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

TL;DR

The paper investigates whether the tree-level $w_{1+\infty}$ asymptotic symmetry generated by positive-helicity soft gravitons remains exact after quantum corrections. In a toy model of quantum self-dual gravity in Klein space, where all-plus amplitudes are one-loop exact, the authors show the $w_{1+\infty}$ algebra is undeformed, with collinear splitting supporting perturbative exactness. The loop-corrected celestial OPEs reproduce the same $w^p_m$ current algebra, leading to an undeformed celestial commutator $[w^p_m, w^q_n] = [m(q-1) - n(p-1)] w^{p+q-2}_{m+n}$ and preserved $w_{1+\infty}$ structure. This establishes perturbative exactness of soft symmetry towers in a controlled quantum gravity setting, while noting the special nature of the all-plus/self-dual sector.

Abstract

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 $w_{1+\infty}$ 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 $w_{1+\infty}$ algebra persists in quantum self-dual gravity without corrections.