Vertex operator algebra

A vertex algebra is a collection of data that satisfy certain axioms.

• a vector space ${\displaystyle V}$, called the space of states. The underlying field is typically taken to be the complex numbers, although Borcherds's original formulation allowed for an arbitrary commutative ring.
• an identity element ${\displaystyle 1\in V}$, sometimes written ${\displaystyle |0\rangle }$ or ${\displaystyle \Omega }$ to indicate a vacuum state.
• an endomorphism ${\displaystyle T:V\rightarrow V}$, called "translation". (Borcherds's original formulation included a system of divided powers of ${\displaystyle T}$, because he did not assume the ground ring was divisible.)
• a linear multiplication map ${\displaystyle Y:V\otimes V\rightarrow V((z))}$, where ${\displaystyle V((z))}$ is the space of all formal Laurent series with coefficients in ${\displaystyle V}$. This structure has some alternative presentations:
  • as an infinite collection of bilinear products ${\displaystyle \cdot _{n}:u\otimes v\mapsto u_{n}v}$ where ${\displaystyle n\in \mathbb {Z} }$ and ${\displaystyle u_{n}\in \mathrm {End} (V)}$, so that for each ${\displaystyle v}$, there is an ${\displaystyle N}$ such that ${\displaystyle u_{n}v=0}$ for ${\displaystyle n<N}$.
  • as a left-multiplication map ${\displaystyle V\rightarrow \mathrm {End} (V)[[z^{\pm 1}]]}$. This is the 'state-to-field' map of the so-called state-field correspondence. For each ${\displaystyle u\in V}$, the endomorphism-valued formal distribution ${\displaystyle Y(u,z)}$ is called a vertex operator or a field, and the coefficient of ${\displaystyle z^{-n-1}}$ is the operator ${\displaystyle u_{n}}$. In the context of vertex algebras, a field is more precisely an element of ${\displaystyle \mathrm {End} (V)[[z^{\pm 1}]]}$, which can be written ${\displaystyle A(z)=\sum _{n\in \mathbb {Z} }A_{n}z^{n},A_{n}\in \mathrm {End} (V)}$ such that for any ${\displaystyle v\in V,A_{n}v=0}$ for sufficiently small ${\displaystyle n}$ (which may depend on ${\displaystyle v}$). The standard notation for the multiplication is

$${\displaystyle u\otimes v\mapsto Y(u,z)v=\sum _{n\in \mathbf {Z} }u_{n}vz^{-n-1}.}$$

These data are required to satisfy the following axioms:

• Identity. For any ${\displaystyle u\in V\,,\,Y(1,z)u=u}$ and ${\displaystyle \,Y(u,z)1\in u+zV[[z]]}$.[lower-alpha 1]
• Translation. ${\displaystyle T(1)=0}$, and for any ${\displaystyle u,v\in V}$,

${\displaystyle [T,Y(u,z)]v=TY(u,z)v-Y(u,z)Tv={\frac {d}{dz}}Y(u,z)v}$

• Locality (Jacobi identity, or Borcherds identity). For any ${\displaystyle u,v\in V}$, there exists a positive integer N such that:

${\displaystyle (z-x)^{N}Y(u,z)Y(v,x)=(z-x)^{N}Y(v,x)Y(u,z).}$

The Locality axiom has several equivalent formulations in the literature, e.g., Frenkel–Lepowsky–Meurman introduced the Jacobi identity:

${\displaystyle \forall u,v,w\in V:\qquad z^{-1}\delta \left({\frac {y-x}{z}}\right)Y(u,x)Y(v,y)w-z^{-1}\delta \left({\frac {-y+x}{z}}\right)Y(v,y)Y(u,x)w=y^{-1}\delta \left({\frac {x+z}{y}}\right)Y(Y(u,z)v,y)w,}$

where we define the formal delta series by:

${\displaystyle \delta \left({\frac {y-x}{z}}\right):=\sum _{s\geq 0,r\in \mathbf {Z} }{\binom {r}{s}}(-1)^{s}y^{r-s}x^{s}z^{-r}.}$

Borcherds[1] initially used the following two identities: for any vectors u, v, and w, and integers m and n we have

${\displaystyle (u_{m}(v))_{n}(w)=\sum _{i\geq 0}(-1)^{i}{\binom {m}{i}}\left(u_{m-i}(v_{n+i}(w))-(-1)^{m}v_{m+n-i}(u_{i}(w))\right)}$

and

${\displaystyle u_{m}v=\sum _{i\geq 0}(-1)^{m+i+1}{\frac {T^{i}}{i!}}v_{m+i}u}$.

He later gave a more expansive version that is equivalent but easier to use: for any vectors u, v, and w, and integers m, n, and q we have

${\displaystyle \sum _{i\in \mathbf {Z} }{\binom {m}{i}}\left(u_{q+i}(v)\right)_{m+n-i}(w)=\sum _{i\in \mathbf {Z} }(-1)^{i}{\binom {q}{i}}\left(u_{m+q-i}\left(v_{n+i}(w)\right)-(-1)^{q}v_{n+q-i}\left(u_{m+i}(w)\right)\right)}$

Finally, there is a formal function version of locality: For any ${\displaystyle u,v,w\in V}$, there is an element

${\displaystyle X(u,v,w;z,x)\in V[[z,x]]\left[z^{-1},x^{-1},(z-x)^{-1}\right]}$

such that ${\displaystyle Y(u,z)Y(v,x)w}$ and ${\displaystyle Y(v,x)Y(u,z)w}$ are the corresponding expansions of ${\displaystyle X(u,v,w;z,x)}$ in ${\displaystyle V((z))((x))}$ and ${\displaystyle V((x))((z))}$.

A vertex operator algebra is a vertex algebra equipped with a conformal element ${\displaystyle \omega \in V}$, such that the vertex operator ${\displaystyle Y(\omega ,z)}$ is the weight two Virasoro field ${\displaystyle L(z)}$:

${\displaystyle Y(\omega ,z)=\sum _{n\in \mathbf {Z} }\omega _{n}{z^{-n-1}}=L(z)=\sum _{n\in \mathbf {Z} }L_{n}z^{-n-2}}$

and satisfies the following properties:

• ${\displaystyle [L_{m},L_{n}]=(m-n)L_{m+n}+{\frac {1}{12}}\delta _{m+n,0}(m^{3}-m)c\,\mathrm {Id} _{V}}$, where ${\displaystyle c}$ is a constant called the central charge, or rank of ${\displaystyle V}$. In particular, the coefficients of this vertex operator endow ${\displaystyle V}$ with an action of the Virasoro algebra with central charge ${\displaystyle c}$.
• ${\displaystyle L_{0}}$ acts semisimply on ${\displaystyle V}$ with integer eigenvalues that are bounded below.
• Under the grading provided by the eigenvalues of ${\displaystyle L_{0}}$, the multiplication on ${\displaystyle V}$ is homogeneous in the sense that if ${\displaystyle u}$ and ${\displaystyle v}$ are homogeneous, then ${\displaystyle u_{n}v}$ is homogeneous of degree ${\displaystyle \mathrm {deg} (u)+\mathrm {deg} (v)-n-1}$.
• The identity ${\displaystyle 1}$ has degree 0, and the conformal element ${\displaystyle \omega }$ has degree 2.
• ${\displaystyle L_{-1}=T}$.

A homomorphism of vertex algebras is a map of the underlying vector spaces that respects the additional identity, translation, and multiplication structure. Homomorphisms of vertex operator algebras have "weak" and "strong" forms, depending on whether they respect conformal vectors.

In vertex algebra theory, due to associativity, we can abuse notation to write, for ${\displaystyle A,B,C\in V,}$

$${\displaystyle Y(A,z)Y(B,w)C=\sum _{n\in \mathbb {Z} }{\frac {Y(A_{(n)}\cdot B,w)}{(z-w)^{n+1}}}C.}$$

This is the operator product expansion. Equivalently,

$${\displaystyle Y(A,z)Y(B,w)=\sum _{n\geq 0}{\frac {Y(A_{(n)}\cdot B,w)}{(z-w)^{n+1}}}+:Y(A,z)Y(B,w):.}$$

Since the normal ordered part is regular in ${\displaystyle z}$ and ${\displaystyle w}$, this can be written more in line with physics conventions as

$${\displaystyle Y(A,z)Y(B,w)\sim \sum _{n\geq 0}{\frac {Y(A_{(n)}\cdot B,w)}{(z-w)^{n+1}}},}$$

where the equivalence relation ${\displaystyle \sim }$ denotes equivalence up to regular terms.

Here some OPEs frequently found in conformal field theory are recorded.[3]

OPEs

1st distribution	2nd distribution	Commutation relations	OPE	Name	Notes
${\displaystyle a(z)=\sum a_{m}z^{-m-1}}$	${\displaystyle b(w)=\sum b_{n}z^{-n-1}}$	${\displaystyle [a_{m},b_{n}]=c_{m+n}}$	${\displaystyle a(z)b(w)\sim {\frac {\sum c_{n}w^{-n-1}}{z-w}}}$	Generic OPE
${\displaystyle a(z)=\sum a_{m}z^{-m-1}}$	${\displaystyle b(w)=\sum b_{n}z^{-n-1}}$	${\displaystyle [a_{m},b_{n}]=m\delta _{m+n,0}}$	${\displaystyle a(z)b(w)\sim {\frac {1}{(z-w)^{2}}}}$	Free boson OPE	Invariance under ${\displaystyle z\leftrightarrow w}$ shows 'bosonic' nature of this OPE.
${\displaystyle L(z)=\sum L_{m}z^{-m-2}}$	${\displaystyle a(w)=\sum a_{n}w^{-n-\Delta }}$	${\displaystyle [L_{m},a_{n}]=((\Delta -1)m-n)a_{m+n}}$	${\displaystyle L(z)a(w)\sim {\frac {\Delta a(w)}{(z-w)^{2}}}+{\frac {\partial a(w)}{z-w}}}$	Primary field OPE	Primary fields are defined to be fields a(z) satisfying this OPE when multiplied with the Virasoro field. These are important as they are the fields which transform 'like tensors' under coordinate transformations of the worldsheet in string theory.
${\displaystyle L(z)=\sum L_{m}z^{-m-2}}$	${\displaystyle L(w)=\sum L_{n}z^{-n-2}}$	${\displaystyle [L_{m},L_{n}]=(m-n)L_{m+n}+{\frac {m^{3}-m}{12}}\delta _{m+n,0}c}$	${\displaystyle L(z)L(w)\sim {\frac {c/2}{(z-w)^{4}}}+{\frac {2L(w)}{(z-w)^{2}}}+{\frac {\partial L(w)}{z-w}}}$	TT OPE	In physics, the Virasoro field is often identified with the stress-energy tensor and labelled T(z) rather than L(z).

A basic example of a noncommutative vertex algebra is the rank 1 free boson, also called the Heisenberg vertex operator algebra. It is "generated" by a single vector b, in the sense that by applying the coefficients of the field b(z) := Y(b,z) to the vector 1, we obtain a spanning set. The underlying vector space is the infinite-variable polynomial ring ${\displaystyle \mathbb {C} [b_{-1},b_{-2},\cdots ]}$, where for positive ${\displaystyle n}$, ${\displaystyle b_{-n}}$ acts obviously by multiplication, and ${\displaystyle b_{n}}$ acts as ${\displaystyle n\partial _{b_{-n}}}$. The action of b₀ is multiplication by zero, producing the "momentum zero" Fock representation V₀ of the Heisenberg Lie algebra (generated by b_n for integers n, with commutation relations [b_n,b_m]=n δ_n,–m), induced by the trivial representation of the subalgebra spanned by b_n, n ≥ 0.

The Fock space V₀ can be made into a vertex algebra by the following definition of the state-operator map on a basis ${\displaystyle b_{j_{1}}b_{j_{2}}...b_{j_{k}}}$ with each ${\displaystyle j_{i}<0}$,

${\displaystyle Y(b_{j_{1}}b_{j_{2}}...b_{j_{k}},z):={\frac {1}{(-j_{1}-1)!(-j_{2}-1)!\cdots (-j_{k}-1)!}}:\partial ^{-j_{1}-1}b(z)\partial ^{-j_{2}-1}b(z)...\partial ^{-j_{k}-1}b(z):}$

where ${\displaystyle$ :{\mathcal {O}}:} denotes normal ordering of an operator ${\displaystyle {\mathcal {O}}}$. The vertex operators may also be written as a functional of a multivariable function f as:

${\displaystyle Y[f,z]\equiv :f\left({\frac {b(z)}{0!}},{\frac {b'(z)}{1!}},{\frac {b''(z)}{2!}},...\right):}$

if we understand that each term in the expansion of f is normal ordered.

The rank n free boson is given by taking an n-fold tensor product of the rank 1 free boson. For any vector b in n-dimensional space, one has a field b(z) whose coefficients are elements of the rank n Heisenberg algebra, whose commutation relations have an extra inner product term: [b_n,c_m]=n (b,c) δ_n,–m.

The Heisenberg vertex operator algebra has a one-parameter family of conformal vectors with parameter ${\displaystyle \lambda \in \mathbb {C} }$ of conformal vectors ${\displaystyle \omega _{\lambda }}$ given by

${\displaystyle \omega _{\lambda }={\frac {1}{2}}b_{-1}^{2}+\lambda b_{-2},}$

with central charge ${\displaystyle c_{\lambda }=1-12\lambda ^{2}}$.[4]

When ${\displaystyle \lambda =0}$, there is the following formula for the Virasoro character:

${\displaystyle Tr_{V}q^{L_{0}}:=\sum _{n\in \mathbf {Z} }\dim V_{n}q^{n}=\prod _{n\geq 1}(1-q^{n})^{-1}}$

This is the generating function for partitions, and is also written as q^1/24 times the weight −1/2 modular form 1/η (the reciprocal of the Dedekind eta function). The rank n free boson then has an n parameter family of Virasoro vectors, and when those parameters are zero, the character is q^n/24 times the weight −n/2 modular form η⁻ⁿ.

Virasoro vertex operator algebras are important for two reasons: First, the conformal element in a vertex operator algebra canonically induces a homomorphism from a Virasoro vertex operator algebra, so they play a universal role in the theory. Second, they are intimately connected to the theory of unitary representations of the Virasoro algebra, and these play a major role in conformal field theory. In particular, the unitary Virasoro minimal models are simple quotients of these vertex algebras, and their tensor products provide a way to combinatorially construct more complicated vertex operator algebras.

The Virasoro vertex operator algebra is defined as an induced representation of the Virasoro algebra: If we choose a central charge c, there is a unique one-dimensional module for the subalgebra C[z]∂_z + K for which K acts by cId, and C[z]∂_z acts trivially, and the corresponding induced module is spanned by polynomials in L_–n = –z^−n–1∂_z as n ranges over integers greater than 1. The module then has partition function

${\displaystyle Tr_{V}q^{L_{0}}=\sum _{n\in \mathbf {R} }\dim V_{n}q^{n}=\prod _{n\geq 2}(1-q^{n})^{-1}}$.

This space has a vertex operator algebra structure, where the vertex operators are defined by:

${\displaystyle Y(L_{-n_{1}-2}L_{-n_{2}-2}...L_{-n_{k}-2}|0\rangle ,z)\equiv {\frac {1}{n_{1}!n_{2}!..n_{k}!}}:\partial ^{n_{1}}L(z)\partial ^{n_{2}}L(z)...\partial ^{n_{k}}L(z):}$

and ${\displaystyle \omega =L_{-2}|0\rangle }$. The fact that the Virasoro field L(z) is local with respect to itself can be deduced from the formula for its self-commutator:

${\displaystyle [L(z),L(x)]=\left({\frac {\partial }{\partial x}}L(x)\right)w^{-1}\delta \left({\frac {z}{x}}\right)-2L(x)x^{-1}{\frac {\partial }{\partial z}}\delta \left({\frac {z}{x}}\right)-{\frac {1}{12}}cx^{-1}\left({\frac {\partial }{\partial z}}\right)^{3}\delta \left({\frac {z}{x}}\right)}$

where c is the central charge.

Given a vertex algebra homomorphism from a Virasoro vertex algebra of central charge c to any other vertex algebra, the vertex operator attached to the image of ω automatically satisfies the Virasoro relations, i.e., the image of ω is a conformal vector. Conversely, any conformal vector in a vertex algebra induces a distinguished vertex algebra homomorphism from some Virasoro vertex operator algebra.

The Virasoro vertex operator algebras are simple, except when c has the form 1–6(p–q)²/pq for coprime integers p,q strictly greater than 1 – this follows from Kac's determinant formula. In these exceptional cases, one has a unique maximal ideal, and the corresponding quotient is called a minimal model. When p = q+1, the vertex algebras are unitary representations of Virasoro, and their modules are known as discrete series representations. They play an important role in conformal field theory in part because they are unusually tractable, and for small p, they correspond to well-known statistical mechanics systems at criticality, e.g., the Ising model, the tri-critical Ising model, the three-state Potts model, etc. By work of Weiqang Wang[5] concerning fusion rules, we have a full description of the tensor categories of unitary minimal models. For example, when c=1/2 (Ising), there are three irreducible modules with lowest L₀-weight 0, 1/2, and 1/16, and its fusion ring is Z[x,y]/(x²–1, y²–x–1, xy–y).

By replacing the Heisenberg Lie algebra with an untwisted affine Kac–Moody Lie algebra (i.e., the universal central extension of the loop algebra on a finite-dimensional simple Lie algebra), one may construct the vacuum representation in much the same way as the free boson vertex algebra is constructed. This algebra arises as the current algebra of the Wess–Zumino–Witten model, which produces the anomaly that is interpreted as the central extension.

Concretely, pulling back the central extension

${\displaystyle 0\to \mathbb {C} \to {\hat {\mathfrak {g}}}\to {\mathfrak {g}}[t,t^{-1}]\to 0}$

along the inclusion ${\displaystyle {\mathfrak {g}}[t]\to {\mathfrak {g}}[t,t^{-1}]}$ yields a split extension, and the vacuum module is induced from the one-dimensional representation of the latter on which a central basis element acts by some chosen constant called the "level". Since central elements can be identified with invariant inner products on the finite type Lie algebra ${\displaystyle {\mathfrak {g}}}$, one typically normalizes the level so that the Killing form has level twice the dual Coxeter number. Equivalently, level one gives the inner product for which the longest root has norm 2. This matches the loop algebra convention, where levels are discretized by third cohomology of simply connected compact Lie groups.

By choosing a basis J^a of the finite type Lie algebra, one may form a basis of the affine Lie algebra using J^a_n = J^a tⁿ together with a central element K. By reconstruction, we can describe the vertex operators by normal ordered products of derivatives of the fields

${\displaystyle J^{a}(z)=\sum _{n=-\infty }^{\infty }J_{n}^{a}z^{-n-1}=\sum _{n=-\infty }^{\infty }(J^{a}t^{n})z^{-n-1}.}$

When the level is non-critical, i.e., the inner product is not minus one half of the Killing form, the vacuum representation has a conformal element, given by the Sugawara construction.[lower-alpha 2] For any choice of dual bases J^a, J_a with respect to the level 1 inner product, the conformal element is

${\displaystyle \omega ={\frac {1}{2(k+h^{\vee })}}\sum _{a}J_{a,-1}J_{-1}^{a}1}$

and yields a vertex operator algebra whose central charge is ${\displaystyle k\cdot \dim {\mathfrak {g}}/(k+h^{\vee })}$. At critical level, the conformal structure is destroyed, since the denominator is zero, but one may produce operators L_n for n ≥ –1 by taking a limit as k approaches criticality.

Given a vertex algebra V with multiplication Y, a V-module is a vector space M equipped with an action Y^M: V ⊗ M → M((z)), satisfying the following conditions:

(Identity) Y^M(1,z) = Id_M

(Associativity, or Jacobi identity) For any u, v ∈ V, w ∈ M, there is an element

${\displaystyle X(u,v,w;z,x)\in M[[z,x]][z^{-1},x^{-1},(z-x)^{-1}]}$

such that Y^M(u,z)Y^M(v,x)w and Y^M(Y(u,z–x)v,x)w are the corresponding expansions of ${\displaystyle X(u,v,w;z,x)}$ in M((z))((x)) and M((x))((z–x)). Equivalently, the following "Jacobi identity" holds:

${\displaystyle z^{-1}\delta \left({\frac {y-x}{z}}\right)Y^{M}(u,x)Y^{M}(v,y)w-z^{-1}\delta \left({\frac {-y+x}{z}}\right)Y^{M}(v,y)Y^{M}(u,x)w=y^{-1}\delta \left({\frac {x+z}{y}}\right)Y^{M}(Y(u,z)v,y)w.}$

The modules of a vertex algebra form an abelian category. When working with vertex operator algebras, the previous definition is sometimes given the name weak ${\displaystyle V}$-module, and genuine V-modules must respect the conformal structure given by the conformal vector ${\displaystyle \omega }$. More precisely, they are required to satisfy the additional condition that L₀ acts semisimply with finite-dimensional eigenspaces and eigenvalues bounded below in each coset of Z. Work of Huang, Lepowsky, Miyamoto, and Zhang has shown at various levels of generality that modules of a vertex operator algebra admit a fusion tensor product operation, and form a braided tensor category.

When the category of V-modules is semisimple with finitely many irreducible objects, the vertex operator algebra V is called rational. Rational vertex operator algebras satisfying an additional finiteness hypothesis (known as Zhu's C₂-cofiniteness condition) are known to be particularly well-behaved, and are called regular. For example, Zhu's 1996 modular invariance theorem asserts that the characters of modules of a regular VOA form a vector-valued representation of ${\displaystyle \mathrm {SL} (2,\mathbb {Z} )}$. In particular, if a VOA is holomorphic, that is, its representation category is equivalent to that of vector spaces, then its partition function is ${\displaystyle \mathrm {SL} (2,\mathbb {Z} )}$-invariant up to a constant. Huang showed that the category of modules of a regular VOA is a modular tensor category, and its fusion rules satisfy the Verlinde formula.

Modules of the Heisenberg algebra can be constructed as Fock spaces ${\displaystyle \pi _{\lambda }}$ for ${\displaystyle \lambda \in \mathbb {C} }$ which are induced representations of the Heisenberg Lie algebra, given by a vacuum vector ${\displaystyle v_{\lambda }}$ satisfying ${\displaystyle b_{n}v_{\lambda }=0}$ for ${\displaystyle n>0}$, ${\displaystyle b_{0}v_{\lambda }=0}$, and being acted on freely by the negative modes ${\displaystyle b_{-n}}$ for ${\displaystyle n>0}$. The space can be written as ${\displaystyle \mathbb {C} [b_{-1},b_{-2},\cdots ]v_{\lambda }}$. Every irreducible, ${\displaystyle \mathbb {Z} }$-graded Heisenberg algebra module with gradation bounded below is of this form.

These are used to construct lattice vertex algebras, which as vector spaces are direct sums of Heisenberg modules, when the image of ${\displaystyle Y}$ is extended appropriately to module elements.

The module category is not semisimple, since one may induce a representation of the abelian Lie algebra where b₀ acts by a nontrivial Jordan block. For the rank n free boson, one has an irreducible module V_λ for each vector λ in complex n-dimensional space. Each vector b ∈ Cⁿ yields the operator b₀, and the Fock space V_λ is distinguished by the property that each such b₀ acts as scalar multiplication by the inner product (b, λ).

Unlike ordinary rings, vertex algebras admit a notion of twisted module attached to an automorphism. For an automorphism σ of order N, the action has the form V ⊗ M → M((z^1/N)), with the following monodromy condition: if u ∈ V satisfies σ u = exp(2πik/N)u, then u_n = 0 unless n satisfies n+k/N ∈ Z (there is some disagreement about signs among specialists). Geometrically, twisted modules can be attached to branch points on an algebraic curve with a ramified Galois cover. In the conformal field theory literature, twisted modules are called twisted sectors, and are intimately connected with string theory on orbifolds.

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 Λ is an even lattice (if the lattice is not even, the structure obtained is instead a vertex superalgebra), the lattice vertex algebra V_Λ decomposes into free bosonic modules as:

${\displaystyle V_{\Lambda }\cong \bigoplus _{\lambda \in \Lambda }V_{\lambda }}$

Lattice vertex algebras are canonically attached to double covers of 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.[1]

The double covers in question are uniquely determined up to isomorphism by the following rule: elements have the form ±e_α for lattice vectors α ∈ Λ (i.e., there is a map to Λ sending e_α to α that forgets signs), and multiplication satisfies the relations e_αe_β = (–1)^(α,β)e_βe_α. Another way to describe this is that given an even lattice Λ, there is a unique (up to coboundary) normalised cocycle ε(α, β) with values ±1 such that (−1)^(α,β) = ε(α, β) ε(β, α), where the normalization condition is that ε(α, 0) = ε(0, α) = 1 for all α ∈ Λ. This cocycle induces a central extension of Λ by a group of order 2, and we obtain a twisted group ring C_ε[Λ] with basis e_α (α ∈ Λ), and multiplication rule e_αe_β = ε(α, β)e_α+β – the cocycle condition on ε ensures associativity of the ring.[6]

The vertex operator attached to lowest weight vector v_λ in the Fock space V_λ is

${\displaystyle Y(v_{\lambda },z)=e_{\lambda }:\exp \int \lambda (z):=e_{\lambda }z^{\lambda }\exp \left(\sum _{n<0}\lambda _{n}{\frac {z^{-n}}{n}}\right)\exp \left(\sum _{n>0}\lambda _{n}{\frac {z^{-n}}{n}}\right),}$

where z^λ is a shorthand for the linear map that takes any element of the α-Fock space V_α to the monomial z^(λ,α). 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 Λ ⊗ C, but the condition that the extra Fock spaces have integer L₀ eigenvalues constrains the choice of s: for an orthonormal basis x_i, the vector 1/2 x_i,1² + s₂ must satisfy (s, λ) ∈ Z for all λ ∈ Λ, i.e., s lies in the dual lattice.

If the even lattice 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 Frenkel–Kac–Segal) construction, and is based on the earlier construction by Sergio Fubini and Gabriele Veneziano of the 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 E₈.[6][7]

The monster vertex algebra ${\displaystyle V^{\natural }}$ (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, ${\displaystyle j(\tau )-744}$, 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 ${\displaystyle V^{\natural }}$ is the unique holomorphic vertex operator algebra with central charge 24, and partition function ${\displaystyle j(\tau )-744}$. This conjecture is still open.

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 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 and Ricci-flat, and a 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.[8]

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 field theory. [9] 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 BRST reduction of a bcβγ system.