Supermanifold

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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.

Physics[edit]

In physics, a supermanifold is a manifold with both bosonic and fermionic coordinates. These coordinates are usually denoted by

(x,\theta,\bar{\theta})

where x is the usual spacetime vector, and \theta\, and \bar{\theta} are Grassmann-valued spinors.

Whether these extra coordinates have any physical meaning is debatable. However this formalism is very useful for writing down supersymmetric Lagrangians.

Supermanifold: a definition[edit]

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 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 C^\infty(\mathbb{R}^p)\otimes\Lambda^\bullet(\xi_1,\dots\xi_q).

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the definition of a supermanifold is similar to that of a differentiable manifold, except that the model space Rp has been replaced by the model superspace Rp|q.

Side comment[edit]

This is different from the alternative definition where, using a fixed Grassmann algebra generated 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 preferable.

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 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 OMOM/I corresponds to an injective map MM; thus M is a submanifold of M.

Examples[edit]

  • 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 EM 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.

Batchelor's theorem[edit]

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[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

\omega = \sum_{i}  d\xi_i \wedge dx_i ,

where x_i are even coordinates, and \xi_i 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

\sum_i dp_i \wedge dq_i+\sum_j \frac{\varepsilon_j}{2}(d\xi_j)^2,

where p_i,q_i are even coordinates, \xi_i odd coordinates and \varepsilon_j 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

\{F,G\}=\frac{\partial_rF}{\partial z^i}\omega^{ij}(z)\frac{\partial_lG}{\partial z^j}.

Here \partial_r and \partial_l 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 {\mathcal{R}}^{n|n} 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

\Delta H=\frac{1}{2\rho}\frac{\partial_r}{\partial z^a}\left(\rho\omega^{ij}(z)\frac{\partial_l H}{\partial z^j}\right).

In Darboux coordinates this definition reduces to

\Delta=\frac{\partial_r}{\partial x^a}\frac{\partial_l}{\partial \theta_a}

where xa and θa are even and odd coordinates such that

\omega=dx^a\wedge d\theta_a.

The Laplacian is odd and nilpotent

\Delta^2=0.

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  P\subset TM. With such a distribution one associates its Frobenius tensor  S^2 P \mapsto TM/P (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 GL(P)\times GL(TM/P), M is called a SUSY-manifold. SUSY-structure in dimension (1, k) is the same as odd contact structure.

See also[edit]

References[edit]

[1] Joseph Bernstein, `Lectures on Supersymmetry` (notes by Dennis Gaitsgory) [1], "Quantum Field Theory program at IAS: Fall Term"

[2] A. Schwarz, `Geometry of Batalin-Vilkovisky quantization`, hep-th/9205088

[3] C. Bartocci, U. Bruzzo, D. Hernandez Ruiperez, The Geometry of Supermanifolds (Kluwer, 1991) ISBN 0-7923-1440-9

[4] A. Rogers, Supermanifolds: Theory and Applications (World Scientific, 2007) ISBN 981-02-1228-3

[5] 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]