Mirror Principle and Modularity

摘要奖：DMS-0072158主要研究者：Bong H.这个项目在镜像对称和二元性的背景下解决了三个密切相关的领域的问题。作为目前与K. Liu和S.T. Yau，Lian提出了推广和专门研究稳定映射模空间上向量丛特征类的理论（“镜像原理”）。首先，这项工作迄今认为凸射影流形。 如果要考虑一般的Calabi-Yaumanifolds，则放弃凸性假设是重要的。本建议的第一部分概述了一种方法，有望导致镜像原理在亏格零中的全面推广。这里主要的新输入是一种将虚拟循环的困难机械和镜像原理的许多成分联合收割机结合起来的方法。第二，镜像原理可以专门应用于曲面和最近在局部镜像对称性中出现的许多新问题，以及曲面上的计数几何。 在前一种情况下，当下垫面为椭圆时，给定亏格的配分函数与模形式有关.在后一种情况下，在曲面中以适当的关联度枚举给定亏格的曲线也会产生模形式。这个项目试图从稳定映射模空间上向量束的特征类的角度来理解模性。对于正亏格，镜像原理还需要另一种推广。 在这个项目的第二部分，细野，连，刘和丘将研究这些新问题。Hosono，Lian和Yau在最近的联合工作中，解决了复曲面流形中“万有”Calabi-Yau超曲面族的普遍大半径极限的构造问题.在第三部分，Lian、Todorov和Yau将研究更一般的族的极限。弦物理是一项雄心勃勃的努力，旨在统一自然界所有的基本力。弦论的一个显著预言是，自然界显然允许许多不同版本的时空。 弦物理学目前的一个主要问题是理解在“弦对偶”的标题下，过多的明显不同的时空是如何联系在一起的，通常是以一种意想不到的和可解释的方式。镜像对称伊萨弦对偶的一种特殊而又非平凡的情形。尽管它们种类繁多，但问题中的时空仍然是高度受限的。 它们原来是一类几何对象，被称为卡-丘流形，数学家已经研究了100多年。 物理学家已经发现，与某些卡拉比-姚曼折叠对（“镜像对”）相关的弦理论是等价的。 本项目旨在从数学的角度理解这些镜像流形的几何。 物理学家和数学家在镜像对称和其他问题上不断交换见解和反馈，这是弦理论在过去20年发展中的一个标志。

AbstractAward: DMS-0072158Principal Investigator: Bong H. LianThis project addresses problems in three closely related areas inthe context of mirror symmetry and duality. As a continuation ofcurrent joint work with K. Liu and S.T. Yau, Lian proposes toboth generalize and specialize their theory ("mirror principle")for studying characteristic classes of vector bundles on a stablemap moduli space. First, this work has thus far considered convexprojective manifolds. Dropping the convexity assumption isimportant if one wishes to consider general Calabi-Yaumanifolds. Part I of this proposal outlines an approach which isexpected to lead to the full generalization of mirror principlein in genus zero. The main new input here is a way to combine thedifficult machinery of virtual cycles and the many ingredients ofthe mirror principle. Second, the mirror principle can bespecialized to surfaces and many new questions which haverecently arisen in local mirror symmetry, as well as enumerativegeometry on surfaces. In the former case partition functions ofa given genus are related to modular forms whenever theunderlying surface is elliptic. In the latter case, enumeratingcurves of a given genus with suitable incidence in a surface alsoyields modular forms. This project seeks to understand modularityfrom the point of view of characteristic classes of vectorbundles on stable map moduli spaces. For positive genus, themirror principle requires yet another generalization. In Part IIof this project, Hosono, Lian, Liu and Yau will examine these newquestions. In recent joint work of Hosono, Lian and Yau, theyhave settled the problem of constructing the ubiquitous largeradius limit for the "universal" family of Calabi-Yauhypersurfaces in a toric manifold. In Part III Lian, Todorov andYau will study this limit for more general families.String physics is an ambitious effort to unify all thefundamental forces of nature. A remarkable prediction of StringTheory is that nature apparently allows for many differentversions of spacetimes. A major current problem in stringphysics is to understand how a plethora of apparently differentspacetimes are related, often in an unexpected and remarkableways, under the rubric of ``String Duality''. Mirror symmetry isa special yet nontrivial case of String Duality. Though they comein vast variety, the spacetimes in questions are still highlyrestricted. They turn out to be a class of geometrical objects,known as Calabi-Yau manifolds, which have been studied bymathematicians for over 100 years. Physicists have discoveredthat string theories associated to certain pairs of Calabi-Yaumanifolds ("mirror pairs") are equivalent. This project aims atunderstanding the geometry of these mirror manifolds from themathematical point of view. A constant exchange of insights andfeedback between physicists and mathematicians on mirror symmetryand other issues has been a hallmark of String Theory in its last20 years of development.