Supersymmetry, a Biased Review

TL;DR

This paper provides a concise review of supersymmetry, covering its mathematical foundations, representations, and geometric aspects, with a focus on superspace, superfields, and nonlinear sigma models.

Contribution

It offers a focused overview of supersymmetry's theoretical structure, emphasizing superspace, superfields, and their geometric interpretations in nonlinear sigma models.

Findings

Superspace generalizes Minkowski space for supersymmetry.

Supersymmetry relates to complex geometry in target space.

Gauging isometries in sigma models involves quotient constructions.

Abstract

This set of lectures contain a brief review of some basic supersymmetry and its representations, with emphasis on superspace and superfields. Starting from the Poincar\'e group, the supersymmetric extensions allowed by the Coleman-Mandula theorem and its generalisation to superalgebras, the Haag, Lopuszanski and Sohnius theorem, are discussed. Minkowski space is introduced as a quotient space and Superspace is presented as a direct generalization of this. The focus is then shifted from a general presentation to the relation between supersymmetry and complex geometry as manifested in the possible target space geometries for N=1 and N=2 supersymmetric nonlinear sigma models in four dimensions. Gauging of isometries in nonlinear sigma models is discussed for these cases, and the quotient construction is described.