Singularities and String Theory

这个项目涉及数学中的代数几何和理论物理中的弦理论的界面研究。代数几何是研究多项式方程的代数解，这些解的图形的几何，以及描述它们的性质的理论结构。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，以及在这个项目中推进的弦理论的应用。弦理论是一个统一场论的物理理论框架，它包含了自然界中所有的力和粒子。弦理论已经在统一场论之外得到了广泛的应用，包括凝聚态物理，黑洞物理，以及本项目正在推进的在数学中的应用。首席研究员将让研究生和本科生参与项目的各个方面，帮助他们的专业发展和对科学工作的发展做出贡献。更具体地说，将探索代数几何中弦理论和奇点理论交界处的一系列相互关联的问题。在代数几何中，有两大类问题。第一部分探讨了三重正则奇点理论中的一些开放方向、它们的初始解以及这些解的计数不变量。另一个方向是利用分解定理研究椭圆纤维Calabi-Yau三折叠的奇性及其与Hodge理论的联系。在弦理论中，还有两组主要的问题。第一个方向是研究五维超共形场理论及其模数的几何。另一个方向是F-理论的性质的发展，直接从它的定义为IIB型弦理论，具有不同的轴伸缩而不引用对偶性。探索和发展正则奇点与5维超共形场论之间的联系，以及霍奇理论与F理论之间的联系。该项目还包括弦理论中的其他主题的调查。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.