Isomonodromic τ-functions and W N conformal blocks

Abstract

Abstract: We study the solution of the Schlesinger system for the 4-point slN isomonodromy problem and conjecture an expression for the isomonodromic τ-function in terms of 2d conformal field theory beyond the known N = 2 Painlevé VI case. We show that this relation can be used as an alternative definition of conformal blocks for the WN algebra and argue that the infinite number of arbitrary constants arising in the algebraic construction of WN 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 W3 algebra and demonstrate its consistency with the conjectured form of the structure constants.