Noncommutative gravity with self-dual variables

Abstract

We build a noncommutative extension of Palatini-Holst theory on a twist-deformed spacetime, generalizing a model that has been previously proposed by Aschieri and Castellani.The twist deformation entails an enlargement of the gauge group, and leads to the introduction of new gravitational degrees of freedom. In particular, the tetrad degrees of freedom must be doubled, thus leading to a bitetrad theory of gravity. The model is shown to exhibit new duality symmetries. The introduction of the Holst term leads to a dramatic simplification of the dynamics, which is achieved when the Barbero-Immirzi parameter takes the value β = −i, corresponding to a self-dual action. We study in detail the commutative limit of the model, focusing in particular on the role of torsion and non-metricity. The effects of spacetime noncommutativity are taken into account perturbatively, and are computed explicitly in a simple example. Connections with bimetric theories and the role of local conformal invariance in the commutative limit are also explored.

On almost commutative Friedmann–Lemaître–Robertson–Walker geometries

“…It is also worth mentioning that some general noncommutative models, with deformed space-time effectively lead to a version of action and metric fields that in the classical limit reduce themselves to a bimetric gravity models [14].…”

T-dualities and Doubled Geometry of the Principal Chiral Model

“…Later on we shall discuss more in detail this important point. noncommutative extensions of Palatini-Holst theory of gravity imply the doubling of the tetrad degrees of freedom, leading to a bi-tetrad theory of gravity, with the manifestation of new duality symmetries [73,74].…”

Gravity with antisymmetric components

“…Moreover, it would be interesting to see whether there is a relation to the non-commutative setup of Ref. [56].…”