superalgebra and (synthetic ) supergeometry
A supermanifold is a space locally modeled on Cartesian spaces and superpoints.
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.
See at geometry of physics -- supergeometry the section Supermanifolds.
We discuss a description of supermanifolds that goes back to (BerezinLeites).
A supermanifold $X$ of dimension $p|q$ is a ringed space $(|X|, O_X)$ where
the topological space $|X|$ is second countable space, Hausdorff space,
$O_X$ is a sheaf of commutative super algebras that is locally on small enough open subsets $U \subset |X|$ isomorphic to one of the form $C^\infty(\mathbb{R}^p) \otimes \wedge^\bullet \mathbb{R}^q$.
A morphism of supermanifolds is a homomorphism of ringed spaces (…).
Forgetting the graded part by projecting out the nilpotent ideal in $O_X$ (i.e. applying the bosonic modality) yields the underlying ordinary smooth manifold $X_{red}$.
One just writes $C^\infty(X)$ for the super algebra $O_X(X)$ of global sections.
With the obvious morphisms of ringed space this forms the category SDiff of supermanifolds.
For $E \to X$ a smooth finite-rank vector bundle the manifold $X$ equipped with the Grassmann algebra over $C^\infty(X)$ of the sections of the dual bundle
is a supermanifold. This is usually denoted by $\Pi E$.
In particular, let $\mathbb{R}^{p+q} \to \mathbb{R}^p$ be the trivial rank $q$ vector bundle on $\mathbb{R}^p$ then one writes
for the corresponding supermanifold.
(Batchelor’s theorem)
Every supermanifold is isomorphic to one of the form $\Pi E$ where $E$ is an ordinary smooth vector bundle.
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:
There is a natural bijection
so the contravariant embedding of supermanifolds into superalgebra is a full and faithful functor.
Composition with the standard coordinate functions on $\mathbb{R}^{p|q}$ yields an isomorphism
The first statement is a direct extension of the classical fact that smooth manifolds embed into formal duals of R-algebras?.
We discuss a desription of supermanifolds that goes back to (DeWitt 92) and (Rogers).
(…)
Let $SuperPoint$ be the category of superpoints. And $Sh(SuperPoint) = PSh(SuperPoint)$ its presheaf topos.
We discuss a definition of supermanifolds that characterizes them, roughly, as manifolds over this base topos. See (Sachse) and the references at super ∞-groupoid.
See also this post at Theoretical Atlas.
Let
be the sheaf topos over superpoints. Let
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 $\mathbb{K}$-module.
This appears as (Sachse, def. 4.6).
We now want to describe supermanifolds as manifolds in $SuperSet$ modeled on superdomains.
Write SmoothMfd for the category of ordinary smooth manifolds.
A supermanifold is a functor $X : SuperPoint^{op} \to SmoothMfd$ equipped with an equivalence class of supersmooth atlases.
A morphism of supermanifolds is a natural transformation $f : X \to X'$, such that for each pair of charts $u : U \to X$ and $u' : U' \to X'$ the pullback
can be equipped with the structture of a Banach superdomain such that $p_1$ and $p_2$ are supersmooth (…)
This appears as (Sachse, def. 4.13, 4.14).
The categories of supermanifolds defined as locally ringed spaces, def. 1 and as manifolds over superpoints, def. 4 are equivalent.
This appears as (Sachse, theorem 5.1). See section 5.2 there for a discussion of the relation to the DeWitt-definition.
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
The observation that the study of super-structures in mathematics is usefully regarded as taking place over the base topos on the site of super points has been made around 1984 in
Albert Schwarz, On the definition of superspace, Teoret. Mat. Fiz. (1984) Volume 60, Number 1, Pages 37–42, (russian original pdf)
Alexander Voronov, Maps of supermanifolds , Teoret. Mat. Fiz. (1984) Volume 60, Number 1, Pages 43–48
and in
A summary/review is in the appendix of
Anatoly Konechny and Albert Schwarz,
On $(k \oplus l|q)$-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 $(k \oplus l|q)$-dimensional supermanifolds Sel. math., New ser. 6 (2000) 471 - 486
Albert Schwarz, I- Shapiro, Supergeometry and Arithmetic Geometry (arXiv:hep-th/0605119)
A review with more emphasis on the relevant category theory/topos theory is in
The site of formal duals not just to Grassmann algebras but to all super-infinitesimally thickened points is discussed in (Konechny-Schwarz) above and also in
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.
Alberto S. Cattaneo, Florian Schaetz, Introduction to supergeometry, arxiv/1011.3401
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