Geodesic
Higher Rank $\hat{Z}$ and $F_K$
Symmetry, integrability and geometry: methods and applications, Tome 16 (2020) Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: $3$-manifold, knot, quantum invariant, complex Chern–Simons theory, TQFT, $q$-series, colored Jones polynomial, colored HOMFLY-PT polynomial. 
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     author = {Sunghyuk Park},
     title = {Higher {Rank} $\hat{Z}$ and $F_K$},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2020_16_a43/}
}
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Sunghyuk Park. Higher Rank $\hat{Z}$ and $F_K$. Symmetry, integrability and geometry: methods and applications, Tome 16 (2020). http://geodesic.mathdoc.fr/item/SIGMA_2020_16_a43/

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