Null Field Construction in Conformal and Superconformal Algebras

Abstract

We present the vertex operator construction of null fields (singular vertex operators) based on the oscillator representation of conformal and superconformal algebras with central charge extention. As a consequence of the null field construction we determine the degenerate highest weights and prove the corresponding Kac determinant formulae for the conformal ($N=0$) and superconformal ($N=1, 2$) algebras with $N$ supercharges. We also present an explicit construction of spin fields in the $N=1$ oscillator representations of the Neveu–Schwarz and Ramond algebras with extended central charges.