Hagedorn strings and correspondence principle in AdS₃

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. 1

On the relevance of the thermal scalar

“…These expression agree with those written down in equation (4.11). 25 For correlators, we can schematically write 26 T ij th.sc. (x)T kl th.sc.…”

“…Here again φ(x)φ * (y) ≈ ψ 0 (x)ψ * 0 (y) λ 0 (7.12) 25 Recall that for D = 0, the logarithm and the D/2 factor should be dropped. The constantC can then be fixed by comparing more closely.…”

“…Let us look at the time-averaged, high energy string charge tensor in the WZW AdS 3 spacetime [22][23][24] [25]. We will not give a self-contained treatment of this model here and the reader is refered elsewhere for details concerning the thermal spectrum.…”

Winding tachyons on a Banados-Teitelboim-Zanelli spacetime

“…Note added.-After this paper was completed, we learned that tachyons in BTZ have also been investigated from a Euclidean perspective in [35,36].…”

“…In this appendix, we will explain why this route is obstructed. Identifying the tachyon from the Euclidean BTZ partition function is also discussed in [35,36].…”