Editing Vertex operator algebra (section)

=== Vertex operator algebra defined by an even lattice ===
The lattice vertex algebra construction was the original motivation for defining vertex algebras. It is constructed by taking a sum of irreducible modules for the Heisenberg algebra corresponding to lattice vectors, and defining a multiplication operation by specifying intertwining operators between them. That is, if {{math|Λ}} is an even lattice (if the lattice is not even, the structure obtained is instead a vertex superalgebra), the lattice vertex algebra {{math|''V''<sub>Λ</sub>}} decomposes into free bosonic modules as:

:<math>V_\Lambda \cong \bigoplus_{\lambda \in \Lambda} V_\lambda</math>

Lattice vertex algebras are canonically attached to double covers of [[unimodular lattice|even integral lattices]], rather than the lattices themselves. While each such lattice has a unique lattice vertex algebra up to isomorphism, the vertex algebra construction is not functorial, because lattice automorphisms have an ambiguity in lifting.{{sfn|Borcherds|1986}}

The double covers in question are uniquely determined up to isomorphism by the following rule: elements have the form {{mvar|±e<sub>α</sub>}} for lattice vectors {{math|''α'' ∈ Λ}} (i.e., there is a map to {{math|Λ}} sending {{mvar|e<sub>α</sub>}} to α that forgets signs), and multiplication satisfies the relations ''e''<sub>α</sub>''e''<sub>β</sub> = (–1)<sup>(α,β)</sup>''e''<sub>β</sub>''e''<sub>α</sub>. Another way to describe this is that given an even lattice {{math|Λ}}, there is a unique (up to coboundary) normalised [[Group cohomology|cocycle]] {{math|''ε''(''α'', ''β'')}} with values {{math|±1}} such that {{math|(−1)<sup>(''α'',''β'')</sup> {{=}} ''ε''(''α'', ''β'') ''ε''(''β'', ''α'')}}, where the normalization condition is that ε(α, 0) = ε(0, α) = 1 for all {{math|''α'' ∈ Λ}}. This cocycle induces a central extension of {{math|Λ}} by a group of order 2, and we obtain a twisted group ring {{math|'''C'''<sub>''ε''</sub>[Λ]}} with basis {{math|''e<sub>α</sub>'' (''α'' ∈ Λ)}}, and multiplication rule {{math|''e<sub>α</sub>e<sub>β</sub>'' {{=}} ''ε''(''α'', ''β'')''e''<sub>''α''+''β''</sub>}} – the cocycle condition on {{mvar|ε}} ensures associativity of the ring.{{sfn|Kac|1998}}

The vertex operator attached to lowest weight vector {{mvar|v<sub>λ</sub>}} in the Fock space {{mvar|V<sub>λ</sub>}} is

:<math>Y(v_\lambda,z) = e_\lambda :\exp \int \lambda(z): = e_\lambda z^\lambda \exp \left (\sum_{n<0} \lambda_n \frac{z^{-n}}{n} \right )\exp \left (\sum_{n>0} \lambda_n \frac{z^{-n}}{n} \right ),</math>

where {{mvar|z<sup>λ</sup>}} is a shorthand for the linear map that takes any element of the α-Fock space {{mvar|V<sub>α</sub>}} to the monomial {{math|''z''<sup>(''λ'',''α'')</sup>}}. The vertex operators for other elements of the Fock space are then determined by reconstruction.

As in the case of the free boson, one has a choice of conformal vector, given by an element ''s'' of the vector space {{math|Λ ⊗ '''C'''}}, but the condition that the extra Fock spaces have integer ''L''<sub>0</sub> eigenvalues constrains the choice of ''s'': for an orthonormal basis {{mvar|x<sub>i</sub>}}, the vector 1/2 ''x''<sub>i,1</sub><sup>2</sup> + ''s''<sub>2</sub> must satisfy {{math|(''s'', ''λ'') ∈ '''Z'''}} for all λ ∈ Λ, i.e., ''s'' lies in the dual lattice.

If the even lattice {{math|Λ}} is generated by its "root vectors" (those satisfying (α, α)=2), and any two root vectors are joined by a chain of root vectors with consecutive inner products non-zero then the vertex operator algebra is the unique simple quotient of the vacuum module of the affine Kac–Moody algebra of the corresponding simply laced simple Lie algebra at level one.  This is known as the Frenkel–Kac (or [[Igor Frenkel|Frenkel]]–[[Victor Kac|Kac]]–[[Graeme Segal|Segal]]) construction, and is based on the earlier construction by [[Sergio Fubini]] and [[Gabriele Veneziano]] of the [[tachyon|tachyonic vertex operator]] in the [[dual resonance model]]. Among other features, the zero modes of the vertex operators corresponding to root vectors give a construction of the underlying simple Lie algebra, related to a presentation originally due to [[Jacques Tits]].  In particular, one obtains a construction of all ADE type Lie groups directly from their root lattices. And this is commonly considered the simplest way to construct the 248-dimensional group ''E''<sub>8</sub>.{{sfn|Kac|1998}}{{sfn|Frenkel|Lepowsky|Meurman|1988}}