Program on Large-N Limit Problems in Kähler Geometry

该项目为参加2015年4月20日至6月19日在西蒙斯几何与物理中心（斯托尼布鲁克大学）举行的“凯勒几何中的大N方法”项目提供部分支持。 项目名称的大N极限是指1/h，其中h是普朗克常数，一个非常小的物理常数，用于测量物理过程中量子力学效应的重要性。 量子效应在原子水平上很重要，但对于像汽车这样的宏观物体就不重要了，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N = 1/h趋于无穷大时，量子力学模型趋于经典模型。类似的情况出现在数学和物理的许多领域，只要有一个大参数N，使N趋于无穷大极限的主要特征，量子力学趋于经典力学的极限。在几何学中，N代表多项式的次数。在Kaehler几何中，有一些特殊的度量称为Bergman度量，类似于N次多项式。任何度量都可以近似为N次的多项式类Bergman度量。会议致力于这种近似在几何和物理学中的许多用途，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一个曲率形式为Kaehler形式的Kaehler流形上的正埃尔米特线丛L的N次方的全纯截面的空间。在Tian-Yau-唐纳森关于极值度量稳定性的研究中，N次Bergman度量对一般Kaehler度量的逼近起着重要作用。N次Bergman度量是由M以L的N次方全纯截面的基在维数为1的复射影空间中的全纯嵌入导出的特殊Kaehler度量。 全纯嵌入本质上是映射z - B（N，w，z），其中B是Bergman核，即，在L的N次幂的全纯截面空间上的正交投影。 伯格曼核和度量出现在物理学和几何学中。例如，曲面上分数量子霍尔效应的劳克林波函数与伯格曼核密切相关，N由表面上的电子数给出。Bergman核和度量也给出了一种新的方法来定义所有Kaehler度量空间上的概率测度，这有望与量子引力有关。 该计划侧重于全纯随机几何和数学物理。会议网站：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