Dipolar BF 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.