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

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.

arXiv comment74 pages, many figures

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Cited by in corpus (19)

• Deformations of colored sl(N) link homologies via foams
• Line and surface defects in Reshetikhin-Turaev TQFT
• Current algebras and categorified quantum groups
• Rigidity in higher representation theory
• Virtual Khovanov homology using cobordisms
• Khovanov homology for virtual tangles and applications
• Symmetric webs, Jones-Wenzl recursions and $q$-Howe duality
• Annular Evaluation and Link Homology
• Cyclicity for categorified quantum groups
• Invariants of 4-manifolds from Khovanov-Rozansky link homology
• Quantum K-theoretic geometric Satake
• ${\mathfrak{gl}}_n$-webs, categorification and Khovanov-Rozansky homologies
• Khovanov polynomials for satellites and asymptotic adjoint polynomials
• Braid group actions from categorical symmetric Howe duality on deformed Webster algebras
• On webs in quantum type $C$
• gl(2) foams and the Khovanov homotopy type
• A categorification of the skew Howe action on a representation category of $U_q(\mathfrak{gl}(m|n))$
• Foams, iterated wreath products, field extensions and Sylvester sums
• A new way to evaluate MOY graphs