  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 super-selection 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 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.
