Singularities and String Theory

本项目涉及数学中的代数几何和理论物理中的弦理论的接口研究。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。它在科学和工程中有许多应用，包括统计学、生物学和几何建模，以及在这个项目中正在推进的弦理论的应用。弦理论是统一场论的物理理论框架，它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本科生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。更具体地说，在弦理论和奇点理论在代数几何的接口相互关联的问题的集合将被探索。在代数几何中，有两大类问题。第一追求一些开放的方向，在理论规范奇点的三倍，他们的cretative决议，并列举这些决议的不变量。另一个方向是利用分解定理研究椭圆纤维Calabi-Yau三折的奇异性及其与Hodge理论的联系。在弦理论中，也有两大类问题。第一部分研究五维超共形场理论及其模的几何。另一个方向是直接从f -理论的定义发展其性质，即不调用对偶性而具有变轴张量的IIB型弦理论。规范奇点与五维超共形场理论之间的联系以及霍奇理论与f理论之间的联系将被探索和发展。该项目还包括研究弦理论的其他主题。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 project concerns research 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 statistics, biology, and geometric modeling, as well as the application to string theory being advanced in this project. String theory is a physical theoretical framework for a unified field theory incorporating all 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 undergraduate students in aspects of the project, assisting their professional development and contributing to the development of the scientific workforce.More specifically, a collection of interrelated questions at the interface of string theory and singularity theory in algebraic geometry will be explored. Within algebraic geometry, there are two major sets of problems. The first pursues some open directions in the theory of canonical singularities of threefolds, their crepant resolutions, and enumerative invariants of these resolutions. The other direction is the study of the singularities of elliptically fibered Calabi-Yau threefolds and their connection to Hodge theory via the decomposition theorem. Within string theory, there are also two major sets of problems. The first pursues the geometry of 5-dimensional superconformal field theories and their moduli. The other direction is the development of the properties of F-theory directly from its definition as Type IIB string theory with varying axiodilaton without invoking duality. Connections between canonical singularities and 5-dimensional superconformal field theories will be explored and developed as well as the connection between Hodge theory and F-theory. The project also includes investigation of additional topics in string theory.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.