| Abstract: |
An important problem in string theory is counting the metastable vacua of the theory, i.e. the local minima of the energy function on the configuration space. An efficient approach is to count critical points and then determine what fraction of critical points are local minima. I will report on advances in this direction based on work with D. Marsh and T. Wrase. We obtain a random matrix model for the Hessian matrix, which is well-approximated by the sum of a Wigner matrix and two Wishart matrices. Computing the eigenvalue spectrum from the free convolution of the constituent spectra, we 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 and $N$ the dimension of the configuration space. We conclude that in supergravity theories with high-dimensional moduli spaces, an overwhelming majority of critical points are saddle points rather than local minima. Our results have significant implications for the counting of cosmological vacua in string theory, but the number of vacua remains vast. Finally, I will explain how our methods can be used to make statistical predictions for the CMB perturbations arising in multifield inflation. |
