Ghost-Free Spectrum of a Quantum String in SL(2,R) Curved Spacetime

TL;DR

This work resolves the long-standing unitarity issue for strings in SL(2,R) curved spacetime by introducing modified SL(2,R) currents with a logarithmic $\ln z$ cut and a new zero mode $\alpha_0^-$, alongside a free-field realization that makes the model exactly solvable. By preserving local current algebras and enforcing monodromy on the physical subspace, the authors obtain a ghost-free spectrum with the total central charge $c = \frac{3k}{k-2}$ and a stress tensor decomposed into free-field pieces, enabling explicit mass-shell and state constructions. The spectrum is shown to consist of principal-series representations, with monodromy quantization $p^+ p^- = -r$ and a mass-shell condition that yields a consistent, unitary theory for both open and closed strings; the construction also recovers flat-space limits as $k \to \infty$. The approach opens avenues for computing correlation functions and generalizing to higher-dimensional curved-spacetime models, including SL(2,R)/U(1) black-hole settings and heterotic variants, thereby broadening the scope of solvable string theories in curved backgrounds. Overall, the paper provides a complete, ghost-free quantum solution for the SL(2,R) WZW model and a robust framework for exploring string dynamics in non-compact curved spacetimes.

Abstract

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