Nonautonomous Hamiltonian systems related to higher Hitchin integrals

We describe nonautonomous Hamiltonian systems derived from the Hitchin integrable systems. The Hitchin integrals of motion depend on W-structures of the basic curve. The parameters of the W-structures play the role of times. In particular, the quadratic integrals depend on the complex structure (the W-2-structure) of the basic curve, and the times are coordinates in the Teichmuller space. The corresponding flows are the monodromy-preserving equations such as the Schlesinger equations, the Painleve VI Equation, and their generalizations. The equations corresponding to the higher integrals are the monodromy-preserving conditions with respect to changing the W-k-structures (k > 2). They are derived by the symplectic reduction of a gauge field theory on the basic curve interacting with the W-k-gravity. As a by-product, we obtain the classical Ward identities in this theory.