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We consider classical, pure Yang-Mills theory in a box. We show how a set of
static electric fields that solve the theory in an adiabatic limit correspond to geodesic motion
on the space of vacua, equipped with a particular Riemannian metric that we identify. The
vacua are generated by spontaneously broken global gauge symmetries, leading to an infinite
number of conserved momenta of the geodesic motion. We show these correspond to the
soft multipole charges of Yang-Mills theory.
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