Workshop on Elliptic Integrable Systems II

2022 Workshop on Elliptic Integrable Systems

March 8th-11th 2022 over Zoom

Organizers: Peter Koroteev, Vyacheslav Spiridonov

This is the second workshop in the series (see Elliptic Workshop in 2021).

The meeeting will be focused on the connection between (elliptic) integrable systems, enumerative algebraic geometry, representation theory, and gauge/string theory. Recently there has been significant progress in understanding elliptic and double elliptic integrable systems, dualities between quantum and classical integrable models (i.e. XXZ chains and trigonometric Ruijsenaars-Schneider models), etc. Other direction to be addressed at the workshop are algebro-geometric and representation-theoretic aspects of integrable systems.
The talks are intended to be informal with sufficient time for discussions (50 minutes per talk including questions). Since the speakers are located at different time zones we should arrange talks/discussion times in mornings/evenings in order to achieve maximal overlap between the participants.

Schedule
Evening sessions last from 5pm till 8pm, morning sessions are between 9am and 12pm (all US Pacific time)
All talks will be recorded.

Tuesday morning (3/8) [Video]
9:10am-10:00am Hjalmar Rosengren (Chalmers University). Deformed Ruijsenaars operators
The Ruijsenaars operators are a commuting family of difference operators with elliptic coefficients, which define an integrable system of relativistic quantum particles. In the trigonometric limit, they have Macdonald polynomials as joint eigenfunctions. Through the work of Chalykh, Feigin, Silantyev, Veselov and others, it has become apparent that even more general "deformed" or "super" operators exist. We will describe how to obtain the main properties of such operators in a direct way, which works also in the elliptic setting. In particular, we can prove that the deformed elliptic Ruijsenaars model is integrable, in the sense of constructing a sufficiently large family of algebraically independent operators that commute with the Hamiltonian. The talk is based on joint work with Martin Hallnäs, Edwin Langmann and Masatoshi Noumi.

10:10am-11:00pm Simon Ruijsenaars (Leeds) Around Razamat's A_2 and A_3 kernel identities

11:10am-12:00pm Jan Felipe van Diejen (Universidad de Talca). Eigenfunctions of a discrete elliptic integrable particle model with hyperoctahedral symmetry [Slides]
I report on joint work with Tamás Görbe (University of Groningen) concerning the eigenfunctions of a finite-dimensional truncation of the $BC_n$ elliptic Ruijsenaars type quantum hamiltonian. In the trigonometric limit the eigenfunctions in question reproduce a well-known $q$-Racah type reduction of the Koornwinder-Macdonald polynomials. When the interaction between the particles degenerates to a Pauli repulsion of free fermions, the orthogonal eigenbasis can be expressed in terms of generalized Schur polynomials on the spectrum that are associated with recently found elliptic Racah polynomials.

Tuesday Evening (3/8) [Video]
5:00pm-6:00pm Pavel Etingof (MIT) Hecke operators over local fields and an analytic approach to the geometric Langlands correspondence.
I will review an analytic approach to the geometric Langlands correspondence, following my work with E. Frenkel and D. Kazhdan, arXiv:1908.09677, arXiv:2103.01509, arXiv:2106.05243. This approach was developed by us in the last couple of years and involves ideas from previous and ongoing works of a number of mathematicians and mathematical physicists, Kontsevich, Langlands, Teschner, and Gaiotto-Witten. One of the goals of this approach is to understand single-valued real analytic eigenfunctions of the quantum Hitchin integrable system. The main method of studying these functions is realizing them as the eigenbasis for certain compact normal commuting integral operators the Hilbert space of L2 half-densities on the (complex points of) the moduli space Bun_G of principal G-bundles on a smooth projective curve X, possibly with parabolic points. These operators actually make sense over any local field, and over non-archimedian fields are a replacement for the quantum Hitchin system. We conjecture them to be compact and prove this conjecture in the genus zero case (with parabolic points) for G=PGL(2). I will first discuss the simplest non-trivial example of Hecke operators over local fields, namely G=PGL(2) and genus 0 curve with 4 parabolic points. In this case the moduli space of semistable bundles Bun_G^{ss} is P^1, and the situation is relatively well understood; over C it is the theory of single-valued eigenfunctions of the Lame operator with coupling parameter -1/2 (previously studied by Beukers and later in a more functional-analytic sense in our work with Frenkel and Kazhdan). I will consider the corresponding spectral theory and then explain its generalization to N>4 points and conjecturally to higher genus curves.

6:00pm-7:00pm Junichi Shiraishi (Tokyo). Quantization of Discrete Sixth Painleve Equation and Shakirov's Conjecture

7:00pm-8:00pm Wenbin Yan (Tsinghua) Tetrahedron Instantons [Slides]
We introduce and analyze tetrahedron instantons, which can be realized in string theory by D0-branes probing a configuration of intersecting D6-branes with a nonzero constant background B-field. Physically they capture instantons in six-dimensional gauge theories with the most general intersecting codimention-two supersymmetric defects. We study the properties of the moduli space of tetrahedron instantons. We compute the instanton partition function, which lies between the higher-rank Donaldson-Thomas invariants and the partition function of the magnificent four model. Remarkably the instanton partition function has a closed-form expression in terms of the plethystic exponential, and matches the index of M-theory on a Calabi-Yau fivefold. Our computations provide evidence of the duality between M-theory and type IIA string theory on more general grounds.

Wednesday Morning (3/9) [Video]
9:00am-10:00am Peter Koroteev (Berkeley, Rutgers) DAHA Representations and Branes[Slides]

10:00am-11:00am Yegor Zenkevich (SISSA) Pentagon identity in DIM algebra

11:00am-12:00am Andrey Zotov (MI RAS) Spin generalization of the elliptic Macdonald-Ruijsenaars operators and elliptic version of q-deformed Haldane-Shastry model [Slides]
We describe anisotropic spin generalization of the elliptic Macdonald-Ruijsenaars operators. Commutativity of the operators follows from a set of R-matrix identities. Then the obtained spin operators are used for constructing elliptic generalization of q-deformed Haldane-Shastry long-range spin chain. It is derived by the freezing trick and elliptic function identities.

Wednesday Evening (3/9) [Video]
5:00pm-6:00pm Igor Krichever (Columbia) Turning points of elliptic integrable systems
The notion of turning points of the elliptic Calogero-Moser system was introduced in the recent work of N.Nekrasov and the author where, in particular, the quasi-holomorphic solutions of the double periodic O(2n+1) – sigma model were constructed. These solutions are expressed in terms of the Riemann theta-functions of the spectral curves corresponding to the turning points. Lately, in our joint works with Zabrodin the notion of turning points was extended to the case of the elliptic Ruijsenaars-Schneider system. As it turned out, the theory of the corresponding finite-dimensional systems is isomorphic to the theory of elliptic solutions of some basic (2+1)-dimensional hierarchies with symmetries. Among them CKP hierarchy, Constrained 2D Toda hierarchy.

6:00pm-7:00pm Andrey Smirnov (UNC) Difference equations arising from elliptic stable envelopes
In my talk I explain how certain class of q-difference equations can be constructed from the elliptic stable envelope. In this approach, the elliptic R-matrices are identified with monodromy of difference equations. The difference equations are arising from the monodromy via various limiting procedures. For quiver varieties such approach leads to quantum difference equations, playing role in enumerative geometry.

7:00pm-8:00pm Hee-Cheol Kim (POSTECH) Elliptic quantum curves of 6d SCFTs

Thursday Morning (3/10)[Video]
9:00am-10:00am Oleg Chalykh (Leeds) Twisted Ruijsenaars models[Slides]
The quantum Ruijsenaars model is a q-analogue of the Calogero—Moser model, described by n commuting partial difference operators (quantum hamiltonians) h_1,..., h_n. It turns out that for each natural number m>1, there exists an integrable system whose quantum hamiltonians loosely resemble the m-th powers of h_1,..., h_n. I will discuss several ways of arriving at this generalisation. In the elliptic case, the deformation parameter (“twisting”) is an arbitrary $m$-torsion point c on the underlying elliptic curve; when c=0 one gets precisely the m-th powers of h_1,..., h_n.

10:00am-11:00am Matthew Bullimore (Durham) 3d Supersymmetric Gauge Theory on an Elliptic Curve
There are beautiful connections between supersymmetric gauge theory in various dimensions and generalised cohomology theories. In this talk, I will discuss the ground states of 3d supersymmetric theories on an elliptic curve, focussing on the mathematical structure of the Berry connection that encodes their dependence on background flat connections for global symmetries. I will explain connections to generalised doubly periodic monopoles, equivariant elliptic cohomology and equivariant K-theory. Based on https://arxiv.org/abs/2109.10907 and work in progress with Daniel Zhang.

11:00am-12:00am Shlomo Razamat (Technion) Comments on Modularity of Schur Indices

Thursday Evening (3/10)[Video]
5:00pm-6:00pm Davide Gaiotto (Perimeter) Algebras from twisted M-theory
I will describe the relations between twisted M-theory, the affine Yangian and Calogero models. These constructions govern many aspects of the theory of W-algebras, including degenerate modules, the Miura transform and Coulomb gas constructions.

6:10pm-7:00pm Ole Warnaar (Queensland) Generalised Elliptic Selberg Integrals
In this talk I will describe some recent joint work with Seamus Albion (Vienna) and Eric Rains (Caltech) on generalisations of the elliptic Selberg integral.

7:00pm-8:00pm Jonathan Heckman (UPenn) 6D SCFTs, 4D SCFTs, Conformal Matter and Spin Chains[Slides]

Friday Morning (3/11)[Video]
9:00am-10:00am Peter Koroteev (Berkeley, Rutgers) q-Opers as Geometrization of N=2 Theories [Slides]

10:00am-11:00am Alexander Gorsky (IITP Moscow) Dualities in the integrable many-body systems and integrable probabilities [Slides]
We shall discuss several dualities in the realm of integrable many-body systems which involve two families. One family involves inhomogeneous XXZ spin chain and its degenerations while the second involves Calogero-Moser-Ruijsenaars-Schneider (CM-RS) integrable models. First type of duality-so called spectral duality acts within families and relates two different many-body systems when the coordinates and action variables get interchanged. The second QC duality acts between the families and maps the quantum spin chain system to the classical CM-RS models with nontrivial identification of variables and parameters. It can be generalized to the QQ duality when both dual systems are quantum. We shall map the dualities observed in the framework of integrable probabilities into the spectral and QQ dualities . As an example, we will show a new duality between the discrete-time inhomogeneous multispecies TASEP model on the circle and the quantum Goldfish model from the RS family.

11:00am-12:00am Vyacheslav Spiridonov (JINR, Dubna, Hse, Moscow) Limits of the elliptic hypergeometric equation and integrable systems [Slides]
We describe some new rational degenerations of the elliptic hypergeometric equation and its solutions. They lead to new rational degenerations of the Ruijsenaars and van Diejen elliptic integrable systems. This is a joint work with Gor Sarkissian.