The Ricci flow conjecture

Let $D$ be the space-time dimension, let $\mathcal{M}$ be a space-time manifold, and let $\mathbf{G}_{\mathcal{M}}$ be the generalized moduli space of metrics on $\mathcal{M}$. Consider a curve

$$g:[s_0,s_1)\longrightarrow\mathbf{G}_{\mathcal{M}},$$

where $s_1$ may be finite or infinite. Assume that $g_0=g(s_0)$ describes a consistent quantum-gravity low-energy effective theory, that $g$ follows Ricci flow up to diffeomorphisms,

$$\frac{\mathrm{d}g}{\mathrm{d}s}=-2\operatorname{Ric}(g)+\mathcal{L}_\xi g,$$

that $g_1=\lim_{s\to s_1}g(s)$ is a Ricci-flow fixed point, and that the geodesic distance satisfies

$$\lim_{s\to s_1}\Delta(s,s_0)=\infty.$$

Ricci flow conjecture. Any Ricci-flow fixed point at infinite distance should be accompanied by an infinite tower of additional fields with a flow-dependent mass threshold $m:[s_0,s_1)\to\mathbb{R}$ obeying

$$m(s)\sim m(s_0)\exp\left(-\alpha\frac{\Delta(s,s_0)}{\sqrt{M_{\mathrm{P}}^{D-2}}}\right),$$

where $\alpha$ is positive and of order one.

This conjecture extends the swampland distance conjecture to displacements in the space of space-time geometries. It is motivated by the Ricci flow of Anti-de Sitter space, which approaches flat space at infinite flow time, but its general validity remains open.