physicsand 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.
In physics, a supermanifold is a
manifoldwith both bosonic and fermionic coordinates. These coordinates are usually denoted by
where "x" is the usual
spacetimevector, and and are Grassmann-valued spinors.
Whether these extra coordinates have any physical meaning is debatable. However this formalism is very useful for writing down
Supermanifold: a definition
Although supermanifolds are special cases of noncommutative manifolds, the local structure of supermanifolds make them better suited to study with the tools of standard
differential geometryand 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 .
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differentiable manifold, except that the model space Rp has been replaced by the "model superspace" Rp|q.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 algebragenerated 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 more 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" ΠT"M" is a supermanifold given by the sheaf Ω("M") of differential forms on "M".
* More generally, let "E" → "M" be a
vector bundle. Then Π"E" is a supermanifold given by the sheaf Γ(ΛE*). In fact, Π is a functorfrom 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 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.
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 odd symplectic structure.All natural geometric objects on a supermanifold aregraded. In particular, the bundle of two-formsis equipped with a grading. An odd symplecticform 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)", becausethe odd symplectic form induces a pairing ofodd and even variables. There is a version of theDarboux theorem for P-manifolds, which allows oneto equip a P-manifold locally with a setof coordinates where the odd symplectic form is writtenas ::(here, are even coordinates, - odd coordinates).
Given an odd symplectic 2-form ω one may define a
Poisson bracketknown 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. Equipt with this bracket, the algebra of functions on a supermanifold becomes an antibracket algebra.
coordinate transformationthat preserves the antibracket is called a P-transformation. If the Berezinianof a P-transformation is equal to one then it is called an SP-transformation.
P and SP-manifolds
Darboux theoremfor 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-manifoldif 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 functionρ such that on each coordinate patchthere exist Darboux coordinatesin which ρ is identically equal to one.
One may define a
Laplacian operatorΔ on an SP-manifold as the operator which takes a function "H" to one half of the divergenceof 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
cohomologyof functions "H" with respect to the Laplacian. In [http://arxiv.org/abs/hep-th/9205088 Geometry of Batalin-Vilkovisky quantization] , Albert Schwarzhas 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"-dimensionaldistribution .With such a distribution one associatesits Frobenius tensor (since "P" is odd, the skew-symmetric Frobeniustensor 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) [http://www.math.ias.edu/QFT/fall/] ,"Quantum Field Theory program at IAS: Fall Term"
 A. Schwarz, `Geometry of Batalin-Vilkovisky quantization`, [http://arxiv.org/abs/hep-th/9205088 hep-th/9205088]
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