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日在石溪大学西蒙斯几何与物理中心举行的“Kaehler几何中的大n方法”项目提供部分支持。项目标题中的大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，用来衡量物理过程中量子力学效应的重要性。量子效应在原子层面上很重要，但对于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，只要有一个大的参数N，使得N趋向于无穷大的极限，就像量子力学趋向于经典力学一样，具有极限的主要特征。在几何中，N表示多项式的次数。在凯勒几何中，有一些特殊的度量称为伯格曼度量，类似于n次多项式。任何度量都可以用n次多项式的伯格曼度量来近似。会议致力于这种近似在几何和物理中的许多应用，从量子引力到量子霍尔效应。在Kaehler几何中，Kaehler流形的几何量化是曲率形式为Kaehler形式的Kaehler流形上的正厄米线束L的n次幂的全纯截面空间。Tian-Yau-Donaldson程序的有关稳定极值指标,中央所扮演的角色是班上一般Kaehler指标的近似程度N伯格曼的伯格曼指标度量程度的N是特殊Kaehler度量诱导的全纯映射进行M到复射影空间维度的基地的N次方的全纯部分l .全纯嵌入地图基本上是z - B(氮、w、z), B是伯格曼内核,即,L. Bergman核和度量的n次幂在全纯截面空间上的正交投影在物理和几何中都有出现。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，其中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度量空间上的概率度量，这有望与量子引力有关。这个程序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