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

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.