D=4 Einstein gravity from higher D CS and BI gravity and an alternative to dimensional reduction

TL;DR

This paper proposes an alternative approach to dimensional reduction in gravity, embedding 4D Einstein gravity with a topological term into higher-dimensional Chern-Simons theories, potentially linking to quantum gravity frameworks.

Contribution

It introduces a novel dimensional reduction mechanism from higher-dimensional Chern-Simons gravity to 4D Einstein gravity with a topological term, offering new insights into quantum gravity embedding.

Findings

4D Einstein gravity embedded in 5D Chern-Simons gravity

Dimensional reduction from higher to lower dimensions analyzed

Potential connection to 11D supergravity and quantum gravity

Abstract

An alternative to usual dimensional reduction for gravity is analyzed, in the vielbein-spin connection formulation. Usual 4d Einstein gravity plus a topological term (the "Born-Infeld" Lagrangian for gravity), is shown to be obtained by a generalized dimensional reduction from 5d Chern-Simons gravity. Chern-Simons gravity in d=2n+1 is dimensionally reduced to CS gravity in d=2n-1 via a mechanism similar to descent equations. The consistency of the dimensional reduction in both cases is analyzed. The dimensional reduction of d=2n+2 Born-Infeld gravity to d=2n BI gravity, as well as d=2n BI gravity to d=2n-1 CS gravity is hard to achieve. Thus 4d gravity (plus a topological term) can be embedded into d=2n+1 CS gravity, including 11d CS, whose supersymmetric version could possibly be related to usual 11d supergravity. This raises the hope that maybe 4d quantum Einstein gravity could be embedded in a well defined quantum theory, similar to Witten's treatment of 3d quantum Einstein gravity as a CS theory.