Elliptic Fibrations and Applications to Particle Physics

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AbstractAward: DMS-1007414Principal Investigator: David R. MorrisonString theory provides consistent high-energy extrapolations of theoretical models of particle physics, as well as naturally unifying those models with gravity. One promising type of string theory model for particle physics known as "F-theory" has seen a resurgence of interest in the past two years. The research funded by this grant will investigate the mathematical underpinnings of F-theory models (which are known as "elliptic fibrations" in the mathematical world), with the goal of answering some key mathematical questions which will clarify the physical properties of the models. The goals are the determination of the extent to which the F-theory constructions can be studied locally; determining how the mathematical structure of the local Picard group allows the hypercharge gauge symmetry in the physical theory to survive to low energy; and studying various specific aspects of elliptic fibrations (codimension three phenomena, canonical bundle formulae) which may affect how numerous the F-theory models are.Particle physicists have employed a wide variety of sophisticated mathematical tools to help explain the behavior and structure of our world at a subatomic level. In preparation for the data which will soon be available from the Large Hadron Collider at the European Organization for Nuclear Research (CERN), theoretical particle physicists have been refining their predictions; in some cases, these refinements can only be made if the mathematical tools themselves are improved. The research funded by this grant aims to improve one of the important mathematical tools currently being used, a tool known as ``elliptic fibrations'' which comes from the geometric study of solutions of polynomial equations, a part of the mathematical field of algebraic geometry. Recent uses of this tool by theoretical particle physicists have focussed attention on some areas where it needs improvement in order to be able to make the necessarily calculations for particle physics; this research will make those improvements.