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Elliptic Integrable Systems and Hypergeometric Functions

James Atkinson A
symmetric generalisation of the polynomial that
defines the fundamental integrable quadgraph model
Q4 and its multiquadratic 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
3space, 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 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 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 CalogeroMoser problem at integer coupling
parameters Let
L be the quantum elliptic CalogeroMoser operator
associated to a root system R and Winvariant 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 YangBaxter
equation, parameter permutations, and the elliptic beta integral We
construct the general solution of the YangBaxter 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 Loperators associated with the Sklyanin algebra, an elliptic deformation of sl(2).
This Roperator 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 Martin Hallnas A
recursive construction of joint eigenfunctions for
the hyperbolic CalogeroMoser 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) CalogeroMoser Nparticle
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_N1 HeckmanOpdam 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 KacMoody 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 KacMoody algebras. Such structures have appeared in the
investigation the XXZ model with general boundary conditions. Moreover, they
are expected to be related to differenceelliptic CalogeroMoser
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 KacMoody
algebras due to Kac and Wang. Hitoshi Konno The
Uq,p(ˆg) is an elliptic
analogue of the quantum affine algebra Uq(ˆg) in the Drinfeld realization. It is known that the level1 Uq,p(A(1) N ) provides an algebra
of screening currents of the deformed Walgebra. 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 LepowskyWilson’s Zalgebras.
Some examples on the level1 irreducible representations lead us to a
conjecture that there exists a deformation of the coset
type Walgebras associated with ˆg, which contains FateevLukyanov’s
WBlalgebra. In addition, we introduce a coalgebra
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 Walgebras. Edwin Langmann Kernel
functions and quantum systems of CalogeroMoserSutherland
type I
review the use of kernel functions to solve quantum systems of CalogeroMoserSutherland type, including elliptic such
systems. I also describe methods allowing to find such
kernel functions. Vladimir Mangazeev 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 FrenkelTuraev
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) LandauLifschitz
equation, the KricheverNovikov 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
AskeyWilson functions and associated elliptic Schur functions In
this talk I discuss a family of elliptic functions that generalize AskeyWilson 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 Gbundles 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 StiefelWhitney
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
nonautonomous CalogeroInozemtsev system for
trivial bundles; 2) as the nonautonomous ZhukovslyVolterra SL(2) hyrostat
for nontrivial 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 spacetime
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 RuijsenaarsSchneider 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 6jsymbols, which can
be viewed as matrix elements of intertwiners between corepresentations of Felder's quantum group (a Hopf algebroid based on the
Rmatrix of the eightvertexsolidonsolid 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 CalogeroMoser
type Abstract:
In the first part of this seminar, we present a bird's eye view on the area of integrable Nparticle systems of CalogeroMoser
type with elliptic interactions. In the second part, we survey various results
involving socalled 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 DingIohara
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
EulerGauss hypergeometric function. Their structure
leads to the twoindex biorthogonality concept, the
elliptic modular doubling principle, an integral operators
calculus (integral Bailey chains), etc. Their first physical interpretation
emerged in the integrable Nbody 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 manyvariable generalization of Heun's differential equation. We present kernel functions for Inozemtsev
Hamiltonians, which induce integral transformations for them. By restricting to the onevariable 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
blowingups, 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 QuispelRobertsThompson 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 blowingup 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. [Back] 