Higher Depth in Representation Theory, Number Theory, and Quantum Topology

表示论是通过线性代数来研究代数对象的对称性。模形式是定义在上半平面上的一类重要的特殊函数，它服从一定的变换性质。它们被广泛用于数论中，以导出有趣的算术关系。量子拓扑是数学的一个分支，它将量子场论的思想与低维拓扑联系起来。本研究项目位于这些领域的接口，并关注几种类型的分区或计数函数的属性。 该项目的主要目的是开发新的工具来研究它们的算术性质和分析行为，并使用它们来解决具体问题。游戏中的技术引起了广泛的数学家和理论物理学家的兴趣。除了研究之外，本项目还将支持研究生的培训。用更专业的术语来说，本项目将加深我们对顶点代数表示特征的理解，特别是那些表现出“更高深度”现象的顶点代数。 为此，PI将介绍和研究更高深度的伪模形式和更高深度的量子模形式。这些是量子模形式（Zagier之后）和密切相关的更高深度模拟模形式的推广。PI将研究多个变量的亚纯Jacobi形式的系数及其傅立叶系数，推广了单变量的现有结果。 在量子拓扑学中，PI将进一步研究Gukov，Pei，Putrov和Vafa的“同调块”（或Z-hat不变量）的性质。特别是，PI将研究Gukov关于垂直3-流形的Z-hat不变量的更高深度量子模块性的猜想。在不同的方向，也是由物理学的动机，PI将证明严格的公式舒尔指数的某些4d N=2 SCFT的启发计算公式的BPS粒子的库仑分支。 相关的方法将被应用于代数几何中与某些弧空间的希尔伯特-庞加莱级数有关的问题。其他方向包括W-代数的阿盖尔-道格拉斯理论，主子空间，parafermionic顶点代数，置换orbifolds。这个奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

Representation theory is the study of symmetries of algebraic objects by means of linear algebra. Modular forms are important special functions defined on the upper half-plane that obey certain transformation properties. They are extensively used in number theory to derive interesting arithmetic relations. Quantum topology is a branch of mathematics that connects ideas of quantum field theory with low-dimensional topology. This research project lies at the interface of these areas and is concerned with properties of several types of partition or counting functions. A primary aim of the project is to develop new tools for studying their arithmetic properties and analytic behaviors and use them to solve concrete problems. The techniques in play are of interest to a broad range of mathematicians and theoretical physicists. In addition to research, this project will also support training of graduate students.In more technical terms, this project will deepen our understanding of characters of representation of vertex algebras, specifically those exhibiting "higher depth" phenomena. For this purpose the PI will introduce and study higher depth false modular forms and higher depth quantum modular forms. These are generalizations of quantum modular forms (after Zagier) and closely related higher depth mock modular forms. The PI will investigate coefficients of meromorphic Jacobi forms in several variables and their Fourier coefficients, generalizing the existing results for a single variable. In quantum topology, the PI will further investigate properties of "homological blocks" (or Z-hat invariants) of Gukov, Pei, Putrov and Vafa. In particular, the PI will study Gukov's conjecture on higher depth quantum modularity of Z-hat invariants of plumbed 3-manifolds. In a different direction, also motivated by physics, the PI will prove rigorous formulas for Schur's indices of certain 4d N=2 SCFTs inspired by counting formulas for BPS particles on the Coulomb branch. Related methods will be applied to problems in algebraic geometry pertaining to Hilbert-Poincare series of certain arc spaces. Other directions include W-algebras for Argyres-Douglas theories, principal subspaces, parafermionic vertex algebras, and permutation orbifolds.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.