Holonomy operator for spin connection and spatial scalar curvature operator in loop quantum gravity

Abstract: In this article we propose a new construction of the spatial scalar curvature operator in (1+3)-dimensional LQG based on the twisted geometry. The starting point of the construction is to express the holonomy of the spin connection on a graph in terms of the twisted geometry variables, and we check that this expression reproduces the spin connection in terms of triads in a certain continuum limit. The spatial scalar curvature in terms of twisted geometry is obtained by considering the composition of the holonomy of the spin connection on the loops. With the twisted geometry parametrization of the holonomy-flux phase space, we further express the holonomy of the spin connection and the spatial scalar curvature on a graph in terms of fluxes. Finally, they are promoted as well-defined operators by replacing the fluxes with ordered flux operators.