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N = 4 Supersymmetric Yang–Mills Theory
''N'' = 4 supersymmetric Yang–Mills (SYM) theory is a mathematical and physical model created to study particles through a simple system, similar to string theory, with conformal symmetry. It is a simplified toy theory based on Yang–Mills theory that does not describe the real world, but is useful because it can act as a proving ground for approaches for attacking problems in more complex theories. It describes a universe containing bosonic field, boson fields and fermion fields which are related by four Supersymmetry, supersymmetries (this means that swapping boson, fermion and scalar fields in a certain way leaves the predictions of the theory invariant). It is one of the simplest (because it has no free parameters except for the gauge group) and one of the few finite quantum field theories in 4 dimensions. It can be thought of as the most symmetric field theory that does not involve gravity. Meaning of ''N'' and numbers of fields In ''N'' supersymmetric Yang– ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physical Model
A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure. Models can be divided into physical models (e.g. a model plane) and abstract models (e.g. mathematical expressions describing behavioural patterns). Abstract or conceptual models are central to philosophy of science, as almost every scientific theory effectively embeds some kind of model of the physical or human sphere. In commerce, "model" can refer to a specific design of a product as displayed in a catalogue or show room (e.g. Ford Model T), and by extension to the sold product itself. Types of models include: Physical model A physical model (most commonly referred to simply as a model but in this context distinguished from a conceptual model) is a smaller or larger physical copy of an object. The object being modelled may be small ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coupling Constant
In physics, a coupling constant or gauge coupling parameter (or, more simply, a coupling), is a number that determines the strength of the force exerted in an interaction. Originally, the coupling constant related the force acting between two static bodies to the "charges" of the bodies (i.e. the electric charge for electrostatic and the mass for Newtonian gravity) divided by the distance squared, r^2, between the bodies; thus: G in F=G m_1 m_2/r^2 for Newtonian gravity and k_\text in F=k_\textq_1 q_2/r^2 for electrostatic. This description remains valid in modern physics for linear theories with static bodies and massless force carriers. A modern and more general definition uses the Lagrangian \mathcal (or equivalently the Hamiltonian \mathcal) of a system. Usually, \mathcal (or \mathcal) of a system describing an interaction can be separated into a ''kinetic part'' T and an ''interaction part'' V: \mathcal=T-V (or \mathcal=T+V). In field theory, V always contains 3 fields ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
D-brane
In string theory, D-branes, short for ''Dirichlet membrane'', are a class of extended objects upon which open strings can end with Dirichlet boundary conditions, after which they are named. D-branes were discovered by Jin Dai, Leigh, and Polchinski, and independently by Hořava, in 1989. In 1995, Polchinski identified D-branes with black p-brane solutions of supergravity, a discovery that triggered the Second Superstring Revolution and led to both holographic and M-theory dualities. D-branes are typically classified by their spatial dimension, which is indicated by a number written after the ''D.'' A D0-brane is a single point, a D1-brane is a line (sometimes called a "D-string"), a D2-brane is a plane, and a D25-brane fills the highest-dimensional space considered in bosonic string theory. There are also instantonic D(–1)-branes, which are localized in both space and time. Theoretical background The equations of motion of string theory require that the endpoints of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Type IIB String Theory
In theoretical physics, type II string theory is a unified term that includes both type IIA strings and type IIB strings theories. Type II string theory accounts for two of the five consistent superstring theories in ten dimensions. Both theories have the maximal amount of supersymmetry — namely 32 supercharges — in ten dimensions. Both theories are based on oriented closed strings. On the worldsheet, they differ only in the choice of GSO projection. Type IIA string theory At low energies, type IIA string theory is described by type IIA supergravity in ten dimensions which is a non-chiral theory (i.e. left–right symmetric) with (1,1) ''d''=10 supersymmetry; the fact that the anomalies in this theory cancel is therefore trivial. In the 1990s it was realized by Edward Witten (building on previous insights by Michael Duff, Paul Townsend, and others) that the limit of type IIA string theory in which the string coupling goes to infinity becomes a new 11-dimensional ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supercharges
In theoretical physics, a supercharge is a generator of supersymmetry transformations. It is an example of the general notion of a charge in physics. Supercharge, denoted by the symbol Q, is an operator which transforms bosons into fermions, and vice versa. Since the supercharge operator changes a particle with spin one-half to a particle with spin one or zero, the supercharge itself is a spinor that carries one half unit of spin. Depending on the context, supercharges may also be called ''Grassmann variables'' or ''Grassmann directions''; they are generators of the exterior algebra of anti-commuting numbers, the Grassmann numbers. All these various usages are essentially synonymous; they refer to the \mathbb_2 grading between bosons and fermions, or equivalently, the grading between ''c-numbers'' and ''a-numbers''. Calling it a charge emphasizes the notion of a symmetry at work. Commutation Supercharge is described by the Super-Poincaré algebra. Supercharge commutes with ... [...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 man ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superconformal Algebra
In theoretical physics, the superconformal algebra is a graded Lie algebra or superalgebra that combines the conformal algebra and supersymmetry. In two dimensions, the superconformal algebra is infinite-dimensional. In higher dimensions, superconformal algebras are finite-dimensional and generate the superconformal group (in two Euclidean dimensions, the Lie superalgebra does not generate any Lie supergroup). Superconformal algebra in dimension greater than 2 The conformal group of the (p+q)-dimensional space \mathbb^ is SO(p+1,q+1) and its Lie algebra is \mathfrak(p+1,q+1). The superconformal algebra is a Lie superalgebra containing the bosonic factor \mathfrak(p+1,q+1) and whose odd generators transform in spinor representations of \mathfrak(p+1,q+1). Given Kač's classification of finite-dimensional simple Lie superalgebras, this can only happen for small values of p and q. A (possibly incomplete) list is * \mathfrak^*(2N, 2,2) in 3+0D thanks to \mathfrak(2,2)\simeq\m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonrenormalization Theorems
In theoretical physics a nonrenormalization theorem is a limitation on how a certain quantity in the classical description of a quantum field theory may be modified by renormalization in the full quantum theory. Renormalization theorems are common in theories with a sufficient amount of supersymmetry, usually at least 4 supercharges. Perhaps the first nonrenormalization theorem was introduced by Marcus T. Grisaru, Martin Rocek and Warren Siegel in their 1979 papeImproved methods for supergraphs Nonrenormalization in supersymmetric theories and holomorphy Nonrenormalization theorems in supersymmetric theories are often consequences of the fact that certain objects must have a holomorphic dependence on the quantum fields and coupling constants. In this case the nonrenormalization theory is said to be a consequence of holomorphy. The more supersymmetry a theory has, the more renormalization theorems apply. Therefore a renormalization theorem that is valid for a theory with \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
R-symmetry
In theoretical physics, the R-symmetry is the symmetry transforming different supercharges in a theory with supersymmetry into each other. In the simplest case of the ''N''=1 supersymmetry, such an R-symmetry is isomorphic to a global U(1) group or its discrete subgroup (for the Z2 subgroup it is called R-parity). For extended supersymmetry In theoretical physics, extended supersymmetry is supersymmetry whose infinitesimal generators Q_i^\alpha carry not only a spinor index \alpha, but also an additional index i=1,2 \dots \mathcal where \mathcal is integer (such as 2 or 4). Extended ..., the R-symmetry group becomes a global U(N) non-abelian group. In a model that is classically invariant under both ''N''=1 supersymmetry and conformal transformations, the closure of the superconformal algebra (at least on-shell) needs the introduction of a further bosonic generator that is associated to the R-symmetry. References * Supersymmetry {{quantum-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauge Covariant Derivative
The gauge covariant derivative is a variation of the covariant derivative used in general relativity, quantum field theory and fluid dynamics. If a theory has gauge transformations, it means that some physical properties of certain equations are preserved under those transformations. Likewise, the gauge covariant derivative is the ordinary derivative modified in such a way as to make it behave like a true vector operator, so that equations written using the covariant derivative preserve their physical properties under gauge transformations. Overview There are many ways to understand the gauge covariant derivative. The approach taken in this article is based on the historically traditional notation used in many physics textbooks. Another approach is to understand the gauge covariant derivative as a kind of connection, and more specifically, an affine connection.Alexandre Guay, Geometrical aspects of local gauge symmetry' (2004) The affine connection is interesting because it do ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pauli Matrices
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used in connection with isospin symmetries. \begin \sigma_1 = \sigma_\mathrm &= \begin 0&1\\ 1&0 \end \\ \sigma_2 = \sigma_\mathrm &= \begin 0& -i \\ i&0 \end \\ \sigma_3 = \sigma_\mathrm &= \begin 1&0\\ 0&-1 \end \\ \end These matrices are named after the physicist Wolfgang Pauli. In quantum mechanics, they occur in the Pauli equation which takes into account the interaction of the spin of a particle with an external electromagnetic field. They also represent the interaction states of two polarization filters for horizontal/vertical polarization, 45 degree polarization (right/left), and circular polarization (right/left). Each Pauli matrix is Hermitian, and together with the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weyl Fermion
Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is associated with the University of Göttingen tradition of mathematics, represented by Carl Friedrich Gauss, David Hilbert and Hermann Minkowski. His research has had major significance for theoretical physics as well as purely mathematical disciplines such as number theory. He was one of the most influential mathematicians of the twentieth century, and an important member of the Institute for Advanced Study during its early years. Weyl contributed to an exceptionally wide range of mathematical fields, including works on space, time, matter, philosophy, logic, symmetry and the history of mathematics. He was one of the first to conceive of combining general relativity with the laws of electromagnetism. Freeman Dyson wrote that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
