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Virasoro Algebra
In mathematics, the Virasoro algebra is a complex Lie algebra and the unique nontrivial central extension of the Witt algebra. It is widely used in two-dimensional conformal field theory and in string theory. It is named after Miguel Ángel Virasoro. Structure The Virasoro algebra is spanned by generators for and the central charge . These generators satisfy ,L_n0 and The factor of \frac is merely a matter of convention. For a derivation of the algebra as the unique central extension of the Witt algebra, see derivation of the Virasoro algebra or Schottenloher, Thm. 5.1, pp. 79. The Virasoro algebra has a presentation in terms of two generators (e.g. 3 and −2) and six relations. The generators L_ are called annihilation modes, while L_ are creation modes. A basis of creation generators of the Virasoro algebra's universal enveloping algebra is the set : \mathcal = \Big\_ For L\in \mathcal, let , L, = \sum_^k n_i, then _0,L= , L, L. Representation theory In ... [...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] |
Gram Matrix
In linear algebra, the Gram matrix (or Gramian matrix, Gramian) of a set of vectors v_1,\dots, v_n in an inner product space is the Hermitian matrix of inner products, whose entries are given by the inner product G_ = \left\langle v_i, v_j \right\rangle., p.441, Theorem 7.2.10 If the vectors v_1,\dots, v_n are the columns of matrix X then the Gram matrix is X^\dagger X in the general case that the vector coordinates are complex numbers, which simplifies to X^\top X for the case that the vector coordinates are real numbers. An important application is to compute linear independence: a set of vectors are linearly independent if and only if the Gram determinant (the determinant of the Gram matrix) is non-zero. It is named after Jørgen Pedersen Gram. Examples For finite-dimensional real vectors in \mathbb^n with the usual Euclidean dot product, the Gram matrix is G = V^\top V, where V is a matrix whose columns are the vectors v_k and V^\top is its transpose whose rows are the vectors ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Worldsheet
In string theory, a worldsheet is a two-dimensional manifold which describes the embedding of a string in spacetime. The term was coined by Leonard Susskind as a direct generalization of the world line concept for a point particle in special and general relativity. The type of string, the geometry of the spacetime in which it propagates, and the presence of long-range background fields (such as gauge fields) are encoded in a two-dimensional conformal field theory defined on the worldsheet. For example, the bosonic string in 26 dimensions has a worldsheet conformal field theory consisting of 26 free scalar bosons. Meanwhile, a superstring worldsheet theory in 10 dimensions consists of 10 free scalar fields and their fermionic superpartners. Mathematical formulation Bosonic string We begin with the classical formulation of the bosonic string. First fix a d-dimensional flat spacetime (d-dimensional Minkowski space), M, which serves as the ambient space for the string. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conformal Bootstrap
The conformal bootstrap is a non-perturbative mathematical method to constrain and solve Conformal field theory, conformal field theories, i.e. models of particle physics or statistical physics that exhibit similar properties at different levels of resolution. Overview Unlike more traditional techniques of quantum field theory, conformal bootstrap does not use the Lagrangian (field theory), Lagrangian of the theory. Instead, it operates with the general axiomatic parameters, such as the scaling dimensions of the local operators and their operator product expansion coefficients. A key axiom is that the product of local operators must be expressible as a sum over local operators (thus turning the product into an algebra over a field, algebra); the sum must have a non-zero radius of convergence. This leads to decompositions of correlation functions into structure constants and Virasoro conformal block, conformal blocks. The main ideas of the conformal bootstrap were formulated in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conformal Map
In mathematics, a conformal map is a function (mathematics), function that locally preserves angles, but not necessarily lengths. More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-preserving) at a point u_0\in U if it preserves angles between directed curves through u_0, as well as preserving orientation. Conformal maps preserve both angles and the shapes of infinitesimally small figures, but not necessarily their size or curvature. The conformal property may be described in terms of the Jacobian matrix and determinant, Jacobian derivative matrix of a coordinate transformation. The transformation is conformal whenever the Jacobian at each point is a positive scalar times a rotation matrix (Orthogonal matrix, orthogonal with determinant one). Some authors define conformality to include orientation-reversing mappings whose Jacobians can be written as any scalar times any orthogonal matrix. For mappings in two dimensions, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimal Model (physics)
In 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 List of natural phenomena, natural phenomena. This is in contrast to experimental p ..., a minimal model or Virasoro minimal model is a two-dimensional conformal field theory whose spectrum is built from finitely many irreducible representations of the Virasoro algebra. Minimal models have been classified, giving rise to an ADE classification. Most minimal models have been solved, i.e. their 3-point structure constants have been computed analytically. The term minimal model can also refer to a rational CFT based on an algebra that is larger than the Virasoro algebra, such as a W-algebra. Relevant representations of the Virasoro algebra Representations In minimal models, the central charge of the Virasoro algebra takes values of the type : c_ = 1 - 6 \ . where p, q are coprime integers s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dedekind Eta Function
In mathematics, the Dedekind eta function, named after Richard Dedekind, is a modular form of weight 1/2 and is a function defined on the upper half-plane of complex numbers, where the imaginary part is positive. It also occurs in bosonic string theory. Definition For any complex number with , let ; then the eta function is defined by, :\eta(\tau) = e^\frac \prod_^\infty \left(1-e^\right) = q^\frac \prod_^\infty \left(1 - q^n\right) . Raising the eta equation to the 24th power and multiplying by gives :\Delta(\tau)=(2\pi)^\eta^(\tau) where is the modular discriminant. The presence of 24 can be understood by connection with other occurrences, such as in the 24-dimensional Leech lattice. The eta function is holomorphic on the upper half-plane but cannot be continued analytically beyond it. The eta function satisfies the functional equations :\begin \eta(\tau+1) &=e^\frac\eta(\tau),\\ \eta\left(-\frac\right) &= \sqrt\, \eta(\tau).\, \end In the second equation the b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kac–Moody Algebra
In mathematics, a Kac–Moody algebra (named for Victor Kac and Robert Moody, who independently and simultaneously discovered them in 1968) is a Lie algebra, usually infinite-dimensional, that can be defined by generators and relations through a generalized Cartan matrix. These algebras form a generalization of finite-dimensional semisimple Lie algebras, and many properties related to the structure of a Lie algebra such as its root system, irreducible representations, and connection to flag manifolds have natural analogues in the Kac–Moody setting. A class of Kac–Moody algebras called affine Lie algebras is of particular importance in mathematics and theoretical physics, especially two-dimensional conformal field theory and the theory of exactly solvable models. Kac discovered an elegant proof of certain combinatorial identities, the Macdonald identities, which is based on the representation theory of affine Kac–Moody algebras. Howard Garland and James Lepowsky demonstrated ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
GKO 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 Adrian Kent is a British theoretical physicist, Professor of Quantum Physics at the University of Cambridge, member of the Centre for Quantum Information and Foundations, and Distinguished Visiting Research Chair at the Perimeter Institute for The ... and David Olive (1986). The construction produces the complete discrete series 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] |
David Olive
David Ian Olive ( ; 16 April 1937 – 7 November 2012) was a British theoretical physicist. Olive made fundamental contributions to string theory and duality theory, he is particularly known for his work on the GSO projection and Montonen–Olive duality. He was professor of physics at Imperial College, London, from 1984 to 1992. In 1992 he moved to Swansea University to help set up the new theoretical physics group. He was awarded the Dirac Prize and Medal of the International Centre for Theoretical Physics in 1997. He was a Founding Fellow of the Learned Society of Wales. He was elected as a fellow of the Royal Society in 1987, and appointed CBE in 2002. Biography Early life and education David Olive was born in Middlesex in 1937 and educated at the Royal High School, Edinburgh and Edinburgh University. He then moved to St John's College, Cambridge, obtaining his PhD under the supervision of John Taylor in 1963. He married Jenny in 1963; they had 2 daughters and a gran ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adrian Kent
Adrian Kent is a British theoretical physicist, Professor of Quantum Physics at the University of Cambridge, member of the Centre for Quantum Information and Foundations, and Distinguished Visiting Research Chair at the Perimeter Institute for Theoretical Physics. His research areas are the quantum foundations, foundations of quantum theory, quantum information science and quantum cryptography. He is known as the inventor of relativistic quantum cryptography. In 1999 he published the first Information-theoretic security, unconditionally secure protocols for bit commitment and quantum coin flipping, coin tossing, which were also the first relativistic cryptographic protocols. He is a co-inventor of quantum tagging, or quantum position authentication, providing the first schemes for position-based quantum cryptography. In 2005 he published with Lucien Hardy and Jonathan Barrett the first security proof of quantum key distribution based on the no-communication theorem, no-signalling pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
