Lorentz Center - Elliptic Integrable Systems and Hypergeometric Functions from 15 Jul 2013 through 19 Jul 2013
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    Elliptic Integrable Systems and Hypergeometric Functions
    from 15 Jul 2013 through 19 Jul 2013



James Atkinson
A higher analogue of the Addition law, integrable quad-graph models and Fano 3-space.

A symmetric generalisation of the polynomial that defines the fundamental integrable quad-graph model Q4 and its multi-quadratic counterpart Q4* will be given. The natural integrability feature of these  models is the consistency on a cube, and the corresponding consistency property of the more general polynomial is in Fano 3-space, or PG(3,2). It can be interpreted as a higher analogue of the (Euler form of the) polynomial addition law for elliptic functions. This work is part of the Australian Research Council funded project "Algebraic interpretations of discrete integrable equations" DP110104151.


Dan Betea 
Elliptic combinatorics and Schur-like processes on boxed plane partitions and the Aztec diamond

We provide elliptic generalizations to enumerative combinatorial results in the theory of boxed plane partitions and Aztec diamonds. Integrable elliptic weights can be assigned on such objects via their correspondence with lattice paths, some of which have been previously studied by Schlosser, Borodin/Gorin/Rains and the author. We do this via a suitable generalization of the Schur process of Okounkov/Reshetikhin using Rains' elliptic analogues of skew Macdonald polynomials. In the determinantal ($q=t$) case, this generalization leads to efficient exact sampling algorithms from said distributions. As with the rational and trigonometric limits, such models exhibit new phase transitions as the lattice spacing goes to 0.


Fokko van de Bult
Bilateral series as limits of the elliptic beta integral

By taking the limit $p to 0$ elliptic hypergeometric identities become basic hypergeometric identities. By changing the behavior of parameters (other than $p$ and $q$) as $p to 0$ we can obtain multiple different basic hypergeometric identities as limit of a single elliptic hypergeometric identity. We would like to connect every basic hypergeometric identity with an elliptic hypergeometric identity of which it is a limit. In this talk I give an overview of the project to obtain all known basic hypergeometric identities as limits of elliptic hypergeometric identities. In particular I will also discuss my recent work which determines ``all'' bilateral series that can appear as limits of the elliptic beta integral.


Oleg Chalykh

Quantum elliptic Calogero-Moser problem at integer coupling parameters


Let L be the quantum elliptic Calogero-Moser operator associated to a root system R and W-invariant coupling parameters. I will explain what is special about the case of integer coupling parameters: in particular, why in this case the Bloch eigenfunctions of L are parameterised by points of an algebraic variety, how one can calculate them, and how to obtain the discrete spectrum eigenstates for L.



Sergey Derkachov

Yang-Baxter equation, parameter permutations, and the elliptic beta integral


We construct the general solution of the Yang-Baxter equation which is represented as an integral operator with an elliptic hypergeometric kernel acting in the space of functions of two complex variables. It intertwines the product of two standard L-operators associated with the Sklyanin algebra, an elliptic deformation of sl(2). This R-operator is constructed from three basic operators S_1, S_2$ and S_3 generating the permutation group of four parameters. Validity of the key Coxeter relations is based on the elliptic beta integral evaluation formula and the Bailey lemma associated with an elliptic Fourier transformation.

The operators S_j are determined uniquely with the help of an elliptic modular double.


Giovanni Felder
Lamé polynomials as orthogonal polynomials

Polynomial eigenvectors of several classical differential operators,
such as Hermite, Legendre, Laguerre, ... polynomials
form families of orthogonal polynomials: they are uniquely determined
up to normalization by their degree and orthogonality property.
Other classes of polynomial eigenvectors, such as the Lamé polynomials
don't follow this rule: different eigenvalues correspond to the same degree.
We show that Lamé polynomials are uniquely characterized
up to normalization by being jointly orthogonal with respect to two
inner products. I will discuss more generally jointly orthogonal systems
for multiple inner products and discuss examples. This is joint work
with Thomas Willwacher.


Martin Hallnas

A recursive construction of joint eigenfunctions for the hyperbolic Calogero-Moser Hamiltonians


This talk will address recent and ongoing joint work with S. Ruijsenaars on the construction of symmetric joint eigenfunctions for the commuting PDOs associated to the hyperbolic (non relativistic) Calogero-Moser N-particle system. We construct these eigenfunctions via a recursion scheme, which leads to representations by multidimensional integrals whose integrands are elementary functions. We shall also indicate how these eigenfunctions can be tied in with the A_N-1 Heckman-Opdam hypergeometric function, and how the construction can be generalised to the relativistic setting.



Nalini Joshi

Geometry and Asymptotics of Discrete Painlev\'e Equations


Critical solutions of the Painlev\'e equations arise as universal limits in many nonlinear systems. This talk focusses on my geometric approach to describing their asymptotic properties, which was initiated in collaboration with Duistermaat. I will focus in this talk on extensions of this approach to discrete Painlev\'e equations, including a special case of the elliptic Painlev\'e equation discovered by Sakai.


Much of the activity in this field has been concentrated on deducing integrable discrete versions of the Painlev\'e equations, finding transformations and other algebraic properties and describing special solutions that can be expressed in terms of earlier known functions, such as $q$-hypergeometric and elliptic hypergeometric functions. In contrast, in this talk, I focus on finding properties of solutions that cannot be expressed in terms of earlier known functions.



Stefan Kolb

Quantum symmetric Kac-Moody pairs


In this talk I will outline a general theory of quantum group analogs of symmetric pairs for involutive automorphisms of the second kind of symmetrizable Kac-Moody algebras. Such structures have appeared in the investigation the XXZ model with general boundary conditions. Moreover, they are expected to be related to difference-elliptic Calogero-Moser systems via radial part calculations. 


The construction presented in this talk follows G. Letzter's theory of quantum symmetric pairs for semisimple Lie algebras. The main additional ingredient is the classification of involutive automorphisms of the second kind of symmetrizable Kac-Moody algebras due to Kac and Wang.



Hitoshi Konno
Elliptic Quantum Group Uq,p(ˆg) and Deformed W-algebras


The Uq,p(ˆg) is an elliptic analogue of the quantum affine algebra Uq(ˆg) in the Drinfeld realization. It is known that the level-1 Uq,p(A(1) N ) provides an algebra of screening currents of the deformed W-algebra. We discuss some recent results extending such relationship. After introducing Uq,p(ˆg) as a topological algebra over the ring of formal power series in p, we discuss infinite dimensional highest weight representations of Uq,p(ˆg) by means of a quantum dynamical analogue of Lepowsky-Wilson’s Z-algebras. Some examples on the level-1 irreducible representations lead us to a conjecture that there exists a deformation of the coset type W-algebras associated with ˆg, which contains Fateev-Lukyanov’s WBl-algebra. In addition, we introduce a co-algebra structure of Uq,p(ˆg) given by the Drinfeld coproduct. We then discuss a possible role of the vertex operators in a relationship to the deformed




Edwin Langmann

Kernel functions and quantum systems of Calogero-Moser-Sutherland type


I review the use of kernel functions to solve quantum systems of Calogero-Moser-Sutherland type, including elliptic such systems. I also describe methods allowing to find such kernel functions.



Vladimir Mangazeev
The higher spin XXZ chain: a 3D approach

We study a quantum XXZ spin chain from a 3D perspective. Starting
from the trigonometric solution of the tetrahedron equation we derive
a new explicit expression for the $U_q(sl(2))$ $R$-matrix acting in the
tensor product of two representations with spins $s_1$ and $s_2$.
We also construct a new representation for the Q-operator of the XXZ chain
of spin $s$. Our construction of the Q-operator can be naturally
extended to arbitrary real values of spin and the limit to (half)-integer values
of $s$ is non-singular. I shall also briefly discuss a possibility to extend
our results to the elliptic case.

Yasuho Masuda

A duality transformation formula arising from the Ruijsenaars difference operator of type C


In this talk, I present a duality transformation formula for multiple elliptic hypergeometric series of type C. As a special case, this formula gives a multiple generalization of the Frenkel-Turaev summation formula, due to H. Rosengren. I also derive some summation formulas from this formula.

This talk is based on a joint work with Y. Komori and M. Noumi.



Frank Nijhoff

Elliptic integrable systems on the lattice and associated continuous systems


Among the large number of integrable soliton type systems there exist a number of important equations which can be considered to be "elliptic" in the sense that they involve an elliptic curve either through the parameters of the equation, or because the dependent variables live on an elliptic curve, Examples comprise the (discrete and continuous) Landau-Lifschitz equation, the Krichever-Novikov equation and its lattice analogue (Adler's equation, or "Q4"), an elliptic generalization of the lattice KdV and an elliptic generalization of the KP system. In the talk I will review aspects of the underlying structures of those systems and elliptic soliton type solutions. (Parts of this work are in collaboration with J. Atkinson and P. Jennings.)



Masatoshi Noumi

Elliptic Askey-Wilson functions and associated elliptic Schur functions


In this talk I discuss a family of elliptic functions that generalize Askey-Wilson polynomials, with emphasis on their difference equations. Also, I investigate a class of multivariable elliptic functions of Schur type built up from them by determinants. This class of functions can be regarded as an elliptic extension of Koornwinder polynomials with t=q, and carries various characteristic properties. I will describe in particular difference equations for this class, and an explicit formula for rectangle cases.



Mikhail Olshanetsky

Topological classification of isomonodromy problems over elliptic curves


We consider the isomonodromy problems for flat G-bundles over elliptic curves Στ. The bundles are classified by their characteristic classes. The characteristic classes are elements of the second cohomology group H 2(Στ, Z (G)), where Z (G)) is the center of G. For example, for G=Spin the characteristic classes are the Stiefel-Whitney classes. For any G and arbitrary classes we define the moduli space of flat bundles, construct the monodromy preserving equations in the Hamiltonian form and their Lax representations. There exists a symplectomorphisms (the Hecke transformation) between systems related to different classes. In particular, the Painleve VI equation can be described in terms of SL(2) bundles. Since Z (SL(2))  = Z2, Painleve VI has two representations: 1) as the non-autonomous Calogero-Inozemtsev system for trivial  bundles;

2) as the non-autonomous Zhukovsly-Volterra SL(2) hyrostat for non-trivial bundles.



Hugh Osborn

The index in supersymmetric quantum field theories and non trivial integral identities.


Dualities between supersymmetric quantum field theories were proposed in the 1990s. Later it was realised that it is possible to construct an index which is a topological invariant and can be calculated in terms of integrals over the associated gauge groups. The equality of the index for dual theories leads to non trivial integral identities. In the simplest case these identities are identical to results of Spiridonov and Rains.

For other examples the required identities are not yet proven.



Eric Rains

Formal elliptic hypergeometric functions


One of the trickier aspects of the theory of elliptic hypergeometric functions is the fact that infinite series have serious convergence issues.  This can be avoided by working instead with contour integrals, but this is in general not a simple analytic continuation from the finite case.  It turns out that one can fix this, allowing one to express certain elliptic hypergometric integrals as {\em convergent} elliptic hypergeometric series: simply treat p as a formal variable.  As an application, I'll prove a number of multivariate quadratic transformations inspired by representation theory.



Shlomo Razamat

4d CFTs, Riemann surfaces, and elliptic integrable models: a 6d story


We will discuss a large class of functions which can be labeled by punctured Riemann surfaces.

At least some of the functions in this class can be explicitly defined as contour integrals of elliptic Gamma functions.

Making certain assumptions motivated by physics of gauge theories with extended

supersymmetry in four space-time dimensions one can systematically derive numerous

identities of these functions and their contour integral representations. We will

directly relate our construction to elliptic integrable models of Ruijsenaars-Schneider type

and generalizations thereof.  



Hjalmar Rosengren

Felder's elliptic quantum group and and elliptic hypergeometric series on the type A root system


We describe a generalization of elliptic 6j-symbols, which can be viewed as matrix elements of intertwiners between corepresentations of Felder's quantum group (a Hopf algebroid based on the R-matrix of the eight-vertex-solid-on-solid model). For special parameter values, they can be expressed in terms of elliptic hypergeometric series related to the root system of type A. This allows us to obtain algebraic proofs of known and new results for such series, such as transformation formulas and biorthogonality relations.



Simon Ruijsenaars

Elliptic integrable systems of Calogero-Moser type


Abstract: In the first part of this seminar, we present a bird's eye view on the area of integrable N-particle systems of Calogero-Moser type with elliptic interactions. In the second part, we survey various results involving so-called kernel functions, in particular as regards their role in obtaining a Hilbert space version of the elliptic systems.



Junichi Shiraishi

Elliptic hypergeometric series, Ruijsenaars operator and Heine's transformation formula


A difference equation for an elliptic hypergeometric series is presented in termes of the Ruijsenaars operator and a certain difference operator.

By applying the representation theory of the Ding-Iohara algebra, we obtain a conjecture about an elliptic analogue of Heine's transformation for the elliptic hypergeometric series.



Vyacheslav Spiridonov

The beauty of elliptic hypergeometric integrals


General elliptic hypergeometric functions are determined by the elliptic hypergeometric integrals, with the key examples being the elliptic beta integral and elliptic analogue of the Euler-Gauss hypergeometric function. Their structure leads to the two-index biorthogonality concept, the elliptic modular doubling principle, an integral operators calculus (integral Bailey chains), etc. Their first physical interpretation emerged in the integrable N-body models of Ruijsenaars and van Diejen. Dolan and Osborn have shown that they describe superconformal indices of four dimensional supersymmetric gauge field theories, which is the major known application. Properties of these integrals encode many important elements of  the Seiberg duality, among which one has the 't Hooft anomaly matching conditions and chiral symmetry breaking.



Kouichi Takemura

Integral transformations for Inozemtsev systems and Heun's equations


The Inozemtsev Hamiltonian is an elliptic generalization of the differential operator defining the BC_N trigonometric quantum Calogero- Sutherland model, and its eigenvalue equation is a natural many-variable generalization of Heun's differential equation.

We present kernel functions for Inozemtsev Hamiltonians, which induce integral transformations for them.

By restricting to the one-variable case, we recover Euler's integral transformations for Heun's equations.

We also explain applications to properties of monodromy of the equations.

This talk is mainly based on a joint work with Edwin Langmann.



Tomoyuki Takenawa

Dynamical systems on rational elliptic surfaces


Autonomous birational dynamical systems on the complex surface C^2 are classified in terms of degree growth into finite, linear, quadratic or exponential. In the quadratic case, the dynamical system can be lifted to an analytically stable mapping on a rational elliptic surface by successive blowing-ups, which is not always relatively minimal however. In this talk, I will present a method to obtain relatively minimal rational elliptic surface where the dynamical system becomes automorphism. I also present a classification of such dynamical automophisms on rational elliptic surfaces. While the simplest example is the case of the Quispel-Roberts-Thompson systems, the equations of the systems become complicated if the surface is a Halphen pencil of a higher index or the system exchanges the elliptic fibration. We also show an equivalent condition when a rational surface whose Picard rank is 10 has a Halphen pencil of an integer index. This is an extension of a classical result that a surface obtained by blowing-up at 9 points on a smooth elliptic curve on a plane becomes an elliptic surface if and only if an integer multiple of the sum of 9 points is zero for the addition law on the elliptic curve.



Yasuhiko Yamada

A simple expression for the elliptic Painlev\'e equation and its Lax pair


The elliptic Painlev\'e equation discovered by H.Sakai is a very complicated nonlinear difference equation. In this talk, I will give a simple and explicit form of the equation together with its Lax pair. This result is based on a joint work with M.Noumi and S.Tsujimoto.