Abelian TQFTS and Schrödinger local systems

In this paper, 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 . We prove that, for 3-cobordisms compatible with Lagrangian correspondences, there is a normalization of the associated Schrödinger 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 Schrödinger representation, which is linearizable at odd root of 1. Motivated by homology of surface configurations with Schrödinger representation as local coefficients, we define another projective action of \mathrm{Mod}(\Sigma) on Schrödinger representations. We show that the latter is not linearizable by identifying the associated 2-cocycle.