Dipolar background field theory and dipolar braiding statistics

We analyze the recently proposed dipolar background field (BF) theory with couplings to charge and dipole currents. The quasiparticles of the theory are either chargelike or dipolelike, and the mutual braiding statistics between chargelike and dipolelike quasiparticles are dipolar, meaning that it depends on the position of the quasiparticle being encircled. The braiding statistics between two dipolelike quasiparticles is that of ordinary anyons. We further prove that the dipolar BF theory is equivalent to the rank-2 tensor BF theory developed earlier as an effective theory for the rank-2 toric code. Although the two theories are equivalent, the dipolar BF formulation embodies the dipole symmetry explicitly and gives a clean insight into the way the dipole symmetry manifests itself in various conservation laws and the dipolar braiding statistics.