Inphysics andmathematics,supermanifolds are generalizations of themanifold concept based on ideas coming fromsupersymmetry. Several definitions are in use, some of which are described below.

An informal definition is commonly used in physics textbooks and introductory lectures. It defines asupermanifold as amanifold with bothbosonic andfermionic coordinates. Locally, it is composed ofcoordinate charts that make it look like a "flat", "Euclidean"superspace. These local coordinates are often denoted by

${\displaystyle (x,\theta ,\overline{\theta })}$

wherex is the (real-number-valued)spacetime coordinate, and${\displaystyle \theta }$ and${\displaystyle \overline{\theta }}$ areGrassmann-valued spatial "directions".

The physical interpretation of the Grassmann-valued coordinates are the subject of debate; explicit experimental searches forsupersymmetry 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 offunctional integrals, the proper treatment of ghosts inBRST quantization, the cancellation of infinities inquantum field theory, Witten's work on theAtiyah-Singer index theorem, and more recent applications tomirror symmetry.

The use of Grassmann-valued coordinates has spawned the field ofsupermathematics, wherein large portions of geometry can be generalized to super-equivalents, including much ofRiemannian geometry and most of the theory ofLie groups andLie algebras (such asLie superalgebras,etc.) However, issues remain, including the proper extension ofde Rham cohomology to supermanifolds.

Three different definitions of supermanifolds are in use. One definition is as a sheaf over aringed 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 acoarsetopology 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 abase topos of asuperpoint. This approach remains the topic of active research.^[3]

Although supermanifolds are special cases ofnoncommutative manifolds, their local structure makes them better suited to study with the tools of standarddifferential geometry andlocally ringed spaces.

A supermanifoldM of dimension (p,q) is atopological spaceM with asheaf ofsuperalgebras, usually denotedO_M or C^∞(M), that is locally isomorphic to${\displaystyle C^{\mathrm{\infty }}({\mathbb{R}}^p)\otimes {\mathrm{\Lambda }}^{\bullet }({\xi }_1,\dots {\xi }_q)}$, where the latter is a Grassmann (Exterior) algebra onq generators.

A supermanifoldM of dimension (1,1) is sometimes called asuper-Riemann surface.

Historically, this approach is associated withFelix Berezin,Dimitry Leites, andBertram Kostant.

A different definition describes a supermanifold in a fashion that is similar to that of asmooth manifold, except that the model space${\displaystyle {\mathbb{R}}^p}$ has been replaced by themodel superspace${\displaystyle {\mathbb{R}}_c^p\times {\mathbb{R}}_a^q}$.

To correctly define this, it is necessary to explain what${\displaystyle {\mathbb{R}}_c}$ and${\displaystyle {\mathbb{R}}_a}$ are. These are given as the even and odd real subspaces of the one-dimensional space ofGrassmann numbers, which, by convention, are generated by acountably infinite number of anti-commuting variables: i.e. the one-dimensional space is given by${\displaystyle \mathbb{C}\otimes \mathrm{\Lambda }(V),}$ whereV is infinite-dimensional. An elementz is termedreal if${\displaystyle z=z^{\ast }}$; real elements consisting of only an even number of Grassmann generators form the space${\displaystyle {\mathbb{R}}_c}$ ofc-numbers, while real elements consisting of only an odd number of Grassmann generators form the space${\displaystyle {\mathbb{R}}_a}$ ofa-numbers. Note thatc-numbers commute, whilea-numbers anti-commute. The spaces${\displaystyle {\mathbb{R}}_c^p}$ and${\displaystyle {\mathbb{R}}_a^q}$ are then defined as thep-fold andq-fold Cartesian products of${\displaystyle {\mathbb{R}}_c}$ and${\displaystyle {\mathbb{R}}_a}$.^[4]

Just as in the case of an ordinary manifold, the supermanifold is then defined as a collection ofcharts glued together with differentiable transition functions.^[4] This definition in terms of charts requires that the transition functions have asmooth structure and a non-vanishingJacobian. This can only be accomplished if the individual charts use a topology that isconsiderably coarser than the vector-space topology on the Grassmann algebra. This topology is obtained by projecting${\displaystyle {\mathbb{R}}_c^p}$ down to${\displaystyle {\mathbb{R}}^p}$ and then using the natural topology on that. The resulting topology isnotHausdorff, 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,${\displaystyle {\mathbb{R}}_c^p\times {\mathbb{R}}_a^q}$ with the coarse topology is essentially isomorphic^[1][2] to${\displaystyle {\mathbb{R}}^p\otimes {\mathrm{\Lambda }}^{\bullet }({\xi }_1,\dots {\xi }_q)}$

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 supermanifoldM is contained in its sheafO_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 thefunctor of points.

IfM is a supermanifold of dimension (p,q), then the underlying spaceM inherits the structure of adifferentiable manifold whose sheaf of smooth functions is${\displaystyle O_M/I}$, where${\displaystyle I}$ is theideal generated by all odd functions. ThusM is called the underlying space, or the body, ofM. The quotient map${\displaystyle O_M\to O_M/I}$ corresponds to an injective mapM →M; thusM is a submanifold ofM.

• LetM be a manifold. Theodd tangent bundle ΠTM is a supermanifold given by the sheaf Ω(M) of differential forms onM.
• More generally, letE →M be avector bundle. Then ΠE is a supermanifold given by the sheaf Γ(ΛE^*). In fact, Π is afunctor from thecategory of vector bundles to thecategory of supermanifolds.
• Lie supergroups are examples of supermanifolds.

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 anequivalence of categories. It was published byMarjorie Batchelor in 1979.^[5]

Theproof of Batchelor's theorem relies in an essential way on the existence of apartition of unity, so it does not hold for complex or real-analytic supermanifolds.

In many physical and geometric applications, a supermanifold comes equipped with an Grassmann-oddsymplectic 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 onTM. Such a supermanifold is called aP-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 oneto equip a P-manifold locally with a set of coordinates where the odd symplectic form ω is written as

${\displaystyle \omega =\sum _{i}d{\xi }_i\wedge d{x}_i,}$

where${\displaystyle x_i}$ are even coordinates, and${\displaystyle {\xi }_i}$ odd coordinates. (An odd symplectic form should not be confused with a Grassmann-evensymplectic form on a supermanifold. In contrast, the Darboux version of an even symplectic form is

${\displaystyle \sum _{i}d{p}_i\wedge d{q}_i+\sum _j\frac{{\epsilon }_j}{2}(d{\xi }_j{)}^2,}$

where${\displaystyle p_i,q_i}$ are even coordinates,${\displaystyle {\xi }_i}$ odd coordinates and${\displaystyle {\epsilon }_j}$ are either +1 or −1.)

Given an odd symplectic 2-form ω one may define aPoisson bracket known as theantibracket of any two functionsF andG on a supermanifold by

${\displaystyle \{F,G\}=\frac{{\mathrm{\partial }}_{r}F}{\mathrm{\partial }{z}^i}{\omega }^{ij}(z)\frac{{\mathrm{\partial }}_{l}G}{\mathrm{\partial }{z}^j}.}$

Here${\displaystyle {\mathrm{\partial }}_r}$ and${\displaystyle {\mathrm{\partial }}_l}$ are the right and leftderivatives respectively andz are the coordinates of the supermanifold. Equipped with this bracket, the algebra of functions on a supermanifold becomes anantibracket algebra.

Acoordinate 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.

Using theDarboux theorem for odd symplectic forms one can show that P-manifolds are constructed from open sets of superspaces${\displaystyle {\mathcal{R}}^{n|n}}$ 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.

One may define aLaplacian operator Δ on an SP-manifold as the operator which takes a functionH to one half of thedivergence of the correspondingHamiltonian vector field. Explicitly one defines

${\displaystyle \mathrm{\Delta }H=\frac{1}{2\rho }\frac{{\mathrm{\partial }}_r}{\mathrm{\partial }{z}^a}\left(\rho {\omega }^{ij}(z)\frac{{\mathrm{\partial }}_{l}H}{\mathrm{\partial }{z}^j}\right).}$

In Darboux coordinates this definition reduces to

${\displaystyle \mathrm{\Delta }=\frac{{\mathrm{\partial }}_r}{\mathrm{\partial }{x}^a}\frac{{\mathrm{\partial }}_l}{\mathrm{\partial }{\theta }_a}}$

wherex^a andθ_a are even and odd coordinates such that

${\displaystyle \omega =d{x}^a\wedge d{\theta }_a.}$

The Laplacian is odd and nilpotent

${\displaystyle {\mathrm{\Delta }}^2=0.}$

One may define thecohomology of functionsH with respect to the Laplacian. InGeometry of Batalin-Vilkovisky quantization,Albert Schwarz has proven that the integral of a functionH over aLagrangian submanifoldL depends only on the cohomology class ofH and on thehomology class of the body ofL in the body of the ambient supermanifold.

A pre-SUSY-structure on a supermanifold of dimension(n,m) is an oddm-dimensionaldistribution${\displaystyle P\subset TM}$.With such a distribution one associatesits Frobenius tensor${\displaystyle S^{2}P\mapsto TM/P}$(sinceP is odd, the skew-symmetric Frobeniustensor is a symmetric operation).If this tensor is non-degenerate,e.g. lies in an open orbit of${\displaystyle GL(P)\times GL(TM/P)}$,M is calleda SUSY-manifold.SUSY-structure in dimension (1,k)is the same as oddcontact structure.

• Superspace
• Supersymmetry
• Supergeometry
• Graded manifold
• Batalin–Vilkovisky formalism

• 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)

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

• Supersymmetry
• Generalized manifolds
• Structures on manifolds
• Mathematical physics

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