Unifying Mirror Symmetry

统一镜面对称

• 批准号: 0072504
• 负责人: Eric Zaslow
• 金额: $ 8.51万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-09-01 至 2004-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0072504&HistoricalAwards=false
• 关键词: Unifying; Mirror; Symmetry

AbstractAward: DMS-0072504Principal Investigator: Eric ZaslowZaslow proposes research directed towards a unified understandingof mirror symmetry. Current ideas -- including ``classical''mirror symmetry; Kontsevich's conjecture; the work of Vafa; andthe conjecture of Strominger, Yau, and Zaslow -- are only looselyconnected and involve both perturbative and non-perturbativestring reasoning. An understanding of how non-perturbativestring theory relates to the classical mirror symmetry pictureand Gromov-Witten invariants will be an important step towardsunification. Five projects are proposed towards achieving thisgoal: 1) Developing a mathematical formulation of the newinvariants obtained by Gopakumar and Vafa from BPS statecounting. 2) Understanding the multiple-cover formulas whichyield integers from higher-genus Gromov-Witten invariants. 3)Resolving the holomorphic ambiguity by determining it from thestructure of singularities of Calabi-Yau moduli space. 4)Developing a physical understanding of Kontsevich's enlargementof Calabi-Yau moduli space to include A-infinity structures. 5)Defining a geometric Fourier-Mukai-like functor relatingspecial-Lagrangian cycles of one manifold to Hermitian-Yang-Millsconnections on bundles over the mirror. This functor could leadto a geometric proof of Kontsevich's formulation of mirrorsymmetry. All these projects aim to unify our still disparateapproaches to mirror symmetry.This project is directed towards unifying our mathematical andphysical understanding of the phenomenon of mirror symmetry,discovered by theoretical physicists working in string theory.String theory is a proposed physical theory with the promise ofincorporating Einstein's understanding of space and gravity intothe quantum theory. Mirror symmetry is a duality symmetry instring theory, whereby two very different physical theories areactually equivalent. When one of the theories is easilycomputable and the other hard, this leads to predictions ofanswers to difficult calculations. From the mathematical pointof view, this can lead to conjectures relating parallelstructures -- the structures involved in describing thedifferent, but equivalent, mathematical models of physicaltheories. The relationships unearthed by mirror symmetry aredeep and novel. A unified understanding of them may join notonly fields of research but different disciplines -- math andphysics -- as well. This award is partially funded by theprogram in Mathematical Physics of the Division of Physics.

摘要奖：DMS-0072504首席研究员：埃里克·扎斯洛·扎斯洛提出了一项旨在统一理解镜像对称性的研究。目前的想法--包括“经典的”镜像对称性；康采维奇的猜想；瓦法的工作；以及斯特罗明格、尤和扎斯洛的猜想--只是松散地联系在一起，同时涉及微扰和非微扰的弦推理。理解非微扰弦理论如何与经典镜像对称图像和Gromov-Witten不变量相联系，将是迈向太阳化的重要一步。为实现这一目标提出了五个项目：1)建立Gopakumar和Vafa从BPS状态计算中获得的新不变量的数学公式。2)从高亏格Gromov-Witten不变量中理解整数的多重覆盖公式。3)通过由Calabi-Yau模空间的奇点结构确定全纯歧义来解决全纯歧义。4)发展对Kontsevich将Calabi-Yau模空间扩大到包括A-无穷结构的物理理解。5)定义了一个几何Fourier-Mukai函子，它将一个流形上的特殊拉格朗日圈与镜像上丛上的Hermitian-Yang-Mills联络联系联系起来。这个函子可能导致对康采维奇的镜像对称性公式的几何证明。所有这些项目都旨在统一我们仍然不同的镜像对称方法。这个项目旨在统一我们对镜像对称现象的数学和物理理解，这是由从事弦理论的理论物理学家发现的。弦理论是一种提出的物理理论，承诺将爱因斯坦对空间和引力的理解融入到量子理论中。镜像对称是弦理论中的对偶对称，即两个非常不同的物理理论实际上是等价的。当其中一个理论很容易计算，而另一个理论很难计算时，这就导致了对困难计算的答案的预测。从数学的角度来看，这可能导致与并行结构相关的猜想--描述不同但等价的物理理论的数学模型所涉及的结构。镜面对称揭示出的关系是深刻而新颖的。对它们的统一理解不仅可能涉及研究领域，也可能涉及不同的学科--数学和物理--。该奖项的部分资金来自物理系的数学物理项目。