$\Delta$ Invariants of Plumbed Manifolds

TL;DR

This paper investigates the minimal q-exponent invariant in BPS q-series of negative definite plumbed 3-manifolds, relating it to singularity theory and comparing it with Heegaard-Floer correction terms.

Contribution

It introduces a formula for the Δ invariant of Seifert manifolds and explores its behavior in non-Seifert cases, linking different topological invariants.

Findings

Δ invariant expressed in terms of singularity theory invariants

Behavior of Δ in non-Seifert manifolds illustrated through examples

Comparison between Δ invariants and Heegaard-Floer correction terms

Abstract

We study the minimal $q$-exponent $\Delta$ in the BPS $q$-series $\widehat{Z}$ of negative definite plumbed 3-manifolds equipped with a spin$^{\rm c}$-structure. We express $\Delta$ of Seifert manifolds in terms of an invariant commonly used in singularity theory. We provide several examples illustrating the interesting behaviour of $\Delta$ for non-Seifert manifolds. Finally, we compare $\Delta$ invariants with correction terms in Heegaard-Floer homology.