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

TL;DR

This paper explores the connection between isomonodromic $ au$-functions and $W_N$ conformal blocks, proposing a new way to define conformal blocks using monodromy data and verifying it for the $W_3$ case.

Contribution

It introduces a novel expression for the isomonodromic $ au$-function in terms of 2d conformal field theory, extending known results beyond the $N=2$ case and relating algebraic constants to monodromy parameters.

Findings

Proposes an alternative definition of $W_N$ conformal blocks via monodromy data.

Verifies the definition explicitly for the $W_3$ algebra.

Shows consistency with known structure constants.

Abstract

We study the solution of the Schlesinger system for the 4-point $\mathfrak{sl}_N$ isomonodromy problem and conjecture an expression for the isomonodromic $\tau$-function in terms of 2d conformal field theory beyond the known $N=2$ Painlev\'e 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.