Shape Deformations and Modular Perturbation Theory

A shape deformation changes the region and therefore changes the observable algebra, its causal domain, and the regulator boundary. To differentiate modular data, one must first identify the moving algebra with a reference algebra. Stress-tensor fluxes, displacement operators, and contact terms then record the physical change; they cannot be recovered by treating the deformation as an ordinary state variation.

Required background. Ward-identity contact terms supply distributional corrections, modular Hamiltonian domains supply the unbounded generator, and entanglement shape dependence supplies geometric variations.

Helpful background. Nonlocal modular generators explain why perturbative locality need not persist.

Identifying a moving region

Let a reference entangling surface be embedded as $X^\mu(y)$ and deform it by

X^\mu(y)\longmapsto X^\mu(y)+\lambda\zeta^\mu(y).

Tangential $\zeta^\mu$ is locally a reparametrization; normal components change the shape. The spatial region $A_\lambda$ and its domain of dependence $D(A_\lambda)$ both move. A diffeomorphism $f_\lambda$ that maps $A_0$ to $A_\lambda$ provides a pullback

\alpha_\lambda:\mathcal A(A_\lambda)\longrightarrow\mathcal A(A_0).

Only after choosing $\alpha_\lambda$ can $K_{A_\lambda}$ be differentiated as an operator or quadratic form on one Hilbert-space representation. Different identifications differ by an infinitesimal unitary or gauge transformation and shift the connection-like part of the response, while invariant expectation values agree after all terms are included.

The structural map places Shape Deformations and Modular Perturbation Theory along the perturbative sequence from normalized state or shape variations to information metrics and modular transport.

A normalized family yields the fixed-reference first law and a positive quadratic response, which branches into state susceptibility, shape kernels, metric choices, and modular holonomy. — At fixed comparison algebra, normalization removes the linear term in relative entropy. The second response supports several inequivalent constructions: state and shape tangents, stress-tensor kernels, monotone metrics, and zero-mode-projected transport. The diagram is schematic and not to scale.

Deformed half-space as the controlled model

Take the vacuum half-space with known boost modular Hamiltonian and a small deformation supported near the entangling plane. After pulling the deformed algebra back to the reference, the first-order calculation of Faulkner, Leigh, Parrikar, and Wang 2016, §2 has the structure

\delta K =\delta K_{\mathcal H_+} +\delta K_{\mathcal H_-} +i[G_\zeta,K_0] +\delta K_{\rm contact}.

Here $G_\zeta$ generates the chosen identification. In a common null-coordinate and modular-flow convention, the horizon terms are

\delta K_{\mathcal H} =-2\pi\int_{\mathcal H_+}\zeta^+ T_{++} +2\pi\int_{\mathcal H_-}\zeta^- T_{--}.

The signs reverse with the orientation of the null normals or with inverse modular flow. The future and past horizons belong together: dropping one can violate Hermiticity or causal support. The commutator term describes the change of coordinates on the reference Cauchy surface; it is not an optional interaction.

This formula is first order in a smooth deformation under the hypotheses of the half-space analysis. It does not establish an exact local generator for the deformed region.

Displacement operators in replica geometry

In a replica construction, the twist defect sits on the entangling surface. Broken translations normal to the defect obey a Ward identity of the form

\partial_\mu T^{\mu a}(x) =D^a(y)\,\delta_\Sigma(x),

where $a$ labels normal directions and $D^a$ is the displacement operator. Choosing the sign of $D^a$ by this identity fixes the first variation of the replicated partition function:

\delta\log Z_n =-\int_\Sigma d^{d-2}y\; \zeta_a(y)\langle D^a(y)\rangle_n.

Second shape derivatives involve displacement two-point functions plus contact terms from varying the defect, measure, normal frame, and counterterms. The separated-point correlator alone is not the complete Hessian.

The replica and modular-flow descriptions are complementary. The former organizes defect data and Rényi derivatives; the latter exposes causal-horizon stress fluxes. Agreement requires matching analytic continuation in $n$ , normal-vector conventions, and local counterterms.

State variation versus shape variation

Suppose both the global state and the region depend on $\lambda$ . After a pullback to $\mathcal A(A_0)$ , the total derivative separates schematically as

\frac{d}{d\lambda} =\left.\frac{d}{d\lambda}\right|_{\rm state} +\left.\frac{d}{d\lambda}\right|_{\rm shape,\alpha}.

The first term is a normal functional tangent on a fixed algebra. The second differentiates the embedding and the chosen identification. Mixed derivatives depend on ordering and can carry additional contact terms. A first-law calculation that changes both at once cannot attribute the result to a state susceptibility without this decomposition.

Regulator and boundary motion

A lattice cut, brick wall, split collar, or mode truncation has geometric support. Deforming the physical surface while holding that support fixed describes a different family from moving the regulator with the surface. Boundary counterterms can therefore contribute to the shape derivative even when they cancel from a fixed-shape relative entropy.

A controlled calculation states:

which geometric data move;
how operators are pulled back;
whether the UV regulator co-moves;
which null normals and surface measure are used;
where contact and boundary terms enter;
the norm or correlator topology controlling $O(\zeta^2)$ .

Topology-changing deformations are not infinitesimal shape tangents. Pinching off a component or creating a corner can introduce nonanalytic terms and lies outside this perturbative expansion.

Common pitfalls

Differentiating operators on different algebras as if they shared a domain. Supply a pullback or unitary identification first.

Keeping only the visible horizon integral. Coordinate-identification, past-horizon, and contact terms can be required by Hermiticity and Ward identities.

Moving the surface but not declaring regulator motion. The resulting boundary term is scheme dependent and can be mistaken for universal shape response.

Before interpreting this response coefficient, use the validity map to check normalization, tangent domains, contact and regulator terms, metric choice, and analyticity independently.

A reported susceptibility must pass separate checks for normalized families, common tangent domains, contact and regulator terms, and an explicit metric and analytic strip before receiving a physical interpretation. — Normalization, domain, contact-term, metric, and analyticity checks are independent. A failed gate narrows or withdraws the physical claim; it is not a change of notation. The decision map is schematic and not to scale.

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, no. 9 (2016): 038. DOI; arXiv.