The wasteland of random supergravities

A bstract We show that in a general supergravity with N ≫ 1 scalar fields, an exponentially small fraction of the de Sitter critical points are metastable vacua. Taking the superpotential and Kähler potential to be random functions, we construct a random matrix model for the Hessian matrix, which is...

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Bibliographic Details
Published in:The journal of high energy physics 2012-03, Vol.2012 (3), Article 102
Main Authors: Marsh, David, McAllister, Liam, Wrase, Timm
Format: Article
Language:English
Subjects:
Classical and Quantum Gravitation
Convolution
Critical point
Eigenvalues
Elementary Particles
Hessian matrices
High energy physics
Physics
Physics and Astronomy
Quantum Field Theories
Quantum Field Theory
Quantum Physics
Relativity Theory
Scalars
String Theory
Supergravity
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Summary:A bstract We show that in a general supergravity with N ≫ 1 scalar fields, an exponentially small fraction of the de Sitter critical points are metastable vacua. Taking the superpotential and Kähler potential to be random functions, we construct a random matrix model for the Hessian matrix, which is well-approximated by the sum of a Wigner matrix and two Wishart matrices. We compute the eigenvalue spectrum analytically from the free convolution of the constituent spectra and find that in typical configurations, a significant fraction of the eigenvalues are negative. Building on the Tracy-Widom law governing fluctuations of extreme eigenvalues, we determine the probability P of a large fluctuation in which all the eigenvalues become positive. Strong eigenvalue repulsion makes this extremely unlikely: we find P ∝ exp(− c N p ), with c, p being constants. For generic critical points we find p ≈ 1 . 5, while for approximately-supersymmetric critical points, p ≈ 1 . 3. Our results have significant implications for the counting of de Sitter vacua in string theory, but the number of vacua remains vast.
ISSN:1029-8479
1029-8479
DOI:10.1007/JHEP03(2012)102