A proposal for the geometry of W_n gravity

TL;DR

This paper explores the geometric structure of W_n gravity by relating Hitchin's Teichmuller spaces to W-gravity, demonstrating how W-diffeomorphisms emerge from flat connections and vielbeins without matter fields.

Contribution

It introduces a geometric framework connecting Hitchin's spaces to W-gravity, explicitly demonstrating the emergence of W-diffeomorphisms from flat connections and vielbeins.

Findings

W-diffeomorphisms derived geometrically

Relation established between Hitchin's spaces and W-gravity

Provides measure for W-string path integral

Abstract

We relate the Teichmuller spaces obtained by Hitchin to the Teichmuller spaces of $WA_{n}$-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 $WA_2=W_3$ 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.