Dimensional aspects of Lovelock-Lanczos gravity

There has recently been an increasing interest in regularizations of Lovelock-Lanczos gravity (LLG) in four dimensions, in which dimensional poles and possibly counter-terms are introduced to compensate the vanishing of the Lovelock field equations in critical and lower dimensions. In this paper, we review and extend some of these results. We first find a class of LLG theories whose perturbative expansion around a given (A)dS vacuum can be regularized up to arbitrary order, the simplest one being close to Lovelock gravities with a unique vacuum. If well-defined, these models might be interpreted as effective field theories of gravitons in four dimensions, or might be combined with other regularization approaches. Among those, we establish the general procedure to obtain $4$D covariant and background independent regularizations from metric transformations. In the conformal (and critical) case, we generalize previous results obtained for the Gauss-Bonnet theory to the full Lovelock series. Similarly, the regularization of Gauss-Bonnet gravity from the breaking of $4$D-covariance down to $3$D is generalized to arbitrary curvature order, seemingly resulting in new Minimally Modified gravities propagating solely the two degrees of freedom of the graviton. Finally, we present general results regarding the minisuperspace regularization of specific sectors of LLG. Non-perturbative (in curvature) regularized theories admitting non-singular black holes as well as non-singular past-dS$_4$ and cyclic closed cosmologies are found. We conclude with the non-uniqueness of these background regularizations by finding inequivalent regularizations of the Bianchi I sector of Lovelock-Lanczos gravity in four dimensions.