Symplectic maps and hyperKähler moment map geometry

We obtain a correspondence between the group of symplectic diffeomorphisms of a $4$-dimensional real torus and the vanishing locus of a certain hyperKähler 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.