ScalingStacks

There have been several proposed approaches to constructing homology theories for links in 3-manifolds, or 4-manifold invariants, which either intended to categorify $\mathfrak{sl}_{2}$ or $\mathfrak{gl}_{N}$ quantum invariants or to directly generalize Khovanov–Rozansky homology. These include

1. (1)

   categorifying Witten–Reshetikhin–Turaev invariants at roots of unity, see e.g. Khovanov [Kho16], Qi [Qi14], Elias–Qi [EQ16] and Qi–Sussan [QS17],

2. (2)

   using 2-representations of categorified quantum groups in the sense of Rouquier [Rou08] and Khovanov–Lauda [KL10] to construct a 4-category that can be integrated over 4-manifolds, see e.g. Webster [Web17] for categorified tensor products,

3. (3)

   categorifying skein algebras and 3-manifold skein modules, see Asaeda–Przytycki–Sikora [APS04] Thurston [Thu14] and Queffelec–Wedrich [QW18a, QW21], starting from the thickened annulus, see Grigsby–Licata–Wehrli [GLW18], Beliakova–Putyra–Wehrli [BPW19] and Queffelec–Rose [QR18], or connect sums of $S^{1}\times S^{2}$, see Rozansky [Roz10] and Willis [Wil21].

4. (4)

   giving a mathematically rigorous construction of the BPS spectra (“relative Gromov–Witten invariants”) proposed by Gukov–Putrov–Vafa [GPV17] and Gukov–Pei–Putrov–Vafa [GPPV20] based on Gukov–Schwarz–Vafa [GSV05], see e.g. Gukov–Manolescu [GM21] and Ekholm–Shende [ES19],

5. (5)

   extending Witten’s gauge-theoretic interpretation of Khovanov homology [Wit12] from ${\mathbb{R}}^{3}$ to other 3-manifolds, see also Taubes [Tau13, Tau18].

Comparing these approaches with the invariants defined here may be an interesting topic for further research. We expect a close relationship with approach (2) already at the level of 4-categories, and with approach (3) since it uses the same underlying combinatorics. The latter is especially appealing since (3) is, on the one hand, computationally well-developed for thickened surfaces, but, on the other hand, poses many open questions about the categorification of skein algebras and related quantum cluster algebras, onto which our invariants might shed new light.

While this article was under review, Manolescu–Neithalath [MN20] have shown that the values of $\mathcal{S}^{N}_{0}$ on 2-handlebodies can be computed from the Khovanov–Rozansky homology of cables of attaching links. The procedure takes the form of evaluating the Khovanov–Rozansky homology of the attaching link colored by a categorical Kirby color, a structure developed in the prototypical case $N=2$ by Hogancamp–Rose–Wedrich in [HRW3]. More generally, the values of $\mathcal{S}^{N}_{0}$ can be computed for general 4-manifolds from a handle decomposition, see Manolescu–Walker–Wedrich [MWW2].