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An Isaac Newton Institute Workshop

Painleve Equations and Monodromy Problems: Recent Developments

Solving Painleve connection problems using 2-dimensional integrable quantum field theory

20th September 2006

Author: Doyon, B (Oxford)


The Painlev\'e 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 Painlev\'e equations. Indeed, short-distance and large-distance expansions can be obtained in QFT from conformalperturbation theory and form factors, respectively. These expansions areunambiguous once the normalisations of the fields have been fixed, and fully calculable. In turn, they give expansions, including the normalisation, for Painlev\'e 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 Poincar\'e disk, giving in particular predictions concerning connection problems in certain degenerate cases of Painlev\'e VI that are excluded from the general formula of M. Jimbo of 1982.