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Holographic Derivation of Entanglement Entropy from AdS/CFT

Shinsei Ryu, Tadashi Takayanagi

TL;DR

Proposes a holographic prescription equating entanglement entropy in a d+1 CFT with the area of a d-dimensional minimal surface in AdS_{d+2}, S_A = Area(γ_A) / (4 G_N^{(d+2)}). The paper demonstrates exact agreement with the known 2D CFT result via AdS_3 geodesics and extends the construction to higher dimensions with belt and disk regions, including AdS_5×S^5. It also analyzes finite-temperature cases using AdS black holes, showing a thermal (horizon-wrapping) contribution that links entanglement entropy to the Bekenstein-Hawking entropy. Overall, this work provides a unified geometric framework for entanglement in holographic CFTs and clarifies how horizons encode entanglement structure.

Abstract

A holographic derivation of the entanglement entropy in quantum (conformal) field theories is proposed from AdS/CFT correspondence. We argue that the entanglement entropy in d+1 dimensional conformal field theories can be obtained from the area of d dimensional minimal surfaces in AdS_{d+2}, analogous to the Bekenstein-Hawking formula for black hole entropy. We show that our proposal perfectly reproduces the correct entanglement entropy in 2D CFT when applied to AdS_3. We also compare the entropy computed in AdS_5 \times S^5 with that of the free N=4 super Yang-Mills.

Holographic Derivation of Entanglement Entropy from AdS/CFT

TL;DR

Proposes a holographic prescription equating entanglement entropy in a d+1 CFT with the area of a d-dimensional minimal surface in AdS_{d+2}, S_A = Area(γ_A) / (4 G_N^{(d+2)}). The paper demonstrates exact agreement with the known 2D CFT result via AdS_3 geodesics and extends the construction to higher dimensions with belt and disk regions, including AdS_5×S^5. It also analyzes finite-temperature cases using AdS black holes, showing a thermal (horizon-wrapping) contribution that links entanglement entropy to the Bekenstein-Hawking entropy. Overall, this work provides a unified geometric framework for entanglement in holographic CFTs and clarifies how horizons encode entanglement structure.

Abstract

A holographic derivation of the entanglement entropy in quantum (conformal) field theories is proposed from AdS/CFT correspondence. We argue that the entanglement entropy in d+1 dimensional conformal field theories can be obtained from the area of d dimensional minimal surfaces in AdS_{d+2}, analogous to the Bekenstein-Hawking formula for black hole entropy. We show that our proposal perfectly reproduces the correct entanglement entropy in 2D CFT when applied to AdS_3. We also compare the entropy computed in AdS_5 \times S^5 with that of the free N=4 super Yang-Mills.

Paper Structure

This paper contains 5 sections, 25 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Entanglement Entropy in $\mathrm{AdS}_3/\mathrm{CFT}_2$
  3. Higher Dimensional Cases
  4. Yang-Mills at Finite Temperature
  5. Conclusion

Figures (3)

  • Figure 1: (a) AdS$_3$ space and CFT$_2$ living on its boundary and (b) a geodesics $\gamma_A$ as a holographic screen.
  • Figure 2: Minimal surfaces in AdS$_{d+2}$: (a) $A_S$ and (b) $A_D$.
  • Figure 3: (a) Minimal surfaces $\gamma_A$ for various sizes of $A$. (b) $\gamma_A$ and $\gamma_B$ wrap the different parts of the horizon.