An algebraic bootstrap for dimensionally reduced quantum gravity

Cylindrical gravitational waves of Einstein gravity are described by an integrable system (Ernst system) whose quantization is a long standing problem. We propose to bootstrap the quantum theory along the following lines: the quantum theory is described in terms of matrix elements, e.g,, of the metric operator between spectral-transformed multi-vielbein configurations. These matrix elements are computed exactly as solutions of a recursive system of functional equations, which in turn is derived from an underlying quadratic algebra. The Poisson algebra emerging in its classical limit links the spectral-transformed vielbein and the nonlocal conserved charges and can be derived from first principles within the Ernst system. Among the noteworthy features of the quantum theory are: (i) the issue of (non-)renormalizability is sidestepped and (ii) there is an apparently unavoidable "spontaneous" breakdown of the SL(2, R) symmetry that is a remnant of the 4D diffeomorphism invariance in the compactified dimensions. (C) 2000 Elsevier Science B.V. All rights reserved.