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∞-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 book , 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] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Object
In category theory, a branch of mathematics, group objects are certain generalizations of group (mathematics), groups that are built on more complicated structures than Set (mathematics), sets. A typical example of a group object is a topological group, a group whose underlying set is a topological space such that the group operations are continuity (topology), continuous. Definition Formally, we start with a category (mathematics), category ''C'' with finite products (i.e. ''C'' has a terminal object 1 and any two objects of ''C'' have a product (category theory), product). A group object in ''C'' is an object ''G'' of ''C'' together with morphisms *''m'' : ''G'' × ''G'' → ''G'' (thought of as the "group multiplication") *''e'' : 1 → ''G'' (thought of as the "inclusion of the identity element") *''inv'' : ''G'' → ''G'' (thought of as the "inversion operation") such that the following properties (modeled on the group axioms – more precisely, on the Universal algebra#G ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NLab
The ''n''Lab is a wiki for research-level notes, expositions and collaborative work, including original research, in mathematics, physics, and philosophy, with a focus on methods from type theory, category theory, and homotopy theory. The ''n''Lab espouses the "''n''-point of view" (a deliberate pun on Wikipedia's "neutral point of view") that type theory, homotopy theory, category theory, and higher category theory provide a useful unifying viewpoint for mathematics, physics and philosophy. The ''n'' in ''n''-point of view could refer to either ''n''-categories as found in higher category theory, ''n''-groupoids as found in both homotopy theory and higher category theory, or ''n''-types as found in homotopy type theory. Overview The ''n''Lab was originally conceived to provide a repository for ideas (and even new research) generated in the comments on posts at the ''n''-Category Café, a group blog run (at the time) by John C. Baez, David Corfield and Urs Schreiber. Eventua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jim Stasheff
James Dillon Stasheff (born January 15, 1936, New York City) is an American mathematician, a professor emeritus of mathematics at the University of North Carolina at Chapel Hill. He works in algebraic topology and algebra as well as their applications to physics. Biography Stasheff did his undergraduate studies in mathematics at the University of Michigan, graduating in 1956. Stasheff then began his graduate studies at Princeton University; his notes for a 1957 course by John Milnor on characteristic classes first appeared in mimeographed form and later in 1974 in revised form book with Stasheff as a co-author. After his second year at Princeton, he moved to Oxford University on a Marshall Scholarship. Two years later in 1961, with a pregnant wife, needing an Oxford degree to get reimbursed for his return trip to the US, and yet still feeling attached to Princeton, he split his thesis into two parts (one topological, the other algebraic) and earned two doctorates, a D.Phil. from Ox ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Urs Schreiber
Urs Schreiber (born 1974) is a mathematician specializing in the connection between mathematics and theoretical physics (especially string theory) and currently working as a researcher at New York University Abu Dhabi. He was previously a researcher at the Czech Academy of Sciences, Institute of Mathematics, Department for Algebra, Geometry and Mathematical Physics. Education Schreiber obtained his doctorate from the University of Duisburg-Essen in 2005 with a thesis supervised by Robert Graham and titled ''From Loop Space Mechanics to Nonabelian Strings''. Work Schreiber's research fields include the mathematical foundation of quantum field theory. Schreiber is a co-creator of the ''n''Lab, a wiki for research mathematicians and physicists working in higher category theory In mathematics, higher category theory is the part of category theory at a ''higher order'', which means that some equalities are replaced by explicit morphism, arrows in order to be able to explicitly stu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
∞-Chern–Weil Theory
In mathematics, ∞-Chern–Weil theory is a generalized formulation of Chern–Weil theory from differential geometry using the formalism of higher category theory. The theory is named after Shiing-Shen Chern and André Weil, who first constructed the Chern–Weil homomorphism in the 1940s, although the generalization was not developed by them. Generalization There are three equivalent ways to describe the k-th Chern class of complex vector bundles of rank n, which is as a: * (1-categorical) natural transformation ,\operatorname(n)Rightarrow ,K(\mathbb,2k)/math> * homotopy class of a continuous map \operatorname(n)\rightarrow K(\mathbb,2k) * singular cohomology class in H^(\operatorname(n),\mathbb) \operatorname(n) is the classifying space for the unitary group \operatorname(n) and K(\mathbb,2k) is an Eilenberg–MacLane space, which represent the set of complex vector bundles of rank n with ,\operatorname(n)cong\operatorname_\mathbb^n(-) and singular cohomology with ,K(\m ... [...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 Multilinear form, 1-form \mathbf over it, we can define a family of Multilinear form, ''p''-forms: In one dimension, the Chern–Simons Multilinear form, 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 \w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jim Simons
James Harris Simons (April 25, 1938 – May 10, 2024) was an American hedge fund manager, investor, mathematician, and philanthropist. At the time of his death, Simons's net worth was estimated to be $31.4 billion, making him the 55th-richest person in the world. He was the founder of Renaissance Technologies, a quantitative hedge fund based in East Setauket, New York. He and his fund are known to be quantitative investors, using mathematical models and algorithms to make investment gains from market inefficiencies. Due to the long-term aggregate investment returns of Renaissance and its Medallion Fund, Simons was called the "greatest investor on Wall Street" and more specifically "the most successful hedge fund manager of all time". Simons developed the Chern–Simons form (with Shiing-Shen Chern), and contributed to the development of string theory by providing a theoretical framework to combine geometry and topology with quantum field theory. In 1994, Simons and his wif ... [...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] |
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 Shiing-Shen Chern (; , ; October 26, 1911 – December 3, 2004) was a Chinese American mathematician and p ... [...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] |
