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Communications in Mathematical Physics

, Volume 252, Issue 1–3, pp 325–358 | Cite as

Critical Points and Supersymmetric Vacua I

  • Michael R. DouglasEmail author
  • Bernard Shiffman
  • Steve Zelditch
Article

Abstract

Supersymmetric vacua (‘universes’) of string/M theory may be identified with certain critical points of a holomorphic section (the ‘superpotential’) of a Hermitian holomorphic line bundle over a complex manifold. An important physical problem is to determine how many vacua there are and how they are distributed, as the superpotential varies over physically relevant ensembles. In several papers over the last few years, M. R. Douglas and co-workers have studied such vacuum statistics problems for a variety of physical models at the physics level of rigor [Do,AD,DD]. The present paper is the first of a series by the present authors giving a rigorous mathematical foundation for the vacuum statistics problem. It sets down basic results on the statistics of critical points ∇s=0 of random holomorphic sections of Hermitian holomorphic line bundles with respect to a metric connection ∇, when the sections are endowed with a Gaussian measure. The principal results give formulas for the expected density and number of critical points of fixed Morse index of Gaussian random sections relative to ∇. They are particularly concrete for Riemann surfaces. In our subsequent work, the results will be applied to the vacuum statistics problem and to the purely geometric problem of studying metrics which minimize the expected number of critical points.

Keywords

Manifold Riemann Surface Quantum Computing Physical Problem Basic Result 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Michael R. Douglas
    • 1
    Email author
  • Bernard Shiffman
    • 2
  • Steve Zelditch
    • 2
  1. 1.Department of Physics and Astronomy and IHES and CaltechRutgers UniversityPiscatawayUSA
  2. 2.Department of MathematicsJohns Hopkins UniversityBaltimoreUSA

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