期刊
JOURNAL OF HIGH ENERGY PHYSICS
卷 -, 期 3, 页码 -出版社
SPRINGER
DOI: 10.1007/JHEP03(2012)102
关键词
Flux compactifications; Superstring Vacua
资金
- NSF [PHY-0757868]
- Alfred P. Sloan Foundation
- Galo foundation
- German Research Foundation (DFG) [WR 166/1-1]
- Direct For Mathematical & Physical Scien
- Division Of Physics [757868] Funding Source: National Science Foundation
We show that in a general N = 1 supergravity with N >> 1 scalar fields, an exponentially small fraction of the de Sitter critical points are metastable vacua. Taking the superpotential and Kahler 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 proportional to exp(-c N-p), with c, p being constants. For generic critical points we find p approximate to 1.5, while for approximately-supersymmetric critical points, p approximate to 1.3. Our results have significant implications for the counting of de Sitter vacua in string theory, but the number of vacua remains vast.
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