Link Homologies and the Refined Topological Vertex

TL;DR

This paper establishes a direct connection between the refined topological vertex and sl(N) homological invariants of the Hopf link, providing exact calculations and insights into categorification of quantum invariants.

Contribution

It introduces a direct mapping between refined topological vertex and sl(N) link homologies for arbitrary representations, advancing the categorification of quantum group invariants.

Findings

Exact formula for homological invariants of Hopf link with arbitrary representations

Verification of conjectures linking link homologies to BPS state Hilbert spaces

Relation between physical brane charges and mathematical gradings

Abstract

We establish a direct map between refined topological vertex and sl(N) homological invariants of the of Hopf link, which include Khovanov-Rozansky homology as a special case. This relation provides an exact answer for homological invariants of the of Hopf link, whose components are colored by arbitrary representations of sl(N). At present, the mathematical formulation of such homological invariants is available only for the fundamental representation (the Khovanov-Rozansky theory) and the relation with the refined topological vertex should be useful for categorifying quantum group invariants associated with other representations (R_1, R_2). Our result is a first direct verification of a series of conjectures which identifies link homologies with the Hilbert space of BPS states in the presence of branes, where the physical interpretation of gradings is in terms of charges of the branes ending on Lagrangian branes.