Drinfeld--Sokolov Gravity

Abstract

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 $SL(2,\mathbb{C})$ subgroup $S$. The basic fields are a hermitian fiber metric $H$ of $K$ and a $(0,1)$ Koszul gauge field $A^{\ast }$ of $K$ valued in a certain negative graded subalgebra $\goth x$ of $\goth g$ related to $\goth s$. The action governing the $H$ and $A^{\ast }$ dynamics is the effective action of a DS field theory in the geometric background specified by $H$ and $A^{\ast }$. Quantization of $H$ and $A^{\ast }$ 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 $A^{\ast }$ configurations, the DS moduli space. The model has a residual gauge symmetry associated to the DS gauge transformations leaving a given field $A^{\ast }$ invariant. This is the DS counterpart of conformal symmetry. Conformal invariance and certain non perturbative features of the model are discussed in detail.

48 pages, Plain TeX, no figures, requires AMS font files AMSSYM.DEF and AMSSYM.TEX

References in corpus (10)

• W-symmetry in Conformal Field Theory
• Quantum $R^2$ Gravity in Two Dimensions
• Light Cone $W_n$ Geometry and its Symmetries and Projective Field Theory
• The Covariant W_3 Action
• A proposal for the geometry of W_n gravity
• Classical Solution of Two Dimensional $R^2$-Gravity and Cross-Over Phenomenon
• Deformation Theory of Holomorphic Vector Bundles, Extended Conformal Symmetry and Extensions of 2D Gravity
• A Krichever--Novikov Formulation of W--Algebras on Riemann Surfaces
• Liouville and Toda field theories on Riemann surfaces
• Extrinsic Hermitian Geometry of Functional Determinants for Vector Subbundles and the Drinfeld--Sokolov Ghost System