Synonyms for supermanifold or Related words with supermanifold


Examples of "supermanifold"
Every symplectic supermanifold is a Poisson supermanifold but not vice versa.
A pre-SUSY-structure on a supermanifold of dimension
A supermanifold M of dimension "(1,1)" is sometimes called a super-Riemann surface.
Here formula_29 and formula_30 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.
The fundamental strings of string theory are two-dimensional surfaces. A quantum field theory known as the "N" = (1,1) sigma model is defined on each surface. This theory consist of maps from the surface to a supermanifold. Physically the supermanifold is interpreted as spacetime and each map is interpreted as the embedding of the string in spacetime.
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
orthogonal-symplectic supersubgroup formula_32, one can think of the corresponding global section of the quotient superbundle formula_33 as being a supermetric on a supermanifold formula_29.
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" 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.
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.
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 algebra is all we have), formula_1 is equipped with a bilinear map called the Poisson superbracket turning it into a Poisson superalgebra.
A third approach describes a supermanifold as a base topos of a superpoint. This approach remains the topic of active research.
In mathematics, a Mumford measure is a measure on a supermanifold constructed from a bundle of relative dimension 1|1. It is named for David Mumford.
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
where formula_22 are even coordinates, and formula_23 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
A supermanifold M of dimension "(p,q)" is a topological space "M" with a sheaf of superalgebras, usually denoted "O" or C(M), that is locally isomorphic to formula_4, where the latter is a Grassmann algebra on "q" generators.
A different definition describes a supermanifold in a fashion that is similar to that of a smooth manifold, except that the model space formula_5 has been replaced by the "model superspace" formula_6.
A third usage of the term "superspace" is as a synonym for a supermanifold: a supersymmetric generalization of a manifold. Note that both super Minkowski spaces and super vector spaces can be taken as special cases of supermanifolds.
An informal definition is commonly used in physics textbooks and introductory lectures. It defines a supermanifold as a manifold with both bosonic and fermionic coordinates. Locally, it is composed of coordinate charts that make it look like a "flat", "Euclidean" superspace. These local coordinates are often denoted by
The appropriate superlanguage of a Lie algebroid "A" is "ΠA", the supermanifold whose space of (super)functions are the "A"-forms. On this space the Lie algebroid can be encoded via its Lie algebroid differential, which is just an odd vector field.
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.