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Massless Chiral Supermultiplets of Higher Spins and the $θ$-Twistor

Recently N. Berkovits, motivated by the supertwistor description of ${\cal N}=4 D=4$ super Yang-Mills, considered the generalization of the ${\cal N}=1 D=4$ $θ$-twistor construction to D=10 and applied it for a compact covariant description of ${\cal N}=1 D=10$ super Yang-Mills. This supports the relevance of the $θ$-twistor as a supersymmetric twistor alternative to the well-known supertwistor. The minimal breaking of superconformal symmetry is an inherent property of the $θ$-twistor received from its fermionic components, described by a Grassmannian vector instead of a Grassmannian scalar in the supertwistor. The $θ$-twistor description of the ${\cal N}=1 D=4$ massless chiral supermultiplets $(S, S + 1/2)$ with spins $S=0,1/2,1,3/2,2,...$ is considered here. The description permits to restore the auxiliary $F$ fields of the chiral supermultiplets absent in the supertwistor approach. The proposed formalism is naturally generalized to ${\cal N}=4 D=4 $ and can be used for an off-shell description of the corresponding super Yang-Mills theory.