On extensions of superconformal algebras

Abstract

Starting from vector fields that preserve a differential form on a Riemann sphere with Grassmann variables, one can construct a superconformal algebra by considering central extensions of the algebra of vector fields. In this paper, the N=4 case is analyzed closely, where the presence of weight zero operators in the field theory forces the introduction of noncentral extensions. How this modifies the existing field theory, representation theory, and Gelfand–Fuchs constructions is discussed. It is also discussed how graded Riemann sphere geometry can be used to give a geometrical description of the central charge in the N=1 theory.