nLab Moonshine

Idea

In mathematics, Moonshine refers to mysterious connections between the Monster simple group and the modular function $j$, 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 $j(\tau)$ has the Fourier series expansion

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

where $q=\exp(i2\pi\tau)$, 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 $V\natural$).

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”.

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):

• The Conway group $Co_{0}$ is the group of automorphisms of a super VOA of the unique chiral N=1 super vertex operator algebra of central charge $c = 12$ without fields of conformal weight $1/2$

  (Duncan 05, see also Paquette-Persson-Volpato 17, p. 9)

• similarly, there is a super VOA, the Monster vertex operator algebra, whose group of of automorphisms of a VOA is the monster group

  (Frenkel-Lepowski-Meurman 89, Griess-Lam 11)

Related concepts

• Monster

• moonshine

  • Mathieu moonshine

  • umbral moonshine

  • O'Nan moonshine

• automorphism of a vertex operator algebra

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)

• 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 $J$ 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 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)

• Scott Carnahan, Monstrous Moonshine over Z? (arXiv:1804.04161)

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:

• S. Chaudhuri, D.A. Lowe, Monstrous String-String Duality, Nucl. Phys. B469 : 21-36, 1996 (arXiv:hep-th/9512226)

• John F. Duncan, Super-moonshine for Conway’s largest sporadic group (arXiv:math/0502267)

• Natalie Paquette, Daniel Persson, Roberto Volpato, Monstrous BPS-Algebras and the Superstring Origin of Moonshine (arXiv:1601.05412)

• 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)

• Shamit Kachru, Arnav Tripathy, The hidden symmetry of the heterotic string (arXiv:1702.02572)

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

• Miranda Cheng, Sarah M. Harrison, Roberto Volpato, Max Zimet, K3 String Theory, Lattices and Moonshine (arXiv:1612.04404, doi:10.1007/s40687-018-0150-4)

Possible relation to bosonic M-theory:

• Alessio Marrani, Michael Rios, David Chester, Monstrous M-theory, Symmetry 15 2 (2023) 490; [doi:10.3390/sym15020490, arXiv:2008.06742]

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]