Supersymmetry and Supergeometry

Dissertantenkolloquium at the university of Vienna on 29 November 2007. Some references are given, without attemtping to be complete or correct. 1. On the title and the motivation The title of this talk consists of a mathematical part and of a physical part. Supergeometry is mathematical and originated from the works of Berezin, Leites, Kostant (see e.g. in [1, 2, 3, 4]) and many others from 19601970. Supersymmetry is a tool in theoretical physics developped by Salam, Strathdee, Wess, Zumino (see e.g. [5, 6, 7, 8, 9]) and many many others. Since their origin both worlds have contributed to each other and one can say in a certain sense that one without the other does not make sense. I am from the physics world; I am just a simple theoretical physicist by education. The reason I am studying supergeometry now is that I am interested in supersymmetry from a mathematical point of view. Supersymmetry is an interesting symmetry that has many facets and surprises, by my humble opinion. Also from a pure mathematical point of view supersymmetry is interesting; choosing the right supersymmetrical model might give a grip on some fundamental mathematical problem. Among others, supersymmetry helped in giving insight into some index theorems and helped in proving other theorems [10] [11, 12]. In that area I am not an expert, but you might try to look at the work of Witten to see what I mean. Nonetheless, since the tight interplay between physics and mathematics in the playground area of supersymmetry, I will start the talk with saying some words on supersymmetry. After the supersymmetry part, I will give a definition of a superalgebra, give examples and give an example of a supergroup. Then, if time permits, I will say some words on supervarieties. In this talk I will avoid the rich area of super differential geometry see for example the book by Varadarajan[13], the notes of Nelson [14], Deligne and Morgan[15], the book of Tuynman[16],. . .