Infinite families of quantum modular 3-manifold invariants

One of the first key examples of a quantum modular form, which unifies the Witten-Reshetikhin-Turaev (WRT) invariants of the Poincaré 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.

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Received 27 September 2023

Accepted 21 March 2024

Published 7 June 2024