Instantons in conformal gravity

We study extrema of the general conformally invariant action:

$\text{S}{}_{\mathrm{c}}\text{= ∫}\text{1}\text{α}{}^2\text{C}{}^{\mathrm{abcd}}\text{C}{}_{\mathrm{abcd}}\text{+γR}{}^{\mathrm{abcd}}{}^1\text{R}{}_{\mathrm{abcd}}{}^1\text{+iθR}{}^{\mathrm{abcd}}\text{1R}{}_{\mathrm{abcd}}$

.We find the first examples in four dimensions of asymptotically euclidean gravitational instantons. These have arbitrary Euler number and Hirzebruch signature. Some of these instantons represent tunneling between zero-curvature vacua that are not related by small gauge transformations. Others represent tunneling between flat space and topologically non-trivial zero-energy initial data. A general formula for the one-loop determinant is derived in terms of the renormalization group invariant masses, the volume of space-time, the Euler number and the Hirzebruch signature.