Critical points and supersymmetric vacua, II: Asymptotics and extremal metrics
| dc.creator | Douglas, Michael R. | |
| dc.creator | Shiffman, Bernard | |
| dc.creator | Zelditch, Steve | |
| dc.date | 2004-06-05 | |
| dc.date | 2005-12-10 | |
| dc.date.accessioned | 2026-07-07T08:42:27Z | |
| dc.date.available | 2026-07-07T08:42:27Z | |
| dc.description | Motivated by the vacuum selection problem of string/M theory, we study a new geometric invariant of a positive Hermitian line bundle $(L, h)\to M$ over a compact Kähler manifold: the expected distribution of critical points of a Gaussian random holomorphic section $s \in H^0(M, L)$ with respect to the Chern connection $\nabla_h$. It is a measure on $M$ whose total mass is the average number $\mathcal{N}^{crit}_h$ of critical points of a random holomorphic section. We are interested in the metric dependence of $\mathcal{N}^{crit}_h$, especially metrics $h$ which minimize $\mathcal{N}^{crit}_h$. We concentrate on the asymptotic minimization problem for the sequence of tensor powers $(L^N, h^N)\to M$ of the line bundle and their critical point densities $\mathcal{K}^{crit}_{N,h}(z)$. We prove that $\mathcal{K}^{crit}_{N,h}(z)$ has a complete asymptotic expansion in $N$ whose coefficients are curvature invariants of $h$. The first two terms in the expansion of $\mathcal{N}^{crit}_{N,h}$ are topological invariants of $(L, M)$. The third term is a topological invariant plus a constant $β_2(m)$ (depending only on the dimension $m$ of $M$) times the Calabi functional $\int_M ρ^2 dV_h$, where $ρ$ is the scalar curvature of the Kähler metric $ω_h:=\frac i2 Θ_h$. We give an integral formula for $β_2(m)$ and show, by a computer assisted calculation, that $β_2(m)>0$ for $m\leq 5$, hence that $\mathcal{N}^{crit}_{N,h}$ is asymptotically minimized by the Calabi extremal metric (when one exists). We conjecture that $β_2(m)>0$ in all dimensions, i.e. the Calabi extremal metric is always the asymptotic minimizer. | |
| dc.description | 35 pages; 2 Maple worksheets. Final version for Journal of Differential Geometry. This is the second paper in a three-part series including math.CV/0402326 and math-ph/0506015. This revision contains additional conjectures and numerical results, as well as stylistic changes and minor corrections | |
| dc.identifier | https://arxiv.org/abs/math/0406089 | |
| dc.identifier | http://arxiv.org/abs/math/0406089 | |
| dc.identifier | J. Diff. Geometry 72 (2006), 381-427. | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/141963 | |
| dc.subject | Complex Variables | |
| dc.subject | High Energy Physics - Theory | |
| dc.subject | Algebraic Geometry | |
| dc.title | Critical points and supersymmetric vacua, II: Asymptotics and extremal metrics | |
| dc.type | text |