Synonyms for supermanifold or Related words with supermanifold
NotFoundErrorExamples of "supermanifold" |
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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. |