Holomorphic G-bundles On Elliptic Fibrations

Columbia University, New York NY

Investigators

Abstract

ABSTRACT DMS - 0200810. One motivation for this project comes from F theory/heterotic string duality. In this duality, given the data of an elliptically fibered Calabi-Yau manifold, a principal bundle over it with the appropriate structure group and connection, and a B-field, which is roughly like a holomorphic 2-form, one expects to find a new Calabi-Yau manifold fibered in elliptic K3 surfaces over the same base. A second motivation is to understand some of the mathematical issues raised by the first question, in a purely mathematical framework. These issues lead to a rich set of questions concerning two or more commuting elements in compact Lie groups, conjugacy classes in semisimple algebraic groups or complex Lie algebras, the moduli space of semistable principal bundles over elliptic curves, and the geometry of del Pezzo surfaces and K3 surfaces. This project is concerned with the study of certain geometric structures arising in mathematics and physics. One goal of this study is a mathematical understanding of the relation between two physical theories of elementary particles arising from string theory. The relationship is best expressed mathematically by relationships between several geometric structures. These relationships, and various generalizations, are also very interesting from a mathematical point of view, because they are connected with the possible symmetries of spaces in any number of dimensions and with geometric structures connected with these symmetries.