The interaction between geometry in the adjacent dimensions $2$, $3$ and $4$ is a theme which runs through a great deal of the work of mathematicians over the past few years. Investigations are following three intertwining threads: DG, SG and AG - Differential Geometry, Symplectic Geometry and Algebraic Geometry. The picture we will find pulls together various ideas which have been touched on in the mathematical literature but striking constructions which emerge do not seem to be well-known. We will try to fill this gap.