Advances in Theoretical and Mathematical Physics

Volume 25 (2021)

Number 1

Higher Kac–Moody algebras and symmetries of holomorphic field theories

Pages: 129 – 239



Owen Gwilliam (Department of Mathematics and Statistics, University of Massachusetts, Amherst, Mass., U.S.A.)

Brian R. Williams (School of Mathematics, University of Edinburgh, United Kingdom)


We introduce a higher dimensional generalization of the affine Kac–Moody algebra using the language of factorization algebras. In particular, on any complex manifold there is a factorization algebra of “currents” associated to any Lie algebra. We classify local cocycles of these current algebras, and compare them to central extensions of higher affine algebras recently proposed by Faonte–Hennion–Kapranov. A central goal of this paper is to witness higher Kac–Moody algebras as symmetries of a class of holomorphic quantum field theories. In particular, we prove a generalization of the free field realization of an affine Kac–Moody algebra and also develop the theory of $q$-characters for this class of algebras in terms of factorization homology. Finally, we exhibit the “large N” behavior of higher Kac–Moody algebras and their relationship to symmetries of non-commutative field theories.

Published 28 September 2021