# Kac-Moody algebra The Kac-Moody algebra at level $k$ is given by$\left[J_{m}^{a}, J_{n}^{b}\right]=i f^{a b}{}_c\, J_{m+n}^{c}+k\, m \,\delta^{a b} \delta_{m+n, 0}.$Notice that Kac-Moody always contains a copy of [[0032 Virasoro algebra|Virasoro]]. Here $k$ is called the level, an the second term is called the central extension. The Kac-Moody algebra is an important structure in the the study of [[0481 Conformal field theory|CFT]]s. For example, it forms the chiral algebra of [[0601 Weiss-Zumino-Witten models|WZW]] models. However, it is not the symmetry algebra of the CFT: $J^a_ms do not commute with the Hamiltonian except for the $m=0$ modes, which are the conserved charges. The $m=0$ modes form a symmetry algebra whereas the affine Kac-Moody algebra is sometimes called the spectrum-generating algebra. ## Refs - reviews - [[1986#Goddard, Olive (Review)]]: physics oriented - celestial - [[2015#He, Mitra, Strominger]] - [[2016#McLoughlin, Nandan]] ## Applications - [[2015#He, Mitra, Strominger]] suggests that 4D [[0071 Yang-Mills|YM]] has a Kac-Moody algebra (with level 0) as the [[0060 Asymptotic symmetry|asymptotic symmetry]] algebra