Geometry from Physics: Calabi-Yau Spaces and the Positive Mass Conjecture

Duke University, Durham NC

Investigators

Abstract

Abstract Award: DMS-0505767 Principal Investigator: Anda Degeratu A variety of questions in geometry arise from physics. Consistency requirements in superstring theory suggest that we live in a 10-dimensional world, where the extra six dimensions must take the shape of a Calabi-Yau geometry. For the past 20 years, this has led to a tremendous effort on the part of mathematicians and physicists to understand Calabi-Yau spaces, and the various questions which emerge from their role in string theory. On the other hand, more classical areas of physics such as general relativity continue to motivate interesting questions in Riemannian geometry. In particular, extending the Positive Mass Conjecture has important implications for geometric analysis and string theory. The first two projects in this proposal examine the geometry of Calabi-Yau spaces. In studying such objects, one has to take into consideration singular Calabi-Yaus. One way to deal with singularities is to resolve them. As a result, the geometrical structure undergoes changes. The goal of the first research project is to give a complete characterization of the changes which can occur under crepant resolutions. In contrast to more traditional algebraic methods, the proposed techniques for this study involve an interplay between algebraic and analytical methods. Another way to deal with Calabi-Yau singularities is to analyze them from the perspective of various string theory dualities. In the second project, the PI proposes to study the taxonomy of Calabi-Yau spaces for which heterotic/F-theory dualities hold. Along the way, it is hoped that this study will give a geometrical interpretation to a new conjecture of McKay which relates the exceptional simply-laced Dynkin diagrams to the Monster group and its offspring. The focus of the third project is the Positive Mass Conjecture. In the context of general relativity it has been proved that the total mass of an isolated system is never negative, provided that the sources of the gravitational field consist of matter with nonnegative local mass density and that spacetime is asymptotically flat. Witten's proof is based on the analysis of the Dirac operator along three-dimensional spacelike submanifolds in spacetime. The goal of this research is to extend Witten's approach to prove the Positive Mass Conjecture in any dimension, in particular for non-spin manifolds. To summarize, an assortment of questions in mathematics lie at the interface between geometry and physics. This proposal focuses on geometry problems which do not fit nicely into one particular field, but require a variety of techniques from different branches of geometry. Many of these problems come from physics, where ad-hoc methods need further development.