Manchester Geometry Seminar 2006/2007

16 November 2006. Room M12, Mathematics and Social Sciences building, Sackville Street. 4pm

Solving Painlevé Connection Problems Using Two-Dimensional Integrable Quantum Field Theory

Benjamin Doyon (University of Oxford)

The Painlevé equations are related to two-point correlation functions of certain "interacting" spinless scaling fields in free fermionic models of 2-dimensional quantum field theory (QFT). This relation leads to non-trivial predictions for the solutions to some of the connection problems associated to Painlevé equations. Indeed, short-distance and large-distance expansions can be obtained in QFT from conformal perturbation theory and form factors, respectively. These expansions are unambiguous once the normalisations of the fields have been fixed, and fully calculable. In turn, they give expansions, including the normalisation, for Painlevé transcendents near some critical points, as well as the relative normalisation of the associated tau-functions near these critical points. As an example, I will explain how this works in the Dirac theory on the Poincaré disk, giving in particular predictions concerning connection problems in certain degenerate cases of Painlevé VI that are excluded from the general formula of M. Jimbo of 1982.