Quasi-Jacobi forms, Appell-Lerch functions, and false theta functions as q-brackets of functions on partitions
Kathrin Bringmann, Jan-Willem van Ittersum, Jonas Kaszian
TL;DR
The paper develops a partition-based $q$-bracket framework to connect combinatorial generating functions with theta-type modular objects. By introducing algebras $\mathcal{A}$, $\mathcal{S}$, $\mathcal{T}$, and $\Lambda^*$ and analyzing their $q$-brackets, it shows when brackets yield modular theta functions or (quasi-)Jacobi forms, and how products with $s^*$ produce meromorphic quasi-Jacobi forms. A key result is that $\langle t_N\otimes s^*\rangle_q$ decomposes into a Jacobi theta, a false theta, and an Appell–Lerch sum, with explicit completions $\widehat{T}$ and $\widehat{f}_N$ that satisfy modular-type transformation laws. Overall, the work bridges partition statistics with classical modular objects, offering a framework for modularity phenomena in enumerative geometry and related areas.
Abstract
We study certain algebras of theta-like functions on partitions, for which the corresponding generating functions give rise to theta functions, quasi-Jacobi forms, Appell-Lerch sums, and false theta functions.