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 ) -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 supersymmetric gauge theories and conformal field theory.