Singularities and String Theory

奇点和弦理论

• 批准号: 2201203
• 负责人: Sheldon Katz
• 金额: $ 26.49万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-08-01 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2201203&HistoricalAwards=false
• 关键词: Singularities; String; Theory

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.

该项目涉及数学中代数几何与理论物理学中弦理论的接口研究。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。弦理论在统一场论之外有着广泛的应用，包括凝聚态物理学、黑洞物理学，以及在这个项目中正在推进的数学应用。首席研究员将邀请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，并为科学人才的发展做出贡献。更具体地说，将探索代数几何中弦理论和奇点理论界面上的一系列相互关联的问题。在代数几何中，有两组主要的问题。第一个追求一些开放的方向在理论上的规范奇点的threefolds，他们的crepant决议，并枚举不变量的这些决议。另一个方向是研究椭圆纤维卡-丘三重的奇异性及其通过分解定理与霍奇理论的联系。在弦理论中，也有两大类问题。第一个追求的几何5维超共形场理论和它们的模。另一个方向是F理论的性质的发展，直接从它的定义为具有变化的公理扩张的IIB型弦理论，而不需要调用对偶。正则奇点和5维超共形场论之间的联系将被探索和发展，以及霍奇理论和F理论之间的联系。这个奖项反映了NSF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。