Ghost-free spectrum of a quantum string in SL(2, R) curved spacetime

The unitarity problem in curved spacetime is solved for the string described by the SL(2, R) Wess-Zumino-Witten (WZW) model. The spectrum is computed exactly and demonstrated to be ghost-free. The new features include (i) SL(2, R) left-right symmetry currents that have logarithmic cuts on the world sheet but that satisfy the usual local operator products or commutation rules, (ii) physical states consistent with the monodromy condition of closed strings despite the logarithmic singularity in the currents, and (iii) a new free boson realization for these currents which render the SL(2, R) WZW model completely solvable.