Isomonodromic τ-functions and WN conformal blocks

We study the solution of the Schlesinger system for the 4-point sl(N) isomonodromy problem and conjecture an expression for the isomonodromic tau-function in terms of 2d conformal field theory beyond the known N = 2 Painleve VI case. We show that this relation can be used as an alternative definition of conformal blocks for the W-N algebra and argue that the infinite number of arbitrary constants arising in the algebraic construction of W-N conformal block can be expressed in terms of only a finite set of parameters of the monodromy data of rank N Fuchsian system with three regular singular points. We check this definition explicitly for the known conformal blocks of the W-3 algebra and demonstrate its consistency with the conjectured form of the structure constants.