Electromagnetic duality in dyonic Reissner-Nordström/conformal-field-theory correspondence

Abstract

The area law of Bekenstein-Hawking entropy of the black hole suggests that the black hole should have a lower-dimensional holographic description. It has been found recently that such holographic pictures could be set up from the study of the thermodynamics of both outer and inner horizons for a large class of rotating and charged black holes. For a four-dimensional dyonic Reissner-Nordstr\"om black hole, its thermodynamics indicates that it has multiple holographic pictures---not only the electric and magnetic ones corresponding to the conserved electric and magnetic charges carried by the black hole, but also the more generic ones generated by the $SL(2,Z)$ transformations. We show that this $SL(2,Z)$ group originates from the underlying electromagnetic duality symmetry in the Einstein-Maxwell theory. It turns out that the thermodynamics of the black hole not only encodes the information of holographic pictures; moreover, it could reflect the symmetries of the underlying theory.