The swampland distance conjecture

Let $\Phi=(\phi^1,\dots,\phi^N)$ be fields in a low-energy effective field theory coupled to dynamical $D$-dimensional spacetime, with vacuum state $|\Omega\rangle$. Let $\mathbf{M}_\Phi$ be the $N$-dimensional moduli space parametrized by the vacuum expectation values

$$\varphi^i=\langle\Omega|\phi^i|\Omega\rangle,\qquad i=1,\dots,N.$$

For points $\bar\varphi_A,\bar\varphi_B\in\mathbf{M}_\Phi$, write their geodesic distance as $\Delta_{AB}=\Delta(\bar\varphi_A,\bar\varphi_B)$. Swampland distance conjecture. Every infinite-distance limit in $\mathbf{M}_\Phi$ should be accompanied by an infinite tower of additional fields with mass threshold $m:\mathbf{M}_\Phi\to\mathbb{R}$ satisfying

$$m(\bar\varphi_B)\sim m(\bar\varphi_A)\exp\left(-\alpha\frac{\Delta_{AB}}{\sqrt{M_{\mathrm{P}}^{D-2}}}\right)$$

when $\Delta_{AB}\gtrsim M_{\mathrm{P}}$, where $\alpha$ is positive and of order one.

This is the standard distance conjecture for scalar moduli spaces, motivated by towers arising in string compactifications. Its broad general validity remains open.