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N = 2 Superconformal Algebra
In mathematical physics, the 2D ''N'' = 2 superconformal algebra is an infinite-dimensional Lie superalgebra, related to supersymmetry, that occurs in string theory and two-dimensional conformal field theory. It has important applications in mirror symmetry. It was introduced by as a gauge algebra of the U(1) fermionic string. Definition There are two slightly different ways to describe the ''N'' = 2 superconformal algebra, called the ''N'' = 2 Ramond algebra and the ''N'' = 2 Neveu–Schwarz algebra, which are isomorphic (see below) but differ in the choice of standard basis. The ''N'' = 2 superconformal algebra is the Lie superalgebra with basis of even elements ''c'', ''L''''n'', ''J''''n'', for ''n'' an integer, and odd elements ''G'', ''G'', where r\in (for the Ramond basis) or r\in + (for the Neveu–Schwarz basis) defined by the following relations: ::''c'' is in the center :: _m,L_n= \left(m-n\right) L_ + \left(m^3-m\r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Physics
Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories". An alternative definition would also include those mathematics that are inspired by physics (also known as physical mathematics). Scope There are several distinct branches of mathematical physics, and these roughly correspond to particular historical periods. Classical mechanics The rigorous, abstract and advanced reformulation of Newtonian mechanics adopting the Lagrangian mechanics and the Hamiltonian mechanics even in the presence of constraints. Both formulations are embodied in analytical mechanics and lead to understanding the deep interplay of the notions of symmetry and conserved quantities during the dynamical evoluti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SU(2)
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1. The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special case. The group operation is matrix multiplication. The special unitary group is a normal subgroup of the unitary group , consisting of all unitary matrices. As a compact classical group, is the group that preserves the standard inner product on \mathbb^n. It is itself a subgroup of the general linear group, \operatorname(n) \subset \operatorname(n) \subset \operatorname(n, \mathbb ). The groups find wide application in the Standard Model of particle physics, especially in the electroweak interaction and in quantum chromodynamics. The groups are important in quantum computing, as they represent the possible quantum logic gate operations in a quantum circuit with n qubits and thus 2^n basis states. (Alternatively, the more genera ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Algebras
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identity. The Lie bracket of two vectors x and y is denoted [x,y]. The vector space \mathfrak g together with this operation is a non-associative algebra, meaning that the Lie bracket is not necessarily associative property, associative. Lie algebras are closely related to Lie groups, which are group (mathematics), groups that are also smooth manifolds: any Lie group gives rise to a Lie algebra, which is its tangent space at the identity. Conversely, to any finite-dimensional Lie algebra over real or complex numbers, there is a corresponding connected space, connected Lie group unique up to finite coverings (Lie's third theorem). This Lie group–Lie algebra correspondence, correspondence allows one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conformal Field Theory
A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometimes be exactly solved or classified. Conformal field theory has important applications to condensed matter physics, statistical mechanics, quantum statistical mechanics, and string theory. Statistical and condensed matter systems are indeed often conformally invariant at their thermodynamic or quantum critical points. Scale invariance vs conformal invariance In quantum field theory, scale invariance is a common and natural symmetry, because any fixed point of the renormalization group is by definition scale invariant. Conformal symmetry is stronger than scale invariance, and one needs additional assumptions to argue that it should appear in nature. The basic idea behind its plausibility is that ''local'' scale invariant theories have thei ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
String Theory
In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and interact with each other. On distance scales larger than the string scale, a string looks just like an ordinary particle, with its mass, charge, and other properties determined by the vibrational state of the string. In string theory, one of the many vibrational states of the string corresponds to the graviton, a quantum mechanical particle that carries the gravitational force. Thus, string theory is a theory of quantum gravity. String theory is a broad and varied subject that attempts to address a number of deep questions of fundamental physics. String theory has contributed a number of advances to mathematical physics, which have been applied to a variety of problems in black hole physics, early universe cosmology, nuclear physics, and ... [...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] |
Coset Construction
In mathematics, the coset construction (or GKO construction) is a method of constructing unitary highest weight representations of the Virasoro algebra, introduced by Peter Goddard, Adrian Kent and David Olive (1986). The construction produces the complete discrete series In mathematics, a discrete series representation is an irreducible unitary representation of a locally compact topological group ''G'' that is a subrepresentation of the left regular representation of ''G'' on L²(''G''). In the Plancherel meas ... of highest weight representations of the Virasoro algebra and demonstrates their unitarity, thus establishing the classification of unitary highest weight representations. References * * * * Conformal field theory Lie algebras {{quantum-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Super Virasoro Algebra
In mathematical physics, a super Virasoro algebra is an extension of the Virasoro algebra (named after Miguel Ángel Virasoro) to a Lie superalgebra. There are two extensions with particular importance in superstring theory: the Ramond algebra (named after Pierre Ramond) and the Neveu–Schwarz algebra (named after André Neveu and John Henry Schwarz). Both algebras have ''N'' = 1 supersymmetry and an even part given by the Virasoro algebra. They describe the symmetries of a superstring in two different sectors, called the Ramond sector and the Neveu–Schwarz sector. The ''N'' = 1 super Virasoro algebras There are two minimal extensions of the Virasoro algebra with ''N'' = 1 supersymmetry: the Ramond algebra and the Neveu–Schwarz algebra. They are both Lie superalgebras whose even part is the Virasoro algebra: this Lie algebra has a basis consisting of a central element ''C'' and generators ''L''''m'' (for integer ''m'') satisfying L_m , L_n = ( m - n ) L_ + \frac m ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermitian Symmetric Space
In mathematics, a Hermitian symmetric space is a Hermitian manifold which at every point has an inversion symmetry preserving the Hermitian structure. First studied by Élie Cartan, they form a natural generalization of the notion of Riemannian symmetric space from real manifolds to complex manifolds. Every Hermitian symmetric space is a homogeneous space for its isometry group and has a unique decomposition as a product of irreducible spaces and a Euclidean space. The irreducible spaces arise in pairs as a non-compact space that, as Borel showed, can be embedded as an open subspace of its compact dual space. Harish Chandra showed that each non-compact space can be realized as a bounded symmetric domain in a complex vector space. The simplest case involves the groups SU(2), SU(1,1) and their common complexification SL(2,C). In this case the non-compact space is the unit disk, a homogeneous space for SU(1,1). It is a bounded domain in the complex plane C. The one-point compacti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximal Torus
In the mathematical theory of compact Lie groups a special role is played by torus subgroups, in particular by the maximal torus subgroups. A torus in a compact Lie group ''G'' is a compact, connected, abelian Lie subgroup of ''G'' (and therefore isomorphic to the standard torus T''n''). A maximal torus is one which is maximal among such subgroups. That is, ''T'' is a maximal torus if for any torus ''T''′ containing ''T'' we have ''T'' = ''T''′. Every torus is contained in a maximal torus simply by dimensional considerations. A noncompact Lie group need not have any nontrivial tori (e.g. R''n''). The dimension of a maximal torus in ''G'' is called the rank of ''G''. The rank is well-defined since all maximal tori turn out to be conjugate. For semisimple groups the rank is equal to the number of nodes in the associated Dynkin diagram. Examples The unitary group U(''n'') has as a maximal torus the subgroup of all diagonal matrices. That is, : T = \left\. ''T'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Lie Group
In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural generalization of finite groups with the discrete topology and have properties that carry over in significant fashion. Compact groups have a well-understood theory, in relation to group actions and representation theory. In the following we will assume all groups are Hausdorff spaces. Compact Lie groups Lie groups form a class of topological groups, and the compact Lie groups have a particularly well-developed theory. Basic examples of compact Lie groups include * the circle group T and the torus groups T''n'', * the orthogonal group O(''n''), the special orthogonal group SO(''n'') and its covering spin group Spin(''n''), * the unitary group U(''n'') and the special unitary group SU(''n''), * the compact forms of the exceptional Lie groups: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weyl-Kac Character Formula
In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights. It was proved by . There is a closely related formula for the character of an irreducible representation of a semisimple Lie algebra. In Weyl's approach to the representation theory of connected compact Lie groups, the proof of the character formula is a key step in proving that every dominant integral element actually arises as the highest weight of some irreducible representation. Important consequences of the character formula are the Weyl dimension formula and the Kostant multiplicity formula. By definition, the character \chi of a representation \pi of ''G'' is the trace of \pi(g), as a function of a group element g\in G. The irreducible representations in this case are all finite-dimensional (this is part of the Peter–Weyl theorem); so the notion of trace is the usual one from linear algebra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
