An index for the Dirac operator on D3 branes with background fluxes

TL;DR

This work provides a direct IIB framework to assess instanton-generated superpotentials from Euclidean D3 branes in Calabi–Yau orientifolds with background fluxes by formulating and solving the Dirac equation on the D3 worldvolume. Central to the approach is the flux-adjusted index $\chi_{D3}=\tfrac{1}{2}(N_+-N_-)$, which must equal 1 for a nonperturbative superpotential to arise; fluxes can lift extra zero modes and alter the count, enabling new stabilization channels. The authors compute $\chi_{D3}$ in several concrete geometries (e.g., $K3\times T^2/\mathbb{Z}_2$, D3 on Fano bases, and $T^6/\mathbb{Z}_2$) and consistently find results in line with M-theory and F-theory duals, thereby validating the method. The findings have significant implications for moduli stabilization, de Sitter model-building, and inflationary scenarios in string compactifications, by clarifying when instanton effects can contribute to the 4d superpotential. Overall, the paper delivers a robust criterion for flux-enabled D3-instanton corrections and demonstrates its reliability across diverse compactifications.

Abstract

We study the problem of instanton generated superpotentials in Calabi-Yau orientifold compactifications directly in type IIB string theory. To this end, we derive the Dirac equation on a Euclidean D3 brane in the presence of background fluxes. We propose an index which governs whether the generation of a superpotential in the effective 4d theory by D3 brane instantons is possible. Applying the formalism to various classes of examples, including the K3 x T^2/Z_2 orientifold, in the absence and presence of fluxes, we show that our results are consistent with conclusions attainable via duality from an M-theory analysis.