Microlocalization and Mirror Symmetry

这个项目的目标是扩大我们对镜像对称的理解，并说明它如何在数学中引入新的对偶。特别是，孔采维奇的同调镜像对称猜想将这种物理现象表达为膜范畴的等价。 一个这样的范畴是福谷范畴，由拉格朗日子流形构造。这个项目的一个中心焦点将是最近由主要研究者与D。Nadler，在流形的余切丛的福谷范畴和基上的可构造层之间。这种构造将用于：1）研究余切丛的镜像对称; 2）根据其相应的福谷对象来表征反常层; 3）根据Kapustin-Witten的建议，在几何Langlands程序中构造Hecke本征层，以观察希钦纤维化的环面纤维。 另一方面的建议集中在构建镜像映射的模纯粹从技术的同调镜像对称，从而桥梁孔采维奇的观点与历史的方法镜像对称。在过去的几十年里，数学和理论物理学已经成为联系在一起的深刻的方式。 弦理论领域最能说明这种相互依赖性。 在弦理论中，镜像对称是物理学中的一个现象如何将看似不同的数学领域联系起来的最好例子。这项提议将通过创造和利用最新进展的研究路线来拓宽镜像对称的数学范围。特别是，拓扑学、表示论和几何朗兰兹纲领之间的联系已经出现，从同调代数中构造镜像对称例子的新方法也已经出现。因此，这项研究可能会导致几个长期目标的进展：理解镜像对称在其最一般的设置，并构建几何朗兰兹计划中的赫克运算的本征层。

The goal of this project is to expand our understanding of mirror symmetry and illustrate how it incorporates new dualities in mathematics. In particular, the homological mirror symmetry conjecture of Kontsevich expresses this physical phenomenon as an equivalence of brane categories. One such category is the Fukaya category, constructed from Lagrangian submanifolds. A central focus of this project will be the relationship, found recently by the Principal Investigator with D. Nadler, between the Fukaya category of the cotangent bundle of a manifold and constructible sheaves on the base. This construction will be used: 1) to study mirror symmetry for the cotangent bundle; 2) to characterize perverse sheaves in terms of their corresponding Fukaya objects; and 3) to construct Hecke eigensheaves in the geometric Langlands program, following the proposal of Kapustin-Witten to look at torus fibers of the Hitchin fibration. Another aspect of the proposal focusses on constructing the mirror map of moduli purely from techniques of homological mirror symmetry, thereby bridging Kontsevich's point of view with the historical approach to mirror symmetry.Over the past few decades, mathematics and theoretical physics have become linked in a profound way. The field of string theory best illustrates this interdependence. Within string theory, mirror symmetry is the prime example of how one phenomenon in physics can link seemingly disparate fields of mathematics. This proposal will broaden the mathematical scope of mirror symmetry through lines of research that both create, and capitalize on, recent advances. In particular, connections between topology, representation theory, and the geometric Langlands program have emerged, as have new methods for constructing examples of mirror symmetry from homological algebra. This research could thus lead to progress in several longstanding goals: understanding mirror symmetry in its most general setting, and constructing eigensheaves for Hecke operations in the geometric Langlands program.