Supermanifold

 ( Supersymmetry ) 

A third approach describes a supermanifold as a base topos of a superpoint. This approach remains the topic of active research.

A supermanifold M of dimension ( p, q) is a topological space M with a sheaf of , usually denoted O _M or C^∞( M), that is locally isomorphic to $C^\backslash infty(\backslash mathbb\{R\}^p)\backslash otimes\backslash Lambda^\backslash bullet(\backslash xi\_1,\backslash dots\backslash xi\_q)$, 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.

To correctly define this, it is necessary to explain what $\backslash mathbb\{R\}\_c$ and $\backslash mathbb\{R\}\_a$ are. These are given as the even and odd real subspaces of the one-dimensional space of , which, by convention, are generated by a countably infinite number of anti-commuting variables: i.e. the one-dimensional space is given by $\backslash mathbb\{C\}\backslash otimes\backslash Lambda(V),$ where V is infinite-dimensional. An element z is termed real if $z=z^*$; real elements consisting of only an even number of Grassmann generators form the space $\backslash mathbb\{R\}\_c$ of c-numbers, while real elements consisting of only an odd number of Grassmann generators form the space $\backslash mathbb\{R\}\_a$ of a-numbers. Note that c-numbers commute, while a-numbers anti-commute. The spaces $\backslash mathbb\{R\}^p\_c$ and $\backslash mathbb\{R\}^q\_a$ are then defined as the p-fold and q-fold Cartesian products of $\backslash mathbb\{R\}\_c$ and $\backslash mathbb\{R\}\_a$.Bryce DeWitt, Supermanifolds, (1984) Cambridge University Press (See chapter 2.)

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. 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 $\backslash mathbb\{R\}^p\_c$ down to $\backslash mathbb\{R\}^p$ and then using the natural topology on that. The resulting topology is not Hausdorff space, but may be termed "projectively Hausdorff".

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, $\backslash mathbb\{R\}^p\_c\backslash times\backslash mathbb\{R\}^q\_a$ with the coarse topology is essentially isomorphic to $\backslash mathbb\{R\}^p\backslash otimes\backslash Lambda^\backslash bullet(\backslash xi\_1,\backslash dots\backslash xi\_q)$

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 $O\_M\backslash to\; O\_M/I$ corresponds to an injective map M → M; thus M is a submanifold of M.

• 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.

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.

$\backslash omega\; =\; \backslash sum\_\{i\}\; d\backslash xi\_i\; \backslash wedge\; dx\_i\; ,$

$\backslash sum\_i\; dp\_i\; \backslash wedge\; dq\_i+\backslash sum\_j\; \backslash frac\{\backslash varepsilon\_j\}\{2\}(d\backslash xi\_j)^2,$

:$\backslash \{F,G\backslash \}=\backslash frac\{\backslash partial\_rF\}\{\backslash partial\; z^i\}\backslash omega^\{ij\}(z)\backslash frac\{\backslash partial\_lG\}\{\backslash partial\; z^j\}.$

Here $\backslash partial\_r$ and $\backslash partial\_l$ are the right and left 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.

::$\backslash Delta\; H=\backslash frac\{1\}\{2\backslash rho\}\backslash frac\{\backslash partial\_r\}\{\backslash partial\; z^a\}\backslash left(\backslash rho\backslash omega^\{ij\}(z)\backslash frac\{\backslash partial\_l\; H\}\{\backslash partial\; z^j\}\backslash right).$

In Darboux coordinates this definition reduces to

:::$\backslash Delta=\backslash frac\{\backslash partial\_r\}\{\backslash partial\; x^a\}\backslash frac\{\backslash partial\_l\}\{\backslash partial\; \backslash theta\_a\}$

where x ^a and θ _a are even and odd coordinates such that

:::$\backslash omega=dx^a\backslash wedge\; d\backslash theta\_a.$

The Laplacian is odd and nilpotent

:::$\backslash Delta^2=0.$

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.

• 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)
• L. Mangiarotti, G. Sardanashvily, Connections in Classical and Quantum Field Theory (World Scientific, 2000) ()

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