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


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.


Manifold Riemann Surface Quantum Computing Physical Problem Basic Result 
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  1. 1.
    Arnold, V.I., Gusein-Zade, S.M., Varchenko, A.N.: Singularities of differentiable maps. Vol. II. Monodromy and asymptotics of integrals. Monographs in Math. 83, Boston: Birkhäuser, 1988Google Scholar
  2. 2.
    Ashok, S., Douglas, M.R.: Counting Flux Vacua. J. High Energy Phys. 01, 060 (2004)CrossRefGoogle Scholar
  3. 3.
    Banks, T., Dine, M., Gorbatov, E.: Is there a string theory landscape. J. High Energy Phys. 04, 058 (2004)CrossRefGoogle Scholar
  4. 4.
    Bleher, P., Shiffman, B., Zelditch, S.: Universality and scaling of correlations between zeros on complex manifolds. Invent. Math. 142(2), 351–395 (2000)CrossRefzbMATHGoogle Scholar
  5. 5.
    Bleher, P., Shiffman, B., Zelditch, S.: Universality and scaling of zeros on symplectic manifolds. In: P. Bleher, A.R. Its, (eds.), Random matrix models and their applications, MSRI publications 40, Cambridge: Cambridge Univ. Press, 2001, pp. 31–70Google Scholar
  6. 6.
    Bott, R.: On a theorem of Lefschetz. Michigan Math. J. 6, 211–216 (1959)CrossRefzbMATHGoogle Scholar
  7. 7.
    Bousso, R., Polchinski, J.: Quantization of Four-form Fluxes and Dynamical Neutralization of the Cosmological Constant. J. High Energy Phys. 06, 006 (2000)zbMATHGoogle Scholar
  8. 8.
    Denef, F., Douglas, M.R.: Distributions of flux vacua. J. High Energy Phys. 05, 072 (2004)CrossRefGoogle Scholar
  9. 9.
    Douglas, M.R.: The statistics of string/M theory vacua. J. High Energy Phys. 05, 046 (2003)CrossRefGoogle Scholar
  10. 10.
    Douglas, M.R.: Basic results in vacuum statistics., 2004. To appear in Comptes Rendu Physiques
  11. 11.
    Douglas, M.R., Shiffman, B., Zelditch, S.: Critical points and supersymmetric vacua, II: asymptotics and extremal metrics., 2004
  12. 12.
    Douglas, M.R., Shiffman, B., Zelditch, S.: Critical points and supersymmetric vacua, III: String/M models. In preparationGoogle Scholar
  13. 13.
    Ferrara, S., Gibbons, G.W., Kallosh, R.: Black holes and critical points in moduli space. Nucl. Phys. B 500(1–3), 75–93 (1997)Google Scholar
  14. 14.
    Federer, H.: Geometric Measure Theory. New York: Springer, 1969Google Scholar
  15. 15.
    Milnor, J.: Singular points of complex hypersurfaces. Annals of Mathematics Studies, No. 61, Princeton, NJ: Princeton University Press; Tokyo: University of Tokyo Press, 1968Google Scholar
  16. 16.
    Moore, G.: Les Houches lectures on Strings and Arithmetic, 2004
  17. 17.
    Moore, G.: Arithmetic and attractors., 1998
  18. 18.
    Shiffman, B., Zelditch, S.: Distribution of zeros of random and quantum chaotic sections of positive line bundles. Commun. Math. Phys. 200, 661–683 (1999)CrossRefzbMATHGoogle Scholar
  19. 19.
    Susskind, L.: The anthropic landscape of string theory., 2003

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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