Superpotentials from singular divisors
Naomi Gendler, Manki Kim, Liam McAllister, Jakob Moritz, Mike Stillman
Abstract
A bstract 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 $$ {\mathcal{O}}_D $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>O</mml:mi> <mml:mi>D</mml:mi> </mml:msub> </mml:math> 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 $$ {\mathcal{O}}_{\overline{D}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>O</mml:mi> <mml:mover> <mml:mi>D</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:msub> </mml:math> of the normalization $$ \overline{D} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>D</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> 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, $$ {h}_{+}^{\bullet}\left({\mathcal{O}}_{\overline{D}}\right)=\left(1,0,0\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>h</mml:mi> <mml:mo>+</mml:mo> <mml:mo>•</mml:mo> </mml:msubsup> <mml:mfenced> <mml:msub> <mml:mi>O</mml:mi> <mml:mover> <mml:mi>D</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:msub> </mml:mfenced> <mml:mo>=</mml:mo> <mml:mfenced> <mml:mn>1</mml:mn> <mml:mn>0</mml:mn> <mml:mn>0</mml:mn> </mml:mfenced> </mml:math> and $$ {h}_{-}^{\bullet}\left({\mathcal{O}}_{\overline{D}}\right)=\left(0,0,0\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>h</mml:mi> <mml:mo>−</mml:mo> <mml:mo>•</mml:mo> </mml:msubsup> <mml:mfenced> <mml:msub> <mml:mi>O</mml:mi> <mml:mover> <mml:mi>D</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:msub> </mml:mfenced> <mml:mo>=</mml:mo> <mml:mfenced> <mml:mn>0</mml:mn> <mml:mn>0</mml:mn> <mml:mn>0</mml:mn> </mml:mfenced> </mml:math> 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.