BPS Geometry, Singularities, and String Theory

BPS 几何、奇点和弦理论

• 批准号: 1802242
• 负责人: Sheldon Katz
• 金额: $ 31万
• 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: 2018
• 资助国家: 美国
• 起止时间: 2018-05-15 至 2023-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1802242&HistoricalAwards=false
• 关键词: BPS; Geometry; Singularities; String; Theory

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.

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上的持续研究。 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，包括几何建模，机器人，控制理论，编码理论，遗传学，以及在这个项目中正在推进的弦理论的应用。 弦论是一种物理理论，它为统一场论提供了一个框架，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还有广泛的应用，包括凝聚态物理学、黑洞物理学以及该项目正在推进的数学应用。 主要研究者将让研究生和博士后参与项目的各个方面，帮助他们的专业发展，并为科学劳动力的发展做出贡献。用更专业的术语来说，该项目将关注三个相互关联的领域：BPS几何，奇点和弦理论。 BPS态是一类超对称态，在许多超对称物理理论中起着重要的作用。 几何技术将应用于推进相关的BPS不变量及其改进的理论，包括在Calabi-Yau三重上的相干层的导出类别上的定向，作为物理理论的全局对称性的傅立叶-向井变换，对数BPS不变量的新理论，以及推进数学中的Bridgeland稳定性和物理学中的Pi稳定性之间的联系。 奇点的几何学研究至少以两种方式被纳入这个项目：正则三重奇点可以用来设计5维超共形场论，某些余维2奇点可以用来设计类似于希钦系统的规范理论。 在弦理论方面的建议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在椭圆纤维化和Jacobi形式的BPS不变量方面的早期工作。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

项目成果

Dimensional reduction of B-fields in F-theory

F 理论中 B 场的降维

• 发表时间: 2022
• 期刊: Pure and applied mathematics quarterly
• 影响因子: 0.7
• 作者: Katz, S.;Taylor, W.
• 通讯作者: Taylor, W.

Towards refining the topological strings on compact Calabi-Yau 3-folds

• DOI: 10.1007/jhep03(2021)266
• 发表时间: 2020-10
• 期刊: Journal of High Energy Physics
• 影响因子: 5.4
• 作者: Min-xin Huang;S. Katz;A. Klemm

Log BPS Numbers of Log Calabi-Yau Surfaces

Log Calabi-Yau 表面的 Log BPS 数

• DOI: 10.1090/tran/8234
• 发表时间: 2020
• 期刊: Transactions of the American Mathematical Society
• 影响因子: 1.3
• 作者: Choi, J;van Garrel, M;Katz, S;and Takahashi, N
• 通讯作者: and Takahashi, N

Swampland constraints on 5d N = 1 supergravity

沼泽地对 5d N = 1 超重力的约束

• DOI: 10.1007/jhep07(2020)080
• 发表时间: 2020
• 期刊: Journal of high energy physics
• 影响因子: 5.4
• 作者: Katz, S;Kim, H;Tarazi, H;Vafa, C
• 通讯作者: Vafa, C

Local BPS Invariants: Enumerative Aspects and Wall-Crossing

局部 BPS 不变量：枚举方面和跨墙

• DOI: 10.1093/imrn/rny171
• 发表时间: 2018
• 期刊: International Mathematics Research Notices
• 影响因子: 1
• 作者: Choi, Jinwon;van Garrel, Michel;Katz, Sheldon;Takahashi, Nobuyoshi
• 通讯作者: Takahashi, Nobuyoshi