BPS Geometry, Singularities, and String Theory

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

This award supports the continuing research of the Principal Investigator at the interface of algebraic geometry in mathematics and string theory in theoretical physics. Algebraic geometry is the investigation of algebraic solutions of polynomial equations, the geometry of the graphs of these solutions, and the theoretical structure describing their properties. It has numerous applications in science and engineering, including geometric modeling, robotics, control theory, coding theory, phylogenetics, as well as the application to string theory being advanced in this project. String theory is a physical theory which provides a framework for a unified field theory incorporating all of the forces and particles found in nature. String theory has found wide application beyond unified field theory, including condensed matter physics, black hole physics, and the application to mathematics being advanced in this project. The Principal Investigator will involve graduate students and postdocs in aspects of the project, assisting their professional development and contributing to the development of the scientific workforce. In more technical terms, the project will concern itself with three interrelated areas: BPS Geometry, Singularities, and String Theory. BPS states are certain supersymmetric states which play a fundamental role in many supersymmetric physical theories. Geometric techniques will be applied to advance the theory of the associated BPS invariants and their refinements, including orientations on the derived category of coherent sheaves on a Calabi-Yau threefold, Fourier-Mukai transforms as global symmetries of physical theories, a new theory of log BPS invariants, and advancing the connection between Bridgeland stability in mathematics and Pi-stability in physics. The geometric study of singularities is incorporated into this project in at least two ways: canonical threefold singularities can be used to engineer 5 dimensional superconformal field theories, and certain codimension 2 singularities can be used to engineer gauge theories analogous to the Hitchin system. Proposed research in string theory proper includes a more careful analysis of the holomorphic anomaly equation with particular attention to the Principal Investigator's earlier work on BPS invariants of elliptic fibrations and Jacobi forms. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.