State–operator correspondence

The state–operator correspondence maps a local operator of scaling dimension $\Delta$ to a state on $S^{d-1}\times \mathbb{R}$ whose cylinder energy is $\Delta /r$ for sphere radius $r$.

For a primary operator, $T(z)\mathcal{O}(0)\sim h\mathcal{O}(0)/z^2+\partial \mathcal{O}(0)/z$. Taking the residue against $z^{n+1}$ gives the weight for $n=0$, the derivative descendant for $n=-1$, and zero for every $n>0$. Conversely the absence of positive-mode actions excludes higher poles in this operator product and gives the primary transformation law. The antiholomorphic argument is identical. Thus primary states and primary operators correspond under radial quantization.

If the only light field in AdS4 is the graviton, the dual CFT3 spectrum begins with the vacuum at dimension zero, the spin-two stress tensor at dimension three, and its non-null level-one descendants of spins two and three at dimension four.