developed by employing certain orthogonal basis of the Hilbert space.
With the XXZ spin torus model and the open XXX spin-1/2 chain as
examples, we show that for a given inhomogeneous T-Q relation and the
associated Bethe Ansatz equations, the constructed Bethe-type eigenstate
has a well-defined homogeneous limit.

pdf file of the presentation

**CAUDRELIER Vincent **
* Dual Hamiltonian structures in an integrable hierarchy and out-of-equilibrium integrable systems *

The classical and quantum versions of the R matrix are the cornerstones in classical and quantum integrable systems. However, they concentrate all the attention on only one half of the Lax pair: the half that describes the space "evolution" of the system. In classical field theory, this is justified traditionally by the fact that the second half encoding time evolution yields simple time evolution of the scattering data under appropriate assumptions on the boundary conditions. In the quantum case, there is no second half to speak of since one works implicitely in a time-independent or stationary picture, whereby time evolution is already diagonalized over common eigenstates, the task being to find these, using algebraic Bethe ansatz for instance. Motivated originally by the understanding of Liouville integrability in the presence of defects or (not necessarily integrable) boundary conditions in classical field theory, we will show how a dual Hamiltonian structure naturally emerges which gives a fully fledged r-matrix structure to the second/time Lax operator. The interplay between the standard classical r-matrix structure and the new one is what we call dual Hamiltonian structure. This raises many questions but in particular, the issue of the quantum counterpart of this picture. Given that the second half of the classical Lax pair controls time evolution, one may argue that a thorough understanding of out-of-equilibrium integrable quantum field theory should elucidate the quantum counterpart of our findings.

pdf file of the presentation

**DERKACHOV Sergey **
* Iterative construction of eigenfunctions of the monodromy matrix*

Eigenfunctions of the matrix elements of the monodromy matrix provide a convenient
basis for studies of spin chain models. We present an iterative method for constructing the
eigenfunctions.
We derive an explicit integral representation
for the eigenfunctions and calculate the corresponding scalar products (Sklyanin's measure).

pdf file of the presentation

**FELDER Giovanni **
* The space of Gaudin subalgebras *

(joint work with L. Aguirre and A. P. Veselov) Gaudin subalgebras for the root system A_n are Lie subalgebras of the Kohno-Drinfeld Lie algebra of maximal dimension spanned by generators. They form a smooth projective variety isomorphic to the moduli space of stable rational curves with n+1 marked points. A similar result holds for general Coxeter systems if one restricts oneself to the space of principal Gaudin subalgebras, the closure in the Grassmannian of the spans of Gaudin Hamiltonians. The moduli space is replaced by the wonderful compactification of De Concini-Procesi.

pdf file of the presentation

**FINN Caley **
*Mixed boundary qKZ equations and Koornwinder polynomials*

We study solutions of the qKZ equations associated with the mixed boundary Temperley-Lieb loop model. We show that the normalisation of the qKZ solution is a specialised non-symmetric Koornwinder polynomial. At this special point, the non-symmetric Koorwinder polynomials and the components of the qKZ solution are distinguished bases of an irreducible representation of the type B Hecke algebra. We give a graphical construction of the Koornwinder basis analogous to that used to construct the solutions of the qKZ equation, helping to clarify the relation between the two bases.

pdf file of the presentation

**GAINUTDINOV Azat **
* Algebraic Bethe ansatz for the quantum group invariant open XXZ chain at roots of unity*

For generic values of q, all the eigenvectors of the transfer matrix of
the $U_q(sl(2))$-invariant open spin-1/2 XXZ chain with finite length N can
be constructed using the algebraic Bethe ansatz (ABA) formalism of
Sklyanin. However, when q is a root of unity (q=exp(i π/p) with integer
p>1), the Bethe equations acquire continuous solutions, and the
transfer matrix develops Jordan cells. Hence, there appear eigenvectors
of two new types: eigenvectors corresponding to continuous solutions
(exact complete p-strings), and generalized eigenvectors. I will present
our recent general ABA construction for these two new types of
eigenvectors (a work with R. Nepomechie). We present many explicit
examples, and we construct complete sets of (generalized) eigenvectors
for various values of p and N.

pdf file of the presentation

**GOMES Jose Francisco **
*Miura and Generalized Backlund Transformations for KdV Hierarchy*

Using the fact that Miura transformation can be expressed in the form of gauge transformation
connecting the KdV and mKdV equations, we discuss the derivation of
the Backlund transformation connecting both hierarchies.

pdf file of the presentation

**GONCHAROV Alexander **
*Bipartite graphs on torus and cluster integrable systems*

**GORBE Tamas F. **
*Lax representation of the hyperbolic van Diejen system with two coupling parameters*

In his 1994 thesis, Jan Felipe van Diejen proved the quantum integrability of the hyperbolic Ruijsenaars-Schneider model attached to the BC(n) root system. This led to explicit formulas for a complete set of Poisson commuting functions in the classical limit, but a Lax matrix generating these Hamiltonians as its spectral invariants was lacking ever since*. In a recent joint work with B.G. Pusztai [arXiv:1603.06710], we constructed a Lax pair for the classical hyperbolic BC(n) system with two independent couplings. We showed that the dynamics can be solved by a projection method and worked out the asymptotic form of the solutions. The equivalence of the first integrals provided by the eigenvalues of our Lax matrix and van Diejen's commuting Hamiltonians was also demonstrated.

* Except for the 1-coupling cases obtained from the standard A(m) models by 'folding'.

pdf file of the presentation

**ISAEV Alexey **
*Bethe subalgebras in affine Birman--Murakami--Wenzl algebras and flat connections for q-KZ equations*

(in collaboration with A.N. Kirillov and V.O. Tarasov)
Commutative sets of Jucys-Murphy elements for affine braid groups of
A(1), B(1), C(1), D(1) types were defined. Construction of R-matrix
representations of the affine braid group of type C(1) and its
distinguish commutative subgroup generated by the C(1)-type
Jucys--Murphy elements are given. We describe a general method to
produce flat connections for the two-boundary quantum
Knizhnik-Zamolodchikov equations as necessary conditions for Sklyanin's
type transfer matrix associated with the two-boundary multicomponent
Zamolodchikov algebra to be invariant under the action of the C(1)-type
Jucys--Murphy elements. We specify our general construction to the case
of the Birman--Murakami--Wenzl algebras. As an application we suggest a
baxterization of the Dunkl--Cherednik elements Y′s in the double affine
Hecke algebra of type A.

pdf file of the presentation

**JIMBO Michio **
* Finite type modules and Bethe Ansatz for quantum toroidal gl(1)*

A systematic construction of Integrable hierarchies is proposed in
We introduce and study `finite type' modules of the Borel subalgebra of
the quantum toroidal gl(1) algebra. These modules are infinite dimensional
in general but the Cartan like generator ψ^{+}(z) has a finite number
of eigenvalues.
We use them to diagonalize the transfer matrices T(u;p) analogous to
those of the six vertex model, wherein the role of C^{2} is played by
infinite dimensional Fock spaces.
We construct an auxiliary operator Q(u;p) and a new operator T(u;p)
(different from the transfer matrix) satisfying a two-term TQ relation,
from which the Bethe Ansatz equation is derived. We give an expression
for the eigenvalues of T(u;p) in terms of Q(u;p).

pdf file of the presentation

**KASHAEV Rinat **
*Generalized Faddeev-Volkov models*

There is notion of a quantum dilogarithm over a self-dual locally
compact abelian group. For such object, one automatically obtains a
solution of the Yang-Baxter equation which is a direct analog of the
solution associated with the Faddeev-Volkov model. I will discuss those
solutions for few particular known examples of quantum dilogarithms.

pdf file of the presentation

**KONNO Hitoshi **
*Elliptic q-KZ equation and the Weight Functions*

It is know that the representation theory of the elliptic quantum group $U_{q,p}({\mathfrak{g}})$ yields a systematic construction of the integral expressions of the solutions to the face type (dynamical) elliptic q-KZ equation. There the vertex operators and the elliptic (half) currents (screening operators) play important roles. On the other hand there are a lot of significant works by Felder, Tarasov and Varchenko [FTV] on the same solutions for the case ${\mathfrak{g}}=\widehat{\mathfrak{sl}}_2$. However no systematic study on a comparison between the two. In this talk we address this issue. We will present a simple rule of deriving the weight functions, whose transition property with the transition matrix given by the elliptic dynamical $R$ matrix is manifest, as well as their dual functions w.r.t the hypergeometric pairing in the terminology by FTV. We also recover the recursive construction of the weight functions which is the basis in the works by FTV. As examples, we present the expressions of the weight functions associated with the representations of $U_{q,p}(\widehat{\mathfrak{sl}}_2)$ of level $k\in \mathbb{Z}_{>0})$ as well as of $U_{q,p}(\widehat{\mathfrak{sl}}_N)$ of level 1. The case $U_{q,p}(\widehat{\mathfrak{sl}}_2)$ of level $k$ with the finite dim. rep. being the tensor product of the $k+1$-dim.reps. coincides with the one obtained by FTV. The higher rank result is new. If time allows we will make some comments on the relation between the weight functions and the stable envelopes by Okounkov as well as on a connection of the solution to the elliptic $q$-KZ eq. with the conjectural Nekrasov partition function of the 6-dim. SUSY gauge theory.

**KORFF Christian **
*Discrete time evolution and Baxter's Q-operator *

We discuss the quantisation of the Ablowitz-Ladik chain and of its
Darboux transformations. The latter can be linked explicitly to Baxter's
Q-operator in terms of difference equations which describe the discrete
time evolution of q-bosons. Iteration of these Darboux transformations
leads to a 2D TQFT previously constructed by the author and its fusion
coefficients solve the difference equations.

pdf file of the presentation

**KOZLOWSKI Karol **
*Condensation properties of Bethe roots in the XXZ chain*

With the Bethe Ansatz approach, the eigenvectors of the L-site XXZ spin
1/2 chain are parametrised by solutions
to high degree algebraic equations in many variables: the Bethe
equations. In order to compute the thermodynamic limit $L\to +\infty$ of
the per-site ground state energy, Hulten conjectured in 1938 that the
Bethe roots describing the model's ground state are real and form a
dense distribution in the thermodynamic limit. In fact, the whole
description of the thermodynamic limit of the chain (form of the
excitations, conformal structure of the low-energy spectrum, integral
representation for the zero temperature correlation functions, large-L
behaviour of the form factors of local operators)
is based on some variant of Hulten's conjecture. In 2009, Dorlas and
Samsonov managed to prove this conjecture in a regime of the anisotropy
where it is possible to build the analysis on convexity arguments ŕ la
Yang and Yang.
After having recalled the history of the problem and its various
applications, I will present the main ideas of the methodthat I
developed so as to prove condensation properties of Bethe roots
corresponding to certain classes of solutions to the Bethe equations.
The method works independently of the value taken by the anisotropy and
appears to be generalisable to many other quantum integrable models.
This talk is based on the work K. K. Kozlowski, "On densification
properties of Bethe roots", math-ph:1508.05741.

pdf file of the presentation

**KUNIBA Atsuo **
*Tetrahedron equation and matrix product method*

I shall explain how the factorization of
quantum R matrices via the 3DL/3DR-operators
satisfying the tetrahedron equation leads to the matrix product steady states in the
1D totally asymmetric simple-exclusion/zero-range processes of multispecies particle system.
The quadatic algebra of the matrix product operators turns out to be
a far-reaching consequece of the bilinear relations among
layer-to-layer transfer matrices in integrable 3D lattice models.

pdf file of the presentation

**MAILLET Jean-Michel **
*Correlation functions of quantum integrable models: recent advances*

I will review recent advances in the computation of form factors and
correlation functions of quantum integrable lattice models.

pdf file of the presentation

**MALYSHEV Cyril **
*Correlation functions of integrable models, combinatorics, and random walks*

Certain quantum integrable models solvable by the Quantum Inverse
Scattering Method demonstrate a relationship with
combinatorics, partition theory, and random walks.
Special thermal correlation functions of the models in question
are related with the generating functions for plane partitions and
self-avoiding lattice paths. The correlation functions of the integrable
models admit interpretation in terms of nests of the lattice paths made
by vicious walkers.

pdf file of the presentation

**NEKRASOV Nikita **
* Both Q-operators from gauge theory*

The simplest (A_{1}) T-Q equation has two solutions. We explain the gauge theoretic meaning of these solutions as partition functions of supersymmetric gauge theory on a folded spacetime (two R^{4}'s intersecting along an R^{2})

pdf file of the presentation

**NIJHOFF Frank **
* Lagrangian Multiform Theory *

Lagrangian multiform theory (also referred to as "pluri-Lagrangian
structures" in follow-up work by others) was proposed by S. Lobb and the
speaker in 2009 as a novel variational approach to integrable systems
based on the idea of multidimensional consistency. In this theory
Lagrangians are differential- or difference p-forms and the geometry in
the space of independent variables becomes a variational quantity. Most
importantly, the Lagrangian components, rather than being postulated by
separate arguments, are themselves solutions of the system of
variational equations. Since the original paper a large amount of
progress has been made, and the talk will highlight some recent
advances. Furthermore, the implications for quantum theory are
discussed.

pdf file of the presentation

**NIROV Khazret **
*Quantum loop algebras and functional relations: oscillator vs. prefundamental representations*

We consider various representations of quantum groups to specify
different integrability objects and functional relations. In particular,
we present a comparative analysis of the q-oscillator and
prefundamental representations of the Borel subalgebras of quantum loop
algebras.

pdf file of the presentation

**OGIEVETSKY Oleg **
*Diagonal reduction algebra and reflection equation*

We describe the diagonal reduction algebra of the Lie algebra gl(n) in
the R-matrix formalism. As a byproduct we present two families of
central elements and the
braided bialgebra structure of the diagonal reduction algebra.

**PATU Ovidiu **
*Efficient thermodynamic description of the Gaudin-Yang model*

We address the problem of computing the thermodynamic properties of the
repulsive one-dimensional two-component Fermi or Bose gas with contact
interaction, also known as the Gaudin-Yang model. Making use of the
connection with the q=3 Perk-Schultz spin chain and the quantum transfer
matrix we derive an exact system of only two nonlinear integral
equations for the thermodynamics of the homogeneous model which is
valid for all temperatures and values of the chemical potential,
magnetic field and coupling strength. This system allows for an easy
and extremely accurate calculation of the thermodynamic properties
(densities, polarization, magnetic susceptibility, specific heat,
interaction energy, Tan contact and local correlation function of
opposite spin) circumventing the difficulties associated with the
truncation of the thermodynamic Bethe ansatz system of equations.
In the fermionic case we show that at low and intermediate temperatures
the experimentally accessible contact is a
non-monotonic function of the coupling strength. As a function of the
temperature the contact presents a pronounced local minimum in the
Tonks-Girardeau regime signaling an abrupt change of the momentum
distribution in a small interval of temperature which should be
experimentally accessible.

pdf file of the presentation

**PIMENTA A. Rodrigo **
*Universal Bethe ansatz solution for the Temperley-Lieb spin chain *

We discuss some spectral properties of the Temperley-Lieb open quantum
spin chain with free boundary conditions associated with the spin-s
representation of quantum-deformed sl(2). We argue that many of these
properties are universal, in the sense that they are independent of the
value of the spin. For the s=1 case, we show how the reflection algebra
can be used in order to prove the off-shell equation satisfied by the
Bethe vectors. (Based on joint work with R. Nepomechie, arXiv:1601.04378
and arXiv:1601.04328).

pdf file of the presentation

**POZSGAY Balázs **
*Quantum quenches and excited state correlations of the XXZ and XXX spin chains*

Recently there has been considerable interest in deriving exact solutions for quantum quench (or general out-of-equilibrium) situations of the XXZ spin chain. The simplest problems are when at t=0 the system is prepared in a simple product state (for example the Néel or dimer states), and for t>0 it is evolved with the XXZ Hamiltonian. At present no exact solution is known for the full time evolution, but the asymptotic values of the local correlations can be calculated using Bethe Ansatz methods. There are two main steps: Finding the Bethe states that populate the system after the quench, and calculating the correlations in these states. In this talk we concentrate on the second problem and present conjectured exact formulas for the excited states of the XXZ and XXX spin chain, both at finite size and in the thermodynamic limit. The main idea is to use the theory of factorized correlations, which has been established for the ground state or finite temperature situations. We argue that the algebraic part of the construction holds for arbitrary excited states, and present new formulas for the so-called physical part. In finite size the formulas are simple algebraic expressions that use the exact Bethe roots, whereas in the thermodynamic limit they lead to a TBA-like system of equations.

pdf file of the presentation

**ROSSI Paolo **
*Quantization of integrable systems and the moduli space of curves. *

With A. Buryak, we recently invented a very explicit procedure to construct quantizations for a vast class of integrable systems of Hamiltonian eveolutionary PDEs (including KdV, Toda, ILW and all integrable hierarchies associated to Frobenius manifolds via the construction of Dubrovin-Zhang). The construction uses the intersection theory in the cohomology of the moduli space of stable curves. I will present this technique and give explicit examples together with general results on the nature of such quantum integrable systems.

pdf file of the presentation

**ROUBTSOV Vladimir **
*Quantization of Painleve monodromy variety *

We introduce the concept of decorated character variety for the Riemann
surfaces arising in the theory of the Painlevé differential equations.
Since all Painlevé differential equations (apart from the sixth one)
exhibit Stokes phenomenon, it is natural to consider Riemann spheres
with holes and bordered cusps on such holes. The decorated character is
defined as complexification of the bordered cusped Teichmüller space
introduced by Chekhov and Mazzocco. We show that the decorated
character variety of a Riemann sphere with $s$ holes and $n>1$
cusps is a Poisson manifold of dimension $3 s+ 2 n-6$ and we explicitly
compute the Poisson brackets. We also show how to obtain the confluence
procedure of the Painlevé differential equations in geometric terms and
how to quantize them. This is a joint work with Chekhov and Mazzocco.

pdf file of the presentation

**SERBAN Didina **
* Clustering in the N=4 SYM three point function*

I shall present a joint work with Y. Jiang, S. Komatsu and I. Kostov on the re-summation of certain contributions to the three point function of the maximally supersymmetric 4d gauge theory. We consider only the simplest rank-one sectors su(2) and sl(2). The method uses a representation as a multiple integral for the sum-over-partitions expression put forward by Basso, Komatsu and Vieira. The key step allowing to take the limit of large number of magnons is to consider the contribution of the poles in the integration measure. We call clustering the coalescence of multiple integrals into simpler integrals, a phenomenon akin to the formation of bound states. Two main contributions can be handled by clustering. The first is the so-called asymptotic contribution, which we are able to compute to for any value of the coupling constant. The second contribution corresponds to virtual particles in one of the channels, at strong coupling. The results for both the asymptotic and virtual contributions agree with strong coupling computations using the classical string sigma model.

pdf file of the presentation

**SLAVNOV Nikita **
* Form factors of the monodromy matrix entries in the models with gl(2|1) symmetry *

We apply the nested algebraic Bethe ansatz to the models with
gl(2|1) symmetry. We obtain explicit representations for the Bethe
vectors and their scalar products. In some particular cases a we find
determinant formulas for the scalar products. Starting from these
formulas and using the zero modes method we obtain compact determinant
representations for the form factors of the monodromy matrix entries.

pdf file of the presentation

**SUZUKI Junji **
*Form factor expansions of XXZ model*

Recent progress on the study of the correlation functions of XXZ model
by form factor expansions will be discussed. Thermal correlations
function will be treated using the Fredholm determinants. This talk is
based on collaboration with
M. Dugave, F. Goehmann and K. Kozlowski

pdf file of the presentation

**VANICAT Matthieu **
*Integrable dissipative exclusion process.*

I will present a one-parameter generalization of the symmetric simple
exclusion process on a one dimensional lattice. In addition to the usual
dynamics (where particles can hope with equal rate to the left or to
the right with an exclusion constraint), annihilation and creation of
pairs can occur. The system is driven out of equilibrium by two
reservoirs at the boundaries. In this setting the model is still
integrable: it is related to the open XXZ spin chain through a gauge
transformation. This allows us to compute the full spectrum of the
Markov matrix using Bethe equations. Then, we deduce the spectral gap in
the thermodynamical limit.
We also show that the stationary state can be expressed in a matrix
product form permitting to compute the multi-points correlation
functions as well as the mean value of the lattice current and of the
creation-annihilation current. Finally the variance of the lattice
current is exactly computed for a finite size system. In the
thermodynamical limit, it matches perfectly the value obtained from the
associated macroscopic fluctuation theory. It provides a confirmation of
the macroscopic fluctuation theory for dissipative system from a
microscopic point of view.

pdf file of the presentation

**WHEELER Michael **
*Stochastic vertex models and generalized Macdonald polynomials*

I will discuss a new family of stochastic, higher-rank vertex models, which generalize the stochastic six-vertex model. Aside from their probabilistic uses, these models give rise to a new family of symmetric functions, which simultaneously generalize Macdonald polynomials and a family of rational functions introduced by Borodin and Petrov.

Based on collaboration with Luigi Cantini, Alexandr Garbali and Jan de Gier.

pdf file of the presentation