Holomorphic G-bundles On Elliptic Fibrations

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ABSTRACT DMS - 0200810.One motivation for this project comes from F theory/heterotic stringduality. In this duality, given the data of an elliptically fiberedCalabi-Yau manifold, a principal bundle over it with the appropriatestructure group and connection, and a B-field, which is roughly like aholomorphic 2-form, one expects to find a new Calabi-Yau manifold fibered inelliptic K3 surfaces over the same base. A second motivation is tounderstand some of the mathematical issues raised by the first question, ina purely mathematical framework. These issues lead to a rich set ofquestions 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, andthe geometry of del Pezzo surfaces and K3 surfaces.This project is concerned with the study of certain geometric structuresarising in mathematics and physics. One goal of this study is a mathematicalunderstanding of the relation between two physical theories of elementaryparticles arising from string theory. The relationship is best expressedmathematically by relationships between several geometric structures. Theserelationships, and various generalizations, are also very interesting from amathematical point of view, because they are connected with the possiblesymmetries of spaces in any number of dimensions and with geometricstructures connected with these symmetries.