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We comprehensively investigate the existence of gauge transformationsconnecting distinct boundary conditions (BCs) in $SU(N)$ gauge theory on the$S^1/Z_2$ orbifold embedded in five-dimensional Minkowski space-time. We provethat the global transformations make a set of the BCs invariant and there is anew type of BCs-changing local transformations. It is shown that the localtransformations behave differently on the UV-brane from the bulk and IR-brane.We find that possible equivalence classes are $[p,q,r,s]\sim [p\pm 1, q\mp 1,r\mp 1, s\pm1]$ on the UV-brane. However, on the bulk and IR-brane, all sets ofBCs are connected by local transformations with a kink near the fixed points.

Comprehensive Analysis of Equivalence Classes in 5D $SU(N)$ gauge theory on $S^1/Z_2$ Orbifold