F-Theory and G-bundles over Elliptic Curves

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Proposal: DMS-9970437Principal Investigator: Robert D. FriedmanAbstract:This project involves the study of holomorphic G-bundles over an elliptic curve, where G is a reductive algebraic group, together with various generalizations. If G is the complexification of a compact group K, a closely related question is the classification of two commuting elements in K up to simultaneous conjugation. In joint work with J. Morgan and E. Witten, Friedman has classified holomorphic G-bundles over an elliptic curve. The relative situation for G-bundles over an elliptic fibration has also been studied. The next stage of the project is to work out the relation between these constructions, in the case where G is one of the exceptional groups E6, E7, E8, and corresponding families of K3 or del Pezzo surfaces. In terms of families of commuting elements in compact Lie groups, in joint work with A. Borel and J. Morgan, Friedman has given normal forms for commuting pairs and triples, and the method extends in principle to handle N mutually commuting elements. Compact Lie groups are the symmetry groups of nature, and the possible such groups have been classified and understood for one hundred years. Aside from four infinite series of such groups, there are also a small number of exceptional groups. Recently, such groups have become important in particle physics, because the possible subatomic particles and their interactions seem to be described by such symmetry groups. In particular string theory, which is an attempt to unify quantum mechanics with gravity, has suggested that the universe is 10-dimensional and that the symmetry group involved is one of the exceptional groups. Moreover, in some versions of string theory, the missing 6 dimensions of the universe are accounted for by a so-called elliptic fibration. Thus, the project attempts to study a part of mathematics which is relevant to string theory as well as being an interesting piece of mathematics in its own right.