A proposal for the geometry of Wn-gravity

We relate the Teichmuller spaces obtained by Hitchin to the Teichmuller spaces of WAn-gravity. The relationship of this space to W-gravity is obtained by identifying the flat PSL(n + 1, R) connections of Hitchin to generalised vielbeins and connections. This is explicitly demonstrated for WA2 = W3 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.