- Supermanifold
In

physics andmathematics ,**supermanifolds**are generalizations of the manifold concept based on ideas coming fromsupersymmetry . Several definitions are in use, some of which are described below.**Physics**In physics, a

**supermanifold**is amanifold with bothboson ic andfermion ic 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 Lagrangian s.**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 andlocally ringed space s.A supermanifold

**M**of dimension "(p,q)" is atopological space "M" with a sheaf ofsuperalgebra s, usually denoted "O_{M}" 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differentiable manifold , except that the model space**R**^{p}has been replaced by the "model superspace"**R**^{p|q}.**Side comment**This is "different" from the alternative definition where, using a fixed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 "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_{M}/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 afunctor 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 aP-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

derivative s respectively and "z" are the coordinates of the supermanifold. Equipt with this bracket, the algebra of functions on a supermanifold becomes anantibracket algebra .A

coordinate transformation that preserves the antibracket is called aP-transformation . If theBerezinian of a P-transformation is equal to one then it is called anSP-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 anSP-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 adensity function ρ such that on eachcoordinate patch there existDarboux 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 thedivergence of the correspondingHamiltonian 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 θ_{a}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 aLagrangian 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.*

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