Vertex operator algebra

In mathematics, a vertex operator algebra is an algebraic structure that plays an important role in two-dimensional conformal field theory and string theory. In addition to physical applications, vertex operator algebras have proven useful in purely mathematical contexts such as monstrous moonshine and the geometric Langlands correspondence.
The related notion of vertex algebra was introduced by Richard Borcherds in 1986, motivated by a construction of an infinite-dimensional Lie algebra due to Igor Frenkel. In the course of this construction, one employs a Fock space that admits an action of vertex operators attached to elements of a lattice. Borcherds formulated the notion of vertex algebra by axiomatizing the relations between the lattice vertex operators, producing an algebraic structure that allows one to construct new Lie algebras by following Frenkel's method.
The notion of vertex operator algebra was introduced as a modification of the notion of vertex algebra, by Frenkel, James Lepowsky, and Arne Meurman in 1988, as part of their project to construct the moonshine module. They observed that many vertex algebras that appear 'in nature' carry an action of the Virasoro algebra, and satisfy a bounded-below property with respect to an energy operator. Motivated by this observation, they added the Virasoro action and bounded-below property as axioms.
We now have post-hoc motivation for these notions from physics, together with several interpretations of the axioms that were not initially known. Physically, the vertex operators arising from holomorphic field insertions at points in two-dimensional conformal field theory admit operator product expansions when insertions collide, and these satisfy precisely the relations specified in the definition of vertex operator algebra. Indeed, the axioms of a vertex operator algebra are a formal algebraic interpretation of what physicists call chiral algebras or "algebras of chiral symmetries", where these symmetries describe the Ward identities satisfied by a given conformal field theory, including conformal invariance. Other formulations of the vertex algebra axioms include Borcherds's later work on singular commutative rings, algebras over certain operads on curves introduced by Huang, Kriz, and others, D-module-theoretic objects called chiral algebras introduced by Alexander Beilinson and Vladimir Drinfeld and factorization algebras, also introduced by Beilinson and Drinfeld.
Important basic examples of vertex operator algebras include the lattice VOAs, VOAs given by representations of affine Kac–Moody algebras, the Virasoro VOAs, which are VOAs corresponding to representations of the Virasoro algebra, and the moonshine module V^♮, which is distinguished by its monster symmetry. More sophisticated examples such as affine W-algebras and the chiral de Rham complex on a complex manifold arise in geometric representation theory and mathematical physics.

Formal definition

Vertex algebra

A vertex algebra is a collection of data that satisfy certain axioms.

Data

a vector space, called the space of states. The underlying field is typically taken to be the complex numbers, although Borcherds's original formulation allowed for an arbitrary commutative ring.
an identity element, sometimes written or to indicate a vacuum state.
an endomorphism, called "translation".
a linear multiplication map, where is the space of all formal Laurent series with coefficients in. This structure has some alternative presentations:
* as an infinite collection of bilinear products where and, so that for each, there is an such that for.
* as a left-multiplication map. This is the 'state-to-field' map of the so-called state-field correspondence. For each, the endomorphism-valued formal distribution is called a vertex operator or a field, and the coefficient of is the operator. In the context of vertex algebras, a field is more precisely an element of, which can be written such that for any for sufficiently small . The standard notation for the multiplication is
Axioms

These data are required to satisfy the following axioms:

Identity. For any and.
Translation., and for any,
Locality. For any, there exists a positive integer such that:
Equivalent formulations of locality axiom

The locality axiom has several equivalent formulations in the literature, e.g., Frenkel–Lepowsky–Meurman introduced the Jacobi identity:,
where we define the formal delta series by:
Borcherds initially used the following two identities: for any and integers we have
and
He later gave a more expansive version that is equivalent but easier to use: for any and integers we have
This identity is the same as the Jacobi identity by expanding both sides in all formal variables. Finally, there is a formal function version of locality: For any, there is an element
such that and are the corresponding expansions of in and.

Vertex operator algebra

A vertex operator algebra is a vertex algebra equipped with a conformal element, such that the vertex operator is the weight two Virasoro field :
and satisfies the following properties:

, where is a constant called the central charge, or rank of. In particular, the coefficients of this vertex operator endow with an action of the Virasoro algebra with central charge.
acts semisimply on with integer eigenvalues that are bounded below.
Under the grading provided by the eigenvalues of, the multiplication on is homogeneous in the sense that if and are homogeneous, then is homogeneous of degree.
The identity has degree 0, and the conformal element has degree 2.
.

A homomorphism of vertex algebras is a map of the underlying vector spaces that respects the additional identity, translation, and multiplication structure. Homomorphisms of vertex operator algebras have "weak" and "strong" forms, depending on whether they respect conformal vectors.

Commutative vertex algebras

A vertex algebra is commutative if all vertex operators commute with each other. This is equivalent to the property that all products lie in, or that. Thus, an alternative definition for a commutative vertex algebra is one in which all vertex operators are regular at.
Given a commutative vertex algebra, the constant terms of multiplication endow the vector space with a commutative and associative ring structure, the vacuum vector is a unit and is a derivation. Hence the commutative vertex algebra equips with the structure of a commutative unital algebra with derivation. Conversely, any commutative ring with derivation has a canonical vertex algebra structure, where we set, so that restricts to a map which is the multiplication map with the algebra product. If the derivation vanishes, we may set to obtain a vertex operator algebra concentrated in degree zero.
Any finite-dimensional vertex algebra is commutative.

Proof

This follows from the translation axiom. From and expanding the vertex operator as a power series one obtains
Then
From here, we fix to always be non-negative. For, we have.
Now since is finite dimensional, so is, and all the are elements of. So a finite number of the span the vector subspace of spanned by all the. Therefore there's an such that for all. But also,
and the left hand side is zero, while the coefficient in front of is non-zero. So. So is regular.

Thus even the smallest examples of noncommutative vertex algebras require significant introduction.

Basic properties

The translation operator in a vertex algebra induces infinitesimal symmetries on the product structure, and satisfies the following properties:

, so is determined by.

For a vertex operator algebra, the other Virasoro operators satisfy similar properties:

for all.
: For any, the element

given in the definition also expands to in.
The associativity property of a vertex algebra follows from the fact that the commutator of and is annihilated by a finite power of, i.e., one can expand it as a finite linear combination of derivatives of the formal delta function in, with coefficients in.
Reconstruction: Let be a vertex algebra, and let be a set of vectors, with corresponding fields. If is spanned by monomials in the positive weight coefficients of the fields, then we may write the operator product of such a monomial as a normally ordered product of divided power derivatives of fields. Specifically,
More generally, if one is given a vector space with an endomorphism and vector, and one assigns to a set of vectors a set of fields that are mutually local, whose positive weight coefficients generate, and that satisfy the identity and translation conditions, then the previous formula describes a vertex algebra structure.

Operator product expansion

In vertex algebra theory, due to associativity, we can abuse notation to write, for
This is the operator product expansion. Equivalently,
Since the normal ordered part is regular in and, this can be written more in line with physics conventions as
where the equivalence relation denotes equivalence up to regular terms.

Commonly used OPEs

Here some OPEs frequently found in conformal field theory are recorded.

1st distribution	2nd distribution	Commutation relations	OPE	Name	Notes
				Generic OPE
				Free boson OPE	Invariance under shows 'bosonic' nature of this OPE.
				Primary field OPE	Primary fields are defined to be fields a satisfying this OPE when multiplied with the Virasoro field. These are important as they are the fields which transform 'like tensors' under coordinate transformations of the worldsheet in string theory.
				TT OPE	In physics, the Virasoro field is often identified with the stress-energy tensor and labelled T rather than L.

Vertex operator algebra

Formal definition

Vertex algebra

Data

Axioms

Equivalent formulations of locality axiom

Vertex operator algebra

Commutative vertex algebras

Basic properties

Operator product expansion

Commonly used OPEs