Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 16 (2020), 044, 17 pages      arXiv:1909.13002      https://doi.org/10.3842/SIGMA.2020.044

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

Sunghyuk Park
California Institute of Technology, Pasadena, USA

Received January 15, 2020, in final form May 11, 2020; Published online May 24, 2020

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.

Key words: 3-manifold; knot; quantum invariant; complex Chern-Simons theory; TQFT; $q$-series; colored Jones polynomial; colored HOMFLY-PT polynomial.

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