Dynamical Covariant Quantum Spacetime with Fuzzy Extra Dimensions in the IKKT model

We consider general $k=-1$ FLRW covariant quantum spacetimes $\mathcal{M}^{3,1} \times \mathcal{K}$ with fuzzy extra dimensions $\mathcal{K}$ as classical solutions of the IKKT matrix model. The coupled equations of motion are recast in terms of conservation laws, which allow to determine the evolution of spacetime in a transparent way. We show that $\mathcal{K}$ is stabilized as a classical solution in the presence of a large $R$ charge, corresponding to internal angular momentum. This provides a mechanism to maintain a large hierarchy between UV and IR scales. We also argue that the evolution of spacetime is determined by a balance between classical and quantum effects, leading to a cosmic scale factor $a(t) \sim t$ and constant dilaton at late times. On such a background, the undeformed IKKT model leads to a higher-spin gauge theory including gravity.