On the ${\mathrm{\Delta }}_a$ invariants in non-perturbative complex Chern-Simons theory

Recently a set of $q$-series invariants, labelled by ${\mathrm{Spin}}^c$ 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 ${\mathrm{\Delta }}_a$. In this paper we further analyze the structure of these ${\mathrm{\Delta }}_a$ invariants. We review some of the foundations of the ${\mathrm{\Delta }}_a$ invariants and analyze their structure for a subclass of integer homology spheres. In particular, we provide a complete description of the ${\mathrm{\Delta }}_0$ invariants for Brieskorn spheres. Along the way we show that the ${\mathrm{\Delta }}_a$ invariants are not homology cobordism invariants, thereby answering an open question in the literature.