  We further study the elliptic genus of the cigar SL(2,R)/U(1) coset
superconformal field theory. We find that, even in the small curvature,
infinite level limit, there are holomorphic and non-holomorphic parts that are
due to the discrete states and a mismatch in the spectral densities of the
continuum, respectively. The mismatch in the continuum is universal, in the
sense that it is fully determined by the asymptotic cylindrical topology of the
cigar's throat. Since modularity of the elliptic genus requires both the
holomorphic and non-holomorphic parts, the holomorphic term is universal as
well. The contribution of the discrete states is thus present even for
perturbative strings propagating in the background of large Schwarzschild black
holes. We argue that the discrete states live at a stringy distance from the
tip of the cigar both from the conformal field theory wave-function analysis
and from a holonomy space perspective. Thus, the way string theory takes care
of its self-consistency seems to have important consequences for the physics
near horizons, even for parametrically large black holes.
