Tuesday, August 16th, 2016

# RAQIS'16: Titles and abstracts

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.
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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.
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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.
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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.
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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).
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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.
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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.
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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.
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GONCHAROV Alexander     Bipartite graphs on torus and cluster integrable systems

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.
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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 C2 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).
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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.
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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 à 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.
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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.
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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.
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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.
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NEKRASOV Nikita     Both Q-operators from gauge theory
The simplest (A1) 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 R4's intersecting along an R2)
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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
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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.
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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

## Poster Session

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

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

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