Phases of Quantum Gravity in AdS3 and Linear Dilaton Backgrounds
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We show that string theory in AdS3 has two distinct phases depending on the radius of curvature R_{AdS}=\sqrt{k}l_s. For k>1 (i.e. R_{AdS}>l_s), the SL(2,C) invariant vacuum of the spacetime conformal field theory is normalizable, the high energy density of states is given by the Cardy formula with c_{eff}=c, and generic high energy states look like large BTZ black holes. For k<1, the SL(2,C) invariant vacuum as well as BTZ black holes are non-normalizable, c_{eff}<c, and high energy states correspond to long strings that extend to the boundary of AdS3 and become more and more weakly coupled there. A similar picture is found in asymptotically linear dilaton spacetime with dilaton gradient Q=\sqrt{2/k}. The entropy grows linearly with the energy in this case (for k>\half). The states responsible for this growth are two dimensional black holes for k>1, and highly excited perturbative strings living in the linear dilaton throat for k<1. The change of behavior at k=1 in the two cases is an example of a string/black hole transition. The entropies of black holes and strings coincide at k=1.
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