Ghost-free and Modular Invariant Spectra of a String in SL(2,R) and Three Dimensional Black Hole Geometry

TL;DR

This paper presents a new ghost-free, modular invariant string spectrum in SL(2,R) and BTZ black hole backgrounds using a novel free field realization that differs from standard approaches.

Contribution

It introduces a different free field realization for SL(2,R) that ensures ghost-free, modular invariant spectra in both the string and black hole models, expanding understanding of string behavior in curved spacetimes.

Findings

Spectra are ghost-free in SL(2,R) and BTZ black hole backgrounds.

The realization avoids logarithmic cuts in currents.

The spectra are modular invariant and applicable to non-trivial curved backgrounds.

Abstract

Spectra of a string in $ SL(2,R) $ and three dimensional (BTZ) black hole geometry are discussed. We consider a free field realization of ^sl (2,R) different from the standard ones in treatment of zero-modes. Applying this to the string model in SL(2,R), we show that the spectrum is ghost-free. The essence of the argument is the same as Bars' resolution to the ghost problem, but there are differences; for example, the currents do not contain logarithmic cuts. Moreover, we obtain a modular invariant partition function. This realization is also applicable to the analysis of the string in the three dimensional black hole geometry, the model of which is described by an orbifold of the SL(2,R) WZW model. We obtain ghost-free and modular invariant spectra for the black hole theory as well. These spectra provide examples of few sensible spectra of a string in non-trivial backgrounds with curved time and, in particular, in a black hole background with an infinite number of propagating modes.