## Journal of Differential Geometry

### 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 Ncrith of critical points of a random holomorphic section. We are interested in the metric dependence of Ncrith, especially metrics h which minimize Ncrith. We concentrate on the asymptotic minimization problem for the sequence of tensor powers (LN, hN) → M of the line bundle and their critical point densities KcritN,h(z). We prove that KcritN,h(z) has a complete asymptotic expansion in N whose coefficients are curvature invariants of h. The first two terms in the expansion of NcritN,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 ∫Mρ2dVolh, where ρ is the scalar curvature of the Kähler metric ωh := (i/2)Θh. We give an integral formula for β2(m) and show, by a computer assisted calculation, that β2(m) > 0 for m ≤ 5, hence that NcritN,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.

#### Article information

Source
J. Differential Geom., Volume 72, Number 3 (2006), 381-427.

Dates
First available in Project Euclid: 28 March 2006

https://projecteuclid.org/euclid.jdg/1143593745

Digital Object Identifier
doi:10.4310/jdg/1143593745

Mathematical Reviews number (MathSciNet)
MR2219939

Zentralblatt MATH identifier
1236.32013

#### Citation

Douglas, Michael R.; Shiffman, Bernard; Zelditch, Steve. Critical points and supersymmetric vacua, II: Asymptotics and extremal metrics. J. Differential Geom. 72 (2006), no. 3, 381--427. doi:10.4310/jdg/1143593745. https://projecteuclid.org/euclid.jdg/1143593745