Gravity, Superselection Rules and Axions

It is generally accepted that consistent coupling to quantum gravity implies that any low energy free parameter is in reality the vacuum expectation value of some local quantum field. In this note we present a modest attempt to prove this general claim using as key example the theta parameter of QCD and the Peccei-Quinn axion solution of the strong CP problem. The gravitational emergence of the local axion field associated with the superselection theta parameter is worked out using as basic ingredients the quantum representation of classical geometries in terms of coherent states and the interpretation of the theta QCD superselection rule in quantum information terms as the lack of a quantum reference frame for topological charge. The formal connection with the model independent axion in string theory is briefly discussed.