Canonical description of Pontryagin and Euler classes with a Barbero-Immirzi parameter

TL;DR

This work presents a comprehensive canonical analysis of the Pontryagin and Euler topological invariants with a Barbero-Immirzi parameter $\gamma$, recasting them in Holst-like variables and deriving their complete constraint structure. The authors show that the $\gamma$-dependent constraint algebra remains first-class and that the theory has zero physical degrees of freedom, as expected for a topological theory, with reducibility relations between constraints. Special cases $\gamma = \pm i$ reproduce the self-dual description, while real $\gamma$ yields a Barbero-type formulation with a nonpolynomial Hamiltonian constraint and a new Gauss-like constraint, altering the canonical structure. When coupled to the Holst action, the BI parameter continues to influence the constraint algebra, and the results illuminate how $\gamma$ affects quantization approaches for topological gravity and its relation to known self-dual and Barbero formulations.

Abstract

A detailed canonical analysis for Pontryagin and Euler classes with a Barbero-Immirzi [BI] parameter is developed. We rewrite the topological invariants by introducing a set of Holst-like variables, and then study the set of all constraints. We report the complete canonical structure and the symmetries of the theory; we count the physical degrees of freedom and identify reducibility conditions among the constraints. In addition, in our results, if we consider the $BI$ parameter takes the value of $γ= \pm i $, then the self-dual representation of these invariants is reproduced. Finally, we couple the invariants to the Holst action and explore the canonical analysis.