Canonical and symplectic analysis of the Holst action in the $G\to 0$ limit

By using the Dirac and symplectic formalisms, the analysis of the Holst action in the $G\to 0$ limit is performed. We study the theory’s full canonical structure and symmetries. In particular, we focus on the cases $\gamma =i$ and $\gamma \mathrm{\ne }i$, aiming to obtain Smolin’s auto-self-dual model in terms of the Ashtekar variables and a Barbero-type formulation. We then perform the symplectic analysis and discuss the principal insight of these formalisms.