  We consider the moduli space of isotropic maps from a closed surface $\Sigma$
to a symplectic affine space and construct a K\"ahler moment map geometry, on a
space of differential forms on $\Sigma$, such that the isotropic maps
correspond to certain zeroes of the moment map. The moment map geometry induces
a modified moment map flow, whose fixed point set correspond to isotropic maps.
This construction can be adapted to the polyhedral setting. In particular, we
prove that the polyhedral modified moment map flow induces a strong deformation
retraction from the space of polyhedral maps onto the space of polyhedral
isotropic maps.
