Seiberg-Witten maps for SO(1,3) gauge invariance and deformations of gravity

Abstract

A family of diffeomorphism-invariant Seiberg-Witten deformations of gravity is constructed. In a first step Seiberg-Witten maps for an SO(1,3) gauge symmetry are obtained for constant deformation parameters. This includes maps for the vierbein, the spin connection, and the Einstein-Hilbert Lagrangian. In a second step the vierbein postulate is imposed in normal coordinates and the deformation parameters are identified with the components theta(mu nu)(x) of a covariantly constant bivector. This procedure gives for the classical action a power series in the bivector components which by construction is diffeomorphism invariant. Explicit contributions up to second order are obtained. For completeness a cosmological constant term is included in the analysis. Covariant constancy of theta(mu nu)(x), together with the field equations, imply that, up to second order, only four dimensional metrics which are direct sums of two two dimensional metrics are admissible, the two-dimensional curvatures being expressed in terms of theta(mu nu). These four-dimensional metrics can be viewed as a family of deformed emergent gravities.