Mathematics

Using arguments from Conformal Field Theory, Gamayun Iorgov and Lisovyy provided in 2012 an explicit expression for the tau function of the sixth Painlevé equation as a Fourier transform of Virasoro conformal blocks. This "Kiev formula" has been later generalized to more general isomonodromic problems on the sphere, with both regular and irregular punctures. Further, by using the AGT correspondence, one can see that the isomonodromic tau function is the dual partition function of an appropriate class S theory, for which the Painlevé equations (and generalizations thereof) can be thought of as renormalization group or, for the conformal case, conformal manifold deformation equations.
In this talk we will first recap the role of Hitchin systems in the context of class S theories, and what the isomonodromy deformations mean from this perspective. Then, we will show how one can generalize the connection between gauge theory and isomonodromic deformations to the genus one case, by using an approach based on free fermions. This will provide a formula connecting the tau function to the dual partition function of a circular quiver gauge theory with both bifundamental and adjoint hypermultiplets, which has new features absent in the genus zero case