Einsteinian gravity from a topological action

On the macroscopical level, Einstein’s general relativity has passed every test “with flying colors", see Will [67] for a recent review. However, Einstein’s theory sofar has resisted any attempt of quantization, e.g. it is known to be perturbatively nonrenormalizable, partially due to its dimensional coupling constant2. String theory or brane scenarios with extra dimensions have been proposed as a rescue, some of which are implying, however, deviations of standard gravity in the sub-millimeter range. Recent torsion balance experiments [28] have probed the inverse square law and found no deviation even below the hypothetical dark energy (DE) scale of λDE = (h̄c/ρDE) ≃ 85 μm. Thus this ‘window’ of possibly new gravitational physics seems also to be closing. In 1974 Yang proposed an affine gauge theory gravity [70] which, due to its scale invariance, can be regarded as a rather promising fundamental theory of (quantum) gravity in the high-energy limit [25], without invoking extra dimensions or supersymmetry, cf. Ref. [30]. An additional duality constraint on the curvature could be extremely important for the path integral approach to quantum gravity. Then instanton type configurations [22] near the classical ones, i.e. Einstein spaces, are more probable then the ‘spurious’ Thomson spaces, as one would expect naively. For the modified duality with a breaking of scale invariance, the transition amplitude peaks at classical Einstein spaces only. Alternatively, in a four-dimensional Yang-Mills theory gauging the de Sitter group [37, 54], scale invariance gets spontaneously broken by a pseudo-Goldstone type ‘radius vector’ [62], odd under CP, in order to recover the Hilbert-Einstein action plus the Euler term. From the work of Stelle [59] we know that the curvature squared gravity in Riemannian spacetime is perturbatively renormalizable but unfortunately plagued with physical ghost, i.e. negative residues in the graviton propagator [36]. This finding has diminished