To understand in detail the contribution of a world-sheet instanton to the
superpotential in a heterotic string compactification, one has to understand
the moduli dependence (bundle and complex structure moduli) of the one-loop
determinants from the fluctuations, which accompany the classical exponential
contribution (involving K\"ahler moduli) when evaluating the world-volume
partition function. Here we use techniques to describe geometrically these
Pfaffians for spectral bundles over rational base curves in elliptically
fibered Calabi-Yau threefolds, and provide a (partially exhaustive) list of
cases involving {\em factorising} (or vanishing) superpotential. This gives a
conceptual explanation and generalisation of the few previously known cases
which were obtained just experimentally by a numerical computation.Comment: 57 pages; minor changes, discussion section 1.3 adde