6-valent vertex in the ${\mathfrak{gl}}_N$ web category and its categorification

Abstract

We define a $2\pi /3$-rotationally invariant 6-valent vertex in the ${\mathfrak{gl}}_N$ web category. When $N=4,5$, we provide a categorification of the 6-valent vertex using ${\mathfrak{gl}}_N$ foams and decompose the hexagon web into a direct sum of indecomposables. A similar decomposition is conjectured for $N\ge 6$.

arXiv comment36 pages, many figures, comments welcome!