Hagedorn Strings and Correspondence Principle in AdS(3)

TL;DR

This paper investigates the Hagedorn behavior of bosonic strings in AdS_3, exploring the string/black hole correspondence, and finds how the Hagedorn temperature and black hole size depend on the AdS radius.

Contribution

It derives the Hagedorn temperature behavior in AdS_3 from the SL(2,R) WZW model and proposes a string/black hole correspondence based on thermodynamics.

Findings

Hagedorn temperature increases as AdS radius shrinks.

Density of states near Hagedorn resembles flat space strings.

Black hole size varies with AdS radius, forming larger black holes or string-sized black holes.

Abstract

Motivated by the possibility of formulating a strings/black hole correspondence in AdS space, we extract the Hagedorn behavior of thermal AdS_3 bosonic string from 1-loop partition function of SL(2,R) WZW model. We find that the Hagedorn temperature is monotonically increasing as the AdS radius shrinks, reaches a maximum of order of string scale set by the unitarity bound of the CFT for internal space. The resulting density of states near the Hagedorn temperature resembles the form as for strings in flat space and is dominated by the space-like long string configurations. We then argue a conjectured strings/black hole correspondence in AdS space by applying the Hagedorn thermodynamics. We find the size of the corresponding black hole is a function of the AdS radius. For large AdS radius a black hole far bigger than the string scale will form. On the contrary, when the AdS and string scales are comparable a string size black hole will form. We also examine strings on BTZ background obtained through SL(2,Z) transformation. We find a tachyonic divergence for a BTZ black hole of string scale size.