The 2+1 charged black hole in topologically massive Electrodynamics

The 2+1 black hole coupled to a Maxwell field can be charged in two different ways. On the one hand, it can support a Coulomb field whose potential grows logarithmically in the radial coordinate. On the other, due to the existence of a non-contractible cycle, it also supports a topological charge whose value is given by the corresponding Abelian holonomy. Only the Coulomb charge, however, is given by a constant flux integral with an associated continuity equation. The topological charge does not gravitate and is somehow decoupled from the black hole. This situation changes abruptly if one turns on the Chern-Simons term for the Maxwell field. First, the flux integral at infinity becomes equal to the topological charge. Second, demanding regularity of the black hole horizon, it is found that the Coulomb charge (whose associated potential now decays by a power law) must vanish identically. Hence, in 2+1 topologically massive electrodynamics coupled to gravity, the black hole can only support holonomies for the Maxwell field. This means that the charged black hole, as the uncharged one, is constructed from the vacuum by means of spacetime identifications.