Isomonodromic tau-functions from Liouville conformal blocks

Iorgov, N.; Lisovyy, O.; Teschner, J.

Preprint/Internal Report

DESY-2014-02606

Isomonodromic tau-functions from Liouville conformal blocks

Iorgov, N. (Corresponding Author) ; Lisovyy, O. ; Teschner, J.DESY*

2014

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Report No.: DESY-14-012; arXiv:1401.6104

Abstract: The goal of this note is to show that the Riemann-Hilbert problem to find multivalued analytic functions with $SL(2,\mathbb{C})$-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 \cite{GIL}. We briefly discuss a possible application of our results to the study of relations between certain $\mathcal{N}=2$ supersymmetric gauge theories and conformal field theory.

Keyword(s): gauge field theory: supersymmetry ; field theory: Liouville ; field theory: conformal ; conformal block ; tau-function ; Riemann surface ; monodromy