A First Look at Supersymmetry

TL;DR

This paper provides an introductory overview of supersymmetry and supergeometry, covering foundational concepts and mathematical structures without delving into detailed proofs or phenomenological models.

Contribution

It offers a beginner-friendly exposition of supersymmetry and supergeometry aimed at students and researchers with basic mathematical background.

Findings

Clarifies mathematical issues in classical supersymmetry

Introduces supermanifolds and super-Minkowski space-times

Highlights conceptual understanding without detailed proofs

Abstract

These are expanded notes for a short series of lectures, presented at the University of Luxembourg in 2017, giving an introduction to some of the ideas of supersymmetry and supergeometry. In particular, we start from some motivating facts in physics, pass to the theory of supermanifolds, then to spinors, ending up at super-Minkowski space-times. We examine some salient mathematical issues with understanding supersymmetry in a classical setting and make no attempt to discuss phenomenologically interesting models. Moreover, the presentation is "light" in the sense that nothing is carefully proved. The audience of the seminars ranged from Ph.D. students to more experienced postdocs in mathematics. The audience was assumed to have a working knowledge of differential geometry, elementary category theory, and basic ideas from classical & quantum mechanics. No knowledge of supersymmetry and supergeometry is assumed.