Topological parameters in classical and quantum gravity

Gravity theory in four dimensions has three independent topological parameters. In the Lagrangian, these show up as coefficients of three topological densities, namely Nieh-Yan, Pontryagin and Euler. We analyse the Hamiltonian theory based on this generalized Lagrangian. Our analysis indicates that these parameters might affect the quantum geometry nontrivially, and the quantum area may depend on these. We also study how the topological parameters affect the gravity action for generic manifolds with boundaries, and in particular, for asymptotically locally AdS spacetimes. In this case, the Barbero-Immirzi parameter is shown to appear as the only independent topological coupling constant in the action principle. Subsequently, we make a few remarks on the potential implications in the quantum theory in this context.