# AdS/CFT Dictionary, Made Operational

## Overview

RU is a certificate engine for universality and obstructions in constrained nonnegative propagation. Host structures supply finite inputs. Outputs are audited certificates or finite obstruction witnesses.

## RU fit

- **Fit level:** Low

## RU instantiation hooks

- Operational dictionary context; RU applies only after an additional discretization or finite-sector extraction that yields cone-preserving templates.
- This doc supports the motivation for looking for such templates on the boundary side.

## Purpose

Make the AdS/CFT dictionary usable as a method, not just a statement. The emphasis is on what you actually compute, what inputs you need, and what limits make the computation trustworthy.

Wray summarizes the core idea in exactly the “host suitability for RU instantiation” form: gravity in (d+1) dimensions is related to a super Yang–Mills theory on the d-dimensional boundary, and boundary data provides a hologram of bulk physics. (Wray PDF around pages 255–259; curated excerpts `holographic_principle_conjecture` and `ads_cft_dictionary_kinematics_and_dynamics`.)

## The basic dictionary move: sources and generating functionals

The canonical starting point is the GKPW relation:

- choose a bulk field ϕ that approaches a boundary value ϕ₀
- interpret ϕ₀ as a source for a boundary operator O
- equate the bulk partition function with boundary generating functional

Primary anchors for this formulation are Gubser–Klebanov–Polyakov and Witten.

## Kinematics you use constantly

### Symmetry matching

- Isometries of AdS map to conformal symmetries of the boundary theory.
- Global symmetries of the boundary often correspond to gauge symmetries in the bulk.

This matching is the first sanity check for any proposed dual pair.

### The radial direction as an energy scale

A reliable operational rule is:

- moving toward the boundary corresponds to the UV of the boundary theory
- moving deeper into the bulk corresponds to the IR

This is why the dictionary naturally exports RG-flow intuition into geometry.

## Dynamics: how computations are done in practice

### Correlators from bulk equations

In the classical gravity limit:

- solve the bulk field equations for ϕ with specified boundary behavior
- evaluate the on-shell bulk action as a functional of boundary sources
- take functional derivatives with respect to sources to obtain correlation functions

This is the workhorse for spectral functions, susceptibilities, and response.

### Normalizable vs non-normalizable modes

In typical use:

- non-normalizable modes encode turning on sources (deforming the boundary theory)
- normalizable modes encode states (expectation values)

### Controlled limits and corrections

The usual control regime for a gravity computation is:

- large N: suppresses quantum gravity loops (1/N effects)
- large ’t Hooft coupling: suppresses stringy corrections (α′ effects) and makes curvature small

Leaving this regime is not forbidden, but you must track:

- 1/N corrections: bulk quantum effects
- α′ corrections: higher-derivative terms

## A minimal worked example (structural, not technical)

Suppose a scalar field ϕ in AdS is dual to an operator O of dimension Δ.

The operational content is:

- the asymptotic falloff of ϕ near the boundary contains two coefficients
- one coefficient is interpreted as the source for O
- the other is interpreted as the expectation value ⟨O⟩

The precise power laws encode Δ, and the dictionary tells you how to read Δ from the bulk mass.

This example matters because the same structure repeats for:

- conserved currents ↔ bulk gauge fields
- stress tensor ↔ bulk metric perturbations

## High-yield translations (the ones people actually reuse)

### Thermodynamics

Boundary thermal state ↔ bulk black hole or black brane geometry.

This turns questions about free energy and phase structure into geometry and horizon regularity.

### Entanglement and geometry

Entanglement entropy in the boundary theory is computed, in leading order, by an extremal surface area in the bulk (Ryu–Takayanagi and its extensions).

This translation is now used well beyond the original context of AdS/CFT because it made “entanglement in gravity” computable.

### The information problem pathway

A key reason AdS/CFT is used in the information problem is that unitarity is manifest in the boundary description, while the bulk semiclassical story can be corrected by including the right saddles. Maldacena’s discussion of eternal AdS black holes is a classic orientation point.

## Practical gate checks before claiming a result

- Is the bulk geometry weakly curved in string units, and is the effective coupling small?
- Are you working at large N, or are 1/N effects essential for the phenomenon you claim?
- Is the boundary theory in a regime where the bulk description is expected to exist and be local?
- Are you computing a protected quantity, or do you need uncontrolled extrapolation?

## The takeaway

AdS/CFT is reducible into RU certificates because it is not just a correspondence claim. It is a recipe:

- define sources, solve bulk equations, evaluate an action, differentiate
- translate hard many-body quantities into geometric objects

When the control conditions are stated explicitly, this recipe becomes a reusable method that has migrated into multiple subfields.