Homological Mirror Symmetry for Calabi-Yau Hypersurfaces

Calabi-Yau 超曲面的同调镜像对称

• 批准号: 1104779
• 负责人: Eric Zaslow
• 金额: $ 24.7万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-06-01 至 2015-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1104779&HistoricalAwards=false
• 关键词: Homological; Mirror; Symmetry; Calabi; Yau

AbstractAward: DMS-1104779Principal Investigator: Eric ZaslowMirror symmetry in string theory has, quite famously, linkeddisparate fields of mathematics. The central theorem in thesubject is Kontsevich's homological mirror symmetry (HMS)conjecture. The principal investigator proposes to prove thisconjecture using his recent work with collaborators. Connectionsbetween mathematical physics and topology, representation theory,and combinatorics have been revealed through the PI's researchwith Nadler and with Fang, Liu and Treumann. What emerges is alanguage for studying mirror symmetry and other advances inmathematical physics from a simple, geometric viewpoint whicheasily lends itself to computation. This perspective renderssome formidable hurdles of HMS rather tractable. The PI will,with collaborators and graduate students, aim to prove HMS inseveral stages. First, by defining a category, the ConstructiblePlumbing Model (CPM), which models the Fukaya category of aCalabi-Yau manifold at its large radius limit by defining aformal Lagrangian skeleton and gluing together categories ofconstructible sheaves made from pieces of the skeleton. Second,by proving that CPM is equivalent to the category of perfectcomplexes on the mirror Calabi-Yau at its large complex limitpoint. After these steps, a deformation-of-categories argumentcan be made to establish a mirror map and prove HMS.The aim of string theory is to merge the two pillars of modernphysics: Einstein's theory of gravity and the quantum theory ofparticles. Models of the universe from string theory rely on aclass of geometric spaces called Calabi-Yau manifolds.Calculations in these models are quite formidable, but are oftenmade tractable through the phenomenon of mirror symmetry. Theidea of mirror symmetry is that one theory can look totallydifferent from another theory, but the two lead to the samepredictions. Hard calculations using one Calabi-Yau manifold canbecome easy calculations in the completely different "mirror"Calabi-Yau manifold. But to be truly useful, one must havecomplete confidence in the equivalence, namely that thecalculations in the mirror theory can be trusted. This requiresa rigorous mathematical formulation of the model, a rigorousstatement of how to apply the equivalence, and a rigorous proofthat the equivalence is, in fact, true. The statements in mirrorsymmetry have been made rigorous by Fields Medal laureate MaximKontsevich. What is still lacking is a general proof ofKontsevich's conjecture. The principal investigator proposes toprove this conjecture in several steps, using a simple geometricmodel which easily lends itself to calculations. The model canalso serve as a framework for exploring other predications andphenomena of modern theoretical physics.

AbstractAward：DMS-1104779首席研究员：Eric Zaslow弦论中的镜像对称性，非常著名，将不同的数学领域联系起来。本课题的中心定理是Kontsevich的同调镜像对称（HMS）猜想。 主要研究者提议用他最近与合作者的工作来证明这个猜想。数学物理与拓扑学、表象论和组合学之间的联系已经通过PI与Nadler、Fang、Liu和Treumann的研究揭示出来。 出现的是一种语言，用于研究镜像对称性和其他数学物理学的进展，从一个简单的几何观点，这很容易使其适合于计算。 这种观点使HMS的一些难以克服的障碍变得相当容易处理。PI将与合作者和研究生一起，旨在分几个阶段证明HMS。 首先，通过定义一个类别，可构造管道模型（CPM），它通过定义一个正式的拉格朗日骨架和粘合在一起的类别的可构造层从骨架的碎片在其大半径极限的卡-丘流形的福谷类别模型。 其次，证明了CPM在其大复极限点等价于镜像Calabi-Yau上的完全复形范畴。 在这些步骤之后，我们就可以用范畴变形论证来建立镜像映射并证明HMS。弦理论的目标是融合现代物理学的两大支柱：爱因斯坦的引力理论和粒子的量子理论。 弦理论的宇宙模型依赖于一类叫做卡-丘流形的几何空间。在这些模型中的计算相当困难，但通常由于镜像对称现象而变得容易处理。 镜像对称的概念是，一个理论可以看起来与另一个理论完全不同，但两者导致相同的预测。 利用一个Calabi-Yau流形的困难计算可以在完全不同的“镜像“Calabi-Yau流形上变得容易计算。 但要真正有用，人们必须对等价性有完全的信心，也就是说，镜像理论中的计算是可信的。 这就需要对模型进行严格的数学表述，对如何应用等价性进行严格的陈述，并严格证明等价性实际上是正确的。 菲尔兹奖获得者马克西姆·孔采维奇（MaximKontsevich）对镜像对称的陈述进行了严格的阐述。 目前尚缺乏的是对孔采维奇猜想的一般性证明。 主要研究者建议用几个步骤来证明这个猜想，使用一个简单的几何模型，很容易使自己的计算。该模型也可以作为探索现代理论物理的其他预言和现象的框架。