Multivariate hypergeometric cascades, isomonodromy problems and Ward ansätze

Abstract

The Ward ansätze in twistor theory generate solutions of the anti-self-dual Yang–Mills equations from solutions of the wave equation in spacetime. The theory has a straightforward generalization in which the ‘spacetime’ is an open set in the Grassmannian Gr(2, N). The linear ‘wave equation’ in this case has special solutions, called the generalized confluent hypergeometric functions, which are equivariant under the natural action of Jordan groups on spacetime. Using the generalized Penrose–Ward transform, Ward ansätze of increasing weight arising from such hypergeometric functions give a cascade of solutions to isomonodromy problems for systems of ordinary differential equations, generally with irregular singularities. The extended construction is explored in detail, and two examples are given. In the first, solutions of the Schlesinger equations are constructed from the Lauricella FD functions; in the second, solutions of the isomonodromy problem for systems with two double poles and any number of simple poles are obtained from the generalized Bessel functions.