Shape Deformations and Displacement Response

Deforming an entangling surface changes both the boundary modular data and the bulk extremal surface. At linear order, the two sides are related by stress-tensor response and the Jacobi equation for normal surface displacements. The comparison is meaningful only with a matched regulator, a fixed surface branch, and all corner or contact terms included.

Required background. Shape Deformations and Modular Perturbation Theory supplies the boundary variation. Covariant Extremal Surfaces and HRT supplies the bulk surface.

Helpful background. Second Variation, Hessians, and Jacobi Operators supplies the geometric operator. Modular Response Kernels and Stress-Tensor Insertions supplies the response distribution. Shape Dependence and Entanglement Variations, Corners, Cusps, and Defect Data, and Contact Terms, Operator Domains, and Perturbative Error Budgets supply the regulator-sensitive QFT data.

First application. Perturb a planar or spherical entangling surface and match the boundary modular response to the linearized displacement of its bulk surface.

Normal deformations and the Jacobi operator

Let the undeformed extremal surface have embedding $X^a(y)$ and orthonormal normals $n_I^a$. A first-order displacement is

$$\delta X^a=\eta^I(y)n_I^a+\text{tangential reparameterization}.$$

Tangential terms are gauge on a closed smooth surface; boundary anchoring fixes their endpoint role. Linearizing the vanishing mean-curvature condition gives

$$\mathcal J^I{}_J\eta^J =\mathcal S^I[h,\delta x_{\partial A}],$$

where $\mathcal J$ contains the normal-bundle Laplacian, ambient curvature, and extrinsic-curvature terms. The source contains the bulk metric perturbation and the displaced boundary anchor. Regularity and Dirichlet anchoring select a Green function only if $\mathcal J$ has no uncontrolled zero mode.

The first area variation from an embedding displacement vanishes for a fixed-anchor extremal surface. Shape response enters through the moved anchor, metric variation, and at second order through the Jacobi solution. Near a surface phase transition, a small eigenvalue of $\mathcal J$ makes the response large and fixed-branch perturbation theory can fail.

Boundary modular response

For a half-space in the vacuum, the undeformed modular generator is a boost. A small null deformation of its boundary produces a first-order modular-Hamiltonian correction involving an integral of $T_{kk}$ along the Rindler horizon, together with commutator and endpoint prescriptions. This controlled result underlies the deformation analysis of Faulkner, Leigh, Parrikar, and Wang 2016, §§2–4.

For a generic region or state, shape dependence is encoded in modular perturbation theory rather than a local generator. Schematically,

$$\delta K_A =\int_{-\infty}^{\infty}ds\, g(s)\,\sigma_A^{-is/2\pi} (\delta\sigma_A)\sigma_A^{is/2\pi},$$

with a distributional kernel and a prescription around its singularities. Contact terms at the entangling surface and stress-tensor improvements depend on the regulator. Only matched entropy differences or universal coefficients should be compared across schemes.

Planar and spherical checks

For a planar vacuum cut in Poincaré AdS, a Fourier deformation $\delta x_{\partial A}(k)$ gives a linear elliptic boundary-value problem for $\eta(z,k)$. The regular solution decays into the bulk and reproduces the expected nonlocal momentum dependence of the entropy shape Hessian. At $k=0$, a rigid translation is a geometric zero mode rather than a physical entropy change.

For a sphere, conformal symmetry maps the problem to hyperbolic space. Translation, dilation, and special-conformal deformations provide exact modes with known bulk images. They are valuable normalization checks before treating arbitrary shapes.

Adversarial controls

Move the cutoff but not the surface. A coordinate deformation can change the intersection with the regulator. Holding one cutoff fixed and moving the other creates spurious area terms.

Ignore a Jacobi zero mode. Inverting $\mathcal J$ across a surface-branch transition produces a divergent or nonunique displacement. Competing extremal surfaces must be compared nonperturbatively.

Discard corner terms. Nonsmooth entangling surfaces contain universal corner or cusp data. A smooth-surface formula cannot be extrapolated through the singular limit without resolving those terms.

Evidence ceiling

Matched modular and Jacobi response determines the linear displacement and entropy variation for a smooth surface in a fixed semiclassical branch. It does not establish a unique response at a QES transition, remove regulator-dependent contact terms, or make a generic modular Hamiltonian local.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Faulkner, Thomas, Robert G. Leigh, Onkar Parrikar, and Huajia Wang. “Modular Hamiltonians for Deformed Half-Spaces and the Averaged Null Energy Condition.” Journal of High Energy Physics 2016, 038 (2016). DOI; arXiv:1605.08072.
• Mezei, Márk. “Entanglement Entropy across a Deformed Sphere.” Physical Review D 91, 045038 (2015). DOI; arXiv:1411.7011.