Baby Steps in Quantum Ring Theory: towards a background independent framework for Quantum Gravity

Abstract

We investigate gravity as a gauge theory in the language of fiber bundles with tools from algebraic geometry. Compelled by the construction of the Eilenberg-MacLane classifying space via Fox derivations in an integral group ring, the origin of locally curved spacetime as a an abelian group extension is sought algebraically by means of a suitable derivation rule that conforms with the algebraic content of Einstein's equation. Accordingly, a peculiar flavor of background independence is advanced, curved spacetime should be thought of as an abelian proxy for the Lorentz group, causal and spin throughout. Thus, diffeomorphism invariance amounts to "choosing" an appropriate algebro-geometric--differential tool as to make algebraic sense of Einstein's equation or of a sensible evolution of such an equation. In this regard, it is suggested that a left-$\alpha$ derivation rule may facilitate the construction. On the other hand, we are led to interpret "spacetime curvature" in the broad sense of the exterior covariant derivative of a Cartan connection which includes both the torsion of the vierbein as well as the standard curvature of the spin connection. Making manifest use of the index structure in the standard Hilbert action--in particular, the distinction between internal and external indices--inclusion of the torsion 2-form is found to be inconsistent with contraction of the spacetime curvature as a scalar density. Aiming to erect a geometrically justified gravitational action that incorporates torsion, we lay out several fundamental geometric, differential and algebraic building blocks while highlighting descent relations among the different pieces. The groundwork is thus laid out for the search of the sought after action functional.