Self-dual gravity with topological terms

Abstract

The canonical analysis of the (anti-)self-dual action for gravitysupplemented with the (anti-)self-dual Pontrjagin term is carried out.The effect of the topological term is to add a `magnetic' term to theoriginal momentum variable associated with the self-dual action leavingthe Ashtekar connection unmodified. In the new variables, the Gaussconstraint retains its form, while both vector and Hamiltonian constraintsare modified. This shows that the contribution of the Euler and Pontrjaginterms is not the same as that coming from the term associated with theBarbero-Immirzi parameter, and thus the analogy between theθ-angle in Yang-Mills theory and the Barbero-Immirzi parameterof gravity is not appropriate.