Geometry of loop quantum gravity on a graph

We discuss the meaning of geometrical constructions associated to loop quantum gravity states on a graph. In particular, we discuss the “twisted geometries” and derive a simple relation between these and Regge geometries.

Figure 1

A data set and various interpolating functions: polynomial (cf. mode expansion in cosmology), piecewise linear (cf. Regge geometries), and piecewise-flat (discontinuous, cf. twisted geometries for generic holonomy-fluxes).