W-geometry from chiral embeddings

Recent work by Y. Matsuo and the present author is summarized. It is shown that, classically, the conformal Toda equations associated with the simple lie algebras A_n sl (n + 1) follow from the embedding of particular 2D surfaces in CPⁿ. Since these Toda theories provide Noether realizations of W-symmetries, this gives the geometrical interpretation of the corresponding two-dimensional physics.