Geometry of W-algebras from the affine Lie algebra point of view

To classify the classical field theories with W-symmetry one has to classify the symplectic leaves of the corresponding W-algebra, which are the intersection of the defining constraint and the coadjoint orbit of the affine Lie algebra if the W-algebra in question is obtained by reducing a Wess-Zumino-Novikov-Witten (WZNW) model. The fields that survive the reduction will obey nonlinear Poisson bracket (or commutator) relations in general. For example, the Toda models are well known theories which possess such a nonlinear W-symmetry and many features of these models can only be understood if one investigates the reduction procedure. In this paper we analyse the SL(n, R) case from which the so-called W(n)-algebras can be obtained. One advantage of the reduction viewpoint is that it gives a constructive way to classify the symplectic leaves of the W-algebra-for the n = 2 case corresponding to the coadjoint orbits of the Virasoro algebra and for the n = 3 case which gives rise to the Zamolodchikov algebra. Our method, in principle, is capable of constructing explicit representatives on each leaf. Another attractive feature of this approach is the fact that the global nature of the W-transformations can be explicitly described. The reduction method also enables one to determine the classical highest-weight (HW) states which are the stable minima of the energy on a W-leaf. These are important as only to those leaves can a HW representation space of the W-algebra be associated which contains a classical HW state.