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## Isomonodromic $\tau$-functions and $W_N$ conformal blocks

We study the solution of the Schlesinger system for the 4-point $\mathfrak{sl}_N$ 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 *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.