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Graded Manifold
In algebraic geometry, graded manifolds are extensions of the concept of manifolds based on ideas coming from supersymmetry and supercommutative algebra. Both graded manifolds and supermanifolds are phrased in terms of sheaves of graded commutative algebras. However, graded manifolds are characterized by sheaves on smooth manifolds, while supermanifolds are constructed by gluing of sheaves of supervector spaces. Graded manifolds A graded manifold of dimension (n,m) is defined as a locally ringed space (Z,A) where Z is an n-dimensional smooth manifold and A is a C^\infty_Z-sheaf of Grassmann algebras of rank m where C^\infty_Z is the sheaf of smooth real functions on Z. The sheaf A is called the structure sheaf of the graded manifold (Z,A), and the manifold Z is said to be the body of (Z,A). Sections of the sheaf A are called graded functions on a graded manifold (Z,A). They make up a graded commutative C^\infty(Z)-ring A(Z) called the structure ring of (Z,A). The well-known Ba ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Superalgebra
In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z2 grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry. In most of these theories, the ''even'' elements of the superalgebra correspond to bosons and ''odd'' elements to fermions (but this is not always true; for example, the BRST supersymmetry is the other way around). Definition Formally, a Lie superalgebra is a nonassociative Z2-graded algebra, or ''superalgebra'', over a commutative ring (typically R or C) whose product ·, · called the Lie superbracket or supercommutator, satisfies the two conditions (analogs of the usual Lie algebra axioms, with grading): Super skew-symmetry: : ,y-(-1)^ ,x\ The super Jacobi identity: :(-1)^ ,_z">,_[y,_z<_a>_+_(-1)^[y,_[z,_x.html" ;"title=",_z.html" ;"title=", [y, z">, [y, z + (-1)^[y, [z, x">,_z.html" ;"title=", [y, z">, [y, z + (-1)^[y, [z, x + (-1)^[z, [x, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supersymmetry
In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of supersymmetric theories exist. Supersymmetry is a spacetime symmetry between two basic classes of particles: bosons, which have an integer-valued spin and follow Bose–Einstein statistics, and fermions, which have a half-integer-valued spin and follow Fermi–Dirac statistics. In supersymmetry, each particle from one class would have an associated particle in the other, known as its superpartner, the spin of which differs by a half-integer. For example, if the electron exists in a supersymmetric theory, then there would be a particle called a ''"selectron"'' (superpartner electron), a bosonic partner of the electron. In the simplest supersymmetry theories, with perfectly " unbroken" supersymmetry, each pair of superpartners would share the same mass and internal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supermanifold
In physics and mathematics, supermanifolds are generalizations of the manifold concept based on ideas coming from supersymmetry. Several definitions are in use, some of which are described below. Informal definition 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 :(x,\theta,\bar) where ''x'' is the ( real-number-valued) spacetime coordinate, and \theta\, and \bar are Grassmann-valued spatial "directions". The physical interpretation of the Grassmann-valued coordinates are the subject of debate; explicit experimental searches for supersymmetry have not yielded any positive results. However, the use of Grassmann variables allow for the tremendous simplification of a number of important mathematical results. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supergeometry
Supergeometry is differential geometry of modules over graded commutative algebras, supermanifolds and graded manifolds. Supergeometry is part and parcel of many classical and quantum field theories involving odd fields, e.g., SUSY field theory, BRST theory, or supergravity. Supergeometry is formulated in terms of \mathbb Z_2-graded modules and sheaves over \mathbb Z_2-graded commutative algebras (supercommutative algebras). In particular, superconnections are defined as Koszul connections on these modules and sheaves. However, supergeometry is not particular noncommutative geometry because of a different definition of a graded derivation. Graded manifolds and supermanifolds also are phrased in terms of sheaves of graded commutative algebras. Graded manifolds are characterized by sheaves on smooth manifolds, while supermanifolds are constructed by gluing of sheaves of supervector spaces. There are different types of supermanifolds. These are smooth supermanifolds (H^\inft ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graded (mathematics)
In mathematics, the term “graded” has a number of meanings, mostly related: In abstract algebra, it refers to a family of concepts: * An algebraic structure X is said to be I-graded for an index set I if it has a gradation or grading, i.e. a decomposition into a direct sum X = \bigoplus_ X_i of structures; the elements of X_i are said to be "homogeneous of degree ''i'' ". ** The index set I is most commonly \N or \Z, and may be required to have extra structure depending on the type of X. ** Grading by \Z_2 (i.e. \Z/2\Z) is also important; see e.g. signed set (the \Z_2-graded sets). ** The trivial (\Z- or \N-) gradation has X_0 = X, X_i = 0 for i \neq 0 and a suitable trivial structure 0. ** An algebraic structure is said to be doubly graded if the index set is a direct product of sets; the pairs may be called "bidegrees" (e.g. see Spectral sequence). * A I-graded vector space or graded linear space is thus a vector space with a decomposition into a direct sum V = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connection (algebraic Framework)
Geometry of quantum systems (e.g., noncommutative geometry and supergeometry) is mainly phrased in algebraic terms of modules and algebras. Connections on modules are generalization of a linear connection on a smooth vector bundle E\to X written as a Koszul connection on the C^\infty(X)-module of sections of E\to X. Commutative algebra Let A be a commutative ring and M an ''A''- module. There are different equivalent definitions of a connection on M. First definition If k \to A is a ring homomorphism, a k-linear connection is a k-linear morphism : \nabla: M \to \Omega^1_ \otimes_A M which satisfies the identity : \nabla(am) = da \otimes m + a \nabla m A connection extends, for all p \geq 0 to a unique map : \nabla: \Omega^p_ \otimes_A M \to \Omega^_ \otimes_A M satisfying \nabla(\omega \otimes f) = d\omega \otimes f + (-1)^p \omega \wedge \nabla f. A connection is said to be integrable if \nabla \circ \nabla = 0, or equivalently, if the curvature \nabla^2: M \to \Ome ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
BRST Formalism , from its stock ticker, now Democrasoft
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BRST may refer to: * BRST Films, a Serbian video production company * BRST algorithm, an optimization algorithm suitable for finding the global optimum of black box functions * BRST quantization in Yang-Mills theories, a way to quantize a gauge-symmetric field theory * BrasĂlia Summer Time * Burst.com Democrasoft, Inc., formerly Burst.com, is a public company, public software company located in Santa Rosa, California, Santa Rosa, California. History In 2010, the company launched Collaborize, an online communications tool, at the Spring DEMO ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classical Field Theory
A classical field theory is a physical theory that predicts how one or more physical fields interact with matter through field equations, without considering effects of quantization; theories that incorporate quantum mechanics are called quantum field theories. In most contexts, 'classical field theory' is specifically intended to describe electromagnetism and gravitation, two of the fundamental forces of nature. A physical field can be thought of as the assignment of a physical quantity at each point of space and time. For example, in a weather forecast, the wind velocity during a day over a country is described by assigning a vector to each point in space. Each vector represents the direction of the movement of air at that point, so the set of all wind vectors in an area at a given point in time constitutes a vector field. As the day progresses, the directions in which the vectors point change as the directions of the wind change. The first field theories, Newtonian grav ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Variational Bicomplex
In mathematics, the Lagrangian theory on fiber bundles is globally formulated in algebraic terms of the variational bicomplex, without appealing to the calculus of variations. For instance, this is the case of classical field theory on fiber bundles ( covariant classical field theory). The variational bicomplex is a cochain complex of the differential graded algebra of exterior forms on jet manifolds of sections of a fiber bundle. Lagrangians and Euler–Lagrange operators on a fiber bundle are defined as elements of this bicomplex. Cohomology of the variational bicomplex leads to the global first variational formula and first Noether's theorem. Extended to Lagrangian theory of even and odd fields on graded manifolds, the variational bicomplex provides strict mathematical formulation of classical field theory in a general case of reducible degenerate Lagrangians and the Lagrangian BRST theory. See also *Calculus of variations *Lagrangian system In mathematics, a Lagr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Calculus Over Commutative Algebras
In mathematics the differential calculus over commutative algebras is a part of commutative algebra based on the observation that most concepts known from classical differential calculus can be formulated in purely algebraic terms. Instances of this are: # The whole topological information of a smooth manifold M is encoded in the algebraic properties of its \R-algebra of smooth functions A = C^\infty (M), as in the Banach–Stone theorem. # Vector bundles over M correspond to projective finitely generated modules over A, via the functor \Gamma which associates to a vector bundle its module of sections. # Vector fields on M are naturally identified with derivations of the algebra A. # More generally, a linear differential operator of order k, sending sections of a vector bundle E\rightarrow M to sections of another bundle F \rightarrow M is seen to be an \R-linear map \Delta : \Gamma (E) \to \Gamma (F) between the associated modules, such that for any k + 1 elements f_0, \ldots, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jet (mathematics)
In mathematics, the jet is an operation that takes a differentiable function ''f'' and produces a polynomial, the truncated Taylor polynomial of ''f'', at each point of its domain. Although this is the definition of a jet, the theory of jets regards these polynomials as being abstract polynomials rather than polynomial functions. This article first explores the notion of a jet of a real valued function in one real variable, followed by a discussion of generalizations to several real variables. It then gives a rigorous construction of jets and jet spaces between Euclidean spaces. It concludes with a description of jets between manifolds, and how these jets can be constructed intrinsically. In this more general context, it summarizes some of the applications of jets to differential geometry and the theory of differential equations. Jets of functions between Euclidean spaces Before giving a rigorous definition of a jet, it is useful to examine some special cases. One-dimensional ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
