Teaching the complete algebraic holography course

Edition 2.2, 1 October 2026. The 32 notes form a written draft for 30 teaching weeks: 60 lecture hours in Semester I and 45 in Semester II, with a separate weekly problem session if available. These allocations have not been rehearsed. The opening four lectures retain their detailed meeting guide.

The pedagogical reference is the canonical AQFT course. Preserve its distinction between a calculation students can reconstruct, the fuller instructor account, and a research extension. A stated theorem is not a failed proof: it becomes a problem only when students cannot identify its hypotheses or how it is used.

The recurring classroom method

Begin by asking students to identify the system, state, and accessible algebra. Ask for a prediction before introducing the main formula. Work the first nontrivial case at the board, pausing where a trace, a support condition, a regulator, or a change of representation enters. Finish by asking what additional input would be needed for a geometric interpretation.

For a four-hour unit, use two meetings. A provisional first meeting is 15 minutes of retrieval, 25 of setup, 40 of calculation, 20 of a paired check, and 20 of discussion. The second can use 15 minutes of reconstruction, 45 of the next calculation, 25 of interpretation, 25 of problems, and 10 of synthesis. Adjust this after teaching; the numbers are a preparation tool, not evidence that a dense derivation fits.

For a three-hour unit, a possible allocation is 15 minutes of retrieval, 35 of setup, 45 of the worked example, a 10-minute break, 30 of interpretation, 30 of problems, and 15 of synthesis. Do not sacrifice the worked example to cover every optional paragraph.

Lectures 5–7 retain the longer routed lists used in the opening dossier. Lectures 8–32, rewritten to the AQFT standard, have fifteen routed problems each: six core, six self-study and three research extensions. Their answer checkpoints expose the decisive step but leave room for students to write full solutions.

Semester I: the board calculation and its diagnostic question

Lectures	Calculation to retain in class	Diagnostic question before moving on
1–4	Restricted Bell states, any-pair qutrit decoding, modular matrix units, central entropy decomposition	Can a student distinguish a logical operator from its physical representative and from its compression?
5	The eight physical basis vectors and the two parity blocks; one coherent state’s entropy	Why is the flux label central for regional operations but not for every global physical operation?
6	The four-leg tensor and one two-vertex Schmidt decomposition; the authorized four-leaf region	Why does a bond dimension bound entropy without always determining it?
7	The pair-12 dephasing coefficient and the trace-distance obstruction	Which statement concerns one decoder, and which rules out every exact decoder?
8	The reduced kernel of two oscillators and its Mehler match; Williamson’s form in the block-diagonal case; the table of §6, where the entropy grows by $\frac{1}{3}\log 2$ per halving of $a$ and the mutual information converges	Why is restricting a covariance different from deleting couplings, and why do the endpoint divergences cancel in the mutual information?
9	Separating from cyclicity and locality; the norm cost of a rare Schmidt component; the Powers modular spectrum; the unitary-excitation formula for relative entropy; the split inclusion	Why do neither a center nor a divergent entropy characterize type III, and what does a buffer restore?
10	The KMS strip of the two-point function on an orbit; the detector rate by a contour shift to the middle of the strip; the coherent-state relative entropy as boost energy	Which of modular time, boost rapidity, and proper time is dimensionful, and where does the Boltzmann factor come from?
11	The wedge-to-diamond map on the slice $X^0=0$; the flow of $\zeta$ in null coordinates; the ball energy kernel; the exact thermal relative entropy of an interval	Why is the first law only first order, while relative entropy is nonnegative at finite separation?
12	The two-point selection rule from inversion; the second-level norm; the Schwarzian on the cylinder and Cardy’s formula	Which CFT data are fixed by symmetry, and why does the central charge control both the Casimir energy and the high-temperature entropy?
13	The conformal compactification; the indicial equation and the norm test; the hypergeometric quantization $\omega =\Delta +2n+\ell$ and its degeneracies	Why can a negative mass squared be stable, and why does one bulk field give exactly one conformal family?
14	The three pairings of $\mathrm{Tr}{\Phi }^4$; the cumulants of $\mathrm{Tr}{\Phi }^2$; the derivation of $L^3/G_5=2N^2/\pi$	Which parameter controls loops, which controls string corrections, and what does large $N$ leave undetermined?
15	The conformally coupled mode and its counterterm; then the general finite parts that give $2\Delta -d$, and the propagator normalization	Why does the counterterm change the finite response, and why is discarding the divergence written in terms of $J$ a different prescription?
16	The twist dimension from the Schwarzian; the Poincaré and BTZ geodesics and their regulated lengths	Which step calculates a length, which identifies it with entropy, and why does the geodesic run along the horizon for long intervals?
17	The four answers of the three-qutrit code and the first law with a varying central term; the first-order change of the BTZ geodesic against the modular energy of the thermal interval, in two gauges	Can the student say which term cancels in relative entropy, in the code and for the interval, and why the integrands of the geodesic first law agree point by point in Fefferman–Graham gauge but not in BTZ coordinates?

The oscillator unit can absorb an additional problem session if the class has little Gaussian-state experience. The lattice tables of Lectures 8 and 10 are reproducible from the course scripts; the high-precision coherent-state computation of Lecture 10 takes minutes and is best shown, not run, in class. The CFT unit now contains the two-dimensional toolkit that Lectures 16 and 23 use: the Virasoro algebra, the Schwarzian and Cardy’s formula. A class without any prior exposure to conformal field theory may still need a problem session on radial quantization before Lecture 13.

Semester II: retain one argument per meeting

Lecture	Classroom calculation	Material to defer if time is tight
18	The AdS${}_2$ kernel for any $\Delta$ checked by Poisson’s integral; the three arcs of global AdS${}_3$ and the code of Lecture 2	Complexified kernels, black-hole obstructions and interacting reconstruction
19	Both two-interval candidates and the crossing; the bridge that lies in the entanglement wedge and in no causal wedge	The general inclusion theorem, quantum extremal surfaces, exact twist correlators
20	The four steps of the data-processing proof; the equality theorem; the Petz map of the three-qutrit code	Universal recovery channels and rotated Petz maps
21	The sector logarithm cancellation; the first-law derivation of the modular-Hamiltonian identity from a central-area formula	The remaining directions of Harlow’s theorem; the gravitational renormalization
22	The static AdS${}_3$ identity: length variation, the one-form and Stokes’ theorem; injectivity by two radial derivatives	The Iyer–Wald form in general and the boosted frames
23	The modular operator of the thermofield double; the Euclidean action by volume subtraction and the Hawking–Page temperature	The BTZ modular exchange, late-time correlators and the time-averaged bound
24	The continuation of the outgoing mode across the horizon; the Haar purity and its first correction; the Hayden–Preskill bound for an old black hole	Page’s exact formula, the young black hole, and the firewall argument
25	The JT equations, the Schwarzian and its thermodynamics; the Weyl factor in the island functional	The Euclidean welding in detail, the evaporating model, one-loop exactness
26	The zero-temperature quadratic with both Hessian directions; the Kruskal distances and the late-time island equation	The bosonization step, the finite-time extremization, and islands in higher dimensions
27	The envelope argument for the cosmic brane; the Riemann–Hurwitz count and the $2S_0$	The West Coast resolvent, general-$n$ replica geometries and the ensemble interpretation
28	The region map of AdS${}_3$ through the shared geodesic of complementary arcs; the transfer of locality; the three-arc intersection under Haag duality and strong additivity	Rehren’s theorem in general dimension, the conformal Bisognano–Wichmann theorem, interacting boundary limits
29	The c-number commutator of a Gaussian family; the type I algebra of thermal AdS from the convergence of $z(q)$; the proof that a decaying correlator excludes a type I factor	The BTZ spectral density in detail, the converse criterion of Araki and Woods, mirror operators
30	Two dilations composed into a translation; the chiral current with $K$, $P$ and the KMS check; Wiesbrock’s product and the finite-dimensional obstruction	The analytic argument of Borchers’ theorem, standard inclusions and the Möbius group, null-plane modular Hamiltonians
31	The band algebras of the outgoing sector and their half-sided inclusion; the product of modular unitaries and the crossing of the future horizon	The ingoing sector and the Kruskal plane, the crossed product and observers
32	Three entropy formulas side by side; one cancellation in two settings; the two orientations of the modular composition	The final projects and the open questions; this meeting is for synthesis

Deferring a theorem’s proof does not remove its hypotheses from class. In Lecture 22, for example, students should explicitly identify the all-balls assumption, the two dictionary entries, and the analyticity hypothesis of the injectivity step, which the note proves only for static perturbations of AdS${}_3$. Otherwise the argument will sound like an elementary proof of nonlinear Einstein dynamics, which it is not. In Lectures 25–27 the corresponding list is the transparent coupling, the choice of matter state, the island rule itself, and, in Lecture 27, replica symmetry and the continuation in $n$; the free-fermion formula of Lecture 26 is exact for that matter and its late-time limit holds for any conformal field theory. In Lectures 28–31 it is Haag duality, strong additivity and factoriality in the three-arc proposition, the vacuum in the conformal Bisognano–Wichmann theorem, the KMS state and the factor property in the proposition of Lecture 29, the common cyclic and separating vector and the sign of the compression in Wiesbrock’s theorem, and, in Lecture 31, the limit $G_N\to 0$ and the Hartle–Hawking state.

Bridges for students with strong quantum mechanics and information theory

Use partial trace as the first example of restriction, then explain why a functional remains meaningful when a regional trace does not. Use the effective thermal oscillator to introduce an entanglement spectrum before discussing type III. Use the classical recovery channel before the Petz map. Use changes of Schmidt spectrum before graph cuts. Use two ordinary affine maps before abstract modular inclusions.

Repeat an example when the mathematical language changes. The qutrit decoder reappears in entropy, modular flow, distinguishability, approximate recovery, and the final synthesis. Students should recognize the same calculation in a more general setting.

In gravity units, draw the distinction between a fixed-background wave equation, a boundary dictionary, and a gravitational field equation. The scalar calculation belongs to the first two. The entanglement-first-law argument adds the third under stated assumptions.

Four points that need deliberate discussion

Type III. Finite $N$ does not mean finite-dimensional Hilbert space. Continuum local algebras and emergent algebras of selected large-$N$ observables are different constructions. The tracial infinite-product counterexample should remain in the classroom route.

Reconstruction. Equality of one expectation value is weaker than an operator intertwiner. Tests on basis-state populations miss quantum coherence. Near a surface transition, the reconstructed algebra and the chosen code family need to be specified together.

Islands. The written JT example is static at zero temperature and thermal in its two-sided extension. Its thermal entropy saturates; it is not a falling evaporation curve. Its extremum is spatially minimizing and temporally maximizing. Ask students to show both facts.

Emergent time. An affine group is an exact algebraic output under the theorem’s hypotheses. A null-coordinate interpretation and an infalling proper-time interpretation are different additional claims. The one-particle group representation alone does not provide a cyclic separating vacuum and a net.

Review before classroom use

Have the instructor or a peer reconstruct the scalar counterterm calculation, the time-dependent island endpoint variation, and the modular-inclusion sign conventions without consulting the answers. Try a representative meeting with the intended audience. Record which prerequisite required repair and which material was deferred.

The validation record separates executable checks from mathematical exposition and teaching readiness. A clean reader and passing finite-model checks are useful evidence; they do not substitute for a first teaching trial.