nLab vertex operator algebra





algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)



field theory:

Lagrangian field theory


quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization



States and observables

Operator algebra

Local QFT

Perturbative QFT



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 zz-parameterized product operation in the vertex algebras as being the 3-point function of the 2d CFT with field insertions at the points 0, zz 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 nn-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 02Cob_{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

V:2Cob conf 0Vect V : 2Cob_{conf}^0 \to Vect

such that its component V 1V_1 is a holomorphic function on the moduli space of conformal punctured spheres.

Standard definition

(under construction) A vertex algebra consists of the following data:

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

This data satisfy three axioms:


(translation axiom)


In practice, the most important case is when the shift TT is related to a conformal vector “the energy momentum tensor” ω\omega whose Fourier components are related to the generators L iL_i of the Virasoro Lie algebra VirVir. 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 AA is said to be rational if every AA-module is completely reducible. Then it follows that there are only finitely many nonisomorphic simple AA-modules.


Category of vertex operator algebras

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).

Modular category of modules over a VOA

The category of modules/representations over a given vertex operator algebra is a modular tensor category, (Huang)

Goddard-Thorn theorem

Relation to A A_\infty-algebras and RG fixed points

A functor from the category of BRST-VOAs to that of A-infinity algebras is described in

  • Anton Zeitlin, Quasiclassical Lian-Zuckerman Homotopy Algebras, Courant Algebroids and Gauge Theory (arXiv)

and argued to algebraically encode the effective string theory background encoded by the CFT given by the VOA.

Relation to conformal nets

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

  • Igor Frenkel, James Lepowsky, Arne Meurman, Vertex operator algebras and the monster, Pure and Applied Mathematics 134, Academic Press, New York 1998.

says that the Moonshine VOA is, up to isomorphism the unique VOA with these properties. See at monster vertex algebra.




Brief review:

See also:

Relation to sporadic groups:

  • John Duncan, Vertex operator algebras and sporadic groups, pdf; Moonshine for Rudvalis’s sporadic group I, pdf

Vertex operator algebras over non-archimedean fields:

Classification of strongly rational holomorphic vertex operator algebras of central charge 24\leq 24 (of relevance in heterotic string theory and monstrous moonshine):

As algebras over the holomorphic sphere operad

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

  • Yi-Zhi Huang, Two-dimensional conformal geometry and vertex operator algebras, Progr. in Math. Birkhauser (1997) gbooks

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:

  • Bojko Bakalov, Alessandro D’Andrea, Victor Kac, Theory of finite pseudoalgebras, Adv. Math. 162 (2001), no. 1, 1–140, MR2003c:17020

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

As factorization algebras

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

  • Owen Gwilliam, Factorization algebras and free field theories PhD thesis (pdf)

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.

As coalgebras over a comonad

On vertex operator algebras as coalgebras over a comonad:

Relation to modular tensor categories

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

As chiral algebras

An algebrogeometric version is due Beilinson and Drinfel’d and called the chiral algebra.

Much algebraic insight to algebaric structures in CFT is in unfinished notes

  • A. Beilinson, B. Feigin, B. Mazur, Notes on conformal field theory, <>

Relation to 2d conformal field theory

Relation to 2d CFT:

For orbifolds:

Relation specifically to conformal nets:

and for super vertex operator algebras:

See also

  • James E. Tener, Representation theory in chiral conformal field theory: from fields to observables (arXiv:1810.08168)

Relation of the corresponding ribbon categories:


There is an interesting theory of deformation quantization of VOAs from

Deformation quantization of chiral algebras are studied by

  • Dmitry Tamarkin, Deformations of chiral algebras, Proceedings of the ICM, Beijing 2002, vol. 2, 105–118

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.