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

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 $G=\mathrm{SU}(2)$ 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 $G=\mathrm{SU}(2)$ 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.

References in corpus (7)

• Chern-Simons theory, matrix integrals, and perturbative three-manifold invariants
• Fivebranes and 3-manifold homology
• Super-A-polynomial for knots and BPS states
• 3d Modularity
• Large color $R$-matrix for knot complements and strange identities
• 3d-3d correspondence for mapping tori
• Higher depth quantum modular forms and plumbed $3$-manifolds

Cited by in corpus (13)

• $\hat{Z}$ at large $N$: from curve counts to quantum modularity
• Large color $R$-matrix for knot complements and strange identities
• 3d-3d correspondence for mapping tori
• Quivers for 3-manifolds: the correspondence, BPS states, and 3d $\mathcal{N}$=2 theories
• Branches, quivers, and ideals for knot complements
• Revisiting the Melvin-Morton-Rozansky Expansion, or There and Back Again
• Knot Complement, ADO-Invariants and their Deformations for Torus Knots
• Quantum Modular $\hat{Z}{}^G$-Invariants
• On knots, complements, and 6j-symbols
• Inverted state sums, inverted Habiro series, and indefinite theta functions
• Permutohedra for knots and quivers
• Towards a q-series for osp(2|2n)
• Refined and Generalized $\hat{Z}$ Invariants for Plumbed 3-Manifolds