Critical points and supersymmetric vacua

Details Critical points and supersymmetric vacua Critical points and supersymmetric vacua

2004-02-20

arxiv.org/abs/math/0402326v2

Commun.Math.Phys. 252 (2004) 325-358

Mathematics

Complex Variables

38 pages. One theorem was removed; it will be included soon in a new posting

Scientific paper

10.1007/s00220-004-1228-y

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. The present paper initiates the study of the statistics of critical points $\nabla s = 0$ of Gaussian random holomorphic sections with respect to a connection $\nabla$. Even the expected number of critical points depends on the curvature of $\nabla$. The principal results give formulas for the expected density and number of critical points of Gaussian random sections relative to $\nabla$ in a variety of settings. The results are particularly concrete for Riemann surfaces. Analogous results on the density of critical points of fixed Morse index are given.

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