Editing Vertex operator algebra (section)

=== 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'').