Tools for supersymmetry

This is a set of lecture notes, not a research paper. It gathers, in one document, the calculational furniture that supersymmetry practitioners otherwise piece together from scattered papers and appendices. Four blocks are covered. First, bosonic spacetime symmetry: the Coleman-Mandula statement, the anti-de Sitter algebra as a deformation of Poincare, rigid and local conformal symmetry, and a theorem about how covariant derivatives transform that saves a lot of index work. Second, gauge theory and gravity: how the spin connection and the special-conformal gauge field become dependent fields once conventional constraints are imposed, and how Poincare gravity emerges from a gauge choice in conformal gravity. Third, spinors in arbitrary dimension and signature: gamma matrix representations, the charge conjugation matrix, and a systematic derivation of when Majorana, Weyl, Majorana-Weyl and symplectic Majorana spinors exist, summarized in a table indexed by dimension and number of time directions. Fourth, supersymmetry algebras: the Haag-Lopuszanski-Sohnius result, central charges and the BPS bound, super-anti-de Sitter algebras, the classification of simple Lie superalgebras and their real forms, and the Nahm classification of superconformal algebras. The notes include worked exercises and explicit sign conventions, which is where much of the practical value sits: sign errors in this material are endemic and the author is careful to pin them down.