Ryu-Takayanagi Formula

Definition

The Ryu-Takayanagi (RT) formula expresses the entanglement entropy of a boundary subregion $A$ in a holographic CFT as $S(A)=\text{Area}({\gamma }_A)/(4G_N)$, where ${\gamma }_A$ is the bulk minimal-area codimension-2 surface anchored at $\partial A$ on the boundary and homologous to $A$. The covariant generalisation (HRT, Hubeny-Rangamani-Takayanagi) replaces ‘minimal-area’ with ‘extremal-area’ in time-dependent settings. The quantum extremal surface (QES) prescription further generalises to include bulk-matter contributions.

Role in Research

The RT formula is covered in Lecture 16 of the holography course and re-encountered in Lectures 17–22, where its quantum and algebraic forms are developed. It is central to subregion-subalgebra-duality and to the modern algebraic understanding of holographic entropy, connecting the area of bulk surfaces to the entanglement structure of boundary subalgebras.

Relations

• holographic-dictionary — the RT formula is the entropy entry in the holographic dictionary.
• subregion-subalgebra-duality — the RT surface bounds the entanglement wedge.
• replica-trick-gravity — the Lewkowycz-Maldacena derivation of RT uses the gravitational replica trick.
• quantum-extremal-surfaces — the QES prescription generalises the RT formula to include bulk matter.

Papers

• Ryu–Takayanagi 2006, arXiv:hep-th/0603001.
• Hubeny–Rangamani–Takayanagi 2007, arXiv:0705.0016.
• Lewkowycz–Maldacena 2013, arXiv:1304.4926.

Notes

Course coverage: Lecture 16, Lecture 19, Lecture 21 and Lecture 22.