Self-dual gravity with topological terms

Abstract

The canonical analysis of the (anti-) self-dual action for gravity supplemented with the (anti-) self-dual Pontrjagin term is carried out. The effect of the topological term is to add a `magnetic' term to the original momentum variable associated with the self-dual action leaving the Ashtekar connection unmodified. In the new variables, the Gauss constraint retains its form, while both vector and Hamiltonian constraints are modified. This shows, the contribution of the Euler and Pontrjagin terms is not the same as that coming from the term associated with the Barbero-Immirzi parameter, and thus the analogy between the theta-angle in Yang-Mills theory and the Barbero-Immirzi parameter of gravity is not appropriate.