Principal Investigator: David R. Morrison
String 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.
In the first half of the twentieth century, Einstein's theory of general relativity changed the way we view gravity (especially when gravitational fields are strong, near very massive objects), and the theory of quantum mechanics changed the way we view the microscopic world. Both theories have been scientifically validated through extensive experimental testing, but when they are extrapolated to regimes where they both have something to say (i.e., at small distance scales near very massive objects), they are in contradiction. The quest for a quantum theory of gravity which would mediate between these two theories has been one of the primary challenges of theoretical physics in the last fifty years. Our current best hope for a quantum theory of gravity is string theory and some theories related to it known as M-theory and F-theory. The basic idea behind string theory is to modify physics at very small distances by representing particles not as points, but as one-dimensional objects (open or closed "strings"). This modification at small distances turns out to produce gravity automatically in a quantum context, which is why it forms an appealing theoretical framework. M-theory arises when the strength of the interaction between the strings becomes very large, and F-theory exploits an unusual symmetry of string theory to produce a variant type of quantum gravity. These theories require very sophisticated mathematics for their formulation and study, and the past twenty years has seen a very fruitful collaboration between mathematicians and theoretical physicists in carrying out those studies. This project was once such collaboration: the Principal Investigator was trained as a mathematician but has spent much of his research career investigating the interface between the two fields. This particular project focussed on F-theory and the key mathematical tool which is used to study F-theory, known as the theory of elliptic fibrations. Imagine an under-inflated inner tube. The shape made by the rubber tube can be changed by squeezing in various places, and if sufficient air has been let out, one can even constrict the tube with a tight band which pinches off a circle, squeezing it down to a point, somewhere along the tube. An elliptic fibration can be visualized as a collection of such inner tubes in which the shape changes from point to point. Those variable shapes help to determine the properties of the physical model, but they are analyzed mathematically. Mathematically, the inner tube is described by an equation in two complex variables, and the shape is changed by varying the coefficients in the equation. This mathematical theory of elliptic fibrations began in the middle of the nineteenth century in a study of functions of a complex variable, but became thoroughly a part of algebraic geometry in the early 1960's through work of Néron, Kodaira, Tate, and others. The research in this project combined algebraic methods and geometric methods to make further advances in our understanding of these fibrations, including an improvement of an algorithm originally developed by Tate for determining the types of pinched-off tubes which can occur. Direct applications of the results were made back to the physics application of F-theory. In particular, a new way of limiting the types of F-theory configurations which occur was developed, which points the way to a more complete analysis of how F-theory models could be studied in their entirety.
