Seeking the Loop Quantum Gravity Barbero-Immirzi Parameter and Field in 4D, $\cal N$ = 1 Supergravity

TL;DR

This paper embeds the Barbero-Immirzi parameter within 4D, $\\mathcal{N}$=1 supergravity, linking Loop Quantum Gravity and superstring theory through a supersymmetric axion-like field, and explores its implications for quantum gravity and string theory.

Contribution

It introduces a supersymmetric extension of the Barbero-Immirzi parameter as a chiral superfield in supergravity, suggesting a connection between Loop Quantum Gravity and string theory.

Findings

The Barbero-Immirzi parameter can be promoted to a dynamical superfield.

This superfield couples to a supersymmetric Holst term in supergravity.

Potential link between Loop Quantum Gravity and superstring theory via the axion field.

Abstract

We embed the Loop Quantum Gravity Barbero-Immirzi parameter and field within an action describing 4D, $\cal N$ = 1 supergravity and thus within a Low Energy Effective Action of Superstring/M-Theory. We use the fully gauge-covariant description of supergravity in (curved) superspace. The gravitational constant is replaced with the vacuum expectation value of a scalar field, which in local supersymmetry is promoted to a complex, covariantly chiral scalar superfield. The imaginary part of this superfield couples to a supersymmetric Holst term. The Holst term also serves as a starting point in the Loop Quantum Gravity action. This suggest the possibility of a relation between Loop Quantum Gravity and supersymmetric string theory, where the Barbero-Immirzi parameter and field of the former play the role of the supersymmetric axion in the latter. Adding matter fermions in Loop Quantum Gravity may require the extension of the Holst action through the Nieh-Yan topological invariant, while in pure, matter-free supergravity their supersymmetric extensions are the same. We show that, when the Barbero-Immirzi parameter is promoted to a field in the context of 4D supergravity, it is equivalent to adding a dynamical complex chiral (dilaton-axion) superfield with a non-trivial kinetic term (or K\"ahler potential), coupled to supergravity.