  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.
