Marta Mazzocco (University of Manchester)
marta@maths.manchester.ac.uk
The Schlesinger equations describe moduli spaces of flat logarithmic connections on rank m vector bundles on CP1. These moduli spaces possess a Poisson structure. In this talk we study symplectic reductions under the action of the Lie group GL(m).
Symplectic reductions of the Schlesinger equations play an important role in the famous problem of characterizing higher order analogues of the celebrated Painlevé equations. These are second order ODEs in complex variable such that their solutions satisfy two properties: 1) they can be analytically continued to meromorphic functions on the Riemann sphere with at most 3 punctures; 2) They define some new special functions, the so-called Painlevé transcendents. The most general of the Painlevé equations is the Painlevé sixth equation.
The Schlesinger equations reduce to the Painlevé sixth equation when m=2 and the number of logarithmic singularities of the flat connection is 4. The question that will be answered in this talk is: what do the general Schlesinger equations reduce to? Are these reductions some higher order analogues of the sixth Painlevé equation?