Braiding statistics and classification of two-dimensional charge-$2m$ superconductors

We study braiding statistics between quasiparticles and vortices in two-dimensional charge-$2m$ (in units of $e$) superconductors that are coupled to a $\mathbb Z_{2m}$ dynamical gauge field, where $m$ is any positive integer. We show that there exist $16m$ types of braiding statistics when $m$ is odd, but only $4m$ types when $m$ is even. Based on the braiding statistics, we obtain a classification of topological phases of charge-$2m$ superconductors---or formally speaking, a classification of symmetry-protected topological phases, as well as invertible topological phases, of two-dimensional gapped fermions with $\mathbb Z_{2m}^f$ symmetry. Interestingly, we find that there is no nontrivial fermionic symmetry-protected topological phase with $\mathbb Z_4^f$ symmetry.