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

TL;DR

This paper provides a comprehensive canonical analysis of Pontryagin and Euler classes with a Barbero-Immirzi parameter, revealing their structure, symmetries, and special cases like the self-dual representation.

Contribution

It introduces Holst-like variables for these topological invariants, details their canonical structure, and explores the effects of the Barbero-Immirzi parameter, including the self-dual case.

Findings

Complete canonical structure and symmetries identified.

Physical degrees of freedom counted and reducibility conditions found.

Self-dual representation recovered at specific Barbero-Immirzi parameter values.

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 $\gamma = \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.