  We analyze the Euclidean geometry near non-extremal NS5-branes in string
theory, including regions beyond the horizon and beyond the singularity of the
black brane. The various regions have an exact description in string theory, in
terms of cigar, trumpet and negative level minimal model conformal field
theories. We study the worldsheet elliptic genera of these three superconformal
theories, and show that their sum vanishes. We speculate on the significance of
this curious sum rule for black hole physics.
