# A Shortest distance scales that a string can resolve

1. May 30, 2017

### Afonso Campos

On page 5 of the notes (https://arxiv.org/abs/1501.00007) 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 $\ell_{s}$?

2. May 30, 2017

### haushofer

Well, what happens if you go from $R<\ell_{s}$ to $R>\ell_{s}$, bearing in mind that $R\sim \ell_{s}^2 / R$?

3. May 30, 2017

### Afonso Campos

The theory looks the same, but the string is probing length scales shorter than $\ell_s$, right?

It's just that the long distance and short distance physics are the same.