Some notes on the Kodama state, maximal symmetry, and the isolated horizon boundary condition

We recall some well and some less known results about the Kodama state and the related $\theta$ ambiguity in defining canonical variables. Based on them, we make some comments highlighting that the Kodama state for real connection variables can be given a precise meaning and that it implements a vacuum peaked on a (in a suitable sense) maximally symmetric geometry. We also highlight the similarity of this construction with the isolated horizon boundary condition $F \propto \Sigma$ and stress that it is, in agreement with earlier work, inadequate to define the notion of a quantum horizon.