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

Michael R. Douglas, Bernard Shiffman, Steve Zelditch

Research output: Contribution to journalArticlepeer-review

31 Scopus citations


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 inter- ested 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 co- efficients 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) (depend- ing only on the dimension m of M) times the Calabi functional ∫Mρ2dVolh, where ρ is the scalar curvature of the Kähler metric ωh :=i/2Θh. 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 ex- tremal metric (when one exists). We conjecture that β2(m) > 0 in all dimensions, i.e., the Calabi extremal metric is always the asymptotic minimizer.

Original languageEnglish (US)
Pages (from-to)381-427
Number of pages47
JournalJournal of Differential Geometry
Issue number3
StatePublished - 2006

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory
  • Geometry and Topology


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