Critical Points and Supersymmetric Vacua, III: String/M Models
Abstract
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 CalabiYau 3fold X with flux. In particular, complete proofs of the counting formulas in AshokDouglas [AD] and DenefDouglas [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 WeilPetersson 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].
 Publication:

Communications in Mathematical Physics
 Pub Date:
 August 2006
 DOI:
 10.1007/s0022000600037
 arXiv:
 arXiv:mathph/0506015
 Bibcode:
 2006CMaPh.265..617D
 Keywords:

 Mathematical Physics;
 High Energy Physics  Theory;
 Mathematics  Algebraic Geometry;
 Mathematics  Complex Variables
 EPrint:
 Final revision for publication in Commun. Math. Phys. Minor corrections and editorial changes