Light cone W_n geometry and its symmetries and projective field theory

It is shown that the generalized Beltrami differentials and projective connections which appear naturally in induced light cone W_n gravity are geometrical fields parametrizing in one-to-one fashion generalized projective structures on a fixed base Riemann surface. It is also shown that W_n symmetries are nothing but gauge transformations of the flat SL(n,c) vector bundles canonically associated with the generalized projective structures. This provides an original formulation of classical light cone W_n geometry. From the knowledge of the symmetries, the full BRS algebra is derived. Inspired by the results of literature, it is argued that quantum W_n gravity may be formulated as an induced gauge theory of generalized projective connections. This leads to projective field theory. The possible anomalies arising at the quantum level are analysed by solving Wess-Zumino consistency conditions. The implications for induced covariant W_n gravity are briefly discussed. The results presented, valid for arbitrary n, reproduce those obtained for n=2,3 by different methods.