On divergences in non-minimal 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 φ 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 (Butter et al 2017 Phys. Rev. Lett. 118 081602)) depending on an arbitrary holomorphic function f (φ). The special case of the ‘non-minimal’ conformal supergravity with f = e2φ 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.