This paper by W. Nahm explores the classification and representation of supersymmetries in more than 1+1 dimensions, including those with conformal or de Sitter space-time symmetry. It determines the structure of all representations of supersymmetries in flat space and provides formulae for effective computation. The paper shows that for masses m² = 0, 1, 2, the states of the spinning string form supersymmetry multiplets. In Section 2, the paper classifies all manifest supersymmetries in more than 1+1 dimensions. It discusses the structure of the supersymmetry algebra and its representations, and provides a classification of simple supersymmetry algebras. The paper also considers the case of de Sitter space and shows that certain algebras can be contracted to flat space algebras. In Section 3, the paper discusses the little groups of supersymmetries and their representations. It shows that the representations can be induced from the representations of the little group. The paper also discusses the decomposition of the supersymmetry algebra into its even and odd parts and the properties of the little group representations. In Section 4, the paper discusses the representations of the supersymmetry algebra and provides a method for calculating them explicitly. It shows that the representations of the supersymmetry algebra can be decomposed into products of representations of the little group and the irreducible representation of the odd part of the algebra. The paper also discusses the properties of the representations and their relation to the supersymmetry algebra. In Section 5, the paper provides examples of supersymmetry representations and discusses the implications for supersymmetric Yang-Mills and gravity theories. It shows that the low-energy mass levels of the spinning string can be regarded as supersymmetry representations, thus confirming the conjecture that a restricted version of the Neveu-Schwarz-Ramond string yields a renormalizable supersymmetric Yang-Mills and gravity theory in 9+1 dimensions. The paper also discusses the implications of these results for supergravity theories in higher dimensions.This paper by W. Nahm explores the classification and representation of supersymmetries in more than 1+1 dimensions, including those with conformal or de Sitter space-time symmetry. It determines the structure of all representations of supersymmetries in flat space and provides formulae for effective computation. The paper shows that for masses m² = 0, 1, 2, the states of the spinning string form supersymmetry multiplets. In Section 2, the paper classifies all manifest supersymmetries in more than 1+1 dimensions. It discusses the structure of the supersymmetry algebra and its representations, and provides a classification of simple supersymmetry algebras. The paper also considers the case of de Sitter space and shows that certain algebras can be contracted to flat space algebras. In Section 3, the paper discusses the little groups of supersymmetries and their representations. It shows that the representations can be induced from the representations of the little group. The paper also discusses the decomposition of the supersymmetry algebra into its even and odd parts and the properties of the little group representations. In Section 4, the paper discusses the representations of the supersymmetry algebra and provides a method for calculating them explicitly. It shows that the representations of the supersymmetry algebra can be decomposed into products of representations of the little group and the irreducible representation of the odd part of the algebra. The paper also discusses the properties of the representations and their relation to the supersymmetry algebra. In Section 5, the paper provides examples of supersymmetry representations and discusses the implications for supersymmetric Yang-Mills and gravity theories. It shows that the low-energy mass levels of the spinning string can be regarded as supersymmetry representations, thus confirming the conjecture that a restricted version of the Neveu-Schwarz-Ramond string yields a renormalizable supersymmetric Yang-Mills and gravity theory in 9+1 dimensions. The paper also discusses the implications of these results for supergravity theories in higher dimensions.