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

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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.