65 Chapter 2: Motivation Therefore Eq. (2.50) is exactly AdS$_{5}$ × S $^{5}$. While there is no proof yet of the AdS/CFT conjecture a number of tests have found quantities calculated from the field theory side and the string theory side agree. These are nontrivial in the sense that they must be independent of coupling. These tests include checking the equivalence of symmetries in both theories; some correlation functions, usually associated with anomalies, that are independent of quantum corrections and λ = g 2 $_{SY M}$N ; the spectrum of chiral operators match for those that are currently calculable; see [322] and references therein for a more complete list. Having used the above picture to motivate the correspondence we now throw away that geometry and simply take our whole space to be AdS$_{5}$ × S $^{5}$. It is useful to know that the boundary at asymptotic r → ∞ is the usual 3 + 1 D Minkowski. To see this first change coordinates to y = R/r . Then Eq. (2.50) becomes where for the second step we Weyl rescaled the metric by an overall factor of y $^{2}$/R$^{2}$. Now r → ∞ corresponds to y → 0, and we see that at the boundary the radius of the sphere shrinks to 0 and usual 3 + 1 D Minkowski is recovered. ds 2 = R 2 y 2 ( − dt 2 + d⃗x $^{2}$) + R 2 y 2 dy 2 + R $^{2}$d Ω 2 5 (2.59) → − dt 2 + d⃗x 2 + dy 2 + y $^{2}$d Ω 2 $_{5}$, (2.60)