Towards a Complete Topological Quantum Field Theory Description for Bosonic Symmetry-Protected-Topological Phases with Abelian Symmetry in Three Dimensions

Topological quantum field theory (TQFT) plays a very important role in understanding topological phases of quantum matter. For example, Chern-Simons theory reveals the key mechanism of charge-flux attachment for fractional quantum hall effect (FQHE). It also completely describes all the essential topological data, e.g., fractionalized statistics, fractionalized charge of quasiparticles in FQHE sytems. Very recently, a new class of topological phases -- symmetry-protected topological (SPT) phases in interacting bosonic systems has been proposed based on the (extended) group cohomology theory. In two dimensions, it has been shown that bosonic SPT phases with Abelian symmetry can be well understood in terms of Chern-Simons theory. In this paper, we attempt to achieve a complete TQFT description for all bosonic SPT phases with Abelian group symmetry in three dimensions. The TQFT description reveals the key mechanism for three dimensional bosonic SPT phases in a simple and intuitive way.