A proposal for the geometry of W_n gravity

Abstract

We relate the Teichmuller spaces obtained by Hitchin to the Teichmuller spaces of $W{A}_n$-gravity. The relationship of this space to $W$-gravity is obtained by identifying the flat $PSL(n+1,{\BR})$ connections of Hitchin to generalised vielbeins and connections. This is explicitly demonstrated for $W{A}_2=W_3$ gravity. We show how $W$-diffeomorphisms are obtained in this formulation. We find that particular combinations of the generalised connection play the role of projective connections. We thus obtain $W$-diffeomorphisms in a geometric fashion without invoking the presence of matter fields. This description in terms of vielbeins naturally provides the measure for the gravity sector in the Polyakov path integral for $W$-strings.

17 pages, LaTeX, TIFR/TH/94-14 = IMSc--94/28. revised version (minor changes and references added)

References in corpus (9)

• Classical A_n--W-Geometry
• Covariant $W$ Gravity \& its Moduli Space from Gauge Theory
• W Gravity From Chern--Simons Theory
• Light Cone $W_n$ Geometry and its Symmetries and Projective Field Theory
• W-Gravity
• The Covariant W_3 Action
• Geometry of Higgs and Toda Fields on Riemann Surfaces
• Classical and Quantum W-Gravity
• The Geometric Structure of $W_N$-Gravity

Cited by in corpus (7)

• Anti-self-dual Yang-Mills equations on noncommutative spacetime
• Extended Gauge Invariance in Geometrical Particle Models and the Geometry of W-Symmetry
• Higher dimensional uniformisation and W-geometry
• On W_3-morphisms and the Geometry of Plane Curves
• Goldman Algebra, Opers and the Swapping Algebra
• Extrinsic Hermitian Geometry of Functional Determinants for Vector Subbundles and the Drinfeld--Sokolov Ghost System
• Drinfeld--Sokolov Gravity