Vertex operator algebra

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. They 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. Important examples include lattice VOAs (modeling lattice CFTs), 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. 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.

Motivation

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 a point added at infinity. V is taken to be the space of all operators at $z=0$. 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 $mathbb\{C\}/\{0\}$ as a formal Laurent series. This is denoted by the notation V((z)). A holomorphic map over $mathbb\{C\}$ is given by a Taylor series and as a formal power series, this is denoted by Vz.

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.

Formal definition

A vertex algebra is a vector space "V", together with an identity element 1 , an endomorphism "T", and a multiplication map

:$Y:\; V\; otimes\; V\; o\; V((z))$

written:

:$(a,\; b)\; mapsto\; Y(a,z)b\; =\; sum\_\{n\; in\; mathbb\{Z\; a\_n\; b\; z^\{-n-1\}$

satisfying the following axioms:

1. (Identity) For any "a" ∈ "V",

   :$Y(1,z)a\; =\; a\; =\; az^0$ and $Y(a,z)1\; in\; a\; +\; zVz$

2. (Translation) "T(1) = 0", and for any "a", "b" ∈ "V",

   :$Y(a,z)Tb\; -\; TY(a,z)b\; =\; frac\{d\}\{dz\}Y(a,z)b$

3. (Four point function) For any "a", "b", "c" ∈ "V", there is an element

   :$X(a,b,c;z,w)\; in\; Vz,w[z^\{-1\},\; w^\{-1\},\; (z-w)^\{-1\}]$

   such that "Y(a,z)Y(b,w)c", "Y(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.

The multiplication map is often written as a state-field correspondence

:$Y:\; V\; o\; (operatorname\{End\},\; V)z^\{pm\; 1\}$

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"_-2"1", so "T" is determined by "Y".

A vertex algebra"V" is Z₊-graded if

:

such that if "a" ∈ V_k and "b" ∈ V_m, then "a"_n "b" ∈ V_k+m-n-1.

A vertex operator algebra is a Z₊-graded vertex algebra equipped with a Virasoro element ω ∈ "V"₂, such that the vertex operator

:$Y(omega,\; z)\; =\; sum\_\{ninmathbb\{Z\; omega\_\{(n)\}\; \{z^\{-n-1\; =\; sum\_\{ninmathbb\{Z\; L\_n\; z^\{-n-2\}$

satisfies for any "a" ∈ "V"_n, the relations:

• $L\_0\; a\; =\; na$
• $Y(L\_\{-1\}\; a,\; z)\; =\; frac\{d\}\{dz\}\; Y(a,\; z)\; =\; [Y(a,z),T]$
• $[L\_m,\; L\_n]\; a\; =\; (m\; -\; n)L\_\{m\; +\; n\}a\; +\; delta\_\{m\; +\; n,\; 0\}\; frac\{m^3-m\}\{12\}ca$

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.

References

* Richard Borcherds, "Vertex algebras, Kac-Moody algebras, and the Monster", "Proc. Natl. Acad. Sci. USA." 83 (1986) 3068-3071
* Igor Frenkel, James Lepowsky, Arne Meurman, "Vertex operator algebras and the Monster". "Pure and Applied Mathematics, 134." Academic Press, Inc., Boston, MA, 1988. liv+508 pp. ISBN 0-12-267065-5
* Victor Kac, "Vertex algebras for beginners". "University Lecture Series, 10." American Mathematical Society, 1998. viii+141 pp. ISBN 0-8218-0634-2
* Edward Frenkel, David Ben-Zvi, "Vertex algebras and Algebraic Curves". "Mathematical Surveys and Monographs, 88." American Mathematical Society, 2001. xii+348 pp. ISBN 0-8218-2894-0