On Extensions of Superconformal Algebras

TL;DR

This paper explores extensions of superconformal algebras, focusing on the N=4 case, and discusses how non-central extensions influence field theory, representation theory, and geometric interpretations on graded Riemann spheres.

Contribution

It provides a detailed analysis of N=4 superconformal algebra extensions, highlighting the role of non-central extensions and their impact on theoretical frameworks.

Findings

Non-central extensions are necessary for N=4 superconformal algebra.

Graded Riemann sphere geometry offers a geometric view of the central charge.

Modifications to field and representation theories are discussed.

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 note, the N=4 case is analyzed closely, where the presence of weight zero operators in the field theory forces the introduction of non-central 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.