Theta ambiguity, maximal symmetry, and the isolated horizon boundary condition

In a paper from last week, I discussed the relation between the three above concepts, which at first may seem unrelated. The bottom line is that there exists a 1-parameter family of vacua for LQG which seem to be largely unknown. The vacua are induced by the standard Ashtekar-Lewandowski functional, however the underlying holonomy-flux-algebra is based on connection variables where the densitised triad is shifted as $P^{ai} = E^{ai} + \theta \epsilon^{abc} F_{bc}^i$. This actually corresponds to a rigorous implementation of the so called Kodama state in the context of real variables. Since all fluxes annihilate the Ashtekar-Lewandowski vacuum, the condition $P^{ai}=0$ is implemented by the vacuum and equivalent to
\[q_{c[a} q_{b]d} = \beta \theta F_{abcd} = \beta \theta R^{(3)}_{ab cd} - 2 \beta^2 \theta \epsilon_{cde} \sqrt{q} \nabla_{[a} K_{b]} {}^e - 2 \beta^3 \theta K_{c[a} K_{b]d} \]
with $\beta$ being the Barbero-Immirzi parameter. In case of a vanishing extrinsic curvature $K_{ab}$, this condition is nothing else than maximal symmetry for the spatial geometry. A vacuum based on variables with $\theta \neq 0$ is thus very interesting from the point of view of constructing semi-classical states, since it is peaked on a non-degenerate homogeneous geometry. $0 < \theta < \infty$ thus seems to interpolate between the Ashtekar-Lewandowski vacuum ($\theta=0$) and the recently introduced Dittrich-Geiller vacuum, which implements $F = 0$.

People familiar with isolated horizons and their application in LQG will also notice the strong similarity of the condition $P^{ai}=0$ with the isolated horizon boundary condition which is imposed in the black hole entropy computation in LQG (a pullback of $P^{ai} = 0$ to a 2-surface with $\theta$ depending on the total black hole area). It was shown before that computations along the original lines are actually applicable to general boundaries. Here, we also see that trying to impose the isolated horizon boundary condition directly on the full LQG Hilbert space is only enforcing a part of maximal symmetry condition, as opposed to selecting horizons (in contrast to what is often suggested). Thus, also less mainstream works along these lines, while very interesting, do not provide suitable definitions of quantum horizons, but rather quantum symmetric surfaces.