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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-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 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 full field algebra.
The traditional definition of vertex operator algebra (VOA) is long and tends to be somewhat unenlightening. But due to (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 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 cobordisms.
As shown by theorems by Yi-Zhi Huang and Liang Kong, a vertex operator algebra is precisely a holomorphic representation? of this category, or algebra over an operad for this operad i.e. an genus-0 conformal FQFT, hence a monoidal functor
such that its component $V_1$ is a holomorphic function on the moduli space of conformal punctured spheres.
(under construction) A vertex algebra consists of the following data:
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 conformal vertex algebra.
There is a notion of a module over a vertex algebra. A conformal vertex algebra $A$ is said to be rational if every $A$-module is completely reducible. Then it follows that there are only finitely many nonisomorphic simple $A$-modules.
Vertex operator algebras naturally form a category (see section 2.4 of (FrenkenHuangLepowsky). This is naturally a monoidal category with respect to tensor product of VOAs (section 2.5).
This is equivalent to
the category of algebras over the holomorphic punctured sphere operad (Huang);
the category of vertex operator coalgebras (Hubbard).
The category of modules/representations over a given vertex operator algebra is a modular tensor category, (Huang)
A functor from the category of BRST-VOAs to that of A-infinity algebras is described in
and argued to algebraically encode the effective string theory background encoded by the CFT given by the VOA.
Subject to some conditions, from a vertex operator algebra one may induce a conformal net and conversely (Carpi-Kawahigahshi-Longo-Weiner 15, Carpi 16) also (Gui 18).
A class of examples are current algebras .
A database of examples is given by (Gannon-Höhn).
The Moonshine module over the Griess algebra admits the structure of a vertex operator algebra, which has
rank 24;
is a self-dual object in the category of VOAs;
has trivial degree-1 subspaces.
A conjecture due to
says that the Moonshine VOA is, up to isomorphism the unique VOA with these properties. See at monster vertex algebra.
Victor Kac, Vertex algebras for beginners, Amer. Math. Soc., Univ. Lecture Series 10 (1998) [ams:ulect-10-r, ZMATH entry]
Edward Frenkel, David Ben-Zvi: Vertex algebras and algebraic curves, Math. Surveys and Monographs 88, AMS (2001) xii+348 pp. [review, ZMATH entry]
Igor Frenkel, Yi-Zhi Huang, James Lepowsky, On Axiomatic approaches to Vertex Operator Algebras and Modules, Memoirs of the AMS Vol 104, No 494 (1993)
Brief review:
See also:
what-is-the-motivation-for-a-vertex-algebra - a MathOverflow discussion on the motivation
The online database of Vertex Operator Algebras and Modular Categories
Keith Hubbard, The duality between vertex operator algebras and coalgebras, modules and comodules, [pdf]
Relation to sporadic groups:
Vertex operator algebras over non-archimedean fields:
Classification of strongly rational holomorphic vertex operator algebras of central charge $\leq 24$ (of relevance in heterotic string theory and monstrous moonshine):
Adrian Norbert Schellekens, Meromorphic $c = 24$ Conformal Field Theories, Commun. Math. Phys. 153 (1993) 159-186 [doi:10.1007/BF02099044]
Chongying Dong, Geoffrey Mason, Holomorphic vertex operator algebras of small central charges, Pacific Journal of Mathematics 213 2 (2004) 253–266 [arXiv:math/0203005, doi:10.2140/pjm.2004.213.253]
Philip Boyle Smith, Ying-Hsuan Lin, Yuji Tachikawa, Yunqin Zheng, Classification of chiral fermionic CFTs of central charge $\leq 16$ [arXiv:2303.16917]
Brandon C. Rayhaun, Bosonic rational conformal field theories in small genera, chiral fermionization, and symmetry/subalgebra duality [arXiv:2303.16921]
Gerald Höhn, Sven Möller, Classification of self-dual vertex operator superalgebras of central charge at most 24 [arXiv:2303.17190]
The original article with the interpretation of vertex operator algebras as holomorphic algebras over the holomorphic punctured sphere operad is
A standard textbook summarizing these results is
As mentioned in the acknowledgements there, Todd Trimble and Jim Stasheff had a hand in making the operadic picture manifest itself here.
See also:
Other operadic approaches are known, e.g. an earlier one in:
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
Discussion of vertex operator algebras as factorization algebras of observables is in section 6.3 and section 6.5 of
We introduce categories of weak factorization algebras and factorization spaces, and prove that they are equivalent to the categories of ordinary factorization algebras and spaces, respectively. This allows us to define the pullback of a factorization algebra or space by an 'etale morphism of schemes, and hence to define the notion of a universal factorization space or algebra. This provides a generalization to higher dimensions and to non-linear settings of the notion of a vertex algebra.
On vertex operator algebras as coalgebras over a comonad:
The representation categories of (rational) vertex operator algebras (modular tensor categories) are discussed in
Yi-Zhi Huang, Vertex operator algebras, the Verlinde conjecture and modular tensor categories, Proc. Nat. Acad. Sci. 102 (2005) 5352-5356 $[$arXiv:math/0412261, doi:10.1073/pnas.0409901102$]$
Terry Gannon, Gerald Höhn, Hiroshi Yamauchi, The online database of Vertex Operator Algebras and Modular Categories
An algebrogeometric version is due Beilinson and Drinfel’d and called the chiral algebra.
E. Frenkel, N. Reshetikhin, Towards deformed chiral algebras, q-alg/9706023
Ruthi Hortsch, Igor Kriz, Ales Pultr, A universal approach to vertex algebras, arxiv/1006.0027
Much algebraic insight to algebaric structures in CFT is in unfinished notes
Relation to 2d CFT:
For orbifolds:
Relation specifically to conformal nets:
Sebastiano Carpi, Yasuyuki Kawahigashi, Roberto Longo, Mihály Weiner, From vertex operator algebras to conformal nets and back, Memoirs of the American Mathematical Society 254 1213 (2018) [doi:10.1090/memo/1213, arXiv:1503.01260]
Sebastiano Carpi, Operator algebras and vertex operator algebras, Contribution to the Proceedings of the 14th Marcel Grossmann Meeting - MG14 (Rome, 2015) [arXiv:1603.06742, doi:10.1142/9789813226609_0508]
and for super vertex operator algebras:
See also
Relation of the corresponding ribbon categories:
There is an interesting theory of deformation quantization of VOAs from
Selecta Math. (N.S.) 6 (2000), no. 1, 105–130, MR2002i:17022
Deformation quantization of chiral algebras are studied by
A class of “free” vertex algebras are also quantized using Batalin-Vilkovisky formalism, with results important for understanding mirror symmetry, in
Last revised on August 15, 2023 at 08:53:36. See the history of this page for a list of all contributions to it.