BPS Geometry, Singularities, and String Theory

该奖项支持首席研究员在数学中的代数几何和理论物理中的弦理论的接口方面的持续研究。代数几何是对多项式方程的代数解的研究，这些解的图形的几何，以及描述其性质的理论结构。它在科学和工程上有许多应用，包括几何建模、机器人、控制理论、编码理论、系统发育，以及在这个项目中正在推进的弦理论的应用。弦理论是一种物理理论，它为统一场论提供了一个框架，统一场论包含了自然界中发现的所有力和粒子。弦理论在统一场论之外有广泛的应用，包括凝聚态物理、黑洞物理，以及本项目正在推进的数学应用。首席研究员将让研究生和博士后参与项目的各个方面，协助他们的专业发展，并为科研队伍的发展做出贡献。用更专业的术语来说，该项目将涉及三个相互关联的领域：BPS几何、奇点和弦理论。BPS态是某些超对称态，在许多超对称物理理论中起着重要作用。将应用几何技术来推进相关BPS不变量的理论及其改进，包括Calabi-Yau三倍上相干束的派生范畴的取向、作为物理理论全局对称的Fourier-Mukai变换、对数BPS不变量的新理论，以及推进数学中的bridgeeland稳定性与物理中的pi -稳定性之间的联系。奇点的几何研究至少以两种方式纳入本项目：正则三重奇点可用于构建5维超共形场理论，某些余维2奇点可用于构建类似于希钦系统的规范理论。建议在弦理论中进行的研究包括对全纯异常方程的更仔细的分析，特别注意首席研究员早期关于椭圆纤维和Jacobi形式的BPS不变量的工作。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.