Conflicting symmetries in topologically ordered surface states of three-dimensional bosonic symmetry protected topological phases

We study the Z2 topologically ordered surface state of three-dimensional bosonic SPT phases with the discrete symmetries G1×G2. It has been argued that the topologically ordered state cannot be realized on a purely two-dimensional lattice model. We carefully examine the statement and show that the surface state should break G2 if the symmetry G1 is gauged on the surface. This manifests the conflict of the symmetry G1 and G2 on the surface of the three-dimensional SPT phase. Given that there is no such phenomena in the purely two-dimensional model, it signals that the symmetries are encoded anomalously on the surface of the three-dimensional SPT phases and that the surface state can never be realized on the purely two-dimensional models.