Non-perturbative F-terms across lines of BPS stability

TL;DR

This paper analyzes how non-perturbative F-terms from D-brane instantons behave as moduli are varied and instantons cross lines of BPS stability. It demonstrates that BPS instantons contributing to the superpotential cannot become non-BPS; only lines of threshold stability allow splitting into mutually BPS constituents. Instantons with extra zero modes that generate higher F-terms can cross genuine lines of marginal stability, becoming non-BPS and generating D-terms locally, but globally their effect is captured by Beasley-Witten cohomology which preserves holomorphy of the 4d action. The authors also connect zero-mode lifting and 4d SUSY breaking, and illustrate the framework through explicit brane constructions of SQCD-like systems, highlighting the global structure of non-perturbative effects in string compactifications.

Abstract

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.

Figures (3)

• Figure 1: A rigid isolated $U(1)$ instanton (a) on the BPS locus (with respect to the ${\cal N}=1$ supersymmetry preferred by the orientifold plane, shown as a dashed line), and (b) when non-BPS due to a misalignment of its BPS phase. As usual, black and white dots denote the degenerations of the double $\bf C^*$ fibration in the geometries discussed in Appendix \ref{['oovafa']}.
• Figure 2: D-brane realization of the $N_f=N_c$ SQCD theory in a geometry of the kind in Appendix \ref{['oovafa']}. The construction shows that instanton on its BPS locus (a) and away from it (b). Notice that gauge D-branes recombine and remain BPS through the process.
• Figure 3: D-brane realization of the $N_f=N_c-1$ SQCD theory in a geometry of the kind in Appendix \ref{['oovafa']}. The construction shows that instanton on its BPS locus (a) and away from it (b). The gauge D-branes cannot all recombine and define a non-supersymmetric background for the instanton.