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 $SL(2,\mathbb{R})_k/U(1)$ 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, $A_c=\exp(-\gamma/2)$, that admits a critical solution. In string theory, the exact $SL(2,\mathbb{R})_k/U(1)$ 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.