Non-perturbative superpotentials across lines of marginal stability

TL;DR

The paper demonstrates that non-perturbative superpotentials in 4d ${ m N}=1$ Type II compactifications stay holomorphically continuous across lines of marginal stability, despite jumps in the instanton spectrum. It reveals that multi-instanton processes can reconstruct the single-instanton contribution after decay by saturating extra fermion zero modes through intra-instanton interactions, and that $U(1)$ instantons can contribute via non-perturbative lifting mechanisms. The authors provide explicit examples for both non-gauge and gauge D-brane instantons, including Seiberg-duality-related transitions and exotic-to-gauge instanton changes, and connect these phenomena to F-theory topology changes. The results imply a richer and more subtle instanton calculus in string compactifications, with potential implications for moduli stabilization and phenomenology.

Abstract

We discuss the behaviour of non-perturbative superpotentials in 4d N=1 type II compactifications (and orientifolds thereof) near lines of marginal stability, where the spectrum of contributing BPS D-brane instantons changes discontinuously. The superpotential is nevertheless continuous, in agreement with its holomorphic dependence on the closed string moduli. The microscopic mechanism ensuring this continuity involves novel contributions to the superpotential: As an instanton becomes unstable against decay to several instantons, the latter provide a multi-instanton contribution which reconstructs that of the single-instanton before decay. The process can be understood as a non-perturbative lifting of additional fermion zero modes of an instanton by interactions induced by other instantons. These effects provide mechanisms via which instantons with U(1) symmetry can contribute to the superpotential. We provide explicit examples of these effects for non-gauge D-brane instantons, and for D-brane gauge instantons (where the motions in moduli space can be interpreted as Higgsing, or Seiberg dualities).

Figures (10)

• Figure 1: Example of an $O(1)$ instanton $A$ (figure a) splitting into an $O(1)$ instanton $B$ and a $U(1)$ instanton $C$ and its image $C'$ (figure b).
• Figure 2: Schematic picture of a multi-instanton configuration contributing to the superpotential. A number of additional fermion zero modes are saturated against each other, due to interaction terms in the world-volume effective action of the 2-instanton system. The two left-over fermion zero modes are the Goldstinos of the overall BPS D-brane instanton system, and are saturated against the $d^2\theta$ integration in the induced 4d effective action superpotential term.
• Figure 3: Configuration of an $O(1)$ instanton splitting as a $U(1)$ instanton (and its orientifold image). Interpreted as a HW setup, the dots $b$, $a_{\pm \theta}$, denote the locations in the 67 plane for an unrotated NS-brane, and NS5-branes rotated by angles $\pm \theta$ in the 4589 directions. Interpreted as D1-brane instantons in a threefold geometry, the dots $a_{\pm \theta}$, $b$ denote a projection of the degenerations loci of a $\bf C^*$ fiber. D1-brane instantons wrap 2-cycles obtained by fibering the latter over segments defined by such degenerations, and are supersymmetric when the segments lie horizontally.
• Figure 4: a) Marginally stable configuration. b) Moving $a_1$ away from the horizontal axis renders the configuration nonsupersymmetric, so it can c) decay to a supersymmetric configuration by brane recombination.
• Figure 5: Schematic picture of the multi-instanton configuration discussed in the text.