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

围绕，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出现的，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。这种现象被比喻为“镜像对称”，它导致了对此类理论中所涉及的几何结构(“卡拉比-尤流形”)的非常有先见之明的数学预测。1994年，马克西姆·康采维奇在他的“同源镜像对称”(HMS)猜想中为镜像对称的多个数学方面提出了一个组织框架，他说，这将“揭开镜像对称的奥秘”。当时，HMS引用的两个主要数学概念之一--“辛流形的Fukaya范畴”是新的、未开发的，很难证明康采维奇猜想的实例并推导出它的结果。Fukaya范畴的数学最近的发展改变了这一点。这个职业项目的中心是一个实现Kontsevich愿景的计划。PI及其合作者提出了“核心同源镜像对称性”的概念，该程序旨在证明，如果核心HMS对特定的镜像对成立，那么镜像对称性的许多其他方面自然会作为逻辑结果而存在。该项目包括支持研究生与PI一起工作，这将帮助他们开始研究生涯，并支持为数学本科生制定连贯的荣誉计划。研究计划的主要内容是镜像对称，主要是Calabi-Yau(CY)流形。该程序(PI及其合作者Sheridan和Ganatra正在进行中)研究了给定的一对极化CY流形的“核心同调镜像对称性”的结果，该流形粗略地说，一个可以以尊重范畴结构的方式将这两个流形中的一个上的极化线丛的张量幂与另一个的拉格朗日子流形的集合相匹配。基于Strominger-Yau-Zaslow哲学的镜对几何方法，这一假设是非常自然的，尽管目前只在少数情况下得到了证实。我们的目的是证明核心HMS蕴含全HMS和闭弦镜像对称性的几个方面，特别是全纯体积形式的正规化和镜像映射的精确形式。鉴于最近的发展，这已经变得可行，主要是由于Abouzaid将从Fukaya范畴的量子上同调到Hochschild同调的几何“开闭弦映射”与Fukaya范畴的“同调光滑性”的范畴性质联系起来。一个关键的目的是给出五次三次有理曲线的个数的母函数等于它的镜面的汤川耦合的“概念性”证明--这个证明不依赖于计算两边。这一研究对辛拓扑、伪全纯曲线和辛Floer上同调起到了关键作用。这些技巧在第二条线上也同样重要，即在类似于Heegaard Floer理论的直线上发展Floer理论的3-流形不变量，但不是基于Heegaard曲面(被视为复杂曲线)的对称乘积，而是基于具有此类曲线上截面的2阶全纯丛的模空间。这样的不变量可以用来在三维流形的弗洛尔理论的Heegaard和瞬子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.