Unifying Mirror Symmetry

摘要：dms -0072504首席研究员：Eric ZaslowZaslow提出对镜像对称的统一理解的研究方向。当前的观点——包括“经典”镜像对称；Kontsevich猜想;瓦法的作品；以及Strominger、Yau和Zaslow的猜想——只是松散连接的，并且涉及微扰和非微扰弦推理。理解非摄动弦理论与经典镜像对称图和Gromov-Witten不变量的关系将是迈向统一的重要一步。为了实现这一目标，提出了五个项目：1)开发Gopakumar和Vafa从BPS状态计算中获得的新不变量的数学公式。2)理解从高属Gromov-Witten不变量得到整数的多重覆盖公式。3)利用Calabi-Yau模空间的奇异点结构确定全纯歧义，从而解决全纯歧义问题。4)发展对Kontsevich对Calabi-Yau模空间的扩展以包含a -∞结构的物理理解。5)定义一个几何类傅里叶函子，将一个流形的特殊拉格朗日循环与镜面上束上的hermitian - yang - mills连接联系起来。这个函子可以为孔采维奇的镜像对称公式提供几何证明。所有这些项目都旨在统一我们仍然不同的镜像对称方法。该项目旨在统一我们对镜像对称现象的数学和物理理解，该现象是由从事弦理论研究的理论物理学家发现的。弦理论是一种被提出的物理理论，有望将爱因斯坦对空间和引力的理解纳入量子理论。镜像对称是弦理论中的一种对偶对称，即两种截然不同的物理理论实际上是等价的。当其中一种理论容易计算而另一种理论很难计算时，这就会导致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有关的猜想——平行结构涉及描述物理理论的不同但等效的数学模型。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。该奖部分经费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.