{"id":"03063774-9bee-4dd6-acf8-9a96d0c69c11","arxiv_id":"2506.17382","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A mixed-correlator bootstrap for 2D boundary CFTs produces new rigorous bounds on boundary entropy, bulk-to-boundary OPE coefficients, and gap spectra, tested on Ising and free boson and applied to su(2)_2 WZW.","lead":"A new conformal bootstrap setup combines three crossing equations to constrain boundary conditions in two-dimensional conformal field theories. It yields new numerical bounds on boundary entropy and spectrum gaps, including for the su(2)_2 Wess-Zumino-Witten model at central charge 3/2.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Uncontrolled truncation of the boundary-channel blocks (N_trunc=21, |rho_bry|~0.71) leaves the 'rigorous' bound h_2pt^gap<0.5 without a valid certificate; a spurious exclusion is possible.","rationale":"The paper's central claim is that the three crossing equations plus positivity form a valid SDP yielding rigorous bounds on boundary CFT data. The derivation is standard and the method is validated on the Ising model and free boson, so the overall approach is sound. The weakest link is the numerical implementation of the conformal blocks: the SDP uses truncated hypergeometric series with N_trunc = 21 at N = 35, and the boundary-channel radial coordinate at the chosen xi* is approximately 0.708, close to the radius of convergence. The paper reports that this truncation order was selected for numerical stability, not because it preserves positivity with a controlled error. Without interval-arithmetic certificates, a functional that is positive on the truncated blocks can violate positivity on the exact blocks, potentially making excluded regions spurious. This directly affects the word 'rigorous' in the strongest claim. The reported numerical instabilities at small h_ann^gap are a related symptom. A concrete computational check that evaluates the exact blocks and the dual functional from the SDPB output, or reruns with a larger truncation order, would settle the issue. Given that the physical results are likely correct and the method is novel, the appropriate verdict is conditional acceptance: the rigorous claims should either be backed by error-controlled block evaluation or reworded as numerical bounds.","tokens_in":47472,"tokens_out":15574,"duration_ms":161955,"concrete_test":"Re-run the Ising exclusion at h_2pt^gap = 0.5 using the same functional normalization but evaluate the boundary two-point blocks G_bry_h(xi) = xi^{Delta_phi} F(h, rho_bry(xi)) and their first 35 derivatives at xi* with a certified high-precision hypergeometric evaluator (e.g., Arb with interval arithmetic) instead of the N_trunc = 21 truncated series. If the dual functional from the SDPB output satisfies alpha.V^{2pt}_h >= 0 for all h >= 0.5 on the exact blocks, the bound is rigorous; if any violation appears, the truncated series overstates positivity and the excluded point may be allowed. Additionally, rerun the same SDP with N_trunc = 40 and check whether the exclusion boundary shifts; a stable boundary would weaken the concern.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The paper's central claim is that the mixed-correlator SDP yields rigorous numerical bounds, including the first rigorous bound on the two-point boundary gap, h_2pt^gap < 0.5 for the Ising model (fig. 2). The SDP positivity conditions (2.44) are evaluated on conformal blocks approximated by truncating the hypergeometric series in the radial coordinate rho at N_trunc = 21, 27, 36 for N = 35, 45, 55 derivatives (Appendix A.2). At the chosen evaluation point xi* = 0.03010 (eq. A.43), the boundary-channel radial coordinate is rho_bry = 1 + 2 xi* - 2 sqrt(xi*(1+xi*)) ~ 0.708, close to 1. The truncated series is therefore slowly convergent, and the authors state that N_trunc is the lowest order that empirically gives a stable SDP, not an order that guarantees the approximation preserves positivity. A functional that is positive on the truncated blocks may be negative on the exact blocks; if so, the exclusion of, e.g., h_2pt^gap = 0.5 would be spurious. No error bounds or interval-arithmetic certificate checks are provided, so the 'rigorous' label is not currently supported. The reported numerical instabilities for h_ann^gap < 0.2 (c=1) and < 0.12 (c=3/2) are a further symptom that the dual certificates are not robust in parts of parameter space, though the main new bounds (Ising, figs. 14-16) lie outside these unstable regions.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a numerical conformal bootstrap for two-dimensional boundary CFTs by combining crossing symmetry of three observables: the bulk four-point function on the plane, the annulus partition function, and the two-point function in the presence of a boundary. These are assembled into a positive semi-definite program whose dual functionals yield bounds on boundary gaps, the boundary entropy g^2, and bulk-to-boundary OPE coefficients. The method is validated against the Ising model and the free boson, and then applied to c = 3/2 theories, with special focus on the su(2)_2 WZW model. The main new results include the claimed first rigorous bound on the boundary OPE gap h_2pt^gap (e.g., h_2pt^gap < 0.5 for the Ising model, Fig. 2) and new bounds on g^2 and a_phi^2 for c = 3/2 (Figs. 13-16).","tokens_in":47860,"tokens_out":13685,"duration_ms":140115,"significance":"If the bounds are genuinely rigorous, this is a significant methodological advance: it introduces a mixed bulk/boundary SDP that accesses previously unavailable boundary data and sometimes tightens existing annulus-only bounds. The derivation of the SDP is careful and standard, the method is tested on exactly solvable theories where it reproduces or improves known results, and the paper makes its numerical data and SDPB parameters publicly available on Zenodo. The analytic solution of the su(2)_k Knizhnik-Zamolodchikov equation in Appendix B is a useful independent contribution. The main caveat is the uncontrolled truncation of the two-point conformal blocks, which currently weakens the 'rigorous' characterization of the bounds.","major_comments":[{"comment":"The central claim that the mixed-correlator bootstrap yields rigorous bounds, in particular the new bound h_2pt^gap < 0.5 for the Ising model (Fig. 2), is not fully supported because the two-point function blocks are evaluated using a truncated hypergeometric series without error control. The paper states (Appendix A.2) that N_trunc = 21, 27, 36 for N = 35, 45, 55 derivatives is 'the lowest number of terms observed to give us a stable SDP', which is an empirical stability criterion rather than a guarantee that the truncated block preserves the positivity properties required by the SDP. At the evaluation point xi* ≈ 0.03010 (eq. A.43), the boundary-channel radial coordinate is rho_bry ≈ 0.708, so the truncation error is not negligible. A functional that is non-negative on the truncated blocks can in principle be negative on the exact blocks, making the exclusion of h_2pt^gap = 0.5 spurious. The reported numerical instabilities for h_ann^gap < 0.2 (c=1) and < 0.12 (c=3/2) are consistent with this concern. The authors should either provide rigorous error bounds (e.g., interval arithmetic or a certificate that the positivity margin dominates the truncation error) or soften the 'rigorous' characterization of the affected bounds.","section":"Appendix A.2 (eq. A.22); Section 3.1 (Fig. 2)"}],"minor_comments":[{"comment":"The claim of 'first rigorous bootstrap result' for h_2pt^gap should be qualified with respect to the existing boundary bootstrap literature that already uses two-point function crossing equations, e.g., ref. [31] and the follow-ups cited in refs. [30-44].","section":"Abstract / Section 1"},{"comment":"The spelling 'Knizhnik-Zamalodchikov' should be 'Knizhnik-Zamolodchikov'.","section":"Appendix B"},{"comment":"Reference [46] is listed as 'To Appear' without a title or year; please complete the citation.","section":"References"},{"comment":"The objective in eq. (3.18) uses the relation b_{Δφ,h=2} = (Δφ/√(2c)) a_{Δφ}; a short derivation or citation for this relation in the general case would improve readability.","section":"Section 3.2.2, eq. (3.18)"},{"comment":"The sets I_a, I_2, I_4, and I_{2×2} are used in the inequalities (2.44) before their formal definitions in the following paragraph; reordering would make the presentation cleaner.","section":"Section 2.2, eq. (2.44)"}],"recommendation":"major_revision","confidential_remarks":"The paper is a solid and well-executed contribution, but the rigorousness claim needs to be addressed before publication. I recommend asking the authors to either supply rigorous error control for the block truncation or to moderate the 'rigorous' language in the abstract and main text. The 'first' claim in the abstract should also be checked against the existing boundary bootstrap literature."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper delivers what its title promises: it feeds the annulus partition function, the two-point function with a boundary, and the bulk four-point function into one SDP, and gets new numerical bounds on bulk-to-boundary OPE data and, for the first time, on the two-point boundary gap. The Ising and free-boson tests are real validation, not tuned fits, and the c=3/2 exploration is genuinely new territory. The appendices are careful, the SDPB parameters are spelled out, and the Zenodo data makes the plots reproducible. That is good, solid work.\n\nThe soft spot is the 'rigorous' label, and the stress-test note lands. The SDP is built from conformal blocks truncated at N_trunc=21,27,36, evaluated at a point where the boundary-channel radial coordinate is about 0.71. The authors say this is the lowest order that empirically gives a stable SDP, not an order with a proven error bound. That means a functional positive on truncated blocks could be negative on exact blocks, and the claimed exclusion of h_2pt^gap=0.5 for Ising lacks a valid certificate in the strict sense. This is a common gap in numerical bootstrap papers, which makes it no less real: 'rigorous' should be replaced by 'numerical up to controlled truncation' unless tail bounds are supplied.\n\nThat said, the issue is not load-bearing. The central argument holds: the method is new, the validations reproduce known physics, and the c=3/2 bounds are novel even if some regions near small h_ann^gap are numerically unstable. The authors state those instabilities openly. Their use of global blocks rather than Virasoro blocks is a documented simplification, not a hidden assumption. The input bulk spectrum for the WZW application comes from the independent KZ solution, so circularity is not a concern. The non-elementary brane discussion is a genuine strength, not a flaw.\n\nWho should read this? Anyone working on boundary or defect bootstrap, and anyone who wants to see how to combine different crossing equations into one optimization problem. It deserves a serious referee. My recommendation: send it to peer review, and ask the authors to either add tail bounds for the truncated blocks or soften the rigor claim. With that change, I would be happy to see it published.","headline":"Real step forward: first numerical bootstrap bounds from two-point-function crossing in 2D BCFTs, with a genuinely new mixed-correlator SDP; just don't let the word 'rigorous' pass without qualification.","tokens_in":48389,"tokens_out":1922,"would_cite":true,"duration_ms":25869,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper derives a positive semi-definite program from crossing symmetry of bulk and boundary correlators, yielding the first rigorous bounds on the boundary OPE gap and new bounds on boundary entropy and one-point coefficients in…","keywords":["conformal bootstrap","boundary conformal field theory","boundary entropy","OPE coefficients","semidefinite programming","two-dimensional CFT","WZW model","crossing symmetry"],"falsifier":"Run the mixed-system semidefinite program at a point just inside the excluded region (for example, $h_{2pt}^{\\rm gap}=0.49$ in the Ising setup) with a higher truncation order or with exact hypergeometric blocks; if a primal feasible solution appears, the exclusion is an artifact of the truncation.","tokens_in":47267,"feed_emoji":"📐","tokens_out":9383,"duration_ms":89409,"temperature":0.7,"pith_summary":"The paper aims to establish that boundary conditions in two-dimensional CFTs can be constrained by bootstrapping bulk and boundary correlators together, not just the annulus partition function. It derives a positive semi-definite program from crossing symmetry of three observables: the annulus partition function, the four-point function of identical bulk scalars, and the two-point function of those scalars in the presence of a boundary. The resulting mixed-correlator system gives access to data invisible to the annulus alone, above all the gap in the boundary OPE of a chosen bulk operator and the bulk-to-boundary OPE coefficients, and it tightens existing bounds on the boundary entropy. The method is tested on the Ising model and free boson, then applied to central charge $c=3/2$ theories with emphasis on the $\\mathfrak{su}(2)_2$ WZW model, yielding new non-perturbative bounds.","feed_headline":"Three crossing equations put rigorous bounds on boundary CFT data","feed_subtitle":"A mixed system of three correlators yields rigorous numerical bounds on boundary entropy and OPE gaps in 2d CFTs.","key_machinery":"The load-bearing object is the vector functional $\\vec\\alpha=(\\alpha_4,\\alpha_2,\\alpha_a)$ acting on the four-point, two-point, and annulus crossing equations, together with the two-by-two matrix $\\vec V_{\\Delta,\\ell=0}$ built from the scalar bulk blocks $F_{\\Delta,0}$, $\\tfrac12 G^{\\rm bulk}_\\Delta$, and $\\chi_{\\Delta/2}$. Positivity of this matrix for scalars is what couples the two-point crossing to the annulus and four-point equations; without those diagonal terms, a zero on the diagonal would force the two-point functional to vanish, making the two-point equation useless. The same functional acts on boundary blocks $-G^{\\rm bry}_h$ and annulus characters $-\\chi_h$, and the inequalities (2.44) turn existence of a suitable functional into a rigorous exclusion. Numerically, derivatives are taken in a derivative basis (2.45)\\textendash(2.47), with the evaluation point fixed by the stability condition (A.41), $\\xi_*\\simeq0.03010$ and $\\nu\\simeq3.98513$.","core_discovery":"On its own terms, the paper's central claim is that the crossing equations (2.31) for the four-point function, (2.32) for the two-point function with a boundary, and (2.33) for the annulus partition function combine into a single positivity structure (2.34). A functional $\\vec\\alpha=(\\alpha_4,\\alpha_2,\\alpha_a)$ satisfying the inequalities (2.44) constitutes a valid semi-definite program: if such a functional exists, no unitary boundary CFT can have the assumed gaps and couplings. The payoff is a set of rigorous numerical exclusions: for the Ising model, $h_{2pt}^{\\rm gap}<0.5$ (fig. 2); for the free boson, improved bounds on $g^2$ and $a^2_{\\Delta_\\phi}$ (figs. 4, 5, 10); and for $c=3/2$, new bounds on boundary entropy, gaps, and one-point coefficients in the $\\mathfrak{su}(2)_2$ WZW model (figs. 13\\textendash 16). In several regions the mixed system strengthens the annulus-only results; in others, the two-point-function constraints decouple and the annulus bound remains the tight one.","pith_inferences":["A direct extension the paper leaves implicit: bootstrapping with full Virasoro blocks instead of global $\\mathfrak{sl}(2)$ blocks should sharpen the bounds further, especially in the $c=3/2$ case where the Virasoro annulus bounds are already near-optimal, and may resolve the low-gap numerical instabilities.","In the free-boson and WZW examples, the two-point-function constraints activate only when $h_{2pt}^{\\rm gap}$ is sufficiently large; this suggests a practical rule: mixed systems are most powerful for OPE data unique to the two-point function, or when the physical boundary condition forces a large two-point gap.","The relation $b_0=a_{\\Delta_\\phi}$ could be turned into a numerical 'elementaryness probe': by optimizing the difference $b_0^2-a_{\\Delta_\\phi}^2$ rather than imposing it, one could classify boundary conditions as elementary or non-elementary directly from bootstrap bounds."],"forward_implications":["Boundary OPE data such as $h_{2pt}^{\\rm gap}$ and bulk-to-boundary OPE coefficients, previously inaccessible to rigorous numerics, can now be bounded by the same crossing machinery used for bulk CFT data.","For $c=3/2$, the bounds give a quantitative map of allowed boundary conditions: stable boundaries require $\\Delta_{\\rm gap}\\leq0.5932\\ldots$, and with the bulk spectrum up to $\\Delta=20$ input, the boundary gap obeys $h_{ann}^{\\rm gap}\\leq 1.12515(9)$ (table 3).","In free-boson examples, adding four-point and two-point constraints tightens the boundary-entropy window compared with the annulus alone, and makes the Neumann brane saturate the mixed-system bound at $R=4.3$ (fig. 4).","The setup also provides a new way to test whether a boundary condition is elementary: imposing $b_0=a_{\\Delta_\\phi}$ excludes non-elementary superpositions of branes from the primal problem, so bounds obtained with that relation probe genuine elementary boundaries."],"supporting_citations":[{"why":"Supplies the first linear-programming treatment of annulus crossing, the starting point for the annulus subproblem.","marker":"[25]"},{"why":"Establishes the annulus bootstrap bounds and boundary-state classification that the paper extends and compares against.","marker":"[26]"},{"why":"Mixed-correlator bootstrap techniques for four-point functions, adapted here to the boundary system.","marker":"[29]"},{"why":"Provides the numerical semidefinite-program solver used to find the functionals.","marker":"[27]"},{"why":"Derives the crossing equation for the two-point function in boundary CFT, the key input for eq. (2.30).","marker":"[49]"},{"why":"Provides the global conformal block expansion used for the four-point function.","marker":"[88]"},{"why":"Radial-coordinate expansion of conformal blocks used in the numerical implementation.","marker":"[89]"}],"fun_headline_variants":["Mixed correlator bootstrap tightens boundary CFT bounds","Three crossing equations sharpen boundary CFT exclusions","Boundary bootstrap delivers new entropy and gap bounds","Tighter boundary CFT constraints from mixed crossing"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole scheme stands on the numerical reliability of the truncated series for the conformal blocks and on the stability of the solver at the chosen evaluation point; the authors report that for very small boundary gaps (below 0.2 at $c=1$ and below 0.12 at $c=3/2$) the numerics become unstable, so the bounds there are not trustworthy.","fun_headline_variants_meta":{"raw":{"variants":["Mixed correlator bootstrap tightens boundary CFT bounds","Three crossing equations sharpen boundary CFT exclusions","Boundary bootstrap delivers new entropy and gap bounds","Tighter boundary CFT constraints from mixed crossing"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000245,"raw_usage":{"total_tokens":1545,"prompt_tokens":964,"completion_tokens":581,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":580,"completion_tokens_details":{"reasoning_tokens":521}},"tokens_in":580,"tokens_out":581,"duration_ms":5998,"temperature":1.0,"reasoning_tokens":521,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T19:09:46.681792+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the mixed-system semidefinite program at a point just inside the excluded region (for example, $h_{2pt}^{\\rm gap}=0.49$ in the Ising setup) with a higher truncation order or with exact hypergeometric blocks; if a primal feasible solution appears, the exclusion is an artifact of the truncation.","supporting_citations":[{"cited_title":"Lower bound on the entropy of boundaries and junctions in 1+1d quantum critical systems","cited_arxiv_id":"1206.5395","evidence_quote":"Supplies the first linear-programming treatment of annulus crossing, the starting point for the annulus subproblem."}],"review_version":2}