# What sets the AdS radius $\ell$ in the CFT dual description?

In the context of AdS/CFT there should be a mapping between parameters for any given duality. But AdS has at least one dimensionful parameter, $$\ell$$:

$$ds^2 = \frac{\ell^2}{z^2} ( -dt^2 + d\vec x^2 + dz^2).$$

If it comes from a stringy limit there is also the string length which may be related to $$\ell$$. However, a CFT such as $$\mathcal N=4$$ SYM has no dimensionful parameters. What is $$\ell$$ in terms of the parameters of the dual CFT?

When you have two scales you can consider their dimensionless ratio i.e. $$l/l_s$$. It is this ratio that is related to the 't Hooft coupling $$\lambda$$ of the SYM.
You also have the third scale the Planck scale $$l_P$$. With three scales you can get two independent dimensionless parameters with the second one related to $$1/N$$ in SYM.
• Ok, so $\lambda$ and $N$ in SYM set the ratios of mass scales in the bulk. Is it correct to say that the overall mass scale, however, is not specified by AdS/CFT? Or perhaps it's more subtle. – Dwagg Oct 25 '19 at 15:38