In physics and mathematics, supermanifolds are generalizations of the manifold concept based on ideas coming from supersymmetry.
An informal definition is commonly used in physics textbooks and introductory lectures.
Locally, it is composed of coordinate charts that make it look like a "flat", "Euclidean" superspace.
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.
However, issues remain, including the proper extension of de Rham cohomology to supermanifolds.
One definition is as a sheaf over a ringed space; this is sometimes called the "algebro-geometric approach".
[1] This approach has a mathematical elegance, but can be problematic in various calculations and intuitive understanding.
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.
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
Historically, this approach is associated with Felix Berezin, Dimitry Leites, and Bertram Kostant.
A different definition describes a supermanifold in a fashion that is similar to that of a smooth manifold, except that the model space
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
; real elements consisting of only an even number of Grassmann generators form the space
of c-numbers, while real elements consisting of only an odd number of Grassmann generators form the space
[4] 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.
[4] This definition in terms of charts requires that the transition functions have a smooth structure and a non-vanishing Jacobian.
[4] 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.
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".
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
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.
[5] 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.
In many physical and geometric applications, a supermanifold comes equipped with an Grassmann-odd symplectic structure.
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
Using the Darboux theorem for odd symplectic forms one can show that P-manifolds are constructed from open sets of superspaces
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.
Explicitly one defines 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.