Dilaton gravity coupled to a nonlinear $\sigma$ model in 2+$\varepsilon$ dimensions

Quantum theory of dilaton gravity coupled to a nonlinear sigma model with a maximally symmetric target space is studied in 2+\epsilon dimensions. The ultraviolet stable fixed point for the curvature of the nonlinear sigma model demands a new fixed point theory for the dilaton coupling function. The fixed point of the dilaton coupling is a saddle point similarly to the previous case of the flat target space