nLab Moonshine




In mathematics, Moonshine refers to 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 in FLM 89). Nowadays there is also Moonshine for other simple groups, by the work of J. Duncan. Eventually there should 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 group 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-Lepowski-Meurman 89 constructed 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 VV\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”.

Automorphism groups of vertex operator algebras

Realizations of sporadic finite simple groups as automorphism groups of vertex operator algebras in heterotic string theory and type II string theory (mostly on K3-surfaces, see HET - II duality):



  • 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)

  • Robert Griess Jr., Ching Hung Lam, A new existence proof of the Monster by VOA theory (arXiv:1103.1414)

  • Igor Frenkel, James Lepowsky, Arne Meurman, Vertex operator algebras and the monster, Pure and Applied Mathematics 134, Academic Press, New York 1989. 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/0402345 [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]

  • Vassilis Anagiannis, Miranda Cheng, TASI Lectures on Moonshine (arXiv:1807.00723)

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

Possible relation to elliptic cohomology (topological modular forms):

  • Jan Albert, Justin Kaidi, Ying-Hsuan Lin, Topological modularity of Supermoonshine [[arXiv:2210.14923]]

Realization in superstring theory

Discussion of possible realizations in superstring theory (specifically heterotic string theory and type II string theory in K3-surfaces, see HET - II) via automorphisms of super vertex operator algebras:

Specifically in relation to KK-compactifications of string theory on K3-surfaces (duality between heterotic and type II string theory)

Possible relation to bosonic M-theory:

Relation to Seiberg-Witten theory:

  • Shun’ya Mizoguchi, Seiberg-Witten Theory and Monstrous Moonshine [arXiv:2208.10301]

  • Shun’ya Mizoguchi, Takumi Oikawa, Hitomi Tashiro, Shotaro Yata, More on Seiberg-Witten Theory and Monstrous Moonshine [arXiv:2211.07251]

Last revised on February 13, 2023 at 07:24:10. See the history of this page for a list of all contributions to it.