Equivalent and Alternative Forms for BF Gravity with Immirzi Parameter

TL;DR

This paper reformulates the BF approach to general relativity with the Immirzi parameter, deriving an equivalent action involving two BF terms and Lorentz invariants, and connects it to spin foam models and cosmological constant coupling.

Contribution

It introduces a transformed BF action for gravity that explicitly includes the Immirzi parameter and relates to existing spin foam models.

Findings

Recovered the Engle-Pereira-Rovelli action principle

Derived a coupling of the cosmological constant in BF formalism

Presented an equivalent formulation involving two BF terms

Abstract

A detailed analysis of the BF formulation for general relativity given by Capovilla, Montesinos, Prieto, and Rojas is performed. The action principle of this formulation is written in an equivalent form by doing a transformation of the fields of which the action depends functionally on. The transformed action principle involves two BF terms and the two Lorentz invariants that appear in the original action principle generically. As an application of this formalism, the action principle used by Engle, Pereira, and Rovelli in their spin foam model for gravity is recovered and the coupling of the cosmological constant in such a formulation is obtained.