[Paper Review] sl(3)-Foams and the Khovanov-Lauda categorification of quantum sl(k)

Marco Mackaay|ArXiv.org|May 13, 2009

Algebraic structures and combinatorial models4 references14 citations

TL;DR

This paper introduces a 2-functor called 'foamation' that maps Khovanov-Lauda's categorified quantum sl(k) 2-category to a 2-category of universal sl(3) foams with corners, establishing a link between categorified quantum groups and foam models. The key contribution is proving that this foamation functor preserves all defining relations of the Khovanov-Lauda 2-category over F2, providing a concrete realization of the categorification via foams and suggesting a unifying framework via the Kapustin-Li formula.

ABSTRACT

In this paper I define certain interesting 2-functors from the Khovanov-Lauda 2-category which categorifies quantum sl(k), for any k>1, to a 2-category of universal sl(3) foams with corners. For want of a better name I use the term "foamation" to indicate those 2-functors. I conjecture the existence of similar 2-functors to the 2-category of sl(n) foams with corners, for any n>1.

Motivation & Objective

• To establish a 2-functor (called 'foamation') from the Khovanov-Lauda 2-category Uk to a 2-category of universal sl(3) foams with corners.
• To verify that this foamation functor preserves all defining relations of the Khovanov-Lauda 2-category, particularly the nilHecke and bubble relations, over the field F2.
• To conjecture the existence of analogous foamation functors for sl(n) foams for all n ≥ 2, extending the framework beyond sl(3).
• To explore the potential of the Kapustin-Li formula as a unifying mechanism for deriving all relations in the categorified quantum group setting.

Proposed method

• Define a 2-functor from the Khovanov-Lauda 2-category Uk to a 2-category of universal sl(3) foams with corners, mapping 1-morphisms (sequences of dots and crossings) to foam cobordisms.
• Represent 2-morphisms (generators and relations) as foam diagrams with labeled regions, singular curves, and dots, using the Kapustin-Li formula for foam evaluation.
• Verify that all relations in Uk—such as the nilHecke relations, bubble relations, and R-matrix relations—are preserved under the foamation functor via isotopy and foam calculus.
• Use graphical calculus to map each KL-generator (e.g., crossings, dots) to a corresponding foam structure, with degree shifts and labels determined by the bilinear form on weights.
• Apply the infinite Grassmannian relation to handle 'fake bubbles' with negative labels, ensuring degree positivity and consistency.
• Prove preservation of relations by analyzing foam isotopies and applying known foam relations like (DR), (RD), and (SqR) in the target 2-category.

Experimental results

Research questions

• RQ1 Does there exist a 2-functor from the Khovanov-Lauda 2-category Uk to the 2-category of universal sl(3) foams that preserves all defining relations of Uk over F2?
• RQ2 Can the Kapustin-Li formula be used to systematically derive all relations in the categorified quantum group setting via foam models?
• RQ3 Are the foamation functors faithful for sufficiently large n, allowing all relations in the categorified quantum sl(k) to be recovered from foam calculus?
• RQ4 How do the relations in the Khovanov-Lauda 2-category, such as the nilHecke and R-matrix relations, translate into foam isotopies and algebraic identities in the target 2-category of sl(3) foams?

Key findings

• The foamation 2-functor preserves all relations in the Khovanov-Lauda 2-category Uk over F2, including the nilHecke relations, bubble relations, and R-matrix relations.
• The first and second nilHecke relations are preserved due to isotopy invariance and the structure of foam maps.
• The third nilHecke relation corresponds to the (DR) foam relation and is preserved via direct computation from the foamation map.
• The R(ν)-relations are preserved: for |i−j|>1, isotopy suffices; for adjacent i,j, the (RD) relation applies to discs bounded by singular curves.
• The (SqR) relation is verified in the case s−λ1=2, where both A and B are non-zero, showing that A−B=C holds via singular curve analysis.
• The foamation functor is conjectured to extend to sl(n) foams for all n≥2, with the Kapustin-Li formula providing a unifying mechanism for deriving all relations in the categorification.