[1][2][3] 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.
Conway and Norton computed the lower-order terms of such graded traces, now known as McKay–Thompson series Tg, and found that all of them appeared to be the expansions of Hauptmoduln.
In other words, if Gg is the subgroup of SL2(R) which fixes Tg, then the quotient of the upper half of the complex plane by Gg is a sphere with a finite number of points removed, and furthermore, Tg generates the field of meromorphic functions on this sphere.
The Frenkel–Lepowsky–Meurman construction starts with two main tools: Frenkel, Lepowsky, and Meurman then showed that the automorphism group of the moonshine module, as a vertex operator algebra, is M. Furthermore, they determined that the graded traces of elements in the subgroup 21+24.Co1 match the functions predicted by Conway and Norton (Frenkel, Lepowsky & Meurman (1988)).
Borcherds was later quoted as saying "I was over the moon when I proved the moonshine conjecture", and "I sometimes wonder if this is the feeling you get when you take certain drugs.
Cummins and Gannon showed that the recursion relations automatically imply the McKay-Thompson series are either Hauptmoduln or terminate after at most 3 terms, thus eliminating the need for computation at the last step.
Conway and Norton suggested in their 1979 paper that perhaps moonshine is not limited to the monster, but that similar phenomena may be found for other groups.
[a] While Conway and Norton's claims were not very specific, computations by Larissa Queen in 1980 strongly suggested that one can construct the expansions of many Hauptmoduln from simple combinations of dimensions of irreducible representations of sporadic groups.
This conjecture asserts that there is a rule that assigns to each element g of the monster, a graded vector space V(g), and to each commuting pair of elements (g, h) a holomorphic function f(g, h, τ) on the upper half-plane, such that: This is a generalization of the Conway–Norton conjecture, because Borcherds's theorem concerns the case where g is set to the identity.
Like the Conway–Norton conjecture, Generalized Moonshine also has an interpretation in physics, proposed by Dixon–Ginsparg–Harvey in 1988 (Dixon, Ginsparg & Harvey (1989)).
They interpreted the vector spaces V(g) as twisted sectors of a conformal field theory with monster symmetry, and interpreted the functions f(g, h, τ) as genus one partition functions, where one forms a torus by gluing along twisted boundary conditions.
In particular, Ryba conjectured in 1994 that for each prime factor p in the order of the monster, there exists a graded vertex algebra over the finite field Fp with an action of the centralizer of an order p element g, such that the graded Brauer character of any p-regular automorphism h is equal to the McKay-Thompson series for gh (Ryba (1996)).
In 1996, Borcherds and Ryba reinterpreted the conjecture as a statement about Tate cohomology of a self-dual integral form of
This integral form was not known to exist, but they constructed a self-dual form over Z[1/2], which allowed them to work with odd primes p. The Tate cohomology for an element of prime order naturally has the structure of a super vertex algebra over Fp, and they broke up the problem into an easy step equating graded Brauer super-trace with the McKay-Thompson series, and a hard step showing that Tate cohomology vanishes in odd degree.
In 2007, E. Witten suggested that AdS/CFT correspondence yields a duality between pure quantum gravity in (2 + 1)-dimensional anti de Sitter space and extremal holomorphic CFTs.
Pure gravity in 2 + 1 dimensions has no local degrees of freedom, but when the cosmological constant is negative, there is nontrivial content in the theory, due to the existence of BTZ black hole solutions.
Extremal CFTs, introduced by G. Höhn, are distinguished by a lack of Virasoro primary fields in low energy, and the moonshine module is one example.
Under Witten's proposal (Witten (2007)), gravity in AdS space with maximally negative cosmological constant is AdS/CFT dual to a holomorphic CFT with central charge c=24, and the partition function of the CFT is precisely j-744, i.e., the graded character of the moonshine module.
By assuming Frenkel-Lepowsky-Meurman's conjecture that moonshine module is the unique holomorphic VOA with central charge 24 and character j-744, Witten concluded that pure gravity with maximally negative cosmological constant is dual to the monster CFT.
Part of Witten's proposal is that Virasoro primary fields are dual to black-hole-creating operators, and as a consistency check, he found that in the large-mass limit, the Bekenstein-Hawking semiclassical entropy estimate for a given black hole mass agrees with the logarithm of the corresponding Virasoro primary multiplicity in the moonshine module.
In the low-mass regime, there is a small quantum correction to the entropy, e.g., the lowest energy primary fields yield ln(196883) ~ 12.19, while the Bekenstein–Hawking estimate gives 4π ~ 12.57.
Witten had speculated that the extremal CFTs with larger cosmological constant may have monster symmetry much like the minimal case, but this was quickly ruled out by independent work of Gaiotto and Höhn.
Furthermore, they conjectured the existence of a family of twisted chiral gravity theories parametrized by elements of the monster, suggesting a connection with generalized moonshine and gravitational instanton sums.
In 2010, Tohru Eguchi, Hirosi Ooguri, and Yuji Tachikawa observed that the elliptic genus of a K3 surface can be decomposed into characters of the N = (4,4) superconformal algebra, such that the multiplicities of massive states appear to be simple combinations of irreducible representations of the Mathieu group M24.
In 2012, Gannon proved that all but the first of the multiplicities are non-negative integral combinations of representations of M24, and Gaberdiel–Persson–Ronellenfitsch–Volpato computed all analogues of generalized moonshine functions,[7] strongly suggesting that some analogue of a holomorphic conformal field theory lies behind Mathieu moonshine.
Also in 2012, Cheng, Duncan, and Harvey amassed numerical evidence of an umbral moonshine phenomenon where families of mock modular forms appear to be attached to Niemeier lattices.
The special case of the A241 lattice yields Mathieu Moonshine, but in general the phenomenon does not yet have an interpretation in terms of geometry.
The term "monstrous moonshine" was coined by Conway, who, when told by John McKay in the late 1970s that the coefficient of
(namely 196884) was precisely one more than the degree of the smallest faithful complex representation of the monster group (namely 196883), replied that this was "moonshine" (in the sense of being a crazy or foolish idea).