Editing Vertex operator algebra (section)

{{Short description|Algebra used in 2D conformal field theories and string theory}}
{{String theory}}
{{Technical|date=May 2025}} 

In mathematics, a '''vertex operator algebra''' ('''VOA''') 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 [[unimodular lattice|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 [[Hamiltonian (quantum mechanics)|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 expansion]]s 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 (not to be confused with the more precise notion with the same name in mathematics) or "algebras of chiral symmetries", where these symmetries describe the [[Ward identity|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 ring]]s, algebras over certain operads on curves introduced by Huang, Kriz, and others, [[D-module]]-theoretic objects called [[chiral algebra]]s introduced by [[Alexander Beilinson]] and [[Vladimir Drinfeld]] and [[factorization algebra]]s, also introduced by Beilinson and Drinfeld. 

Important basic examples of vertex operator algebras include the lattice VOAs (modeling lattice conformal field theories), VOAs given by representations of affine [[Kac–Moody algebra]]s (from the [[Wess–Zumino–Witten model|WZW model]]), the Virasoro VOAs, which are VOAs corresponding to representations of the [[Virasoro algebra]], and the [[moonshine module]] ''V''<sup>♮</sup>, which is distinguished by its [[monster group|monster]] [[automorphism group|symmetry]].  More sophisticated examples such as [[affine W-algebra]]s and the [[chiral de Rham complex]] on a [[complex manifold]] arise in geometric [[representation theory]] and [[mathematical physics]].