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
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.
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.
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 , where the latter is a Grassmann algebra on q generators.
A supermanifold M of dimension (1,1) is sometimes called a super-Riemann surface.
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 has been replaced by the model superspace .
To correctly define this, it is necessary to explain what and 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 where V is infinite-dimensional. An element z is termed real if ; 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 of a-numbers. Note that c-numbers commute, while a-numbers anti-commute. The spaces and are then defined as the p-fold and q-fold Cartesian products of and .
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 down to 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, with the coarse topology is essentially isomorphic to
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.
- Alice Rogers, Supermanifolds: Theory and Applications, World Scientific, (2007) ISBN 978-981-3203-21-1 (See Chapter 1)
- Rogers, Op. Cit. (See Chapter 8.)
- supermanifold in nLab
- Bryce DeWitt, Supermanifolds, (1984) Cambridge University Press ISBN 0521 42377 5 (See chapter 2.)
- A. Jadczyk, K. Pilch, "Superspaces and supersymmetries". Comm. Math. Phys. 78 (1980), no. 3, pp373--390.
- Joseph Bernstein, "Lectures on Supersymmetry (notes by Dennis Gaitsgory)", Quantum Field Theory program at IAS: Fall Term
- A. Schwarz, "Geometry of Batalin-Vilkovisky quantization", ArXiv hep-th/9205088
- C. Bartocci, U. Bruzzo, D. Hernandez Ruiperez, The Geometry of Supermanifolds (Kluwer, 1991) ISBN 0-7923-1440-9
- L. Mangiarotti, G. Sardanashvily, Connections in Classical and Quantum Field Theory (World Scientific, 2000) ISBN 981-02-2013-8 (arXiv:0910.0092)
- Super manifolds: an incomplete survey at the Manifold Atlas.