In the deconfined quantum criticality literatures like https://doi.org/10.1103/PhysRevX.9.041037, some equations are usually given: $$SO(3)\times Z_{4}\to SO(5),\\U(1)\times Z_{4}\to O(4),\\SO(3)\times Z_2 \to O(4),$$ but how to derive such relations?
It seems like the symmetries in the left hand side of above equations represent the symmetries of the phases in both sides of the transition points, such as the first line describe Neel-VBS transition, the order parameter of Neel phase has $SO(3)$ symmetry, and the VBS has $Z_4$ symmetry.
Question: Could anyone tell me how to derive above equations, and for example $U(1)\times Z_2$ ?
