Discrete $Z_4$ symmetry in quantum gravity

We consider the discrete $Z_4$ symmetry ${\hat {\bf i}}$, which takes place in the scenario of quantum gravity where the gravitational tetrads emerge as the order parameter. Under this symmetry operation $\hat {\bf i}$, the emerging tetrads are multiplied by the imaginary unit, $\hat {\bf i}\,e^a_\mu = - i e^a_\mu$. 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 $^3$He-B. The order parameter $^3$He-B is the analogue of the tetrad field, but it has complex values. The $\hat {\bf i}$-symmetry operation changes the phase of the complex order parameter by $\pi/2$, which corresponds to the $Z_4$ discrete symmetry in quantum gravity. We also considered the alternative scenario of the breaking of this $Z_4$ symmetry, in which the $\hat {\bf i}$-operation changes sign of the scalar curvature, $\hat {\bf i} \,{\cal R} = - {\cal R}$, and thus the Einstein-Hilbert action violates the $\hat {\bf i}$-symmetry. In the alternative scenario of symmetry breaking, the gravitational coupling $K=1/16\pi G$ plays the role of the order parameter, which changes sign under $\hat {\bf i}$-transformation.