Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 354 / 1 / c / Solution

The state–operator correspondence maps $\mathcal{O}$ to the lowest one-particle state on $S^4\times \mathbb{R}$. On a sphere of radius $r$, cylinder energy equals scaling dimension divided by $r$. The primary has $\Delta =5$.

A translation generator raises the dimension by one and transforms as a vector of $SO(5)$. States with no angular momentum arise from scalar descendant pairs $P^2$, so the allowed descendants areTheir dimensions are $5+2n$, and hence the one-quantum, zero-angular-momentum energies areThese are the $l=0$ normal-mode energies of a massless scalar in global Anti-de Sitter spacetime.