Quantum Groups, Support Theory, and Conformal Field Theories

量子群、支持理论和共形场论

基本信息

批准号：
2149817
负责人：
Cris Negron
金额：
$ 15.6万
依托单位：
University of Southern California
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-07-15 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2149817&HistoricalAwards=false
关键词：
Quantum Groups Support Theory Conformal

项目摘要

This project will establish a new framework that connects the physics of certain two-dimensional physical systems, called conformal field theories (CFTs), with the mathematics of certain well-studied algebraic objects, called quantum groups. Establishing such a connection will greatly advance our understanding of both subjects, and provide novel tools for studying two-dimensional CFTs. In implementing this project, new research opportunities will be created for undergraduates and early graduate students. These research opportunities will be supported by grant funding. The PI will also provide direct support for preexisting programs which seek to increase access to mathematics among women and underrepresented minorities.In more detail, one component of the project is to find an equivalence of (ribbon tensor) categories between the representation category of the so-called triplet vertex algebra, and the representation category of small quantum SL(2). Such an equivalence was conjectured by mathematical physicists in the mid-2000’s, and some explicit progress has been made towards its resolution in recent works of the PI and others. A positive resolution to this conjecture will provide the first direct link between logarithmic CFTs and quantum groups, a phenomenon which should be endemic among the most fundamental classes of logarithmic CFTs. A second component of the project is the computation of the Balmer spectrum for small quantum groups associated to arbitrary simple algebraic groups. This computation will employ and establish new links between support theory and geometric representation 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.

该项目将建立一个新的框架，该框架将某些二维物理系统的物理学（称为保形场理论（CFTS））与某些被称为量子群的某些良好代数对象的数学联系在一起。建立这种联系将大大提高我们对这两个主题的理解，并提供研究二维CFT的新工具。在实施该项目时，将为大学生和早期研究生创建新的研究机会。这些研究机会将得到赠款的支持。 PI还将为曾经存在的计划提供直接的支持，这些计划旨在增加妇女和代表性不足的少数群体的数学访问权限。更详细的是，该项目的一个组成部分是在所谓的Triplet Vertex elgebra的表示类别之间找到等效的（丝带张量）类别，以及小量子的代表性类别。该项目的第二个组成部分是对小量子组的Balmer频谱的计算。与任意简单的代数组相关。该计算将采用并建立支持理论与几何表示理论之间的新联系。该奖项反映了NSF的法定使命，并使用基金会的知识分子优点和更广泛的影响审查标准，被认为是通过评估而被视为珍贵的。