On the $Δ_a$ invariants in non-perturbative complex Chern-Simons theory

Abstract: Recently a set of $q$-series invariants, labelled by $\operatorname{Spin}^c$ structures, for weakly negative definite plumbed $3$-manifolds called the $\widehat{Z}_a$ invariants were discovered by Gukov, Pei, Putrov and Vafa. The leading rational power of the $\widehat{Z}_a$ invariants are invariants themselves denoted by $\Delta_a$. In this paper we further analyze the structure of these $\Delta_a$ invariants. We review some of the foundations of the $\Delta_a$ invariants and analyze their structure for a subclass of integer homology spheres. In particular, we provide a complete description of the $\Delta_0$ invariants for Brieskorn spheres. Along the way we show that the $\Delta_a$ invariants are not homology cobordism invariants, thereby answering an open question in the literature.