  We analyze the string spectrum of flat space in polar coordinates, following
the small curvature limit of the $SL(2,\mathbb{R})/U(1)$ cigar CFT. We first
analyze the partition function of the cigar itself, making some clarifications
of the structure of the spectrum that have escaped attention up to this point.
The superstring spectrum (type 0 and type II) is shown to exhibit an involution
symmetry, that survives the small curvature limit. We classify all marginal
states in polar coordinates for type II superstrings, with emphasis on their
links and their superconformal structure. This classification is confirmed by
an explicit large $\tau_2$ analysis of the partition function. Next we compare
three approaches towards the type II genus one entropy in Rindler space: using
a sum-over-fields strategy, using a Melvin model approach and finally using a
saddle point method on the cigar partition function. In each case we highlight
possible obstructions and motivate that the correct procedures yield a
vanishing result: $S=0$. We finally discuss how the QFT UV divergences of the
fields in the spectrum disappear when computing the free energy and entropy
using Euclidean techniques.
