Root lattices and invariant series for plumbed 3-manifolds

Abstract

We study formal series which are invariants of plumbed 3-manifolds twisted by root lattices. These series extend the BPS $q$-series $\widehat{Z}(q)$ recently defined in Gukov-Pei-Putrov-Vafa, Gukov-Manolescu, Park, and further refined in Ri. We show that the series $\widehat{Z}(q)$ is unique in an appropriate sense and decomposes as the average of related series which are themselves invariant under the five Neumann moves amongst plumbing trees. Explicit computations are presented in the case of Brieskorn spheres and a non-Seifert manifold.

Figures (6)

• Figure 1: The five Neumann moves on plumbing trees. Here, $\epsilon \in \{+, -\}$.
• Figure 2: The lattice points of length at most $12$ in the support of $K(z)$ with the corresponding coefficient in the root lattice $A_2$. Here the three positive roots have length $2$ and are marked with dots.
• Figure 3: For a lattice point $\mu$ in the root lattice $A_2$, the figure shows $\mu'$ and the lattice points $\mu'+2w(\rho)$ with $w\neq \iota$.
• Figure 4: The nonzero coefficients $k(\alpha)$ for the lattice points of length at most $6$ in the root lattice $A_2$.
• Figure 5: The Brieskorn sphere $\Sigma(2,3,7)$.

Theorems & Definitions (36)

• Theorem 1
• Theorem 2
• Theorem 3
• Remark 1.1
• Proposition 1.2
• Remark 1.3
• Proposition 1.4
• proof
• Lemma 1.5
• proof