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

TL;DR

This work analyzes the string spectrum of flat space in polar coordinates as the small-curvature limit of the cigar CFT, revealing an involution symmetry in the type II spectrum and a complete classification of marginal states. By examining the large-τ2 and large-k limits, as well as conical deformations and multiple entropy-derivation approaches (sum-over-fields, Melvin regularization, and saddle-point methods), the authors demonstrate a consistent vanishing of the genus-one entropy S=0 in Rindler space, rooted in supersymmetry and modular-domain effects. They also clarify how UV divergences from QFT are resolved in string theory through cancellations among bosons and fermions and by reducing the modular domain, with careful attention to the interpretation of marginal states (thermal scalar, discrete dilaton) and their winding properties. The results illuminate the near-horizon string thermodynamics and provide a framework for understanding black hole entropy in highly symmetric backgrounds, while highlighting subtleties in non-supersymmetric settings and conical geometries.

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 $τ_2$ 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.

Figures (12)

• Figure 1: Scheme of the different conformal models and their link to flat space. Top row (from left to right): cigar model, cigar orbifold model, Melvin regularized cigar model. Bottom row (from left to right): flat space, flat space orbifold model, Melvin model. To get from the Melvin models to the other models as $R\to0$, an additional flat dimension emerges that is not depicted here.
• Figure 2: Geometry of the $SL(2,\mathbb{R})_k/U(1)$ model.
• Figure 3: Left figure: plane identified under a $\mathbb{Z}_N$ rotation ($N=3$ in this example). Right figure: resulting flat cone with angular periodicity $2\pi/N$.
• Figure 4: Table of states. Equally colored regions correspond to the same conformal weights in the spectrum. The $r=\bar{r}=0$ sector is absent, and the off-diagonal blocks correspond to secondaries of the Virasoro algebra.
• Figure 5: The cigar geometry with its winding along the angular coordinate. The asymptotic geometry specifies the winding number number and the winding-dependent thermal GSO projection. As $k$ increases, the cigar flattens out and the tip approaches flat space, but described in polar coordinates. This procedure splits the thermal cigar modes in several classes, depending on their radial spread. Modes that are always sensitive to the asymptotic geometry should be scaled out in the large $k$ limit, if one is interested in the theory at the tip. The winding number transforms into winding around the polar origin.