Unifying Mirror Symmetry

摘要奖：DMS-0072504首席研究员：Eric Zaslow Zaslow提出了针对镜像对称统一理解的研究。 目前的想法--包括“经典”镜像对称; Kontsevich猜想; Vafa的工作; Strominger、Yau和Zaslow的猜想--只是松散地联系在一起，涉及微扰和非微扰弦推理。 理解非微扰弦理论与经典镜像对称图像和Gromov-Witten不变量的关系将是走向统一的重要一步。 为实现这一目标，提出了五个项目：1）发展Gopakumar和Vafa从BPS统计中获得的新变量的数学公式。 2)理解从高亏格Gromov-Witten不变量中得到整数的多重覆盖公式。 3)从Calabi-Yau模空间的奇点结构出发，解决了全纯二义性问题。 4)发展对Kontsevich的Calabi-Yau模空间的扩展以包括A-无穷结构的物理理解。 5)定义了一个几何Fourier-Mukai函子，它将流形的特殊拉格朗日圈与镜像上丛的Hermitian-Yang-Mills联络联系起来. 这个函子可以导致几何证明孔采维奇的制定mirrorsymmetry。 所有这些项目的目标是统一我们对镜像对称性的仍在探索中的方法。这个项目的目标是统一我们对镜像对称现象的数学和物理理解，镜像对称现象是由从事弦理论研究的理论物理学家发现的。弦理论是一种提出的物理理论，它有望将爱因斯坦对空间和引力的理解融入量子理论。 镜像对称是弦理论中的一种对偶对称，两种截然不同的物理理论实际上是等价的。 当其中一个理论容易计算而另一个很难计算时，这就导致了对困难计算的答案的预测。 从数学的角度来看，这可能导致与平行结构相关的结构--这些结构涉及描述物理理论的不同但等价的数学模型。 镜像对称揭示的关系是深刻而新颖的。 对它们的统一理解不仅可以连接研究领域，还可以连接不同的学科--数学和物理学。 该奖项部分由物理系数学物理项目资助。

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.