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

表征理论是用线性代数的方法研究代数对象的对称性。模形式是定义在上半平面上的重要的特殊函数，它遵循一定的变换性质。它们在数论中被广泛应用于推导有趣的算术关系。量子拓扑学是将量子场论思想与低维拓扑学联系起来的一个数学分支。本研究项目是在这些领域的接口，并涉及到几种类型的划分或计数函数的性质。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，并利用它们来解决具体问题。许多数学家和理论物理学家都对这些技术感兴趣。除了研究之外，该项目还将支持研究生的培训。用更专业的术语来说，这个项目将加深我们对顶点代数的表征特征的理解，特别是那些表现出“更高深度”现象的顶点代数。为此，PI将引入和研究更高深度的假模形式和更高深度的量子模形式。这些是量子模形式的推广（在Zagier之后）和密切相关的更高深度模拟模形式。PI将研究几个变量的亚纯雅可比形式的系数及其傅里叶系数，推广现有的单变量结果。在量子拓扑中，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。特别地，PI将研究Gukov关于垂直3流形的Z-hat不变量的高深度量子模性的猜想。在另一个方向上，同样受到物理学的启发，PI将证明某些4d N=2 SCFTs的舒尔指数的严格公式，这些公式受到库仑分支上BPS粒子计数公式的启发。相关的方法将应用于代数几何中关于某些弧空间的希尔伯特-庞加莱级数的问题。其他方向包括阿格雷斯-道格拉斯理论的w代数、主子空间、准费米子顶点代数和置换轨道。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。

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.