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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 O D $$ {\mathcal{O}}_D $$ 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 O D ¯ $$ {\mathcal{O}}_{\overline{D}} $$ of the normalization D ¯ $$ \overline{D} $$ 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 + • O D ¯ = 1 0 0 $$ {h}_{+}^{\bullet}\left({\mathcal{O}}_{\overline{D}}\right)=\left(1,0,0\right) $$ and h − • O D ¯ = 0 0 0 $$ {h}_{-}^{\bullet}\left({\mathcal{O}}_{\overline{D}}\right)=\left(0,0,0\right) $$ give a sufficient condition for a contribution. The examples that we present feature infinitely many isomorphic geometric phases, with corresponding infinite-order monodromy groups Γ. We use the action of Γ 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. |
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