The Ashtekar-Barbero-Immirzi formulation of General Relativity is extended to include spinor matter fields. Our formulation applies to generic values of the Immirzi parameter and reduces to the Ashtekar-Romano-Tate approach when the Immirzi parameter is taken equal to the imaginary unit. The dynamics of the gravity-fermions coupled system is described by the Holst plus Dirac action with a non-minimal coupling term. The non-minimal interaction together with the Holst modification to the Hilbert-Palatini action reconstruct the Nieh-Yan invariant, so that the effective action coming out is the one of Einstein-Cartan theory with a typical Fermi-like interaction term: in spite of the presence of spinor matter fields, the Immirzi parameter plays no role in the classical effective dynamics and results to be only a multiplicative factor in front of a total divergence.We reduce the total action of the theory to the sum of dynamically independent Ashtekar-RomanoTate actions for self and anti-self dual connections, with different weights depending on the Immirzi parameter. This allows to calculate the constraints of the complete theory in a simple way, it is only necessary to realize that the Barbero-Immirzi connection is a weighted sum of the self and anti-self dual Ashtekar connections. Finally the obtained constraints for the separated action result to be polynomial in terms of the self and anti-self dual connections, this could have implications in the inclusion of spinor matter in the framework of non-perturbative quantum gravity.

The Barbero-Immirzi (BI) parameter is promoted to a field and a canonical analysis is performed when it is coupled with a Nieh-Yan topological invariant. It is shown that, in the effective theory, the BI field is a canonical pseudoscalar minimally coupled with gravity. This framework is argued to be more natural than the one of the usual Holst action. Potential consequences in relation with inflation and the quantum theory are briefly discussed.Comment: 10 page

A general argument provides the motivation to consider the Barbero-Immirzi parameter as a field. The specific form of the geometrical effective action allows one to relate the value of the Barbero-Immirzi parameter to other quantum ambiguities through the analog of the Peccei-Quinn mechanism.

In loop quantum gravity the classical point of departure is the Einstein-Hilbert action modified by the addition of the so-called Holst term. Classically, this term does not affect the equations of motion, but it induces a well-known quantization ambiguity in the quantum theory, parametrized by the BarberoImmirzi parameter. Recently, it has been suggested to promote the Barbero-Immirzi parameter to a field. The resulting theory, obtainable starting from the usual Holst action, is general relativity coupled to a pseudoscalar field. However, this theory turns out to have an unconventional kinetic term for the BarberoImmirzi field and a rather unnatural coupling with fermions. The main goal of this work is twofold: First, to propose a further generalization of the Holst action, which yields a theory of gravity and matter with a more natural coupling to the Barbero-Immirzi field; second, to study the possible implications for cosmology correlated to the existence of this new pseudoscalar field.