Constant Curvature and Non-Perturbative W3 Gravity

We show that the new classical action for two dimensional gravity (the Jackiw-Teitelboim model) possesses a $W_3$ algebra. We quantise the resulting $W_3$ gravity in the presence of matter fields with arbitrary central charges and obtain the critical exponents. The auxiliary field of the model, expressing the constancy of the scalar curvature, can be interpreted as one of the physical degrees of freedom of the $W_3$ gravity. Our expressions are corrections to some previously published results for this model where the $W_3$ symmetry was not accounted for.