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