|
Moonshine Module
The monster vertex algebra (or moonshine module) is a vertex algebra acted on by the monster group that was constructed by Igor Frenkel, James Lepowsky, and Arne Meurman. R. Borcherds used it to prove the monstrous moonshine conjectures, by applying the Goddard–Thorn theorem of string theory to construct the monster Lie algebra, an infinite-dimensional generalized Kac–Moody algebra acted on by the monster. The Griess algebra is the same as the degree 2 piece of the monster vertex algebra, and the Griess product is one of the vertex algebra products. It can be constructed as conformal field theory describing 24 free bosons compactified on the torus induced by the Leech lattice and orbifold In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space which is locally a finite group quotient of a Euclidean space. D ...ed by the two-element reflection group. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex Algebra
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. The related notion of vertex algebra was introduced by Richard Borcherds in 1986, motivated by a construction of an infinite-dimensional Lie algebra due to Igor Frenkel. In the course of this construction, one employs a Fock space that admits an action of vertex operators attached to lattice vectors. Borcherds formulated the notion of vertex algebra by axiomatizing the relations between the lattice vertex operators, producing an algebraic structure that allows one to construct new Lie algebras by following Frenkel's method. The notion of vertex operator algebra was introduced as a modification of the notion of vertex algebra, by Frenke ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monster Group
In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group, having order 2463205976112133171923293141475971 = 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 ≈ 8. The finite simple groups have been completely classified. Every such group belongs to one of 18 countably infinite families, or is one of 26 sporadic groups that do not follow such a systematic pattern. The monster group contains 20 sporadic groups (including itself) as subquotients. Robert Griess, who proved the existence of the monster in 1982, has called those 20 groups the ''happy family'', and the remaining six exceptions ''pariahs''. It is difficult to give a good constructive definition of the monster because of its complexity. Martin Gardner wrote a popular account of the monster group in his June 1980 Mathematical Games column in ''Scientific ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Igor Frenkel
Igor Borisovich Frenkel (russian: Игорь Борисович Френкель; born April 22, 1952) is a Russian-American mathematician at Yale University working in representation theory and mathematical physics. Frenkel emigrated to the United States in 1979. He received his PhD from Yale University in 1980 with a dissertation on the "Orbital Theory for Affine Lie Algebras". He held positions at the IAS and MSRI, and a tenured professorship at Rutgers University, before taking his current job of tenured professor at Yale University. He was elected to the National Academy of Sciences in 2018. He is also a Fellow of the American Academy of Arts and Sciences. Mathematical work In collaboration with James Lepowsky and Arne Meurman, he constructed the monster vertex algebra, a vertex algebra which provides a representation of the monster group. Around 1990, as a member of the School of Mathematics at the Institute for Advanced Study, Frenkel worked on the mathematical the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
James Lepowsky
James "Jim" Lepowsky (born July 5, 1944, in New York City) is a professor of mathematics at Rutgers University, New Jersey. Previously he taught at Yale University. He received his Ph.D. from M.I.T. in 1970 where his advisors were Bertram Kostant and Sigurdur Helgason. Lepowsky graduated from Stuyvesant High School in 1961, 16 years after Kostant. His current research is in the areas of infinite-dimensional Lie algebras and vertex algebras. He has written several books on vertex algebras and related topics. In 1988, in a joint work with Igor Frenkel and Arne Meurman, he constructed the monster vertex algebra (also known as the Moonshine module). His PhD students include Stefano Capparelli, Yi-Zhi Huang, Haisheng Li, Arne Meurman, and Antun Milas. In 2012, he became a fellow of the American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and ser ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arne Meurman
Arne Meurman (born 6 April 1956) is a Swedish mathematician working on finite groups and vertex operator algebras. Currently, he is a professor at Lund University. He is best known for constructing the monster vertex algebra The monster vertex algebra (or moonshine module) is a vertex algebra acted on by the monster group that was constructed by Igor Frenkel, James Lepowsky, and Arne Meurman. R. Borcherds used it to prove the monstrous moonshine conjectures, by ap ... together with Igor Frenkel and James Lepowsky. He is interested in chess. Publications * Igor Frenkel, James Lepowsky, Arne Meurman, "Vertex operator algebras and the Monster". ''Pure and Applied Mathematics, 134.'' Academic Press, Inc., Boston, MA, 1988. liv+508 pp. * Arne Meurman, Mirko Primc, "Annihilating fields of standard modules of sl (2,C) and combinatorial identities", Memoirs AMS 1999 References External links *Homepage 1956 births Living people People connected to Lund University 20th- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monstrous Moonshine
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. The monstrous moonshine is now known to be underlain by a vertex operator algebra called the moonshine module (or monster vertex algebra) constructed by Igor Frenkel, James Lepowsky, and Arne Meurman in 1988, which has the monster group as its group of symmetries. This vertex operator algebra is commonly interpreted as a structure underlying a two-dimensional conformal field theory, allowing physics to form a bridge between two mathematical areas. The conjectures made by Conway and Norton were proven by Richard Borcherds for the moonshine module in 1992 using the no-ghost theorem from string theory and the theory of vertex operator algebras and generalized Kac–Moody algebras. History In 1978, John McKay found that the first few ter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Goddard–Thorn Theorem
In mathematics, and in particular in the mathematical background of string theory, the Goddard–Thorn theorem (also called the no-ghost theorem) is a theorem describing properties of a functor that quantizes bosonic strings. It is named after Peter Goddard and Charles Thorn. The name "no-ghost theorem" stems from the fact that in the original statement of the theorem, the natural inner product induced on the output vector space is positive definite. Thus, there were no so-called ghosts ( Pauli–Villars ghosts), or vectors of negative norm. The name "no-ghost theorem" is also a word play on the no-go theorem of quantum mechanics. Formalism There are two naturally isomorphic functors that are typically used to quantize bosonic strings. In both cases, one starts with positive-energy representations of the Virasoro algebra of central charge 26, equipped with Virasoro-invariant bilinear forms, and ends up with vector spaces equipped with bilinear forms. Here, "Virasoro-invar ... [...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 conde ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monster Lie Algebra
In mathematics, the monster Lie algebra is an infinite-dimensional generalized Kac–Moody algebra acted on by the monster group, which was used to prove the monstrous moonshine conjectures. Structure The monster Lie algebra ''m'' is a ''Z2''-graded Lie algebra. The piece of degree (''m'', ''n'') has dimension ''c''''mn'' if (''m'', ''n'') ≠ (0, 0) and dimension 2 if (''m'', ''n'') = (0, 0). The integers ''cn'' are the coefficients of ''q''''n'' of the ''j''-invariant as elliptic modular function ::j(q) -744 = + 196884 q + 21493760 q^2 + \cdots. The Cartan subalgebra is the 2-dimensional subspace of degree (0, 0), so the monster Lie algebra has rank 2. The monster Lie algebra has just one real simple root, given by the vector (1, −1), and the Weyl group has order 2, and acts by mapping (''m'', ''n'') to (''n'', ''m''). The imaginary simple roots are the vectors (1, ''n'') for ''n'' = 1, 2, 3, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Kac–Moody Algebra
In mathematics, a generalized Kac–Moody algebra is a Lie algebra that is similar to a Kac–Moody algebra, except that it is allowed to have imaginary simple roots. Generalized Kac–Moody algebras are also sometimes called GKM algebras, Borcherds–Kac–Moody algebras, BKM algebras, or Borcherds algebras. The best known example is the monster Lie algebra. Motivation Finite-dimensional semisimple Lie algebras have the following properties: * They have a nondegenerate symmetric invariant bilinear form (,). * They have a grading such that the degree zero piece (the Cartan subalgebra) is abelian. * They have a (Cartan) involution ''w''. * (''a'', ''w(a)'') is positive if ''a'' is nonzero. For example, for the algebras of ''n'' by ''n'' matrices of trace zero, the bilinear form is (''a'', ''b'') = Trace(''ab''), the Cartan involution is given by minus the transpose, and the grading can be given by "distance from the diagonal" so that the Cartan subalgebra is the diagonal elements ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Griess Algebra
In mathematics, the Griess algebra is a commutative non-associative algebra on a real vector space of dimension 196884 that has the Monster group In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group, having order 246320597611213317192329314147 ... ''M'' as its automorphism group. It is named after mathematician R. L. Griess, who constructed it in 1980 and subsequently used it in 1982 to construct ''M''. The Monster fixes (vectorwise) a 1-space in this algebra and acts absolutely irreducibly on the 196883-dimensional orthogonal complement of this 1-space. (The Monster preserves the standard inner product on the 196884-space.) Griess's construction was later simplified by Jacques Tits and John H. Conway. The Griess algebra is the same as the degree 2 piece of the monster vertex algebra, and the Griess product is one of the vertex alge ... [...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 their ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
