Integral Representations of Rank Two False Theta Functions and Their Modularity Properties

TL;DR

This paper explores the modular properties of rank two false theta functions, introducing a new approach using iterated Eichler-type integrals, with applications to vertex algebra characters, superconformal indices, and quantum invariants.

Contribution

It develops a novel method to analyze modularity of rank two false theta functions via iterated integrals, extending understanding of their transformation properties.

Findings

Rank two false theta functions are characterized by iterated Eichler-type integrals.

Application to parafermion characters of type A2 and B2 vertex algebras.

Analysis of modularity in superconformal Schur indices and Gukov-Pei-Putrov-Vafa invariants.

Abstract

False theta functions form a family of functions with intriguing modular properties and connections to mock modular forms. In this paper, we take the first step towards investigating modular transformations of higher rank false theta functions, following the example of higher depth mock modular forms. In particular, we prove that under quite general conditions, a rank two false theta function is determined in terms of iterated, holomorphic, Eichler-type integrals. This provides a new method for examining their modular properties and we apply it in a variety of situations where rank two false theta functions arise. We first consider generic parafermion characters of vertex algebras of type $A_2$ and $B_2$. This requires a fairly non-trivial analysis of Fourier coefficients of meromorphic Jacobi forms of negative index, which is of independent interest. Then we discuss modularity of rank two false theta functions coming from superconformal Schur indices. Lastly, we analyze $\hat{Z}$-invariants of Gukov, Pei, Putrov, and Vafa for certain plumbing ${\tt H}$-graphs. Along the way, our method clarifies previous results on depth two quantum modularity.