A Puncture in the Euclidean Black Hole

We consider the backreaction of the winding zero mode on the cigar geometry. 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 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 CFT description of the $SL(2,R)_k/U(1)$ cigar, 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 comment on the Lorentzian interpretation of the solution.