All $S$ invariant gluon OPEs on the celestial sphere

$S$ algebra is an infinite dimensional Lie algebra which is known to be the symmetry algebra of some gauge theories. It is a “coloured version” of the $w_{1+\mathrm{\infty }}$. In this paper we write down all possible $S$ invariant (celestial) OPEs between two positive helicity outgoing gluons and also find the Knizhnik-Zamolodchikov type null states for these theories. Our analysis hints at the existence of an infinite number of $S$ invariant gauge theories which include the (tree-level) MHV-sector and the self-dual Yang-Mills theory.