Supermanifold

In physics and mathematics, supermanifolds are generalizations of the manifold concept based on ideas coming from supersymmetry. Several definitions are in use, some of which are described below.
Informal definition.
An informal definition is commonly used in physics textbooks and introductory lectures. It defines a supermanifold as a manifold with both bosonic and fermionic coordinates. Locally, it is composed of coordinate charts that make it look like a "flat", "Euclidean" superspace. These local coordinates are often denoted by
where "x" is the (real-number-valued) spacetime coordinate, and formula_2 and formula_3 are Grassmann-valued spatial "directions".
The physical interpretation of the Grassmann-valued coordinates are the subject of debate; explicit experimental searches for supersymmetry have not yielded any positive results. However, the use of Grassmann variables allow for the tremendous simplification of a number of important mathematical results. This includes, among other things a compact definition of functional integrals, the proper treatment of ghosts in BRST quantization, the cancellation of infinities in quantum field theory, Witten's work on the Atiyah-Singer index theorem, and more recent applications to mirror symmetry.
The use of Grassmann-valued coordinates has spawned the field of supermathematics, wherein large portions of geometry can be generalized to super-equivalents, including much of Riemannian geometry and most of the theory of Lie groups and Lie algebras (such as Lie superalgebras, "etc.") However, issues remain, including the proper extension of deRham cohomology to supermanifolds.
Definition.
Three different definitions of supermanifolds are in use. One definition is as a sheaf over a ringed space; this is sometimes called the "algebro-geometric approach". This approach has a mathematical elegance, but can be problematic in various calculations and intuitive understanding. A second approach can be called a "concrete approach"; as it is capable of simply and naturally generalizing a broad class of concepts from ordinary mathematics. It requires the use of an infinite number of supersymmetric generators in its definition; however, all but a finite number of these generators carry no content, as the concrete approach requires the use of a coarse topology that renders almost all of them equivalent. Surprisingly, these two definitions, one with a finite number of supersymmetric generators, and one with an infinite number of generators, are equivalent.
A third approach describes a supermanifold as a base topos of a superpoint. This approach remains the topic of active research.
Algebro-geometric: as a sheaf.
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 formula_4, where the latter is a Grassmann (Exterior) algebra on "q" generators.
A supermanifold M of dimension "(1,1)" is sometimes called a super-Riemann surface.
Historically, this approach is associated with Felix Berezin, Dimitry Leites, and Bertram Kostant.
Concrete: as a smooth manifold.
A different definition describes a supermanifold in a fashion that is similar to that of a smooth manifold, except that the model space formula_5 has been replaced by the "model superspace" formula_6.
To correctly define this, it is necessary to explain what formula_7 and formula_8 are. These are given as the even and odd real subspaces of the one-dimensional space of Grassmann numbers, which, by convention, are generated by a countably infinite number of anti-commuting variables: i.e. the one-dimensional space is given by formula_9 where "V" is infinite-dimensional. An element "z" is termed "real" if formula_10; real elements consisting of only an even number of Grassmann generators form the space formula_7 of "c-numbers", while real elements consisting of only an odd number of Grassmann generators form the space formula_8 of "a-numbers". Note that c-numbers commute, while "a-numbers" anti-commute. The spaces formula_13 and formula_14 are then defined as the "p"-fold and "q"-fold Cartesian products of formula_7 and formula_8.
Just as in the case of an ordinary manifold, the supermanifold is then defined as a collection of charts glued together with differentiable transition functions. This definition in terms of charts requires that the transition functions have a smooth structure and a non-vanishing Jacobian. This can only be accomplished if the individual charts use a topology that is considerably coarser than the vector-space topology on the Grassmann algebra. This topology is obtained by projecting formula_13 down to formula_5 and then using the natural topology on that. The resulting topology is "not" Hausdorff, but may be termed "projectively Hausdorff".
That this definition is equivalent to the first one is not at all obvious; however, it is the use of the coarse topology that makes it so, by rendering most of the "points" identical. That is, formula_6 with the coarse topology is essentially isomorphic to formula_20
Historically, this approach is associated with Alice Rogers, Bryce DeWitt and work by Jadczyk and Pilch.
Properties.
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.
Batchelor's theorem.
Batchelor's theorem states that every supermanifold is noncanonically isomorphic to a supermanifold of the form Π"E". 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. It was published by Marjorie Batchelor in 1979.
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 formula_22 are even coordinates, and formula_23 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 formula_25 are even coordinates, formula_23 odd coordinates and formula_27 are either +1 or -1.)
Antibracket.
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 formula_29 and formula_30 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.
A coordinate transformation that preserves the antibracket is called a P-transformation. If the Berezinian of a P-transformation is equal to one then it is called an SP-transformation.
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 formula_31 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.
Laplacian.
One may define a Laplacian operator Δ on an SP-manifold as the operator which takes a function "H" to one half of the divergence of the corresponding Hamiltonian vector field. Explicitly one defines
In Darboux coordinates this definition reduces to
where "x"a 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.
SUSY.
A pre-SUSY-structure on a supermanifold of dimension
"(n,m)" is an odd "m"-dimensional
distribution formula_36.
With such a distribution one associates
its Frobenius tensor formula_37
(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 
formula_38,
"M" is called "a SUSY-manifold".
SUSY-structure in dimension "(1, k)"
is the same as odd contact structure.