$(2+1)$-Dimensional Charged Black Hole in Topologically Massive Electrodynamics

The $2+1$ black hole coupled to a Maxwell field can be charged in two different ways. Besides a Coulomb field, whose potential grows logarithmically in the radial coordinate, there also exists a topological charge due to the existence of a noncontractible cycle. The topological charge does not gravitate and is somehow decoupled from the black hole. This situation changes 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, the Coulomb charge must vanish identically. Hence, in $2+1$ topologically massive electrodynamics coupled to gravity, the black hole can support holonomies only for the Maxwell field. This means that the charged black hole is constructed from the vacuum by means of spacetime identifications.