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The Wasteland of Random Supergravities

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Mathematics and Applications of Branes in String and M-theory

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. Using Coulomb gas techniques, we then 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 propto exp(-c N^2), with c a constant, for generic critical points. Our results have significant implications for the counting of de Sitter vacua in string theory, but the number of vacua remains vast.

This talk is part of the Isaac Newton Institute Seminar Series series.

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