Perspectives on Pfaffians of heterotic world-sheet instanton

To fix the bundle moduli of a heterotic compactification one has to understand the Pfaffian one-loop prefactor of the classical instanton contribution. For compactifications on elliptically fibered Calabi-Yau spaces X this can be made explicit for spectral bundles and world-sheet instantons supported on rational base curves b: one can express the Pfaffian in a closed algebraic form as a polynomial, or it may be understood as a theta-function expression. We elucidate the connection between these two points of view via the respective perception of the relevant spectral curve, related to its extrinsic geometry in the ambient space (the elliptic surface in X over b) or to its intrinsic geometry as abstract Riemann surface. We identify, within a conceptual description, general vanishing loci of the Pfaffian, and derive bounds on the vanishing order, relevant to solutions of the equations W = dW = 0 for the superpotential.