Characteristic Classes and Hitchin Systems. General Construction

We consider topologically non-trivial Higgs G-bundles over Riemann surfaces ?? _g with marked points and the corresponding Hitchin systems. We show that if G is not simply-connected, then there exists a finite number of different sectors of the Higgs bundles endowed with the Hitchin Hamiltonians. They correspond to different characteristic classes of the underlying bundles defined as elements of ${H^{2}(\Sigma_g, \mathcal{Z}(G))}$ , ( ${\mathcal{Z}(G)}$ is a center of G). We define the conformal version CG of G - an analog of GL(N) for SL(N), and relate the characteristic classes with degrees of CG-bundles. We describe explicitly bundles in the genus one (g =? 1) case. If ??₁ has one marked point and the bundles are trivial then the Hitchin systems coincide with Calogero-Moser (CM) systems. For the nontrivial bundles we call the corresponding systems the modified Calogero-Moser (MCM) systems. Their phase space has the same dimension as the phase space of the CM systems with spin variables, but less number of particles and greater number of spin variables. Starting with these bundles we construct Lax operators, quadratic Hamiltonians, and define the phase spaces and the Poisson structure using dynamical r-matrices. The latter are completion of the classification list of Etingof-Varchenko corresponding to the trivial bundles. 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 subsystems related to some (unbroken) subalgebras. The configuration space of the CM particles is the moduli space of the stable holomorphic bundles with non-trivial characteristic classes.