Swampland, Gradient Flow and Infinite Distance

In the first part of this paper we will work out a close and so far not yet noticed correspondence between the swampland approach in quantum gravity and geometric flow equations in general relativity, most notably the Ricci flow. We conjecture that following the gradient flow towards a fixed point, which is at infinite distance in the space of background metrics, is accompanied by an infinite tower of states in quantum gravity. In case of the Ricci flow, this conjecture is in accordance with the generalized distance and AdS distance conjectures, which were recently discussed in the literature, but it should also hold for more general background spaces. We argue that the entropy functionals of gradient flows provide a useful definition of the generalized distance in the space of background fields. In particular we give evidence that for the Ricci flow the distance $\Delta$ can be defined in terms of the mean scalar curvature of the manifold, $\Delta\sim\log \bar R$. For a more general gradient flow, the distance functional also depends on the string coupling constant. In the second part of the paper we will apply the generalized distance conjecture to gravity theories with higher curvature interactions, like higher derivative $R^2$ and $W^2$ terms. We will show that going to the weak coupling limit of the higher derivative terms corresponds to the infinite distance limit in metric space and hence this limit must be accompanied by an infinite tower of light states. For the case of the $R^2$ or $W^2$ couplings, this limit corresponds to the limit of a small cosmological constant or, respectively, to a light additional spin-two field in gravity. In general we see that the limit of small higher curvature couplings belongs to the swampland in quantum gravity, just like the limit of a small $U(1)$ gauge coupling belongs to the swampland as well.