Some geometrical aspects of W - algebras in string theory 1

In this talk, the focus will mainly be on geometrical apects of W algebras as they appear in string theories, rather than in integrable systems and so on. In string theory, W algebras can play a role in two ways. Either they can appear as a ‘global’ symmetry algebra in ordinary string theory, or they can be used as a local symmetry algebra to construct generalizations of string theory usually called W -strings. In the first case the physical states are given by the BRST cohomology of a BRST operator involving only the Virasoro algebra, in the second case of a BRST operator involving the complete W algebra. We will first look briefly at the first possibility, and then in somewhat more detail at the second. Several important open problems will be mentioned; the resolution of some of these seems to be necessary before serious new progress in the field can be made. To see how W algebras can appear as global symmetry algebras and what consequences their presence can have, we consider a non-linear supersymmetric sigma model