It kind of irks me that they basically assume 3+1- space-time.
Ehrenfest and Weyl showed this as early as 1920 and 1922, respectively.
3+1 is also baked into the Minkowski metric by the latter's definition, which leads to the Poincaré group being the isometry group on local patches of spacetime; the Poincaré group is a subgroup of the Standard Model, and is extremely well tested in controlled, laboratory settings (and supported by an abundance of observational evidence too).
Extra dimensions of spacetime have to be small compared to the Planck length in order not to be obvious today. For all practical purposes, if such tiny dimensions exist, we can safely omit them from effective theoretical descriptions of everyday physics, just like we can ignore things like very large extra dimensions where the smallest step one can take is much bigger than the Hubble diameter.