Editing Vertex operator algebra (section)

== Additional examples ==
=== 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}}

=== Monster vertex algebra ===
The [[monster vertex algebra]] <math>V^\natural</math> (also called the "moonshine module") is the key to Borcherds's proof of the [[Monstrous moonshine]] conjectures. It was constructed by Frenkel, Lepowsky, and Meurman in 1988. It is notable because its character is the [[j-invariant]] with no constant term, <math>j(\tau) - 744</math>, and its automorphism group is the [[monster group]]. It is constructed by orbifolding the lattice vertex algebra constructed from the [[Leech lattice]] by the order 2 automorphism induced by reflecting the Leech lattice in the origin. That is, one forms the direct sum of the Leech lattice VOA with the twisted module, and takes the fixed points under an induced involution. Frenkel, Lepowsky, and Meurman conjectured in 1988 that <math>V^\natural</math> is the unique holomorphic vertex operator algebra with central charge 24, and partition function <math>j(\tau) - 744</math>. This conjecture is still open.
=== Chiral de Rham complex ===
Malikov, Schechtman, and Vaintrob showed that by a method of localization, one may canonically attach a bcβγ (boson–fermion superfield) system to a smooth complex manifold. This complex of [[sheaf (mathematics)|sheaves]] has a distinguished differential, and the global cohomology is a vertex superalgebra. Ben-Zvi, Heluani, and Szczesny showed that a [[Riemannian metric]] on the manifold induces an ''N''=1 superconformal structure, which is promoted to an ''N''=2 structure if the metric is [[Kähler manifold|Kähler]] and [[Ricci-flat manifold|Ricci-flat]], and a [[hyperkähler manifold|hyperkähler structure]] induces an ''N''=4 structure. Borisov and Libgober showed that one may obtain the two-variable [[elliptic genus]] of a compact complex manifold from the cohomology of the Chiral de Rham complex. If the manifold is [[Calabi–Yau]], then this genus is a weak [[Jacobi form]].{{sfnp|Borisov|Libgober|2000}}

=== Vertex algebra associated to a surface defect === 
A vertex algebra can arise as a subsector of higher dimensional quantum field theory which localizes to a two real-dimensional submanifold of the space on which the higher dimensional theory is defined. A prototypical example is the construction of Beem, Leemos, Liendo, Peelaers, Rastelli, and van Rees which associates a vertex algebra to any 4d ''N''=2 [[Superconformal algebra|superconformal]] field theory. <ref name="Beemetal2015">{{cite journal |last1=Beem |last2=Leemos|last3=Liendo|last4=Peelaers|last5=Rastelli|last6=van Rees|title=Infinite chiral symmetry in four dimensions. |journal=Communications in Mathematical Physics |date=2015 |volume=336 |issue=3 |pages=1359–1433|doi=10.1007/s00220-014-2272-x |arxiv=1312.5344 |bibcode=2015CMaPh.336.1359B |s2cid=253752439 }}</ref> This vertex algebra has the property that its character coincides with the Schur index of the 4d superconformal theory. When the theory admits a weak coupling limit, the vertex algebra has an explicit description as a [[Vertex operator algebra#Additional constructions |BRST reduction]] of a bcβγ system.