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

职业：深谷范畴、镜像对称和低维拓扑

• 批准号: 1455265
• 负责人: Timothy Perutz
• 金额: $ 40万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-09-15 至 2021-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1455265&HistoricalAwards=false
• 关键词: CAREER; Fukaya; categories; mirror; symmetry

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.

1989年左右，许多数学物理学家研究了宇宙的假设字符串理论模型，发现某些此类模型成对成对，基于不同方程式的理论的“ A”和“ B”，但共享相同的物理观察到的数量。 这种现象获得了隐喻名称“镜像对称性”，导致了有关这种理论所涉及的几何（“ Calabi-yau歧管”）的极为有先见的数学预测。 1994年，Maxim Kontsevich在他的“同源镜对称性”（HMS）猜想中提出了一个组织框架，该框架针对镜子对称的多个数学方面，他说，这将“揭示镜面对称的奥秘”。 当时，HMS引用的两个主要数学概念之一，即“夫妻歧管的福卡亚类别”，这是新的且未开发的，而且很难证明Kontsevich的猜想的实例并推论了它的后果。 福卡亚类别数学的最新发展改变了这一点。 这个职业项目的中心是实现Kontsevich的愿景的计划。 PI和合作者已经提出了“核心同源镜像对称性”的概念，该程序的目的是表明，如果Core 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而然地跟随逻辑后果。 该项目包括对与PI一起工作的研究生的支持，这将在他们的研究职业开始时有助于他们，并支持开发针对数学本科生的连贯荣誉计划。 该计划（在PI和合作者Sheridan和Ganatra的工作中正在进行中）研究了给定的一对两极化的CY歧管的“核心同源镜像对称性”的后果，该歧管的后果大致说，一个大致说，一个人可以将两者限制线的张量符合lagrangian submanifolds的张力量符合，从而与lagrangian submanifold集合在一起，该方式与另一个方面的构造相匹配。鉴于基于Strominger-Yau-Zaslow哲学的几何方法，该假设是很自然的，尽管目前仅在少数情况下才证明。我们的目的是表明核心HMS意味着“封闭式”镜像对称性的完整HMS和几个方面，尤其是全态体积形式的归一化以及镜像图的精确形式。鉴于重新开发的发展，这已成为可行的，主要是由于abouzaid，从量子共同体到福卡亚类别的Hochschild同源性，再到福卡亚类别的“同源平滑性”的分类特性。一个关键目的是给出一个“概念”证明，即3倍五重的有理曲线数量的生成函数等于其镜像的Yukawa耦合 - 这一证明不依赖于计算双方。这项研究赋予了共生拓扑，伪形态曲线和符号浮雕的共同体的关键作用。这些技术在第二链中同样重要，即在与Heegaard Floer理论相似的线条上开发浮雕的3个manifold不变性，但并非基于Heegaard表面的对称产品（被视为复杂曲线），而是基于等级2 Holomorphic holomorphic Bundles loce curves curves curves curves curves的模量架。这种不变的人可能会在Heegaard和Instanton版本的3型Manifolds中进行调解。