Ghost-free and modular invariant spectra of a string in SL(2, R) and three-dimensional black hole geometry

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)over cap>(2, R) different from the standard ones in the treatment of zero-modes. Applying this to the string model in SL(2,PS), 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 a 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. (C) 1998 Elsevier Science B.V.