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We analyze the string spectrum of flat space in polar coordinates, followingthe small curvature limit of the $SL(2,\mathbb{R})/U(1)$ cigar CFT. We firstanalyze the partition function of the cigar itself, making some clarificationsof 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 involutionsymmetry, that survives the small curvature limit. We classify all marginalstates in polar coordinates for type II superstrings, with emphasis on theirlinks and their superconformal structure. This classification is confirmed byan explicit large $\tau_2$ analysis of the partition function. Next we comparethree approaches towards the type II genus one entropy in Rindler space: usinga sum-over-fields strategy, using a Melvin model approach and finally using asaddle point method on the cigar partition function. In each case we highlightpossible obstructions and motivate that the correct procedures yield avanishing result: $S=0$. We finally discuss how the QFT UV divergences of thefields in the spectrum disappear when computing the free energy and entropyusing Euclidean techniques.

String Theory in Polar Coordinates and the Vanishing of the One-Loop Rindler Entropy