Integrable systems associated with elliptic algebras

Abstract

Abstract. We construct some new Integrable Systems (IS) both classical and quantum associated with elliptic algebras. Our constructions are partly based on the algebraic integrability mechanism given by the existence of commuting families in skew fields and partly -on the internal properties of the elliptic algebras and their representations. We give some examples to make an evidence how these IS are related to previously studied.Introduction. This paper is an attempt to establish a direct connection between two close subjects of modern Mathematical Physics -the theory of Integrable Systems (IS) and the Elliptic Algebras. The aim of this connection is two-fold: we clarify some our recent results in the both domains and fill the natural gap proving that some large class of the Elliptic Algebras carries in fact the structure of an IS.We will start with a short account in the subject of the story and will describe briefly a type of the IS's under cosideration.In [12], B.Enriquez and second author had proposed a construction of commuting families of elements in skew fields. They explained how to use this construction in Poisson fraction fields to give an another proof of the integrability of Beauville -Mukai integrable systems associated with a K3 surface S ([1]).The Beauville-Mukai systems had appeared as the Lagrangian fibrations which have the formis the Hilbert scheme of g points of S, equipped with a symplectic structure introduced in [19], and L is a line bundle on S. Later, the authors of [8] explained that these systems are natural deformations of the "separated" (in the sense of [13]) versions of Hitchin's integrable systems, more precisely, of their description in terms of spectral curves (already present in [15]). Beauville-Mukai systems can be generalized to surfaces with a Poisson structure (see [3]). When S is the canonical cone Cone(C) of an algebraic curve C these systems coincide with the separated version of Hitchin's systems. The paper [12] shows how the commuting families construction provides a quantization of these systems on the canonical cone.A quantization of Hitchin's system was proposed in [2]. It seems interesting to construct quantization of Beauville-Mukai systems and to compare it with Beilinson -Drinfeld quantization. We had conjectured the correspondence between the results of [2] and a quantization of fraction fields in [9]. A part of this program

Spectral duality in elliptic systems, six-dimensional gauge theories and topological strings

“…At the level of integrable systems, it was established by K.Hasegawa [133][134][135] (see also a trigonometric version of the correspondence in [143]) and claims an equivalence of the elliptic multi-point spin Ruijsenaars system given by (N, k, l) and the elliptic spin chain on k sites, given by the l-orbit of the SklyaninOdesskii-Feigin gl N [144][145][146][147][148][149]. In the particular case of l = 1 (the orbit of minimal dimension), one obtains the duality between SU(N ) k theory with fundamental matter in 6d and SU(k + 1) N necklace quiver theory in 5d.…”

Hitchin and Beauville-Mukai System: Classical and Quantum Correspondence

“…where we are easily recognizing a "Plucker" form of the commutativity conditions from [15] for the case n = 3 and an appropriate choice of a linear relation between 4 points a, b, c, d and z 1 , z 2 , z 3 , η/3 (modulo an irrelevant exponential factor entering in the relations between the theta-functions in different normalizations).…”

“…Let us describe an elliptic version of the Beauville-Mukai integrable system following [15]. We will start with a classical analog of the Weyl-like algebra V n given by the Poisson brackets…”

“…2 in [15] may be expressed as a kind of 4-Riemann identity (known also as the trisecant Fay's identity):…”

Integer Solutions of Integral Inequalities and H‐InvariantJacobian Poisson Structures

“…Therefore α 0 , α 1 , · · · α n−1 are completely determined by the set of nonnegative integer sequences (s 0 , s 1 , · · · , s n−1 ) satisfying the constraints c i : s n−i−1 − s n−i + r ≥ 0, i ∈ Z/nZ (19) and such that s 0 + s 1 + · · · + s n−1 = (n − 1)n 2 r − l.…”