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.

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National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Christopher Stark
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Northwestern University at Chicago
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