Vertex operator algebra

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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.

[edit] Formal definition

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

$Y: V \otimes V \to V((z))$ written $(a, b) \mapsto Y(a,z)b = \sum_{n \in \mathbb{Z}} a_n b z^{-n-1}$

satisfying the following axioms:

(Identity) For any a ∈ V,

$Y (1, z) a = a = a z 0$ and $Y(a,z)1 \in a + zV[[z]]$
(Translation) T(1) = 0, and for any a, b ∈ V,

$Y(a,z)Tb - TY(a,z)b = \frac{d}{dz}Y(a,z)b$
(Four point function) For any a, b, c ∈ V, there is an element

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

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

$Y: V \to (End V)[[z^{\pm 1}]]$

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_-21, so T is determined by Y.

A vertex algebraV is Z₊-graded if

$V = \bigoplus_{n=0}^\infty V_n\,$

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 ∈ V₂, such that the vertex operator

$Y(\omega, z) = \sum_{n\in\mathbb{Z}} \omega_{(n)} {z^{-n-1}} = \sum_{n\in\mathbb{Z}} L_n z^{-n-2}$

satisfies for any a ∈ V_n, the relations:

$L 0 a = n a$
$Y(L_{-1} a, z) = \frac{d}{dz} Y(a, z) = [Y(a,z),T]$
$[L_m, L_n]a = (m - n)L_{m + n}a + \delta_{m + n, 0} \frac{m^3-m}{12}ca$

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.

[edit] 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
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

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Categories: Nonassociative algebra | Lie algebras | Conformal field theory