  In chiral Einstein-Cartan gravity, a new gauge fixing procedure is
implemented recently, leading to a very economical perturbation expansion of
the action. Using this formulation and the relevant gauge-fixing, we develop
the ghost Lagrangian on an arbitrary Einstein background using the BRST
formalism. The novelty is the appearance of a new term quadratic in the tetrad
field. We next compute the heat-kernel coefficients and understand the
divergences arising in the gravitational one-loop effective action. In our
computation the arising heat kernel coefficients depend only on the self-dual
part of the Weyl curvature. We make a comparison between our results and what
has been obtained for metric GR.
