Properties of the Abelian Projection Fields in $SU(N)$ Lattice Gluodynamics

‘t~Hooft’s abelian projection of $SU(N)$ gauge theory yields $N$ mutually constrained, compact abelian fields which are permutationally equivalent. We formulate the notion of ``species permutation’’ symmetry of the $N$ abelian projection fields and discuss its consequences for cross-species correlators. We show that at large $N$ cross-species interactions are $\frac{1}{N}$ suppressed relative to same-species interactions. Numerical simulations at $N=3$ support our symmetry arguments and reveal the existence of inter-species interactions of size $\mathcal{O}\mathcal{\text{/}}\left(\frac{1}{N-1}\right)$ as analytically predicted.