CAREER: Fukaya categories, mirror symmetry, and low-dimensional topology

大约在1989年，一些数学物理学家在研究假设的宇宙弦理论模型时，发现某些这样的模型是成对的，理论的“a”和“B”版本基于不同的方程，但共享相同的物理可观测量。这种现象获得了“镜像对称”的隐喻名称，导致了对这种理论中涉及的几何（“卡拉比-丘流形”）的极具先见之明的数学预测。1994年，Maxim Kontsevich在他的“同调镜像对称”（homological mirror symmetry， HMS）猜想中提出了镜像对称的多个数学方面的组织框架，他说，这将“揭开镜像对称的神秘面纱”。当时，HMS引用的两个主要数学概念之一，“辛流形的深谷范畴”，是新的和未发展的，并且很难证明Kontsevich猜想的实例并推导出它的结果。深谷分类数学的最新发展改变了这一点。这个CAREER项目的核心是一个实现Kontsevich愿景的项目。PI和合作者已经制定了一个“核心同源镜像对称”的概念，该计划旨在表明，如果核心HMS适用于特定的镜像对，那么镜像对称的许多其他方面自然会作为逻辑结果随之而来。该项目包括支持与PI一起工作的研究生，这将帮助他们开始他们的研究生涯，并支持为数学本科生制定连贯的荣誉计划。研究项目的主要内容是镜面对称，主要是针对Calabi-Yau （CY）流形。该计划（PI和合作者Sheridan和Ganatra正在进行的工作）研究了给定一对极化CY流形的“核心同调镜像对称”的结果，大致表示可以以一种尊重范畴结构的方式将其中一个极化线束的张量幂与另一个的拉格朗日子流形的某些集合相匹配。鉴于基于strominger - you - zaslow哲学的镜像对的几何方法，这个假设是很自然的，尽管目前只在少数情况下得到了证明。我们的目标是证明核心HMS隐含着全HMS和“闭弦”镜像对称的几个方面，特别是全纯体积形式的归一化和镜像映射的精确形式。这在最近的发展中变得可行，主要是由于Abouzaid，将几何“开闭弦映射”，从量子上同调到Fukaya范畴的Hochschild同调，再到Fukaya范畴的“同调平滑”的范畴性质联系起来。一个关键的目标是给出一个“概念性的”证明，即五次三次曲线上有理曲线数量的生成函数等于其镜像的汤川耦合——一个不依赖于计算两边的证明。本文给出了辛拓扑、伪全纯曲线和辛花上同调的关键作用。这些技术在第二链中同样重要，即在与Heegaard Floer理论相似的直线上开发花理论的3流形不变量，但不是基于Heegaard曲面的对称积（视为复杂曲线），而是基于具有此类曲线上截面的2阶全纯束的模空间。这些不变量可以在3流形的花理论的heegard和瞬时版本之间起到中介作用。

Around 1989, a number of mathematical physicists, studying hypothetical string-theoretical models of the universe, discovered that certain such models come in pairs, with versions "A" and "B" of the theory based on different equations yet sharing the same physically observable quantities. This phenomenon, which acquired the metaphorical name "mirror symmetry," led to extraordinarily prescient mathematical predictions about the geometries ("Calabi-Yau manifolds") involved in such theories. In 1994, Maxim Kontsevich proposed an organizing framework for multiple mathematical aspects of mirror symmetry in his "homological mirror symmetry" (HMS) conjecture, which, he said, would "unveil the mystery of mirror symmetry." At the time, one of the two main mathematical notions invoked by HMS, the "Fukaya category of a symplectic manifold," was new and undeveloped, and it was very difficult both to prove instances of Kontsevich's conjecture and to deduce consequences of it. Recent developments in the mathematics of Fukaya categories have changed that. At the center of this CAREER project is a program to realize Kontsevich's vision. The PI and collaborators have formulated a notion of "core homological mirror symmetry," and the program aims to show that if core HMS holds for a particular mirror pair then many other facets of mirror symmetry naturally follow as logical consequences. The project includes support for graduate students working with the PI, which will assist them at the beginning of their research careers, and support for the development of a coherent honors program for mathematics undergraduates.The main strand of the research program concerns mirror symmetry, primarily for Calabi-Yau (CY) manifolds. The program (underway in work of the PI and collaborators Sheridan and Ganatra) studies the consequences of "core homological mirror symmetry" for a given pair of polarized CY manifolds, which roughly says that one can match the tensor powers of the polarizing line bundle over one of these two with some collection of Lagrangian submanifolds of the other in a way that respects categorical structures. This hypothesis is quite natural in light of geometric approaches to mirror pairs based on the Strominger-Yau-Zaslow philosophy, though currently only proven in a few cases. We aim to show that core HMS implies full HMS and several aspects of "closed-string" mirror symmetry, notably the normalization of the holomorphic volume form and the precise form of the mirror map. This has become feasible in light of recents developments, due principally to Abouzaid, relating the geometric "open-closed string map," from quantum cohomology to Hochschild homology of the Fukaya category, to the categorical property of "homological smoothness" of the Fukaya category. A key aim is to give a "conceptual" proof that the generating function for the numbers of rational curves on a quintic 3-fold equals the Yukawa coupling of its mirror -- a proof that does not rely on calculating the two sides. This research gives key roles to symplectic topology, pseudo-holomorphic curves, and symplectic Floer cohomology. Those techniques are equally important in a second strand, which is to develop Floer-thoeretic 3-manifold invariants on similar lines to Heegaard Floer theory but based not on symmetric products of a Heegaard surface (viewed as a complex curve), but rather on moduli spaces of rank 2 holomorphic bundles with section over such curves. Such invariants may serve to mediate between Heegaard and instanton versions of Floer theory for 3-manifolds.