We advance a class of unitary higher derivative theories of gravity that realize an ultraviolet completion of Einstein general relativity in any dimension. This range of theories is marked by an entire function, which averts extra degrees of freedom (including poltergeists) and improves the high energy behaviour of the loop amplitudes. It is proved that only one-loop divergences survive and the theory is super-renormalizable regardless of the spacetime dimension. Furthermore, in odd dimensions there are no counter terms for pure gravity and the theory turns out to be ``finite". Moreover, using the Pauli-Villars regularization procedure introduced by Diaz-Troost-van Nieuwenhuizen-van Proeyen and applied to Einstein's gravity by Anselmi, we are able to remove also the one loop divergences.
Recently we obtained the most general gravitational action introducing a local potential of the Riemann tensor and we explicitly showed in D=4 that all the one-loop beta functions can be make to vanish. The outcome is a finite quantum gravity even at one-loop. The result can be easily extended in extra dimensions and likely the higher dimensional theory will be finite too. In these days we were able to properly defined the theory even for the class of form factors exponentially growing at high energy.
Comparing these theories with string field theories, a unique form factor is singled out. In view of this, we can modify the 10-dimensional supergravity (including the massive states) and finally get a ultraviolet completion of 11-dimensional supergravity by an ``oxidation process". The result is a proposal for a "finite M-theory in field theory".
Additionally, we propose a maximal extension of the theory which admits one single extra degree of freedom (gravi-scalar). This generalization turns out to be a unitary and super-renormalizable (or finite) completion of the Starobinsky theory. Currently we are working on gravity coupled to matter: a preliminary power counting analysis is presented.
