2022/12/13 by Matsusaka, Toshiki
#11F27 #57K16 #Complex Variables (math.CV) #FOS: Mathematics #FOS: Physical sciences #Geometric Topology (math.GT) #Mathematical Physics (math-ph) #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2212.06337
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 Poincaré homology sphere. In 2007, Hikami observed its discontinuity at roots of unity. More precisely, the value of H(ζ) at a root of unity is 1/2 times the limit value of H(q) as q tends towards ζ 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.