Self-dual gravity with topological terms

TL;DR

This paper performs a canonical analysis of self-dual gravity with topological terms, revealing how these terms modify the constraints and clarifying the distinct roles of Euler, Pontrjagin, and Barbero-Immirzi parameters.

Contribution

It provides a detailed canonical analysis showing how topological terms affect the constraints in self-dual gravity, distinguishing their effects from the Barbero-Immirzi parameter.

Findings

Topological terms modify vector and Hamiltonian constraints.

The Gauss constraint remains unchanged.

The analogy between theta-angle and Barbero-Immirzi parameter is invalid.

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.