TY - JOUR

T1 - Critical points and supersymmetric vacua, III

T2 - String/M models

AU - Douglas, Michael R.

AU - Shiffman, Bernard

AU - Zelditch, Steve

N1 - Copyright:
Copyright 2006 Elsevier B.V., All rights reserved.

PY - 2006/8

Y1 - 2006/8

N2 - 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 of full rank in a 2 b 3(X)-dimensional real subspace S ⊂ H 0(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].

AB - 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 of full rank in a 2 b 3(X)-dimensional real subspace S ⊂ H 0(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].

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U2 - 10.1007/s00220-006-0003-7

DO - 10.1007/s00220-006-0003-7

M3 - Article

AN - SCOPUS:33745326098

VL - 265

SP - 617

EP - 671

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 3

ER -