Isomonodromic Tau-Functions from Liouville Conformal Blocks

The goal of this note is to show that the Riemann–Hilbert problem to find multivalued analytic functions with SL(2,C)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm SL}(2,\mathbb{C})}$$\end{document}-valued monodromy on Riemann surfaces of genus zero with n punctures can be solved by taking suitable linear combinations of the conformal blocks of Liouville theory at c = 1. This implies a similar representation for the isomonodromic tau-function. In the case n = 4 we thereby get a proof of the relation between tau-functions and conformal blocks discovered in Gamayun et al. (J High Energy Phys, 10:038, 2012). We briefly discuss a possible application of our results to the study of relations between certain N=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{N}=2}$$\end{document} supersymmetric gauge theories and conformal field theory.