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Horrocks Construction
In mathematics, the Horrocks construction is a method for constructing vector bundles, especially over projective spaces, introduced by . His original construction gave an example of an indecomposable rank 2 vector bundle over 3-dimensional projective space, and generalizes to give examples of vector bundles of higher ranks over other projective spaces. The Horrocks construction is used in the ADHM construction to construct instantons over the 4-sphere In mathematics, an -sphere or a hypersphere is a topological space that is homeomorphic to a ''standard'' -''sphere'', which is the set of points in -dimensional Euclidean space that are situated at a constant distance from a fixed point, call .... References * * Vector bundles {{topology-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every point x of the space X we associate (or "attach") a vector space V(x) in such a way that these vector spaces fit together to form another space of the same kind as X (e.g. a topological space, manifold, or algebraic variety), which is then called a vector bundle over X. The simplest example is the case that the family of vector spaces is constant, i.e., there is a fixed vector space V such that V(x)=V for all x in X: in this case there is a copy of V for each x in X and these copies fit together to form the vector bundle X\times V over X. Such vector bundles are said to be ''trivial''. A more complicated (and prototypical) class of examples are the tangent bundles of smooth (or differentiable) manifolds: to every point of such a manifold w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally, an affine space with points at infinity, in such a way that there is one point at infinity of each direction of parallel lines. This definition of a projective space has the disadvantage of not being isotropic, having two different sorts of points, which must be considered separately in proofs. Therefore, other definitions are generally preferred. There are two classes of definitions. In synthetic geometry, ''point'' and ''line'' are primitive entities that are related by the incidence relation "a point is on a line" or "a line passes through a point", which is subject to the axioms of projective geometry. For some such set of axioms, the projective spaces that are defined have been shown to be equivalent to those resulting from the fol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
ADHM Construction
In mathematical physics and gauge theory, the ADHM construction or monad construction is the construction of all instantons using methods of linear algebra by Michael Atiyah, Vladimir Drinfeld, Nigel Hitchin, Yuri I. Manin in their paper "Construction of Instantons." ADHM data The ADHM construction uses the following data: * complex vector spaces ''V'' and ''W'' of dimension ''k'' and ''N'', * ''k'' × ''k'' complex matrices ''B''1, ''B''2, a ''k'' × ''N'' complex matrix ''I'' and a ''N'' × ''k'' complex matrix ''J'', * a real moment map \mu_r = _1,B_1^\dagger _2,B_2^\daggerII^\dagger-J^\dagger J, * a complex moment map \displaystyle\mu_c = _1,B_2IJ. Then the ADHM construction claims that, given certain regularity conditions, * Given ''B''1, ''B''2, ''I'', ''J'' such that \mu_r=\mu_c=0, an anti-self-dual instanton in a SU(''N'') gauge theory with instanton number ''k'' can be constructed, * All anti-self-dual instantons c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Instanton
An instanton (or pseudoparticle) is a notion appearing in theoretical and mathematical physics. An instanton is a classical solution to equations of motion with a finite, non-zero action, either in quantum mechanics or in quantum field theory. More precisely, it is a solution to the equations of motion of the classical field theory on a Euclidean spacetime. In such quantum theories, solutions to the equations of motion may be thought of as critical points of the action. The critical points of the action may be local maxima of the action, local minima, or saddle points. Instantons are important in quantum field theory because: * they appear in the path integral as the leading quantum corrections to the classical behavior of a system, and * they can be used to study the tunneling behavior in various systems such as a Yang–Mills theory. Relevant to dynamics, families of instantons permit that instantons, i.e. different critical points of the equation of motion, be related to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
4-sphere
In mathematics, an -sphere or a hypersphere is a topological space that is homeomorphic to a ''standard'' -''sphere'', which is the set of points in -dimensional Euclidean space that are situated at a constant distance from a fixed point, called the ''center''. It is the generalization of an ordinary sphere in the ordinary three-dimensional space. The "radius" of a sphere is the constant distance of its points to the center. When the sphere has unit radius, it is usual to call it the unit -sphere or simply the -sphere for brevity. In terms of the standard norm, the -sphere is defined as : S^n = \left\ , and an -sphere of radius can be defined as : S^n(r) = \left\ . The dimension of -sphere is , and must not be confused with the dimension of the Euclidean space in which it is naturally embedded. An -sphere is the surface or boundary of an -dimensional ball. In particular: *the pair of points at the ends of a (one-dimensional) line segment is a 0-sphere, *a circle, which is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
