REVIEW 2 major objections 4 minor 35 references
Does Boundary Distinguish Complexities?
T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read The boundary does not distinguish the main holographic complexity measures, with one exception.
desk verdict Solid new BCFT complexity computations in the path-integral and CV sectors, but the d=2 CA exception depends on an unshown log-zero null-joint subtraction that the authors themselves flag. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central object is the boundary complexity $\Delta C^{\mathrm{bdy}} = C^{\mathrm{BCFT}} - \tfrac{1}{2} C^{\mathrm{CFT}}$, the increment caused by placing the CFT on a half-space. In the path-integral method the work is done by the boundary Liouville action, whose optimization tilts the boundary to $x = -\alpha z$ and yields $\Delta C^{\mathrm{bdy}}_L = \frac{c}{6\pi}\alpha \log(z_\infty/\epsilon)$, with $\alpha = \mu_B L/\sqrt{1-\mu_B^2 L^2}$. In the holographic computations, the load-bearing machinery is the Wheeler-DeWitt action with its joint terms: the null joints where the null surfaces meet the brane have $\log 0$ divergences because the normal and null vectors are orthogonal, and the final $d=2$ result comes from those divergent terms cancelling against the half-AdS subtraction while the corner and timelike-joint terms leave the finite $\sqrt{1+\alpha^2}-1$ factor.
What would settle it
Compute the null-joint contribution $J_{n,1}$ with a small regulator that resolves the orthogonality between the brane normal and the null tangent, for example by using a near-null direction or a corner smoothing, and check whether the regularized logarithmic term produces an $\alpha$-dependent finite remainder as the regulator is removed. If any $\alpha$-dependent piece survives, equation (3.50) is regulator-dependent and the paper's central conclusion for $d=2$ fails.
Extended reading notes
Core claim
On the paper's own terms, the central discovery is a comparison result for boundary complexity. Defining the boundary complexity as the difference between the BCFT complexity and half the parent CFT complexity, the optimized path-integral (Liouville) complexity and the CV complexity both produce a boundary contribution that diverges logarithmically in two dimensions, with coefficients fixed by the boundary tilt parameter $\alpha$, and power-law divergences in higher dimensions. The CA complexity instead gives a non-vanishing boundary contribution in $d>2$ that shares the CV power-law divergence, but in $d=2$ the logarithmic term vanishes and the boundary complexity is the finite constant $\Delta C^{\mathrm{bdy}}_A = \frac{L}{4\pi G_N}(\sqrt{1+\alpha^2}-1)$. Hence, apart from the $\mathrm{AdS}_3/\mathrm{BCFT}_2$ case, the boundary increment does not pick out a unique complexity conjecture; boundaries and defects do not generically distinguish action from volume.
Load-bearing premise
The CA result rests on assuming that the $\log 0$ divergences at the null joints are independent of the boundary parameter $\alpha$ and cancel when the half-AdS complexity is subtracted; if they do not cancel exactly, the finite $d=2$ boundary complexity is an artifact of the subtraction convention.
Editorial extensions
If this is right
- In $d=2$, the CA boundary complexity is a finite universal constant, so the boundary increment has a different divergence structure from both the path-integral and CV complexities.
- In $d>2$, the CA boundary complexity does not vanish and shares the CV divergence structure, so the action/volume distinction found for defects does not extend to boundaries in general.
- The path-integral optimized complexity and the CV complexity produce the same logarithmic boundary scaling in $d=2$, differing only by an overall factor, which the paper reads as quantitative agreement between the two approaches.
- The boundary complexity is a monotonic function of the boundary entropy parameter, so the $g$-theorem implies monotonic decrease of the boundary complexity under boundary renormalization-group flow.
- A complete definition of CA complexity in spacetimes with a brane boundary must handle the $\log 0$ null-joint divergences, since they are not cured by the usual counterterms.
Reading between the lines
- An implication the authors leave implicit is that the $d=2$ CA constant could serve as a sharper diagnostic than the divergent pieces: a numerical tensor-network computation of boundary complexity in a critical chain could look for a finite boundary term rather than a logarithmic one.
- A natural extension is to finite-temperature or finite-interval versions of the same setup; if the finite constant persists there, it would survive as an unambiguous signature of the CA prescription.
- The fact that the corner angle between the brane and the cutoff surfaces appears in the $d>2$ CA result suggests that non-smooth joints, not just null joints, carry the distinction between holographic complexity proposals.
- If the boundary complexity is truly monotonic along boundary RG flow, it provides a candidate complexity analogue of the $g$-theorem that could be tested independently in integrable boundary CFTs.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies whether the presence of a boundary distinguishes different notions of holographic complexity. The authors define a "boundary complexity" as the increment ΔC_bdy = C_BCFT − ½C_CFT and compute it in three approaches: the path-integral (Liouville) complexity in two-dimensional BCFT, the complexity=volume (CV) conjecture, and the complexity=action (CA) conjecture in Takayanagi's AdS/BCFT model. Their main results are that the Liouville and CV boundary complexities diverge logarithmically in d=2, whereas the CA boundary complexity in AdS3/BCFT2 is a finite α-dependent constant (Eq. 3.50); for d>2 the CA boundary complexity diverges as 1/ε^{d−2} (Eq. 3.49). The paper concludes that the boundary does not distinguish the complexities in general, with the CA complexity in AdS3/BCFT2 as an exception, thereby qualifying the earlier defect-complexity argument of Chapman et al.
Significance. The paper addresses a timely and actively studied question, namely whether complexity proposals are mutually consistent in the presence of boundaries or defects. Its path-integral and CV computations are clean, and the boundary-entropy matching in Section 2.3 provides a useful cross-check that identifies the boundary slope α with the brane tension. The paper is also honest about its main weakness: the d=2 CA result relies on a log-zero null-joint subtraction that is not derived in detail, as acknowledged in Section 4 and footnote 6. If that subtraction can be justified with an explicit regulator, the result would be a significant constraint on holographic complexity proposals. As written, however, the advertised central exception for AdS3/BCFT2 is not fully established.
major comments (2)
- [3.3, Eqs. (3.39)–(3.41) and (3.50)] The evaluation of the null joints Jn,1 and Jn,2 is not a complete derivation. Equation (3.41) states that a = log|k·s| = log 0, and the text following Eq. (3.40) asserts that the joint term is independent of the boundary parameter α and is removed by the subtraction of the half-AdS reference. However, the induced metric in Eq. (3.40) contains an α-dependent term (L² α² dz²/z²), so the integrated joint contribution generally inherits α-dependence unless a separate cancellation is proven. No regulator is specified for log 0, and no evaluation of ∫ d^{d−1}X √h log|k·s| is shown. This is load-bearing because the d=2 CA boundary complexity in Eq. (3.50) and the 1/ε^{d−2} coefficient in d>2 both depend on this subtraction. Please provide a regulated evaluation of the null joint (for example, by taking a null normal with a small angle and then taking the limit), and prove that the α-dependence cancels after including the half-AdS subtraction.
- [3.3, Eq. (3.49)] The d>2 CA boundary complexity depends explicitly on the arbitrary counterterm scale l_ct through the term 2 log(l_ct(d−2)/L) arcsinh α. Thus the coefficient of the leading 1/ε^{d−2} divergence is scheme-dependent. The statement that the boundary complexities show "the same divergent structures" in d>2 is therefore weaker than a parameter-free comparison. The authors should specify precisely which quantities are compared (the power of the divergence, the coefficient, or only whether the divergence is present) and discuss whether the l_ct dependence cancels in any physically meaningful difference of complexities.
minor comments (4)
- [2.2, title and text] The word "Liuouville" appears in the heading of Section 2.2; it should read "Liouville".
- [3.3, after Eq. (3.35)] The statement that the M-dependence of the null-surface counterterm cancels with the joint terms is asserted but not demonstrated. A few intermediate lines showing the cancellation would improve readability.
- [3.3, Eq. (3.14)] The signs ϵκ, ϵa, and ϵφ are introduced in words but not summarized in one place. A short table or explicit assignment for each joint would help the reader follow the lengthy computation.
- [References] Reference [23] is listed as "In preparation"; if the paper by Braccia, Cotrone, and Tonni has appeared by publication time, the reference should be updated and the overlap discussed in the note added.
Circularity Check
No circularity: the three complexity computations are independent, with only non-load-bearing reliance on the authors' earlier path-integral proposal.
full rationale
The paper's three computations are independent derivations from separate definitions. The path-integral boundary complexity (2.16) follows from minimizing the boundary Liouville action (2.5), not from CV or CA. The CV boundary complexity (3.11)-(3.12) is a direct volume integral. The CA boundary complexity (3.49)-(3.50) sums bulk, brane, surface, null, and joint contributions; each term is evaluated from the geometry, and the final subtraction is an explicit convention: 'When we subtract the complexity without boundary, we include null joint terms and timelike joint terms to the half of the complexity, CCFT_A/2, such that the boundary complexity vanishes for α=0.' The use of the authors' earlier path-integral optimization proposal [11,12] is a framework choice, not a result forced by that citation; the paper explicitly treats it as one of three definitions and checks it against boundary entropy in Sec. 2.3, finding agreement with Takayanagi's AdS/BCFT model. The log 0 null-joint issue in Sec. 3.3 is acknowledged by the authors ('unavoidable divergences due to log 0') and is a regularization/correctness gap, not a circular reduction: no fitted parameter is renamed as a prediction and no equation is defined in terms of the conclusion. Thus no circular step can be exhibited.
Assumptions & free parameters
free parameters (2)
- α (brane slope, or μ_B L) =
N/A (model parameter)
- l_ct (null counterterm length scale) =
unspecified
assumptions (5)
- domain assumption CV conjecture: complexity equals the maximal volume of a codimension-one surface divided by G_N L
- domain assumption CA conjecture: complexity equals the WDW action divided by πℏ
- domain assumption Takayanagi's AdS/BCFT model: brane Q with Neumann boundary condition and tension T = (d-1)/L α/√(1+α^2)
- domain assumption Path-integral optimization as a measure of complexity: on-shell Liouville action gives the complexity
- domain assumption Boundary Liouville action (2.5) correctly captures the Weyl transformation of the path integral in BCFT
Cite this review
Pith. "Pith review of Does Boundary Distinguish Complexities?." pith.science (2026). https://pith.science/paper/VIY34M5Z
@misc{pith2026190811094,
author = {Pith},
title = {Pith review of: Does Boundary Distinguish Complexities?},
year = {2026},
howpublished = {\url{https://pith.science/paper/VIY34M5Z}},
note = {Machine review of arXiv:1908.11094}
}
abstract
Recently, Chapman et al. argued that holographic complexities for defects distinguish action from volume. Motivated by their work, we study complexity of quantum states in conformal field theory with boundary. In generic two-dimensional BCFT, we work on the path-integral optimization which gives one of field-theoretic definitions for the complexity. We also perform holographic computations of the complexity in Takayanagi's AdS/BCFT model following by the "complexity $=$ volume" conjecture and "complexity $=$ action" conjecture. We find that increments of the complexity due to the boundary show the same divergent structures in these models except for the CA complexity in the AdS$_3$/BCFT$_2$ model as the argument by Chapman et al. Thus, we conclude that boundary does not distinguish the complexities in general.
Reference graph
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Reviewed August 14, 2026 · model on record in the stance chip above.
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