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 Kahler manifold: the expected distribution K crit (z) of critical points dlog |s(z)|h = 0 of a Gaussian random holomorphic section s ∈ H 0 (M,L) with respect to 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 critN (z). We prove that K critN (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 hN are topological invariants of (L,M). The third term is a topological invariant plus a constantm 2 (depending only on the dimension m of M) times the Calabi functional R M � 2 dV olh, whereis the scalar curvature of the curvature form of h. We give an integral formula form 2 and show, by a computer assisted calculation, thatm 2 > 0 for m ≤ 3, hence that N crit hN is asymptotically minimized by the Calabi extremal metric (when one exists). We conjecture thatm 2 > 0 in all dimensions, i.e. that the Calabi extremal metric is always the asymptotic minimizer.