Infinite Families of Quantum Modular 3-Manifold Invariants

TL;DR

This paper introduces an infinite family of quantum modular invariants for 3-manifolds, extending known examples and providing new calculations using advanced lattice cohomology and BPS series theories.

Contribution

It presents the first calculation of AJK series for an infinite family of 3-manifolds and introduces new quantum modular invariants with proven modularity properties.

Findings

Constructed an infinite family of quantum modular invariants for negative definite plumbed 3-manifolds.

Extended lattice cohomology and BPS q-series to new classes of 3-manifolds.

Demonstrated quantum modularity and deformation of WRT invariants in these families.

Abstract

One of the first key examples of a quantum modular form, which unifies the Witten-Reshetikhin-Turaev (WRT) invariants of the Poincar\'e homology sphere, appears in work of Lawrence and Zagier. We show that the series they construct is one instance in an infinite family of quantum modular invariants of negative definite plumbed 3-manifolds whose radial limits toward roots of unity may be thought of as a deformation of the WRT invariants. We use a recently developed theory of Akhmechet, Johnson, and Krushkal (AJK) which extends lattice cohomology and BPS $q$-series of 3-manifolds. As part of this work, we provide the first calculation of the AJK series for an infinite family of $3$-manifolds. Additionally, we introduce a separate but related infinite family of invariants which also exhibit quantum modularity properties.