A puncture in the Euclidean black hole

We consider the backreaction of the winding condensate on the cigar background. We focus on the case of the SL(2, ℝ)_k/U(1) cigar associated with, e.g., the near-horizon limit of k NS5 black-branes. We solve the equations of motion numerically in the large k limit as a function of the amplitude, A, of the winding mode at infinity. We find that there is a critical amplitude, A_c = exp(−γ/2), that admits a critical solution. In string theory, the exact SL(2, ℝ)_k/U(1) cigar CFT fixes completely the winding amplitude, A_s, at infinity. We find that in the large k limit there is an exact agreement, A_c = A_s. The critical solution is a cigar with a puncture at its tip; consequently, the black-hole entropy is carried entirely by the winding condensate. We argue that, in the Lorentzian case, the information escapes the black hole through this puncture.