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Scalar solitons and the microscopic entropy of hairy black holes in three dimensions. (English) Zbl 1214.83016
Summary: General Relativity coupled to a self-interacting scalar field in three dimensions is shown to admit exact analytic soliton solutions, such that the metric and the scalar field are regular everywhere. Since the scalar field acquires slow fall-off at infinity, the soliton describes an asymptotically AdS spacetime in a relaxed sense as compared with the one of Brown and Henneaux. Nevertheless, the asymptotic symmetry group remains to be the conformal group, and the algebra of the canonical generators possesses the standard central extension. For this class of asymptotic behavior, the theory also admits hairy black holes which raises some puzzles concerning a holographic derivation of their entropy à la Strominger. Since the soliton is devoid of integration constants, it has a fixed (negative) mass, and it can be naturally regarded as the ground state of the “hairy sector”, for which the scalar field is switched on. This assumption allows to exactly reproduce the semiclassical hairy black hole entropy from the asymptotic growth of the number of states by means of Cardy formula. Particularly useful is expressing the asymptotic growth of the number of states only in terms of the spectrum of the Virasoro operators without making any explicit reference to the central charges.

##### MSC:
 83C57 Black holes 83E30 String and superstring theories in gravitational theory 83C15 Exact solutions to problems in general relativity and gravitational theory 83C30 Asymptotic procedures (radiation, news functions, $$\mathcal{H}$$-spaces, etc.) in general relativity and gravitational theory 35Q51 Soliton equations 94A17 Measures of information, entropy 81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, $$W$$-algebras and other current algebras and their representations
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