2023/05/22 by Mina Aganagic, Aganagic, Mina, Elise LePage +3 · 1 citation
Mathematics · #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Algebraic structures and combinatorial models
paper · pdf · doi:10.48550/arxiv.2305.13480
There is a generalization of Heegaard-Floer theory from \mathfrakgl1|1 to other Lie (super)algebras L\mathfrakg. The corresponding category of A-branes is solvable explicitly and categorifies quantum Uq(L\mathfrakg) link invariants. The theory was discovered in \citeA1,A2, using homological mirror symmetry. It has novel features, including equivariance and, if L\mathfrakg ≠ \mathfrakgl1|1, coefficients in categories. In this paper, we describe the theory and how it is solved in detail in the two simplest cases: the \mathfrakgl1|1 theory itself, categorifying the Alexander polynomial, and the \mathfraksu2 theory, categorifying the Jones polynomial. Our approach to solving the theory is new, even in the familiar \mathfrakgl1|1 case.