Self-dual gravity with topological terms (original) (raw)

2001, Classical and Quantum Gravity

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 θ-angle in Yang-Mills theory and the Barbero-Immirzi parameter of gravity is not appropriate. PACS: 04.60.Ds diag(−1, +1, +1, +1). ω I J is a Lorentz connection 1-form and R IJ (ω) = 1 2 R µνIJ (ω)dx µ ∧ dx ν is its curvature, R µνI J = ∂ µ ω νI J − ∂ ν ω µI J + ω µI K ω νK J − ω νI K ω µK J . The definition of the dual operator is * T IJ = 1 2 ǫ IJ KL T KL with ǫ 0123 = +1. As far as I know, the full canonical analysis of action (1) has not been carried out. When α 2 = α 3 = α 4 = 0, action (1) reduces to the standard Hilbert-Palatini action whose canonical analysis is already reported in the literature . There, the canonical variables are the 3-dimensional extrinsic curvature K i a and the densitizied inverse triad field E a