  In contrast to the Einstein-Hilbert action, the action for self-dual gravity
contains vierbeins. They are eleminated at the level of observables by an
$SL(2,\mathbb{C})$ gauge condition implied by the action. We argue that despite
this condition, new "edge" or superselected state vectors corresponding to maps
of the spheres $S^2_{\infty}$ at infinity to $SL(2, \mathbb{C})$ arise. They
are characterised by new quantum numbers and they lead to mixed states. For
black holes, they arise both at the horizon and the spatial infinity and may be
relevant for the black hole information paradox. Similar comments can be made
about the Einstein-Palatini action which uses vierbeins.

High Energy Physics - Theory;General Relativity and Quantum Cosmology