Ho\v{r}ava models as palladium of unitarity and renormalizability in quantum gravity

We give a review of UV renormalization of Ho\v{r}ava gravity (HG) models introduced as a remedy against violation of unitarity in quantum gravity theory. Projectable and non-projectable low-dimensional HG models and the spectra of their physical degrees of freedom are considered in the linearized approximation on flat spacetime background. The problem of regularity of their propagators depending on the choice of gauge fixing procedure is discussed along with the role of this regularity in UV renormalization. With the choice of a special class of quasi-relativistic gauge conditions perturbative renormalizability of projectable HG models is proven in any spacetime dimension and the status of renormalization in non-projectable models is briefly discussed. We show how the covariance of counterterms is provided within the class of background covariant gauge conditions and show asymptotic freedom of $(2+1)$-dimensional HG. We also present the calculation of beta-functions in $(3+1)$-dimensional theory by the generalized Schwinger-DeWitt technique of universal functional traces and obtain fixed points of its renormalization group flow, some of them being good candidates for asymptotic freedom.