The aim of this paper is two-fold. First, we define symplectic maps between Hitchin systems related to holomorphic bundles of different degrees. We call these maps the Symplectic Hecke Correspondence (SHC) of the corresponding Higgs bundles. They are constructed by means of the Hecke correspondence of the underlying holomorphic bundles. SHC allows to construct Bäcklund transformations in the Hitchin systems defined over Riemann curves with marked points. We apply the general scheme to the elliptic Calogero-Moser (CM) system and construct SHC to an integrable SL(N, C) Euler-Arnold top (the elliptic SL(N, C)-rotator). Next, we propose a generalization of the Hitchin approach to 2d integrable theories related to the Higgs bundles of infinite rank. The main example is an integrable two-dimensional version of the two-body elliptic CM system. The previous construction allows to define SHC between the two-dimensional elliptic CM system and the Landau-Lifshitz equation.

We show that the Painlevé VI equation has an equivalent form of the non-autonomous Zhukovsky-Volterra gyrostat. This system is a generalization of the Euler top in C 3 and include the additional constant gyrostat momentum. The quantization of its autonomous version is achieved by the reflection equation. The corresponding quadratic algebra generalizes the Sklyanin algebra. As by product we define integrable XYZ spin chain on a finite lattice with new boundary conditions. June 4, 2018 PVI was discovered by B.Gambier [10] in 1906. He accomplished the Painlevé classification program of the second order differential equations whose solutions have not movable critical points. This equation and its degenerations P V − P I have a lot of applications in Theoretical and Mathematical Physics. (see, for example [41]).We prove here that PVI can be write down in a very simple form as ODE with a quadratic non-linearity. It is a non-autonomous version of the SL(2, C) Zhukovsky-Volterra gyrostat (ZVG) [42,39]. The ZVG generalizes the standard Euler top in the space C 3 by adding an external constant rotator momentum. The ZVG equation describes the evolution of the momentum vector S = (S 1 , S 2 , S 3 ) lying on a SL(2, C) coadjoint orbit. We consider Non-Autonomous Zhukovsky-Volterra gyrostat (NAZVG)where J(τ ) · S = (J 1 S 1 , J 2 S 2 , J 3 S 3 ). Three additional constants ν ′ = (ν ′ 1 , ν ′ 2 , ν ′ 3 ) form the gyrostat momentum vector, and the vector J = {J α (τ )}, (α = 1, 2, 3) is the inverse inertia vector CRDF RM1-2545. The work of A.Z. was also partially supported by the grant MK-2059MK- .2005.2. We are grateful for hospitality to the Max Planck Institute of Mathematics at Bonn where two of us (A.L. and M.O.) were staying during preparation of this paper. We would like to thank a referee for valuable remarks, that allows us to improve the paper.

We describe relationships between integrable systems with N degrees of freedom arising from the AGT conjecture. Namely, we prove the equivalence (spectral duality) between the N -cite Heisenberg spin chain and a reduced gl N Gaudin model both at classical and quantum level. The former one appears on the gauge theory side of the AGT relation in the Nekrasov-Shatashvili (and further the SeibergWitten) limit while the latter one is natural on the CFT side. At the classical level, the duality transformation relates the Seiberg-Witten differentials and spectral curves via a bispectral involution. The quantum duality extends this to the equivalence of the corresponding Baxter-Schrödinger equations (quantum spectral curves). This equivalence generalizes both the spectral self-duality between the 2 × 2 and N × N representations of the Toda chain and the famous AHH duality.

We consider topologically non-trivial Higgs bundles over elliptic curves with marked points and construct corresponding integrable systems. In the case of one marked point we call them the modified Calogero-Moser systems (MCM systems). Their phase space has the same dimension as the phase space of the standard CM systems with spin, but less number of particles and greater number of spin variables. Topology of the holomorphic bundles are defined by their characteristic classes. Such bundles occur if G has a non-trivial center, i.e. classical simply-connected groups, E 6 and E 7 . We define the conformal version CG of G -an analog of GL(N) for SL(N), and relate the characteristic classes with degrees of CGbundles. Starting with these bundles we construct Lax operators, quadratic Hamiltonians, define the phase spaces and the Poisson structure using dynamical r-matrices. To describe the systems we use a special basis in the Lie algebras that generalizes the basis of t'Hooft matrices for sl(N). We find that the MCM systems contain the standard CM systems related to some (unbroken) subalgebras. The configuration space of the CM particles is the moduli space of the holomorphic bundles with non-trivial characteristic classes.

In our recent paper we described relationships between integrable systems inspired by the AGT conjecture. On the gauge theory side an integrable spin chain naturally emerges while on the conformal field theory side one obtains some special reduced Gaudin model. Two types of integrable systems were shown to be related by the spectral duality. In this paper we extend the spectral duality to the case of higher spin chains. It is proved that the N -site GL k Heisenberg chain is dual to the special reduced k + 2-points gl N Gaudin model. Moreover, we construct an explicit Poisson map between the models at the classical level by performing the Dirac reduction procedure and applying the AHH duality transformation.

In this paper we clarify the relationship between inhomogeneous quantum spin chains and classical integrable many-body systems. It provides an alternative (to the nested Bethe ansatz) method for computation of spectra of the spin chains. Namely, the spectrum of the quantum transfer matrix for the inhomogeneous gl n -invariant XXX spin chain on N sites with twisted boundary conditions can be found in terms of velocities of particles in the rational N -body Ruijsenaars-Schneider model. The possible values of the velocities are to be found from intersection points of two Lagrangian submanifolds in the phase space of the classical model. One of them is the Lagrangian hyperplane corresponding to fixed coordinates of all N particles and the other one is an N -dimensional Lagrangian submanifold obtained by fixing levels of N classical Hamiltonians in involution. The latter are determined by eigenvalues of the twist matrix. To support this picture, we give a direct proof that the eigenvalues of the Lax matrix for the classical Ruijsenaars-Schneider model, where velocities of particles are substituted by eigenvalues of the spin chain Hamiltonians, calculated through the Bethe equations, coincide with eigenvalues of the twist matrix, with certain multiplicities. We also prove a similar statement for the gl n Gaudin model with N marked points (on the quantum side) and the Calogero-Moser system with N particles (on the classical side). The realization of the results obtained in terms of branes and supersymmetric gauge theories is also discussed.

The Painlevé-Calogero correspondence is extended to auxiliary linear problems associated with Painlevé equations. The linear problems are represented in a new form which has a suggestive interpretation as a "quantized" version of the Painlevé-Calogero correspondence. Namely, the linear problem responsible for the time evolution is brought into the form of non-stationary Schrödinger equation in imaginary time, ∂ t ψ = ( 1 2 ∂ 2 x + V (x, t))ψ, whose Hamiltonian is a natural quantization of the classical Calogero-like Hamiltonian H = 1 2 p 2 + V (x, t) for the corresponding Painlevé equation.

Motivated by recent progress in the study of supersymmetric gauge theories we propose a very compact formulation of spectral duality between XXZ spin chains. The action of the quantum duality is given by the Fourier transform in the spectral parameter. We investigate the duality in various limits and, in particular, prove it for q → 1, i.e. when it reduces to the XXX/Gaudin duality. We also show that the universal difference operators are given by the normal ordering of the classical spectral curves.Integrable systems provide a key to understanding N = 2 supersymmetric gauge theories. The Seiberg-Witten theory [1] is naturally formulated in terms of integrable systems [2], and ǫ-deformation of the former corresponds to quantization of the latter. Another view on this connection is provided by the AGT correspondence [3], in which a (quantization of) Hitchin integrable system arises in a natural way.Interestingly, it turns out that for a large class of gauge theories there are two different integrable systems, associated with each of them [4]. The equivalence between the two systems is given by the spectral duality. This duality was proposed in [5,6,7] and further explored in different ways in [8, 9, 10] and [4]. On the classical level the duality exchanges the two coordinates in the equation for the spectral curve of the system. The classical Hamiltonians of the two systems, therefore, coincide. On the quantum level the situation is more subtle: the duality comes in several variants. In the weakest version, one claims that some subset of all quantum Hamiltonians is shared by the two systems [8]. In a stronger version [9], the whole Bethe subalgebras, i.e. all the Hamiltonians, of the two systems are identified. In the third version [10] one identifies the orbits of solutions to the Bethe