There are different approaches to the definition and theory of supermanifolds in the literature. The definition
is popular. The definition
has been argued to have advantages, see also the references at super ∞-groupoid.
We discuss a description of supermanifolds that goes back to (BerezinLeites).
A supermanifold of dimension is a ringed space where
One just writes for the super algebra of global sections.
is a supermanifold. This is usually denoted by .
In particular, let be the trivial rank vector bundle on then one writes
for the corresponding supermanifold.
Nevertherless, the category of supermanifolds is far from being equivalent to that of vector bundles: a morphism of vector bundles translates to a morphism of supermanifolds that is strictly homogeneous in degrees, while a general morphism of supermanifolds need not be of this form.
But we have the following useful characterization of morphisms of supermanifolds:
The first statement is a direct extension of the classical fact that smooth manifolds embed into formal duals of R-algebras?.
See also this post at Theoretical Atlas.
be the canonical continuum real line under the restricted Yoneda embedding of supermanifolds and equipped with its canonical internal algebra structure, hence by prop. 2 the presheaf of algebras which sends a Grassmann algebra to its even subalgebra, as discussed at superalgebra.
A superdomain is an open subfunctor (…) of a locally convex -module.
This appears as (Sachse, def. 4.6).
We now want to describe supermanifolds as manifolds in modeled on superdomains.
A supermanifold is a functor equipped with an equivalence class of supersmooth atlases.
can be equipped with the structture of a Banach superdomain such that and are supersmooth (…)
This appears as (Sachse, def. 4.13, 4.14).
A brief survey is in
Discussion with an eye towards supergravity is in
Discussion with an eye on integration over supermanifolds is in
Global properties are discussed in
Felix Berezin, D. A. Leĭtes, Supermanifolds, (Russian) Dokl. Akad. Nauk SSSR 224 (1975), no. 3, 505–508; English transl.: Soviet Math. Dokl. 16 (1975), no. 5, 1218–1222 (1976).
I. L. Buchbinder, S. M. Kuzenko, Ideas and methods of supersymmetry and supergravity; or A walk through superspace
A more general variant of this in the spirit of locally algebra-ed toposes is in
Alexander Voronov, Maps of supermanifolds , Teoret. Mat. Fiz. (1984) Volume 60, Number 1, Pages 43–48
A summary/review is in the appendix of
Anatoly Konechny and Albert Schwarz,
On -dimensional supermanifolds, in: Julius Wess, V. Akulov (eds.) Supersymmetry and Quantum Field Theory (D. Volkov memorial volume) Springer-Verlag, 1998 , Lecture Notes in Physics, 509 (arXiv:hep-th/9706003)
Theory of -dimensional supermanifolds Sel. math., New ser. 6 (2000) 471 - 486
Bryce DeWitt, Supermanifolds, Cambridge Monographs on Mathematical Physics, 1984
Alice Rogers, Supermanifolds: Theory and Applications, World Scientific, (2007)
Alice Rogers claims, in Chapter 1, that the smooth-manifold-of-(infinite-dimensional)-Grassmann-algebras approach (the “concrete approach”) is identical to the sheaf-of-ringed-spaces approach (the “algebro-geometric” approach) and that this equivalence is shown in Chapter 8. DeWitt seems unsure of this, but is writing more than 20 years earlier, before the ringed-space approach has been fully developed.
Yuri Manin, Topics in noncommutative geometry, Princeton Univ. Press 1991.
Pierre Deligne, P. Etingof, Daniel Freed, L. Jeffrey, D. Kazhdan, J. Morgan, D.R. Morrison and Edward Witten (eds.) Quantum Fields and Strings, A course for mathematicians, 2 vols. Amer. Math. Soc. Providence 1999. (web version)
V. S. Varadarajan, Supersymmetry for mathematicians: an introduction, AMS and Courant Institute, 2004.
There are many books in physics on supersymmetry (most well known is by Wess and Barger from 1992), but they emphasise more on the supersymmetry algebras rather than on (the superspace and) supermanifolds. They should therefore rather be listed under entry supersymmetry.
See also pdf