ScalingStacks

In the following we give an outline of the construction. We start with the Khovanov–Rozansky link homologies, which are categorifications of the $\mathfrak{gl}_{N}$ quantum link invariants of Reshetikhin–Turaev [RT90]. These link homologies take the shape of functors

which were constructed by Ehrig–Tubbenhauer–Wedrich in [ETW18] following earlier work on functoriality by Bar-Natan [BN05], Clark–Morrison–Walker [CMW09], and Blanchet [Bla10], and using technology developed by Robert–Wagner [RW20] and Rose–Wedrich [RW16] following Mackaay–Stošić–Vaz [MSV09], Lauda–Queffelec–Rose [LQR15], and Queffelec–Rose [QR16].

It is worth emphasizing that the functors $\mathrm{KhR}_{N}$ considered here are defined combinatorially and normalized to be sensitive to framing changes, in contrast to earlier incarnations of Khovanov–Rozansky homology. In the following, all links are oriented and framed and all link cobordisms are oriented.

The first step in our construction is to show that Khovanov–Rozansky homologies make sense as functorial invariants of links in $S^{3}$, rather than just in ${\mathbb{R}}^{3}$. From the point of view of link embeddings and link cobordisms, there is not much difference between these two cases. A generic link embedding will miss the point $\infty$ if we consider ${\mathbb{R}}^{3}=S^{3}\setminus\{\infty\}$ and a generic link cobordism embedded in $S^{3}\times{[0,1]}$ will miss $\{\infty\}\times{[0,1]}$. However, the analogous statement is no longer true for isotopies of link cobordisms. While link embeddings and their cobordisms can be represented by link diagrams in ${\mathbb{R}}^{2}$ and movies between them, there are additional isotopies of link cobordisms in $S^{3}\times{[0,1]}$, that do not exist in ${\mathbb{R}}^{3}\times{[0,1]}$. In addition to the standard Carter–Rieger–Saito movie moves [CS93, CRS97], a link homology theory that is functorial in $S^{3}$ additionally has to satisfy the so-called sweep-around move (1.1), which encodes a small isotopy of a sheet of link cobordism through $\infty\times{[0,1]}$. The central technical result that we prove in §3 is the following.

This move is significantly more complex than any of the Carter–Saito movie moves because it lacks any locality after the projection to ${\mathbb{R}}^{2}$, and thus has to be checked for any tangle $T$ with two endpoints. We do this in §3 and thereby also demonstrate how computable cobordism maps in Khovanov–Rozansky homology have become.

The main tool in constructing the 4-manifold invariants $\mathcal{S}^{N}_{0}$ is a family of 4-categories with sufficiently well-behaved duals. This is in analogy with the case of quantum invariants of 3-manifolds, which—in one way or another—all depend on a suitable 3-category, such as the ribbon category $\Rep(U_{q}(\mathfrak{gl}_{N}))$ of finite-dimensional representations of quantum $\mathfrak{gl}_{N}$.

In fact, the 4-categories we construct should be thought of as categorified representation categories¹¹ 1 These are related, but not identical, to categories of higher representations of categorified quantum $\mathfrak{gl}_{N}$. of quantum $\mathfrak{gl}_{N}$. They are defined to have unique 0- and 1-morphisms and

• •

  2-morphisms are indexed by finite sets of points in a disk,

• •

  3-morphisms are indexed by tangles in a ball,

• •

  4-morphisms between two tangles $T_{1}$ and $T_{2}$ are elements of the Khovanov–Rozansky homology $\mathrm{KhR}_{N}(T_{1}\sqcup\overline{T_{2}})$ of the link obtained by reflecting $T_{2}$ and gluing it with $T_{1}$ along their corresponding endpoints.

The various ways of composing $k$-morphisms are purely geometric for $k\leq 3$ and use certain cobordism maps between Khovanov–Rozansky homologies to define composition of $4$-morphisms. We give two constructions of such 4-categories, following the axioms of a disklike 4-category in §5 and of a braided monoidal 2-category in §6.

We invite the reader to use Khovanov–Rozansky link homology to build interesting examples of 4-categories following different axiomatizations, and to explore the appropriate incarnations of the sweep-around property in these settings.

The construction of the 4-manifold invariant $\mathcal{S}^{N}_{0}$ is most straightforward when using the setting of a disklike 4-category or the related notion of a lasagna algebra, a 4-dimensional analog of a planar algebra which we introduce in §5. Indeed, the bigraded abelian group $\mathcal{S}^{N}_{0}(W;L)$ is constructed as a skein module (inspired by the 3-dimensional analogs of Conway, Przytycki [Prz91] and Turaev [Tur91]) spanned by certain decorated surfaces in $W$ bounding $L$, which we call lasagna fillings, modulo skein relations imposed by the operad structure of the lasagna algebra.

More generally, there are bigraded abelian groups $\mathcal{S}^{N}_{i}(W;L)$ for $i\geq 1$ that arise as homology groups of the blob complex defined in [MW12] and can be thought of as higher derived analogs of $\mathcal{S}^{N}_{0}(W;L)$. In fact, the construction of the blob complex was motivated by the idea of using the $\mathcal{S}^{N}_{i}(W;L)$ as tools for computing $\mathcal{S}^{N}_{0}(W;L)$. We can think of $\mathcal{S}^{N}_{i}(W;L)$ as analogous to the $i$-th Hochschild homology $\mathrm{HH}_{i}$, where the input algebra of the Hochschild construction has been replaced by the 4-category derived from $\mathrm{KhR}_{N}$ and the implicit circle in the Hochschild construction has been replaced by the 4-manifold $W$. In particular, when $W$ is the standard 4-ball, so $L$ is a link in the 3-sphere, then $\mathcal{S}^{N}_{0}(B^{4};L)$ is isomorphic to the usual Khovanov–Rozansky homology of $L$, and for $i>0$ the abelian groups are zero.

The Khovanov–Rozansky 4-categories can be fed into the general machinery of [Wal, MW12] to produce fully extended $4{+}\epsilon$-dimensional TQFTs. One consequence of this is that the invariants $\mathcal{S}^{N}_{*}$ satisfy a gluing formula [MW12, Theorem 7.2.1] expressed in terms of a tensor product over a category associated to the gluing locus. In particular, we expect that $\mathcal{S}^{N}_{*}(B^{3}\times S^{1};\{2n\text{ points}\}\times S^{1})$ is related to the Hochschild homology of the $\mathfrak{gl}_{N}$ analog of Khovanov’s arc algebra. For applications of the latter to link homology see Rozansky [Roz10] and Willis [Wil21] $(N=2)$ and Gorsky–Hogancamp–Wedrich [GHW21] ($N=\infty$).

We would like to emphasize that $\mathcal{S}^{N}_{*}$ should be thought of as a categorified analog of the $3{+}\epsilon$-dimensional skein module TQFTs, see Walker [Wal], or the 3-dimensional layers of Crane–Yetter–Kauffman TQFTs [CKY97] at generic $q$, but not the $2{+}1$-dimensional Witten–Reshetikhin–Turaev TQFTs [Wit89, RT91].

Precise relationships along these lines and calculations based on the gluing formula appear in later papers [MN20, MWW2, HRW3].

Current constructions of Khovanov–Rozansky link homologies proceed via a functorial invariant of tangles and tangle cobordisms up to isotopy, taking values in the bounded homotopy category of an additive category; see §2. Our proof of Theorem 1.1 is stronger than necessary in the sense that it shows that a certain equivalent reformulation of the sweep-around move holds on the chain level (i.e. not just up to homotopy) provided the tangle $T$ is presented as a partial braid closure.

It is an open question whether the Khovanov–Rozansky homologies are truncations of homotopy-coherent versions with values in chain complexes over the same additive category. If this is indeed the case, then it is plausible that our method of proof would be suitable for an analog of Theorem 1.1 in this setting. Given a fully homotopy-coherent invariant of links in $S^{3}$, we could construct a disklike 4-category enriched in chain complexes (rather than abelian groups), and then use a homotopy colimit construction to extend this invariant to 4-manifolds [MW12, §7]. The result would be a well-defined-up-to-coherent-homotopy chain complex assigned to a 4-manifold $W$ and a boundary condition $L$. At the end of this process we could take homology of this chain complex to produce an abelian group. The invariants $\mathcal{S}^{N}_{*}(W;L)$ can be thought of as an approximation to the latter, given by taking homology (too) early in the construction. One would then expect the two theories to be related by a spectral sequence.

The results here hold for the ordinary Khovanov–Rozansky $\mathfrak{gl}_{N}$ link homologies as well as for their $\mathrm{GL(N)}$-equivariant and deformed versions [Lee05, Kho06, BNM06, Wu12, ETW18]. In the case of links in $S^{3}=\partial B^{4}$, the passage from the ordinary to deformed settings gives rise to spectral sequences that were studied in [Gor04, Ras15, Wu09, RW16]. Lobb and Wu [Lob09, Wu09], following pioneering work of Rasmussen [Ras10], showed that the associated filtrations for the generically deformed knot homologies in $S^{3}=\partial B^{4}$ contain lower bounds on the slice genus, i.e. the minimal genus of smooth surfaces in $B^{4}$ bounding the knot. Using such invariants, Freedman–Gompf–Morrison–Walker have outlined a strategy for testing counterexamples to the smooth 4-dimensional Poincaré conjecture [FGMW10]. One motivation for studying 4-manifold invariants from Khovanov–Rozansky homologies is that analogous spectral sequences might give rise to lower bounds on the genera of smooth surfaces in 4-manifolds $W^{4}$ bounding knots in $M^{3}=\partial W^{4}$.

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].

The authors would like to thank Ian Agol, Chris Douglas, Mike Freedman, Marco Mackaay, Anton Mellit, Stephen Morgan, and Hoel Queffelec for helpful conversations. Scott Morrison was partially supported by Australian Research Council grants ‘Low dimensional categories’ DP160103479 and ‘Quantum symmetries’ FT170100019. Paul Wedrich was supported by Australian Research Council grants ‘Braid groups and higher representation theory’ DP140103821 and ‘Low dimensional categories’ DP160103479.