Field theories on Newton-Cartan space-time

Newton-Cartan space-times have an amazingly rich symmetry structure, described by an infinite dimensional Lie group. These space-times are not Riemannian manifolds, indicating non-compliance with relativistic principle. However the notion of Galilean relativity can be comfortably expressed through these geometries. We are in process to develop consistent quantum field theories on these geometries in an attempt to see exact realization of the infinite symmetries in field theory models. In this talk I would rather informally try to elaborate on these ideas and the discuss the added mileage one expects out of this large amount of symmetry in the context of field theories.