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融入新的二元性。特别是，康采维奇的同调镜像对称猜想将这一物理现象表述为膜范畴的等价性。Fukaya范畴就是这样一个范畴，它由拉格朗日子流形构造而成。该项目的一个中心焦点将是首席调查员最近与D.Nadler发现的流形的余切丛的Fukaya类别与基座上的可构造滑轮之间的关系。这一结构将被用于：1)研究余切丛的镜像对称性；2)根据对应的Fukaya物体来刻画弯曲的鞘；以及3)按照Kapustin-Witten的建议，在几何朗兰兹计划中构造Hecke本征鞘，以观察Hitchin纤维的环面纤维。该提议的另一个方面是纯粹从同调镜像对称的技术来构造模的镜像映射，从而将康采维奇的观点与镜像对称的历史方法联系起来。在过去的几十年里，数学和理论物理已经以一种深刻的方式联系在一起。弦理论领域最好地说明了这种相互依存关系。在弦理论中，镜像对称性是一个物理现象如何将看似完全不同的数学领域联系在一起的最好例子。这一提议将通过既创造并利用最新进展的研究路线来扩大镜像对称的数学范围。特别是，拓扑学、表示理论和几何朗兰兹程序之间的联系已经出现，并且有了从同调代数构造镜像对称例子的新方法。因此，这项研究可能会在几个长期目标上取得进展：在最一般的背景下理解镜像对称性，以及在几何朗兰兹计划中为Hecke运算构建本征层。

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.