Data

  1. Title: Singular TQFTs, foams and type $\mathrm{D}$ arc algebras
  2. Authors: Michael Ehrig, Daniel Tubbenhauer and Arik Wilbert
  3. Status: Preprint. Last update: Tue, 22 Nov 2016 18:17:57 GMT
  4. ArXiv link: https://arxiv.org/abs/1611.07444

Abstract

We combinatorially describe the $2$-category of singular cobordisms, called (rank one) foams, which governs the functorial version of Khovanov homology. As an application we topologically realize the type $\mathrm{D}$ arc algebra (which e.g. controls parabolic category $\mathcal{O}$ of type $\mathrm{D}$ with respect to a maximal parabolic of type A) using this singular cobordism construction.
 

A few extra words

In this paper we study the web algebra $\mathfrak{W}$ attached to $\mathfrak{gl}_2$.
The algebra $\mathfrak{W}$ naturally appears in the setup of singular TQFTs in the sense that its bimodule $2$-category is equivalent to the $2$-category of certain singular surfaces called foams, and $\mathfrak{W}$ algebraically controls the functorial version of Khovanov's link homology.
The $2$-category of foams is a sign modified version of Bar-Natan's original cobordism (``$\mathfrak{sl}_2$-foam'') $2$-category attached to Khovanov's link and tangle invariant. The signs are crucial for making Khovanov's link homology functorial, but very delicate to compute in practice. Our first main result can be seen as a combinatorial way to compute these signs, i.e. we define a combinatorial, planar model $c\mathfrak{W}$ of $\mathfrak{W}$.
Next, we know that $\mathfrak{W}$ topologically controls the principal block of the parabolic BGG category $\mathcal{O}$ of type $\mathrm{A}_m$ with parabolic of type $\mathrm{A}_{p}\times\mathrm{A}_{q}$ for $p+q=m$. This follows since Khovanov's original arc algebra is a subalgebra of $\mathfrak{W}$.
There is a type $\mathrm{D}$ generalization $\mathfrak{A}$ of Khovanov's arc algebra, called the type $\mathrm{D}$ arc algebra. The algebra $\mathfrak{A}$ controls the principal block of the parabolic BGG category $\mathcal{O}$ of type $\mathrm{D}_n$ with parabolic of type $\mathrm{A}_{n-1}$.
Our second main result is then that, surprisingly, the type $\mathrm{D}$ arc algebra is also a subalgebra of $\mathfrak{W}$.
Thus, the representation theory of the web algebra (and hence, the cobordism $2$-category of foams) controls the functoriality of Khovanov homology and (certain) parabolic category $\mathcal{O}$ in types $\mathrm{A}_n$ and $\mathrm{D}$.
The picture the reader should keep in mind how these three ``worlds'', represented by elements from the algebras $\mathfrak{W}$, $c\mathfrak{W}$ and $\mathfrak{A}$ is:
This is an illustration how to go from foams (left) to their combinatorial model (middle) to arc diagrams (right).