Principal Investigator: Bong H. Lian
This project addresses problems in three closely related areas in the context of mirror symmetry and duality. As a continuation of current joint work with K. Liu and S.T. Yau, Lian proposes to both generalize and specialize their theory ("mirror principle") for studying characteristic classes of vector bundles on a stable map moduli space. First, this work has thus far considered convex projective manifolds. Dropping the convexity assumption is important if one wishes to consider general Calabi-Yau manifolds. Part I of this proposal outlines an approach which is expected to lead to the full generalization of mirror principle in in genus zero. The main new input here is a way to combine the difficult machinery of virtual cycles and the many ingredients of the mirror principle. Second, the mirror principle can be specialized to surfaces and many new questions which have recently arisen in local mirror symmetry, as well as enumerative geometry on surfaces. In the former case partition functions of a given genus are related to modular forms whenever the underlying surface is elliptic. In the latter case, enumerating curves of a given genus with suitable incidence in a surface also yields modular forms. This project seeks to understand modularity from the point of view of characteristic classes of vector bundles on stable map moduli spaces. For positive genus, the mirror principle requires yet another generalization. In Part II of this project, Hosono, Lian, Liu and Yau will examine these new questions. In recent joint work of Hosono, Lian and Yau, they have settled the problem of constructing the ubiquitous large radius limit for the "universal" family of Calabi-Yau hypersurfaces in a toric manifold. In Part III Lian, Todorov and Yau will study this limit for more general families.
String physics is an ambitious effort to unify all the fundamental forces of nature. A remarkable prediction of String Theory is that nature apparently allows for many different versions of spacetimes. A major current problem in string physics is to understand how a plethora of apparently different spacetimes are related, often in an unexpected and remarkable ways, under the rubric of ``String Duality''. Mirror symmetry is a special yet nontrivial case of String Duality. Though they come in vast variety, the spacetimes in questions are still highly restricted. They turn out to be a class of geometrical objects, known as Calabi-Yau manifolds, which have been studied by mathematicians for over 100 years. Physicists have discovered that string theories associated to certain pairs of Calabi-Yau manifolds ("mirror pairs") are equivalent. This project aims at understanding the geometry of these mirror manifolds from the mathematical point of view. A constant exchange of insights and feedback between physicists and mathematicians on mirror symmetry and other issues has been a hallmark of String Theory in its last 20 years of development.