ISOMONODROMIC QUANTIZATION OF DIMENSIONALLY REDUCED GRAVITY

We present a detailed account of the isomonodromic quantization of dim ensionally reduced Einstein gravity with two commuting Killing vectors . This theory constitutes an integrable ''midi-superspace'' version of quantum gravity with infinitely many interacting physical degrees of freedom. The canonical treatment is based on the complete separation o f variables in the isomonodromic sectors of the model. The Wheeler-DeW itt and diffeomorphism constraints are thereby reduced to the Knizhnik -Zamolodchikov equations for SL(2,R). The physical states are manifest ly invariant under the full diffeomorphism group. An infinite set of i ndependent observables a la Dirac exists both at the classical and the quantum level. Using the discrete unitary representations of SL(2,R), we construct explicit quantum states. However, the problem of satisfy ing the additional constraints associated with the coset space SL(2,R) /SO(2) remains open. We briefly discuss the possible implications of o ur results for string theory.