Random walks and the Hagedorn transition

TL;DR

This paper investigates the approach to the Hagedorn temperature in string theory, analyzing the thermal scalar, random walks, and instabilities in various backgrounds, including AdS spaces, and relates these to black hole physics.

Contribution

It provides a detailed analysis of the thermal scalar near the Hagedorn transition, connecting random walks to modular transformations and exploring instabilities in AdS backgrounds.

Findings

The partition function for a single string at finite temperature is a torus amplitude with unit winding.

The sum over random walks of the thermal scalar is a modular transform of spatial configurations of excited strings.

Winding modes indicate an instability in AdS spaces, with the negative mass squared decreasing as AdS radius shrinks.

Abstract

We study details of the approach to the Hagedorn temperature in string theory in various static spacetime backgrounds. We show that the partition function for a {\it single} string at finite temperature is the torus amplitude restricted to unit winding around Euclidean time. We use the worldsheet path integral to derive the statement that the the sum over random walks of the thermal scalar near the Hagedorn transition is precisely the image under a modular transformation of the sum over spatial configurations of a single highly excited string. We compute the radius of gyration of thermally excited strings in $AdS_D\times S^n$. We show that the winding mode indicates an instability despite the AdS curvature at large radius, and that the negative mass squared decreases with decreasing AdS radius, much like the type 0 tachyon. We add further arguments to statements by Barbon and Rabinovici, and by Adams {\it et. al.}, that the Euclidean AdS black hole can thought of as a condensate of the thermal scalar. We use this to provide circumstantial evidence that the condensation of the thermal scalar decouples closed string modes.