Mathematical Laboratory and Theoretical Physics

UniversitÃ© de Tours

UniversitÃ© de Tours

I will start with an introduction to Painlevé equations and explain how their general solutions can be constructed in terms of conformal blocks of the Virasoro algebra. The 2d-4d correspondence of Alday-Gaiotto-Tachikawa relates such conformal blocks to partition functions of certain supersymmetric gauge theories and ultimately leads to explicit representations of Painlevé transcendents in the form of combinatorial series over pairs of Young diagrams. I am going to explain how these series representations can be found directly, without reference to CFT and gauge theory, by associating a tau function to "any" Riemann-Hilbert problem on a circle.