Topological invariants for gauge theories and symmetry-protected topological phases

We study the braiding statistics of particlelike and looplike excitations in two-(2D) and three-dimensional (3D) gauge theories with finite, Abelian gauge group. The gauge theories that we consider are obtained by gauging the symmetry of gapped, short-range entangled, lattice boson models. We define a set of quantities, called topological invariants, that summarize some of the most important parts of the braiding statistics data for these systems. Conveniently, these invariants are always Abelian phases, even if the gauge theory supports excitations with non-Abelian statistics. We compute these invariants for gauge theories obtained from the exactly soluble group cohomology models of Chen, Gu, Liu, and Wen, and we derive two results. First, we find that the invariants take different values for every group cohomology model with finite, Abelian symmetry group. Second, we find that these models exhaust all possible values for the invariants in the 2D case, and we give some evidence for this in the 3D case. The first result implies that every one of these models belongs to a distinct symmetry-protected topological (SPT) phase, while the second result suggests that these models may realize all SPT phases. These results support the group cohomology classification conjecture for SPT phases in the case where the symmetry group is finite, Abelian, and unitary.