Solution of the SL(2,R) string in curved spacetime

TL;DR

This paper corrects previous misconceptions about the SL(2,R) WZW model by including zero modes, restoring unitarity, and provides a complete solution using a novel free field quantization approach.

Contribution

It demonstrates the importance of zero modes in the SL(2,R) WZW model and introduces a new Hermitean free field formalism for its exact solution.

Findings

Unitarity is restored when zero modes are included.

The mass-shell condition is modified by zero modes.

The model is solved exactly using a Hermitean free field formalism.

Abstract

The SL(2,R) WZW model, one of the simplest models for strings propagating in curved space time, was believed to be non-unitary in the algebraic treatment involving affine current algebra. It is shown that this was an error that resulted from neglecting a zero mode that must be included to describe the correct physics of non-compact WZW models. In the presence of the zero mode the mass-shell condition is altered and unitarity is restored. The correct currents, including the zero mode, have logarithmic cuts on the worldsheet. This has physical consequences for the spectrum because a combination of zero modes must be quantized in order to impose periodic boundary conditions on mass shell in the physical sector. To arrive at these results and to solve the model completely, the SL(2,R) WZW model is quantized in a free field formalism that differs from previous ones in that the fields and the currents are Hermitean, there are cuts, and there is a new term that could be present more generally, but is excluded in the WZW model.