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Bogomol'nyi–Prasad–Sommerfield State
In theoretical physics, massive representations of an extended supersymmetry algebra called BPS states have mass equal to the supersymmetry central charge ''Z''. Quantum mechanically, if the supersymmetry remains unbroken, exact equality to the modulus of ''Z'' exists. Their importance arises as the supermultiplets shorten for generic massive representations, with stability and mass formula exact. ''d'' = 4 ''N'' = 2 The generators for the odd part of the superalgebra have relations: : \begin \ & = 2 \sigma_^m P_m \delta^A_B\\ \ & = 2 \epsilon_ \epsilon^ \bar\\ \ & = -2 \epsilon_ \epsilon_ Z\\ \end where: \alpha \dot are the Lorentz group indices, A and B are R-symmetry indices. Take linear combinations of the above generators as follows: : \begin R_\alpha^A & = \xi^ Q_\alpha^A + \xi \sigma_^0 \bar^\\ T_\alpha^A & = \xi^ Q_\alpha^A - \xi \sigma_^0 \bar^\\ \end Consider a state ψ which has 4 momentum (M,0,0,0). Applying the following operator to this state gives: : \begin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theoretical Physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experimental tools to probe these phenomena. The advancement of science generally depends on the interplay between experimental studies and theory. In some cases, theoretical physics adheres to standards of mathematical rigour while giving little weight to experiments and observations.There is some debate as to whether or not theoretical physics uses mathematics to build intuition and illustrativeness to extract physical insight (especially when normal experience fails), rather than as a tool in formalizing theories. This links to the question of it using mathematics in a less formally rigorous, and more intuitive or heuristic way than, say, mathematical physics. For example, while developing special relativity, Albert Einstein was concerned wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 supersymmetry is also called \mathcal=2, \mathcal=4 supersymmetry, for example. Extended supersymmetry is very important for analysis of mathematical properties of quantum field theory and superstring theory Superstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modeling them as vibrations of tiny supersymmetric strings. 'Superstring theory' is a shorthand for supersymmetric string t .... The more extended supersymmetry is, the more it constrains physical observables and parameters. See also * Supersymmetry algebra * Harmonic superspace * Projective superspace Supersymmetry {{Quantum-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Central Charge
In theoretical physics, a central charge is an operator ''Z'' that commutes with all the other symmetry operators. The adjective "central" refers to the center of the symmetry group—the subgroup of elements that commute with all other elements of the original group—often embedded within a Lie algebra. In some cases, such as two-dimensional conformal field theory, a central charge may also commute with all of the other operators, including operators that are not symmetry generators. Overview More precisely, the central charge is the charge that corresponds, by Noether's theorem, to the center of the central extension of the symmetry group. In theories with supersymmetry, this definition can be generalized to include supergroups and Lie superalgebras. A central charge is any operator which commutes with all the other supersymmetry generators. Theories with extended supersymmetry typically have many operators of this kind. In string theory, in the first quantized formali ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Short Supermultiplet
In theoretical physics, a short supermultiplet is a supermultiplet i.e. a representation of the supersymmetry algebra whose dimension is smaller than 2^ where N is the number of real supercharges. The representations that saturate the bound are known as the long supermultiplets. The states in a long supermultiplet may be produced from a representative by the action of the lowering and raising operators, assuming that for any basis vector, either the lowering operator or its conjugate raising operator produce a new nonzero state. This is the reason for the dimension indicated above. On the other hand, the short supermultiplets admit a subset of supercharges that annihilate the whole representation. That is why the short supermultiplets contain the BPS states, another description of the same concept. The BPS states are only possible for objects that are either massless or massive extremal, i.e. carrying a maximum allowed value of some central charge In theoretical physics, a centra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superalgebra
In mathematics and theoretical physics, a superalgebra is a Z2-graded algebra. That is, it is an algebra over a commutative ring or field with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading. The prefix ''super-'' comes from the theory of supersymmetry in theoretical physics. Superalgebras and their representations, supermodules, provide an algebraic framework for formulating supersymmetry. The study of such objects is sometimes called super linear algebra. Superalgebras also play an important role in related field of supergeometry where they enter into the definitions of graded manifolds, supermanifolds and superschemes. Formal definition Let ''K'' be a commutative ring. In most applications, ''K'' is a field of characteristic 0, such as R or C. A superalgebra over ''K'' is a ''K''-module ''A'' with a direct sum decomposition :A = A_0\oplus A_1 together with a bilinear multiplication ''A'' × ''A'' → ''A'' such t ... [...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, the R-symmetry group becomes a global U(N) non-abelian group In mathematics, and specifically in group theory, a non-abelian group, sometimes called a non-commutative group, is a group (''G'', ∗) in which there exists at least one pair of elements ''a'' and ''b'' of ''G'', such that ''a'' ∗ ' .... 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] |
Bogomol'nyi–Prasad–Sommerfield Bound
The Bogomol'nyi–Prasad–Sommerfield bound (named after Evgeny Bogomolny, M.K. Prasad, and Charles Sommerfield) is a series of inequalities for solutions of partial differential equations depending on the homotopy class of the solution at infinity. This set of inequalities is very useful for solving soliton equations. Often, by insisting that the bound be satisfied (called "saturated"), one can come up with a simpler set of partial differential equations to solve the Bogomolny equations. Solutions saturating the bound are called "BPS states" and play an important role in field theory and string theory. Example In a theory of non-abelian Yang–Mills–Higgs, the energy at a given time ''t'' is given by :E=\int d^3x\, \left frac\pi^T \pi + V(\varphi) + \frac\operatorname\left[\vec\cdot\vec+\vec\cdot\vec\rightright] where \pi is the covariant derivative of the Higgs field and ''V'' is the potential. If we assume that ''V'' is nonnegative and is zero only for the Higgs vacu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wall-crossing
In algebraic geometry and string theory, the phenomenon of wall-crossing describes the discontinuous change of a certain quantity, such as an integer geometric invariant, an index or a space of BPS state, across a codimension-one wall in a space of stability conditions, a so-called wall of marginal stability A wall is a structure and a surface that defines an area; carries a load; provides security, Shelter in place, shelter, or soundproofing; or, is decorative. There are many kinds of walls, including: * Walls in buildings that form a fundamental .... References * Kontsevich, M. and Soibelman, Y. "Stability structures, motivic Donaldson–Thomas invariants and cluster transformations" (2008). . * M. Kontsevich, Y. Soibelman, "Motivic Donaldson–Thomas invariants: summary of results", * Joyce, D. and Song, Y. "A theory of generalized Donaldson–Thomas invariants," (2008). . * Gaiotto, D. and Moore, G. and Neitzke, A. "Four-dimensional wall-crossing via three-dimensio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
