Higher rank $\hat{Z}$ and $F_K$

TL;DR

This paper extends the study of $q$-series invariants of 3-manifolds to arbitrary root systems $G$, defining new invariants $\uhat{Z}^G$ and $F_K^G$, and exploring their properties and relations to quantum invariants.

Contribution

It introduces definitions of $^G_K$ and $^G_K$ for general root systems and establishes their relations to surgery formulas and quantum $A$-polynomials.

Findings

Defined $^G_K$ and $^G_K$ for negative definite plumbed 3-manifolds and torus knot complements.

Established a surgery formula relating $F_K^G$ to $^G$ of Dehn surgeries.

Showed that $F_K^G$ satisfies a recurrence relation via the quantum $A$-polynomial.

Abstract

We study $q$-series-valued invariants of 3-manifolds that depend on the choice of a root system $G$. This is a natural generalization of the earlier works by Gukov-Pei-Putrov-Vafa [arXiv:1701.06567] and Gukov-Manolescu [arXiv:1904.06057] where they focused on $G={\rm SU}(2)$ case. Although a full mathematical definition for these ''invariants'' is lacking yet, we define $\hat{Z}^G$ for negative definite plumbed 3-manifolds and $F_K^G$ for torus knot complements. As in the $G={\rm SU}(2)$ case by Gukov and Manolescu, there is a surgery formula relating $F_K^G$ to $\hat{Z}^G$ of a Dehn surgery on the knot $K$. Furthermore, specializing to symmetric representations, $F_K^G$ satisfies a recurrence relation given by the quantum $A$-polynomial for symmetric representations, which hints that there might be HOMFLY-PT analogues of these 3-manifold invariants.