CFT approach to the $q$-Painlevé VI equation

Abstract: Iorgov, Lisovyy, and Teschner established a connection between isomonodromic deformation of linear differential equations and Liouville conformal field theory at $c=1$. In this paper we present a $q$ analog of their construction. We show that the general solution of the $q$-Painlev\'e VI equation is a ratio of four tau functions, each of which is given by a combinatorial series arising in the AGT correspondence. We also propose conjectural bilinear equations for the tau functions.