Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 354 (Gauge/Gravity Duality)

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.

The horizon is the largest positive root of $f(r_H)=0$. With $x=r_H^2$,Choosing the positive root gives

After Wick rotation $t=-i{t}_E$, the near-horizon metric hasRegularity at the origin of this polar plane requires the Euclidean black-hole regularity condition ${\beta }_H=4\pi /f^{\prime }(r_H)$. Sincewe obtain

Keep the smooth $\beta ={\beta }_H$ geometry fixed away from the horizon but identify Euclidean time with arbitrary period $\beta$. The horizon then has deficit angle $2\pi (1-\beta /{\beta }_H)$. Its delta-function curvature contributesand henceUsing $\log Z=-I_{\mathrm{grav}}$,

All smooth bulk terms, including the cosmological-constant volume term, are proportional to the Euclidean time period and are annihilated by $1-\beta {\partial }_{\beta }$. The asymptotic Gibbons–Hawking–York boundary term and holographic counterterms are likewise smooth and linear in $\beta$. Only the curvature singularity at the fixed point of the Euclidean time circle survives.

Varying the boundary period and filling it by the corresponding smooth Euclidean saddle computes the same canonical partition function. Because each bulk metric obeys the Einstein equation, the implicit first-order metric variation of the on-shell action reduces to boundary terms; regularity relates the varied horizon radius to the varied period. The thermodynamic identitythen yields the same Bekenstein-Hawking entropy. The conical method is an off-shell way to isolate the local horizon term, while the smooth-saddle method packages that term into the variation of the entire solution.

The two-sided eternal black hole is dual to the thermofield double stateTracing out the left CFT givesThus the geometric period $\beta$ is the boundary inverse temperature, and $S_{\mathrm{BH}}$ is the Von Neumann entropy $S_R=-\mathrm{Tr}({\rho }_R\log {\rho }_R)$, equivalently the entanglement entropy between the two CFTs.

The modular Hamiltonian of this thermal state isFor any first-order state variation with $\mathrm{Tr}\delta \rho =0$,This is the first law of entanglement entropy.

The shell is created by a unitary $U_R$ acting only on the right CFT. Therefore ${\rho }_R\mapsto U_R{\rho }_R{U}_R^{†}$ and its eigenvalues, hence the exact left--right entanglement entropy, do not change. The Hubeny–Rangamani–Takayanagi surface homologous to the complete right boundary remains the old extremal bifurcation surface in the portion of the bulk preceding the shell, behind the enlarged late-time event horizon. Its entropy isnot $A({\mu }^{\prime })/(4G_N)$.

The larger late-time horizon area $A({\mu }^{\prime })$ instead gives the coarse-grained thermodynamic entropy of the final equilibrium black hole. It counts the entropy obtained after discarding detailed information about the coherent unitary excitation. The distinction between the unchanged HRT area and the increased final horizon area is the bulk counterpart of fine-grained entropy conservation under unitary evolution alongside thermodynamic entropy production after coarse graining.