Isomonodromic Tau-Functions from Liouville Conformal Blocks

By N. Iorgov, O. Lisovyy, J. Teschner. 2014

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The goal of this note is to show that the Riemann–Hilbert problem to find multivalued analytic functions with $${{\rm SL}(2,\mathbb{C})}$$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 $${\mathcal{N}=2}$$N=2 supersymmetric gauge theories and conformal field theory.

Subfields: Geometry and Topology, Mathematical Physics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/a781999d1d22e5a94b6904e80cb94140a6d465a1