Hikami's observations on unified WRT invariants and false theta functions

TL;DR

This paper explores the properties of certain $q$-series related to quantum invariants, explaining Hikami's observations on their discontinuities at roots of unity and connecting them to false theta functions.

Contribution

It generalizes Hikami's observations by using Bailey's lemma and false theta functions to explain the discontinuity phenomena of $q$-series at roots of unity.

Findings

Hikami's $H(q)$ exhibits a discontinuity at roots of unity.

The $1/2$-factor in the limit value is explained.

Generalization of Hikami's observations using Bailey's lemma and false theta functions.

Abstract

The object of this article is a family of $q$-series originating from Habiro's work on the Witten-Reshetikhin-Turaev invariants. The $q$-series usually make sense only when $q$ is a root of unity, but for some instances, it also determines a holomorphic function on the open unit disc. Such an example is Habiro's unified WRT invariant $H(q)$ for the Poincar\'{e} homology sphere. In 2007, Hikami observed its discontinuity at roots of unity. More precisely, the value of $H(\zeta)$ at a root of unity is $1/2$ times the limit value of $H(q)$ as $q$ tends towards $\zeta$ radially within the unit disc. In this article, we explain the appearance of the $1/2$-factor and generalize Hikami's observations by using Bailey's lemma and the theory of false theta functions.