Superpotentials from Singular Divisors

Abstract

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. Analytically continuing the superpotential through the flops, we find that singular divisors whose normalizations are rigid can contribute to the superpotential: specifically, $h_{+}^{\bullet }({\mathcal{O}}_{\overline{D}})=(1,0,0)$ and $h_{-}^{\bullet }({\mathcal{O}}_{\overline{D}})=(0,0,0)$ give a sufficient condition for a contribution. The examples that we present feature infinitely many isomorphic geometric phases, with corresponding infinite-order monodromy groups $\Gamma$. We use the action of $\Gamma$ on effective divisors to determine the exact effective cones, which have infinitely many generators. The resulting nonperturbative superpotentials are Jacobi theta functions, whose modular symmetries suggest the existence of strong-weak coupling dualities involving inversion of divisor volumes.

31 pages, 4 figures

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Cited by in corpus (3)

• Moduli Space Reconstruction and Weak Gravity
• Symmetries of Calabi-Yau Prepotentials with Isomorphic Flops
• Symmetry Points of $\mathcal{N}=1$ Modular Geometry