Painlevé functions and conformal blocks
The authors discuss recent developments in explicit combinatorial series representations of the tau-functions for the classical sixth and third Painlevé equations coming from a plausible connection of the isomonodromy systems with the 2D Conformal Field Theories. The authors' idea is to identify the isomonodromic tau-function and a four-point chiral correlation function of the monodromy fields (Virasoro primaries) associated with the regular singular points of the linear system. The expansion of the tau-function of the authors' interest is the expansion of the four-point correlator in terms of the three-point functions and certain universal functions called ``conformal blocks'' since they do not depend on a specific CFT model. Using the AGT correspondence, the authors find the coefficients of the series explicitly. For the PVI case, the initial terms of the expansion for arbitrary parameter values are verified analytically. Numerical experiments with random parameter values show that the critical expansions at different branch points glue together to a smooth global tau-function. Also the authors show that such conformal expansions allow them to solve the notoriously difficult connection problem for the tau-function between \(0\) and \(1\). The authors also discuss the irregular limit in the conformal expansion leading to the series representation of the tau-function of PIII.
scientific article; zbMATH DE number 6385016
scientific article