  We construct an action of 3-cobordisms on the finite dimensional
Schr\"odinger 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.
