Stringy Instantons at Orbifold Singularities

TL;DR

This work analyzes how D-brane (stringy) instantons at orbifold and orientifold singularities affect holomorphic quantities in four-dimensional gauge theories. By carefully tracking neutral and charged zero-modes, the authors show that extra fermionic zero-modes can suppress exotic instanton contributions, but orientifold projections can lift these modes and unleash new superpotential terms. They demonstrate ADS-type superpotentials in a Z2×Z2 orbifold, vanishings in exotic configurations without orientifold lifting, and explicit exotic contributions to the prepotential in a Z3 orientifold with N=2 dynamics. The findings illuminate when stringy instantons modify low-energy dynamics and offer mechanisms for moduli stabilization and non-perturbative structure in string phenomenology.

Abstract

We study the effects produced by D-brane instantons on the holomorphic quantities of a D-brane gauge theory at an orbifold singularity. These effects are not limited to reproducing the well known contributions of the gauge theory instantons but also generate extra terms in the superpotential or the prepotential. On these brane instantons there are some neutral fermionic zero-modes in addition to the ones expected from broken supertranslations. They are crucial in correctly reproducing effects which are dual to gauge theory instantons, but they may make some other interesting contributions vanish. We analyze how orientifold projections can remove these zero-modes and thus allow for new superpotential terms. These terms contribute to the dynamics of the effective gauge theory, for instance in the stabilization of runaway directions.

Figures (6)

• Figure 1: Quiver diagram for the $\mathbf{Z}_2 \times \mathbf{Z}_2$ orbifold theory. Round circles correspond to $\mathrm{SU}(N_\ell)$ gauge factors while the lines connecting quiver nodes represent the bi-fundamental chiral superfields $\Phi_{\ell m}$.
• Figure 2: Quiver diagram describing an ordinary instanton in a $\mathrm{SU}(N_c) \times \mathrm{SU}(N_f)$ theory. Gauge theory nodes are represented by round circles, instanton nodes by squares. The ED1 is wrapped on the same cycle as the color branes. All zero-modes are included except the $\theta$'s and the $x^\mu$'s, which only contribute to the measure for the integral over chiral superspace.
• Figure 3: Quiver diagram describing an exotic instanton in a $\mathrm{SU}(N_c) \times \mathrm{SU}(N_f)$ theory. Gauge theory nodes are represented by round circles, instanton nodes by squares. The ED1 is wrapped on a different cycle with respect to both sets of quiver branes.
• Figure 4: The generalized $\mathbf{Z}_2 \times \mathbf{Z}_2$ orientifold quiver and the exotic instanton contribution.
• Figure 5: The $\mathbf{Z}_3$ (un-orientifolded) theory. The lines with both ends on a single node represent adjoint chiral multiplets which, together with the vector multiplets at each node constitute the $\mathcal{N} = 2$ vector multiplets. Similarly, lines between nodes represent chiral multiplets which pair up into hyper-multiplets, in $\mathcal{N} =2$ language.