Two-Dimensional Higher Derivative Quantum Gravity with Constant-Curvature Constraint

The quantization of the two-dimensional R2-gravity coupled with conformal matters under the constraint of constant curvature is discussed. The partition function for this system is found for arbitrary genus. It is shown that no critical dimension exists as in the case of the Jackiw·Teitelboim model. In the case of the R2-gravity coupled with fermions described by the Gross-Neveu model a semiclassical solution is constructed which represents a modified version of the CGHS-Witten black hole. Current activity in the study of two-dimensional quantum gravity is motivated by several reasons: It is extremely difficult to clarify the physics of black holes and of the early universe in realistic models of four-dimensional quantum gravity. Solvable toy models such as two-dimensional quantum gravity help simplifying the situation so that one can get an insight into the realistic physics. Moreover two-dimensional quantum gravity is interesting by itself since it is a good laboratory for developing formal procedures in quantizing the theory of gravity. The two-dimensional quan tum gravity has also deep connection with string models. The local covariant two-dimensional action!)