Anyons, ’t Hooft loops, and a generalized connection in three dimensions

We consider a generalized connection in three dimensions and show that it emerges in Chern-Simons-Maxwell theories when one studies the disorder instanton operator. We generalize this construction to non-Abelian theories and find that the disorder operator (the ’t Hooft operator) is equivalent to a generalized Wilson loop in a representation that depends on the Chern-Simons term. We speculate about the effective action of the disorder operator and its applications to the possible phases of the theory in the infrared. We also show that fractional statistics can emerge in gauge theories without a Chern-Simons term if the generalized connection rather than the ordinary connection is used to couple charged particles.