Pure connection gravity at one loop: Instanton background

TL;DR

This paper computes one-loop divergences in pure connection formulation of General Relativity on an instanton background, revealing differences from metric formulation and highlighting the relation between the two approaches.

Contribution

It provides the first calculation of one-loop divergences in the pure connection formalism of GR, showing how it differs from and relates to metric-based quantum gravity.

Findings

Divergent contributions involve volume and Euler terms with different constants.

The connection and metric formulations differ by an integer in their divergence constants.

The Euclidean action in the connection formalism has a definite sign, unlike in metric GR.

Abstract

In the "pure connection" formulation General Relativity becomes a particular diffeomorphism invariant SL(2) gauge theory. Using this formalism, we compute the divergent contributions to the gravitational one-loop effective action. Calculations of the on-shell effective action simplify greatly if one specialises to an instanton background where only the anti-self-dual part of the Weyl curvature is non-vanishing. One of the most striking features of the connection formulation is that the (linearised) Euclidean action has a definite sign, unlike in the metric case. As in the metric GR, we find the logarithmically divergent contribution to consist of the volume and Euler character terms, but the arising numerical constants are different. However, the difference between the two results turns out to be always an integer. We explain this by noting that at one loop the connection and metric based quantum theories are closely related, the only difference being in a finite number of scalar modes.