Hikami's observations on unified WRT invariants 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.