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

TL;DR

This paper investigates the string spectrum in polar coordinates and demonstrates that the one-loop Rindler entropy vanishes, providing insights into the spectrum's structure and the nature of quantum divergences.

Contribution

It classifies the superstring spectrum in polar coordinates, analyzes the cigar partition function, and shows the vanishing of the one-loop Rindler entropy using multiple approaches.

Findings

Superstring spectrum exhibits involution symmetry in the small curvature limit.

All marginal states in polar coordinates are classified and confirmed by partition function analysis.

The one-loop Rindler entropy is shown to vanish across different computational methods.

Abstract

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.