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\begin{document}

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\section*{automorphism of a vertex operator algebra}

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

\hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra}

[[!include higher algebra - contents]]

\hypertarget{string_theory}{}\paragraph*{{String theory}}\label{string_theory}

[[!include string theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{in_perturbative_string_vacua_and_conneslott_models}{In perturbative string vacua and Connes-Lott models}\dotfill \pageref*{in_perturbative_string_vacua_and_conneslott_models} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{moonshine}{Moonshine}\dotfill \pageref*{moonshine} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{AsSymmetriesOfStringCompactifications}{As symmetries of non-geometric string compactifications}\dotfill \pageref*{AsSymmetriesOfStringCompactifications} \linebreak
\noindent\hyperlink{moonshine_automorphism_groups}{Moonshine automorphism groups}\dotfill \pageref*{moonshine_automorphism_groups} \linebreak


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

Given a [[vertex operator algebra]] $\mathcal{V}$ ([[VOA]]) or [[super vertex operator algebra]] ([[sVOA]]), or more generally a full [[2d CFT]] (of which the [[VOA]] is the local and chiral data) or [[2d SCFT]] (hence a ``[[2-spectral triple]]'') one may ask (as for any [[object]] in any [[category]]) for its [[automorphisms]], hence the [[homomorphisms]]

\begin{displaymath}
\mathcal{V} 
   \overset{\simeq}{\longrightarrow}
  \mathcal{V}
\end{displaymath}
in the corresponding [[category]] of sVOA-s/[[2d SCFTs]], from $\mathcal{V}$ to itself, which are [[invertible morphism|invertible]] and hence constitute a \emph{[[symmetry]]} of $\mathcal{V}$.

\hypertarget{in_perturbative_string_vacua_and_conneslott_models}{}\subsection*{{In perturbative string vacua and Connes-Lott models}}\label{in_perturbative_string_vacua_and_conneslott_models}

If $\mathcal{V}$ is a [[direct sum|direct summand]] of a [[2d CFT]]/[[2-spectral triple]] encoding a [[perturbative string theory vacuum]], such as, typically, a [[rational 2d CFT]] such as a [[Gepner model]] encoding a ``[[non-geometric string vacuum|non-geometric]]'' [[KK-compactification]], then the automorphisms of $\mathcal{V}$ are the [[formal dual|formal duals]] to [[symmetries]] of that [[KK-compactification]]-[[fiber]] space (see the references \hyperlink{AsSymmetriesOfStringCompactifications}{below})

In the point-particle limit where the [[2d SCFT]]/[[2-spectral triple]] becomes an ordinary [[spectral triple]] (see \href{2-spectral+triple#References}{there}) this hence reduces to the automorphisms of internal algebras as discussed in [[Connes-Lott-Chamseddine-Barrett models]].

(\ldots{})

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{moonshine}{}\subsubsection*{{Moonshine}}\label{moonshine}

[[moonshine]]-examples:

\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{references}{}\subsection*{{References}}\label{references}

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

(\ldots{})

\hypertarget{AsSymmetriesOfStringCompactifications}{}\subsubsection*{{As symmetries of non-geometric string compactifications}}\label{AsSymmetriesOfStringCompactifications}

Automorphisms of [[vertex operator algebras]] regarded as [[symmetries]] of non-geometric [[perturbative string theory vacua]] (e.g. [[Gepner models]]):

\begin{itemize}%
\item [[Chris Hull]], Dan Israel, Alessandra Sarti, \emph{Non-geometric Calabi-Yau Backgrounds and K3 automorphisms}, JHEP11(2017)084 (\href{https://arxiv.org/abs/1710.00853}{arXiv:1710.00853})

\end{itemize}
\hypertarget{moonshine_automorphism_groups}{}\subsubsection*{{Moonshine automorphism groups}}\label{moonshine_automorphism_groups}

\begin{itemize}%
\item [[Igor Frenkel]], [[James Lepowsky]], Arne Meurman, \emph{Vertex operator algebras and the monster}, Pure and Applied Mathematics \textbf{134}, Academic Press, New York 1998. liv+508 pp. \href{http://www.ams.org/mathscinet-getitem?mr=996026}{MR0996026}


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


\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 [[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{Monstrous BPS-Algebras and the Superstring Origin of Moonshine} (\href{http://arxiv.org/abs/1601.05412}{arXiv:1601.05412})


\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})


\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}
[[!redirects automorphisms of a vertex operator algebra]] [[!redirects automorphisms of vertex operator algebras]]

[[!redirects automorphism of a VOA]] [[!redirects automorphisms of a VOA]] [[!redirects automorphisms of VOAs]]

[[!redirects automorphism of a super vertex operator algebra]] [[!redirects automorphisms of a super vertex operator algebra]] [[!redirects automorphisms of super vertex operator algebras]]

[[!redirects automorphism of a super VOA]] [[!redirects automorphisms of a super VOA]] [[!redirects automorphisms of super VOAs]]

[[!redirects automorphism of an sVOA]] [[!redirects automorphisms of an sVOA]] [[!redirects automorphisms of sVOAs]]

[[!redirects automorphism of a 2d conformal field theory]] [[!redirects automorphisms of a 2d conformal field theory]] [[!redirects automorphisms of 2d conformal field theories]]

[[!redirects automorphism of a 2d CFT]] [[!redirects automorphisms of a 2d CFT]] [[!redirects automorphisms of 2d CFT]]

[[!redirects automorphism of a 2d super-conformal field theory]] [[!redirects automorphisms of a 2d super-conformal field theory]] [[!redirects automorphisms of 2d super-conformal field theories]]

[[!redirects automorphism of a 2d sCFT]] [[!redirects automorphisms of a 2d sCFT]] [[!redirects automorphisms of 2d sCFT]]

[[!redirects automorphism of a 2-spectral triple]] [[!redirects automorphisms of a 2-spectral triple]] [[!redirects automorphisms of 2-spectral triples]]



\end{document}