Quantum Groups, Support Theory, and Conformal Field Theories

该项目将建立一个新的框架，将某些二维物理系统（称为共形场论）的物理学与某些被充分研究的代数对象（称为量子群）的数学联系起来。建立这种联系将极大地促进我们对这两个主题的理解，并为研究二维CFTs提供新的工具。在实施这个项目的过程中，将为本科生和早期研究生创造新的研究机会。这些研究机会将由拨款资助。PI还将为旨在增加妇女和代表性不足的少数民族学习数学的现有项目提供直接支持。更详细地说，该项目的一个组成部分是在所谓的三重顶点代数的表示范畴和小量子SL(2)的表示范畴之间找到（带状张量）范畴的等价。数学物理学家在2000年代中期就推测出了这种等价，在PI和其他人最近的工作中，在解决这一问题方面取得了一些明显的进展。对这一猜想的积极解决将提供对数cft和量子群之间的第一个直接联系，这一现象应该是对数cft最基本类别中特有的。该项目的第二个组成部分是计算与任意简单代数群相关的小量子群的Balmer谱。这种计算将在支撑理论和几何表示理论之间建立新的联系。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 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.