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The geometry of Calogero-Moser systems. (English) Zbl 1135.37023

Summary: We give a geometric construction of the phase space of the elliptic Calogero-Moser system for arbitrary root systems, as a space of Weyl invariant pairs (bundles, Higgs fields) on the \(r\)-th power of the elliptic curve, where \(r\) is the rank of the root system. The Poisson structure and the Hamiltonians of the integrable system are given natural constructions. We also exhibit a curious duality between the spectral varieties for the system associated to a root system, and the Lagrangian varieties for the integrable system associated to the dual root system. Finally, the construction is shown to reduce to an existing one for the \(A_n\) root system.

MSC:

37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory)
70G55 Algebraic geometry methods for problems in mechanics
70H06 Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics
81R12 Groups and algebras in quantum theory and relations with integrable systems
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