Gauge invariant effective Lagrangian for Kaluza-Klein modes

We construct a manifestly gauge invariant Lagrangian in $3+1$ dimensions for N Kaluza-Klein modes of an $\mathrm{SU}(m)$ gauge theory in the bulk. For example, if the bulk is $4+1,$ the effective theory is ${\ensuremath{\Pi}}_{i=1}^{N+1}{\mathrm{SU}(m)}_{i}$ with N chiral $(\overline{m},m)$ fields connecting the groups sequentially. This can be viewed as a Wilson action for a transverse lattice in ${x}^{5},$ and is shown explicitly to match the continuum $4+1$ compactified Lagrangian truncated in momentum space. Scale dependence of the gauge couplings is described by the standard renormalization group technique with threshold matching, leading to effective power law running. We also discuss the unitarity constraints, and chiral fermions.