ov 1 99 2 QMW - 92 - 6 November 1992 hep - th / 9211113 W - GEOMETRY

The geometric structure of theories with gauge fields of spins two and higher should involve a higher spin generalisation of Riemannian geometry. Such geome-tries are discussed and the case of W ∞-gravity is analysed in detail. While the gauge group for gravity in d dimensions is the diffeomorphism group of the space-time, the gauge group for a certain W-gravity theory (which is W ∞-gravity in the case d = 2) is the group of symplectic diffeomorphisms of the cotangent bundle of the space-time. Gauge transformations for W-gravity gauge fields are given by requiring the invariance of a generalised line element. Densities exist and can be constructed from the line element (generalising det g µν) only if d = 1 or d = 2, so that only for d = 1, 2 can actions be constructed. These two cases and the corresponding W-gravity actions are considered in detail. In d = 2, the gauge group is effectively only a subgroup of the symplectic diffeomorphism group. Some of the constraints that arise for d = 2 are similar to equations arising in the study of self-dual four-dimensional geometries and can be analysed using twistor methods , allowing contact to be made with other formulations of W-gravity. While the twistor transform for self-dual spaces with one Killing vector reduces to a Legen-dre transform, that for two Killing vectors gives a generalisation of the Legendre transform.