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\section*{Moonshine}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{exceptional_structures}{}\paragraph*{{Exceptional structures}}\label{exceptional_structures}

[[!include exceptional structures -- contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{AutomorphismGroupsOfVertexOperatorAlgebras}{Automorphism groups of vertex operator algebras}\dotfill \pageref*{AutomorphismGroupsOfVertexOperatorAlgebras} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{historical_references}{Historical References}\dotfill \pageref*{historical_references} \linebreak
\noindent\hyperlink{FurtherDevelomentsReferences}{Further developments}\dotfill \pageref*{FurtherDevelomentsReferences} \linebreak
\noindent\hyperlink{realization_in_superstring_theory}{Realization in superstring theory}\dotfill \pageref*{realization_in_superstring_theory} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Moonshine usually refers to the 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 \hyperlink{FrenkelLepowskiMeurman89}{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

\begin{displaymath}
j(\tau) = q^{-1} + 744 + 196884q + 21493760q^{2} + \dots
\end{displaymath}
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 was merely \emph{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\ldots{}).

\hyperlink{FrenkelLepowskiMeurman89}{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 theorem|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''.

\hypertarget{AutomorphismGroupsOfVertexOperatorAlgebras}{}\subsection*{{Automorphism groups of vertex operator algebras}}\label{AutomorphismGroupsOfVertexOperatorAlgebras}

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]]):

\begin{itemize}%
\item The [[Conway group]] $Co_{0}$ is the [[automorphism group|group of]] [[automorphisms of a super VOA]] of the unique chiral [[number of supersymmetries|N=1]] [[super vertex operator algebra]] of [[central charge]] $c = 12$ without fields of [[conformal weight]] $1/2$

(\hyperlink{Duncan05}{Duncan 05}, see also \hyperlink{PaquettePerssonVolpato17}{Paquette-Persson-Volpato 17, p. 9})


\item similarly, there is a super VOA, the \emph{[[Monster vertex operator algebra]]}, whose [[automorphism group|group of]] of [[automorphisms of a VOA]] is the [[monster group]]

(\hyperlink{FrenkelLepowskiMeurman89}{Frenkel-Lepowski-Meurman 89}, \hyperlink{GriessLam11}{Griess-Lam 11})



\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Monster]]


\item moonshine

\begin{itemize}%
\item [[Mathieu moonshine]]


\item [[umbral moonshine]]


\item [[O'Nan moonshine]]



\end{itemize}

\item [[automorphism of a vertex operator algebra]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

\begin{itemize}%
\item [[Richard Borcherds]], \emph{What is Moonshine?, Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998).}Doc. Math.\_ 1998, Extra Vol. I, 607--615 (electronic). \href{http://www.ams.org/mathscinet-getitem?mr=1660657}{MR1660657} \href{http://arxiv.org/abs/math/9809110}{arXiv:math/9809110v1} math.QA


\item John F. R. Duncan, Michael J. Griffin, Ken Ono, \emph{Moonshine} (\href{http://arxiv.org/abs/1411.6571}{arXiv:1411.6571})


\item [[Robert Griess]] Jr., Ching Hung Lam, \emph{A new existence proof of the Monster by VOA theory} (\href{https://arxiv.org/abs/1103.1414}{arXiv:1103.1414})


\item [[Igor Frenkel]], [[James Lepowsky]], Arne Meurman, \emph{Vertex operator algebras and the monster}, Pure and Applied Mathematics \textbf{134}, Academic Press, New York 1989. liv+508 pp. \href{http://www.ams.org/mathscinet-getitem?mr=996026}{MR0996026}


\item [[Terry Gannon]], ``Monstrous moonshine: the first twenty-five years.'' \emph{Bull. London Math. Soc.} \textbf{38} (2006), no. 1, 1--33. \href{http://www.ams.org/mathscinet-getitem?mr=2201600}{MR2201600} \href{http://arxiv.org/abs/math/0402345}{arXiv:math/0402345v2} math.QA


\item Terry Gannon, \emph{Moonshine beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics}, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, Massachusetts 2006. \href{http://www.ams.org/mathscinet-getitem?mr=2257727}{MR2257727}


\item Koichiro Harada, \emph{``Moonshine'' of finite groups}. EMS Series of Lectures in Mathematics. European Mathematical Society (EMS), Z\"u{}rich, 2010. viii+76 pp. \href{http://www.ams.org/mathscinet-getitem?mr=2722318}{MR2722318}


\item Griess, Robert L., Jr.; Lam, Ching Hung ``A moonshine path from E8 to the Monster.'' \emph{J. Pure Appl. Algebra} 215 (2011), no. 5, 927--948 \href{http://www.ams.org/mathscinet-getitem?mr=2747229}{MR2747229} \href{http://arxiv.org/abs/0910.2057v2}{arXiv:0910.2057v2} math.GR


\item Jae-Hyun Yang ``Kac-Moody algebras, the Monstrous Moonshine, Jacobi forms and infinite products.'' \emph{Number theory, geometry and related topics} (Iksan City, 1995), 13--82, Pyungsan Inst. Math. Sci., Seoul, 1996. \href{http://www.ams.org/mathscinet-getitem?mr=1404967}{MR1404967} \href{http://arxiv.org/abs/math/0612474}{arXiv:math/0612474v2} math.NT


\item Vassilis Anagiannis, Miranda C. N. Cheng, \emph{TASI Lectures on Moonshine} (\href{https://arxiv.org/abs/1807.00723}{arXiv:1807.00723})



\end{itemize}
\hypertarget{historical_references}{}\subsubsection*{{Historical References}}\label{historical_references}

\begin{itemize}%
\item John Conway and Simon Norton, ``Monstrous moonshine.'' \emph{Bull. London Math. Soc.} \textbf{11} (1979), no. 3, 308--339; \href{http://www.ams.org/mathscinet-getitem?mr=554399}{MR0554399} (81j:20028)


\item [[Igor Frenkel]], [[James Lepowsky]], Arne Meurman, ``A natural representation of the Fischer-Griess Monster with the modular function $J$ as character.'' \emph{Proc. Nat. Acad. Sci. U.S.A.} \textbf{81} (1984), no. 10, Phys. Sci., 3256--3260. \href{http://www.ams.org/mathscinet-getitem?mr=747596}{MR0747596} (85e:20018)


\item [[Robert Griess]], ``The friendly giant.'' \emph{Invent. Math.} \textbf{69} (1982), no. 1, 1--102. \href{http://www.ams.org/mathscinet-getitem?mr=671653}{MR671653} (84m:20024)


\item John G. Thompson, ``Some numerology between the Fischer-Griess Monster and the elliptic modular function.'' \emph{Bull. London Math. Soc.} \textbf{11} (1979), no. 3, 352--353. \href{http://www.ams.org/mathscinet-getitem?mr=554402}{MR0554402} (81j:20030)



\end{itemize}
\hypertarget{FurtherDevelomentsReferences}{}\subsubsection*{{Further developments}}\label{FurtherDevelomentsReferences}

\begin{itemize}%
\item Miranda C. N. Cheng, John F. R. Duncan, Jeffrey A. Harvey, \emph{Umbral Moonshine} (\href{http://arxiv.org/abs/1204.2779}{arXiv:1204.2779})


\item John F. R. Duncan, Michael J. Griffin and Ken Ono, \emph{Proof of the Umbral Moonshine Conjecture} (\href{http://arxiv.org/abs/1503.01472}{arXiv:1503.01472})


\item [[Scott Carnahan]], \emph{Monstrous Moonshine over Z?} (\href{https://arxiv.org/abs/1804.04161}{arXiv:1804.04161})



\end{itemize}
\hypertarget{realization_in_superstring_theory}{}\subsubsection*{{Realization in superstring theory}}\label{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]]:

\begin{itemize}%
\item S. Chaudhuri, D.A. Lowe, \emph{Monstrous String-String Duality}, Nucl.Phys.B469:21-36, 1996 (\href{https://arxiv.org/abs/hep-th/9512226}{arXiv:hep-th/9512226})


\item John F. Duncan, \emph{Super-moonshine for Conway's largest sporadic group} (\href{https://arxiv.org/abs/math/0502267}{arXiv:math/0502267})


\item [[Natalie Paquette]], Daniel Persson, Roberto Volpato, \emph{Monstrous BPS-Algebras and the Superstring Origin of Moonshine} (\href{http://arxiv.org/abs/1601.05412}{arXiv:1601.05412})


\item [[Shamit Kachru]], [[Natalie Paquette]], Roberto Volpato, \emph{3D String Theory and Umbral Moonshine} (\href{http://arxiv.org/abs/1603.07330}{arXiv:1603.07330})


\item [[Natalie Paquette]], Daniel Persson, Roberto Volpato, \emph{BPS Algebras, Genus Zero, and the Heterotic Monster} (\href{https://arxiv.org/abs/1701.05169}{arXiv:1701.05169})


\item [[Shamit Kachru]], Arnav Tripathy, \emph{The hidden symmetry of the heterotic string} (\href{https://arxiv.org/abs/1702.02572}{arXiv:1702.02572})



\end{itemize}
Specifically in relation to [[KK-compactifications]] of [[string theory]] on [[K3-surfaces]] ([[duality between heterotic and type II string theory]])

\begin{itemize}%
\item [[Miranda Cheng]], Sarah M. Harrison, Roberto Volpato, Max Zimet, \emph{K3 String Theory, Lattices and Moonshine} (\href{https://arxiv.org/abs/1612.04404}{arXiv:1612.04404}, \href{https://doi.org/10.1007/s40687-018-0150-4}{doi:10.1007/s40687-018-0150-4})

\end{itemize}
[[!redirects moonshine]] [[!redirects monstrous moonshine]]

[[!redirects Moonshine vertex algebra]] [[!redirects moonshine vertex algebra]]

[[!redirects Moonshine vertex algebras]] [[!redirects moonshine vertex algebras]]



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