Characteristic Classes and Integrable Systems. General Construction

Abstract

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 CG-bundles. 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.

52 pages, 8 tables

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Cited by in corpus (12)

• Relativistic Classical Integrable Tops and Quantum R-matrices
• Spectrum of Quantum Transfer Matrices via Classical Many-Body Systems
• Characteristic Classes of SL(N)-Bundles and Quantum Dynamical Elliptic R-Matrices
• Noncommutative extensions of elliptic integrable Euler-Arnold tops and Painleve VI equation
• Three-particle Integrable Systems with Elliptic Dependence on Momenta and Theta Function Identities
• Hecke Transformations of Conformal Blocks in WZW Theory. I. KZB Equations for Non-Trivial Bundles
• On R-matrix valued Lax pairs for Calogero-Moser models
• Field theory generalizations of two-body Calogero-Moser models in the form of Landau-Lifshitz equations
• GL(NM) quantum dynamical $R$-matrix based on solution of the associative Yang-Baxter equation
• Monopole solutions to the Bogomolny equation as three-dimensional generalizations of the Kronecker series
• Geometry of Higgs bundles over elliptic curves related to automorphisms of simple Lie algebras, Calogero-Moser systems and KZB equations
• Multi-pole extension for elliptic models of interacting integrable tops