On divergences in non-minimal N=4 conformal supergravity

We review the question of quantum consistency of N=4 conformal supergravity in 4 dimensions. The UV divergences and anomalies of the standard (minimal) conformal supergravity where the complex scalar $phi$ is not coupled to the Weyl graviton kinetic term can be cancelled by coupling this theory to N=4 super Yang-Mills with gauge group of dimension 4. The same turns out to be true also for the non-minimal N=4 conformal supergravity with the action (recently constructed in arXiv:1609.09083) depending on an arbitrary holomorphic function $f(phi)$. The special case of the non-minimal conformal supergravity with $f= e^{2phi}$ appears in the twistor-string theory. We show that divergences and anomalies do not depend on the form of the function $f$ and thus can be cancelled just as in the minimal $f=1$ case by coupling the theory to four N=4 vector multiplets.