Workshop on Elliptic Integrable Systems

March 7th-9th 2021 over Zoom

Organizer: Peter Koroteev

The workshop 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.

Sunday evening (3/7) [Videos]
5:05-5:10pm Welcome remarks

5:10-6:00pm Andrey Smirnov (UNC Chapel Hill). Elliptic stable envelope for Hilbert scheme of points in C^2 [Notes]
In this talk I discuss the elliptic stable envelope classes of torus fixed points in the Hilbert scheme of points in the complex plane. I describe the 3D-mirror self-duality of the elliptic stable envelopes. The K-theoretic limits of these classes provide various special bases in the space of symmetric polynomials, including well known bases of Macdonald or Schur functions. The mirror symmetry then translates to new symmetries for these functions. In particular, I outline a proof of conjectures by E.Gorsky and A.Negut on ``Infinitesimal change of stable basis'', which relate the wall R-matrices of the Hilbert scheme with the Leclerc-Thibon involution for $U_q(\frak{gl}_b)$.

6:10-7:00pm Junichi Shiraishi (University of Tokyo). Branching formula for q-Toda function of type B [Notes]
We present a proof of the explicit formula for the asymptotically free eigenfunctions of the $B_N$ $q$-Toda operator which was conjectured by Ayumu Hoshino and J.S. This formula can be regarded as a branching formula from the $B_N$ $q$-Toda eigenfunction restricted to the $A_{N-1}$ $q$-Toda eigenfunctions. The proof is given by a contigulation relation of the $A_{N-1}$ Toda eigenfunctions and a recursion relation of the branching coefficients.

Monday morning (3/8) [Videos]
9:10-10:00am Andrey Zotov (Steklov Mathematical Institute RAS). Characteristic determinant and Manakov triple for double elliptic integrable system [Slides]
Using the intertwining matrix of the IRF-Vertex correspondence we propose a determinant representation for the generating function of the commuting Hamiltonians of the double elliptic integrable system. More precisely, it is a ratio of the normally ordered determinants, which turns into a single determinant in the classical case. This gives expression for the spectral curve and the corresponding L-matrix, which is obtained explicitly as a weighted average of the Ruijsenaars and/or Sklyanin type Lax matrices with the weights as in the theta function series definition. By construction the L-matrix satisfies the Manakov triple representation instead of the Lax equation. Finally, we discuss double elliptic analogue for the Dunkl-Cherednik type construction and its applications. The talk is mainly based on paper 2010.08077. Some ideas and results of 2102.06853 are discussed as well.

10:10-11:00am Simon Ruijsenaars (University of Leeds). Painleve-Calogero correspondence: The elliptic 8-coupling level [Slides]
This seminar is based on joint work with M.Noumi and Y.Yamada [1]. The 8-parameter elliptic Sakai difference Painleve equation [2] admits a Lax pair formulation. We sketch how a suitable specialization of one of the Lax equations gives rise to the time-independent Schrodinger equation for the $BC_1$ 8-coupling relativistic Calogero-Moser Hamiltonian due to van Diejen[3]. This amounts to a generalization of previous results concerning the Painleve-Calogero correspondence to the highest level of the two hierarchies. In both settings, there exists a symmetry under the Weyl group of $E_8$ [2,4].
References:
[1] M.Noumi, S.Ruijsenaars and Y.Yamada, The elliptic Painleve Lax equation vs. van Diejen's 8-coupling elliptic Hamiltonian, SIGMA 16 (2020), 063, 16 pages, 1903.09738.
[2] H.Sakai, Rational surfaces associated with affine root systems and geometry of the Painleve equations, Commun.Math.Phys. 220 (2001), 165-229.
[3] J.F.van Diejen, Integrability of difference Calogero-Moser systems, J.Math.Phys. 35 (1994), 2983-3004.
[4] S.Ruijsenaars, Hilbert-Schmidt operators vs. Integrable systems of elliptic Calogero-Moser type IV. The relativistic Heun (van Diejen) case, SIGMA 11 (2015), 004, 78 pages, 1404.4392.

11:10am-12:00pm Shlomo Razamat (Technion). Three roads to the van Diejen model and beyond [Slides]
First, we will discuss the relation between the E-string theory and the $BC_1$ van Diejen model. In particular we will present three different constructions which lead to the same model. These different constructions can be thought of as integrable models formally corresponding to $A_{N=1}$, $C_{N=1}$ and $(A_1)^{N=1}$ root systems. The constructions will provide three different types of Kernel functions for the $BC_1$ van Diejen model. Second, we will argue that the three roads to the van Diejen model generalize to integrable models associated to $A_{N}$, $C_{N}$ and $(A_1)^{N}$ root systems. We will give details of the construction of the $A_{N}$ models.

Monday evening (3/8) [Video]
7:10-8:00pm Gleb Aminov (Stony Brook). Dell systems and the Seiberg-Witten prepotentials. [Slides]
In this talk we are going to review the approach to the double elliptic systems initially introduced by H.W. Braden, A. Marshakov, A. Mironov, A. Morozov in 1999. In particular, we will focus on the N-particle description by A. Mironov and A. Morozov developed later in 2000. In this description the Hamiltonians are made from the higher-genus theta functions on the Jacobians of the Seiberg-Witten curves. The Poisson commutativity of these Dell Hamiltonians still remains a hypothesis. We will start by describing some theta-function identities that play the role of necessary conditions for the commutativity of the Hamiltonians in the 3- and 4-particle cases. The identities in the 3-particle case are proven to be true for any period matrix and could be connected to the Macdonald identities for some affine root systems. Next, we will consider the general case of N-particle systems and discover that the identities become specific constraints on the period matrices of the corresponding SW curves. We will argue that the most generic curve is given by the Dell system. Reversing the task of proving the commutativity, one can use the constraints to compute the period matrices and the corresponding SW prepotentials. In doing so, we will compute the 6d Dell prepotential and obtain the instanton expansion, that can be rewritten as an expansion in powers of the flat moduli (including the adjoint mass) with coefficients being quasimodular forms of the elliptic parameter. The quasimodular properties of the instanton expansion will lead us to the 6d generalizations of the MNW modular anomaly equation (also known as the Holomorphic anomaly equation). We will conclude by presenting some open questions and relations to the other known descriptions of the Dell systems.

Tuesday morning (3/9) [Video]
9:10-10:00am Matej Penciak (Northeastern). Geometric Action-Angle Coordinates for the spin Ruijsenaars-Schneider system [Notes]
In this talk we will offer a description of a completion of the RS phase space as a moduli space of so-called framed spectral sheaves. An open subset of this moduli space is exactly given by line bundles supported on RS spectral curves, but the full space allows for more general sheaves on singular and non-reduced curves. In this description, the flows of the RS system are the natural flows coming from modifications of sheaves at a point. We begin with a parallel story for the Calogero-Moser system which was worked out by David Ben-Zvi and Tom Nevins. We then move to the main result for the RS system where the identification with the RS phase space goes through the moduli space of multiplicative Higgs bundles. We end with some consequences of these results, and some hopeful speculation about what a similar description of the Dell system would look like.

10:10-11:00am Yegor Zenkevich (ITEP). Elliptic DIM algebra and elliptic integrable systems
We deomnstrate that elliptic Ruijsenaars-Schneider Hamiltonians can be understood as acting on certain set of symmetric polynomials E. Moreover, these polynomials furnish a representation of elliptic Ding-Iohara-Miki (ell-DIM) algebra. It has been conjectured that the pq-duals of elliptic RS Hamiltonians are given by a trigonometric degeneration of quantum Dell Hamiltonians introduced by Koroteev and Shakirov (KS Hamiltonians). We show that the correct statement is in fact more subtle: the eigenfunctions of the degeneration KS Hamiltonians are conjugates of the polynomials E with respect to Schur scalar product.
Finally, we analyze the structure of ell-DIM algebra and show that in a remarkable way it is isomorphic to a direct sum of the ordinary (trigonometric) DIM algebra and an additional Heisenberg algebra. The isomorphism is inspired by the Bogolyubov transformation in the thermo field double formalism. We explore the implications of the isomorphism for representations and commuting subalgebras of ell-DIM algebra.
This talk is based on the joint works 2012.15352 with Mohamed Ghoneim, Can Kozcaz, Kerem Kursun, and 2103.02508 with Andrei Mironov and Alexey Morozov.

11:10am-12:00pm Vyacheslav Spiridonov (JINR, Dubna and HSE, Moscow). From Elliptic Hypergeometric Integrals to Complex Hypergeometric Functions. [Slides]
Elliptic hypergeometric integrals are top transcendental special functions of hypergeometric type. They have found applications in 4d supersymmetric quantum fields theories (superconformal indices), in integrable systems (wave functions in quantum N-body problems) and 2d statistical mechanics (partition functions). I will describe how these integrals can be degenerated in a chain of limits to complex hypergeometric functions related to the representation theory of SL(2,C).