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 along the inclusion 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 , 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 Ja of the finite type Lie algebra, one may form a basis of the affine Lie algebra using Jan = Ja tn together with a central element K. By reconstruction, we can describe the vertex operators by normal ordered products of derivatives of the fields 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. For any choice of dual bases Ja, Ja with respect to the level 1 inner product, the conformal element is and yields a vertex operator algebra whose central charge is . At critical level, the conformal structure is destroyed, since the denominator is zero, but one may produce operators Ln for n ≥ –1 by taking a limit as k approaches criticality. This construction can be altered to work for the rank 1 free boson. In fact, the Virasoro vectors form a one-parameter family ωs = 1/2 x12 + s x2, endowing the resulting vertex operator algebras with central charge 1−12s2. When s = 0, we have the following formula for the graded dimension: This is known as the generating function for partitions, and is also written as q1/24 times the weight −1/2 modular form 1/η (the Dedekind eta function). The rank n free boson