Infinite dimensional geometry and quantum field theory of strings. III. Infinite dimensional W-geometry of a second quantized free string

Abstract

The present paper is devoted to various objects of the infinite dimensional W-geometry of a second quantized free string. Our purpose is to include the W-symmetries into the general infinite dimensional geometrical picture related to the quantum field theory of strings, which was described in the first part of the paper (Algebras Groups Geom.11(1994)[to appear]). It is done by the change of the Lie algebra of all infinitesimal reparametrizations of a string world-sheet on the Lie quasi(pseudo)algebra of classical W-transformations (Gervais-Matsuo quasi(pseudo)algebra) as well as of the Virasoro algebra on the central extended enlarged Gervais-Matsuo quasi(pseudo)algebra. A way to obtain W-algebras from classical W-transformations (i.e. Gervais-Matsuo quasi (pseudo)algebra) is proposed. The relation of Gervais-Matsuo differential W-geometry to the Batalin-Weinstein-Karasev-Maslov approach to nonlinear Poisson brackets as well as to L.V.Sabinin program of "nonlinear geometric algebra" are mentioned.