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 $^L{ \mathfrak{ g}} \neq { \mathfrak{ gl}}_{ 1|1}$, 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.
Categories: hep-th math.AG math.QA math.RT math.SG
