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\section*{vertex operator algebra}

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

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

[[!include higher algebra - contents]]

\hypertarget{aqft}{}\paragraph*{{AQFT}}\label{aqft}

[[!include AQFT and operator algebra contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{standard_definition}{Standard definition}\dotfill \pageref*{standard_definition} \linebreak
\noindent\hyperlink{Properties}{Properties}\dotfill \pageref*{Properties} \linebreak
\noindent\hyperlink{CategoryOfVOAs}{Category of vertex operator algebras}\dotfill \pageref*{CategoryOfVOAs} \linebreak
\noindent\hyperlink{ModularTensorCategories}{Modular category of modules over a VOA}\dotfill \pageref*{ModularTensorCategories} \linebreak
\noindent\hyperlink{goddardthorn_theorem}{Goddard-Thorn theorem}\dotfill \pageref*{goddardthorn_theorem} \linebreak
\noindent\hyperlink{RelationToAInfinityAlgebras}{Relation to $A_\infty$-algebras and RG fixed points}\dotfill \pageref*{RelationToAInfinityAlgebras} \linebreak
\noindent\hyperlink{relation_to_conformal_nets}{Relation to conformal nets}\dotfill \pageref*{relation_to_conformal_nets} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{variants}{Variants}\dotfill \pageref*{variants} \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{AsOperadAlgebras}{As algebras over the holomorphic sphere operad}\dotfill \pageref*{AsOperadAlgebras} \linebreak
\noindent\hyperlink{as_factorization_algebras}{As factorization algebras}\dotfill \pageref*{as_factorization_algebras} \linebreak
\noindent\hyperlink{relation_to_modular_tensor_categories}{Relation to modular tensor categories}\dotfill \pageref*{relation_to_modular_tensor_categories} \linebreak
\noindent\hyperlink{as_chiral_algebras}{As chiral algebras}\dotfill \pageref*{as_chiral_algebras} \linebreak
\noindent\hyperlink{relation_to_2d_conformal_field_theory}{Relation to 2d conformal field theory}\dotfill \pageref*{relation_to_2d_conformal_field_theory} \linebreak
\noindent\hyperlink{category_of_vertex_operator_algebras_2}{Category of vertex operator algebras}\dotfill \pageref*{category_of_vertex_operator_algebras_2} \linebreak
\noindent\hyperlink{relation_to_sporadic_groups}{Relation to sporadic groups}\dotfill \pageref*{relation_to_sporadic_groups} \linebreak
\noindent\hyperlink{deformations}{Deformations}\dotfill \pageref*{deformations} \linebreak
\noindent\hyperlink{further_references}{Further references}\dotfill \pageref*{further_references} \linebreak


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

Vertex operator algebras (or ``vertex algebras'', for short) are [[algebras]] with a product-operation parameterized by points in the [[complex plane]].

Vertex operator algebras equipped with an [[action]] of the [[Virasoro algebra]] encode the local ([[genus of a surface|genus-0]] behaviour) of [[2-dimensional conformal field theories]]. Here one may think of the [[complex plane]] as the [[Riemann sphere]] and of the $z$-parameterized product operation in the vertex algebras as being the [[n-point function|3-point function]] of the [[2d CFT]] with field insertions at the points 0, $z$ and $\infty$. In fact one vertex operator algebra encodes (only) one chiral/holomorphic half of such a genus-0 2d CFT; a full genus-0 2d CFT is given by the combination of two vertex operator algebras called a \emph{[[full field algebra]]}.

The traditional definition of \emph{vertex operator algebra} (VOA) is long and tends to be somewhat unenlightening. But due to (\hyperlink{Huang}{Huang 91}) it is now known that vertex operator algebras have equivalently an [[FQFT]]-type characterization which manifestly captures this relation to [[n-point functions]] in the [[2d CFT]]:

There is a [[monoidal category]] or [[operad]] whose [[morphism|morphisms]] are conformal spheres with $n$-punctures marked as incoming and one puncture marked as outgoing (each puncture equipped with a conformally parametrized annular neighborhood). Composition of such spheres is by gluing along punctures. This can be regarded as a category $2Cob_{conf}^0$ of 2-dimensional genus-0 conformal [[cobordism|cobordisms]].

As shown by theorems by [[Yi-Zhi Huang]] and [[Liang Kong]], a vertex operator algebra is precisely a \emph{[[holomorphic representation]]} of this category, or [[algebra over an operad]] for this [[operad]] i.e. an genus-0 [[conformal field theory|conformal]] [[FQFT]], hence a [[monoidal functor]]

\begin{displaymath}
V : 2Cob_{conf}^0 \to Vect
\end{displaymath}
such that its component $V_1$ is a holomorphic function on the moduli space of conformal punctured spheres.

\hypertarget{standard_definition}{}\subsection*{{Standard definition}}\label{standard_definition}

(under construction) A \textbf{vertex algebra} consists of the following data:

\begin{itemize}%
\item nonnegatively graded complex vector space $V = \oplus_{n =0}^\infty V_n$
\item vacuum vector $|0\rangle\in V_0$
\item a shift operator $T: V\to V$
\item operation $Y = Y(-,z) : V\to End V [ [z,z^{-1}] ]$

\end{itemize}
This data satisfy three axioms:

(vacuum)

(translation axiom)

(locality)

In practice, the most important case is when the shift $T$ is related to a conformal vector ``the energy momentum tensor'' $\omega$ whose Fourier components are related to the generators $L_i$ of the [[Virasoro Lie algebra]] $Vir$. In that case we talk about a \textbf{conformal vertex algebra}.

There is a notion of a module over a vertex algebra. A conformal vertex algebra $A$ is said to be \textbf{rational} if every $A$-module is completely reducible. Then it follows that there are only finitely many nonisomorphic simple $A$-modules.

\hypertarget{Properties}{}\subsection*{{Properties}}\label{Properties}

\hypertarget{CategoryOfVOAs}{}\subsubsection*{{Category of vertex operator algebras}}\label{CategoryOfVOAs}

Vertex operator algebras naturally form a [[category]] (see section 2.4 of (\hyperlink{FrenkenHuangLepowsky}{FrenkenHuangLepowsky}). This is naturally a [[monoidal category]] with respect to [[tensor product]] of VOAs (section 2.5).

This is [[equivalence of categories|equivalent]] to

\begin{itemize}%
\item the category of algebras over the holomorphic punctured sphere operad (\hyperlink{HuangCFT}{Huang});


\item the category of vertex operator coalgebras (\hyperlink{Hubbard}{Hubbard}).



\end{itemize}
\hypertarget{ModularTensorCategories}{}\subsubsection*{{Modular category of modules over a VOA}}\label{ModularTensorCategories}

The [[category]] of [[modules]]/[[representations]] over a given vertex operator algebra is a [[modular tensor category]], (\hyperlink{Huang}{Huang})

\hypertarget{goddardthorn_theorem}{}\subsubsection*{{Goddard-Thorn theorem}}\label{goddardthorn_theorem}

\begin{itemize}%
\item [[Goddard-Thorn theorem]]

\end{itemize}
\hypertarget{RelationToAInfinityAlgebras}{}\subsubsection*{{Relation to $A_\infty$-algebras and RG fixed points}}\label{RelationToAInfinityAlgebras}

A [[functor]] from the category of [[BRST complex|BRST]]-VOAs to that of [[A-infinity algebra]]s is described in

\begin{itemize}%
\item [[Anton Zeitlin]], \emph{Quasiclassical Lian-Zuckerman Homotopy Algebras, Courant Algebroids and Gauge Theory} (\href{http://arxiv.org/abs/0910.3652}{arXiv})

\end{itemize}
and argued to algebraically encode the [[effective QFT|effective]] [[string theory]] background encoded by the [[CFT]] given by the VOA.

\hypertarget{relation_to_conformal_nets}{}\subsubsection*{{Relation to conformal nets}}\label{relation_to_conformal_nets}

Subject to some conditions, from a vertex operator algebra one may induce a [[conformal net]] and conversely (\hyperlink{CarpiKawahigahshiLongoWeiner15}{Carpi-Kawahigahshi-Longo-Weiner 15}, \hyperlink{Carpi16}{Carpi 16}) also (\hyperlink{Gui18}{Gui 18}).

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

A class of examples are \emph{[[current algebras]]} .

A database of examples is given by (\hyperlink{GannonHoehn}{Gannon-H\"o{}hn}).

The [[Moonshine]] module over the [[Griess algebra]] admits the structure of a vertex operator algebra, which has

\begin{itemize}%
\item rank 24;


\item is a self-[[dual object]] in the category of VOAs;


\item has trivial degree-1 subspaces.



\end{itemize}
A conjecture by Frenkel, Lepowsky and Meurman says that the Moonshine VOA is, up to isomorphism the unique VOA with these properties.

See at \emph{[[monster vertex algebra]]}.

\hypertarget{variants}{}\subsection*{{Variants}}\label{variants}

\begin{itemize}%
\item [[Poisson vertex algebra]]


\item [[rational vertex operator algebra]]


\item [[super vertex operator algebra]]


\item [[sheaf of vertex operator algebras]], [[vertex operator algebroid]]



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

\begin{itemize}%
\item [[operator product expansion]], [[conformal bootstrap]]


\item [[affine Lie algebra]]


\item [[conformal field theory]],


\item [[factorization algebra]],


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



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

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

\begin{itemize}%
\item [[Victor Kac]], \emph{Vertex algebras for beginners}, Amer. Math. Soc. (\href{http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0924.17023&format=complete}{ZMATH entry})


\item [[Edward Frenkel]], [[David Ben-Zvi]]: \emph{Vertex algebras and algebraic curves}, Math. Surveys and Monographs \textbf{88}, AMS 2001, xii+348 pp. (Bull. AMS. \href{http://www.ams.org/journals/bull/2002-39-04/S0273-0979-02-00955-2/S0273-0979-02-00955-2.pdf}{review}, \href{http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1106.17035&format=complete}{ZMATH entry})


\item [[Igor Frenkel]], [[Yi-Zhi Huang]], [[James Lepowsky]], \emph{On Axiomatic approaches to Vertex Operator Algebras and Modules} , Memoirs of the AMS Vol 104, No 494 (1993)



\end{itemize}
\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.

\end{itemize}
\hypertarget{AsOperadAlgebras}{}\subsubsection*{{As algebras over the holomorphic sphere operad}}\label{AsOperadAlgebras}

The original article with the interpretation of vertex operator algebras as holomorphic algebras over the holomorphic punctured sphere operad is

\begin{itemize}%
\item [[Yi-Zhi Huang]], \emph{Geometric interpretation of vertex operator algebras}, Proc. Natl. Acad. Sci. USA \textbf{88} (1991) pp. 9964-9968

\end{itemize}
A standard textbook summarizing these results is

\begin{itemize}%
\item [[Yi-Zhi Huang]], \emph{Two-dimensional conformal geometry and vertex operator algebras}, Progr. in Math. Birkhauser 1997, \href{http://books.google.hr/books?isbn=0817638296}{gbooks}

\end{itemize}
As mentioned in the \href{http://books.google.com/books?id=SUUdknTpYjEC&pg=PR11&lpg=PR11&dq=%22vertex+operator+algebras%22+%22trimble%22&source=bl&ots=v3GHx_ra2M&sig=TW5MzAtDi4n_gJjvUHb6eELoAXQ&hl=en&ei=9jdnSoXqKJiCtge6xqDuDw&sa=X&oi=book_result&ct=result&resnum=4}{acknowledgements} there, [[Todd Trimble]] and [[Jim Stasheff]] had a hand in making the operadic picture manifest itself here. Other operadic approaches are known, e.g. an earlier one in

\begin{itemize}%
\item Bojko Bakalov, Alessandro D'Andrea, [[Victor Kac]], \emph{Theory of finite pseudoalgebras}, Adv. Math. \textbf{162} (2001), no. 1, 1--140, \href{http://www.ams.org/mathscinet-getitem?mr=2003c:17020}{MR2003c:17020}

\end{itemize}
and even earlier treatments via coloured [[operads]] and [[generalized multicategories]], for references (Snydal, Soibelman, Beilinson-Drinfeld) see [[relaxed multicategory]].

More recently Huang's student [[Liang Kong]] has been developing these results further, generalizing them to open-closed CFTs and to non-chiral, i.e. completely general CFTs. See

\begin{itemize}%
\item [[Liang Kong]], \emph{Open-closed field algebras} Commun. Math. Physics. 280, 207-261 (2008), \href{http://arxiv.org/abs/math/0610293}{math.QA/0610293}.

\end{itemize}
\hypertarget{as_factorization_algebras}{}\subsubsection*{{As factorization algebras}}\label{as_factorization_algebras}

Discussion of vertex operator algebras as [[factorization algebras of observables]] is in section 6.3 and section 6.5 of

\begin{itemize}%
\item [[Owen Gwilliam]], \emph{Factorization algebras and free field theories} PhD thesis (\href{http://math.berkeley.edu/~gwilliam/thesis.pdf}{pdf})

\end{itemize}
\hypertarget{relation_to_modular_tensor_categories}{}\subsubsection*{{Relation to modular tensor categories}}\label{relation_to_modular_tensor_categories}

The [[representation categories]] of (rational) vertex operator algebras ([[modular tensor categories]]) are discussed in

\begin{itemize}%
\item [[Yi-Zhi Huang]], \emph{Vertex operator algebras, the Verlinde conjecture and modular tensor categories} (\href{http://arxiv.org/abs/math/0412261}{arXiv:math/0412261})

\end{itemize}
\begin{itemize}%
\item [[Terry Gannon]], [[Gerald Höhn]], Hiroshi Yamauchi, \emph{\href{http://www.math.ksu.edu/~gerald/voas/}{The online database of Vertex Operator Algebras and Modular Categories}}

\end{itemize}
\hypertarget{as_chiral_algebras}{}\subsubsection*{{As chiral algebras}}\label{as_chiral_algebras}

An algebrogeometric version is due Beilinson and Drinfel'd and called the [[chiral algebra]].

\begin{itemize}%
\item E. Frenkel, N. Reshetikhin, \emph{Towards deformed chiral algebras}, \href{http://arXiv.org/abs/q-alg/9706023}{q-alg/9706023}


\item Ruthi Hortsch, [[Igor Kriz]], Ales Pultr, \emph{A universal approach to vertex algebras}, \href{http://arxiv.org/abs/1006.0027}{arxiv/1006.0027}



\end{itemize}
Much algebraic insight to algebaric structures in CFT is in unfinished notes

\begin{itemize}%
\item [[A. Beilinson]], [[B. Feigin]], B. Mazur, \emph{Notes on conformal field theory}, $<$http://www.math.sunysb.edu/{\tt \symbol{126}}kirillov/manuscripts.html{\tt \symbol{62}}

\end{itemize}
\hypertarget{relation_to_2d_conformal_field_theory}{}\subsubsection*{{Relation to 2d conformal field theory}}\label{relation_to_2d_conformal_field_theory}

\begin{itemize}%
\item [[Yi-Zhi Huang]], \emph{Two-dimensional conformal geometry and vertex operator algebras} Birkh\"a{}user (1997)

\end{itemize}
Relation specifically to [[conformal nets]] is discussed in

\begin{itemize}%
\item [[Sebastiano Carpi]], [[Yasuyuki Kawahigashi]], [[Roberto Longo]], Mih\'a{}ly Weiner, \emph{From vertex operator algebras to conformal nets and back}, Memoirs of the American Mathematical Society, Volume 254, Number 1213 2018; (\href{https://doi.org/10.1090/memo/1213}{doi:10.1090/memo/1213}, \href{http://arxiv.org/abs/1503.01260}{arXiv:1503.01260})


\item [[Sebastiano Carpi]], \emph{Operator algebras and vertex operator algebras}, Contribution to the Proceedings of the 14th Marcel Grossmann Meeting - MG14 (Rome, 2015) (\href{http://arxiv.org/abs/1603.06742}{arXiv:1603.06742}, \href{https://doi.org/10.1142/9789813226609_0508}{doi:10.1142/9789813226609\_0508})



\end{itemize}
See also

\begin{itemize}%
\item James E. Tener, \emph{Representation theory in chiral conformal field theory: from fields to observables} (\href{https://arxiv.org/abs/1810.08168}{arXiv:1810.08168})

\end{itemize}
Relation of the corresponding [[ribbon categoires]]:

\begin{itemize}%
\item Bin Gui, \emph{Categorical extensions of conformal nets} (\href{https://arxiv.org/abs/1812.04470}{arXiv:1812.04470})

\end{itemize}
\hypertarget{category_of_vertex_operator_algebras_2}{}\subsubsection*{{Category of vertex operator algebras}}\label{category_of_vertex_operator_algebras_2}

\begin{itemize}%
\item [[Keith Hubbard]], \emph{The duality between vertex operator algebras and coalgebras, modules and comodules}, \href{http://www.faculty.sfasu.edu/hubbardke/duality.pdf}{pdf}

\end{itemize}
\hypertarget{relation_to_sporadic_groups}{}\subsubsection*{{Relation to sporadic groups}}\label{relation_to_sporadic_groups}

\begin{itemize}%
\item [[John Duncan]], \emph{Vertex operator algebras and sporadic groups}, \href{http://arxiv.org/PS_cache/arxiv/pdf/0811/0811.1306v1.pdf}{pdf}; \emph{Moonshine for Rudvalis's sporadic group I}, \href{http://arxiv.org/PS_cache/math/pdf/0609/0609449v2.pdf}{pdf}

\end{itemize}
\hypertarget{deformations}{}\subsubsection*{{Deformations}}\label{deformations}

There is an interesting theory of deformation quantization of VOAs from

\begin{itemize}%
\item [[Pavel Etingof]], [[David Kazhdan]], \emph{Quantization of Lie bialgebras. V. Quantum vertex operator algebras}, Selecta Math. (N.S.) 6 (2000), no. 1, 105--130, \href{http://www.ams.org/mathscinet-getitem?mr=2002i:17022}{MR2002i:17022}

\end{itemize}
Deformation quantization of chiral algebras are studied by

\begin{itemize}%
\item [[Dmitry Tamarkin]], \emph{Deformations of chiral algebras}, Proceedings of the ICM, Beijing 2002, vol. 2, 105--118

\end{itemize}
A class of ``free'' vertex algebras are also quantized using [[Batalin-Vilkovisky formalism]], with results important for understanding [[mirror symmetry]], in

\begin{itemize}%
\item Si Li, \emph{Vertex algebras and quantum master equation}, \href{https://arxiv.org/abs/1612.01292}{arxiv/1612.01292}

\end{itemize}
\hypertarget{further_references}{}\subsubsection*{{Further references}}\label{further_references}

\begin{itemize}%
\item \href{http://mathoverflow.net/questions/53988/what-is-the-motivation-for-a-vertex-algebra}{what-is-the-motivation-for-a-vertex-algebra} - a MathOverflow discussion on the motivation
\item \href{http://www.math.ksu.edu/~gerald/voas}{The online database of Vertex Operator Algebras and Modular Categories}
\item Yi-Zhi Huang, \emph{Meromorphic open-string vertex algebras}, J. Math. Phys. \textbf{54}, 051702 (2013) \href{http://dx.doi.org/10.1063/1.4806686}{doi}

\end{itemize}
[[!redirects vertex algebra]] [[!redirects vertex algebras]] [[!redirects Vertex operator algebra]]

[[!redirects vertex operator algebras]]

[[!redirects VOA]] [[!redirects VOAs]]



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