TASI Lectures on the Emergence of the Bulk in AdS/CFT

• The lectures ask how local bulk effective field theory emerges from CFT, which CFTs support it, and what limits constrain its emergence.Source: §1 · p. 2
• They present sufficient conditions for a conformal field theory to have a semiclassical dual.Source: §1 · p. 3
• They develop perturbative reconstruction of bulk fields as operators in the boundary CFT.Source: §1 · p. 3
• They explain AdS/CFT as a quantum error-correcting code and illustrate the reformulation with exactly soluble toy models.Source: §1 · p. 3
• They study the quantum Ryu-Takayanagi formula, its relation to local bulk physics, and entanglement wedge reconstruction behind black hole horizons.Source: §1 · p. 3
• The presentation proceeds from basic AdS/CFT structure through bulk reconstruction, quantum error correction, and quantum Ryu-Takayanagi results.Source: §1 · p. 3

2.1 Anti de Sitter space

• AdS is a homogeneous, isotropic, negatively curved spacetime whose conformal boundary makes signals return in finite time, giving it finite-box-like causal behavior.Standard scalar quantization produces a discrete particle spectrum, reinforcing this finite-box interpretation; asymptotically AdS geometries may instead include black holes, wormholes, or compact factors M.Source: §2.1 · pp. 4–7

2.2 Conformal Field Theory

• CFT primary operators and descendants correspond one-to-one with states on S^(d−1), with an operator of dimension ∆ producing a state of energy ∆+E0.This correspondence follows from the Euclidean-ball path integral and the equivalence between radial dilations and cylinder-time translations.Source: §2.2 · pp. 7–8, Figure 2 · p. 7

2.3 Statement of the Correspondence

• AdS/CFT identifies any CFT on R×S^(d−1) with quantum gravity in asymptotically AdS_(d+1)×M spacetime, with matching Hilbert spaces, symmetries, and observables [19, 20, 21].The dictionary includes the bulk metric–boundary stress-tensor and bulk gauge-field–boundary current pairs, while semiclassical duality near the vacuum requires a finite effective bulk theory reproducing correlators to all orders in 1/(ℓΛ).Source: §2.3 · pp. 8–9
• Vacuum-near semiclassical duality does not by itself capture black holes, so the stronger definition requires correlator matching for arbitrary fixed boundary geometries, including S1×S^(d−1).This extension is designed to reproduce high-temperature CFT entropy as black-hole entropy derived from the bulk effective action.Source: §2.3 · pp. 9–10

2.4 Gapped Large-N Theories

• Claim 2.2 states that any gapped large-N theory has a semiclassical dual, with bulk fields φi corresponding to the single-trace primaries Oi.Source: §2.4 · p. 11
• Gapped large-N theories are assumed to yield bulk effective actions whose finitely many fields have interactions of at most gravitational strength.The evidence uses conformal-bootstrap solutions of four-point crossing equations and Mellin-space analyses to associate parameters with local terms in Seff, although the full argument remains incomplete [3, 28, 29].Source: §2.4 · p. 11
• Known examples generally involve a nontrivial compact manifold M, whose inclusion replaces each bulk field by an infinite large-N tower of primaries; these lectures instead assume M is trivial [30].Trivial-M theories are expected from string compactification on fluxed Calabi–Yau manifolds, but independent dual CFT constructions and a generalized analysis remain unavailable.Source: §2.4 · pp. 11–12
• The gravitational-strength criterion is sufficient but not necessary, because it excludes interesting bulk particle physics present in more sophisticated compactifications.This restriction motivates treating the criterion as non-necessary for semiclassical duals, including AdS vacua in the string landscape.Source: §2.4 · p. 11
• CFT and black-hole thermodynamics match in N-scaling, and special cases reproduce the Bekenstein-Hawking entropy coefficient microscopically [35] [36].For N = 4 Yang–Mills, matrix degrees of freedom and planar diagrams give # = 2 in thermodynamic quantities and the stress-tensor three-point function.Source: §2.4 · p. 12

3. Bulk Reconstruction

• Bulk reconstruction seeks CFT operators for experiments whose outputs lie behind horizons and therefore cannot be related directly to local boundary correlation functions.The lectures motivate reconstructing bulk operators far from the boundary because behind-horizon outputs cannot reach the boundary in the semiclassical picture.Source: §3 · p. 13

3.1 Free Scalar Reconstruction

• Perturbative reconstruction solves bulk equations of motion as operator equations in the CFT, using the extrapolate dictionary to impose boundary conditions.The initial construction begins with a free scalar in AdS and extends to interactions.Source: §3.1 · p. 14
• Global reconstruction expresses a bulk scalar as a smearing integral of the dual CFT operator over spacelike-separated boundary points with support on an entire Cauchy slice.The representation is nonlocal in space and time, although boundary evolution can rewrite it using operators on one slice.Source: §3.1 · p. 15, Figure 5 · p. 16

3.2 Including Interactions

• Interactions are incorporated with a spacelike Green function and iterative corrections, yielding a Feynman-like reconstruction whose nonlocality grows with perturbative order.The expansion brings in additional single-trace operators, including the stress tensor, when the reconstructed field couples to them.Source: §3.2 · pp. 16–17, Figure 6 · p. 17
• Using incorrect bulk equations produces spacelike commutators proportional to C_OOO, while the correctly tuned O(g) correction cancels them.This ties bulk operator algebra to the CFT interaction data.Source: §3.2 · p. 18
• At sufficiently high orders, gauge and diffeomorphism invariance require dressed operators, such as Wilson-line or boundary-attached gravitational dressings.The construction also has unresolved scope: spacelike Green functions for generic asymptotically-AdS geometries remain an unsolved linear-PDE problem.Source: §3.2 · p. 18

3.3 AdS/Rindler Reconstruction

• AdS-Rindler reconstruction represents any bulk scalar inside an AdS-Rindler wedge using Heisenberg operators on the wedge boundary, then evolves them to operators supported in a boundary hemisphere.Conformal transformations extend the construction from hemispheres to arbitrary ball-shaped boundary regions.Source: §3.3 · p. 20, Figure 7 · p. 20
• The causal wedge is a natural upper bound for subregion reconstruction, but alternative methods can reconstruct beyond it and motivate a complete determination of the bulk region dual to a boundary subregion.The causal wedge reconstruction conjecture states that operators in C[R] can be represented using only operators in R, while the lectures proceed toward stronger reconstruction results.Source: §3.3 · pp. 21–22

4.1 Some Puzzles

• Bulk radial locality conflicts with the boundary time-slice axiom, but the conflict is resolved by requiring reconstruction only within a code subspace associated with semiclassical states.AdS-Rindler reconstruction can represent one bulk operator on any two of three boundary regions, while generic states outside the code subspace correspond to large-black-hole microstates; quantum error correction supplies the framework for this restricted duality.Source: §4.1 · pp. 23–25

4.2 Three Qutrit Code

• The three-qutrit code protects one logical qutrit against loss of any one physical qutrit because entanglement enables recovery from every pair while leaving each individual qutrit uninformative.The same redundancy lets any logical operator act on any two physical qutrits, realizing bulk reconstruction and radial locality within the code subspace without contradicting the time-slice axiom.Source: §4.2 · pp. 26–28
• The three-qutrit model captures bulk effective field theory, radial locality, AdS-Rindler reconstruction, and black-hole entropy through one logical qutrit encoded into three boundary qutrits.Its redundancy depends crucially on entanglement, and the orthogonal complement of the code subspace is interpreted as black-hole microstates rather than additional semiclassical bulk states.Source: §4.2 · pp. 27–29

4.3 General Properties of Holographic Codes

• The general code theorem equates logical-operator reconstruction on a region with correctability of its erasure, absence of information leakage, and a standard entanglement-based encoding structure.Applied to n physical qudits with symmetric access determined by region size, the quantum singleton bound requires the smallest reconstructing region to contain more than half the system and constrains the logical code-subspace dimension.Source: §4.3 · pp. 29–31
• The quantum singleton bound requires k log d ≤ S_C ≤ (2m − n) log d, and random code subspaces typically achieve correctability with exponentially small errors as this condition improves.In the three-qutrit code, n = 3, k = 1, and m = 2 saturate the relation 1 + 3 = 2 × 2.Source: §4.3 · p. 31
• When the code subspace grows to k ∼ N^♯, correctability fails because most resulting states become large central black holes, preserving holographic locality by preventing excessive bulk excitations.For k ∼ N^0, central excitations require at least half the boundary; increasing k eventually triggers black-hole formation and breakdown of correctability.Source: §4.3 · pp. 32–33, Figure 15 · p. 33
• Radial position determines erasure protection: larger boundary erasures corrupt deeper bulk operators, while subsystem quantum error correction recovers only operators in the corresponding causal or entanglement wedge.The subsystem formulation accommodates code subspaces containing low-energy bulk operators throughout the interior, unlike encodings that represent every logical operator on one boundary region.Source: §4.3 · pp. 33–34

4.4 A Tensor Network Model of Holography

• The HaPPY code uses a hyperbolic pentagon network of six-qubit perfect tensors, whose operator-pushing property yields an isometric bulk-to-boundary encoding and AdS-Rindler reconstruction.Perfect tensors implement unitary maps across balanced bipartitions; the network provides a volume’s worth of bulk degrees of freedom with radial locality and redundancy against erasures.Source: §4.4 · pp. 35–37
• The HaPPY code reproduces holographic error-correction features but lacks natural boundary dynamics and short-distance bulk locality, so it illustrates rather than fully realizes holography.Removing tensors models black holes and eventually accounts for all boundary states, but the construction remains a discrete toy model.Source: §4.4 · pp. 37, 39
• Random tensor networks remove HaPPY’s discrete recovery artifacts, achieving subsystem quantum error correction with complementary recovery for arbitrary regions in the large contracted-index limit.In HaPPY, some regions leave bulk degrees of freedom unrecoverable on either a region or its complement; random tensors avoid this problem to high accuracy [13].Source: §4.4 · pp. 39–40

5.1 RT Formula and the Entanglement Wedge

• The entanglement wedge W[R] is the domain of dependence of a homology hypersurface HR bounded by R and its HRT surface γR.The HRT surface is extremal, homologous to R, and minimal among surfaces satisfying those conditions; existence and uniqueness are assumed.Source: §5.1 · pp. 40–41
• The entanglement wedge can exceed the causal wedge, and unlike the causal wedge it can contain points behind an event horizon.Figure 24 gives examples in empty AdS3 and an elongated wormhole.Source: §5.1 · p. 41, Figure 23 · p. 41, Figure 24 · p. 42
• The quantum RT formula combines a bulk operator localized on γR with the bulk von Neumann entropy of fields on HR, equivalently the entropy of operators in W[R].Its modern form incorporates dynamical geometries and quantum corrections, while its validity is stated up to terms vanishing with a positive power of G.Source: §5.1 · pp. 41–42
• Entanglement-wedge reconstruction allows the full algebra of operators in W[R] to be reconstructed from CFT operators in R, including operators behind black-hole horizons.This resolves the apparent limitation of reconstruction based only on the causal wedge.Source: §5.1 · pp. 41, 43

5.2 RT formula in toy holography

• In the three-qutrit code, the RT formula holds exactly within the code subspace because the central logical degree of freedom contributes bulk entropy only to the appropriate physical region.The area term records fixed entanglement in |χ⟩, which is required for code redundancy and represents CFT entanglement absent from low-energy bulk degrees of freedom.Source: §5.2 · p. 43
• HaPPY and random tensor networks reproduce an RT formula in which the area term equals the entanglement across cut links, yielding the correct holographic entropy scaling.The proportionality coefficient is the logarithm of the link Hilbert-space dimension, matching the CFT degrees-of-freedom coefficient N♯.Source: §5.2 · p. 44

5.3 Algebraic encoding and a theorem

• Operator-algebra quantum error correction generalizes subsystem codes so that a nontrivial center supports a nontrivial area operator, matching the non-factorization implied by bulk diffeomorphism constraints.The framework accommodates superpositions of geometries and black holes, provided the homology condition and dressed observables are handled appropriately.Source: §5.3 · pp. 44, 48
• Theorem 5.2 establishes equivalence among entanglement-wedge subregion duality, the quantum RT formula with central area operator, and the JLMS relation.Applied to AdS/CFT, the theorem identifies the algebra M with bulk operators in W[R] and shows that these previously distinct holographic properties are equivalent; its statements are either all true or all false.Source: §5.3 · pp. 46–48

Conclusion

• The lectures portray the emergent radial direction as substantially understood through toy models and general results, while major gaps in holography remain.The conclusion frames the listed problems as important open questions for understanding AdS/CFT and holography beyond it.Source: §6 · p. 49

Sub-AdS Locality

• Exactly solvable holographic models generally lack local bulk physics at scales parametrically smaller than the AdS radius, leaving sub-AdS locality unresolved.The notes attribute this limitation to missing structure beyond spatial entanglement and highlight the absence of a tunable parameter corresponding to large ’t Hooft coupling.Source: §6 · p. 49

Time Evolution

• Most code subspaces are not preserved under time evolution, while accumulated reconstruction errors may become important for long-time bulk dynamics.The notes connect this problem to phenomena such as information return from black holes and long wormholes becoming short again.Source: §6 · p. 49

Gravitational Dressing

• Gravitational dressing becomes essential when backreaction is large, especially where causal and entanglement wedges differ.The notes identify this issue as relevant to chaos, traversable wormholes, and the proper description of black-hole interiors.Source: §6 · pp. 49–50

The Black Hole Interior

• The description of black-hole interiors in generic microstates remains unresolved, including whether firewalls occur and whether infalling observers experience a breakdown of quantum mechanics.Entanglement-wedge subregion duality provides grounds for optimism because it shows that horizons are not impenetrable barriers, but a complete theory remains out of reach.Source: §6 · p. 50

What about the real world

• Applying these ideas to the real world remains necessary because the observed cosmological constant in our local universe is not negative.The notes contrast this setting with AdS, where a stable boundary provides a better-defined reference, and suggest that bulk observables may need to be described with some imprecision.Source: §6 · p. 50