Vertex Operator Algebra - Formal Definition

Formal Definition

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:

1. (Identity) For any a ∈ V, Y(1,z)a = a = az0 and .
2. (Translation) T(1) = 0, and for any a, b ∈ V,

3. (Four point function) For any a, b, c ∈ V, there is an element
   such that
   are the corresponding expansions of in

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 along . 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, b are homogeneous, then a_n b is homogeneous of degree deg(a)+deg(b)-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 ∈ V_n, 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.