On divergences in non-minimal { \newcommand{\N}{{{ N}}} \N=4} conformal supergravity

We review the question of quantum consistency of \newcommand{\N}{{{ N}}} \N=4 conformal supergravity in 4 dimensions. The UV divergences and anomalies of the standard (‘minimal’) conformal supergravity where the complex scalar \newcommand{\vp}{\varphi} \vp is not coupled to the Weyl graviton kinetic term can be cancelled by coupling this theory to \newcommand{\N}{{{ N}}} \N=4 super Yang-Mills with gauge group of dimension 4. The same turns out to be true also for the ‘non-minimal’ \newcommand{\N}{{{ N}}} \N=4 conformal supergravity with the action (recently constructed (Butter et al 2017 Phys. Rev. Lett. 118 081602)) depending on an arbitrary holomorphic function \newcommand{\vp}{\varphi} f(\vp) . The special case of the ‘non-minimal’ conformal supergravity with \newcommand{\vp}{\varphi} f= e^2\vp 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 \newcommand{\N}{{{ N}}} \N=4 vector multiplets.