Isotropic maps and moment map flow

We consider the moduli space of isotropic maps from a closed surface $\mathrm{\Sigma }$ to a symplectic affine space and construct a Kähler moment map geometry, on a space of differential forms on $\mathrm{\Sigma }$, 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.