Critical Points and Supersymmetric Vacua, III: String/M Models

A fundamental problem in contemporary string/M theory is to count the number of inequivalent vacua satisfying constraints in a string theory model. This article contains the first rigorous results on the number and distribution of supersymmetric vacua of type IIb string theories compactified on a Calabi-Yau 3-fold X with flux. In particular, complete proofs of the counting formulas in Ashok-Douglas [AD] and Denef-Douglas [DD1] are given, together with van der Corput style remainder estimates. Supersymmetric vacua are critical points of certain holomorphic sections (flux superpotentials) of a line bundle mathcal{L} to mathcal{C} over the moduli space of complex structures on X × T ² with respect to the Weil-Petersson connection. Flux superpotentials form a lattice of full rank in a 2 b ₃( X)-dimensional real subspace mathcal{S} subset H^0(mathcal{C}, mathcal{L}). We show that the density of critical points in mathcal{C} for this lattice of sections is well approximated by Gaussian measures of the kind studied in [DSZ1,DSZ2,AD,DD1].