Homological Link Invariants from Floer Theory

Abstract

There is a generalization of Heegaard-Floer theory from ${\mathfrak{gl}}_{1|1}$ to other Lie (super)algebras ${}^L\mathfrak{g}$. The corresponding category of A-branes is solvable explicitly and categorifies quantum $U_q{(}^L\mathfrak{g})$ link invariants. The theory was discovered in \cite{A1,A2}, using homological mirror symmetry. It has novel features, including equivariance and, if ${}^L\mathfrak{g}\ne {\mathfrak{gl}}_{1|1}$, coefficients in categories. In this paper, we describe the theory and how it is solved in detail in the two simplest cases: the ${\mathfrak{gl}}_{1|1}$ theory itself, categorifying the Alexander polynomial, and the ${\mathfrak{su}}_2$ theory, categorifying the Jones polynomial. Our approach to solving the theory is new, even in the familiar ${\mathfrak{gl}}_{1|1}$ case.

146 pages