We construct an action of 3-cobordisms on the finite dimensional Schrödinger representations of the Heisenberg group by Lagrangian correspondences. In addition, we review the construction of the abelian Topological Quantum Field Theory (TQFT) associated with a q-deformation of U(1) for any root of unity q. Restricting to mapping cylinders, our construction yields two projective representations of the mapping class group. We show that their linearizations do not coincide by analysing the corresponding 2-cocycles.
