The sln foam 2-category: A combinatorial formulation of Khovanov-Rozansky homology via categorical skew Howe duality

We give an elementary construction of colored sln link homology. The invariant takes values in a 2-category where 2-morphisms are given by foams, singular cobordisms between sln 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 [42].