Abelian TQFTS and Schr\"odinger local systems

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 $q$. We prove that for3-cobor\-disms compatible with Lagrangian correspondences, there is a normalization of the associated Schr\"odinger bimodule action that reproduces the abelian TQFT. The full abelian TQFT provides a projective representation of the mapping class group $\mathrm{Mod}(\Sigma)$ on the Schr\"odinger representation,which is linearizable at odd root of 1. Motivated by homology of surface configurations with Schr\"odinger representation as local coefficients, we define another projective action of $\mathrm{Mod}(\Sigma)$ on Schr\"odinger representations. We show that the latter is not linearizable by identifying the associated 2-cocycle.