Superpotentials from Singular Divisors

Understand

We study Euclidean D3-branes wrapping divisors $D$ in Calabi-Yau orientifold compactifications of type IIB string theory.

• Witten's counting of fermion zero modes in terms of the cohomology of the structure sheaf $\mathcal{O}_D$ applies when $D$ is smooth, but we argue that effective divisors of Calabi-Yau threefolds typically have singularities along rational curves.
• We generalize the counting of fermion zero modes to such singular divisors, in terms of the cohomology of the structure sheaf $\mathcal{O}_{\overline{D}}$ of the normalization $\overline{D}$ of $D$.
• We establish this by detailing compactifications in which the singularities can be unwound by passing through flop transitions, giving a physical incarnation of the normalization process.