Non-perturbative F-terms across lines of BPS stability

We consider non-perturbative terms in the 4d effective action due to BPS D-brane instantons, and study their continuity properties in moduli space as instantons cross lines of BPS stability, potentially becoming non-BPS. We argue that BPS instantons contributing to the superpotential cannot become non-BPS anywhere in moduli space, since they cannot account for the required four goldstino fermion zero modes. At most they can reach lines of threshold stability, where they split into mutually BPS multi-instantons, as already discussed in the literature. On the other hand, instantons with additional fermion zero modes, contributing to multi-fermion F-terms, can indeed cross genuine lines of marginal stability, beyond which they lead to non-BPS systems. The non-BPS instanton generates an operator which is a D-term locally in moduli space, but not globally. This is due to a cohomological obstruction localized on the BPS locus, where the D-term must be written as an F-term, thus ensuring the continuity of the 4d contribution to the effective action. We also point out an interesting relation between lifting of fermion zero modes on instantons and 4d supersymmetry breaking.