Higher rank isomonodromic deformations and W-algebras

TL;DR

This paper links higher rank isomonodromic deformations to W_N-algebra conformal blocks, providing explicit solutions and tau functions for Fuchsian systems of rank N, advancing the understanding of their algebraic and analytical structures.

Contribution

It constructs solutions to rank N Fuchsian systems using semi-degenerate conformal blocks of W_N-algebra, generalizing previous vertex operator matrix element results.

Findings

Explicit solutions for Fuchsian systems of rank N

Representation of tau functions via conformal blocks

Generalization of vertex operator matrix element results

Abstract

We construct the general solution of a class of Fuchsian systems of rank $N$ as well as the associated isomonodromic tau functions in terms of semi-degenerate conformal blocks of $W_N$-algebra with central charge $c=N-1$. The simplest example is given by the tau function of the Fuji-Suzuki-Tsuda system, expressed as a Fourier transform of the 4-point conformal block with respect to intermediate weight. Along the way, we generalize the result of Bowcock and Watts on the minimal set of matrix elements of vertex operators of the $W_N$-algebra for generic central charge and prove several properties of semi-degenerate vertex operators and conformal blocks for $c=N-1$.