2021/05/13 by Aganagic, Mina · 6 citations
#Algebraic Geometry (math.AG) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Representation Theory (math.RT) #Symplectic Geometry (math.SG)
paper · doi:10.48550/arxiv.2105.06039
I provide two solutions to the problem of categorifying quantum link invariants, which work uniformly for all gauge groups and originate in geometry and string theory. The first is based on a category of equivariant B-type branes on \cal X which is a moduli space of singular G-monopoles on \mathbb R3. In this paper, I give the second approach, which is based on a category of equivariant A-type branes on Y with potential W. The first and the second approaches are related by equivariant homological mirror symmetry: Y is homological mirror to X, a core locus of \cal X preserved by an equivariant action related to \mathfrakq. The theory of equivariant A-branes on Y is the same as the derived category of modules of an algebra A, which is a cousin of the algebra considered by Khovanov, Lauda, Rouquier and Webster, but simpler. The result is a new, geometric formulation of Khovanov homology, which generalizes to all groups. In part III, I will explain the string theory origin of the two approaches, and the relation to an approach being developed by Witten. The three parts may be read independently.