In mathematics, a vertex operator algebra (VOA) is an algebraic structure that plays an important role in conformal field theory and related areas of physics. Vertex operator algebras have proven useful in purely mathematical contexts such as monstrous moonshine and the geometric Langlands correspondence.
Vertex operator algebras were first introduced by Richard Borcherds in 1986, motivated by the vertex operators arising from field insertions in two dimensional conformal field theory, a framework that is essential to define string theory. The axioms of a vertex operator algebra are a formal algebraic interpretation of what physicists called chiral algebras, whose definition was made mathematically rigorous by Alexander Beilinson and Vladimir Drinfel'd.
Important examples of vertex operator algebras include lattice VOAs (modeling lattice conformal field theories), VOAs given by representations of affine Kac-Moody algebras (from the WZW model), the Virasoro VOAs (i.e., VOAs corresponding to representations of the Virasoro algebra) and the moonshine module V♮, constructed by Frenkel, Lepowsky and Meurman in 1988.
A vertex algebra is a vector space V, together with an identity element 1∈V, an endomorphism T: V → V, and a linear multiplication map
from the tensor product of V with itself to the space V((z)) of all formal Laurent series with coefficients in V, written as:
and satisfying the following axioms:
The multiplication map is often written as a state-field correspondence
associating an operator-valued formal distribution (called a vertex operator) to each vector. Physically, the correspondence is an insertion at the origin, and T is a generator of infinitesimal translations. The four-point axiom combines associativity and commutativity, up to singularities. Note that the translation axiom implies Ta = a-21, so T is determined by Y.
A vertex algebra V is Z+-graded if
such that if a ∈ Vk and b ∈ Vm, then an b ∈ Vk+m-n-1.
A vertex operator algebra is a Z+-graded vertex algebra equipped with a Virasoro element ω ∈ V2, such that the vertex operator
satisfies for any a ∈ Vn, the relations:
where c is a constant called the central charge, or rank of V. In particular, this gives V the structure of a representation of the Virasoro algebra.
The reason for the translation operator, T, and the Virasoro operators, L, come from the idea that we are interested in systems with translational and 2D conformal symmetries respectively. (With both these symmetries the system will also have rotational and scale invariance also.)
The axioms of a vertex algebra are obtained from abstracting away the essentials of the operator product expansion of operators in a 2D Euclidean chiral conformal field theory. The two dimensional Euclidean space is treated as a Riemann sphere with the point at infinity removed. V is taken to be the space of all operators at . The operator product expansion is holomorphic in z and so, we can make a Laurent expansion of it. 1 is the identity operator. We treat an operator valued holomorphic map over as a formal Laurent series. This is denoted by the notation V((z)). A holomorphic map over is given by a Taylor series and as a formal power series, this is denoted by V[[z]].
The operator b(0) is abstracted to b and the operator a(z) to Y(a,z). The derivative a'(z) is abstracted to -Ta.
If one considers only the singular part of the OPE in a Vertex algebra, one arrives at the definition of a Lie conformal algebra. Since one is often only concerned with the singular part of the OPE, this makes Lie conformal algebras a natural object to study.
Another way to think of vertex algebras is as a generalisation of a Lie algebra with the addition of a continuous variable to both the generators of the algebra and the structure constant. (The structure constants define the algebra). So we would have:
which can also be expressed without the y dependence as:
which reveals the Vertex Multiplication as a group operation in which the group constants vary with their position on the conformal sheet.
The vertex algebras satisfy a generalisation of the Jacobi identity given by:
which can also be written as
where the delta function is defined formally by:
(always expanded in terms of the second variable w)
To make things more concrete a trivial example of a commutative vertex algebra is presented. We take the vector space V to consist of 2-vectors, v, with components . If we think of this vector as a complex number, p+qi, the Y function can be seen as taking it from its vector form to a 2x2 matrix form of (p-qz)+qi. The p and q are real numbers.
So that multiplication between two vectors and is defined by:
The identity element so that:
and
and the operator T in this example is:
it is easy to check:
Since the Y matrices commute, locality is satisfied trivially. Non-trivial vertex algebras and vertex operator algebras would require infinite sized matrices to represent them. All the terms with powers of with n<0 must be zero when applied to the identity vector . i.e. of the form:
but adding a single term like this won't commute with the other terms. Hence our example would no longer be trivial since we may need an infinite number of terms to restore locality.
The Heisenberg Lie algebra is defined by the commutation relations:
One representation is to define the operators b, in terms of the dummy variables x as:
and .
This can be made into a vertex algebra by the definition:
where :..: denotes normal ordering (i.e. moving all derivatives in x to the right). Thus Y(1,z) = Id.
T is defined by the conditions T.1 = 0 and
Setting
it follows that
and hence:
The vertex operators may also be written as a functional of a multivariable function f as:
if we understand that each term in the expansion of f is normal ordered.
The Virasoro vertex operator algebra is a conformal vertex algebra. It is defined as follows:
With L(z) defined as above. So we have and with and . We see that this satisfies the axioms of a vertex algebra and includes a representation of the Virasoro algebra. The fact that the Virasoro field is local can be deduced from the formula for its self-commutator:
where c is central charge.
The monster vertex algebra is a conformal vertex operator derived from 26 dimensional bosonic string theory compactified on the hyper-torus induced by the Leech lattice and orbifolded by the two-element reflection group. It is denoted as . It was used to prove the Monstrous moonshine conjectures.
In the string model the vectors a in Y(a,z) are the different states or vibrational modes of the string which correspond to different particles and polarisations and z is a point (or vertex) on the world sheet which corresponds to an ingoing or outgoing string. Hence why it is called a Vertex Algebra.
When the underlying vector space V has a Z2 grading, so that it splits as a sum of even and odd parts
with 1 in V+, the structure of a vertex superalgebra can be defined on V by incorporating the usual rule of signs in the axiom for the four point function:
such that Y(a,z)Y(b,w)c, εY(b,w)Y(a,z)c, and Y(Y(a,z-w)b,w)c are the expansions of X(a,b,c;z,w) in V((z))((w)), V((w))((z)), and V((w))((z-w)), respectively, where ε is -1 if both a and b are odd and 1 otherwise.
If in addition there is a Virasoro element ω in the even part of V2, then V is called a vertex operator superalgebra. One of the simplest examples is the vertex operator superalgebra generated by a single complex fermion.[1]
Let Λ be an integral lattice in Euclidean space X = RN, i.e. a subgroup isomorphic to Zn with m ≤ N and such that (α,β) lies in Z for α,β in Λ . The lattice is said to be even if (α,α) is even for each α in Λ. Setting
there is an essentially unique normalised cocycle ε(α,β) with values ±1 such that
and the cocycle identity
is satisfied along with the normalisation conditions
A cocycle representation can be defined on C[Λ], with basis eα (α in Λ), by
Thus
and
The operators Uα are unitary if the eα are taken to be orthonormal.
There is a bosonic system associated to X, namely operators vn depending linearly on v in X such that
In addition the system has a derivation D satisfying with
There is a unique irreducible representation of this system characterised by the existence of a vacuum vector Ω with vn Ω = 0 for n ≥ 0. The underlying space S has a unique inner product structure for which vn* = v–n. The vector space V of the vertex superalgebra is defined by
The operators vn with n non-zero act on exactly as they act on S,
The operators v0 act as scalars on :
For each v in X define the field
For each α in Λ define
where
If v(i) is an orthonormal basis of X, define
where the normal ordering is given by
Then the vertex operators v(z) and Φα(z) generate a vertex operator superalgebra with underlying space V. The operators D and T are given by L0 and L–1 respectively.[2][3]