Mirror Principle and Modularity

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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.