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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^+) - \operat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and 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 manifolds. A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying structu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fixed Point (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. Specifically, in mathematics, a fixed point of a function is an element that is mapped to itself by the function. In physics, the term fixed point can refer to a temperature that can be used as a reproducible reference point, usually defined by a phase change or triple point. Fixed point of a function Formally, is a fixed point of a function if belongs to both the domain and the codomain of , and . 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, , has no fixed points, since is never equal to for any real number. In graphical terms, a fixed point means the point is on the line , or in other words the graph of has a point in common with that line. Fixed-point iteration In numerical analysis, ''fixed-point iter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Index Theorem
Index (or its plural form 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 a Halo megastructure in the ''Halo'' series of video games 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 Corpora ... [...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 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 In topology, a branch of mathematics, Borel's theorem, due to , says the cohomology ring of a classifying space or a classifying stack is a polynomial ring. See also *Atiyah–Bott formula In algebraic geometry, the Atiyah–Bott formula s ..., 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'') is the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivariant Bundle
In geometry and topology, given a group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ... ''G'', an equivariant bundle is a fiber bundle such that the total space and the base spaces are both ''G''-spaces 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. 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 quantum mechanics, a Dirac operator is a differential operator that is a formal square root, or half-iterate, of a second-order operator such as a Laplacian. The original case which concerned Paul Dirac was to factorise formally an operator for Minkowski space, to get a form of quantum theory compatible with special relativity; to get the relevant Laplacian as a product of first-order operators he introduced spinors. It was first published in 1928. 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 Laplacian of ''V'', then ''D'' is called a Dirac operator. In high-energy physics, this requirement is often relaxed: only the second-order part of ''D''2 must equal the Laplacian. Examples Example 1 ''D'' = −''i'' ∂''x'' is a Dirac operator on the tangent bundle over a line. Example 2 Consider a simple bundle of notable import ... [...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, within representation theory of a finite group. 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, also due to ... [...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 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 defined as :T(\mathbf_j)=\mathbf_i T^i_j.\, The grading of ''T''''i''''j'' is the sum of the gradings of ''T'', e''i'', e''j'' mod 2. A change of basis to e1', ..., ep', e(''p''+1)', ..., e(''p''+''q'')' is given by the supermatrix :\mathbf_=\mathbf_i A^i_ ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kawasaki's Riemann–Roch Formula
In differential geometry, Kawasaki's Riemann–Roch formula, introduced by Tetsuro Kawasaki, is the Riemann–Roch formula for orbifolds. It can compute the Euler characteristic of an orbifold. Kawasaki's original proof made a use of the equivariant index theorem. Today, the formula is known to follow from the Riemann–Roch formula for quotient stack In algebraic geometry, a quotient stack is a stack that parametrizes equivariant objects. Geometrically, it generalizes a quotient of a scheme or a variety by a group: a quotient variety, say, would be a coarse approximation of a quotient stack. Th ...s. References *Tetsuro Kawasaki. The Riemann-Roch theorem for complex V-manifolds. Osaka J. Math., 16(1):151–159, 1979 Theorems in differential geometry Theorems in algebraic geometry See also * Riemann–Roch-type theorem {{differential-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
