Sharp lower bound on the exponent in the distance conjecture
In plain words
When a parameter of string theory is moved through a distance $D$, a tower of particles becomes lighter by a factor $\exp(-\lambda D)$. The conjecture is that $\lambda$ can never be smaller than a fixed number set only by the number of spacetime dimensions.
Precise statement
Distance conjecture (Ooguri and Vafa 2006): along a geodesic of length D in moduli space, measured in Planck units, a tower of states has masses $m \sim \exp(-\lambda D)$. Sharpened version (Etheredge, Heidenreich, Kaya, Qiu, Rudelius 2022): the lightest tower obeys $\lambda \ge 1/\sqrt{d-2}$ in d spacetime dimensions. Prove the bound for all infinite-distance limits of string compactifications, or find a limit with $\lambda < 1/\sqrt{d-2}$.
What would settle it
A proof over all infinite-distance limits of string compactifications in each $d$, or an explicit limit with $\lambda < 1/\sqrt{d-2}$.
Status in the literature
Proposed in 2022 with the bound preserved under dimensional reduction and saturated in many string and M-theory compactifications.