Editing Vertex operator algebra (section)

=== 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.