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Characteristic classes and Hitchin systems. General construction. (English) Zbl 1252.30029

Summary: We consider topologically non-trivial Higgs \(G\)-bundles over Riemann surfaces \(\Sigma _{g }\) 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 \(H^{2}\big(\Sigma_g, \mathcal{Z}(G)\big)\), (\({\mathcal{Z}(G)}\) is the center of \(G\)). We define the conformal version \(CG\) of \(G\) – an analog of \(\mathrm{GL}(N)\) for \(\mathrm{SL}(N)\), and relate the characteristic classes with degrees of \(CG\)-bundles. We describe explicitly bundles in the genus one case (\(g = 1\)). If \(\Sigma_{1}\) has one marked point and the bundles are trivial, then the Hitchin systems coincide with Calogero-Moser (CM) systems. For nontrivial bundles we call the corresponding systems modified Calogero-Moser (MCM) systems. Their phase space has the same dimension as the phase space of the CM systems with spin variables, but a smaller number of particles and a 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 the 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 \[ \mathrm{sl}(N) \] . 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.

MSC:

30F10 Compact Riemann surfaces and uniformization
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
32L05 Holomorphic bundles and generalizations
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