Single Extra Dimension from $\kappa$-Poincar\'e and Gauge Invariance

We show that $\kappa$-Poincar\'e invariant gauge theories on $\kappa$-Minkowski space with physically acceptable commutative (low energy) limit must be 5-d. The gauge invariance requirement of the action fixes the dimension of the $\kappa$-Minkowski space to $d=5$ and selects the unique twisted differential calculus with which the construction can be achieved. We characterize a BRST symmetry related to the 5-d noncommutative gauge invariance though the definition of a nilpotent operation, which is used to construct a gauge-fixed action. We also consider standard scenarios assuming (compactification of) flat extra dimension, for which the 5-d deformation parameter $\kappa$ can be viewed as the bulk 5-d Planck mass. We study physical properties of the resulting 4-d effective theories. Recent data from collider experiments require $\kappa\gtrsim\mathcal{O}(10^{13})\ \text{GeV}$. The use of standard test of in-vacuo dispersion relations of Gamma Ray Burst photons increases this lower bound by 4 orders of magnitude. The robustness of this bound is discussed in the light of possible new features of noncommutative causal structures.