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 [D. Korotkin and H. Nicolai, Phys. Rev. Lett. 74 (1995) 1272] 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.