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Gibbons–Hawking Ansatz
In mathematics, the Gibbons–Hawking ansatz is a method of constructing gravitational instantons introduced by . It gives examples of hyperkähler manifolds in dimension 4 that are invariant under a circle action. Description Suppose that U is an open subset of \mathbb^3, and let * denote the Hodge star operator on \mathbb^3 with respect to the usual (flat) Euclidean metric. V is a harmonic function defined on U such that the cohomology class \left frac*dV\right/math> is integral, i.e. lies in the image of H^2(U;\mathbb) \hookrightarrow H^2(U;\mathbb). Then there is a U(1)-principal bundle \pi : P \to U equipped with a connection 1-form \eta \in \Omega^1(P;\mathfrak(1)) whose curvature form is d\eta = \pi^*(*dV). Then the Riemannian metric g = V\sum_^ dx_j \otimes dx_j + \frac \eta \otimes \eta is hyperkahler, and typically extends to the boundary of U. Examples Quaternions The usual (flat) metric on the quaternions \mathbb \cong \mathbb^2 is hyperkahler. It can be obtain ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gravitational Instanton
In mathematical physics and differential geometry, a gravitational instanton is a four-dimensional complete Riemannian manifold satisfying the vacuum Einstein equations. They are so named because they are analogues in quantum theories of gravity of instantons in Yang–Mills theory. In accordance with this analogy with self-dual Yang–Mills instantons, gravitational instantons are usually assumed to look like four dimensional Euclidean space at large distances, and to have a self-dual Riemann tensor. Mathematically, this means that they are asymptotically locally Euclidean (or perhaps asymptotically locally flat) hyperkähler 4-manifolds, and in this sense, they are special examples of Einstein manifolds. From a physical point of view, a gravitational instanton is a non-singular solution of the vacuum Einstein equations with ''positive-definite'', as opposed to Lorentzian, metric. There are many possible generalizations of the original conception of a gravitational i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperkähler Manifold
In differential geometry, a hyperkähler manifold is a Riemannian manifold (M, g) endowed with three integrable almost complex structures I, J, K that are Kähler with respect to the Riemannian metric g and satisfy the quaternionic relations I^2=J^2=K^2=IJK=-1. In particular, it is a hypercomplex manifold. All hyperkähler manifolds are Ricci-flat and are thus Calabi–Yau manifolds. Hyperkähler manifolds were first given this name by Eugenio Calabi in 1979. Early history Marcel Berger's 1955 paper on the classification of Riemannian holonomy groups first raised the issue of the existence of non-symmetric manifolds with holonomy Sp(''n'')·Sp(1). Interesting results were proved in the mid-1960s in pioneering work by Edmond Bonan and Kraines who have independently proven that any such manifold admits a parallel 4-form \Omega. Bonan's later results include a Lefschetz-type result: wedging with this powers of this 4-form induces isomorphisms ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circle Action
In mathematics, the circle group, denoted by \mathbb T or , is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers \mathbb T = \. The circle group forms a subgroup of , the multiplicative group of all nonzero complex numbers. Since \C^\times is abelian, it follows that \mathbb T is as well. A unit complex number in the circle group represents a rotation of the complex plane about the origin and can be parametrized by the angle measure : \theta \mapsto z = e^ = \cos\theta + i\sin\theta. This is the exponential map for the circle group. The circle group plays a central role in Pontryagin duality and in the theory of Lie groups. The notation \mathbb T for the circle group stems from the fact that, with the standard topology (see below), the circle group is a 1-torus. More generally, \mathbb T^n (the direct product of \mathbb T with itself n times) is geometrically an n-torus. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hodge Star Operator
In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a Dimension (vector space), finite-dimensional orientation (vector space), oriented vector space endowed with a Degenerate bilinear form, nondegenerate symmetric bilinear form. Applying the operator to an element of the algebra produces the Hodge dual of the element. This map was introduced by W. V. D. Hodge. For example, in an oriented 3-dimensional Euclidean space, an oriented plane can be represented by the exterior product of two basis vectors, and its Hodge dual is the normal vector given by their cross product; conversely, any vector is dual to the oriented plane perpendicular to it, endowed with a suitable bivector. Generalizing this to an -dimensional vector space, the Hodge star is a one-to-one mapping of -vectors to -vectors; the dimensions of these spaces are the binomial coefficients \tbinom nk = \tbinom. The Natural transformation, naturalness of the star operator ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonic Function
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f\colon U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is, \frac + \frac + \cdots + \frac = 0 everywhere on . This is usually written as \nabla^2 f = 0 or \Delta f = 0 Etymology of the term "harmonic" The descriptor "harmonic" in the name "harmonic function" originates from a point on a taut string which is undergoing harmonic motion. The solution to the differential equation for this type of motion can be written in terms of sines and cosines, functions which are thus referred to as "harmonics." Fourier analysis involves expanding functions on the unit circle in terms of a series of these harmonics. Considering higher dimensional analogues of the harmonics on the unit ''n''-sphere, one arrives at the spherical harmonics. These functions satisfy Laplace's equation and, over time, "harmon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principal Bundle
In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equipped with # An action of G on P, analogous to (x, g)h = (x, gh) for a product space (where (x, g) is an element of P and h is the group element from G; the group action is conventionally a right action). # A projection onto X. For a product space, this is just the projection onto the first factor, (x,g) \mapsto x. Unless it is the product space X \times G, a principal bundle lacks a preferred choice of identity cross-section; it has no preferred analog of x \mapsto (x,e). Likewise, there is not generally a projection onto G generalizing the projection onto the second factor, X \times G \to G that exists for the Cartesian product. They may also have a complicated topology that prevents them from being realized as a product space even if a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connection (principal Bundle)
In mathematics, and especially differential geometry and gauge theory, a connection is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. A principal ''G''-connection on a principal G-bundle P over a smooth manifold ''M'' is a particular type of connection that is compatible with the action of the group ''G''. A principal connection can be viewed as a special case of the notion of an Ehresmann connection, and is sometimes called a principal Ehresmann connection. It gives rise to (Ehresmann) connections on any fiber bundle associated to ''P'' via the associated bundle construction. In particular, on any associated vector bundle the principal connection induces a covariant derivative, an operator that can differentiate sections of that bundle along tangent directions in the base manifold. Principal connections generalize to arbitrary principal bundles the concept of a linear connection on the fra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gravitational Instanton
In mathematical physics and differential geometry, a gravitational instanton is a four-dimensional complete Riemannian manifold satisfying the vacuum Einstein equations. They are so named because they are analogues in quantum theories of gravity of instantons in Yang–Mills theory. In accordance with this analogy with self-dual Yang–Mills instantons, gravitational instantons are usually assumed to look like four dimensional Euclidean space at large distances, and to have a self-dual Riemann tensor. Mathematically, this means that they are asymptotically locally Euclidean (or perhaps asymptotically locally flat) hyperkähler 4-manifolds, and in this sense, they are special examples of Einstein manifolds. From a physical point of view, a gravitational instanton is a non-singular solution of the vacuum Einstein equations with ''positive-definite'', as opposed to Lorentzian, metric. There are many possible generalizations of the original conception of a gravitational i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eguchi–Hanson Space
In mathematics and theoretical physics, the Eguchi–Hanson space is a non-compact, self-dual, asymptotically locally Euclidean (ALE) metric on the cotangent bundle of the 2-sphere ''T''*''S''2. The holonomy group of this 4-real-dimensional manifold is SU(2). The metric is generally attributed to the physicists Tohru Eguchi and Andrew J. Hanson; it was discovered independently by the mathematician Eugenio Calabi around the same time in 1979. The Eguchi-Hanson metric has Ricci tensor equal to zero, making it a solution to the vacuum Einstein equations of general relativity, albeit with Riemannian rather than Lorentzian metric signature. It may be regarded as a resolution of the ''A''1 singularity according to the ADE classification which is the singularity at the fixed point of the ''C''2/''Z''2 orbifold where the ''Z''2 group inverts the signs of both complex coordinates in ''C''2. The even dimensional space ''C''d/2/''Z''d/2 of (real-)dimension d can be described using co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gibbons–Hawking Space
In mathematical physics, a Gibbons–Hawking space, named after Gary Gibbons and Stephen Hawking, is essentially a hyperkähler manifold with an extra U(1) symmetry. (In general, Gibbons–Hawking metrics are a subclass of hyperkähler metrics.) Gibbons–Hawking spaces, especially ambipolar ones, find an application in the study of black hole A black hole is a massive, compact astronomical object so dense that its gravity prevents anything from escaping, even light. Albert Einstein's theory of general relativity predicts that a sufficiently compact mass will form a black hole. Th ... microstate geometries. See also * Gibbons–Hawking effect References {{DEFAULTSORT:Gibbons-Hawking space Structures on manifolds Complex manifolds Riemannian manifolds Algebraic geometry Stephen Hawking ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ooguri–Vafa Metric
In differential geometry, the Ooguri–Vafa metric is a four-dimensional Hyperkähler metric. The Ooguri–Vafa metric is named after Hirosi Ooguri and Cumrun Vafa, who first described it in 1996 using the Gibbons–Hawking ansatz. Another construction was given by Davide Gaiotto, Gregory Moore and Andrew Neitzke in 2008. Definition The Ooguri–Vafa metric is defined on the four-dimensional total spaces of principal U(1)-bundles over open subsets of the three-dimensional euclidean space \mathbb^3. In particular the whole space results in \mathbb^3\times S^1. Define the elliptical fibers \tau(z)=\frac\log(z) with \tau_1=\operatorname(\tau(z)) and \tau_2=\operatorname(\tau(z)) and let \lambda be the string coupling constant. Further define the scaled spatial coordinate :\mathbf=\left(x,\frac,\frac\right). The metric of Ooguri and Vafa has the form :ds^2=\lambda^2 ^(dt-\mathbf\cdot d\mathbf)^2+V d\mathbf^2/math> where \mathbf=(A_x,A_z,A_) and :A_x=-\tau_1=\frac\log\left(\frac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Advances In Mathematics
''Advances in Mathematics'' is a peer-reviewed scientific journal covering research on pure mathematics. It was established in 1961 by Gian-Carlo Rota. The journal publishes 18 issues each year, in three volumes. At the origin, the journal aimed at publishing articles addressed to a broader "mathematical community", and not only to mathematicians in the author's field. Herbert Busemann writes, in the preface of the first issue, "The need for expository articles addressing either all mathematicians or only those in somewhat related fields has long been felt, but little has been done outside of the USSR. The serial publication ''Advances in Mathematics'' was created in response to this demand." Abstracting and indexing The journal is abstracted and indexed in: * [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
