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

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.

At spacelike separation $x_{12}^2>0$, conformal symmetry fixes the scalar-primary two-point function toApplying the d'Alembertian at $x_2$ givesaway from contact terms. Thus, up to a real multiplicative constant, $F_0\propto (x_{12}^2{)}^{-\Delta -1}$. The scalar conformal unitarity bound makes the coefficient nonnegative and makes it vanish when the bound is saturated, as expected for the null free-field equation of motion.

Withthe time-ordered vacuum correlator can be written, in this source convention, asEquivalent formulas in terms of the connected generating functional differ only by the standard factors of $i$.

A constant source deforms the action by $J\int d^{d}x\mathcal{O}$. Since $[J]=d-\Delta$, a nonzero $J$ introduces no scale only ifThis marginality is necessary; for the deformed theory to remain a CFT for every $J$, the operator must additionally be exactly marginal, with vanishing beta function.

For a bulk scalar with $m^2=\Delta (\Delta -d)$ in unit-radius Poincaré AdS, the AdS scalar-field boundary asymptotics areThus $J$ and $\mathcal{O}$ are, up to normalization and local counterterms, the nonnormalizable and normalizable boundary coefficients respectively.

For $d=3$, $\Delta =1$, and $J=0$, alternative quantization givesThe bulk Klein-Gordon equation issoAt separated points in any state,This contains only radial bulk derivatives, as required.