Vertex operator algebraInmathematics, avertex operator algebra(VOA) is an algebraic structure that plays an important role inconformal field theoryandstring theory. In addition to physical applications, vertex operator algebras have proven useful in purely mathematical contexts such asmonstrous moonshineand thegeometric Langlands correspondence.The related notion ofvertex algebrawas introduced byRichard Borcherdsin 1986, motivated by a construction of an infinite-dimensional Lie algebra due toFrenkel. In the course of this construction, one employs aFock spacethat admits an action of vertex operators attached to lattice vectors. 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,Lepowsky, andMeurmanin 1988, as part of their project to construct themoonshine module. They observed that many vertex algebras that appear in nature have a useful additional structure (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 (i.e., vertices) in two dimensional conformal field theory admitoperator product expansionswhen 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 callchiral 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, andD-module-theoretic objects called chiral algebras introduced byAlexander BeilinsonandVladimir Drinfeld. While related, these chiral algebras are not precisely the same as the objects with the same name that physicists use.Important basic examples of vertex operator algebras include lattice VOAs (modeling lattice conformal field theories), VOAs given by representations of affineKacMoody algebras(from theWZW model), the Virasoro VOAs (i.e., VOAs corresponding to representations of theVirasoro algebra) and themoonshine moduleV, which is distinguished by its monster symmetry. More sophisticated examples such asaffine W-algebrasand thechiral de Rham complexon a complex manifold arise in geometric representation theory andmathematical physics

Vertex AlgebraAvertex algebrais a collection of data that satisfy certain axioms.Data[edit] avector spaceV, 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 1 V, sometimes writtenorto indicate a vacuum state. anendomorphismT:VV, called "translation". (Borcherds's original formulation included a system of divided powers ofT, because he did not assume the ground ring was divisible.) a linear multiplication mapY:VVV((z)), whereV((z))is the space of allformal Laurent serieswith coefficients inV. This structure is alternatively presented as an infinite collection of bilinear productsunv, or as a left-multiplication mapV End(V)[[z1]], called the state-field correspondence. For eachuV, the operator-valued formal distributionY(u,z)is called a vertex operator or a field (inserted at zero), and the coefficient ofzn1is the operatorun. The standard notation for the multiplication is.Axioms[edit]These data are required to satisfy the following axioms: Identity.For anyuV,Y(1,z)u=u=uz0andY(u,z)1 u+zV[[z]]. Translation.T(1) = 0, and for anyu,vV,

Locality (Jacobi identity, or Borcherds identity).For anyu,vV, there exists a positive integerNsuch that:

Equivalent Formulations of Locality Axiom[edit]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 identity: for any vectorsu,v, andw, and integersmandnwe have

He later gave a more expansive version that is equivalent but easier to use: for any vectorsu,v, andw, and integersm,n, andqwe have

Finally, there is a formal function version of locality: For anyu,v,wV, there is an element

such thatY(u,z)Y(v,x)wandY(v,x)Y(u,z)ware the corresponding expansions ofinV((z))((x))andV((x))((z)).

Vertex Operator Algebra

Avertex operator algebrais a vertex algebra equipped with aconformal element, such that the vertex operatorY(,z)is the weight two Virasoro fieldL(z):and satisfies the following properties:[Lm,Ln] = (mn)Lm+n+ (m+n,0/12) (m3m)cIdV, wherecis a constant called thecentral charge, orrankofV. In particular, the coefficients of this vertex operator endowVwith an action of the Virasoro algebra with central chargec.L0acts semisimply onVwith integer eigenvalues that are bounded below.Under the grading provided by the eigenvalues ofL0, the multiplication onVis homogeneous in the sense that ifuandvare homogeneous, thenunvis homogeneous of degreedeg(u) + deg(v) n 1.The identity 1 has degree 0, and the conformal elementhas degree 2.L1=T.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.