  We obtain a correspondence between the group of symplectic diffeomorphisms of
a 4-dimensional real torus and the vanishing locus of a certain hyperK\"ahler
moment map. This observation gives rise to a new flow, called the modified
moment map flow. The construction can be adapted to the polyhedral setting, for
which we prove a Duistermaat type theorem. This paper lays out the ground work
for some effective polyhedral symplectic geometry and for a potential
Morse-Bott theory, with applications to the topology of the space of symplectic
maps of the 4-torus.
