The
Monster group has remained a mysterious symmetry group since its discovery. Currently, the most popular description of it is via the monster vertex algebra, with the Monster acting on the
Griess algebra as its degree 2 piece. The Griess algebra is a commutative, nonassociative algebra on a 196884-dimensional real vector space.
So this begs the question: what is this 196884-dimensional real vector space? What are these vectors?
The answer becomes meaningful through the lens of quantum gravity. Over the years, physics has served as a light that illuminates the path forward in mathematics, and even in the case of the Monster, Borcherds used the
Goddard-Thorn 'no-ghost' theorem in string theory to define the infinite dimensional
Monster Lie algebra. This means the most natural description of the Griess algebra is not only tied to quantum gravity but string/M-theory.
Edward Witten in 2007
studied three-dimensional gravity and found that a monstrous conformal field theory may be the first
in a discrete series of CFT's that are dual to three-dimensional gravity. The construction required an anti-de Sitter space (AdS) in 3D. The
j-invariant contains 196884 as one of its leading coefficients. This originally inspired the concept of
Monstrous moonshine.
Hence, an elementary description of the 196884 is of utmost importance. First, one can note a similarity in dimension between 196884 and the 196560 norm four vectors of the
Leech lattice. This is an enticing observation, as the Conway group Co_0 acts on such Leech lattice vectors, being that it describes the automorphisms of the Leech lattice. The correspondence is more subtle, as in physics, quantum states can be identified up to a phase. Hence, the 196560 norm four vectors are acted on modulo the correspondence v~(-v), in other words, with antipodal vectors identified, reducing the 196560 to 196560/2=98280. In string theory, this is an orbifolding of bosonic string states upon compactification on the Leech lattice. In condensed matter, one sees antipodal states identified in the study of defects in nematic liquid crystals.
The remaining 98604 vectors of the 196884 also have an interesting quantum gravity description. We will leave that for next time.