Supermanifold


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.

Physics

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

:(x, heta,ar{ heta})

where "x" is the usual spacetime vector, and heta, and ar{ heta} 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

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)otimesLambda^ullet(xi_1,dotsxi_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the definition of a supermanifold is similar to that of a differentiable manifold, except that the model space Rp has been replaced by the "model superspace" Rp|q.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

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 more preferable.

Properties

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.

Examples

* 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

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 :sum dx_i wedge dxi_i, :(here, x_i are even coordinates,xi_i - odd coordinates).

Antibracket

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. Equipt 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

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

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 ho}frac{partial_r}{partial z^a}( hoomega^{ij}(z)frac{partial_l H}{partial z^j}).

In Darboux coordinates this definition reduces to::::Delta=frac{partial_r}{partial x^a}frac{partial_l}{partial heta_a}

where "x"a and θa are even and odd coordinates such that

::::omega=dx^awedge d heta_a.

The Laplacian is odd and nilpotent

::::Delta^2=0.

One may define the cohomology of functions "H" with respect to the Laplacian. In [http://arxiv.org/abs/hep-th/9205088 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

A pre-SUSY-structure on a supermanifold of dimension"(n,m)" is an odd "m"-dimensionaldistribution Psubset TM.With such a distribution one associatesits Frobenius tensor S^2 P mapsto TM/P(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 GL(P) imes GL(TM/P),"M" is called "a SUSY-manifold".SUSY-structure in dimension "(1, k)"is the same as odd contact structure.

References

[1] Joseph Bernstein, `Lectures on Supersymmetry` (notes by Dennis Gaitsgory) [http://www.math.ias.edu/QFT/fall/] ,"Quantum Field Theory program at IAS: Fall Term"

[2] A. Schwarz, `Geometry of Batalin-Vilkovisky quantization`, [http://arxiv.org/abs/hep-th/9205088 hep-th/9205088]


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • supermanifold — noun A manifold with both bosonic and fermionic coordinates, used in mathematical models …   Wiktionary

  • Poisson supermanifold — In differential geometry a Poisson supermanifold is a differential supermanifold M such that the supercommutative algebra of smooth functions over it (to clarify this: M is not a point set space and so, doesn t really exist, and really, this… …   Wikipedia

  • Inventions in the modern Islamic world — [ Abdus Salam, the 1979 Nobel Prize in Physics recipient, include the electroweak interaction, electroweak symmetry breaking, magnetic photon, neutral current, preon, W and Z bosons, supergeometry, supermanifold, superspace and superfield.] This… …   Wikipedia

  • Superspace — has had two meanings in physics. The word was first used by John Wheeler to describe the configuration space of general relativity; for example, this usage may be seen in his famous 1973 textbook Gravitation .The second meaning refers to the… …   Wikipedia

  • Topological string theory — In theoretical physics, topological string theory is a simplified version of string theory. The operators in topological string theory represent the algebra of operators in the full string theory that preserve a certain amount of supersymmetry.… …   Wikipedia

  • Supergroup (physics) — The concept of supergroup is a generalization of that of group. In other words, every group is a supergroup but not every supergroup is a group.First, let us define a Hopf superalgebra. A Hopf algebra can be defined category theoretically as an… …   Wikipedia

  • Supergravity — In theoretical physics, supergravity (supergravity theory) is a field theory that combines the principles of supersymmetry and general relativity. Together, these imply that, in supergravity, the supersymmetry is a local symmetry (in contrast to… …   Wikipedia

  • Superalgebra — In mathematics and theoretical physics, a superalgebra is a Z2 graded algebra. That is, it is an algebra over a commutative ring or field with a decomposition into even and odd pieces and a multiplication operator that respects the grading.The… …   Wikipedia

  • Canonical quantization — In physics, canonical quantization is one of many procedures for quantizing a classical theory. Historically, this was the earliest method to be used to build quantum mechanics. When applied to a classical field theory it is also called second… …   Wikipedia

  • Poisson superalgebra — In mathematics, a Poisson superalgebra is a Z2 graded generalization of a Poisson algebra. Specifically, a Poisson superalgebra is an (associative) superalgebra A with a Lie superbracket: [cdot,cdot] : Aotimes A o Asuch that ( A , [ middot;,… …   Wikipedia


Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.