Supermanifold: a definition
Although supermanifolds are special cases of noncommutative manifolds, their local structure makes them better suited to study with the tools of standard differential geometry and locally ringed spaces.
A supermanifold M of dimension (p,q) is a topological space M with a sheaf of superalgebras, usually denoted OM or C∞(M), that is locally isomorphic to , where the latter is surely a Grassmann algebra on q generators.
Note that the definition of a supermanifold is similar to that of a differentiable manifold, except that the model space Rp has been replaced by the model superspace Rp|q.
This is different from the alternative definition where, using a fixed Grassmann algebra generated by a countable number of generators Λ, one defines a supermanifold as a point set space using charts with the "even coordinates" taking values in the linear combination of elements of Λ with even grading and the "odd coordinates" taking values which are linear combinations of elements of Λ with odd grading. This raises the question of the physical meaning of all these Grassmann-valued variables. Many physicists claim that they have none and that they are purely formal; if this is the case, this may make the definition in the main part of the article preferable.
Unlike a regular manifold, a supermanifold is not entirely composed of a set of points. Instead, one takes the dual point of view that the structure of a supermanifold M is contained in its sheaf OM of "smooth functions". In the dual point of view, an injective map corresponds to a surjection of sheaves, and a surjective map corresponds to an injection of sheaves.
An alternative approach to the dual point of view is to use the functor of points.
If M is a supermanifold of dimension (p,q), then the underlying space M inherits the structure of a differentiable manifold whose sheaf of smooth functions is OM/I, where I is the ideal generated by all odd functions. Thus M is called the underlying space, or the body, of M. The quotient map OM → OM/I corresponds to an injective map M → M; thus M is a submanifold of M.
- Let M be a manifold. The odd tangent bundle ΠTM is a supermanifold given by the sheaf Ω(M) of differential forms on M.
- More generally, let E → M be a vector bundle. Then ΠE is a supermanifold given by the sheaf Γ(ΛE*). In fact, Π is a functor from the category of vector bundles to the category of supermanifolds.
- Lie supergroups are examples of supermanifolds.
Batchelor's theorem states that every supermanifold is noncanonically isomorphic to a supermanifold of the form ΠE. The word "noncanonically" prevents one from concluding that supermanifolds are simply glorified vector bundles; although the functor Π maps surjectively onto the isomorphism classes of supermanifolds, it is not an equivalence of categories.
The proof of Batchelor's theorem relies in an essential way on the existence of a partition of unity, so it does not hold for complex or real-analytic supermanifolds.
Odd symplectic structures
Odd symplectic form
In many physical and geometric applications, a supermanifold comes equipped with an Grassmann-odd symplectic structure. All natural geometric objects on a supermanifold are graded. In particular, the bundle of two-forms is equipped with a grading. An odd symplectic form ω on a supermanifold is a closed, odd form, inducing a non-degenerate pairing on TM. Such a supermanifold is called a P-manifold. Its graded dimension is necessarily (n,n), because the odd symplectic form induces a pairing of odd and even variables. There is a version of the Darboux theorem for P-manifolds, which allows one to equip a P-manifold locally with a set of coordinates where the odd symplectic form ω is written as
where are even coordinates, and odd coordinates. (An odd symplectic form should not be confused with a Grassmann-even symplectic form on a supermanifold. In contrast, the Darboux version of an even symplectic form is
where are even coordinates, odd coordinates and are either +1 or -1.)
Given an odd symplectic 2-form ω one may define a Poisson bracket known as the antibracket of any two functions F and G on a supermanifold by
Here and are the right and left derivatives respectively and z are the coordinates of the supermanifold. Equipped with this bracket, the algebra of functions on a supermanifold becomes an antibracket algebra.
P and SP-manifolds
Using the Darboux theorem for odd symplectic forms one can show that P-manifolds are constructed from open sets of superspaces glued together by P-transformations. A manifold is said to be an SP-manifold if these transition functions can be chosen to be SP-transformations. Equivalently one may define an SP-manifold as a supermanifold with a nondegenerate odd 2-form ω and a density function ρ such that on each coordinate patch there exist Darboux coordinates in which ρ is identically equal to one.
In Darboux coordinates this definition reduces to
where xa and θa are even and odd coordinates such that
The Laplacian is odd and nilpotent
One may define the cohomology of functions H with respect to the Laplacian. In Geometry of Batalin-Vilkovisky quantization, Albert Schwarz has proven that the integral of a function H over a Lagrangian submanifold L depends only on the cohomology class of H and on the homology class of the body of L in the body of the ambient supermanifold.
A pre-SUSY-structure on a supermanifold of dimension (n,m) is an odd m-dimensional distribution . With such a distribution one associates its Frobenius tensor (since P is odd, the skew-symmetric Frobenius tensor is a symmetric operation). If this tensor is non-degenerate, e.g. lies in an open orbit of , M is called a SUSY-manifold. SUSY-structure in dimension (1, k) is the same as odd contact structure.
 Joseph Bernstein, `Lectures on Supersymmetry` (notes by Dennis Gaitsgory) , "Quantum Field Theory program at IAS: Fall Term"
 A. Schwarz, `Geometry of Batalin-Vilkovisky quantization`, hep-th/9205088
 C. Bartocci, U. Bruzzo, D. Hernandez Ruiperez, The Geometry of Supermanifolds (Kluwer, 1991) ISBN 0-7923-1440-9
- Super manifolds at the Manifold Atlas.