What a defect extremal surface is

What must be calculated beyond area?

In a static holographic model, the RT prescription computes leading entanglement entropy from the area of a minimal surface. If the bulk contains a defect carrying quantum fields, a surface crossing or ending on it also selects defect degrees of freedom whose entropy contributes. A defect extremal surface (DES) includes that entropy in the variational problem.

Think of the surface endpoint as a movable choice: moving it changes both the surface length and the selected region on the defect. Finding its preferred position requires evaluating both responses to displacement. This is an intuitive picture of the variation; the defect theory and quantum state determine the actual responses.

The two steps in the prescription

In a three-dimensional bulk, the surface's “area” is a curve length. Set ħ and k_B to 1. Let G₃ be the three-dimensional Newton constant, and IΓ the defect region determined jointly by the curve Γ and boundary region A:

Candidate curves must satisfy endpoint anchoring and the relevant homology condition. First find stationary configurations under variations of the curve and its intersection with the defect. Then compare the allowed stationary solutions and select the smaller generalized entropy. Finding a single stationary point leaves this comparison unfinished. The expression focuses on the defect matter correction studied in the original paper, with the rest of the bulk described by classical gravity.

An example in which the endpoint stays put

Section 3.4 of the original DES paper studies the interval [0,L] on a half-line, with the physical boundary at 0. One end of the geodesic is anchored at L; the other can move along the EOW brane. With the paper's conformal state, brane geometry and regulator prescription, the matter entropy of the relevant brane interval is constant as this endpoint moves.

It therefore adds to the total entropy without adding a derivative term to the endpoint variation. The extremal curve coincides with the original RT curve. This example tests the intuition that quantum entropy must move the surface. For more general defect states, endpoint-dependent matter entropy can shift the stationary position. Even a correction constant within one branch can affect the entropy comparison between branches.

How to check a calculation

1. Fix the boundary interval, brane position, quantum state and cutoffs on both sides. Specify which reduced state defines the defect entropy.
2. Express the length and defect-entropy terms separately as functions of the endpoint coordinates. Check their derivatives before finding stationary points of the sum.
3. List all allowed connection patterns, check the parameter range of each solution, and compare their generalized entropies.
4. When comparing with a lower-dimensional island calculation, match the effective Newton constant, matter entropy and cutoff dictionary term by term.

The published AdS₃/BCFT₂ construction finds agreement between DES and the boundary quantum extremal surface. Subsequent work extends the comparison to time-dependent Page curves and JT partial reduction. In higher dimensions, the bulk DES entropy is generally lower than the island result in the boundary low-energy effective theory. Dimension and the scope of the effective description belong beside the equality when reading this sequence of papers.

Original references

• F. Deng, J. Chu and Y. Zhou, Defect extremal surface as the holographic counterpart of Island formula, JHEP 03 (2021) 008, arXiv:2012.07612. Sections 3.3–3.5 give the prescription, constant-correction example and branch comparison.
• J. Chu, F. Deng and Y. Zhou, Page curve from defect extremal surface and island in higher dimensions, JHEP 10 (2021) 149, arXiv:2105.09106. Time-dependent agreement and higher-dimensional differences.
• F. Deng, Y.-S. An and Y. Zhou, JT gravity from partial reduction and defect extremal surface, JHEP 02 (2023) 219, arXiv:2206.09609. Small transverse brane fluctuations, the JT action and entropy matching.

Read the related research

• Geometry and quantum information →

  Which quantum-information quantities can extremal geometry represent, and when does it agree with a lower-dimensional effective theory?

• Boundaries, defects and interfaces →

  After adding a defect, an EOW brane or a finite cutoff, which boundary data determine entropy and spectrum calculations?