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

Abstract

Recently a set of $q$-series invariants, labelled by ${\mathrm{Spin}}^c$ structures, for weakly negative definite plumbed $3$-manifolds called the ${\hat{Z}}_a$ invariants were discovered by Gukov, Pei, Putrov and Vafa. The leading rational power of the ${\hat{Z}}_a$ 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.

arXiv comment27 Pages. Comments are welcome. Removed Proposition 5.7 in previous version of paper which may be erroneous. To prepare for journal submission, the claims in Section 4 of the previous version were removed, but still warrant investigation. Similarly Lemma 5.6 in the previous version was removed as it was redundant

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