Discrete Z₄ symmetry in quantum gravity

We consider the discrete Z₄ symmetry î, which takes place in the scenario of quantum gravity where the gravitational tetrads emerge as the order parameter. Under this symmetry operation î, the emerging tetrads are multiplied by the imaginary unit, î\,eᵃ_μ = - i eᵃ_μ. The existence of such symmetry and the spontaneous breaking of this symmetry are also supported by the consideration of the symmetry breaking scheme in the topological superfluid ³He-B. The order parameter ³He-B is the analogue of the tetrad field, but it has complex values. The î-symmetry operation changes the phase of the complex order parameter by π/2, which corresponds to the Z₄ discrete symmetry in quantum gravity. We also considered the alternative scenario of the breaking of this Z₄ symmetry, in which the î-operation changes sign of the scalar curvature, î \, R = - R, and thus the Einstein-Hilbert action violates the î-symmetry. In the alternative scenario of symmetry breaking, the gravitational coupling K=1/16π G plays the role of the order parameter.