Homological Mirror Symmetry for Calabi-Yau Hypersurfaces

摘要奖：DMS-1104779首席研究员：埃里克·扎斯洛弦理论中的镜像对称性将不同的数学领域联系在一起，这一点非常著名。本课题的中心定理是康采维奇的同调镜像对称性猜想。首席研究员建议用他最近与合作者的工作来证明这一猜想。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以及与方、刘和特鲁曼的研究而被揭示。出现的是一种语言，用于从简单的几何角度研究镜像对称性和数学物理中的其他进展，这有助于计算。这种观点使卫生与公众服务部面临的一些难以克服的障碍变得相当容易。PI将与合作者和研究生一起，致力于分几个阶段证明HMS。首先，通过定义一个范畴，可构造管道模型(CPM)，它通过定义形式的拉格朗日骨架并将由骨架碎片制成的可构造滑轮类别粘合在一起，在其大半径极限上模拟了Calabi-Yau流形的Fukaya范畴。其次，通过证明CPM等价于镜像Calabi-Yau在其大复极限点上的完美复形范畴。在这些步骤之后，可以进行范畴变形的论证来建立镜像图并证明HMS。弦理论的目的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。弦理论中的宇宙模型依赖于一类被称为Calabi-Yau流形的几何空间。这些模型中的计算非常令人敬畏，但由于镜像对称的现象，这些模型往往很容易处理。镜像对称的概念是，一种理论可以看起来与另一种理论完全不同，但这两种理论导致了相同的预测。使用一个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。但要想真正有用，一个人必须对等价性有完全的信心，即镜像理论中的计算是可信的。这需要一个严格的模型数学公式，一个关于如何应用等价性的严格陈述，以及一个严格的证明等价性实际上是真的。菲尔兹奖章获得者马克西姆·康采维奇对镜像对称的声明进行了严谨的处理。目前还缺乏对康采维奇猜想的普遍证明。首席研究人员建议分几步证明这一猜想，使用一个简单的几何模型，这种模型很容易进行计算。该模型还可以作为探索现代理论物理的其他预言和现象的框架。

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.