content_image/313861.jpg
strongly coupled. We can be bolder still, and make the conjecture that when the gravity side is strongly coupled, the field side will be weakly coupled. We propose that the duality is a strong-weak duality . 

This is why this duality is so exciting to those studying strongly cou- pled gauge theories. As explained earlier, solutions to the non-perturbative regime of such theories have evaded capture for decades. This duality gives us hope that instead of having to try and solve the nightmare of a strongly coupled theory, we can instead re-parameterise the theory in terms of a weakly coupled supergravity theory, which we can solve using perturbation theory! 

## 2.5 Energy-radius Duality 

Ignoring for the moment the SU (4)$_{R}$ ∼ = SO (6) part of the duality, we have a 4+1 spacetime dual to a 3+1 field theory. Conventionally the ‘extra’ fifth dimension of the spacetime is denoted with r , and here we consider its physical interpretation. 

The AdS side has the metric 

<img src='content_image/313861.jpg'>

But the CFT is invariant under x µ → e $^{α}$x$^{µ}$. If the CFT side is invariant under this transformation, then so too must be the AdS side. Hence r must scale as r → re − $^{α}$. It transforms as an energy. 

This can be understood from the point of view of an observer at infinity (the boundary of AdS$_{5}$ × S $^{5}$). To her, a photon emitted from the centre of the space is red-shifted. If the observer moves in from infinity to some finite r , the photon is less red-shifted. So the energy-radius duality indicates that 

