ScalingStacks

Foams provide a framework for a combinatorial description of Khovanov–Rozansky link homologies, in a similar way as webs are useful for the type A Reshetikhin-Turaev invariants. We will use $\mathfrak{gl}_{N}$-foams constructed via the combinatorial evaluation formula for closed foams due to Robert–Wagner [RW20]. More precisely, we will organize these $\mathfrak{gl}_{N}$-foams into a monoidal bicategory $\boldsymbol{\mathrm{Foam}}_{N}$ which categorifies the integral form of $\boldsymbol{\mathrm{Web}}_{N}$.

The graded, additive monoidal bicategory $\boldsymbol{\mathrm{Foam}}_{N}$ has objects given by finite sets of points in ${[0,1]}$, each labeled by an element of $\{n,n^{*}|n\in{\mathbb{Z}}_{>0}\}$. The 1-morphisms are (formal direct sums of formal grading shifts of) webs, properly embedded in ${[0,1]}^{2}$ and connecting boundary points of appropriate labels. Note that webs are not considered up to any relations in $\boldsymbol{\mathrm{Foam}}_{N}$. The 2-morphisms are (matrices of degree zero) ${\mathbb{Z}}$-linear combinations of $\mathfrak{gl}_{N}$-foams in ${[0,1]}^{3}$, considered up to isotopy relative to the boundary and certain local relations, as defined in [ETW18, Section 2]. The three compositions are given by (the bilinear extension of) stacking these topological objects along the three interval directions.

Foams are the natural notion of cobordisms between webs and the relations between $2$-morphisms in $\boldsymbol{\mathrm{Foam}}_{N}$ are chosen so that the defining web equalities in $\boldsymbol{\mathrm{Web}}_{N}$ can be lifted to explicit web isomorphisms in $\boldsymbol{\mathrm{Foam}}_{N}$. We refer to [ETW18] for a rigorous definition of $\mathfrak{gl}_{N}$-foams, as well as a complete description of the relations between them, and a survey of various flavors of $\boldsymbol{\mathrm{Foam}}_{N}$. Here we only comment on aspects relevant to the rest of this paper.

Foams are represented by 2-dimensional cell complexes, such that every point has a neighborhood either modelled on ${\mathbb{R}}^{2}$, three half-planes meeting in a line, or the cone on the 1-skeleton of a tetrahedron. Such cone points are called singular vertices of the foam. The points on the line in the second case form a seam of the foam, and the connected components of the set of manifold points are called the facets of the foam. An example of a foam with six singular vertices is shown in Figure 1. The facets are oriented and labeled by positive integers. If three facets meet along a seam, then two of their labels, say $a$ and $b$, sum to the third, $a+b$. The orientation of the seam agrees with the orientation induced by the $a$ and $b$ facets, and disagrees with the $a+b$ facet.

Each facet of a foam in $\boldsymbol{\mathrm{Foam}}_{N}$ admits an action of the algebra of symmetric functions $\Lambda$. This is to say that facets may be decorated by points labeled by symmetric functions, which are allowed to move freely on facets. A point labeled by a product $fg\in\Lambda$ may be split into two points labeled $f$ and $g$ respectively, and a foam with a point labeled $f+g\in\Lambda$ may be split into a sum of foams with points labeled $f$ and $g$ respectively. The $\Lambda$-actions on adjacent facets are compatible in the sense that $f\in\Lambda$ on an $a+b$ facet may be moved across a seam, where it distributes into $\Delta(f)\in\Lambda\otimes\Lambda$ acting on the adjacent $a$ and $b$ facets. The degree of a foam is computed as twice the degree of the symmetric function decoration, minus a weighted Euler characteristic, depending on facet labels.

$\boldsymbol{\mathrm{Foam}}_{N}$ is designed to have finite-dimensional spaces of $2$-morphisms, and in particular, the $\Lambda$-action on each $a$-facet factors through a finite-dimensional quotient, namely $H^{*}(\mathrm{Gr}({\mathbb{C}}^{a}\subset{\mathbb{C}}^{N}))$, the cohomology ring of the Grassmannian of $a$-dimensional subspaces of ${\mathbb{C}}^{N}$, which is obtained as quotient of $\Lambda$ by the ideal $\langle h_{N-a+i}|i>0\rangle$ generated by sufficiently large complete symmetric functions. In the case of a $1$-labeled facet, the symmetric function $e_{1}=h_{1}$ is called the dot.