Critical points and supersymmetric vacua, II: Asymptotics and extremal metrics

Abstract

Motivated by the vacuum selection problem of string/M theory, we study a new geometric invariant of a positive Hermitian line bundle (L, h) → M over a compact Kähler manifold: the expected distribution of critical points of a Gaussian random holomorphic section s ∈ H⁰(M,L) with respect to the Chern connection ∇_h. It is a measure on M whose total mass is the average number N^crit_h of critical points of a random holomorphic section. We are interested in the metric dependence of N^crit_h, especially metrics h which minimize N^crit_h. We concentrate on the asymptotic minimization problem for the sequence of tensor powers (L^N, h^N) → M of the line bundle and their critical point densities K^crit_N,h(z). We prove that K^crit_N,h(z) has a complete asymptotic expansion in N whose coefficients are curvature invariants of h. The first two terms in the expansion of N^crit_N,h are topological invariants of (L,M). The third term is a topological invariant plus a constant β₂(m) (depending only on the dimension m of M) times the Calabi functional ∫_Mρ²dVol_h, where ρ is the scalar curvature of the Kähler metric ω_h := (i/2)Θ_h. We give an integral formula for β₂(m) and show, by a computer assisted calculation, that β₂(m) > 0 for m ≤ 5, hence that N^crit_N,h is asymptotically minimized by the Calabi extremal metric (when one exists). We conjecture that β₂(m) > 0 in all dimensions, i.e., the Calabi extremal metric is always the asymptotic minimizer.