Strings in AdS_3 and the SL(2,R) WZW Model. Part 2: Euclidean Black Hole

TL;DR

This work computes the one-loop partition function for strings on Euclidean BTZ/thermal AdS_3 and shows modular invariance, enabling a direct read-off of the Lorentzian spectrum. It confirms the presence of discrete short-string states and a continuous long-string sector, with a precise density of states for long strings arising from Liouville regularization and SL(2,R) reflection data. By connecting Euclidean path integrals to the previously derived spectrum, the paper clarifies the role of spectral flow and the IR structure near AdS_3 horizons. The results provide a robust Euclidean derivation of AdS_3 string dynamics and its black hole limit, with implications for holography and stringy black hole physics.

Abstract

We consider the one-loop partition function for Euclidean BTZ black hole backgrounds or equivalently thermal AdS_3 backgrounds which are quotients of H_3 (Euclidean AdS_3). The one-loop partition function is modular invariant and we can read off the spectrum which is consistent to that found in hep-th/0001053. We see long strings and discrete states in agreement with the expectations.

Figures (3)

• Figure 1: The sum over $n$ is traded for the sum over copies of the fundamental domain.
• Figure 2: Poles in the $\tau$ plane, shown for $w=1$ to 4.
• Figure 3: Shifting the contour of integration picks up the pole residues corresponding to the short string spectrum.