  We investigate differential geometric aspects of moduli spaces parametrizing
solutions of coupled vortex equations over a compact Kaehler manifold X. These
solutions are known to be related to polystable triples via a Kobayashi-Hitchin
type correspondence. Using a characterization of infinitesimal deformations in
terms of the cohomology of a certain elliptic double complex, we construct a
Hermitian structure on these moduli spaces. This Hermitian structure is proved
to be Kaehler. The proof involves establishing a fiber integral formula for the
Hermitian form. We compute the curvature tensor of this Kaehler form. When X is
a Riemann surface, the holomorphic bisectional curvature turns out to be
semi--positive. It is shown that in the case where X is a smooth complex
projective variety, the Kaehler form is the Chern form of a Quillen metric on a
certain determinant line bundle.
