Representation theory in unoriented and non-semisimple physics

该项目属于表象理论领域，与理论物理有关。表示论是数学的一个分支，主要研究如何使用线性代数中的技巧来研究对称性。几何学中对称的经典例子包括正方形的反射和球体的旋转。现代数学研究自然会遇到更抽象的对称性概念，因此需要为它们的研究开发更先进的技术。这个项目将识别和研究这些抽象对称性的具体实例，并应用结果来促进对二维(通过Landau-Ginzburg模型)和三维(通过Rozansky-Witten模型)量子场论的数学理解。这些量子场论的两个新特征是，它们定义在不可定向的几何上，或者涉及非半简单的线运算符类别；这两个特征使得它们的研究特别具有挑战性。该项目为高中、本科生和研究生提供研究和职业培训机会。更具体地说，PI将参与三个相关的研究项目，所有这些项目都涉及学生。通过第一个项目，PI将发展一个实等变矩阵因式分解理论，使用范畴表示理论的技术，并应用结果来建立Landau-Ginzburg方向折叠的数学理论。在第二个项目中，PI将扩展先前关于上同调Hall代数的表示理论及其在定向Donaldson-Thomas理论中的应用的工作。在第三个项目中，PI将利用量子超群的表示理论来构造全纯辛流形的等变Rozansky-Witten理论。前两个项目涉及没有方向性的物理理论；所有三个项目都不是半简单的。所考虑的代表性理论的物理起源表明了许多不平凡的相互关系。该奖项反映了NSF的法定使命，并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，被认为值得支持。

This project is in the field of representation theory, with connections to theoretical physics. Representation theory is the branch of mathematics concerned with the study of symmetry using techniques from linear algebra. Classical examples of symmetry in geometry include reflections of a square and rotations of a sphere. Modern research in mathematics naturally encounters more abstract notions of symmetry and so requires the development of more advanced techniques for their study. This project will identify and study concrete instances of these abstract symmetries and apply the results to advance the mathematical understanding of quantum field theory in dimensions two (via Landau-Ginzburg models) and three (via Rozansky-Witten models). Two novel features of these quantum field theories are that they are defined on non-orientable geometries or involve non-semisimple categories of line operators; both features make them particularly challenging to study. This project provides research and career training opportunities for high school, undergraduate and graduate students.More specifically, the PI will engage in three related research projects, all of which involve students. Through the first project the PI will develop a theory of real equivariant matrix factorizations, using techniques from categorical representation theory, and apply the results to formulate a mathematical theory of Landau-Ginzburg orientifolds. In the second project the PI will extend prior work on the representation theory of cohomological Hall algebras and its applications to orientifold Donaldson-Thomas theory. In the third project the PI will use the representation theory of quantum supergroups to construct the equivariant Rozansky-Witten theory of a holomorphic symplectic manifold. The first two projects involve physical theories which are unoriented; all three are non-semisimple. The physical origins of the representation theories considered suggest many non-trivial interrelations.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.