The spectrum consistency of fractional quantum Hall effect model

The spectrum consistency of the three fermion family $U(1)\times U(1)$ quantum electrodynamics in three space-time dimensions [Phys. Rev. Lett. 124 (2020) 131601] is analyzed. It has been verified that the originally proposed action violates the unitarity consistency condition in the gauge field sector by exhibiting negative norm states in the spectrum. However, the cure came through the fundamental gauge fields properly settled, consequently, the action rewritten in terms of those fundamental fields shows to be free from any spurious degrees of freedom, and the model now becomes safe for further quantization.