Non-autonomous Hamiltonian systems related to highest Hitchin integrals

TL;DR

This paper explores non-autonomous Hamiltonian systems derived from Hitchin integrable systems, linking W-structures of curves to monodromy preserving equations like Painleve VI, through symplectic reduction and gauge theory interactions.

Contribution

It introduces a new class of monodromy preserving equations associated with higher W_k-structures, extending classical integrable systems and gauge theory connections.

Findings

Derived monodromy preserving equations for higher W_k-structures.

Connected Hitchin integrals with Teichmuller space parameters.

Obtained classical Ward identities from gauge theory interactions.

Abstract

We describe non-autonomous Hamiltonian systems coming from the Hitchin integrable systems. The Hitchin integrals of motion depend on the W-structures of the basic curve. The parameters of the W-structures play the role of times. In particular, the quadratic integrals dependent on the complex structure (W_2-structure) of the basic curve and times are coordinate on 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 highest integrals are monodromy preserving conditions with respect to changing of the W_k-structures (k>2). They are derived by the symplectic reduction from the gauge field theory on the basic curve interacting with W_k-gravity. As by product we obtain the classical Ward identities in this theory.