  This paper presents a new approach to constructing a meromorphic bundle map
between flat vector bundles over a compact Riemann surface having a prescribed
Weil divisor (i.e., having prescribed zeros and poles with directional as well
as multiplicity information included in the vector case). This new formalism
unifies the earlier approach of Ball-Clancey (in the setting of trivial bundles
over an abstract Riemann surface) with an earlier approach of the authors
(where the Riemann surface was assumed to be the normalizing Riemann surface
for an algebraic curve embedded in ${\bold C}^2$ with determinantal
representation, and the vector bundles were assumed to be presented as the
kernels of linear matrix pencils). The main tool is a version of the Cauchy
kernel appropriate for flat vector bundles over the Riemann surface. Our
formula for the interpolating bundle map (in the special case of a single zero
and a single pole) can be viewed as a generalization of the Fay trisecant
identity from the usual line bundle case to the vector bundle case in terms of
Cauchy kernels. In particular we obtain a new proof of the Fay trisecant
identity.
