Drinfeld-Sokolov gravity

A lagrangian euclidean model of Drinfeld-Sokolov (DS) reduction leading to generalW-algebras on a Riemann surface of any genus is presented. The background geometry is given by the DS principal bundleK associated to a complex Lie groupG and anSL(2,) subgroupS. The basic fields are a hermitian fiber metricH ofK and a (0, 1) Koszul gauge fieldA^* ofK valued in a certain negative graded subalgebrar ofg related tos. The action governing theH andA^* dynamics is the effective action of a DS field theory in the geometric background specified byH andA^*. Quantization ofH andA^* implements on one hand the DS reduction and on the other defines a novel model of 2d gravity, DS gravity. The gauge fixing of the DS gauge symmetry yields an integration on a moduli space of DS gauge equivalence classes ofA^* configurations, the DS moduli space. The model has a residual gauge symmetry associated to the DS gauge transformations leaving a given fieldA^* invariant. This is the DS counterpart of conformal symmetry. Conformal invariance and certain non-perturbative features of the model are discussed in detail.