# At what scale does UV/IR mixing happen in string theory?

Imagine, if you will, a string background with an extremely tiny value for the string coupling. The Planck scale is many orders of magnitude smaller in size than the string scale. Is the UV/IR mixing scale the Planck scale, or the string scale? What happens at intermediate scales?

Gravitons are string modes, and as such, their spatial extent is the string scale. They can never resolve anything smaller than that. On the other hand, D-branes are more sensitive.

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## 1 Answer

In perturbative string theory, for $g\ll 1$, it's always the string scale where the UV/IR mixing happens first. It's the lightest mass scale associated with the lightest (possibly extended) objects - and in perturbative string theory regimes, it's always the fundamental strings. For example, the low-energy spectrum at masses below $M_{\rm string}$ is always related to the high-energy spectrum at masses above $M_{\rm string}$ by the modular invariance (of the toroidal world sheet) - and similarly for open strings and cylindrical world sheets.

Different effects may see different forms of UV/IR mixing for which different scales are relevant, however. The term UV/IR mixing is a somewhat vague, umbrella term that covers many effects in string theory or quantum gravity. D-branes may resolve shorter distances than the string scales but there is actually always some sense in which even e.g. D0-branes are linked to the string scale. If they resolve distances $\Delta x$ and times $\Delta t$, then $\Delta x \cdot \Delta t > L_{\rm string}^2$. Yoneya would play with insights like this. D0-branes may resolve sub-stringy distances but only if their velocity is very small, in which case they can only measure the time with much-worse-than-stringy resolution.

There are also - less understood - aspects of the UV/IR mixing that arise at the Planck scale. But some of them are known so incompletely that the arguments are only valid up to the assumption that $g$ is considered to be of order one. The relevant scale where geometry - with degrees of freedom at each "layer" of scales being independent of each other - fails to be applicable is always given by the scale associated with the first higher-order corrections to the Einstein-Hilbert action etc. and it's always the string scale i.e. the scale associated with the tension of the lightest fundamental objects.

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Can you elaborate upon the less understood aspects of UV/IR mixing at the Planck scale? Thanks. – Shankar Apr 6 '11 at 11:22
If UV/IR mixing happens at the string scale, what happens if we have S-duality between two different string theories? e.g. type IIB, or type I with heterotic SO(32)? There are now two different string scales. Surely UV/IR mixings can't happen at both scales simultaneously? The only scale invariant under S-duality is the Planck scale. – Shankar Apr 7 '11 at 14:12