  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.
