The spectrum of strings on BTZ black holes and spectral flow in the SL(2,R) WZW model

TL;DR

This work reframes the spectrum of bosonic strings on rotating BTZ black holes in terms of spectral flow in the $SL(2,R)$ WZW model, showing that twisted sectors arise from spectral flow in the hyperbolic basis. It develops the representation theory and current algebra necessary to describe strings in the BTZ background, including the construction of twisted sectors via vertex operators and the correct projection condition that enforces level matching. A key result is that short strings in BTZ backgrounds yield a continuous spacetime energy spectrum, in contrast to the discrete spectrum found in AdS$_3$, due to the hyperbolic basis diagonalization of $J^2_0$. The paper also clarifies the correct Noether-current generator for the projection in winding sectors, resolving ambiguities that arise in topologically nontrivial sectors and connecting spectral flow to the physical invariant subspace under the BTZ identifications.

Abstract

We study the spectrum of bosonic string theory on rotating BTZ black holes, using a SL(2,R) WZW model. Previously, Natsuume and Satoh have analyzed strings on BTZ black holes using orbifold techniques. We show how an appropriate spectral flow in the WZW model can be used to generate the twisted sectors, emphasizing how the spectral flow works in the hyperbolic basis natural for the BTZ black hole. We discuss the projection condition which leads to the quantization condition for the allowed quantum numbers for the string excitations, and its connection to the anomaly in the corresponding conserved Noether current.