Phases of quantum gravity in AdS₃ and linear dilaton backgrounds

We show that string theory in AdS₃ has two distinct phases depending on the radius of curvature R_AdS = √kl_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 = √2/k. The entropy grows linearly with the energy in this case (for k > 1/2). 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.