The monster vertex algebra is a vertex algebra acted on by the monster group

Monster group

In the mathematical field of group theory, the Monster group M or F1 is a group of finite order:...

 that was constructed by Igor Frenkel

Igor Frenkel

Igor Borisovich Frenkel is a mathematician at Yale University working in representation theory and mathematical physics.-Biography:...

, James Lepowsky

James Lepowsky

James Lepowsky 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 advisor was Bertram Kostant. Lepowsky graduated from Stuyvesant High School in 1961, 16 years after Kostant...

, and Arne Meurman

Arne Meurman

Arne Meurman is a Swedish mathematician working on finite groups and vertex operator algebras. Currently he is a professor at Lund University....

. R. Borcherds used it to prove the monstrous moonshine

Monstrous moonshine

In mathematics, monstrous moonshine, or moonshine theory, is a term devised by John Horton Conway and Simon P. Norton in 1979, used to describe the connection between the monster group M and modular functions .- History :Specifically, Conway and Norton, following an initial observationby John...

 conjectures, by applying the no-ghost theorem of string theory

String theory

String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...

 to construct the monster Lie algebra

Monster Lie algebra

In mathematics, the monster Lie algebra is an infinitedimensional generalized Kac–Moody algebra acted on by the monster group, which was used to prove the monstrous moonshine conjectures.- Structure :...

, an infinite-dimensional generalized Kac–Moody algebra

Generalized Kac–Moody algebra

In mathematics, a generalized Kac–Moody algebra is a Lie algebra that is similar to aKac–Moody algebra, except that it is allowed to have imaginary simple roots.Generalized Kac–Moody algebras are also sometimes called GKM algebras,...

 acted on by the monster.

The Griess algebra

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 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...

 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

Conformal field theory

A conformal field theory is a quantum field theory that is invariant under conformal transformations...

 describing 24 free bosons compactified on the torus induced by the Leech lattice

Leech lattice

In mathematics, the Leech lattice is an even unimodular lattice Λ24 in 24-dimensional Euclidean space E24 found by .-History:Many of the cross-sections of the Leech lattice, including the Coxeter–Todd lattice and Barnes–Wall lattice, in 12 and 16 dimensions, were found much earlier than...

 and orbifold

Orbifold

In the mathematical disciplines of topology, geometry, and geometric group theory, an orbifold is a generalization of a manifold...

ed by the two-element reflection group.