|
Infinite-dimensional Chern–Simons Theory
In mathematics, infinite-dimensional Chern–Simons theory (not to be confused with ∞-Chern–Simons theory) is a generalization of Chern–Simons theory to manifolds with infinite dimensions. These are not modeled with finite-dimensional Euclidean spaces, but infinite-dimensional topological vector spaces, for example Hilbert, Banach and Fréchet spaces, which lead to Hilbert, Banach and Fréchet manifolds respectively. Principal bundles, which in finite-dimensional Chern–Simons theory are considered with (compact) Lie groups as gauge groups, are then fittingly considered with Hilbert Lie, Banach Lie and Fréchet Lie groups as gauge groups respectively, which also makes their total spaces into a Hilbert, Banach and Fréchet manifold respectively. These are called Hilbert, Banach and Fréchet principal bundles respectively. The theory is named after Shiing-Shen Chern and James Simons James Harris Simons (; born 25 April 1938) is an American mathematician, billionaire hed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i.e. that the space not exclude any ''limiting values'' of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval ,1would be compact. Similarly, the space of rational numbers \mathbb is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers \mathbb is not compact either, because it excludes the two limiting values +\infty and -\infty. However, the ''extended'' real number line ''would'' be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other top ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Six-dimensional Holomorphic Chern–Simons Theory
In mathematical physics, six-dimensional holomorphic Chern–Simons theory or sometimes holomorphic Chern–Simons theory is a gauge theory on a three-dimensional complex manifold. It is a complex analogue of Chern–Simons theory, named after Shiing-Shen Chern and James Simons who first studied Chern–Simons forms which appear in the action of Chern–Simons theory. The theory is referred to as six-dimensional as the underlying manifold of the theory is three-dimensional as a complex manifold, hence six-dimensional as a real manifold. The theory has been used to study integrable systems through four-dimensional Chern–Simons theory, which can be viewed as a symmetry reduction of the six-dimensional theory. For this purpose, the underlying three-dimensional complex manifold is taken to be the three-dimensional complex projective space \mathbb^3, viewed as twistor space. Formulation The background manifold \mathcal on which the theory is defined is a complex manifold which has ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Four-dimensional Chern–Simons Theory
In mathematical physics, four-dimensional Chern–Simons theory, also known as semi-holomorphic or semi-topological Chern–Simons theory, is a quantum field theory initially defined by Nikita Nekrasov, rediscovered and studied by Kevin Costello, and later by Edward Witten and Masahito Yamazaki. It is named after mathematicians Shiing-Shen Chern and James Simons who discovered the Chern–Simons 3-form appearing in the theory. The gauge theory has been demonstrated to be related to many integrable systems, including exactly solvable lattice models such as the six-vertex model of Lieb and the Heisenberg spin chain and integrable field theories such as principal chiral models, symmetric space coset sigma models and Toda field theory, although the integrable field theories require the introduction of two-dimensional surface defects. The theory is also related to the Yang–Baxter equation and quantum groups such as the Yangian. The theory is similar to three-dimensional Chern� ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chern–Simons Form
In mathematics, the Chern–Simons forms are certain secondary characteristic classes. The theory is named for Shiing-Shen Chern and James Harris Simons, co-authors of a 1974 paper entitled "Characteristic Forms and Geometric Invariants," from which the theory arose. Definition Given a manifold and a Lie algebra valued 1-form \mathbf over it, we can define a family of ''p''-forms: In one dimension, the Chern–Simons 1-form is given by :\operatorname \mathbf In three dimensions, the Chern–Simons 3-form is given by :\operatorname \left \mathbf \wedge \mathbf-\frac \mathbf \wedge \mathbf \wedge \mathbf \right= \operatorname \left d\mathbf \wedge \mathbf + \frac \mathbf \wedge \mathbf \wedge \mathbf\right In five dimensions, the Chern–Simons 5-form is given by : \begin & \operatorname \left \mathbf\wedge\mathbf \wedge \mathbf-\frac \mathbf \wedge\mathbf\wedge\mathbf\wedge\mathbf +\frac \mathbf \wedge \mathbf \wedge \mathbf \wedge \mathbf \wedge\mathbf \right\\ pt= & \ope ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jim Simons
{{hndis, Simons, James ...
Jim or James Simons may refer to: *Jim Simons (mathematician) (born 1938), mathematician and hedge fund manager *Jim Simons (golfer) (1950–2005), American golfer *Jimmy Simons (born 1970), Dutch footballer *Jimmy Simons, co-winner of 2001 Primetime Emmy Award for Outstanding Comedy Series *James Simons, preceded by John Calhoun Sheppard as speaker of the South Carolina House of Representatives See also *James Simmons (other) *James Simon (other) James or Jim Simon may refer to: * James Simon (composer) (1880–1944), German composer, pianist and musicologist * James Simon (journalist), journalism professor at Fairfield University, Fairfield, Connecticut * James D. Simon (1897–1982), Lou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. 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 (division), or equivalently, the concept of addition and the taking of inverses (subtraction). Combining these two ideas, one obtains a continuous group where multiplying points and their inverses are continuous. If the multiplication and taking of inverses are 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 rotational symmetry in three dimensions (given by the special orthogonal group \text(3)). Lie groups are widely used in many parts of modern mathematics and physics. ... [...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. # A projection onto X. For a product space, this is just the projection onto the first factor, (x,g) \mapsto x. Unlike a product space, principal bundles lack a preferred choice of identity cross-section; they have no preferred analog of (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 number of arbitrary choices are made to try to define such a structure by defining it on smaller pieces of the space. A common example of a princi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
∞-Chern–Simons Theory
In mathematics, ∞-Chern–Simons theory (not to be confused with infinite-dimensional Chern–Simons theory) is a generalized formulation of Chern–Simons theory from differential geometry using the formalism of higher category theory, which in particular studies ∞-categories. It is obtained by taking general abstract analogs of all involved concepts defined in any cohesive ∞-topos, for example that of smooth ∞-groupoids. Principal bundles on which Lie groups act are for example replaced by ∞-principal bundles on with group objects in ∞-topoi act.Definition in Schreiber 2013, 1.2.6.5.2 The theory is named after Shiing-Shen Chern and James Simons, who first described Chern–Simons forms in 1974, although the generalization was not developed by them. See also * ∞-Chern–Weil theory Literature * * * * {{cite arXiv , arxiv=1301.2580 , author1=Domenico Fiorenza , author2=Hisham Sati , author3=Urs Schreiber Urs Schreiber (born 1974) is a mathematician sp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
