An Integrable Model of Quantum Gravity

We present a new quantization scheme for 2D gravity coupled to an SU(2) principal chiral field and a dilaton; this model represents a slightly simplified version of stationary axisymmetric quantum gravity. The analysis makes use of the separation of variables found in our previous work [1] and is based on a two-time Hamiltonian approach. The quantum constraints are shown to reduce to a pair of compatible first order equations, with the dilaton playing the role of a “clock-field”. Exact solutions of the Wheeler-DeWitt equation are constructed via the integral formula for solutions of the Knizhnik-Zamolodchikov equations. Introduction. At present only few models of quantum gravity exist for which the Wheeler-DeWitt equation can be solved exactly. The known examples include pure gravity [2] and supergravity [3] in three dimensions, as well as certain mini-superspace models such Supported by Alexander von Humboldt Foundation. 1 as static spherically symmetric gravity [4] or locally supersymmetric Bianchi-type models [5]. All of these models describe only finitely many physical degrees of freedom and are thus bound to miss essential features of quantum gravity. It is therefore clearly desirable to find models of quantum gravity with infinitely many physical degrees of freedom, which can still be treated exactly. In this paper we present such a model describing 2D gravity coupled to a principal chiral SU(2) model and a dilaton, and demonstrate that it can be quantized exactly by use of methods developed in the context of 2D integrable systems [6, 7]. The construction is based on our previous work [1], where we have proposed a novel canonical formulation of the Ernst equation based on the complete separation of the equations of motion in the isomonodromic sector of the theory and the use of the logarithmic derivative of the related Ψ-function with respect to the spectral parameter as the fundamental canonical variable. Our model can be regarded as a version of the “midisuperspace” approximation of quantum gravity, but differs from dimensionally reduced gravity in that we have replaced the non-compact configuration space SU(1, 1)/U(1) by the compact group manifold SU(2). Apart from this technical modification (the reasons for which will be briefly explained at the end of this paper) the model captures all the essential features of dimensionally reduced quantum gravity, and is a non-trivial example of a completely integrable model of quantum gravity with infinitely many propagating degrees of freedom. A further intriguing aspect of the present work is the possible relevance of quantum groups suggested by the link with the Knizhnik-Zamolodchikov (KZ) equations. Classical model. For definiteness we consider only Euclidean worldsheets with complex local coordinates (ξ, ξ̄) (the extension of our results to Lorentzian worldsheets is com-