  Using the covariant phase space formalism, we construct the phase space for
non-Abelian gauge theories in $(d+2)$-dimensional Minkowski spacetime for any
$d \geq 2$, including the edge modes that symplectically pair to the low energy
degrees of freedom of the gauge field. Despite the fact that the symplectic
form in odd and even-dimensional spacetimes appear ostensibly different, we
demonstrate that both cases can be treated in a unified manner by utilizing the
shadow transform. Upon quantization, we recover the algebra of the vacuum
sector of the Hilbert space and derive a Ward identity that implies the leading
soft gluon theorem in $(d+2)$-dimensional spacetime.
