Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 354 / 1 / a

By the state–operator correspondence, a CFT operator of scaling dimension $\Delta$ creates a cylinder state of energy $E=\Delta /R$. The identity gives the vacuum; the first nontrivial low-energy single-trace operator is the conserved stress-energy tensor, with $\Delta =3$ in three dimensions; and one translation raises the dimension by one. Taking the scheme-dependent vacuum energy to vanish,All other single-trace primaries are heavy by assumption, while the first two-graviton state begins at $6/R$.

The vacuum is unique, so $N_0=1$. The stress tensor is the spin-$2$ irreducible representation of $SU(2)$ and has $2j+1=5$ states. At the next level, a translation of spin $1$ givesStress-tensor conservation makes the spin-$1$ divergence a null conformal descendant, leaving dimensions $7+5$. Hence

The corresponding rotation representations areThe would-be $j=1$ representation at $E_2$ is absent because it is the null descendant expressing conservation of the stress tensor.

The AdS-CFT correspondence maps the vacuum to empty anti-de Sitter spacetime and the stress-tensor conformal family to one-graviton normal modes. ThereforeThe distinction between $E_1$ and $E_2$ is a different excitation within the same one-particle conformal family, not an additional graviton.