Homological Mirror Symmetry for Calabi-Yau Hypersurfaces

摘要：获奖：dms -1104779首席研究员：Eric zaslow弦理论中的镜像对称，众所周知，将不同的数学领域联系在一起。本学科的中心定理是Kontsevich的同调镜像对称（HMS）猜想。首席研究员提议用他最近与合作者的工作来证明这一猜想。通过PI与Nadler、Fang、Liu和Treumann的研究，揭示了数学物理与拓扑学、表示理论和组合学之间的联系。由此产生了一种语言，可以从简单的几何角度研究镜像对称和数学物理的其他进展，这种语言很容易用于计算。这一观点使得HMS的一些巨大障碍变得相当容易处理。PI将与合作者和研究生一起，分几个阶段证明HMS。首先，通过定义一个类别，即可构造管道模型（CPM），该模型通过定义一个正式的拉格朗日骨架并将由骨架碎片组成的可构造轴类粘合在一起，在其大半径极限下模拟了aCalabi-Yau流形的Fukaya类别。其次，通过证明CPM在镜面Calabi-Yau的大复极限点上等价于完美复的范畴。在这些步骤之后，可以使用类别变形论证来建立镜像图并证明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量子粒子理论。弦理论中的宇宙模型依赖于一类称为Calabi-Yau流形的几何空间。在这些模型中进行计算是相当艰巨的，但通常通过镜像对称现象变得容易处理。镜像对称的观点是，一种理论可能看起来与另一种理论完全不同，但这两种理论得出的预测是相同的。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流形中变得容易计算。但要真正有用，就必须对等效性有完全的信心，也就是说，镜像理论中的计算是可信的。这需要对模型进行严格的数学表述，对如何应用等效性进行严格的陈述，并严格证明等效性实际上是正确的。菲尔兹奖得主马克西姆·康茨维奇（MaximKontsevich）对镜像对称的陈述进行了严格的验证。现在还缺少的是对康采维奇猜想的一般证明。首席研究员建议分几个步骤来证明这一猜想，使用一个简单的几何模型，使其易于计算。该模型也可以作为探索现代理论物理的其他预测和现象的框架。

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.