2-Verma modules and the Khovanov-Rozansky link homologies

Abstract

We explain how Queffelec-Sartori's construction of the HOMFLY-PT link polynomial can be interpreted in terms of parabolic Verma modules for ${\mathfrak{gl}}_{2n}$. Lifting the construction to the world of categorification, we use parabolic 2-Verma modules to give a higher representation theory construction of Khovanov-Rozansky's triply graded link homology.

arXiv commentv1, 32 pages, colored figures. v2, 42 pages, Proof of the main result expanded into a new subsection, minor corrections. v3, 24 pages. A gap in the proof of Theorem 4.9 (Dunfield-Gukov-Rasmussen conjecture) was found. As a consequence, it was removed and section 4.6 was substantially reduced. v4, 25 pages, peer reviewed version