Editing Vertex operator algebra (section)

== Examples from Lie algebras ==
The basic examples come from infinite-dimensional Lie algebras.
=== Heisenberg vertex operator algebra ===
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]] <math>\mathbb{C}[b_{-1}, b_{-2}, \cdots]</math>, where for positive <math>n</math>, <math>b_{-n}</math> acts obviously by multiplication, and <math>b_n</math> acts as <math>n\partial_{b_{-n}}</math>.  The action of ''b''<sub>0</sub> is multiplication by zero, producing the "momentum zero" Fock representation ''V''<sub>0</sub> of the Heisenberg Lie algebra (generated by ''b''<sub>n</sub> for integers ''n'', with commutation relations [''b''<sub>n</sub>,''b''<sub>m</sub>]=''n'' δ<sub>n,–m</sub>), induced by the trivial representation of the subalgebra spanned by ''b''<sub>n</sub>, n ≥ 0.

The Fock space ''V''<sub>0</sub> can be made into a vertex algebra by the following definition of the state-operator map on a basis <math>b_{j_1}b_{j_2}...b_{j_k}</math> with each <math>j_i < 0</math>,

:<math>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):</math>

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

:<math> Y[f,z] \equiv :f\left(\frac{b(z)}{0!},\frac{b'(z)}{1!},\frac{b''(z)}{2!},...\right): </math>

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''<sub>n</sub>,''c''<sub>m</sub>]=''n'' (b,c) δ<sub>n,–m</sub>.

The Heisenberg vertex operator algebra has a one-parameter family of conformal vectors with parameter <math>\lambda \in \mathbb{C}</math> of conformal vectors <math>\omega_\lambda</math> given by
:<math>\omega_\lambda = \frac{1}{2}b_{-1}^2 + \lambda b_{-2},</math>
with central charge <math>c_\lambda = 1 - 12\lambda^2</math>.<ref>{{cite book |last1=Ben-Zvi |first1=David |last2=Frenkel |first2=Edward |title=Vertex algebras and algebraic curves |date=2004 |location=[Providence, Rhode Island] |isbn=9781470413156 |page=45 |edition=Second}}</ref>

When <math>\lambda = 0</math>, there is the following formula for the Virasoro [[character theory|character]]:

:<math>Tr_V q^{L_0} := \sum_{n \in \mathbf{Z}} \dim V_n q^n = \prod_{n \geq 1} (1-q^n)^{-1}</math>

This is the [[generating function]] for [[partition (number theory)|partitions]], and is also written as ''q''<sup>1/24</sup> 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''<sup>''n''/24</sup> times the weight −''n''/2 modular form η<sup>−''n''</sup>.

=== Virasoro vertex operator algebra<!--'Virasoro constraint', 'Virasoro vertex operator algebra', 'Virasoro vertex operator algebras' redirect here-->===
'''Virasoro vertex operator algebras'''<!--boldface per WP:R#PLA--> 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]∂<sub>z</sub> + ''K'' for which ''K'' acts by ''c''Id, and '''C'''[z]∂<sub>z</sub> acts trivially, and the corresponding induced module is spanned by polynomials in ''L''<sub>–n</sub> = –z<sup>−n–1</sup>∂<sub>z</sub> as ''n'' ranges over integers greater than 1.  The module then has partition function

:<math>Tr_V q^{L_0} = \sum_{n \in \mathbf{R}} \dim V_n q^n = \prod_{n \geq 2} (1-q^n)^{-1}</math>.

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

:<math>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):</math>

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

<math>[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)</math>

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'')<sup>2</sup>/''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 [[two-dimensional critical Ising model|Ising model]], the [[tri-critical Ising model]], the three-state [[Potts model]], etc.  By work of [[Weiqang Wang]]{{sfn|Wang|1993}} concerning [[fusion rule]]s, 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 ''L''<sub>0</sub>-weight 0, 1/2, and 1/16, and its fusion ring is '''Z'''[''x'',''y'']/(''x''<sup>2</sup>–1, ''y''<sup>2</sup>–''x''–1, ''xy''–''y'').

=== Affine vertex algebra ===
By replacing the [[Heisenberg Lie algebra]] with an untwisted [[affine Lie algebra|affine Kac–Moody Lie algebra]] (i.e., the universal [[central extension (mathematics)|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 (physics)|anomaly]] that is interpreted as the central extension.

Concretely, pulling back the central extension

:<math>0 \to \mathbb{C} \to \hat{\mathfrak{g}} \to \mathfrak{g}[t,t^{-1}] \to 0</math>

along the inclusion <math>\mathfrak{g}[t] \to \mathfrak{g}[t,t^{-1}]</math> 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 <math>\mathfrak{g}</math>, 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 group]]s.

By choosing a basis ''J''<sup>a</sup> of the finite type Lie algebra, one may form a basis of the affine Lie algebra using ''J''<sup>a</sup><sub>''n''</sub> = ''J''<sup>a</sup> ''t''<sup>''n''</sup> together with a central element ''K''.  By reconstruction, we can describe the vertex operators by [[normal order]]ed products of derivatives of the fields

:<math>J^a(z) = \sum_{n=-\infty}^\infty J^a_n z^{-n-1} = \sum_{n=-\infty}^\infty (J^a t^n) z^{-n-1}.</math>

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]].{{efn|The history of the Sugawara construction is complicated, with several attempts required to get the formula correct.[https://mathoverflow.net/q/16406]}} For any choice of dual bases ''J''<sup>a</sup>, ''J''<sub>a</sub> with respect to the level 1 inner product, the conformal element is

:<math>\omega = \frac{1}{2(k+h^\vee)} \sum_a J_{a,-1} J^a_{-1} 1</math>

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