{"id":"e353d0ab-cd0e-4fd8-b0fc-4f1a16928276","arxiv_id":"1908.11094","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"In BCFT, boundary complexity increments show the same divergent structure for volume, action, and path-integral measures in d>2, but in d=2 the action measure gives a finite constant instead of a logarithmic divergence.","lead":"This paper computes several measures of quantum complexity in conformal field theories with a boundary. It finds that, except in one special case, the boundary contribution to complexity looks the same across different complexity measures.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The d=2 CA boundary complexity rests on an unregularized log-zero null-joint subtraction; α-independence of the integrated joint is not shown, so Eq. (3.50) is not established.","rationale":"The reader's weakest assumption is exactly the null-joint subtraction in the CA computation, and I agree that it is the most load-bearing point. The path-integral optimization and CV computations are transparent and internally consistent; the d=2 logarithmic divergence there is supported by the boundary-entropy parameter matching, and I have no substantive objection to those parts. For CA, however, the log-zero joint is not actually computed. The paper's own discussion section concedes that the joint terms contain unavoidable divergences and that the resolution is open. Equations (3.49) and (3.50) therefore rely on the unstated assumption that the divergent joint contribution is α-independent and cancels from the boundary complexity. Because the joint integration measure in Eq. (3.40) depends on α, a finite α-dependent remainder from any regulator would change both the d>2 coefficient and, more importantly, the d=2 finite constant (3.50). Since that constant is the advertised exception to the claim that boundary does not distinguish complexities, the headline conclusion is hostage to this regularization. The l_ct dependence in Eq. (3.49) is a lesser concern: it affects the coefficient of the d>2 power divergence but not the divergence structure, so it is not the deciding issue. I therefore recommend no change to the reader's conditional verdict: the paper is plausible but needs an explicit regulator treatment of the null joints before the CA result can be accepted.","tokens_in":101,"tokens_out":11033,"duration_ms":213648,"concrete_test":"Regularize the null joint explicitly. For example, tilt the brane by a small angle δ so that k·s_δ = δ |k||s| (or set x1 = -α z + δ z), compute Jn(α,δ) = (1/8πG) ∫ √h log|k·s_δ| over the joint with the same UV and IR cutoffs, subtract the α=0 half-AdS reference Jn(0,δ), and take δ→0. If lim_{δ→0}[Jn(α,δ) - Jn(0,δ)] is nonzero or divergent, then the assertion that the log-zero term is α-independent fails and Eq. (3.50) must be revised. A practical variant: repeat the d=2 CA computation with a finite-angle brane boundary and verify that the finite part of ΔC_A is independent of the regulator before accepting (3.50).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is the treatment of the null joints Jn,1 and Jn,2 in §3.3. Eq. (3.41) gives a = log|k·s| = log 0, and the paper asserts that the joint term is independent of the boundary parameter α and is removed when the half-AdS reference is subtracted, without evaluating ∫ d^{d-1}X √h a. This subtraction carries the d=2 conclusion: the final CA boundary complexity (3.50) is a finite α-dependent constant, while the CV and path-integral results are logarithmically divergent, and that contrast is the advertised exception to 'boundary does not distinguish complexities.' A log-zero joint is a prescription-dependent infinity; it is not enough that the logarithm itself is α-independent. Eq. (3.40) shows the induced metric on the joint depends on α through the α² dz² term, so the integrated joint contribution generally inherits α-dependence unless a separate cancellation is proven. The manuscript itself flags this in §4 and footnote 6 ('unavoidable divergences due to log 0') and says a resolution might shed light on holographic complexity. As written, the d=2 CA claim rests on an unverified cancellation; the same subtraction also enters the 1/ε^{d-2} coefficient in d>2. This is an internal gap, not a disagreement with external consensus.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":14558,"tokens_out":3565,"duration_ms":37822,"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":[{"comment":"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.","section":"3.3, Eqs. (3.39)–(3.41) and (3.50)"},{"comment":"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.","section":"3.3, Eq. (3.49)"}],"minor_comments":[{"comment":"The word \"Liuouville\" appears in the heading of Section 2.2; it should read \"Liouville\".","section":"2.2, title and text"},{"comment":"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.","section":"3.3, after Eq. (3.35)"},{"comment":"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.","section":"3.3, Eq. (3.14)"},{"comment":"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.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper's Liouville and CV parts are solid and well cross-checked via boundary entropy. The main difficulty is the CA computation in d=2, where the log-zero null joint is handled by a subtraction whose α-independence is only asserted, not shown. Since the d=2 CA exception is the advertised central result, this needs to be fixed before publication. I believe the authors can address it with an explicit regulator, so major revision rather than rejection is appropriate. There is also acknowledged overlap with the forthcoming Braccia-Cotrone-Tonni work; the final version should clearly state the relationship."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nWhat you need to know: this is a useful, honest paper that computes boundary complexity in BCFTs three ways — path-integral optimization, CV, and CA — and asks whether the boundary distinguishes them. The path-integral and CV sections are solid, and the boundary-entropy cross-check is a nice touch. The CA section is the weak link: the advertised exception in AdS3/BCFT2 (finite boundary complexity, versus logarithmic for the other two) rests on subtracting log-zero null-joint divergences, and that subtraction is asserted, not derived. The authors flag this themselves in Section 4, but flagging a gap is not closing it.\n\nWhat is actually new: the path-integral optimization for BCFT2 with the boundary Liouville action, which produces the tilted brane x = -αz; the CV boundary complexity, which matches the Liouville result in d=2; and the CA computation in Takayanagi's AdS/BCFT, including the WDW patch with the brane. The d>2 story — CV and CA both diverge as 1/ε^{d-2}, so the boundary does not distinguish them — is more robust and is the main takeaway. The paper also candidly notes the overlap with the forthcoming Braccia-Cotrone-Tonni work, which is the right thing to do.\n\nWhere it wobbles: the null joints Jn,1 and Jn,2 between the null surfaces and the brane. Equation (3.41) gives a = log|k·s| = log 0, and the paper says the joint term is α-independent and gets subtracted. But the induced metric on the joint, (3.40), depends on α through the α² dz² term, so the integrated joint will generally inherit an α-dependence unless a separate cancellation is shown. None is demonstrated. That is load-bearing: the d=2 result (3.50) is the one case where CA differs from CV, and the finite constant is exactly what the subtraction arrangement delivers. For d>2, the same subtraction feeds the 1/ε^{d-2} coefficient, though the qualitative conclusion survives. The d>2 coefficient also depends on the counterterm scale l_ct, which the authors acknowledge. These issues are technical and possibly fixable, but they keep the d=2 claim from being established as written.\n\nFor people working on holographic complexity or AdS/BCFT, this is a useful reference for the boundary CV and Liouville results, and the CA section is a good cautionary case study in null-boundary regularization.\n\nBottom line: worth a serious referee. A good referee will ask for a proper regularization of the null joints, not kill the paper. The path-integral and CV parts stand on their own; the CA part asks the right question but is not finished. Send it out.","headline":"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.","tokens_in":15124,"tokens_out":13043,"would_cite":true,"duration_ms":111177,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The boundary does not distinguish the main holographic complexity measures, with one exception.","keywords":["holographic complexity","complexity equals volume","complexity equals action","boundary CFT","AdS/BCFT","path-integral optimization","Liouville action","boundary entropy"],"falsifier":"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.","tokens_in":14080,"feed_emoji":"🧮","tokens_out":8038,"duration_ms":74707,"temperature":0.7,"pith_summary":"The paper asks whether placing a conformal field theory on a half-space with a boundary can tell apart the two main holographic definitions of quantum complexity: 'complexity equals volume' (CV) and 'complexity equals action' (CA). It computes the boundary-induced increment, the BCFT complexity minus half that of the parent CFT, in three approaches: the path-integral optimization definition, a field-theoretic counting of redundant path-integral degrees of freedom, and the CV and CA conjectures in the holographic AdS/BCFT model. In every case the boundary increment is nonzero, and its divergence structure is the same across approaches, with one exception: CA complexity in $\\mathrm{AdS}_3/\\mathrm{BCFT}_2$, where the logarithmic divergence drops out and the boundary contribution is a finite constant. The paper concludes that the boundary does not distinguish the complexities in general, unlike the defect case where action was invisible. If right, this means a boundary, by itself, is not a sharp test between the leading complexity conjectures except in one special dimension.","feed_headline":"Boundary does not distinguish the main complexity conjectures","feed_subtitle":"Volume and action complexities diverge alike in boundary CFTs, except for action in three bulk dimensions.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"The defect-CFT computation this paper tests, which found that action complexity vanishes for defects while volume complexity does not.","marker":"[19]"},{"why":"Introduced the path-integral optimization definition of complexity in CFT that the authors apply to the boundary case.","marker":"[11, 12]"},{"why":"Introduced the AdS/BCFT model with a brane boundary and its Neumann boundary condition, which supplies the holographic spacetime used for the CV and CA computations.","marker":"[21, 22]"},{"why":"Proposed the complexity-equals-volume conjecture that is evaluated in the AdS/BCFT background.","marker":"[4, 5]"},{"why":"Proposed the complexity-equals-action conjecture and the Wheeler-DeWitt action prescription used for the CA computation.","marker":"[6, 7]"},{"why":"Provide the gravitational action terms for null boundaries and joints, including the null-joint and corner contributions that carry the log-zero divergences.","marker":"[30, 31]"},{"why":"Introduced boundary entropy and the g-theorem that links the boundary parameter to monotonic RG flow.","marker":"[27, 28]"}],"fun_headline_variants":["Boundary does not single out action or volume complexity","Except in AdS3, CV and CA boundary increments agree","Boundary complexity agrees, except for AdS3 action","No boundary complexity winner, save AdS3 action","Boundary does not distinguish complexities, mostly"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Boundary does not single out action or volume complexity","Except in AdS3, CV and CA boundary increments agree","Boundary complexity agrees, except for AdS3 action","No boundary complexity winner, save AdS3 action","Boundary does not distinguish complexities, mostly"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000731,"raw_usage":{"total_tokens":3235,"prompt_tokens":875,"completion_tokens":2360,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":491,"completion_tokens_details":{"reasoning_tokens":2283}},"tokens_in":491,"tokens_out":2360,"duration_ms":19934,"temperature":1.0,"reasoning_tokens":2283,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T10:24:09.214117+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}