Pad\'e\ approximants on Riemann surfaces and KP tau functions

Abstract

The paper has two relatively distinct but connected goals; the first is to define the notion of Pad\'e\ approximation of Weyl-Stiltjes transforms on an arbitrary compact Riemann surface of higher genus. The data consists of a contour in the Riemann surface and a measure on it, together with the additional datum of a local coordinate near a point and a divisor of degree $g$. The denominators of the resulting Pad\'e--like approximation also satisfy an orthogonality relation and are sections of appropriate line bundles. A Riemann--Hilbert problem for a square matrix of rank two is shown to characterize these orthogonal sections, in a similar fashion to the ordinary orthogonal polynomial case. The second part extends this idea to explore its connection to integrable systems. The same data can be used to define a pairing between two sequences of line bundles. The locus in the deformation space where the pairing becomes degenerate for fixed degree coincides with the zeros of a "tau" function. We show how this tau function satisfies the Kadomtsev--Petviashvili hierarchy with respect to either deformation parameters, and a certain modification of the 2--Toda hierarchy when considering the whole sequence of tau functions. We also show how this construction is related to the Krichever construction of algebro--geometric solutions.