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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?

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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.

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  • $\begingroup$ 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. $\endgroup$ – Dwagg Oct 25 '19 at 15:38
  • $\begingroup$ @Dwagg yep, no absolute scale is specified. Though it's always the case that ultimately all physically meaningful statements are made about ratios of scales. $\endgroup$ – OON Oct 25 '19 at 15:46

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