Shadows and soft exchange in celestial CFT

We study exponentiated soft exchange in $d+2$ dimensional gauge and gravitational theories using the celestial CFT formalism. These models exhibit spontaneously broken asymptotic symmetries generated by transformations with non-compact support, effective dynamics of associated Goldstone "edge" mode is expected to be $d$-dimensional. The introduction an infrared regulator also explicitly breaks these so edge regulated theory really a $d$-dimensional pseudo-Goldstone boson. Symmetry considerations determine leading terms action, whose coefficients are controlled cutoff. Computations this model reproduce abelian divergences $d=2$, capture re-summed (infrared finite) higher dimensions. reproduces theorems all Interestingly, we find that it shadow transform has local dynamics: action expressed non-local for $d>2$. introduce discuss new magnetic theorems. Our analysis demonstrates symmetry principles suffice calculate gravity.