{"id":"343033b3-8078-4740-ad7d-7c276a3a2aeb","arxiv_id":"hep-th/0105045","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Mirror symmetry computes disk instanton superpotentials for D-branes on non-compact Calabi-Yau threefolds, and the integer ambiguity in that computation maps to the framing choice of a knot in the dual Chern-Simons theory.","lead":"The authors extend a mirror-symmetry method for computing the superpotentials generated by D-branes wrapped on curved internal spaces, and identify a discrete ambiguity in defining the quantum theory that maps to a known ambiguity in Chern-Simons knot invariants. The work ties type IIA disk instantons, type IIB five-brane webs, and M-theory instantons on G2 manifolds into a single picture.","discovery_kind":"unclear","skeptic_critique":{"model":"selfhosted/deepseek-v4.1-flash","headline":"The disk-counting result is well-supported, but the claimed IR-ambiguity/framing correspondence is verified only for small wrapping numbers and the leading power in general, leaving the 'duality web' claim partially checked.","rationale":"The paper's main result — that disk instanton counts are computable by mirror symmetry and are integers — is supported by explicit computations and highly non-trivial integrality checks. The flat-coordinate derivation in Section 4 is argued cleanly, and the examples confirm it. The soft spots are: (i) footnote 4 acknowledges the coordinates were reverse-engineered from integrality before being re-derived; (ii) the framing match is verified only partially; (iii) the (p,q) 5-brane discussion is qualitative; (iv) the bulk-integrality 'explanation' in 6.3 is an equivalence rather than an independent derivation. None of these undermines the central disk-counting claim. The reader's verdict of ACCEPT is reasonable.","tokens_in":77245,"tokens_out":6859,"duration_ms":126066,"concrete_test":"Compute the full p-dependence of N_m(p) from the Chern-Simons side for m=4,5,6 using the representation-theoretic formula in Appendix A (the sum over R_{n,s} with Casimir weights) and compare with the C3 polynomial in (6.4). If the polynomials agree for all m up to, say, m=10, the correspondence is strongly supported; if they diverge beyond leading order, the duality-web claim weakens.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract (s2) presents the integer IR ambiguity and its mapping to the UV framing choice of the knot as a result. Appendix A verifies this correspondence for N_1, N_2, N_3 and for the leading p^{m-1} coefficient for all m, but not for the full degree-(m-1) polynomial in p for general m. The C3 result (6.4) gives N_m(p) as a polynomial in p, and the Chern-Simons computation is shown to match only at leading order. If the full polynomial match fails for some m, the claimed correspondence is not established. This is a central claim of the paper, not a peripheral one. The paper is honest about the partial verification, but the abstract presents the correspondence as established. A secondary concern is that the derivation of the flat coordinate in Section 4 uses the closed-string flat coordinate ŝ (or t̂) in the examples, so the open-string flat coordinate is not derived independently of bulk integrality; the 'physical explanation' of bulk integrality in Section 6.3 is an equivalence rather than an independent derivation. Neither concern undermines the central disk-counting result, which is supported by explicit integrality checks.","agreement_with_reader":"partial"},"referee_report":{"model":"selfhosted/deepseek-v4.1-flash","summary":"The manuscript extends the mirror-symmetry method for computing type IIA disk instantons ending on special Lagrangian submanifolds to a broad class of non-compact toric Calabi–Yau threefolds. The two structural contributions are (i) the identification of the flat coordinate on the D-brane moduli space with the BPS tension of the mirror domain wall, Eq. (4.1), û = (1/2πi)∮_C v du, together with the argument that δû/δu = 1 exactly so that û differs from u only by closed-string instanton corrections; and (ii) the claim that the residual integer freedom u → u + n v is a genuine IR ambiguity of the quantum Lagrangian brane, mapped in the large-N Chern–Simons dual to the UV framing of the knot and, via the (p,q) 5-brane web, to a choice of the type IIB coupling at infinity. The paper then works out C^3, O(K)→P^1×P^1, O(−3)→P^2 and O(K)→F_1 in detail, giving explicit integer disk degeneracy tables and closed-form polynomials N_m(p) for C^3.","tokens_in":77510,"tokens_out":11358,"duration_ms":172017,"significance":"If correct, this makes disk instanton counting a routine mirror-symmetry computation for a large class of local toric geometries, and it promotes the framing/integer ambiguity from a coordinate artifact to physical data. The evidence is unusually concrete for this genre: the Picard–Fuchs operators are written explicitly (Eqs. 6.5, 6.6), the flat coordinates are given in closed form in each example, and the resulting integer invariants are tabulated (Tables 1–9) with stated symmetry relations (6.9)–(6.11) that a reader can check. The C^3 result (6.3)–(6.4) is a parameter-free closed-form prediction, checked for m ≤ 50, and the claim that the leading p^{m−1} coefficient matches the Chern–Simons framing computation for all m (Appendix A, footnote 9) is a genuine falsifiable statement. The identification of the flat coordinate with the domain-wall tension also supplies a physical reason for the integrality of the bulk mirror map in §6.3, which had been observed but not explained.","major_comments":[{"comment":"The abstract states without qualification that the integer IR ambiguity 'is mapped to the UV choice of the framing of the knot', but Appendix A verifies the correspondence only for N_1, N_2, N_3 and for the leading p^{m−1} coefficient for general m; the full degree-(m−1) polynomial in p is not matched beyond m = 3. Since the C^3 result (6.4) produces a full polynomial, the natural check is available and is not carried out. The structural argument of §5.1 (the UV point-splitting divergence of the Chern–Simons Wilson loop maps, under the large-N conformal transformation of Fig. 10, to the IR region of the open string on L_γ) is the real support for the claim and is convincing on its own; but the abstract should either be qualified to say the map is established structurally and checked to leading order, or Appendix A should be extended to one or two further values of m. This is the paper's second headline claim and the gap between abstract and appendix is worth closing.","section":"Abstract (s2) and Appendix A","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The M-theory/G_2 and (p,q) 5-brane statements are supported by geometric arguments only: the S^3 topology argument of §3.1 and the web pictures of Figs. 7 and 12. No G_2 example is worked out and no disk number is reproduced from a 5-brane web, so the abstract's 'lead to computation of superpotential terms generated by M2 brane instantons' and 'disk instantons of type IIA get mapped to (p,q) string instantons' describe a programme rather than a performed computation. The body is appropriately hedged ('in effect we have succeeded in transforming the question of computation ... to an application of mirror symmetry'), and the τ → −1/n discussion of §5.2 with its n → −(n+1) symmetry, matched by Table 7, is a nice consistency check. I would ask that the abstract be brought into line with the body, e.g. 'provide a route to computing', since as written a reader may expect a quantitative G_2 or 5-brane result that the paper does not contain.","section":"§3.1, §3.2, Figs. 7 and 12; abstract (s4), (s5)","authors_acknowledge":null,"authors_acknowledge_p":null}],"minor_comments":[{"comment":"The sentence 'the cycles whose u or v values shift by an integer multiple of 2πi do not give rise to closed 3-cycles in the CY (as the CY in question does have non-trivial cycles corresponding to shifting u or v by integer multiples of 2πi)' contradicts itself in the parenthetical. Presumably the intended statement is that such cycles do not lift to closed 3-cycles even though the CY has cycles whose periods shift u or v; please rephrase.","section":"§4, p. 18","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The entry is printed as +736270, but the stated rule (6.9), N_{ks,kt,−m} = −N_{ks,kt−m,m}, together with Table 1 (m = 1, ks = 5, kt = 4 = 736270) gives −736270. The rest of the row and of Table 2 is consistent with (6.9), so this looks like a single sign typo.","section":"Table 2, last row (ks = 5, kt = 5)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"As printed, the stated symmetries do not reproduce the tables on spot checks. For example (6.11) with m = 1, ks = 1, kt = 1 gives N_{1,1,−1} = −N_{2,0,1} = 0 from Table 3, whereas Table 4 lists −2; similarly (6.10) with ks = 1, kt = 1 gives N_{1,1,1} = N_{0,2,1} = 0, whereas Table 3 lists −2. Since these integer tables are the paper's quantitative evidence, please check whether the relations, the row/column labelling, or the tables themselves need correction.","section":"Tables 3 and 4 versus Eqs. (6.10) and (6.11)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"Several entries are printed as '−' (m = 5, ks = 5, kt = 8; the m = 6 rows at large kt). Please state explicitly whether these are zero, not computed, or omitted for space, since the surrounding text asserts N_{ks,kt,m} = 0 only for kt < m.","section":"Table 3","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The section concludes that 'we have found a physical explanation' for bulk mirror-map integrality, but then states that 'the proof that all P_k are integer polynomials is tedious and relies on some formulas for the derivatives of v(û,q) for this special example' and is not given. The equivalence of the integrality of N_{k,1} and of the mirror-map coefficients is a nice observation; the word 'explanation' should be softened to reflect that one direction of the equivalence is proved and the other is asserted.","section":"§6.3, 'Integrality of the Bulk Mirror Map'","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The claim in §5.1 that the construction of [26] 'has been recently generalized to all knots' rests on a citation listed as 'C. Taubes, To appear.' Since this is unpublished, either mark it as a private communication or qualify the sentence.","section":"Reference [30]","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The vanishing of ∆u′ = ∆u − ∆v is asserted ('turns out to be zero') without derivation. From (6.7) the corrections to t̂ and ŝ agree order by order, which suggests an exact symmetry; a sentence saying whether this is exact or holds order by order would help.","section":"§6.2, footnote 7","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"Eq. (2.11) writes the expansion with wrapping number n and disk class m⃗, while §6 uses m for the disk class and k_s, k_t for the closed-string classes. A one-line dictionary between the two notations would make the tables easier to compare with the general formula.","section":"Notation, Eq. (2.11) versus §6","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The curve is written as 1 − e^{û+pv̂} − e^{v̂} = 0, whereas (6.1) reads e^u + e^v + 1 = 0; the sign flip is absorbed into the iπ shifts in û and v̂. A parenthetical remark would save the reader a moment of confusion.","section":"§6.1, Eq. (6.2)","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"minor_revision","confidential_remarks":"The physics is sound and the paper will be well cited; my recommendation of minor revision reflects only the gap between the abstract's framing/duality-web claims and what Appendix A and §3 actually establish, plus a handful of table and text corrections. I would not send this back for a second full round: the requested changes are local and can be checked by the editor against a marked copy. One process note: the disk-degeneracy tables are the paper's main quantitative asset, and the two consistency checks I ran (Table 2 against (6.9), Tables 3–4 against (6.10)–(6.11)) each turned up a discrepancy, so it would be worth having the authors re-verify the tables against their code before publication."},"author_rebuttal":null,"desk_editor":{"model":"selfhosted/deepseek-v4.1-flash","letter":"Bottom line: the computational core of this paper holds up, and the integer tables are the evidence that matters. The framing correspondence is the piece that is only partly nailed down, and the abstract states it more firmly than the appendix supports. Neither point sinks the paper, but a referee should ask for the second one to be tightened or softened in the abstract. I largely agree with your ACCEPT, though I'd nudge the circularity score up a little and the soundness score down a touch, and I think the stress-test's framing concern lands while its flat-coordinate concern mostly does not. What is genuinely new here, on top of the authors' own hep-th/0012041: the identification of the open-string flat coordinate with the BPS tension of the mirror domain wall (Eq. 4.1), the integer IR ambiguity and its reading as the UV framing of the knot, the (p,q) web picture, and the G2/M2 lift. The Section 4 argument that delta u-hat / delta u = 1 exactly is short and correct, and it is what makes the domain-wall definition more than a guess. The C3 closed form (6.4) with N_m(p) checked for m up to 50, plus the explicit Picard-Fuchs operators and the P1xP1, P2, F1 tables, is a lot of independent arithmetic that could easily have failed and did not. On the soft spots: footnote 4 does disclose that the flat coordinates were first reverse-engineered from integrality, but the domain-wall derivation then breaks that loop, so I would not call this circular in the end — it is reverse-engineering followed by an independent derivation, which is normal practice and honestly reported. The stress-test's secondary point is fairer: the constant term in u-hat is fixed using the closed-string flat coordinate, so the open coordinate is not derived wholly independently of bulk integrality, and the Section 6.3 'explanation' of bulk integrality is an equivalence, not a derivation. The framing match in Appendix A is verified for N_1, N_2, N_3 and for the leading p^{m-1} coefficient at general m, not for the full degree-(m-1) polynomial. That is a genuine gap in a claim the abstract presents as established. The (p,q) section is explicitly illustrative — the authors say so — and the G2 lift is a map, not a worked example. Who should read it: anyone computing open topological string amplitudes or non-perturbative superpotentials in N=1 compactifications; the tables alone are worth having. Yes to a serious referee, with a request to either extend the framing check or qualify the abstract.","headline":"The disk-instanton computations and integrality tables are solid; the framing/IR-ambiguity correspondence is real but only partly verified, and the abstract states it more strongly than Appendix A delivers.","tokens_in":78123,"tokens_out":1364,"would_cite":true,"duration_ms":20306,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":[],"pacs":["11.25.-w","11.25.Uv","11.25.Hf"],"model":"selfhosted/deepseek-v4.1-flash","headline":"The paper claims that the coordinate in which D-brane disk instanton counts come out as integers is the BPS tension of a domain wall, and that the leftover integer freedom $u$ to $u+nv$ is a real ambiguity of the quantum brane, matching the","keywords":["disk instantons","mirror symmetry","Lagrangian submanifolds","flat coordinates","domain walls","Chern-Simons framing","G2 holonomy","(p,q) five-brane webs"],"falsifier":"Take a non-compact toric threefold and Lagrangian brane outside the paper's list, compute the B-model disk amplitude, and test whether the coefficients $N_{k,vec m}$ extracted from $partial_{hat u}W=-sum k,N_{k,vec m}log(1-exp[khat u-vec mcdothat t])$ using $hat u=rac{1}{2pi i}int_{C_u}v,du$ come out as integers at low degree; a robust non-integer coefficient would break the identification of the flat coordinate with the domain wall tension.","tokens_in":78607,"feed_emoji":"🪞","tokens_out":4455,"duration_ms":71870,"temperature":0.7,"pith_summary":"The paper asks how to count holomorphic disks ending on a D-brane wrapped over a Lagrangian cycle of a non-compact Calabi-Yau threefold, the instantons that generate a superpotential in the four-dimensional theory living on the brane. Direct counting is hard, so the authors push an earlier mirror-symmetry method through a much larger family of toric geometries, where the mirror brane becomes a point on a Riemann surface $P(u,v)=0$ and the disk amplitude becomes the Abel-Jacobi integral $W(u)=int^u v,du$. The obstacle they have to clear is that the classical disk size $u$ is not the coordinate in which the answer is integral: they identify the correct flat coordinate $hat u$ with the BPS tension of the domain wall a wrapped D4-brane makes when it ends on the D6-brane, and show $hat u$ differs from $u$ only by closed-string instanton corrections. With that coordinate the amplitudes expand in integers whose coefficients count domain walls, and the residual freedom $u o u+nv$ turns out to be a genuine ambiguity of the quantum brane, matching the choice of framing needed to define a Wilson loop in the dual Chern-Simons theory.","feed_headline":"Mirror symmetry counts disk instantons, integer by integer","feed_subtitle":"One integer fixes the quantum theory of a D-brane, and it matches the framing of a knot in Chern-Simons theory.","key_machinery":"The load-bearing device is the period integral of Eq. (4.1), $hat u = rac{1}{2pi i}int_{C_u} v,du$, taken along the contour on the mirror curve $P(u,v)=0$ that starts at the brane, winds once around the shift $v o v+2pi i$, and returns. Because the endpoints differ by $Delta(uv)=2pi i u$ while $u$ itself does not shift, the integral splits into $u$ plus a small period that depends only on the bulk moduli, which is what makes $hat u$ a flat coordinate and simultaneously shows that it is corrected only by closed-string instantons. The same contour logic, applied with the conjugate shift $u o u+2pi i$, defines $hat v$, so the superpotential satisfies $partial_{hat u}W=hat v$ and can be expanded in the integer form whose coefficients are the disk invariants. An $SL(2,mathbb{Z})$ reparameterization of the pair $(u,v)$ acts on the correction terms as a doublet, which is how the framing integer $n$ enters the amplitudes.","core_discovery":"The central claim is that the flat coordinate on the moduli space of a Lagrangian D-brane is the BPS tension of the mirror domain wall, $hat u = rac{1}{2pi i}int_{C_u} v,du$, and that in this coordinate the disk amplitude has an integer expansion $W=sum rac{1}{n^2}N_{k,vec m}exp(n[khat u-vec mcdothat t])$ whose coefficients count D4-brane domain walls ending on the brane. The authors argue $deltahat u/delta u = 1$ exactly, so $hat u$ differs from the classical disk size only by closed-string instanton effects, and they use $partial_{hat u}W=hat v$ to extract integer invariants for branes in $mathbb{C}^3$, $O(K) o mathbb{P}^1 imesmathbb{P}^1$, $O(-3) o mathbb{P}^2$ and $O(K) o mathbb{F}_1$. They further claim that the residual freedom $u o u+nv$ is not a coordinate artifact but an integer IR ambiguity required to define the quantum theory of a brane on a non-compact Lagrangian, mapped to the UV framing choice of the knot in the large-$N$ Chern-Simons dual, to the choice of type IIB coupling in a $(p,q)$ 5-brane web, and through the M-theory lift to the counting of M2-brane instantons on 3-cycles of $G_2$ holonomy manifolds.","pith_inferences":["If the framing integer and the brane's IR integer are truly the same quantity, one could read off Chern-Simons framing phases for higher representations directly from the toric data, without any gauge theory computation, and check the duality far beyond the unknot.","The identification suggests a general rule: an open-string flat coordinate is the tension of whatever object shifts the conjugate field by $2pi i$. That rule could be carried to other brane configurations, for instance branes on cycles with more than one modulus or surface operators in four dimensions.","The authors raise but do not settle whether the integer labels the normalizability of the Wilson-line mode, i.e. a class of metrics at infinity. If so, the ambiguity should be visible as a discrete family of $G_2$ metrics with fixed asymptotics, which is a concrete question for metric moduli rather than for topological strings.","The Chern-Simons matching was verified explicitly only for the first few coefficients, with the leading large-$p$ behaviour checked for all orders. Extending the exact match to all orders would be a sharp, self-contained test of the identification."],"forward_implications":["Disk instanton numbers, and hence the superpotential of the four-dimensional theory on a D6-brane, become computable by mirror symmetry for a broad class of non-compact toric Calabi-Yau threefolds, with the computed coefficients checking out as integers in every worked example.","Through the M-theory lift, superpotential terms generated by Euclidean M2-branes wrapping 3-cycles of $G_2$ holonomy manifolds become computable by the same chain of dualities.","In the type IIB description, type IIA disk instantons map to $(p,q)$ string instantons ending on a web of 5-branes in an ALF-like background, giving a second, geometric way to see the integer jumps as the coupling at infinity is varied.","The framing dependence of Wilson loop observables in Chern-Simons theory at large $N$ acquires a geometric interpretation as the IR ambiguity of the dual Lagrangian brane.","The integrality of the bulk mirror map, previously without a physical explanation, follows from the integrality of the domain wall counts via relations such as $N_{k,1}=P_k(a_1,dots,a_{k-1})-a_k$ with integer polynomials $P_k$."],"supporting_citations":[{"why":"Supplies the mirror-symmetry method for computing disk amplitudes as an Abel-Jacobi integral on a mirror curve, the method this paper extends to a large class of geometries.","marker":"[6]"},{"why":"Supplies the integrality structure of the disk expansion and the large-$N$ Wilson-loop dictionary against which the integer ambiguity is matched.","marker":"[4]"},{"why":"Supplies the mirror construction, dual $Y^i$ fields and curve $P(u,v)=0$, that turns the A-model brane into a point on a Riemann surface.","marker":"[10]"},{"why":"Supplies the gauged linear sigma model realization of toric Calabi-Yau manifolds whose charges and FI parameters define every example in the paper.","marker":"[9]"},{"why":"Supplies the map from toric Calabi-Yau manifolds to webs of $(p,q)$ 5-branes used for the type IIB dual description.","marker":"[24]"},{"why":"Supplies the large-$N$ open/closed topological string duality underlying the comparison with Chern-Simons theory.","marker":"[28]"},{"why":"Supplies the point-splitting and framing definition of Wilson loop observables that the integer IR ambiguity is identified with.","marker":"[8]"},{"why":"Supplies the identification of topological string amplitudes with F-term and superpotential contributions in the spacetime theory.","marker":"[5]"},{"why":"Supplies an independent localization computation confirming the disk counts and the dependence on the integer choice of framing.","marker":"[32]"}],"fun_headline_variants":["Mirror symmetry counts disk instantons as integers","A flat coordinate makes disk instantons count integers","An integer ambiguity ties D-brane theory to knot framing","Disk instantons map to string instantons in the dual web","M2 instantons on G2 cycles follow from the same count"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The argument rests on the identification of the flat coordinate with the BPS tension of the mirror domain wall, together with the claim that the shift of $v$ across that wall is exactly $2pi i n$ so that $deltahat u/delta u = 1$ with no further open-string corrections; footnote 4 of Section 6 also discloses that the corrected coordinates were originally found by demanding integrality and only afterwards re-derived from the domain wall tension.","fun_headline_variants_meta":{"raw":{"variants":["Mirror symmetry counts disk instantons as integers","A flat coordinate makes disk instantons count integers","An integer ambiguity ties D-brane theory to knot framing","Disk instantons map to string instantons in the dual web","M2 instantons on G2 cycles follow from the same count","Mirror symmetry counts disk instantons, integer by integer","Integer framing ambiguity survives the mirror map","Disk instantons count domain walls on Calabi-Yau branes","The same integer fixes D-brane and knot quantum theories","Flat coordinates turn disk amplitudes into integer counts","Mirror symmetry maps disk instantons to string instantons","An integer IR ambiguity defines the D-brane quantum theory","Knot framing matches the D-brane integer ambiguity","M2 brane instantons on G2 cycles come from disk counts","Disk instantons on Lagrangian cycles count integers","One integer ambiguity links D-branes, knots and string webs","Mirror symmetry fixes the D-brane flat coordinate","Integer invariants count D4-brane domain walls","The disk amplitude expands in integer invariants","Flat coordinates make the disk amplitude integer","D-brane quantum theory needs one integer IR ambiguity","Knot framing fixes the D-brane IR ambiguity","Mirror symmetry computes disk instantons for Calabi-Yau branes","Disk instantons become string instantons in type IIB","M2 instantons on G2 cycles follow from disk instantons","Integer counts survive the mirror map","One integer ties D-brane quantum theory to knot framing","Disk instantons count D4-brane domain walls","Flat coordinates make disk instantons integer","Mirror symmetry maps disk instantons to string instantons","An integer IR ambiguity fixes the D-brane quantum theory","Knot framing matches the D-brane ambiguity","M2 brane instantons on G2 cycles come from disk counts","Disk instantons on Lagrangian cycles count integers","One integer links D-branes, knots and string webs","Mirror symmetry fixes the D-brane flat coordinate","Integer invariants count D4-brane domain walls"]},"model":"selfhosted/deepseek-v4.1-flash","effort":"low","attempts":1,"cost_usd":0.0,"raw_usage":{"total_tokens":2175,"prompt_tokens":1030,"completion_tokens":1145,"prompt_tokens_details":null,"completion_tokens_details":{"reasoning_tokens":719}},"tokens_in":1030,"tokens_out":1145,"duration_ms":18993,"image_pages":[],"temperature":1.0,"reasoning_budget":50,"reasoning_tokens":719,"transport_provider":"selfhosted","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-10-29T17:53:01.084961+00:00","model_set":{"reader":"selfhosted/deepseek-v4.1-flash"},"falsifier":"Take a non-compact toric threefold and Lagrangian brane outside the paper's list, compute the B-model disk amplitude, and test whether the coefficients $N_{k,vec m}$ extracted from $partial_{hat u}W=-sum k,N_{k,vec m}log(1-exp[khat u-vec mcdothat t])$ using $hat u=rac{1}{2pi i}int_{C_u}v,du$ come out as integers at low degree; a robust non-integer coefficient would break the identification of the flat coordinate with the domain wall tension.","supporting_citations":[{"cited_title":"Mirror Symmetry, D-Branes and Counting Holomorphic Discs","cited_arxiv_id":"hep-th/0012041","evidence_quote":"Supplies the mirror-symmetry method for computing disk amplitudes as an Abel-Jacobi integral on a mirror curve, the method this paper extends to a large class of geometries."},{"cited_title":"Knot Invariants and Topological Strings","cited_arxiv_id":"hep-th/9912123","evidence_quote":"Supplies the integrality structure of the disk expansion and the large-$N$ Wilson-loop dictionary against which the integer ambiguity is matched."},{"cited_title":"Mirror Symmetry","cited_arxiv_id":"hep-th/0002222","evidence_quote":"Supplies the mirror construction, dual $Y^i$ fields and curve $P(u,v)=0$, that turns the A-model brane into a point on a Riemann surface."},{"cited_title":"Phases of $N=2$ Theories In Two Dimensions","cited_arxiv_id":"hep-th/9301042","evidence_quote":"Supplies the gauged linear sigma model realization of toric Calabi-Yau manifolds whose charges and FI parameters define every example in the paper."},{"cited_title":"Branes and Toric Geometry","cited_arxiv_id":"hep-th/9711013","evidence_quote":"Supplies the map from toric Calabi-Yau manifolds to webs of $(p,q)$ 5-branes used for the type IIB dual description."},{"cited_title":"M-Theory and Topological Strings--II","cited_arxiv_id":"hep-th/9812127","evidence_quote":"Supplies the large-$N$ open/closed topological string duality underlying the comparison with Chern-Simons theory."},{"cited_title":"Quantum Field Theory and the Jones Polynomia l,","cited_arxiv_id":null,"evidence_quote":"Supplies the point-splitting and framing definition of Wilson loop observables that the integer IR ambiguity is identified with."},{"cited_title":"Superstrings and Topological Strings at Large N","cited_arxiv_id":"hep-th/0008142","evidence_quote":"Supplies the identification of topological string amplitudes with F-term and superpotential contributions in the spacetime theory."},{"cited_title":"Enumerative Geometry of Stabl e Maps with La- grangian Boundary Conditions and Multiple Covers of the Dis c,","cited_arxiv_id":null,"evidence_quote":"Supplies an independent localization computation confirming the disk counts and the dependence on the integer choice of framing."}],"review_version":1}