ar X iv : m at h - ph / 0 50 60 15 v 2 9 J un 2 00 5 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 L → C over the moduli space of complex structures on X × T 2 with respect to the Weil-Petersson connection. Flux superpotentials form a lattice SZ of full rank in a 2b3(X)-dimensional real subspace S ⊂ H0(C,L). We show that the density of critical points in C for this lattice of sections is well approximated by Gaussian measures of the kind studied in [DSZ1, DSZ2, AD, DD1].