An Integrable Model of Quantum Gravity

TL;DR

This paper introduces an integrable quantum gravity model in two dimensions involving an SU(2) field and dilaton, providing exact solutions through a novel quantization scheme and separation of variables.

Contribution

It presents a new quantization approach for 2D quantum gravity coupled with matter fields, simplifying the analysis of the Wheeler-DeWitt equation using integrable systems techniques.

Findings

Reduced quantum constraints to compatible first order equations

Constructed exact solutions using Knizhnik-Zamolodchikov equations

Demonstrated integrability of the model

Abstract

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-Zamolodchiokov equations.