Program on Large-N Limit Problems in Kähler Geometry

该项目为参加于2015年4月20日至6月19日在西蒙斯几何与物理中心(石溪大学)举行的《Kaehler几何中的大N方法》项目提供部分支持。项目标题的大N限制指的是1/h，其中h是普朗克常数，一个非常小的物理常数，衡量量子力学效应在物理过程中的重要性。量子效应在原子水平上很重要，但对于汽车等宏观物体来说并不重要，因为汽车的运动是由经典力学(牛顿定律)描述的。当h趋于0，或者N=1/h趋于无穷大时，量子力学模型趋向于经典模型。类似的情况出现在数学和物理的许多领域，当有一个大的参数N时，使得N趋于无穷大，从而分享极限的主要特征，就像量子力学趋于经典力学一样。在几何学中，N表示多项式的次数。在Kaehler几何中，有一些特殊的度量被称为Bergman度量，它们类似于N次多项式。任何度量都可以用类似于N次多项式的Bergman度量来近似。会议致力于讨论这种近似在几何和物理中的许多用途，从量子引力到量子霍尔效应。在Kaehler几何中，Kaehler流形的几何量子化是曲率形式为Kaehler形式的Kaehler流形上的正Hermite线丛L的N次方全纯截面的空间。在将稳定性与极值度量联系起来的Tian-Yau-Donaldson程序中，N次Bergman度量对类中一般Kaehler度量的逼近起着核心作用。N次Bergman度量是由L的N次方全纯截面基将M全纯嵌入到维复射影空间中而产生的特殊Kaehler度量。全纯嵌入本质上是映射z-B(N，w，z)，其中B是Bergman核的N次方全纯截面空间上的正交投影。例如，曲面上分数量子霍尔效应的Laughlin波函数与Bergman核密切相关，N由表面上的电子数给出。Bergman核和度量还给出了一种在所有Kaehler度量空间上定义概率度量的新方法，这有望与量子引力有关。本课程的重点是全纯随机几何和数学物理。会议网址：scgp.stonybrook.edu/scientific/programs/fall-2014-spring-2015-program-details/large-n-limit-problems-in-kahler-geometry

The project provides partial support for participation in the program "Large-N Methods in Kaehler Geometry" held at the Simons Center for Geometry and Physics (Stony Brook University) from April 20 to June 19, 2015. The large N limit of the project title refers to 1/h where h is Planck's constant, a very small physical constant that measures the importance of quantum mechanical effects in a physical process. Quantum effects are important at the atomic level, but not for macroscopic objects such as cars, whose motion is described by classical mechanics (Newton's laws). As h tends to 0, or N = 1/h tends to infinity, quantum mechanical models tend to classical counterparts. Analogous situations arise in many areas of mathematics and physics whenever there is a large parameter N so that the N tends to infinity limit shares the principal features of the limit as quantum mechanics tends to classical mechanics. In geometry, N represents the degree of a polynomial. In Kaehler geometry, there are special metrics known as Bergman metrics that are analogous to polynomials of degree N. Any metric may be approximated by the polynomial-like Bergman metrics of degree N. The conference is devoted to the many uses of this approximation in geometry and physics, ranging from quantum gravity to the quantum Hall effect. In Kaehler geometry, the geometric quantization of a Kaehler manifold is the space of holomorphic sections of the Nth power of a positive Hermitian line bundle L over a Kaehler manifold whose curvature form is the Kaehler form. In the Tian-Yau-Donaldson program of relating stability to extremal metrics, a central role is played by the approximation of general Kaehler metrics in the class by Bergman metrics of degree N. The Bergman metrics of degree N are special Kaehler metrics induced by holomorphic embeddings of M into complex projective space of dimension by bases of holomorphic sections of the Nth power of L. The holomorphic embedding is essentially the map z - B(N, w, z) where B is the Bergman kernel, i.e., the orthogonal projection onto the space of holomorphic sections of the Nth power of L. Bergman kernels and metrics arise in physics as well as geometry. For instance, the Laughlin wave function of the fractional quantum Hall effect on a curved surface is intimately related to the Bergman kernel, with N given by the number of electrons on the surface. Bergman kernels and metrics also give a new approach to defining probability measures on the space of all Kaehler metrics, which is expected to be related to quantum gravity. This program focuses on holomorphic stochastic geometry and mathematical physics.Conference web site: scgp.stonybrook.edu/scientific/programs/fall-2014-spring-2015-program-details/large-n-limit-problems-in-kahler-geometry