Extended Sugawara construction for the superalgebrasSU(M+1|N+1). II. The third-order Casimir algebra

The machinery developed in paper I is used to compute the operator-product algebra (OPA) of an operator constructed from the third-order Casimir invariant of the superalgebra $\mathrm{SU}\mathrm{}\left(m\right|n)$. The vertex operator construction of $\mathrm{SU}{\mathrm{}\left(m\right|n)}^{\mathrm{}\left(1\right)\mathrm{}}$ is used to find a realization of the OPA for level $k=1\mathrm{}$ in terms of free bosonic fields only. It turns out that in many respects the conformal structure of the affinized Lie superalgebra $\mathrm{SU}{\mathrm{}\left(m\right|n)}^{\mathrm{}\left(1\right)\mathrm{}}$ is similar to that of the Kač-Moody algebra $\mathrm{SU}{\mathrm{}(m-n)\mathrm{}}^{\mathrm{}\left(1\right)\mathrm{}}$. An intermediate result suggests the occurrence of extended conformal symmetries in $\mathrm{bc}$ systems, to which we will devote a separate discussion.