Drinfeld-Sokolov gravity

A lagrangian euclidean model of Drinfeld-Sokolov (DS) reduction leading to general W-algebras on a Riemann surface of any genus is presented. The background geometry is given by the DS principal bundle K associated to a complex Lie group G and an SL(2, ℂ) subgroup S. The basic fields are a hermitian fiber metric H of K and a (0, 1) Koszul gauge field A* of K valued in a certain negative graded subalgebra r of g related to s. The action governing the H and A* dynamics is the effective action of a DS field theory in the geometric background specified by H and A*. Quantization of H and A* 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 of A* configurations, the DS moduli space. The model has a residual gauge symmetry associated to the DS gauge transformations leaving a given field A* invariant. This is the DS counterpart of conformal symmetry. Conformal invariance and certain non-perturbative features of the model are discussed in detail.