Perelomov type coherent states of SO(D + 1) in all dimensional loop quantum gravity

A comprehensive study of the application of SO$(D+1)$ coherent states of Perelomov type to loop quantum gravity in general spacetime dimensions $D+1\geq 3$ is given in this paper. We focus on so-called simple representations of SO$(D+1)$ which solve the simplicity constraint and the associated homogeneous harmonic function spaces. With the harmonic function formulation, we study general properties of the coherent states such as the peakedness properties and the inner product. We also discuss the properties of geometric operators evaluated in the coherent states. In particular, we calculate the expectation value of the volume operator, and the results agree with the ones obtained from the classical label of the coherent states up to error terms which vanish in the limit of large representation labels $N$, i.e. the analogue of the large spin limit in standard $3+1$-dimensional loop quantum gravity.