Editing Current algebra (section)

==Conformal field theory==
For the case where space is a one-dimensional circle, current algebras arise naturally as a [[Lie algebra extension#Central|central extension]] of the [[loop algebra]], known as [[Kac–Moody algebra]]s or, more specifically, [[affine Lie algebra]]s. In this case, the commutator and normal ordering can be given a very precise mathematical definition in terms of integration contours on the complex plane, thus avoiding some of the formal divergence difficulties commonly encountered in quantum field theory.

When the [[Killing form]] of the Lie algebra is contracted with the current commutator, one obtains the [[energy–momentum tensor]] of a [[two-dimensional conformal field theory]].  When this tensor is expanded as a [[Laurent series]], the resulting algebra is called the [[Virasoro algebra]].<ref>{{citation|first=Jurgen|last= Fuchs|title=Affine Lie Algebras and Quantum Groups|year=1992|publisher=Cambridge University Press|isbn=0-521-48412-X}}</ref>  This calculation is known as the [[Sugawara construction]].

The general case is formalized as the [[vertex operator algebra]].