Renormalizable Non-Metric Quantum Gravity?

We argue that four-dimensional quantum gravity may be essentially renormalizable if one relaxes the assumption of metricity of the theory. We work with Plebanski formulation of general relativity in which the metric (tetrad), the connection, and the curvature are all independent variables and the usual relations among these quantities are valid only on-shell. One of the Euler-Lagrange equations of this theory ensures its metricity. We show that quantum corrected action contains a counterterm that destroys this metricity property, and that no other counterterms appear, at least, at the one-loop level. The new term in the action is akin to a curvature-dependent cosmological ``constant''.