Lattice cohomology and $q$-series invariants of $3$-manifolds

TL;DR

This paper introduces a new invariant for negative definite plumbed 3-manifolds with spin^c-structures that unifies lattice cohomology and BPS q-series, connecting surface singularities, quantum invariants, and modularity.

Contribution

It develops a unified invariant that extends lattice cohomology and BPS q-series, linking different theories and providing a 2-variable refinement of the -invariant.

Findings

The invariant unifies lattice cohomology and BPS q-series.

It recovers SU(2) quantum invariants at roots of unity.

Provides a 2-variable refinement of the -invariant.

Abstract

An invariant is introduced for negative definite plumbed $3$-manifolds equipped with a spin$^c$-structure. It unifies and extends two theories with rather different origins and structures. One theory is lattice cohomology, motivated by the study of normal surface singularities, known to be isomorphic to the Heegaard Floer homology for certain classes of plumbed 3-manifolds. Another specialization gives BPS $q$-series which satisfy some remarkable modularity properties and recover ${\rm SU}(2)$ quantum invariants of $3$-manifolds at roots of unity. In particular, our work gives rise to a $2$-variable refinement of the $\widehat Z$-invariant.