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

TL;DR

The notes survey how local bulk physics emerges from boundary CFT data in AdS/CFT, detailing conditions for a semiclassical dual, explicit bulk reconstruction methods, and the quantum error correction perspective that underpins subregion duality. They show how bulk operators can be encoded redundantly across boundary regions, how AdS-Rindler and tensor-network models realize this redundancy, and how the quantum RT formula emerges from entanglement wedge considerations. A key highlight is the algebraic formulation proving the equivalence of subregion duality, the RT formula, and JLMS, unifying geometric and information-theoretic perspectives. The work also outlines major open challenges, including sub-AdS locality, gravitational dressing, black hole interiors, and extending these ideas to cosmological settings.

Abstract

These lectures review recent developments in our understanding of the emergence of local bulk physics in AdS/CFT. The primary topics are sufficient conditions for a conformal field theory to have a semiclassical dual, bulk reconstruction, the quantum error correction interpretation of the correspondence, tensor network models of holography, and the quantum Ryu-Takayanagi formula.

Figures (24)

• Figure 1: The causal structure of $AdS_3$: a photon from the center goes out to the boundary and bounces back in time $\Delta t=\pi$.
• Figure 2: The state-operator correspondence. Doing path integral over the ball $\rho<1$ with boundary condition $\phi$ and an operator $\mathcal{O}$ at $\rho=0$ defines a wave functional $\Psi[\phi]$, which has sphere energy $\Delta+E_0$ if $\mathcal{O}$ has dimension $\Delta$. Moreover given such a state, we can construct an operator that produces it by evolving the state radially inwards assuming no operators are present until we are left with something at the center that must be local.
• Figure 3: Using the extrapolate dictionary to describe a bulk scattering experiment to the CFT.
• Figure 4: A bulk experiment we don't know how to describe using correlation functions of local operators in the CFT. This is the Penrose diagram of the two-sided AdS-Schwarzschild geometry, we have prepared an initial state with two incoming particles, and we would like to know the probability distribution for what they produce behind the horizon.
• Figure 5: Global Reconstruction A bulk scalar field $\phi(x)$ is represented in the CFT (at leading order in the interactions) as an integral of the CFT local operator $\mathcal{O}$ dual to $\phi$, integrated over the set of boundary points $S_x$ that are spacelike separated from $x$, shaded here in green. $S_x$ has nontrivial support on an entire Cauchy slice of the boundary, denoted $\Sigma$.

Theorems & Definitions (14)

• Claim 2.1
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Claim 2.2
• Definition 3.1
• Theorem 4.1
• Definition 5.1
• Definition 5.2
• Definition 5.3