REVIEW
From bulk flows to boundary generators and costs
High Energy Physics - Theory · Submitted 21 Oct 2026 · 69 references
Pith's verdict: Finite-dimension penalties and scalar deformation of volume are a real advance; the York cancellation assumption is soft but acknowledged.
T0 · AI review
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Referee report
The paper
The paper claims
Bulk observables define quadratic cost metrics for boundary state preparation via symplectic transport
The reading
What carries the argument
The key machinery is the equality of the bulk symplectic form Ω_bulk and the boundary Kähler form Ω_∂, which allows the Hamiltonian vector field of a bulk observable (constructed via the Peierls prescription) to be transported to variations of Euclidean preparation sources. The boundary generator H_CFT_W is obtained by integrating the resulting one-form. Its Hessian H at a stationary reference state defines the cost metric G = ½H. The linearised flow A, satisfying Ω_∂(u,Av)=H(v,u), determines a compatible complex structure J_W and a scaling factor |A| that relate the optimal preparation path to the transported flow: the terminal velocity is −|A|⁻¹ J_W δ_W to leading order.
Extended reading notes
Core claim
The paper discovers that for the maximal-volume observable, the Hessian of its boundary generator produces two unequal principal penalties for variations of the real part of the source (field displacement) and the imaginary part (momentum displacement) when the scalar conformal dimension is finite. The ratios approach a common value only in the large-Δ limit, where the cost metric becomes proportional to the Fubini-Study metric. Furthermore, deforming the volume by quadratic scalar field and gradient terms shifts the field-displacement penalties while leaving momentum-displacement penalties unchanged, with the gradient term introducing mode dependence. This demonstrates that the choice of bulk observable directly selects the boundary preparation geometry.
Load-bearing premise
The construction requires that the reference state be a stationary point of the boundary generator with positive-definite Hessian, and that the symplectic transport identification holds for the chosen family of states, which in the examples relies on the cancellation of the York deformation contribution via the boundary Ward identity and the absence of corner contributions.
Editorial extensions
If this is right
- Different bulk observables (volume, action, etc.) will automatically select different boundary penalty factors, organising the landscape of holographic complexity proposals under a single principle.
- The first law of complexity δC = δW_bulk holds to quadratic order around the reference state, providing a way to compute changes in complexity directly from bulk geometry without defining circuits.
- For maximal volume, finite conformal dimension effects break the proportionality between cost and Fubini-Study metric at relative order 1/Δ², which may be testable in holographic setups or tensor network models.
- Adding gradient couplings to the bulk observable introduces mode-dependent penalties, allowing different source profiles to carry different relative costs, which could be compared with circuit complexity calculations in CFT.
Reading between the lines
- The prescription is likely extendable to backgrounds such as two-sided black holes, but stationarity and Hessian positivity must be verified case by case; the authors note that the transport construction is harder for black holes.
- Beyond the quadratic approximation, the cost metric may not remain constant in source coordinates, and the optimal geodesics could deviate from straight lines, connecting the framework to non-linear Nielsen geometry.
- The boundary generator H_CFT_W does not necessarily equal the expectation value of a known CFT operator, but its first variation can be interpreted via symplectic pairing, offering a universal dictionary for any diffeomorphism-invariant observable.
Original abstract
Holographic complexity admits a broad family of bulk observables, yet their relation to boundary state-preparation costs remains unclear. We propose a local prescription that associates suitable bulk observables with quadratic cost geometries on families of states prepared by Euclidean sources. Building on the holographic transport of bulk Hamiltonian flows to Euclidean preparation sources, we use the Hessian of the transported generator at a stationary reference state to define a candidate cost metric. When this Hessian is positive definite, the minimised preparation energy reproduces the change in the bulk observable to quadratic order. We characterise the resulting preparation penalties relative to the Fubini-Study metric and compute them for the maximal volume in scalar perturbations of AdS$_3$. At large-conformal dimension, we recover the known proportionality between cost and Fubini-Study metrics. At finite dimension, we obtain positive but generally unequal penalties for source variations that displace the bulk field and its conjugate momentum, breaking this proportionality even for a single scalar operator. We further study a deformation of volume by quadratic scalar and gradient terms in the large-dimension regime, determining how the couplings modify the penalties and constrain positivity. These additions selectively change field-displacement costs, with the gradient term introducing mode dependence. Our results provide an explicit local connection between the choice of bulk observables and the assignment of boundary preparation penalties.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
Holographic complexity admits a broad family of bulk observables, yet their relation to boundary state-preparation costs remains unclear. We propose a local prescription that associates suitable bulk observables with quadratic cost geometries on families of states prepared by Euclidean sources. Building on the holographic transport of bulk Hamiltonian flows to Euclidean preparation sources, we use the Hessian of the transported generator at a stationary reference state to define a candidate cost metric. When this Hessian is positive definite, the minimised preparation energy reproduces the change in the bulk observable to quadratic order. We characterise the resulting preparation penalties relative to the Fubini-Study metric and compute them for the maximal volume in scalar perturbations of AdS$_3$. At large-conformal dimension, we recover the known proportionality between cost and Fubini-Study metrics. At finite dimension, we obtain positive but generally unequal penalties for source variations that displace the bulk field and its conjugate momentum, breaking this proportionality even for a single scalar operator. We further study a deformation of volume by quadratic scalar and gradient terms in the large-dimension regime, determining how the couplings modify the penalties and constrain positivity. These additions selectively change field-displacement costs, with the gradient term introducing mode dependence. Our results provide an explicit local connection between the choice of bulk observables and the assignment of boundary preparation penalties.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Holographic complexity admits a broad family of bulk observables, yet their relation to boundary state-preparation costs remains unclear. | gap 0.91 | 0.05 | 0.03 | asserted 0.92 |
| s1 | We propose a local prescription that associates suitable bulk observables with quadratic cost geometries on families of states prepared by Euclidean sources. | contribution 0.96 | 0.91 | 0.21 | asserted 0.74 |
| s2 | Building on the holographic transport of bulk Hamiltonian flows to Euclidean preparation sources, we use the Hessian of the transported generator at a stationary reference state to define a candidate cost metric. | method 0.96 | 0.90 | 0.09 | asserted 0.83 |
| s3 | When this Hessian is positive definite, the minimised preparation energy reproduces the change in the bulk observable to quadratic order. | result 1.00 | 0.85 | 0.25 | asserted 0.98 |
| s4 | We characterise the resulting preparation penalties relative to the Fubini-Study metric and compute them for the maximal volume in scalar perturbations of AdS$_3$. | contribution 0.74 | 0.95 | 0.08 | asserted 1.00 |
| s5 | At large-conformal dimension, we recover the known proportionality between cost and Fubini-Study metrics. | result 0.98 | 0.70 | 0.10 | asserted 1.00 |
| s6 | At finite dimension, we obtain positive but generally unequal penalties for source variations that displace the bulk field and its conjugate momentum, breaking this proportionality even for a single scalar operator. | result 0.99 | 0.96 | 0.22 | asserted 1.00 |
| s7 | We further study a deformation of volume by quadratic scalar and gradient terms in the large-dimension regime, determining how the couplings modify the penalties and constrain positivity. | contribution 0.98 | 0.95 | 0.08 | asserted 1.00 |
| s8 | These additions selectively change field-displacement costs, with the gradient term introducing mode dependence. | result 0.97 | 0.92 | 0.07 | asserted 1.00 |
| s9 | Our results provide an explicit local connection between the choice of bulk observables and the assignment of boundary preparation penalties. | interpretation 0.60 | 0.92 | 0.84 | asserted 1.00 |
What was examined, and what was not
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References
Discussion
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