Horava-Lifshitz theory as a Fermionic Aether in Ashtekar gravity

TL;DR

This paper demonstrates that Hořava-Lifshitz gravity naturally arises within Ashtekar's formulation of general relativity when a fermionic field acts as a spacetime-filling aether, linking fermionic matter to preferred foliation and symmetry breaking.

Contribution

It introduces a novel interpretation of HL gravity as a fermionic aether in Ashtekar variables, connecting fermionic fields with the theory's parameters and symmetry properties.

Findings

HL gravity emerges from fermionic coupling in Ashtekar variables.

The fermionic aether explains HL theory's good convergence properties.

The Immirzi parameter and fermionic charge density determine the HL parameter λ.

Abstract

We show how Ho\v{r}ava-Lifshitz (HL) theory appears naturally in the Ashtekar formulation of relativity if one postulates the existence of a fermionic field playing the role of aether. The spatial currents associated with this field must be switched off for the equivalence to work. Therefore the field supplies the preferred frame associated with breaking refoliation (time diffeomorphism) invariance, but obviously the symmetry is only spontaneously broken if the field is dynamic. When Dirac fermions couple to the gravitational field via the Ashtekar variables, the low energy limit of HL gravity, recast in the language of Ashtekar variables, naturally emerges (provided the spatial fermion current identically vanishes). HL gravity can therefore be interpreted as a time-like current, or a Fermi aether, that fills space-time, with the Immirzi parameter, a chiral fermionic coupling, and the fermionic charge density fixing the value of the parameter $\lambda$ determining HL theory. This reinterpretation sheds light on some features of HL theory, namely its good convergence properties.