Real forms of extended Kac–Moody symmetries and higher spin gauge theories

Abstract

We consider the relation between higher spin gauge fields and real Kac–Moody Lie algebras. These algebras are obtained by double and triple extensions of real forms $${\mathfrak{g}_0}$$ of the finite-dimensional simple algebras $${\mathfrak{g}}$$ arising in dimensional reductions of gravity and supergravity theories. Besides providing an exhaustive list of all such algebras, together with their associated involutions and restricted root diagrams, we are able to prove general properties of their spectrum of generators with respect to a decomposition of the triple extension of $${\mathfrak{g}_0}$$ under its gravity subalgebra $${\mathfrak{gl}(D,\mathbb {R})}$$ . These results are then combined with known consistent models of higher spin gauge theory to prove that all but finitely many generators correspond to non-propagating fields and there are no higher spin fields contained in the Kac–Moody algebra.