Link Homologies and the Refined Topological Vertex
Sergei Gukov, Amer Iqbal, Can Kozcaz, Cumrun Vafa

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…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
