Mathematics

The sixth Painlevé equation describes monodromy preserving deformations of rank 2 Fuchsian systems with four regular singular points on the Riemann sphere. The monodromy data play a role of integrals of motion for Painlevé VI, and its solution amounts to constructing explicit inverse of the Riemann-Hilbert map. We will show that this problem can be solved by expressing Painlevé VI tau function and the solution of the auxiliary linear system in terms of Fourier transforms of Virasoro conformal blocks. We will also discuss various generalizations of this isomonodromy/CFT correspondence.