Takeaway

AdS/CFT equates a gravity theory in (d+1)-dimensional Anti–de Sitter space with a conformal field theory on its d-dimensional boundary; bulk geometry ↔ boundary quantum dynamics.

The problem (before → after)

  • Before: Strongly coupled quantum systems resist calculation; quantum gravity lacks a nonperturbative definition.
  • After: Duality maps intractable problems to tractable ones on the other side (strong/weak, gravity/field theory), providing new tools and insights.

Mental model first

Think of a shadow puppet: a rich 3D scene is encoded in a 2D silhouette. In holography, the boundary CFT “silhouette” fully captures the bulk gravitational “scene,” with radial depth mapped to energy scale.

Just-in-time concepts

  • Dictionary: Z_gravity[ϕ|{∂}] = Z_CFT[J=ϕ|] relates bulk fields to boundary operators.
  • Radial/energy: Bulk radial coordinate r corresponds to RG scale in the CFT.
  • Correlators: Bulk propagators with boundary insertions yield CFT correlators.

First-pass solution

In the canonical example, type IIB string theory on AdS₅×S⁵ is dual to 𝒩=4 SU(N) SYM in 4D. At large N and strong 't Hooft coupling on the CFT, the bulk is weakly curved classical gravity.

Iterative refinement

  1. Transport: Black holes in AdS compute viscosities and conductivities via Kubo formulas.
  2. Entanglement: Ryu–Takayanagi area formula relates boundary entanglement entropy to minimal bulk surfaces.
  3. Beyond AdS: Flat space, de Sitter, and nonconformal dualities are active research.

Principles, not prescriptions

  • Dualities relate seemingly different theories; pick the side where the problem is simpler.
  • Geometry encodes quantum information (e.g., entanglement ↔ area).

Common pitfalls

  • Assuming every quantum system has a gravity dual; conditions are specific.
  • Overextending classical gravity beyond its parametric validity.

Connections and contrasts

  • See also: [/blog/black-hole-information], [/blog/renormalization-group] (radial flow ↔ energy scale), [/blog/topological-order].

Quick checks

  1. What does “holography” mean here? — Lower-dimensional theory encodes higher-dimensional gravity.
  2. When is gravity classical? — Large N, strong coupling on the CFT side.
  3. How is entanglement computed? — Minimal surfaces via Ryu–Takayanagi.

Further reading

  • Maldacena, 1997 (source above)
  • Aharony et al., “Large N Field Theories, String Theory and Gravity”
  • Hartnoll et al., “Holographic Quantum Matter”