Towards a gravitational and supergravitational Elko theory

Abstract

We describe a possible and alternative route to connect gravity with Elko theory. Our approach is based on the possibility to introduce a totally antisymmetric gauge field in the generalized Elko field equation, which is an alternative extension of the Dirac field equation. The corresponding totally antisymmetric field strength ([Formula: see text]-form) is then associated with Grassmann–Plücker coordinates and therefore with a decomposable [Formula: see text]-form. We show that such a totally antisymmetric field strength can be considered as the square root of the Riemann tensor associated to a homogeneous space. Motivated by this result we conjecture that a full connection with Riemann geometry and therefore with general relativity must be possible if such a field strength is not decomposable. We also show how a supergravity version could arise from a generalization of the above ideas.