Coherent intertwiner solution of simplicity constraint in all dimensional loop quantum gravity

We propose a new treatment of the quantum simplicity constraints appearing in the general ${SO(D+1)}$ formulation of loop quantum gravity for the ${(1+D)}$-dimensional space-time. Instead of strongly imposing the constraints, we construct a specific form of weak solutions by employing the spin net-work states with specific ${SO(D+1)}$ coherent intertwiners. These states weakly satisfy the quantum simplicity constraint via the vanishing expectation values, and the quantum Gaussian constraints can be imposed strongly. Remarkably, those specific ${SO(D+1)}$ coherent intertwiners used to construct our solutions have natural interpretations of the $D$-dimensional polytopes, commonly viewed as basic units of the discrete spatial geometry. Therefore, while the strong imposition of the quantum simplicity constraints leads to an over-constrained solution space, our weak solution space for the constraints may contain the correct semiclassical degrees of freedom for intrinsic geometry of the spatial hypersurfaces. Moreover, some concrete relations are established between our construction and other existing approaches in solving the simplicity constraints in all dimensional loop quantum gravity, providing valuable insights into this unresolved important issue.