One common idea behind many extensions to the Standard Model (such as String Theory or Kaluza-Klein Theory) are small or hidden "Extra-Dimensions", that are compactified.
According to my understanding of Quantum Physics, this would result in each particle's wavefunction having a component into the direction of these extra-dimensions, and only discrete energy-states would be allowed (similar to electrons in an atom).
Now imagine a photon, which is considered to be a particle without rest mass in the Standard Model. Its wavefunction would also have components into the direction of the extra-dimensions. Consequently, it would have to occupy one of these energy states. So there would be some energy consisting out of the photon's standing wave in the extra dimension, which - according to my understanding - would behave just like a finite rest mass of the photon.
So how can there be particles without any invariant mass in a theory with compact extra dimensions?