Capstone Problems

Why capstones belong in the toolkit

AdS/CFT is learned by doing calculations that force the two sides of the dictionary to agree. A slogan can be memorized; a holographic computation cannot. The point of this page is to turn the course into a set of research rehearsals.

Each capstone asks you to start from a boundary observable, choose a bulk dual object, impose the correct boundary and interior conditions, renormalize or extremize the answer, and then check the result in at least two independent ways. This is the rhythm of real holographic work.

The problems below are intentionally not isolated textbook exercises. They are small projects. A good solution should look like a short research note: definitions first, approximation stated clearly, calculation organized cleanly, final answer boxed, and limitations named honestly.

The capstone problems are organized around the same workflow used in research calculations: identify the CFT observable, formulate the bulk problem, impose boundary and interior conditions, renormalize or extremize, and check the answer. The tracks span local correlators, black holes, transport, Wilson loops, RG flows, finite density, entanglement, numerical gravity, and quantum information.

How to use this page

There are three good ways to use these problems.

As a final exam. Choose five capstones: one from correlators, one from finite temperature or transport, one from nonlocal observables, one from entanglement or information, and one from the toolkit. Write complete solutions.

As a research preparation sequence. Work through the capstones in order. For each one, write a clean notebook or note that another student could reproduce.

As a self-diagnostic. Pick the capstone closest to your research area. If you cannot state the source, vev, boundary conditions, counterterms, and consistency checks, revisit the relevant course pages.

A complete capstone solution should include the following items.

Item	What it should contain
Observable	The precise CFT quantity: correlator, free energy, entropy, Wilson loop, transport coefficient, or spectral function.
State and ensemble	Vacuum, thermal state, grand-canonical ensemble, fixed charge ensemble, TFD state, evaporating setup, or deformed vacuum.
Bulk dual	Field, metric perturbation, string worldsheet, brane embedding, extremal surface, or full geometry.
Approximation	Large $N$, large $\lambda$, classical supergravity, probe limit, hydrodynamic limit, linear response, or semiclassical gravity.
Boundary conditions	Source data at the AdS boundary and regularity, infalling, smoothness, horizon, cap, or DeTurck conditions in the interior.
Renormalization	Counterterms, subtractions, contact terms, scheme dependence, or generalized entropy renormalization.
Final answer	A result with dimensions, normalizations, and regime of validity stated.
Checks	Ward identities, conformal scaling, thermodynamics, limits, positivity, causality, numerical convergence, or known special cases.

Capstone 1: scalar two-point function from Euclidean AdS

Problem

Consider a scalar field in Euclidean AdS$_{d+1}$ with metric

$$ds^2 = \frac{L^2}{z^2}\left(dz^2 + d x^i d x_i\right), \qquad z>0,$$

and action

$$S_\phi = \frac{\mathcal N_\phi}{2} \int d^{d+1}x\sqrt g \left(g^{ab}\partial_a\phi\partial_b\phi+m^2\phi^2\right).$$

Let

$$\Delta(\Delta-d)=m^2L^2, \qquad \Delta>\frac d2.$$

Your task is to compute the separated-point two-point function of the dual scalar primary $\mathcal O$.

1. Derive the near-boundary behavior of $\phi$.
2. Write the bulk-to-boundary solution for boundary source $J(x)$.
3. Reduce the on-shell action to a boundary term.
4. Explain which terms are contact terms and which term determines the separated-point correlator.
5. Obtain the scaling

$$\langle \mathcal O(x)\mathcal O(0)\rangle = \frac{C_{\mathcal O}}{|x|^{2\Delta}}.$$

6. State how $C_{\mathcal O}$ depends on conventions.

Solution

The scalar equation is

$$\frac{1}{\sqrt g}\partial_a\left(\sqrt g g^{ab}\partial_b\phi\right)-m^2\phi=0.$$

Near $z=0$, neglecting $x$-derivatives gives

$$z^{d+1}\partial_z\left(z^{1-d}\partial_z\phi\right)-m^2L^2\phi=0.$$

With $\phi\sim z^\alpha$, this becomes

$$\alpha(\alpha-d)=m^2L^2.$$

Thus

$$\phi(z,x) = z^{d-\Delta}\left(J(x)+\cdots\right) + z^\Delta\left(A(x)+\cdots\right),$$

where $J$ is the source in standard quantization and $A$ is proportional to the one-point function.

The normalized Euclidean bulk-to-boundary propagator is conventionally written

$$K_\Delta(z,x;x') = C_\Delta \left(\frac{z}{z^2+|x-x'|^2}\right)^\Delta,$$

with

$$C_\Delta = \frac{\Gamma(\Delta)}{\pi^{d/2}\Gamma(\Delta-d/2)}.$$

This choice satisfies

$$\lim_{z\to0}z^{\Delta-d}K_\Delta(z,x;x')=\delta^{(d)}(x-x')$$

in the distributional sense. The classical solution with source $J$ is

$$\phi_{\rm cl}(z,x)=\int d^d x'\,K_\Delta(z,x;x')J(x').$$

Using the equation of motion, the bulk action reduces to the cutoff boundary term

$$S_{\rm os}(\epsilon) = \frac{\mathcal N_\phi}{2} \int_{z=\epsilon} d^d x\sqrt\gamma\,\phi n^a\partial_a\phi.$$

In Euclidean Poincaré AdS, $\sqrt\gamma=(L/\epsilon)^d$ and $n^a\partial_a=-(\epsilon/L)\partial_z$ for the outward normal at the cutoff surface. The expansion contains divergent local terms in $J$, such as powers of $\epsilon$ multiplying $J^2$, $J\Box J$, and so on. These are removed by local counterterms and affect only contact terms in position space.

The nonlocal finite term gives

$$S_{\rm ren}[J] = -\frac{\mathcal N_\phi}{2} (2\Delta-d)C_\Delta L^{d-1} \int d^d x\,d^d y\, \frac{J(x)J(y)}{|x-y|^{2\Delta}}$$

up to sign conventions for the Euclidean generating functional and possible local contact terms. Differentiating twice gives

$$\langle\mathcal O(x)\mathcal O(y)\rangle = \mathcal N_\phi L^{d-1}(2\Delta-d)C_\Delta \frac{1}{|x-y|^{2\Delta}} + \text{contact terms},$$

up to the convention for whether $Z=e^{-S_{\rm ren}}$ or $W=-S_{\rm ren}$ is used.

The universal content is the conformal scaling $|x-y|^{-2\Delta}$. The coefficient depends on the normalization of the bulk field, the normalization of $\mathcal O$, the sign convention for $W[J]$, and possible finite local counterterms. A physical comparison between two calculations must keep these conventions fixed.

Research extension. Repeat the computation in momentum space using the regular solution

$$\phi(z,k)=J(k)\,\mathcal C_\nu k^\nu z^{d/2}K_\nu(kz), \qquad \nu=\Delta-\frac d2,$$

and identify how logarithms arise when $\nu$ is an integer.

Capstone 2: black-brane thermodynamics and the $N^2$ plasma

Problem

Consider the planar AdS$_{d+1}$ black brane

$$ds^2 = \frac{L^2}{z^2} \left[ -f(z)dt^2+d\vec x^{\,2}+\frac{dz^2}{f(z)} \right], \qquad f(z)=1-\left(\frac{z}{z_h}\right)^d.$$

Compute its temperature, entropy density, energy density, pressure, and free-energy density. Then specialize to $d=4$ and use

$$\frac{L^3}{G_5}=\frac{2N^2}{\pi}$$

to obtain the thermodynamics of strongly coupled planar $\mathcal N=4$ SYM.

Solution

Near the horizon, set $z=z_h-u$ with small $u$. Since

$$f(z)=1-\left(1-\frac{u}{z_h}\right)^d \simeq \frac{du}{z_h},$$

the Euclidean metric in the $(u,\tau)$ plane is smooth only if

$$T=\frac{|f'(z_h)|}{4\pi}=\frac{d}{4\pi z_h}.$$

The entropy density follows from the Bekenstein-Hawking formula:

$$s=\frac{1}{4G_{d+1}}\left(\frac{L}{z_h}\right)^{d-1}.$$

Using conformal invariance and the first law,

$$\epsilon=(d-1)p, \qquad s=\frac{\partial p}{\partial T},$$

and $z_h=d/(4\pi T)$, one obtains

$$p = \frac{1}{16\pi G_{d+1}} \left(\frac{L}{z_h}\right)^{d-1} \frac{1}{z_h} = \frac{L^{d-1}}{16\pi G_{d+1}}\left(\frac{4\pi T}{d}\right)^d,$$

and therefore

$$f_{\rm free}=-p, \qquad \epsilon=(d-1)p, \qquad Ts=\epsilon+p=dp.$$

For $d=4$,

$$T=\frac{1}{\pi z_h}, \qquad s=\frac{1}{4G_5}\left(\frac{L}{z_h}\right)^3 =\frac{\pi^2}{2}N^2T^3.$$

The pressure, free energy, and energy density are

$$p=\frac{\pi^2}{8}N^2T^4, \qquad f_{\rm free}=-\frac{\pi^2}{8}N^2T^4, \qquad \epsilon=\frac{3\pi^2}{8}N^2T^4.$$

The $N^2$ scaling reflects the adjoint degrees of freedom of the deconfined plasma and, in the bulk, the scaling $L^3/G_5\sim N^2$.

Research extension. Add the leading finite-coupling correction schematically controlled by $\alpha'^3R^4$. Explain why it enters as a power of $\lambda^{-3/2}$ rather than $1/\lambda$.

Capstone 3: Kubo formula and $\eta/s$

Problem

For a translationally invariant thermal state, the shear viscosity is defined by the Kubo formula

$$\eta = -\lim_{\omega\to0}\frac{1}{\omega} \operatorname{Im}G^R_{T_{xy}T_{xy}}(\omega,\vec k=0).$$

In an Einstein-gravity dual, the source for $T_{xy}$ is a metric perturbation $h_{xy}$. Show that the transverse graviton mode obeys the same equation as a minimally coupled massless scalar at zero spatial momentum. Use the membrane argument to derive

$$\frac{\eta}{s}=\frac{1}{4\pi}.$$

Solution

Let

$$h^x{}_y(t,z)=\psi(z)e^{-i\omega t}.$$

For an isotropic black brane governed by two-derivative Einstein gravity, this tensor channel decouples from other perturbations. To quadratic order, its effective action takes the scalar form

$$S^{(2)} = -\frac{1}{32\pi G_{d+1}} \int d^{d+1}x\sqrt{-g}\,g^{ab}\partial_a\psi\partial_b\psi + \text{boundary terms}.$$

The canonical radial momentum is

$$\Pi(z,\omega) = -\frac{1}{16\pi G_{d+1}}\sqrt{-g}\,g^{zz}\partial_z\psi.$$

The retarded function is extracted as

$$G^R_{T_{xy}T_{xy}}(\omega,0) = \lim_{z\to0}\frac{\Pi(z,\omega)}{\psi(z, \omega)} + \text{contact terms}.$$

At small $\omega$, the radial flow of $\Pi/(i\omega\psi)$ becomes trivial, so it can be evaluated at the horizon. Imposing the infalling condition gives, near the horizon,

$$\partial_z\psi \simeq -i\omega\sqrt{\frac{g_{zz}}{-g_{tt}}}\,\psi.$$

Therefore

$$\eta = \lim_{\omega\to0}\frac{\Pi}{i\omega\psi} = \frac{1}{16\pi G_{d+1}} \left.\sqrt{\frac{-g}{-g_{tt}g_{zz}}}\,g^{xx}\right|_{z=z_h}.$$

For the shear graviton in an isotropic metric, this horizon expression is one quarter of the entropy density divided by $\pi$:

$$\eta = \frac{1}{16\pi G_{d+1}}\left(\text{horizon area density}\right) = \frac{s}{4\pi}.$$

Hence

$$\boxed{\frac{\eta}{s}=\frac{1}{4\pi}}.$$

The result assumes a two-derivative Einstein action, isotropy, a regular horizon, no mixing with additional light tensor-channel fields, and the standard infalling prescription. Higher-derivative terms, anisotropy, or nonminimal couplings can change it.

Research extension. Compare this derivation with the absorption-cross-section argument. Which step fails if the graviton has higher-derivative interactions?

Capstone 4: heavy-quark potential from a U-shaped string

Problem

Compute the strong-coupling potential between an external quark and antiquark in the vacuum of planar $\mathcal N=4$ SYM using a fundamental string in Euclidean AdS$_5$.

Use

$$ds^2=\frac{L^2}{z^2}\left(dt_E^2+dx^2+d\vec y^{\,2}+dz^2\right), \qquad \frac{L^2}{\alpha'}=\sqrt\lambda.$$

A rectangular Wilson loop has temporal extent $T_E$ and spatial separation $R$. Use static gauge $t_E=\tau$, $x=\sigma$, $z=z(x)$.

1. Derive the Nambu-Goto action.
2. Find the first integral associated with translation in $x$.
3. Express $R$ in terms of the turning point $z_*$.
4. Subtract the two straight-string masses.
5. Obtain $V(R)\propto-\sqrt\lambda/R$.

Solution

The Nambu-Goto action is

$$S_{\rm NG} = \frac{1}{2\pi\alpha'} \int d\tau d\sigma\sqrt{\det h_{ab}}.$$

For the static embedding,

$$S_{\rm NG} = \frac{T_E L^2}{2\pi\alpha'} \int_{-R/2}^{R/2}dx\, \frac{\sqrt{1+z'(x)^2}}{z^2}.$$

The Lagrangian has no explicit $x$ dependence, so

$$\frac{1}{z^2\sqrt{1+z'^2}}=\frac{1}{z_*^2},$$

where $z_*$ is the maximal depth and $z'(0)=0$. Therefore

$$z'^2=\frac{z_*^4}{z^4}-1.$$

The separation is

$$\frac{R}{2} = \int_0^{z_*}dz\,\frac{z^2}{\sqrt{z_*^4-z^4}} = z_*\int_0^1dv\,\frac{v^2}{\sqrt{1-v^4}}.$$

The regularized energy is obtained from

$$V(R)=\frac{S_{\rm NG}^{\rm conn}-2S_{\rm straight}}{T_E}.$$

After subtracting the divergent straight-string masses, the standard result is

$$\boxed{ V(R) = -\frac{4\pi^2}{\Gamma(1/4)^4}\frac{\sqrt\lambda}{R} }$$

for the conventional normalization of the fundamental string action in AdS$_5\times S^5$.

The $1/R$ behavior is required by conformal invariance. The nontrivial result is the strong-coupling coefficient proportional to $\sqrt\lambda$, reflecting the classical string tension in AdS units.

Research extension. Repeat the analysis in a confining background. Identify the geometric condition under which the large-$R$ potential becomes linear.

Capstone 5: RT surfaces and mutual information for two intervals

Problem

In the vacuum of a large-$c$ holographic CFT$_2$, consider two equal intervals

$$A=[0,\ell], \qquad B=[\ell+x,2\ell+x],$$

with $x>0$. Using the RT formula in AdS$_3$, determine the mutual information

$$I(A:B)=S(A)+S(B)-S(A\cup B)$$

at leading order in $c$.

Your solution should explain the competition between disconnected and connected RT saddles.

Solution

For a single interval of length $L$ in the vacuum CFT$_2$,

$$S(L)=\frac{c}{3}\log\frac{L}{\epsilon}.$$

Thus

$$S(A)+S(B)=\frac{2c}{3}\log\frac{\ell}{\epsilon}.$$

For $A\cup B$, there are two candidate RT pairings. The disconnected one gives

$$S_{\rm disc}(A\cup B) = S(\ell)+S(\ell) = \frac{2c}{3}\log\frac{\ell}{\epsilon}.$$

The connected one gives

$$S_{\rm conn}(A\cup B) = S(x)+S(2\ell+x) = \frac{c}{3} \log\frac{x(2\ell+x)}{\epsilon^2}.$$

RT chooses the smaller value:

$$S(A\cup B)=\min\{S_{\rm disc},S_{\rm conn}\}.$$

Therefore

$$I(A:B) = \max\left\{0, \frac{c}{3}\log\frac{\ell^2}{x(2\ell+x)} \right\}.$$

The phase transition occurs when

$$\ell^2=x(2\ell+x).$$

Writing $y=x/\ell$, this gives

$$y^2+2y-1=0, \qquad y=\sqrt2-1.$$

Thus

$$I(A:B)>0 \quad\Longleftrightarrow\quad \frac{x}{\ell}<\sqrt2-1.$$

At leading classical order, holographic mutual information is either $O(c)$ or exactly zero, because only the dominant RT saddle is retained. Subleading bulk quantum corrections smooth the interpretation but not the leading large-$c$ phase structure.

Research extension. Add a thermal scale by replacing the vacuum interval entropy with the finite-temperature CFT$_2$ expression. Track how the transition changes with $\ell T$ and $xT$.

Capstone 6: relevant deformation, domain wall, and holographic $c$-function

Problem

Consider a $(d+1)$-dimensional Einstein-scalar model

$$S = \frac{1}{16\pi G_{d+1}} \int d^{d+1}x\sqrt{-g}\left[ R-\frac12(\partial\phi)^2-V(\phi) \right],$$

with domain-wall ansatz

$$ds^2=dr^2+e^{2A(r)}\eta_{\mu\nu}dx^\mu dx^\nu, \qquad \phi=\phi(r).$$

Assume the geometry approaches AdS in the UV, where $A(r)\sim r/L_{\rm UV}$. Show that the null energy condition implies a monotonic holographic $c$-function.

Solution

The Einstein equations for the domain-wall ansatz imply

$$A''(r)=-\frac{1}{2(d-1)}\phi'(r)^2.$$

Thus

$$A''(r)\le 0.$$

For flows with $A'(r)>0$, define

$$\mathcal C(r) = \frac{\mathcal N_d}{[A'(r)]^{d-1}},$$

where $\mathcal N_d$ is a positive normalization chosen so that at an AdS fixed point $\mathcal C$ equals the appropriate central quantity, such as $c$ in $d=2$ or $a$ in many $d=4$ Einstein-gravity flows.

Differentiating gives

$$\mathcal C'(r) = -(d-1)\mathcal N_d\frac{A''(r)}{[A'(r)]^d}.$$

Since $A''(r)\le0$, we have

$$\mathcal C'(r)\ge0$$

as $r$ increases toward the UV. Equivalently, $\mathcal C$ decreases along the RG flow from UV to IR.

At an AdS fixed point with radius $L_*$,

$$A'(r)=\frac{1}{L_*}, \qquad \mathcal C_*\propto\frac{L_*^{d-1}}{G_{d+1}}.$$

The monotonicity is a geometric reflection of the null energy condition. It is not, by itself, a proof that every possible QFT monotone is represented by this particular function; the precise identification depends on dimension, theory, and gravity action.

Research extension. Work out the same monotonicity in coordinates $z$ where the metric is written as

$$ds^2=e^{2A(z)}\left(-dt^2+d\vec x^{\,2}\right)+e^{2B(z)}dz^2.$$

Be explicit about how the radial orientation changes the sign of the flow.

Capstone 7: RN-AdS, chemical potential, and the AdS$_2$ throat

Problem

Consider a finite-density holographic CFT dual to an Einstein-Maxwell theory admitting a planar charged black brane. Near extremality, the geometry develops a near-horizon region

$$\mathrm{AdS}_2\times\mathbb R^{d-1}.$$

Your task is to explain how the chemical potential, charge density, and emergent IR scaling appear in the bulk.

1. State the source/vev interpretation of the near-boundary gauge field.
2. Explain why regularity requires a gauge choice with $A_t(r_h)=0$ at the horizon.
3. Derive the general form of the near-horizon extremal expansion that produces AdS$_2$.
4. For a probe scalar of charge $q$ and mass $m$, state the form of the IR scaling exponent $\nu_k$.
5. Explain how violation of the AdS$_2$ BF bound signals an instability.

Solution

Near the AdS boundary, the bulk gauge field behaves as

$$A_t(r)=\mu-\frac{\rho}{\mathcal N_A r^{d-2}}+\cdots,$$

or in $z$ coordinates as $A_t(z)=\mu-\rho z^{d-2}/\mathcal N_A+\cdots$, depending on conventions. The leading coefficient $\mu$ is the chemical potential, while the radial electric flux determines the charge density:

$$\langle J^t\rangle \propto \lim_{r\to\infty}\sqrt{-g}F^{rt}.$$

In Euclidean signature, the thermal circle shrinks at the horizon. The one-form $A=A_t dt$ is regular there only in a gauge for which

$$A_t(r_h)=0.$$

The gauge-invariant chemical potential is therefore the potential difference between boundary and horizon.

At extremality, the blackening factor has a double zero:

$$f(r) \simeq \frac{(r-r_h)^2}{L_{2,\mathrm{eff}}^2}\times\text{constant}.$$

After a suitable rescaling of time and radial coordinate, the near-horizon metric becomes

$$ds^2 \simeq L_2^2\left(-\zeta^2dt^2+\frac{d\zeta^2}{\zeta^2}\right) + \frac{r_h^2}{L^2}d\vec x^{\,2}.$$

The spatial directions do not scale under the AdS$_2$ dilatation; this is why the IR is often called semi-local or locally critical.

For a charged scalar mode with boundary spatial momentum $k$, the AdS$_2$ region sees an effective mass and effective electric field. The IR Green function has scaling

$$\mathcal G_k(\omega)\sim \omega^{2\nu_k},$$

where schematically

$$\nu_k = \sqrt{ \frac{1}{4} +L_2^2\left(m^2+k^2 g^{xx}(r_h)\right) -q^2 e_d^2 }.$$

The precise normalization of $e_d$ depends on the gauge-field convention and the background dimension. If the expression under the square root becomes negative, the AdS$_2$ BF bound is violated and the extremal charged black brane is unstable to forming scalar hair or another ordered phase.

Research extension. Perform the matched-asymptotic logic for a neutral scalar and explain how the UV coefficients combine with the IR Green function to produce the full boundary Green function.

Capstone 8: formulate an Einstein-DeTurck boundary value problem

Problem

Formulate a static numerical holography problem using the Einstein-DeTurck method. You do not need to solve it numerically; the goal is to specify the problem so precisely that it could be implemented.

Choose one of the following examples:

• a static inhomogeneous black brane with a periodic boundary source,
• an AdS soliton with a spatially varying boundary metric,
• a static domain-wall geometry with one inhomogeneous spatial direction.

Your task is to write an ansatz, define the DeTurck vector, specify a reference metric, list all boundary conditions, and state the convergence checks.

Solution

A typical static inhomogeneous black-brane ansatz in four bulk dimensions might be

$$ds^2 =\frac{L^2}{z^2}\left[ -(1-z)P(z)Q_{tt}(z,x)dt^2 +\frac{Q_{zz}(z,x)dz^2}{(1-z)P(z)} +Q_{xx}(z,x)\left(dx+z^2Q_{zx}(z,x)dz\right)^2 +Q_{yy}(z,x)dy^2 \right],$$

where $z=0$ is the AdS boundary, $z=1$ is the horizon, and $x$ is periodic. The functions $Q_i(z,x)$ are the numerical unknowns.

The Einstein-DeTurck equation is

$$R_{ab}+\frac{d}{L^2}g_{ab}-\nabla_{(a}\xi_{b)}=T^{\rm matter}_{ab}-\frac{1}{d-1}T^{\rm matter}g_{ab},$$

with

$$\xi^a =g^{bc}\left(\Gamma^a_{bc}[g]-\bar\Gamma^a_{bc}[\bar g]\right).$$

The reference metric $\bar g$ should have the same asymptotic, horizon, and coordinate structure as the desired solution. A common choice is the homogeneous black brane with the same temperature and boundary metric.

Boundary conditions include:

• at $z=0$, Dirichlet data fixing the boundary metric and sources;
• at $z=1$, regularity in Euclidean or ingoing coordinates, including a fixed temperature condition such as $Q_{tt}(1,x)=Q_{zz}(1,x)$;
• periodicity in $x$;
• any reflection or parity conditions if the unit cell is reduced;
• regularity at axes or caps, if present.

After solving, one must verify:

$$\xi^2=g_{ab}\xi^a\xi^b\to0$$

with increasing resolution. Additional checks include spectral or finite-difference convergence, residual convergence, Ward identities of the boundary stress tensor, agreement with known homogeneous limits, thermodynamic first-law checks, and independence of the reference metric within the same boundary-value problem.

A solution of the Einstein-DeTurck equation with nonzero $\xi^a$ is a Ricci soliton, not the desired Einstein solution. Numerically, this is the trap the method forces you to confront rather than hide.

Research extension. Add a Maxwell field and scalar lattice source. State which boundary data correspond to chemical potential, charge density, lattice amplitude, and scalar vev.

Capstone 9: Page curve and island saddle competition

Problem

Consider a simplified evaporating black-hole setup in which the entropy of radiation is computed by minimizing a generalized entropy over candidate quantum extremal surfaces. Suppose the no-island saddle gives

$$S_{\rm no\ island}(t)=\alpha t$$

for early and intermediate times, while the island saddle gives approximately

$$S_{\rm island}(t)=S_0+\delta S(t), \qquad |\delta S(t)|\ll S_0$$

at late times. Use this model to explain the Page transition.

Your answer should connect the toy model to the general formula

$$S(R) = \min_{I}\operatorname*{ext}_{\partial I} \left[ \frac{\operatorname{Area}(\partial I)}{4G_N}+S_{\rm bulk}(R\cup I) \right].$$

Solution

The generalized entropy prescription says that the fine-grained entropy of the radiation region $R$ is not computed only by the entropy of quantum fields in $R$. One must allow candidate islands $I$ in the gravitating region and extremize

$$S_{\rm gen}(R,I) = \frac{\operatorname{Area}(\partial I)}{4G_N}+S_{\rm bulk}(R\cup I).$$

Then one chooses the dominant saddle, meaning the smallest extremized generalized entropy.

In the toy model, the empty-island saddle gives

$$S_{\rm no\ island}(t)=\alpha t.$$

This is the Hawking-like answer: the radiation entropy grows with time because the emitted quanta appear increasingly entangled with degrees of freedom behind the horizon.

The island saddle gives roughly

$$S_{\rm island}(t)=S_0+\delta S(t),$$

where $S_0$ is set by an area term of order the black-hole entropy. The physical entropy is

$$S_R(t) = \min\left\{\alpha t,\,S_0+\delta S(t)\right\}.$$

The transition occurs when

$$\alpha t_{\rm Page}=S_0+\delta S(t_{\rm Page}).$$

If $\delta S$ is small, then

$$t_{\rm Page}\simeq \frac{S_0}{\alpha}.$$

Before this time, the no-island saddle dominates and the entropy grows. After this time, the island saddle dominates and the entropy is bounded by a quantity of order the black-hole entropy. This produces the Page-curve behavior expected from unitary evaporation.

The important lesson is not the linear toy model. The lesson is the saddle competition. The same semiclassical path integral can contain more than one extremal generalized-entropy saddle, and the dominant saddle can change as the radiation region grows.

Research extension. In a two-dimensional JT-gravity-plus-bath model, write the generalized entropy for a candidate island endpoint and solve the extremality condition explicitly in a late-time approximation.

Capstone 10: write a miniature research proposal

Problem

Choose one topic from the course and write a two-page research proposal. It should be narrow enough that a calculation can begin within a week.

Your proposal must include:

1. a boundary observable,
2. a bulk dual object,
3. a background or class of backgrounds,
4. a regime of validity,
5. a calculation plan,
6. a list of consistency checks,
7. one possible failure mode,
8. one reason the result would be useful even if it is negative.

Good topics include:

Direction	Example proposal
Correlators	Compute how a scalar two-point function changes under a simple relevant deformation.
Transport	Study a low-frequency conductivity in an axion model and compare horizon and Green-function methods.
Wilson loops	Compare screening lengths in two finite-temperature backgrounds.
Entanglement	Track an RT phase transition in a confining geometry.
Finite density	Analyze the onset of an AdS$_2$ BF-bound instability as a function of momentum.
Numerical holography	Formulate and solve a one-dimensional inhomogeneous black-brane DeTurck problem.
Black-hole information	Build a toy generalized-entropy model with two competing island saddles.

Solution

A strong miniature proposal has the following structure.

Title. Make it specific. For example: “Horizon and boundary computations of DC conductivity in a linear-axion black brane.”

Question. State one question, not a field. For example: “How does the DC conductivity depend on the momentum-relaxation scale $k$ at fixed charge density and temperature?”

Observable. Define the boundary observable:

$$\sigma_{\rm DC}=\lim_{\omega\to0}\frac{1}{i\omega}G^R_{J_xJ_x}(\omega,0).$$

Bulk setup. State the action, ansatz, fields, and boundary conditions. For the axion example, one might use fields $\psi_I=kx_I$ to break translations homogeneously.

Approximation. State whether the calculation is classical gravity, probe limit, linear response, hydrodynamic, numerical, or perturbative in a small parameter.

Plan. Break the work into reproducible steps:

1. solve or quote the background;
2. perturb by an electric field;
3. construct a radially conserved current;
4. evaluate it at the horizon using regularity;
5. compare with the small-frequency Green-function extraction.

Checks. Include at least three:

• $k\to0$ should recover the translationally invariant divergence or Drude pole;
• Ward identities should hold;
• horizon and boundary methods should agree;
• dimensions and scaling should match;
• known limits in the literature should be reproduced.

Failure mode. For example: “The simple axion model may not capture a realistic lattice; the result may be model-dependent.”

Value. A negative or model-dependent result can still clarify which parts of the conductivity are universal horizon data and which depend on the chosen translation-breaking mechanism.

A grading rubric for yourself

Use this rubric to evaluate a capstone solution.

Criterion	Excellent	Needs work
Dictionary	Source, vev, operator, and bulk dual are all explicit.	The answer jumps between bulk and boundary language without specifying the map.
Boundary conditions	Boundary and interior conditions are stated before solving.	Conditions are imposed after the fact or confused with gauge choices.
Approximation	Large $N$, large $\lambda$, probe, hydrodynamic, or semiclassical limits are named.	The result is presented as exact when it is not.
Renormalization	Divergences, counterterms, subtractions, or contact terms are handled.	Infinite or scheme-dependent quantities are treated as physical.
Checks	At least two independent checks are performed.	Only algebraic manipulation is shown.
Interpretation	The result is connected to CFT physics.	The calculation remains a bulk exercise with no boundary meaning.
Limitations	Model dependence and failure modes are named.	The answer oversells universality.

Common capstone mistakes

Mistake 1: treating the radial coordinate as literally the RG scale. The radial/scale relation is powerful, but a complete RG statement requires boundary conditions, counterterms, and a definition of sources and vevs.

Mistake 2: forgetting that Euclidean and Lorentzian prescriptions differ. Euclidean regularity computes Euclidean correlators and thermodynamic saddles. Retarded correlators require Lorentzian infalling conditions.

Mistake 3: extracting a vev before renormalizing. Near-boundary coefficients are not automatically expectation values. Counterterms and finite scheme choices matter.

Mistake 4: confusing a saddle with the answer. For Hawking-Page transitions, RT phase transitions, Wilson-loop screening, and island calculations, the physical answer comes from comparing saddles.

Mistake 5: calling every bottom-up result universal. A horizon formula may be robust inside a model class, but changing the matter sector, higher-derivative terms, boundary conditions, or symmetry assumptions can change the result.

Mistake 6: trusting a numerical solution without diagnostics. A plot is not a solution. Residuals, convergence, Ward identities, boundary data, and DeTurck norm checks are part of the calculation.

Suggested capstone sequences

Different readers should not necessarily do the problems in the same order.

Goal	Suggested sequence
Master the original computational dictionary	1, 2, 4, 5
Prepare for holographic transport research	2, 3, 7, 8
Prepare for AdS/QCD or confinement work	4, 6, 8, 10
Prepare for holographic quantum matter	3, 7, 8, 10
Prepare for entanglement and information	5, 9, 10
Prepare for numerical holography	2, 6, 8, then one original extension

What mastery looks like

After completing several capstones, you should be able to look at a new holography paper and identify the calculation almost immediately:

• Which boundary quantity is being computed?
• Which bulk object computes it?
• Which limit makes the computation controllable?
• Which boundary and interior conditions select the answer?
• Which pieces are local scheme-dependent data?
• Which result is universal, and which is model-specific?
• What would you check before believing the result?

That is the practical endpoint of this course. Not “knowing AdS/CFT” as a collection of famous formulas, but being able to formulate, execute, debug, and interpret a holographic calculation.

Further reading

For the capstones, the most useful references are not new. They are the technical anchors used throughout the course.

Capstone	Good starting references
Scalar correlators	Gubser-Klebanov-Polyakov, Witten, D’Hoker-Freedman lectures, Skenderis
Black-brane thermodynamics	Witten on thermal AdS/CFT, Aharony et al. review
Real-time response and viscosity	Son-Starinets, Policastro-Son-Starinets, Iqbal-Liu
Wilson loops	Maldacena, Rey-Yee
Entanglement	Ryu-Takayanagi, Hubeny-Rangamani-Takayanagi, Lewkowycz-Maldacena
Numerical holography	Headrick-Kitchen-Wiseman, Dias-Santos-Way
Islands and Page curves	Penington, Almheiri et al., Penington-Shenker-Stanford-Yang