# 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 ), 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).