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In mathematics, the Virasoro algebra is a complex Lie algebra and the unique central extension of the Witt algebra. It is widely used in two-dimensional conformal field theory and in string theory.
In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group M and modular functions, in particular, the j function. The term was coined by John Conway and Simon P. Norton in 1979.
In mathematics, and in particular in the mathematical background of string theory, the Goddard–Thorn theorem is a theorem describing properties of a functor that quantizes bosonic strings. It is named after Peter Goddard and Charles Thorn.
In mathematics, a vertex operator algebra (VOA) is an algebraic structure that plays an important role in two-dimensional conformal field theory and string theory. In addition to physical applications, vertex operator algebras have proven useful in purely mathematical contexts such as monstrous moonshine and the geometric Langlands correspondence.
In mathematics, an affine Lie algebra is an infinite-dimensional Lie algebra that is constructed in a canonical fashion out of a finite-dimensional simple Lie algebra. Given an affine Lie algebra, one can also form the associated affine Kac-Moody algebra, as described below. From a purely mathematical point of view, affine Lie algebras are interesting because their representation theory, like representation theory of finite-dimensional semisimple Lie algebras, is much better understood than that of general Kac–Moody algebras. As observed by Victor Kac, the character formula for representations of affine Lie algebras implies certain combinatorial identities, the Macdonald identities.
In mathematics, if G is a group and ρ is a linear representation of it on the vector space V, then the dual representationρ* is defined over the dual vector space V* as follows:
In theoretical physics and mathematics, a Wess–Zumino–Witten (WZW) model, also called a Wess–Zumino–Novikov–Witten model, is a type of two-dimensional conformal field theory named after Julius Wess, Bruno Zumino, Sergei Novikov and Edward Witten. A WZW model is associated to a Lie group, and its symmetry algebra is the affine Lie algebra built from the corresponding Lie algebra. By extension, the name WZW model is sometimes used for any conformal field theory whose symmetry algebra is an affine Lie algebra.
Bertram Kostant was an American mathematician who worked in representation theory, differential geometry, and mathematical physics.
In mathematics, the orbit method establishes a correspondence between irreducible unitary representations of a Lie group and its coadjoint orbits: orbits of the action of the group on the dual space of its Lie algebra. The theory was introduced by Kirillov for nilpotent groups and later extended by Bertram Kostant, Louis Auslander, Lajos Pukánszky and others to the case of solvable groups. Roger Howe found a version of the orbit method that applies to p-adic Lie groups. David Vogan proposed that the orbit method should serve as a unifying principle in the description of the unitary duals of real reductive Lie groups.
Victor Gershevich (Grigorievich) Kac is a Soviet and American mathematician at MIT, known for his work in representation theory. He co-discovered Kac–Moody algebras, and used the Weyl–Kac character formula for them to reprove the Macdonald identities. He classified the finite-dimensional simple Lie superalgebras, and found the Kac determinant formula for the Virasoro algebra. He is also known for the Kac–Weisfeiler conjectures with Boris Weisfeiler.
In mathematics, the Harish-Chandra isomorphism, introduced by Harish-Chandra (1951), is an isomorphism of commutative rings constructed in the theory of Lie algebras. The isomorphism maps the center Z(U ) of the universal enveloping algebra U(g) of a reductive Lie algebra g to the elements S(h)W of the symmetric algebra S(h) of a Cartan subalgebra h that are invariant under the Weyl group W.
In mathematical physics, a super Virasoro algebra is an extension of the Virasoro algebra to a Lie superalgebra. There are two extensions with particular importance in superstring theory: the Ramond algebra and the Neveu–Schwarz algebra. 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.
In mathematical physics the Knizhnik–Zamolodchikov equations, or KZ equations, are linear differential equations satisfied by the correlation functions of two-dimensional conformal field theories associated with an affine Lie algebra at a fixed level. They form a system of complex partial differential equations with regular singular points satisfied by the N-point functions of affine primary fields and can be derived using either the formalism of Lie algebras or that of vertex algebras.
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations. The theory of matrices and linear operators is well-understood, so representations of more abstract objects in terms of familiar linear algebra objects helps glean properties and sometimes simplify calculations on more abstract theories.
David Ian Olive 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.
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 M. Ademollo, L. Brink, and A. D'Adda et al. (1976) as a gauge algebra of the U(1) fermionic string.
Nolan Russell Wallach is a mathematician known for work in the representation theory of reductive algebraic groups. He is the author of the 2-volume treatise Real Reductive Groups.
Thomas Jones Enright was an American mathematician known for his work in the algebraic theory of representations of real reductive Lie groups.
Dmitry Borisovich Fuchs is a Russian-American mathematician, specializing in the representation theory of infinite-dimensional Lie groups and in topology.
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 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 unconditionally secure protocols for bit commitment and 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-signalling principle.
References
- Goddard, P.; Kent, A.; Olive, D. (1986). "Unitary representations of the Virasoro and super-Virasoro algebras". Comm. Math. Phys. 103 (1): 105–119. Bibcode:1986CMaPh.103..105G. doi:10.1007/BF01464283. S2CID 91181508.
- Victor Kac (2001) [1994], "Virasoro algebra", Encyclopedia of Mathematics , EMS Press
- Kac, V. G.; Raina, A. K. (1987). Bombay lectures on highest weight representations. World Sci. ISBN 9971-5-0395-6.
- Wassermann, Antony. "Lecture Notes on the Kac-Moody and Virasoro algebras". Archived from the original on 2007-03-22.
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