Faddeev gravity action on the piecewise constant fundamental vector fields

TL;DR

This paper develops a discrete Faddeev gravity action using piecewise constant vector fields on a simplicial spacetime, allowing for non-coincident simplices and introducing an analog of the Barbero-Immirzi parameter.

Contribution

It derives a minisuperspace Faddeev action for piecewise constant fields on a simplicial complex, extending previous formulations of gravity.

Findings

Formulation of a discrete Faddeev gravity action on flat 4-simplices.

Inclusion of an analog of the Barbero-Immirzi parameter.

Relaxation of the requirement for simplices to coincide on faces.

Abstract

In the Faddeev formulation of gravity, the metric is regarded as composite field, bilinear of $d = 10$ 4-vector fields. We derive the minisuperspace (discrete) Faddeev action by evaluating the Faddeev action on the spacetime composed of the (flat) 4-simplices with constant 4-vector fields. This is an analog of the Regge action obtained by evaluating the Hilbert-Einstein action on the spacetime composed of the flat 4-simplices. One of the new features of this formulation is that the simplices are not required to coincide on their common faces. Also an analog of the Barbero-Immirzi parameter $\gamma$ can be introduced in this formalism.