Isomonodromic tau-functions from Liouville conformal blocks

Understand

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.