Virasoro VOAs (i.e., 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". (Borcherds's original formulation included a system of divided powers of , because he did not assume the ground ring was divisible.) a linear multiplication map , where is the space of all formal Laurent series with coefficients in . This structure is alternatively presented as an infinite collection of bilinear products , or as a left-multiplication map , called the state-field correspondence. For each , the operator-valued formal distribution is called a vertex operator or a field (inserted at zero), and the coefficient of is the operator . 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 (Jacobi identity, or Borcherds identity). 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 vectors u, v, and w, and integers m and n we have and . He later gave a more expansive version that is equivalent but easier to use: for any vectors u, v, and w, and integers m, n, and q we have 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 .