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Equivariant Index Theorem
In differential geometry, the equivariant index theorem, of which there are several variants, computes the (graded) trace of an element of a compact Lie group acting in given setting in terms of the integral over the fixed points of the element. If the element is neutral, then the theorem reduces to the usual index theorem. The classical formula such as the Atiyah–Bott formula is a special case of the theorem. Statement Let \pi: E \to M be a clifford module bundle. Assume a compact Lie group ''G'' acts on both ''E'' and ''M'' so that \pi is equivariant. Let ''E'' be given a connection that is compatible with the action of ''G''. Finally, let ''D'' be a Dirac operator on ''E'' associated to the given data. In particular, ''D'' commutes with ''G'' and thus the kernel of ''D'' is a finite-dimensional representation of ''G''. The equivariant index of ''E'' is a virtual character given by taking the supertrace: :\operatorname(g\mid\ker D) = \operatorname(g\mid\ker D^+) - \operator ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as classical antiquity, antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Nikolai Lobachevsky, Lobachevsky. The simplest examples of smooth spaces are the Differential geometry of curves, plane and space curves and Differential geometry of surfaces, surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Group
In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additional properties it must have to be thought of as a "transformation" in the abstract sense, for instance multiplication and the taking of inverses (to allow division), or equivalently, the concept of addition and subtraction. Combining these two ideas, one obtains a continuous group where multiplying points and their inverses is continuous. If the multiplication and taking of inverses are smoothness, smooth (differentiable) as well, one obtains a Lie group. Lie groups provide a natural model for the concept of continuous symmetry, a celebrated example of which is the circle group. Rotating a circle is an example of a continuous symmetry. For an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fixed Point (mathematics)
In mathematics, a fixed point (sometimes shortened to fixpoint), also known as an invariant point, is a value that does not change under a given transformation (mathematics), transformation. Specifically, for function (mathematics), functions, a fixed point is an element that is mapped to itself by the function. Any set of fixed points of a transformation is also an invariant set. Fixed point of a function Formally, is a fixed point of a function if belongs to both the domain of a function, domain and the codomain of , and . In particular, cannot have any fixed point if its domain is disjoint from its codomain. If is defined on the real numbers, it corresponds, in graphical terms, to a curve in the Euclidean plane, and each fixed-point corresponds to an intersection of the curve with the line , cf. picture. For example, if is defined on the real numbers by f(x) = x^2 - 3 x + 4, then 2 is a fixed point of , because . Not all functions have fixed points: for example, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Index Theorem
Index (: indexes or indices) may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on the Halo Array in the ''Halo'' video game series Periodicals and news portals * ''Index Magazine'', a publication for art and culture * Index.hr, a Croatian online newspaper * index.hu, a Hungarian-language news and community portal * ''The Index'' (Kalamazoo College), a student newspaper * ''The Index'', an 1860s European propaganda journal created by Henry Hotze to support the Confederate States of America * ''Truman State University Index'', a student newspaper Other arts, entertainment and media * The Index (band) * ''Indexed'', a Web cartoon by Jessica Hagy * ''Index'', album by Ana Mena Business enterprises and events * Index (retailer), a former UK catalogue retailer * INDEX, a market research fair in Lucknow, India * Index Corporation, a Japanese v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Atiyah–Bott Formula
In algebraic geometry, the Atiyah–Bott formula says the cohomology ring :\operatorname^*(\operatorname_G(X), \mathbb_l) of the moduli stack of principal bundles is a free algebra, free supercommutative algebra, graded-commutative algebra on certain homogeneous generators. The original work of Michael Atiyah and Raoul Bott concerned the integral cohomology ring of \operatorname_G(X). See also *Borel's theorem, which says that the cohomology ring of a classifying stack is a polynomial ring. Notes References * * Theorems in algebraic geometry {{topology-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clifford Module Bundle
In differential geometry, a Clifford module bundle, a bundle of Clifford modules or just Clifford module is a vector bundle whose fibers are Clifford modules, the representations of Clifford algebras. The canonical example is a spinor bundle. In fact, on a Spin manifold, every Clifford module is obtained by twisting the spinor bundle. The notion "Clifford module bundle" should not be confused with a Clifford bundle, which is a bundle of Clifford algebras. Spinor bundles Given an oriented Riemannian manifold ''M'' one can ask whether it is possible to construct a bundle of irreducible Clifford modules over ''Cℓ''(''T''*''M''). In fact, such a bundle can be constructed if and only if ''M'' is a spin manifold. Let ''M'' be an ''n''-dimensional spin manifold with spin structure ''F''Spin(''M'') → ''F''SO(''M'') on ''M''. Given any ''Cℓ''''n''R-module ''V'' one can construct the associated spinor bundle :S(M) = F_(M) \times_\sigma V\, where σ : Spin(''n'') → GL(''V'') i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivariant Bundle
In geometry and topology, given a group ''G'' (which may be a topological or Lie group), an equivariant bundle is a fiber bundle In mathematics, and particularly topology, a fiber bundle ( ''Commonwealth English'': fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a pr ... \pi\colon E\to B such that the total space E and the base space B are both ''G''-spaces (continuous or smooth, depending on the setting) and the projection map \pi between them is equivariant: \pi \circ g = g \circ \pi with some extra requirement depending on a typical fiber. For example, an equivariant vector bundle is an equivariant bundle such that the action of ''G'' restricts to a linear isomorphism between fibres. References *Berline, Nicole; Getzler, E.; Vergne, Michèle (2004), Heat Kernels and Dirac Operators, Berlin, New York: Springer-Verlag Fiber bundles {{differential-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirac Operator
In mathematics and in quantum mechanics, a Dirac operator is a first-order differential operator that is a formal square root, or half-iterate, of a second-order differential operator such as a Laplacian. It was introduced in 1847 by William Hamilton and in 1928 by Paul Dirac. The question which concerned Dirac was to factorise formally the Laplace operator of the Minkowski space, to get an equation for the wave function which would be compatible with special relativity. Formal definition In general, let ''D'' be a first-order differential operator acting on a vector bundle ''V'' over a Riemannian manifold ''M''. If : D^2=\Delta, \, where ∆ is the (positive, or geometric) Laplacian of ''V'', then ''D'' is called a Dirac operator. Note that there are two different conventions as to how the Laplace operator is defined: the "analytic" Laplacian, which could be characterized in \R^n as \Delta=\nabla^2=\sum_^n\Big(\frac\Big)^2 (which is negative-definite, in the sense that \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brauer's Theorem On Induced Characters
Brauer's theorem on induced characters, often known as Brauer's induction theorem, and named after Richard Brauer, is a basic result in the branch of mathematics known as character theory, which is part of the representation theory of finite groups. Background A precursor to Brauer's induction theorem was Artin's induction theorem, which states that , ''G'', times the trivial character of ''G'' is an integer combination of characters which are each induced from trivial characters of cyclic subgroups of ''G.'' Brauer's theorem removes the factor , ''G'', , but at the expense of expanding the collection of subgroups used. Some years after the proof of Brauer's theorem appeared, J.A. Green showed (in 1955) that no such induction theorem (with integer combinations of characters induced from linear characters) could be proved with a collection of subgroups smaller than the Brauer elementary subgroups. Another result between Artin's induction theorem and Brauer's induction theorem, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supertrace
In the theory of superalgebras, if ''A'' is a commutative superalgebra, ''V'' is a free right ''A''- supermodule and ''T'' is an endomorphism from ''V'' to itself, then the supertrace of ''T'', str(''T'') is defined by the following trace diagram: : More concretely, if we write out ''T'' in block matrix In mathematics, a block matrix or a partitioned matrix is a matrix that is interpreted as having been broken into sections called blocks or submatrices. Intuitively, a matrix interpreted as a block matrix can be visualized as the original matrix w ... form after the decomposition into even and odd subspaces as follows, :T=\beginT_&T_\\T_&T_\end then the supertrace :str(''T'') = the ordinary trace of ''T''00 − the ordinary trace of ''T''11. Let us show that the supertrace does not depend on a basis. Suppose e1, ..., ep are the even basis vectors and e''p''+1, ..., e''p''+''q'' are the odd basis vectors. Then, the components of ''T'', which are elements of ''A'', are de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivariant Topological K-theory
In mathematics, equivariance is a form of symmetry for functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry group, and when the function commutes with the action of the group. That is, applying a symmetry transformation and then computing the function produces the same result as computing the function and then applying the transformation. Equivariant maps generalize the concept of invariants, functions whose value is unchanged by a symmetry transformation of their argument. The value of an equivariant map is often (imprecisely) called an invariant. In statistical inference, equivariance under statistical transformations of data is an important property of various estimation methods; see invariant estimator for details. In pure mathematics, equivariance is a central object of study in equivariant topology and its subtopics equivariant cohomology and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kawasaki's Riemann–Roch Formula
Kawasaki disease (also known as mucocutaneous lymph node syndrome) is a syndrome of unknown cause that results in a fever and mainly affects children under 5 years of age. It is a form of vasculitis, in which medium-sized blood vessels become inflamed throughout the body. The fever typically lasts for more than five days and is not affected by antipyretics, usual medications. Other common symptoms include lymphadenopathy, large lymph nodes in the neck, a rash in the genital area, lips, Palm (hands), palms, or soles of the feet, and red eye (medicine), red eyes. Within three weeks of the onset, the skin from the hands and feet may peel, after which recovery typically occurs. The disease is the leading cause of acquired heart disease in children in developed countries, which include the formation of coronary artery aneurysms and myocarditis. While the specific cause is unknown, it is thought to result from an excessive immune response to particular infections in children who are G ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
