TY - JOUR

T1 - Critical points and supersymmetric vacua I

AU - Douglas, Michael R.

AU - Shiffman, Bernard

AU - Zelditch, Steve

N1 - Copyright:
Copyright 2021 Elsevier B.V., All rights reserved.

PY - 2004/12

Y1 - 2004/12

N2 - 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.

AB - 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.

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U2 - 10.1007/s00220-004-1228-y

DO - 10.1007/s00220-004-1228-y

M3 - Article

AN - SCOPUS:11244277103

VL - 252

SP - 325

EP - 358

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 1-3

ER -