Shape Deformations and Modular Perturbation Theory

A shape deformation changes the region, its causal domain, and generally its regulator boundary. Modular generators for two different regions cannot be differentiated until their algebras and domains have been identified. Once that choice is explicit, horizon stress fluxes, pullback terms, displacement operators, and contacts organize the response.

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

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

The overview’s differentiable-family map makes clear why a moving region first needs a common algebra identification, while its independent validity gates apply separately to operator domains, contacts, and regulator motion. The preceding chapter’s governed discussion of smooth modes, cusps, and fixed cutoffs supplies the relevant geometric figure; duplicating it here would not add a new relationship.

Identifying the moving algebra

Let the reference entangling surface have embedding $X^\mu(\mathbf y)$ and deform it by

$$X^\mu(\mathbf y) \longmapsto X^\mu(\mathbf y)+\epsilon\zeta^\mu(\mathbf y).$$

A tangential $\zeta^\mu$ is locally a reparametrization. Its normal part changes the surface, the spatial region $A_\epsilon$, and the causal domain $D(A_\epsilon)$. Choose a diffeomorphism or unitary pullback

$$\alpha_\epsilon: \mathcal A(A_\epsilon)\longrightarrow\mathcal A(A_0).$$

Only $\alpha_\epsilon(K_{A_\epsilon})$ is a family on one representation. Changing the pullback shifts a commutator-like connection term, although invariant matrix elements agree when operator motion and contacts are transformed consistently.

For a smooth deformation of the vacuum half-space, Faulkner, Leigh, Parrikar, and Wang 2016, §1.1, eqs. (9)–(14) find, in their inward-pointing and null-coordinate convention,

$$\alpha_\epsilon(K_A) =K_{A_0} -2\pi\epsilon\int_{\mathcal H_+}\zeta^+T_{++} +2\pi\epsilon\int_{\mathcal H_-}\zeta^-T_{--} +\epsilon\delta K_{\rm pullback} +O(\epsilon^2).$$

The source’s eq. (14) gives $\delta K_{\rm pullback}$ explicitly in its continuation convention. It must not be dropped or rewritten with a guessed factor of $i$. Reversing the null normals, taking an outward deformation, or using inverse modular flow changes signs. The formula is an equality of regulated operators or controlled matrix elements; it is not an operator-norm expansion of arbitrary continuum modular Hamiltonians.

Exact null-cut benchmark

An exact CFT-vacuum family makes the first application particularly clean. Use the global metric convention $(+---)$ and let

$$x^\mu=\lambda\xi^\mu+y^\mu, \qquad \xi^\mu=(1,1,0,\ldots,0),$$

parametrize a null plane. The affine coordinate is $\lambda$ and $\mathbf y\in\mathbb R^{d-2}$ labels its generators. Let the future horizon start at $\lambda=\gamma(\mathbf y)$. With $T_{\lambda\lambda}=T_{\mu\nu}\xi^\mu\xi^\nu$, the vacuum-subtracted modular Hamiltonian on that horizon is

$$H_\gamma =2\pi\int d^{d-2}y \int_{\gamma(\mathbf y)}^\infty d\lambda\, [\lambda-\gamma(\mathbf y)] T_{\lambda\lambda}(\lambda,\mathbf y).$$

Casini, Testé, and Torroba 2017, §1, eqs. (1.5)–(1.7), and §3.2 establish this local null-horizon expression and explain its normalization. Away from the null surface the generator need not be local.

Choose the compact smooth profile

$$\gamma_\epsilon(\mathbf y)=\epsilon f_L(\mathbf y),$$

with

$$f_L(\mathbf y)= \begin{cases} \displaystyle \exp\!\left(1-\frac{1}{1-\lvert\mathbf y\rvert^2/L^2}\right), &\lvert\mathbf y\rvert<L,\\[6pt] 0,&\lvert\mathbf y\rvert\geq L. \end{cases}$$

Here $L>0$, $\epsilon$ has dimensions of length, and $\max f_L=1$. Differentiating the exact formula gives the leading correction

$$\delta H =-2\pi\epsilon \int d^{d-2}y\,f_L(\mathbf y) \int_0^\infty d\lambda\, T_{\lambda\lambda}(\lambda,\mathbf y).$$

This is the requested compact deformation of a vacuum half-space. It is a transverse smearing of a half-sided null-energy operator, not a pointwise stress tensor.

The approximation has a reproducible matrix-element error. Taylor’s theorem applied on each null generator gives the exact remainder

$$R_2 =2\pi\int d^{d-2}y \int_0^{\epsilon f_L(\mathbf y)}du\, [\epsilon f_L(\mathbf y)-u] T_{\lambda\lambda}(u,\mathbf y).$$

At a fixed UV regulator, on a common finite-energy core, define

$$M_{\Psi,\Phi}(\mathbf y) =\sup_{\lvert u\rvert\leq\lvert\epsilon\rvert} \left\lvert \langle\Psi|T_{\lambda\lambda}(u,\mathbf y)|\Phi\rangle \right\rvert.$$

Then

$$\lvert\langle\Psi|R_2|\Phi\rangle\rvert \leq \pi\epsilon^2 \int d^{d-2}y\,f_L(\mathbf y)^2 M_{\Psi,\Phi}(\mathbf y).$$

In the regulated theory the shape-control parameter is

$$\eta=\max\lvert\nabla\gamma_\epsilon\rvert \leq2.171\frac{\lvert\epsilon\rvert}{L}.$$

For example, $\lvert\epsilon\rvert/L=10^{-2}$ gives $\eta<2.18\times10^{-2}$. The error ceiling is a statement about declared regulated matrix elements, not an operator norm. In the continuum, $T_{\lambda\lambda}$ is an operator-valued distribution, so the corresponding bound must be stated in a test-function or energy norm rather than with the pointwise $M_{\Psi,\Phi}$. Longitudinal light-ray integrals may be defined first with a smooth large-$\lambda$ cutoff and then removed on the common domain.

Displacement and contact terms

In a replica description the twist operator is a codimension-two defect. Fix the displacement-operator sign by

$$\partial_\mu T_{\rm tot}^{\mu a}(x) =-\delta_\Sigma(x)D^a(\mathbf y)+\text{derivative contacts},$$

where $a$ is normal to the defect. Billò, Gonçalves, Lauria, and Meineri 2016, §5.1, eqs. (5.21)–(5.24) derive the defect Ward identities and show how the displacement insertion represents shape response.

At second order, a separated-point correlator $\langle D^aD^b\rangle$ is not the whole Hessian. Varying the defect, surface measure, normal frame, stress tensor, and counterterms creates coincident contacts. Replica and horizon calculations agree only after matching normal orientation, analytic continuation in replica number, and the local counterterm scheme.

Adversarial tests

Cusp. Replace the bump by

$$\gamma_\alpha(\mathbf y) =\alpha\lvert y_1\rvert\chi(\mathbf y/L),$$

where $\alpha$ is a dimensionless cusp amplitude and $\chi$ is smooth, compactly supported, and equals one near the origin. This $\alpha$ is distinct from the length-valued smooth-bump amplitude used above. The profile is continuous but not $C^1$, and distributionally

$$\partial_{y_1}^2\gamma_\alpha =2\alpha\delta(y_1)+\text{regular terms near the cusp}.$$

A smooth shape expansion that keeps only separated points misses this localized curvature/contact contribution. The general smooth-pullback remainder is therefore withdrawn. The stronger exact null-plane formula can still survive for a continuous cut on the light-ray domain; this does not promote the cusp to a smooth tangent.

Fixed-coordinate cutoff. Put a short-distance affine cutoff $a>0$ on each null generator. A cutoff co-moving with the cut gives

$$F_{\rm co}(\gamma) =\int_{\gamma+a}^\infty d\lambda\,(\lambda-\gamma)T(\lambda),$$

whereas a coordinate-fixed cutoff gives

$$F_{\rm fixed}(\gamma) =\int_a^\infty d\lambda\,(\lambda-\gamma)T(\lambda).$$

At $\gamma=0$,

$$F_{\rm fixed}'(0)-F_{\rm co}'(0)=aT(a).$$

After transverse smearing, the apparent first-order modular response differs by $2\pi a\int f_LT(a)$. This term cannot be discarded before controlling the near-cut stress matrix element and its counterterms. Thus a statement made with a fixed coordinate cutoff does not automatically describe a physically co-moving regulator.

Common pitfalls

Differentiating generators on different algebras. Supply the pullback and common quadratic-form domain first.

Keeping only one horizon flux. Future, past, and pullback terms are tied to the chosen Cauchy surface and identification.

Calling a cusp a small smooth deformation. Small amplitude does not bound derivatives. Smoothness and amplitude are independent control parameters.

Exercises

1. Show that a purely tangential deformation of a closed entangling surface is a reparametrization to first order.

Solution

Write $\zeta^\mu=v^i\partial_iX^\mu$. Under the coordinate change $y^i\mapsto y^i+\epsilon v^i$, the embedding becomes

$$X^\mu(\mathbf y+\epsilon\mathbf v) =X^\mu(\mathbf y) +\epsilon v^i\partial_iX^\mu+O(\epsilon^2).$$

Thus the displaced points describe the same geometric surface to first order. A nonzero response from this direction alone signals an unmatched pullback, boundary, anomaly, or regulator term.

2. Differentiate the exact null-cut modular Hamiltonian twice with respect to a one-parameter cut $\gamma_s=sf$.

Solution

For one generator define

$$G(a)=\int_a^\infty d\lambda\,(\lambda-a)T(\lambda).$$

Leibniz differentiation gives

$$G'(a)=-\int_a^\infty T(\lambda)d\lambda, \qquad G''(a)=T(a).$$

Therefore

$$\left.\frac{dH_{sf}}{ds}\right|_0 =-2\pi\int d^{d-2}y\,f \int_0^\infty d\lambda\,T_{\lambda\lambda},$$

and

$$\left.\frac{d^2H_{sf}}{ds^2}\right|_0 =2\pi\int d^{d-2}y\,f^2 T_{\lambda\lambda}(0,\mathbf y).$$

The second expression is distributional and requires the stated smearing and common domain.

3. Derive the boundary mismatch between a co-moving affine cutoff and a coordinate-fixed cutoff.

Solution

For the co-moving prescription,

$$F_{\rm co}'(0) =-aT(a)-\int_a^\infty T(\lambda)d\lambda.$$

For the fixed prescription only the weight changes, so

$$F_{\rm fixed}'(0) =-\int_a^\infty T(\lambda)d\lambda.$$

Subtracting gives $aT(a)$. Multiplication by the modular normalization and transverse profile yields $2\pi a\int d^{d-2}y\,f(\mathbf y)T(a,\mathbf y)$. Its limit is controlled only if the near-boundary stress insertion and counterterms are controlled.

References

• Billò, Marco, Vasco Gonçalves, Edoardo Lauria, and Marco Meineri. “Defects in Conformal Field Theory.” Journal of High Energy Physics 2016, no. 4 (2016): 091. DOI; arXiv.
• Casini, Horacio, Eduardo Testé, and Gonzalo Torroba. “Modular Hamiltonians on the Null Plane and the Markov Property of the Vacuum State.” Journal of Physics A: Mathematical and Theoretical 50, no. 36 (2017): 364001. DOI; arXiv.
• 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.