# 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[edit]

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 and 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 de Rham cohomology to supermanifolds.

## Definition[edit]

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".^{[1]} 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",^{[1]} 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.^{[1]}^{[2]}

A third approach describes a supermanifold as a base topos of a superpoint. This approach remains the topic of active research.^{[3]}

### Algebro-geometric: as a sheaf[edit]

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 *O _{M}* or C

^{∞}(

**M**), that is locally isomorphic to , 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[edit]

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 .^{[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. 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".^{[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. That is, with the coarse topology is essentially isomorphic^{[1]}^{[2]} to

## Properties[edit]

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 *O _{M}* 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 *O _{M}/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

*O*→

_{M}*O*corresponds to an injective map

_{M}/I*M*→

**M**; thus

*M*is a submanifold of

**M**.

## Examples[edit]

- 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 functor from the category of vector bundles to the category of supermanifolds. - Lie supergroups are examples of supermanifolds.

## Batchelor's theorem[edit]

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.^{[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.

## Odd symplectic structures[edit]

### Odd symplectic form[edit]

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.)

### Antibracket[edit]

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.

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[edit]

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.

### Laplacian[edit]

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 θ_{a} 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[edit]

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.

## See also[edit]

## References[edit]

- ^
^{a}^{b}^{c}^{d}Alice Rogers,*Supermanifolds: Theory and Applications*, World Scientific, (2007) ISBN 978-981-3203-21-1*(See Chapter 1)* - ^
^{a}^{b}Rogers,*Op. Cit.**(See Chapter 8.)* **^**supermanifold at the*n*Lab- ^
^{a}^{b}^{c}Bryce DeWitt,*Supermanifolds*, (1984) Cambridge University Press ISBN 0521 42377 5*(See chapter 2.)* **^**Batchelor, Marjorie (1979), "The structure of supermanifolds",*Transactions of the American Mathematical Society*,**253**: 329–338, doi:10.2307/1998201, JSTOR 1998201, MR 0536951

- 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)

## External links[edit]

- Super manifolds: an incomplete survey at the Manifold Atlas.