$\theta$ parameter in loop quantum gravity: Effects on quantum geometry and black hole entropy

The precise analog of the $\theta$-quantization ambiguity of Yang-Mills theory exists for the real $SU(2)$ connection formulation of general relativity. As in the former case $\theta$ labels representations of large gauge transformations, which are superselection sectors in loop quantum gravity. We show that unless $\theta =0$, the (kinematical) geometric operators such as area and volume are not well defined on spin network states. More precisely the intersection of their domain with the dense set $Cyl$ in the kinematical Hilbert space $\mathcal{H}$ of loop quantum gravity is empty. The absence of a well-defined notion of area operator acting on spin network states seems at first in conflict with the expected finite black hole entropy. However, we show that the black hole (isolated) horizon area—which in contrast to kinematical area is a (Dirac) physical observable—is indeed well defined, and quantized so that the black hole entropy is proportional to the area. The effect of $\theta$ is negligible in the semiclassical limit where proportionality to area holds.