Covariant $W$ Gravity \& its Moduli Space from Gauge Theory

In this paper we study arbitrary $W$ algebras related to embeddings of $sl_2$ in a Lie algebra $g$. We give a simple formula for all $W$ transformations, which will enable us to construct the covariant action for general $W$ gravity. It turns out that this covariant action is nothing but a Fourier transform of the WZW action. The same general formula provides a geometrical interpretation of $W$ transformations: they are just homotopy contractions of ordinary gauge transformations. This is used to argue that the moduli space relevant to $W$ gravity is part of the moduli space of $G$-bundles over a Riemann surface.