This award funds the research activities of Professor Eric Sharpe at Virginia Polytechnic Institute and State University.
Mirror symmetry is a duality between string theories whose discovery led to ground-breaking advances in both physics and mathematics. In his research, Professor Sharpe aims to develop a generalization of mirror symmetry, known as ``(0,2) mirror symmetry,'' which would be a powerful computational tool for understanding four-dimensional theories with minimal supersymmetry obtained from string theory by rolling up or `compactifying' six dimensions on a compact space. One significant part of that research will be to further develop `quantum sheaf cohomology,' a notion originally developed by Professors Sharpe and Sheldon Katz, which computes quantum corrections to charged matter couplings and is closely tied to the mathematics of these theories.
This project will also have significant broader impacts. Professor Sharpe will involve graduate students and postdocs in his research, and so provide critical training to junior physicists beginning research in this area. In addition to organizing Southeast regional mathematical string theory meetings at Duke University twice annually, he also plans to organize a workshop on the subject of (0,2) mirror symmetry.
This project derived results on compactifications of string theory, an attempt to unify general relativity and quantum field theory, the two primary developments of twentieth-century physics. String theory predicts that the universe is ten-dimensional, whereas we only see four space and time dimensions. A `compactification' of string theory reconciles that issue by taking six of the dimensions to be rolled up on a small compact space, whose geometry and topology is only observed indirectly via, for example, the corresponding set of observable low-energy particles. Over the course of this project we developed some new types of string compactifications, gained insight into some existing compactifications, and also solved some technical problems with existing compactifications. For example, several papers were written on `noncommutative resolutions,' a new type of string compactification, and others concerning compactifications on generalizations of spaces known as stacks. Several papers were written on `quantum sheaf cohomology,' which computes subtle `non-perturbative' contributions to charged matter couplings in low-energy quantum field theories derived from string compactifications. We also derived dualities between existing compactifications, showing how certain a priori distinct compactifications were secretly equivalent. Finally, we worked out how to compute `moduli' in certain compactifications on spaces known as `non-Kahler,' solving a longstanding old problem in those compactifications, which is important as those moduli count gauge neutral scalar fields, analogues of the Higgs field, in the low-energy effective quantum field theory. NSF funds were used to help train graduate students, by providing research support during the summers, as well as travel support for students and faculty to attend conferences and summer schools.
