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 Lie supergroup, a supermanifold and a 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, a graded Lie group and a Lie supergroup.
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.
Both graded manifolds and
supermanifolds are phrased in terms of sheaves of graded commutative algebras.
However, graded manifolds are characterized by sheaves on smooth manifolds, while
supermanifolds are constructed by gluing of sheaves on supervector spaces. In
field theory, treating odd fields on a smooth manifold X, one can follow the well-known Serre--Swan theorem extended to
graded manifolds. It states that, if a graded commutative ring over a ring C(X) of smooth functions on X is generated by a projective C(X)-module of finite rank, it is
isomorphic to a ring of graded functions on a graded manifold whose body is X. Therefore,
odd fields in field theory are described in terms of graded manifolds.
One usually consider 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.
References
G. Sardanashvily, Lectures on supergeometry arXiv: 0910.0092
WikipediA Supergeometry
