Open Access
March 2006 Critical points and supersymmetric vacua, II: Asymptotics and extremal metrics
Michael R. Douglas, Bernard Shiffman, Steve Zelditch
J. Differential Geom. 72(3): 381-427 (March 2006). DOI: 10.4310/jdg/1143593745

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.

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Michael R. Douglas. Bernard Shiffman. Steve Zelditch. "Critical points and supersymmetric vacua, II: Asymptotics and extremal metrics." J. Differential Geom. 72 (3) 381 - 427, March 2006. https://doi.org/10.4310/jdg/1143593745

Information

Published: March 2006
First available in Project Euclid: 28 March 2006

zbMATH: 1236.32013
MathSciNet: MR2219939
Digital Object Identifier: 10.4310/jdg/1143593745

Rights: Copyright © 2006 Lehigh University

Vol.72 • No. 3 • March 2006
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