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Supergeometry
Supergeometry is differential geometry of modules over graded commutative algebras, supermanifolds and graded manifolds. Supergeometry is part and parcel of many classical and quantum field theories involving odd fields, e.g., SUSY field theory, BRST theory, or supergravity. Supergeometry is formulated in terms of \mathbb Z_2-graded modules and sheaves over \mathbb Z_2-graded commutative algebras (supercommutative algebras). In particular, superconnections are defined as Koszul connections on these modules and sheaves. However, supergeometry is not particular noncommutative geometry because of a different definition of a graded derivation. Graded manifolds and supermanifolds also are phrased in terms of sheaves of graded commutative algebras. Graded manifolds are characterized by sheaves on smooth manifolds, while supermanifolds are constructed by gluing of sheaves of supervector spaces. There are different types of supermanifolds. These are smooth supermanifolds (H^\infty-, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connection (algebraic Framework)
Geometry of quantum systems (e.g., noncommutative geometry and supergeometry) is mainly phrased in algebraic terms of modules and algebras. Connections on modules are generalization of a linear connection on a smooth vector bundle E\to X written as a Koszul connection on the C^\infty(X)-module of sections of E\to X. Commutative algebra Let A be a commutative ring and M an ''A''-module. There are different equivalent definitions of a connection on M. First definition If k \to A is a ring homomorphism, a k-linear connection is a k-linear morphism : \nabla: M \to \Omega^1_ \otimes_A M which satisfies the identity : \nabla(am) = da \otimes m + a \nabla m A connection extends, for all p \geq 0 to a unique map : \nabla: \Omega^p_ \otimes_A M \to \Omega^_ \otimes_A M satisfying \nabla(\omega \otimes f) = d\omega \otimes f + (-1)^p \omega \wedge \nabla f. A connection is said to be integrable if \nabla \circ \nabla = 0, or equivalently, if the curvature \nabla^2: M \to \Omeg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graded Manifold
In algebraic geometry, graded manifolds are extensions of the concept of manifolds based on ideas coming from supersymmetry and supercommutative algebra. Both graded manifolds and supermanifolds are phrased in terms of sheaves of graded commutative algebras. However, graded manifolds are characterized by sheaves on smooth manifolds, while supermanifolds are constructed by gluing of sheaves of supervector spaces. Graded manifolds A graded manifold of dimension (n,m) is defined as a locally ringed space (Z,A) where Z is an n-dimensional smooth manifold and A is a C^\infty_Z-sheaf of Grassmann algebras of rank m where C^\infty_Z is the sheaf of smooth real functions on Z. The sheaf A is called the structure sheaf of the graded manifold (Z,A), and the manifold Z is said to be the body of (Z,A). Sections of the sheaf A are called graded functions on a graded manifold (Z,A). They make up a graded commutative C^\infty(Z)-ring A(Z) called the structure ring of (Z,A). The well-known B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supermetric
Supermetric is a mathematical concept used in a number of fields in physics. See also *Supergeometry *Supergravity *Super Minkowski space *Gauge gravitation theory In quantum field theory, gauge gravitation theory is the effort to extend Yang–Mills theory, which provides a universal description of the fundamental interactions, to describe gravity. ''Gauge gravitation theory'' should not be confused with th ... References Further reading * Deligne, P. and Morgan, J. (1999) Notes on supersymmetry (following Joseph Bernstein). In: ''Quantum Field Theory and Strings: A Course for Mathematicians, Vol. 1'' (Providence, RI: Amer. Math. Soc.) pp. 41-97 . * External links * G. Sardanashvily, Lectures on supergeometry, {{arXiv, 0910.0092. Supersymmetry ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graded Manifold
In algebraic geometry, graded manifolds are extensions of the concept of manifolds based on ideas coming from supersymmetry and supercommutative algebra. Both graded manifolds and supermanifolds are phrased in terms of sheaves of graded commutative algebras. However, graded manifolds are characterized by sheaves on smooth manifolds, while supermanifolds are constructed by gluing of sheaves of supervector spaces. Graded manifolds A graded manifold of dimension (n,m) is defined as a locally ringed space (Z,A) where Z is an n-dimensional smooth manifold and A is a C^\infty_Z-sheaf of Grassmann algebras of rank m where C^\infty_Z is the sheaf of smooth real functions on Z. The sheaf A is called the structure sheaf of the graded manifold (Z,A), and the manifold Z is said to be the body of (Z,A). Sections of the sheaf A are called graded functions on a graded manifold (Z,A). They make up a graded commutative C^\infty(Z)-ring A(Z) called the structure ring of (Z,A). The well-known B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of n-dimensional Euclidean space. One-dimensional manifolds include lines and circles, but not lemniscates. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. The concept has applications in computer-graphics given the need to associate pictures with coordinates (e.g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supersymmetry
In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of supersymmetric theories exist. Supersymmetry is a spacetime symmetry between two basic classes of particles: bosons, which have an integer-valued spin and follow Bose–Einstein statistics, and fermions, which have a half-integer-valued spin and follow Fermi–Dirac statistics. In supersymmetry, each particle from one class would have an associated particle in the other, known as its superpartner, the spin of which differs by a half-integer. For example, if the electron exists in a supersymmetric theory, then there would be a particle called a ''"selectron"'' (superpartner electron), a bosonic partner of the electron. In the simplest supersymmetry theories, with perfectly " unbroken" supersymmetry, each pair of superpartners would share the same mass and intern ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supermanifold
In physics and mathematics, supermanifolds are generalizations of the manifold concept based on ideas coming from supersymmetry. Several definitions are in use, some of which are described below. Informal definition An informal definition is commonly used in physics textbooks and introductory lectures. It defines a supermanifold as a manifold with both bosonic and fermionic coordinates. Locally, it is composed of coordinate charts that make it look like a "flat", "Euclidean" superspace. These local coordinates are often denoted by :(x,\theta,\bar) where ''x'' is the ( real-number-valued) spacetime coordinate, and \theta\, and \bar are Grassmann-valued spatial "directions". The physical interpretation of the Grassmann-valued coordinates are the subject of debate; explicit experimental searches for supersymmetry have not yielded any positive results. However, the use of Grassmann variables allow for the tremendous simplification of a number of important mathematical results. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K-theory
In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It is also a fundamental tool in the field of operator algebras. It can be seen as the study of certain kinds of invariants of large matrices. K-theory involves the construction of families of ''K''-functors that map from topological spaces or schemes to associated rings; these rings reflect some aspects of the structure of the original spaces or schemes. As with functors to groups in algebraic topology, the reason for this functorial mapping is that it is easier to compute some topological properties from the mapped rings than from the original spaces or schemes. Examples of results gleaned from the K-theory approach include the Grothendieck–Riemann–Roch theorem, Bott periodicity, the Atiyah ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chern Class
In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau manifolds, string theory, Chern–Simons theory, knot theory, Gromov–Witten invariants, topological quantum field theory, the Chern theorem etc. Chern classes were introduced by . Geometric approach Basic idea and motivation Chern classes are characteristic classes. They are topological invariants associated with vector bundles on a smooth manifold. The question of whether two ostensibly different vector bundles are the same can be quite hard to answer. The Chern classes provide a simple test: if the Chern classes of a pair of vector bundles do not agree, then the vector bundles are different. The converse, however, is not true. In topology, differential geometry, and algebraic geometry, it is often important to count how many l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yuval Ne'eman
Yuval Ne'eman ( he, יובל נאמן, 14 May 1925 – 26 April 2006) was an Israeli theoretical physicist, military scientist, and politician. He was Minister of Science and Development in the 1980s and early 1990s. He was the President of Tel Aviv University from 1971 to 1977. He was awarded the Israel Prize in the field of exact sciences (which he returned in 1992 in protest of the award of the Israel Prize to Emile Habibi), the Albert Einstein Award, the Wigner Medal, and the EMET Prize for Arts, Sciences and Culture. Biography Yuval Ne'eman was born in Tel Aviv during the Mandate era, graduated from high school at the age of 15, and studied mechanical engineering at the Technion. At the age of 15, Ne'eman also joined the Haganah. During the 1948 Arab-Israeli War Ne'eman served in the Israel Defense Forces (IDF) as battalion deputy commander, then as Operations Officer of Tel Aviv, and commander of Givati Brigade. Later (1952–54) he served as Deputy Commander of O ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Daniel Quillen
Daniel Gray "Dan" Quillen (June 22, 1940 – April 30, 2011) was an American mathematician. He is known for being the "prime architect" of higher algebraic ''K''-theory, for which he was awarded the Cole Prize in 1975 and the Fields Medal in 1978. From 1984 to 2006, he was the Waynflete Professor of Pure Mathematics at Magdalen College, Oxford. Education and career Quillen was born in Orange, New Jersey, and attended Newark Academy. He entered Harvard University, where he earned both his AB, in 1961, and his PhD in 1964; the latter completed under the supervision of Raoul Bott, with a thesis in partial differential equations. He was a Putnam Fellow in 1959. Quillen obtained a position at the Massachusetts Institute of Technology after completing his doctorate. He also spent a number of years at several other universities. He visited France twice: first as a Sloan Fellow in Paris, during the academic year 1968–69, where he was greatly influenced by Grothendieck, and the ... [...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 which 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 f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
