Tools for supersymmetry

TL;DR

This work delivers a technical toolkit for supersymmetry, detailing bosonic spacetime symmetries, gauge theories of gravity, spinor calculus in arbitrary dimensions, and the structure of supersymmetry algebras. It combines foundational results (Coleman–Mandula, HLS), AdS and conformal symmetries, and the gauging of conformal gravity with explicit spinor formalisms, including central charges and BPS bounds, as well as a systematic classification of superalgebras and superconformal algebras. The treatment emphasizes concrete algebraic and geometric methods—covariant derivatives, gamma-matrix identities, charge conjugation, and Fierz relations—crucial for constructing SUSY theories and their AdS/CFT applications. Together, these tools enable rigorous formulation and analysis of super-Poincaré, super-AdS, and superconformal theories across multiple dimensions, with practical guidance for handling representations, invariants, and consistency conditions.

Abstract

This is an elementary introduction to basic tools of supersymmetry: the spacetime symmetries, gauge theory and its application in gravity, spinors and superalgebras. Special attention is devoted to conformal and anti-de Sitter algebras.