Disk Instantons, Mirror Symmetry and the Duality Web

Albrecht Klemm; Cumrun Vafa; Mina Aganagic

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Disk Instantons, Mirror Symmetry and the Duality Web

Mina Aganagic, Albrecht Klemm, Cumrun Vafa

High Energy Physics - Theory · Submitted 4 May 2001 · 37 references

Pith's verdict: 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.

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CitationAlbrecht Klemm, Cumrun Vafa, Mina Aganagic. "Disk Instantons, Mirror Symmetry and the Duality Web." arXiv:hep-th/0105045 (2001). https://pith.science/paper/Y27DFU7B

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Editor's note

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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.

Referee report

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 objections (2)

[Abstract (s2) and Appendix A] 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.
[§3.1, §3.2, Figs. 7 and 12; abstract (s4), (s5)] 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.

Minor objections (9)

[§4, p. 18] 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.
[Table 2, last row (ks = 5, kt = 5)] 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.
[Tables 3 and 4 versus Eqs. (6.10) and (6.11)] 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.
[Table 3] 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.
[§6.3, 'Integrality of the Bulk Mirror Map'] 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.
[Reference [30]] 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.
[§6.2, footnote 7] 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.
[Notation, Eq. (2.11) versus §6] 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.
[§6.1, Eq. (6.2)] 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.

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The paper

The paper claims

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

The reading

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.

What carries the argument

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.

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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 $P_k$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

Original abstract

We apply the methods recently developed for computation of type IIA disk instantons using mirror symmetry to a large class of D-branes wrapped over Lagrangian cycles of non-compact Calabi-Yau 3-folds. Along the way we clarify the notion of ``flat coordinates'' for the boundary theory. We also discover an integer IR ambiguity needed to define the quantum theory of D-branes wrapped over non-compact Lagrangian submanifolds. In the large $N$ dual Chern-Simons theory, this ambiguity is mapped to the UV choice of the framing of the knot. In a type IIB dual description involving $(p,q)$ 5-branes, disk instantons of type IIA get mapped to $(p,q)$ string instantons. The M-theory lift of these results lead to computation of superpotential terms generated by M2 brane instantons wrapped over 3-cycles of certain manifolds of $G_2$ holonomy.

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,998 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.20 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.23 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We apply the methods recently developed for computation of type IIA disk instantons using mirror symmetry to a large class of D-branes wrapped over Lagrangian cycles of non-compact Calabi-Yau 3-folds. Along the way we clarify the notion of ``flat coordinates'' for the boundary theory. We also discover an integer IR ambiguity needed to define the quantum theory of D-branes wrapped over non-compact Lagrangian submanifolds. In the large $N$ dual Chern-Simons theory, this ambiguity is mapped to the UV choice of the framing of the knot. In a type IIB dual description involving $(p,q)$ 5-branes, disk instantons of type IIA get mapped to $(p,q)$ string instantons. The M-theory lift of these results lead to computation of superpotential terms generated by M2 brane instantons wrapped over 3-cycles of certain manifolds of $G_2$ holonomy.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We apply the methods recently developed for computation of type IIA disk instantons using mirror symmetry to a large class of D-branes wrapped over Lagrangian cycles of non-compact Calabi-Yau 3-folds.	contribution 0.93	0.92	0.21	asserted 1.00
s1	Along the way we clarify the notion of ``flat coordinates'' for the boundary theory.	contribution 0.89	0.89	0.05	asserted 1.00
s2	We also discover an integer IR ambiguity needed to define the quantum theory of D-branes wrapped over non-compact Lagrangian submanifolds.	result 0.90	0.95	0.24	asserted 1.00
s3	In the large $N$ dual Chern-Simons theory, this ambiguity is mapped to the UV choice of the framing of the knot.	interpretation 0.53	0.67	0.12	asserted 1.00
s4	In a type IIB dual description involving $(p,q)$ 5-branes, disk instantons of type IIA get mapped to $(p,q)$ string instantons.	result 0.85	0.73	0.13	asserted 1.00
s5	The M-theory lift of these results lead to computation of superpotential terms generated by M2 brane instantons wrapped over 3-cycles of certain manifolds of $G_2$ holonomy.	result 0.86	0.85	0.25	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.97 · evidence from proof or derivation 0.95.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
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C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 12.** In the weak coupling limit, we consider an NS (0, 1) 5-brane with an ALF like space ending on it. If this is connected to (1, 0) and (1, 1) 5-branes in a junction, the only instantons allowed involve the worldsheet instanton stretched along the interval shown on the left-hand side of the figure above, times the 5-th circle which shrinks at the attachment point of the ALF space. There is no other instanton allowed in this geometry. However if NS (0, 1) 5-brane is attached to (1, n) and (1, n + 1) 5-branes the geometry of the intersection dramatically changes (as long as n 6= 0, −1)…

**Figure 17.** Three phases of the A−brane on O(−3) → P 2 . Phases I and II are related by Z3 symmetry of the P 2 . As the branes in phases I and II are related by the Z3 symmetry of P2 , the special Lagrangian D-branes of the A-model for this Calabi-Yau have two inequivalent phases, together with a Z-family of choices of framing in each. The mirror B-model geometry (see e.g. [34][35][36]) can be written as xz = e u + e v + e −t−u−v + 1 (6.14) 42

Showing 2 of the paper's 19 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:hep-th/0105045 by the authors.

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Spectral Theory and Mirror Curves of Higher Genus
hep-th 2015-07 conditional novelty 7.0 of 10

A generalized spectral determinant built from topological string amplitudes reproduces, to high numerical precision, the spectra of quantum operators obtained by quantizing genus-two mirror curves.

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hep-th 2014-04 conditional novelty 7.0 of 10

An integral monodromy basis for Calabi-Yau fourfold periods is constructed and cross-checked, then used to argue that integer fluxes can drive F-theory moduli to orbifold and gauge-enhancement loci.
Non-perturbative effects and the refined topological string
hep-th 2013-06 conditional novelty 7.0 of 10

Membrane instanton corrections in ABJM theory equal the quantum periods of local P1xP1, i.e. the refined topological string in the Nekrasov-Shatashvili limit, checked to tenth order but not proved.
Large N Duality, Mirror Symmetry, and a Q-deformed A-polynomial for Knots
hep-th 2012-04 conditional novelty 7.0 of 10

Large N duality plus a generalized SYZ construction assigns to each knot a Q-deformed A-polynomial whose vanishing curve is a distinct mirror of the resolved conifold.
Vortex Counting and Lagrangian 3-manifolds
hep-th 2010-06 conditional novelty 7.0 of 10

Vortex partition functions of 2d N=(2,2) gauge theories on surface operators equal refined open BPS invariants of toric Calabi-Yau geometries and degenerate conformal blocks in 2d CFT.
Remodeling the B-model
hep-th 2007-09 accept novelty 7.0 of 10

A recursive residue calculus on the mirror curve in C* x C* generates all open and closed B-model topological string amplitudes, checked against the topological vertex and against Chern-Simons Wilson loops in lens spaces.
Instanton Geometry and Quantum A-infinity structure on the Elliptic Curve
hep-th 2006-03 conditional novelty 7.0 of 10

The full set of non-vanishing disk and annulus correlators for long-diagonal branes on the elliptic curve is computed and verified to satisfy the classical and quantum A-infinity relations.
Matrix Models, Geometric Engineering and Elliptic Genera
hep-th 2003-10 accept novelty 7.0 of 10

For N=2* theory compactified from six to four dimensions on a torus, matrix models, geometric engineering via the topological vertex, and elliptic-genus instanton calculus all yield the same prepotential, encoded in a...
Universal Correlators on Exponentially Ramified Spectral Curves
math-ph 2026-07 conditional novelty 6.0 of 10

Generalized topological recursion extends to spectral curves with essential singularities by replacing residues at the essential point with residues at ordinary meromorphic points.
Non-Perturbative Real Topological Strings
hep-th 2023-09 unverdicted novelty 6.0 of 10

Extends operator formalism of closed topological strings to derive all-order trans-series solutions for real topological strings, with disk invariants as Stokes constants and numerical checks on local P2.
Feynman Integrals in Dimensional Regularization and Extensions of Calabi-Yau Motives
hep-th 2021-08 conditional novelty 6.0 of 10

Griffiths transversality yields quadratic relations among maximal cuts that reduce l-loop banana integrals in two dimensions to an integral over a Calabi-Yau period, with an extension to dimensional regularization.
Quantum Mirror Map for Del Pezzo Geometries
hep-th 2019-08 conditional novelty 6.0 of 10

The quantum mirror map of the D5 del Pezzo curve is expressed in D5 characters with a signed multi-covering structure, and it reproduces the effective chemical potentials of the (2,2) and (1,1,1,1) super Chern-Simons ...
ABJM Wilson Loops in Arbitrary Representations
hep-th 2013-06 conditional novelty 6.0 of 10

Half-BPS Wilson loop expectation values in ABJM theory obey a Giambelli-type determinant formula in the grand canonical ensemble, reducing every representation to hook representations.
BPS Degeneracies and Superconformal Index in Diverse Dimensions
hep-th 2012-10 conditional novelty 6.0 of 10

Refined topological string amplitudes, which count BPS particles, reproduce the five-dimensional superconformal index for a class of toric Calabi-Yau theories, including sectors with 3d defects.
Factorisation of N = 2 theories on the squashed 3-sphere
hep-th 2011-11 conditional novelty 6.0 of 10

The partition function of abelian N=2 theories on the squashed three-sphere factorises into a product of vortex and antivortex partition functions related by a modular transformation.
Torus knots and mirror symmetry
hep-th 2011-05 accept novelty 6.0 of 10

A single spectral curve, obtained by an Sl(2,Z) transformation of the resolved conifold, generates the colored HOMFLY invariants of all torus knots via topological recursion.
Quantum Geometry of Refined Topological Strings
hep-th 2011-05 conditional novelty 6.0 of 10

Brane wave-functions in refined topological strings satisfy an exact multi-time Schrödinger equation, and their single-valuedness reproduces the Nekrasov-Shatashvili Bethe ansatz equations.
Vertices, Vortices & Interacting Surface Operators
hep-th 2011-02 conditional novelty 6.0 of 10

The vortex partition function of N=(2,2) gauge theories with adjoint and anti-fundamental matter equals the field-theory limit of the topological vertex on a strip with column-diagram boundary conditions, and resums i...
Localization with a Surface Operator, Irregular Conformal Blocks and Open Topological String
hep-th 2010-08 conditional novelty 6.0 of 10

Instanton partition functions with a surface operator agree, order by order in the instanton expansion, with Liouville correlators carrying a degenerate insertion and with open topological string amplitudes on local H...
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hep-th 2010-04 conditional novelty 6.0 of 10

Multiple surface operator insertions in 4d N=2 quiver gauge theories are matched, order by order, to open topological string amplitudes with multiple toric branes and to degenerate-operator insertions in Toda conforma...
Computing Brane and Flux Superpotentials in F-theory Compactifications
hep-th 2009-09 conditional novelty 6.0 of 10

An explicit Calabi-Yau fourfold computation shows that a single four-form flux reproduces the full D5-brane disk instanton superpotential of local P2 in a local limit.
Borel and Stokes Nonperturbative Phenomena in Topological String Theory and c=1 Matrix Models
hep-th 2009-07 conditional novelty 6.0 of 10

In c=1 matrix models and topological strings on the resolved conifold, instanton effects are controlled by Borel poles and arise from eigenvalue tunneling between sheets of a multi-valued effective potential.
Open string amplitudes and large order behavior in topological string theory
hep-th 2006-12 conditional novelty 6.0 of 10

Matrix-model loop equations, run on a spectral curve read off from the disk amplitude, reproduce known open topological string amplitudes on toric Calabi-Yau manifolds and give the D-brane instanton action that sets t...
Supersymmetry of Noncompact MQCD-like Membrane Instantons and Heat Kernel Asymptotics
hep-th 2006-03 conditional novelty 6.0 of 10

Heat kernel Seeley-deWitt coefficients for a Dirac-type operator and one Laplace-type operator match to O(ζ²) for an MQCD-like membrane instanton on a noncompact G₂ three-fold, indicating residual supersymmetry.
Matrix Model as a Mirror of Chern-Simons Theory
hep-th 2002-11 accept novelty 6.0 of 10

Mirror symmetry maps Chern-Simons theory on S^3 and lens spaces to a Hermitian matrix model with unitary measure, and the large-N limit reproduces all-genus topological string amplitudes on local Calabi-Yau geometries.
Les Houches lectures on matrix models and topological strings
hep-th 2004-10 accept novelty 2.0 of 10

Lecture notes showing that open topological B strings on resolved Calabi-Yau singularities are described by a matrix model whose planar spectral curve is the deformed closed-string geometry.
Four-dimensional String Compactifications with D-Branes, Orientifolds and Fluxes
hep-th 2006-10 unverdicted

A comprehensive pedagogical review of four-dimensional orientifold and flux compactifications with D-branes, assembling model-building rules, effective-action methods, and statistical vacuum counts.

How later papers used it

76 later papers cite it

Evidence: substantial. 63 of the citing papers share no author with it, and 8 of those carry a confident label from Pith's citation classifier.

Pith read 24 of the 178 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

3 later papers by other authors are listed below: 1 used its method or tool, 2 compared their own results with it and 1 cited it as support. 1 of them appears under more than one label. 1 more paper by the same authors is listed: 1 compared its own results with it.

Uses its method or tool 1 of 3 papers with this label

Computed the disk amplitude in a similar manner to this paper.

Localization with a Surface Operator, Irregular Conformal Blocks and Open Topological String (2010) · 0.95 · paraphrased: the licence does not allow quoting

Compares with it 3 of 5 papers with this label · 1 by the same authors

Compared its own integers with the disk invariants for framing p = -2 obtained in the formulas of this paper after their Equation 6.4.

The Chromatic Lagrangian: Wavefunctions and Open Gromov-Witten Conjectures (2023) · 0.97 · paraphrased: the licence does not allow quoting
Compared with this paper and found that it only had to take into account trivial open mirror maps.

Bethe/Gauge correspondence in odd dimension: modular double, non-perturbative corrections and open topological strings (2016) · 1.00 · paraphrased: the licence does not allow quoting

By the same authors

Obtained a perfect match of the disk instantons for both brane phases I/II and III considered in this paper.

Five-Brane Superpotentials, Blow-Up Geometries and SU(3) Structure Manifolds (2010) (same authors) · 0.87 · paraphrased: the licence does not allow quoting

Cites it as support 1 of 3 papers with this label

Said that the Harvey-Lawson brane in with its various framings, as studied in this paper, gives further evidence.

The Chromatic Lagrangian: Wavefunctions and Open Gromov-Witten Conjectures (2023) · 1.00 · paraphrased: the licence does not allow quoting

How the literature uses it

72 arXiv papers cite this in 174 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 125 sentences without a confident label

Builds on it 4 papers

This was extended by Aganagic, Klemm and Vafa [ref] to include the effect of framing, and the case of the resolved conifold was treated in [ref] .

Open String Invariants and Mirror Curve of the Resolved Conifold · Introduction · 0.85
These results were generalized in [ref] , where the role of framing was clarified.] For knot conormals which have passed through the conifold transition, these functions $H$ have been studied as a quantization of the A-polynomials of the knot [ref] ,[The definition of the A-polynomial of a knot is essentially as follows.

Wavefunctions for a Class of Branes in Three-space · Branes and Wavefunctions · 0.85
In this appendix, we understand the co-dimension 2 defect partition function as an open refined topological string partition function Z^top_open=Z^pert(1+∑_k=1^∞q_ϕ^k Z_k^def), and extract the refined Ooguri-Vafa (OV) invariants, as a generalisation of the disk invariants [ref] , from the free energy $F^top_open(t,x)=logZ^top_open(t,x)$.

E-string Quantum Curve · Open topological strings and refined Ooguri-Vafa invariants · 0.86

Uses its method or tool 3 papers

The closed and open mirror maps are given by [ref] , [equation] where $X=X(x)$ is the open string moduli on the A model side.

Localization with a Surface Operator, Irregular Conformal Blocks and Open Topological String · Toric brane on local Hirzebruch surface $K_F_0$: pure $SU(2)$ · 0.80
The closed string parameters can be fixed in each large radius patch by the methods of [ref] , which are reviewed in the next section.

Remodeling the B-model · Moving in the moduli space · 0.86
[ref] — and a moduli-dependent analysis of the matching conditions [ref].[A similar analysis of moduli dependent matching conditions is required for building blocks $(Z,S)$ arising from general weak Fano threefold for which Beauville's theorem [ref] is not applicable.]

Effective action from M-theory on twisted connected sum $G_2$-manifolds · Conclusions · 0.92

Compares with it 5 papers

On the other hand, we may compare these integers with the disk invariants for framing $p = 2 - 2·2 = -2$ obtained in the formulas of [ref] after their Equation 6.4, where they find $n^(2)_d = 1, 1, 3, 10$ for these same values of $d$.

The Chromatic Lagrangian: Wavefunctions and Open Gromov-Witten Conjectures · Kac polynomial of a quiver · 0.97
The gauge flux induces an effective D5-charge on the D7-brane and we will be able to compare the D5-brane superpotential of [ref] to the D7-brane superpotential with appropriate $F_2$ in the local limit.

Computing Brane and Flux Superpotentials in F-theory Compactifications · Toric branes and their mirrors · 0.84
Indeed we obtain a perfect match of the disk instantons for both brane phases $I/II$, $III$ considered in [ref] for the choices of superpotential [equation] where we in addition fix the parameters $a_3=a_7=0$ to switch off the contribution of the closed instantons of $Z_3$ through its double logarithmic periods.

Five-Brane Superpotentials, Blow-Up Geometries and SU(3) Structure Manifolds · Brane Superpotential at Large Volume: Disk Instantons · 0.87

Cites it as support 2 papers

The Harvey-Lawson brane in $^3$ with its various framings, as studied in [ref] , gives further evidence.

The Chromatic Lagrangian: Wavefunctions and Open Gromov-Witten Conjectures · Open Gromov-Witten Conjectures · 1.00
An evidence in this direction is that in [ref] disk amplitudes are expressed precisely in terms of hypergeometric functions with parameters fixed in terms of the masses and the strong coupling scale.

Vertices, Vortices & Interacting Surface Operators · Discussion and Open Issues · 1.00

Cites it as background 26 papers

The function $M(Q)$ originally appeared in the work of Aganagic–Klemm–Vafa [ref] as the open mirror map relating Aganagic–Vafa branes [ref] on the canonical bundle $K_^2$ and its mirror, and was used to determine open Gromov–Witten invariants in [ref] .

The proper Landau-Ginzburg potential is the open mirror map · Introduction and Summary · 0.89
We are mainly interested in the mirror Type IIB geometry with Calabi-Yau space $Y$ and D5-branes [ref] . $Y$ is a conic bundle [equation] branched over a Riemann surface $Y$ given by $H(x,y;z)=0$ [ref] .

The D5-brane effective action and superpotential in N=1 compactifications · Non-compact Calabi-Yau spaces · 0.91
In particular, we review the proposal of [ref] that the one-leg, framed topological vertex obeys a similar set of recursion relations by using the mirror geometry first found in [ref] .

Hurwitz numbers, matrix models and enumerative geometry · Introduction · 0.96

Discussion

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