# Shortest distance scales that a string can resolve

On page 5 of the notes by Veronika Hubeny on The AdS/CFT correspondence, we find the following:

Nevertheless, already at this level we encounter several intriguing surprises. Since strings are extended objects, some spacetimes which are singular in general relativity (for instance those with a timelike singularity akin to a conical one) appear regular in string theory. Spacetime topology-changing transitions can likewise have a completely controlled, non-singular description. Moreover, the so-called 'T-duality' equates geometrically distinct spacetimes: because strings can have both momentum and winding modes around compact directions, a spacetime with a compact direction of size $R$ looks the same to strings as spacetime with the compact direction having size $\ell_{s}^{2}/R$, which also implies that strings can’t resolve distances shorter than the string scale $\ell_{s}$. Indeed this idea is far more general (known as mirror symmetry [11]), and exemplifies why spacetime geometry is not as fundamental as one might naively expect.

The penultimate sentence states that

A spacetime with a compact direction of size $R$ looks the same to strings as spacetime with the compact direction having size $\ell_{s}^{2}/R$.

Why does this imply that strings can’t resolve distances shorter than the string scale $\ell_{s}$?

If $R < \ell_s$, then the T-dual spacetime has $R' = \ell_s^2/R > \ell_s$, and since the string theories on both spacetimes are equivalent, you can not meaningfully talk about $R$ being smaller than $\ell_s$ since there's always an equivalent theory where it is greater than $\ell_s$, so there are no phenomena which can only happen for very small values of $R$, and therefore there is no conceivable measurement that could test a hypothesis like $R < \ell_s$ (nor strictly speaking the hypothesis $R > \ell_s$, this is what she means by spacetime geometry not being as fundamental as expected).