Higher rank series and root puzzles for plumbed 3-manifolds

We construct plumbed three-manifold invariants in the form of Laurent series twisted by root lattices. Specifically, given a triple consisting of a weakly negative definite plumbing tree, a root lattice, and a generalized \(\mathrm{Spin}^c\)-structure, we construct a family of Laurent series which are invariant up to Neumann moves between plumbing trees and the action of the Weyl group of the root lattice. We show that there are infinitely many such series for irreducible root lattices of rank at least \(2\), with each series depending on a solution to a combinatorial puzzle defined on the root lattice. Our invariants recover certain related series recently defined by Gukov-Pei-Putrov-Vafa, Gukov-Manolescu, Park, and Ri as special cases. Explicit computations are given for Brieskorn homology spheres, for which the series may be expressed as modified higher rank false theta functions.