628 research outputs found
Three lectures on classical integrable systems and gauge field theories
In these lectures I consider the Hitchin integrable systems and their
relations with the self-duality equations and the twisted super-symmetric
Yang-Mills theory in four dimension follow Hitchin and Kapustin-Witten. I
define the Symplectic Hecke correspondence between different integrable
systems. As an example I consider Elliptic Calogero-Moser system and integrable
Euler-Arnold top on coadjoint orbits of the group GL(N,C) and explain the
Symplectic Hecke correspondence for these systems.Comment: 36 pages, Lectures given at Advanced Summer School on Integrable
Systems and Quantum Symmetries (Prague, June, 2007
Spin Calogero models associated with Riemannian symmetric spaces of negative curvature
The Hamiltonian symmetry reduction of the geodesics system on a symmetric
space of negative curvature by the maximal compact subgroup of the isometry
group is investigated at an arbitrary value of the momentum map. Restricting to
regular elements in the configuration space, the reduction generically yields a
spin Calogero model with hyperbolic interaction potentials defined by the root
system of the symmetric space. These models come equipped with Lax pairs and
many constants of motion, and can be integrated by the projection method. The
special values of the momentum map leading to spinless Calogero models are
classified under some conditions, explaining why the models with two
independent coupling constants are associated with as found by Olshanetsky and Perelomov. In the zero curvature limit our
models reproduce rational spin Calogero models studied previously and similar
models correspond to other (affine) symmetric spaces, too. The construction
works at the quantized level as well.Comment: 26 pages, v3: final version with a remark added after equation (5.3
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