Perturbatively Exact $w_{1+\mathrm{\infty }}$ Asymptotic Symmetry of Quantum Self-Dual Gravity

The infinite tower of positive-helicity soft gravitons in any minimally coupled, tree-level, asymptotically flat four-dimensional (4D) gravity was recently shown to generate a $w_{1+\mathrm{\infty }}$ asymptotic symmetry algebra. It is natural to ask whether this classical algebra acquires quantum corrections at loop level. We explore this in quantum self-dual gravity, whose amplitudes acquire known one-loop exact all-plus helicity quantum corrections. We show using collinear splitting formulae that, remarkably, the $w_{1+\mathrm{\infty }}$ algebra persists in quantum self-dual gravity without corrections.