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Superpotentials from singular divisors

Apr 13, 2022
38 pages
Published in:
  • JHEP 11 (2022) 142
  • Published: Nov 24, 2022
e-Print:
DOI:
Report number:
  • MIT-CTP/5388
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Abstract: (Springer)
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 OD {\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 OD‾ {\mathcal{O}}_{\overline{D}} of the normalizationD‾ \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+∙(OD‾)=(1,0,0) {h}_{+}^{\bullet}\left({\mathcal{O}}_{\overline{D}}\right)=\left(1,0,0\right) and h−∙(OD‾)=(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.
Note:
  • 31 pages, 4 figures
  • D-Branes
  • Flux Compactifications
  • String and Brane Phenomenology
  • Superstring Vacua
  • fermion: zero mode
  • orientifold: Calabi-Yau
  • phase: geometrical
  • superpotential: nonperturbative
  • compactification: orientifold
  • cohomology
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