Constructs an action of the positive Witt algebra on categorified quantum groups for simply-laced Lie algebras, recovering the foam action in type A and inducing the current-algebra action via trace decategorification.
The $\mathfrak{sl}_n$ foam 2-category: a combinatorial formulation of Khovanov-Rozansky homology via categorical skew Howe duality
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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.
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Action of the Witt algebra on categorified quantum groups
Constructs an action of the positive Witt algebra on categorified quantum groups for simply-laced Lie algebras, recovering the foam action in type A and inducing the current-algebra action via trace decategorification.