Moonshine (Rev #19, changes) in nLab

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Contents

Idea

Moonshine usually refers to the mysterious connections between the Monster simple group and the modular function jj, the j-invariant. There were a bunch of conjectures about this connection that were proved by Richard Borcherds, en passant mentioning the existence of the Moonshine vertex algebra (constructed then later by FLM). Nowadays there is also Moonshine for other simple groups, by the work of J. Duncan. Everntually there shoould be an entry for the general moonshine phenomenon.

The whole idea of moonshine began with John McKay’s observation that the Monster group’s first nontrivial irreducible representation has dimension 196883, and the j-invariant j(τ)j(\tau) has the Fourier series expansion

j(τ)=q −1+744+196884q+21493760q 2+… j(\tau) = q^{-1} + 744 + 196884q + 21493760q^{2} + \dots

where q=exp(i2πτ)q=\exp(i2\pi\tau), and famously 196883+1=196884. Thompson observed in (1979) that the other coefficients are obtained from the dimensions of Monster’s irreducible representations.

But the monster was merely conjectured to exist until Griess (1982) explicitly constructed it. The construction is horribly complicated (take the sum of three irreducible representations for the centralizer of an involution of…).

Frenkel, Lepowsky, Meurman (1984) construct (Frenkel-Lepowski-Meurman 89) an infinite-dimensional module for the Monster vertex algebra. This is by a generalized Kac-Moody algebra via bosonic string theory and the Goddard-Thorn "No Ghost" theorem. The Monster group acts naturally on this “Moonshine Module” (denoted by V♮V\natural).

To cut the story short, we end up getting from the Monster group to a module it acts on which is related to “modular stuff” (namely, the modular j-invariant). The idea Terry Gannon pitches is that Moonshine is a generalization of this association, it’s a sort of “mapping” from “Algebraic gadgets” to “Modular stuff”.

• Monster
• Mathieu moonshine
• umbral moonshine

References

General

• Richard Borcherds, What is Moonshine?, Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998).Doc. Math._ 1998, Extra Vol. I, 607–615 (electronic). MR1660657 arXiv:math/9809110v1 [math.QA]

• John F. R. Duncan, Michael J. Griffin, Ken Ono, Moonshine (arXiv:1411.6571)

• Igor Frenkel, James Lepowsky, Arne Meurman, Vertex operator algebras and the monster, Pure and Applied Mathematics 134, Academic Press, New York 1998. liv+508 pp. MR0996026

• Terry Gannon, “Monstrous moonshine: the first twenty-five years.” Bull. London Math. Soc. 38 (2006), no. 1, 1–33. MR2201600 arXiv:math/0402345v2 [math.QA]

• Terry Gannon, Moonshine beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, Massachusetts 2006. MR2257727

• Koichiro Harada, “Moonshine” of finite groups. EMS Series of Lectures in Mathematics. European Mathematical Society (EMS), Zürich, 2010. viii+76 pp. MR2722318

• Griess, Robert L., Jr.; Lam, Ching Hung “A moonshine path from E8 to the Monster.” J. Pure Appl. Algebra 215 (2011), no. 5, 927–948 MR2747229 arXiv:0910.2057v2 [math.GR]

• Jae-Hyun Yang “Kac-Moody algebras, the Monstrous Moonshine, Jacobi forms and infinite products.” Number theory, geometry and related topics (Iksan City, 1995), 13–82, Pyungsan Inst. Math. Sci., Seoul, 1996. MR1404967 arXiv:math/0612474v2 [math.NT]

Historical References

• John Conway and Simon Norton, “Monstrous moonshine.” Bull. London Math. Soc. 11 (1979), no. 3, 308–339; MR0554399 (81j:20028)

• Igor Frenkel, James Lepowsky, Arne Meurman, “A natural representation of the Fischer-Griess Monster with the modular function JJ as character.” Proc. Nat. Acad. Sci. U.S.A. 81 (1984), no. 10, Phys. Sci., 3256–3260. MR0747596 (85e:20018)

• Robert Griess, “The friendly giant.” Invent. Math. 69 (1982), no. 1, 1–102. MR671653 (84m:20024)

• John G. Thompson, “Some numerology between the Fischer-Griess Monster and the elliptic modular function.” Bull. London Math. Soc. 11 (1979), no. 3, 352–353. MR0554402 (81j:20030)

Further developments

• Miranda C. N. Cheng, John F. R. Duncan, Jeffrey A. Harvey, Umbral Moonshine (arXiv:1204.2779)

• John F. R. Duncan, Michael J. Griffin and Ken Ono, Proof of the Umbral Moonshine Conjecture (arXiv:1503.01472)

Realization in superstring theory

Discussion of possible realizations in superstring theory includes (specificallyheterotic string theory) includes

• Shamit Kachru, Natalie Paquette, Roberto Volpato, 3D String Theory and Umbral Moonshine (arXiv:1603.07330)

• Natalie Paquette, Daniel Persson, Roberto Volpato, BPS Algebras, Genus Zero, and the Heterotic Monster (arXiv:1701.05169)