The $\mathfrak{sl}_n$ foam 2-category: a combinatorial formulation of Khovanov-Rozansky homology via categorical skew Howe duality

TL;DR

This paper introduces a purely combinatorial approach to colored rak{sl}_n link homology using foams and skew Howe duality, simplifying evaluations and resolving sign issues in the theory.

Contribution

It provides a new combinatorial construction of rak{sl}_n link homology with enhanced foam facets and uses skew Howe duality to evaluate closed foams combinatorially.

Findings

Introduces enhanced foam facets to fix sign issues.

Uses skew Howe duality for combinatorial evaluation of foams.

Recovers Khovanov-Rozansky homology via a TQFT-like functor.

Abstract

We give a purely combinatorial construction of colored $\mathfrak{sl}_n$ link homology. The invariant takes values in a 2-category where 2-morphisms are given by foams, singular cobordisms between $\mathfrak{sl}_n$ webs; applying a (TQFT-like) representable functor recovers (colored) Khovanov-Rozansky homology. Novel features of the theory include the introduction of `enhanced' foam facets which fix sign issues associated with the original matrix factorization formulation and the use of skew Howe duality to show that (enhanced) closed foams can be evaluated in a completely combinatorial manner. The latter answers a question posed in math.GT/0708.2228.