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

Symmetry Integrability and Geometry-Methods and Applications

article

Sunghyuk Park¹ 

[1]California Institute of Technology

关键词: 3-manifold;    knot;    quantum invariant;    complex Chern–Simons theory;    TQFT;    q-series;    colored Jones polynomial;    colored HOMFLY-PT polynomial;   

DOI  :  10.3842/SIGMA.2020.044

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown