The paper by W. Nahm, titled "Supersymmetries and Their Representations," explores the classification and properties of manifest supersymmetries in dimensions greater than 1+1, including those with conformal or de Sitter space-time symmetry. The author determines the structure of representations for flat space supersymmetries and provides formulae for their effective computation. Specifically, he shows that for masses \( m^2 = 0, 1, 2 \), the states of the spinning string form supersymmetry multiplets. The paper is structured into several sections: 1. **Introduction**: Discusses the known supersymmetries in 3+1 dimensions and the importance of supersymmetries in higher dimensions or de Sitter space-time. It also mentions the conjecture that a restricted version of the Neveu-Schwarz-Ramond string can yield a renormalizable supersymmetric Yang-Mills and gravity theory in 9+1 dimensions. 2. **Classification of Supersymmetries**: Classifies all manifest supersymmetries in more than 1+1 dimensions, focusing on the structure of the corresponding little groups and representations. 3. **Little Groups**: Determines the representations of the little groups for supersymmetries graded according to specific decompositions, considering both massless and massive cases. 4. **Representations**: Provides a detailed analysis of the representations of the little groups, including the universal associative enveloping algebra and the number of fermion and boson states. 5. **Examples**: Discusses specific examples of fundamental representations that allow multiplets with highest spin one, including the spinning string and supersymmetric Yang-Mills theories. Nahm's work is significant for understanding the structure and properties of supersymmetry in higher dimensions, particularly in the context of string theory and supergravity.The paper by W. Nahm, titled "Supersymmetries and Their Representations," explores the classification and properties of manifest supersymmetries in dimensions greater than 1+1, including those with conformal or de Sitter space-time symmetry. The author determines the structure of representations for flat space supersymmetries and provides formulae for their effective computation. Specifically, he shows that for masses \( m^2 = 0, 1, 2 \), the states of the spinning string form supersymmetry multiplets. The paper is structured into several sections: 1. **Introduction**: Discusses the known supersymmetries in 3+1 dimensions and the importance of supersymmetries in higher dimensions or de Sitter space-time. It also mentions the conjecture that a restricted version of the Neveu-Schwarz-Ramond string can yield a renormalizable supersymmetric Yang-Mills and gravity theory in 9+1 dimensions. 2. **Classification of Supersymmetries**: Classifies all manifest supersymmetries in more than 1+1 dimensions, focusing on the structure of the corresponding little groups and representations. 3. **Little Groups**: Determines the representations of the little groups for supersymmetries graded according to specific decompositions, considering both massless and massive cases. 4. **Representations**: Provides a detailed analysis of the representations of the little groups, including the universal associative enveloping algebra and the number of fermion and boson states. 5. **Examples**: Discusses specific examples of fundamental representations that allow multiplets with highest spin one, including the spinning string and supersymmetric Yang-Mills theories. Nahm's work is significant for understanding the structure and properties of supersymmetry in higher dimensions, particularly in the context of string theory and supergravity.