Critical Points and Supersymmetric Vacua, III: String/M Models |
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| Authors: | Michael R Douglas Bernard Shiffman Steve Zelditch |
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| Institution: | (1) NHETC and Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08855–0849, USA;(2) I.H.E.S., Bures-sur-Yvette, France;(3) Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218, USA |
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| 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 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
 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
 . We show that the density of critical points in
 for this lattice of sections is well approximated by Gaussian measures of the kind studied in DSZ1,DSZ2,AD,DD1].Research partially supported by DOE grant DE-FG02-96ER40959 (first author) and NSF grants DMS-0100474 (second author) and DMS-0302518 (third author). |
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