What is meant by supergeometry

SUSY extension of field theory including supergravity is greatly motivated by grand-unification models and contemporary string and brane theories. However, there are different notions of a  supermanifold, Lie supergroup and superbundle.

Let us mention a definition of a super Lie group as a Harish -- Chandra pair of a Lie group and a super Lie algebra. There are graded manifolds, graded Lie groups, and graded bundles. It should be emphasized that graded manifolds are not supermanifolds in a strict sense. However, every graded manifold can be associated to a DeWitt infinity-smooth H-supermanifold, and vice versa.

One usually considers supermanifolds over Grassmann algebras of finite rank. This is the case of infinity-smooth GH-, H-, G-supermanifolds and G-supermanifolds. By analogy with familiar smooth manifolds, infinity-smooth supermanifolds are constructed by gluing of open subsets of supervector spaces endowed with the Euclidean topology. However, if a supervector space is provided with the non-Hausdorff DeWitt topology, we are in the case of DeWitt supermanifolds.

In a more general setting, one considers supermanifolds over the so called Arens--Michael algebras of Grassmann origin. They are R-supermanifolds obeying a certain set of axioms. In the case of a finite Grassmann algebra, the category of R-supermanifolds is equivalent to the category of G-supermanifolds.

The most of theoreticians however ignore these mathematical details.

Reference:

G.Sardanashvily, Lectures on supergeometry, arXiv: 0910.0092