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arxiv: 2606.09054 · v2 · pith:X6DULCYJnew · submitted 2026-06-08 · 🧮 math.SG

From Morse Trees to J-Holomorphic Discs -- Rigid Y-Graphs

Erkao Bao , Ke Zhu This is my paper

Pith reviewed 2026-06-27 14:24 UTC · model grok-4.3

classification 🧮 math.SG
keywords Morse flow treesJ-holomorphic discsY-graphscotangent bundleobstruction bundle gluingLagrangian submanifoldsMorse-Bott gluingsymplectic geometry
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The pith

A rigid Y-shaped Morse flow tree produces at least one J-holomorphic disc in the cotangent bundle for every small enough height ε.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs J-holomorphic discs directly from rigid Y-shaped Morse flow trees by gluing. It applies an obstruction bundle technique to resolve the Morse-Bott obstructions that arise when the pieces are glued inside the cotangent bundle. The result holds for every sufficiently small positive height ε of the boundary Lagrangians. This supplies the base case for a broader correspondence between Morse graphs and holomorphic discs that the authors plan to extend in later work.

Core claim

Given a rigid, transversely cut-out Y-shaped Morse flow tree, for every sufficiently small ε > 0 there exists at least one corresponding J-holomorphic disc in the cotangent bundle, with boundaries inside corresponding Lagrangian submanifolds of height ε.

What carries the argument

obstruction bundle gluing technique applied to the Morse-Bott gluing problem for a Y-graph configuration under the standard almost complex structure on the cotangent bundle

If this is right

  • The construction produces at least one holomorphic disc from each rigid Y-tree.
  • The same gluing method is intended to extend to all ribbon trees and to moduli spaces of all dimensions.
  • The correspondence is set up to be shown injective and surjective in subsequent papers.
  • The approach is designed to generalize to equivariant settings.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method may allow direct comparison of algebraic invariants defined via Morse trees with those defined via holomorphic curve counts.
  • If the full series succeeds, one could replace holomorphic curve moduli spaces in cotangent bundles by purely Morse-theoretic data for certain calculations.
  • The same obstruction bundle framework might apply to other obstructed gluing problems involving graphs of flow lines in symplectic geometry.

Load-bearing premise

The obstruction bundle gluing technique applies directly to the Morse-Bott gluing problem arising from the Y-graph configuration inside the cotangent bundle.

What would settle it

A rigid transversely cut-out Y-tree for which no J-holomorphic disc with boundary on the height-ε Lagrangians exists when ε is taken sufficiently small would falsify the claim.

read the original abstract

The correspondence between Morse flow trees and $J$-holomorphic discs was established by Fukaya--Oh and Ekholm. We revisit this correspondence and present an alternative approach, designed to generalize naturally to the equivariant setting and to certain Morse graph configurations. The central ingredient is a gluing construction that produces $J$-holomorphic discs from Morse flow trees. A well-known difficulty is that this gluing is of Morse--Bott type, equivalently, in an appropriate Fredholm framework, pieces to be glued together are obstructed. We resolve this via the obstruction bundle gluing technique of Hutchings--Taubes. Given a rigid, transversely cut-out Y-shaped Morse flow tree, we show that for every sufficiently small $\epsilon > 0$ there exists at least one corresponding $J$-holomorphic discs in the cotangent bundle, with boundaries inside corresponding Lagrangian submanifolds of height $\epsilon$. This is the first paper in a series; subsequent work will extend the result to all ribbon trees and to moduli spaces of all dimensions and establish the injectivity and surjectivity of the correspondence.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript develops an alternative gluing construction using the obstruction bundle technique of Hutchings-Taubes to produce J-holomorphic discs from rigid Y-shaped Morse flow trees in the cotangent bundle. It asserts that for any such transversely cut-out tree, there exists at least one corresponding J-holomorphic disc with boundaries on height-ε Lagrangians for small ε > 0. This is positioned as the first in a series extending to all ribbon trees and higher-dimensional moduli spaces.

Significance. If the central claim holds, the work provides a foundation for generalizing the Fukaya-Oh-Ekholm correspondence to equivariant settings and more complex configurations, with the obstruction bundle approach addressing Morse-Bott obstructions in a way that may facilitate machine-checkable or more explicit constructions in future papers.

major comments (1)
  1. [Gluing construction and obstruction bundle analysis (the section containing the main existence argument)] The applicability of the Hutchings-Taubes obstruction bundle gluing to the Y-graph configuration with standard J on T*M is load-bearing for the existence result. The manuscript must demonstrate that the cokernel of the linearized CR operator at the approximate glued solution (built from the three Morse trajectories) matches exactly the model obstruction bundle, with no extra kernel/cokernel contributions from the cotangent geometry or the integrable J; this verification of asymptotic operators and vertex matching conditions is required to ensure the section whose zero is asserted to exist is the one analyzed by the cited technique.
minor comments (1)
  1. [Abstract] The abstract states the result for rigid Y-trees but does not specify the expected dimension or index of the moduli space; adding this would clarify the transversality assumption.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for highlighting the need for explicit verification in the obstruction bundle analysis. We address the major comment below and will incorporate the requested clarifications in a revised version.

read point-by-point responses

  1. Referee: [Gluing construction and obstruction bundle analysis (the section containing the main existence argument)] The applicability of the Hutchings-Taubes obstruction bundle gluing to the Y-graph configuration with standard J on T*M is load-bearing for the existence result. The manuscript must demonstrate that the cokernel of the linearized CR operator at the approximate glued solution (built from the three Morse trajectories) matches exactly the model obstruction bundle, with no extra kernel/cokernel contributions from the cotangent geometry or the integrable J; this verification of asymptotic operators and vertex matching conditions is required to ensure the section whose zero is asserted to exist is the one analyzed by the cited technique.

    Authors: We agree that explicit verification of the cokernel matching is essential to justify the direct application of the Hutchings-Taubes technique. The manuscript invokes the standard asymptotic operators for the cotangent bundle equipped with the integrable complex structure (as established in the Fukaya-Oh and Ekholm frameworks for Morse flow trees), which are designed to produce precisely the model obstruction bundle without additional contributions. However, to strengthen the exposition and address the referee's point directly, we will add a dedicated paragraph (or short subsection) in the gluing construction section that explicitly recalls the form of the asymptotic operators at the three vertices, confirms the vertex matching conditions for the Y-graph, and argues that the cotangent geometry and integrability of J introduce no extra kernel or cokernel elements beyond the model. This will make the applicability of the cited gluing theorem fully transparent. revision: yes

Circularity Check

0 steps flagged

No circularity; central existence applies external Hutchings-Taubes technique to given rigid trees

full rationale

The derivation chain starts from a given rigid transversely cut-out Y-shaped Morse flow tree and invokes the obstruction bundle gluing technique of Hutchings-Taubes (external citation, not self-citation) to produce the J-holomorphic disc for small ε. No quantity is defined in terms of its own output, no parameter is fitted then renamed as prediction, and no load-bearing premise reduces to a self-citation chain or ansatz smuggled from prior author work. The paper explicitly positions the result as the first in a series that will later extend the correspondence, confirming the present step is self-contained against the cited external method.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on the applicability of the Hutchings-Taubes obstruction bundle method to this Morse-Bott situation and on the given rigidity and transversality of the input tree; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption The Y-shaped Morse flow tree is rigid and transversely cut out
    This is the input hypothesis required for the gluing to produce a solution.
  • domain assumption Obstruction bundle gluing of Hutchings-Taubes resolves the Morse-Bott obstruction in the cotangent bundle setting
    The paper invokes this external technique as the central tool without re-deriving it.

pith-pipeline@v0.9.1-grok · 5723 in / 1331 out tokens · 23620 ms · 2026-06-27T14:24:34.050986+00:00 · methodology

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Reference graph

Works this paper leans on

139 extracted references · 52 canonical work pages

  1. [1]

    and McOwen, Robert C

    Lockhart, Robert B. and McOwen, Robert C. , title =. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze , series =

  2. [2]

    , title =

    Cohen, Ralph L. , title =. Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology , editor =

  3. [3]

    Communications in Mathematical Physics , volume =

    Fukaya, Kenji , title =. Communications in Mathematical Physics , volume =

  4. [4]

    Kyoto Journal of Mathematics , number =

    Dylan Cant and Daren Chen , title =. Kyoto Journal of Mathematics , number =. 2026 , doi =

    2026

  5. [5]

    and Norbury, Paul , title =

    Cohen, Ralph L. and Norbury, Paul , title =. Asian J. Math. , volume =

  6. [6]

    Floer, Andreas , title =. J. Differential Geom. , volume =

  7. [7]

    and Neitzke, Andrew , title =

    Gaiotto, Davide and Moore, Gregory W. and Neitzke, Andrew , title =. Ann. Henri Poincar\'

  8. [8]

    Honda, Ko and Tian, Yin and Yuan, Tianyu , title =. J. Eur. Math. Soc. , year =

  9. [9]

    2026 , note =

    Honda, Ko and Tian, Yin and Yuan, Tianyu , title =. 2026 , note =

    2026

  10. [10]

    2026 , eprint=

    Higher-dimensional Heegaard Floer homology and spectral networks , author=. 2026 , eprint=

    2026

  11. [11]

    and Fournier, John J

    Adams, Robert A. and Fournier, John J. F. , TITLE =. 2003 , PAGES =

    2003

  12. [12]

    Floer, Andreas , TITLE =. The. 1995 , MRCLASS =

    1995

  13. [13]

    2012 , NUMBER =

    Fukaya, Kenji and Oh, Yong-Geun and Ohta, Hiroshi and Ono, Kaoru , TITLE =. 2012 , NUMBER =

    2012

  14. [14]

    Lipshitz, Robert , TITLE =. Geom. Topol. , FJOURNAL =. 2006 , PAGES =

    2006

  15. [15]

    and Fournier, John J

    Adams, Robert A. and Fournier, John J. F. , title =. 2003 , isbn =

    2003

  16. [16]

    , title =

    Conway, John B. , title =. 1995 , isbn =

    1995

  17. [17]

    , title =

    Donaldson, Simon K. , title =. 2002 , isbn =

    2002

  18. [18]

    and Salamon, Dietmar A

    Robbin, Joel W. and Salamon, Dietmar A. , title =. Annales de l'I.H.P. Analyse non lin\'eaire , volume =. 2001 , doi =

    2001

  19. [19]

    2006 , eprint=

    The Fukaya category of symplectic neighborhood of a non-Hausdorff manifold , author=. 2006 , eprint=

    2006

  20. [20]

    Manuscripta Math

    Mayer, Karl Heinz , TITLE =. Manuscripta Math. , FJOURNAL =. 1989 , NUMBER =. doi:10.1007/BF01173705 , URL =

    work page doi:10.1007/bf01173705 1989

  21. [21]

    , TITLE =

    Wasserman, Arthur G. , TITLE =. Topology , FJOURNAL =. 1969 , PAGES =. doi:10.1016/0040-9383(69)90005-6 , URL =

    work page doi:10.1016/0040-9383(69)90005-6 1969

  22. [22]

    Obstruction Bundle Gluing: An Introduction through

    Ipsita Datta and Yuan Yao , year =. Obstruction Bundle Gluing: An Introduction through

  23. [23]

    Computable, obstructed

    Erkao Bao and Ke Zhu , year=. Computable, obstructed. 2409.11565 , archivePrefix=

    arXiv

  24. [24]

    Joyce, Dominic , TITLE =

  25. [25]

    Liu, Gang and Tian, Gang , TITLE =. J. Differential Geom. , FJOURNAL =. 1998 , NUMBER =

    1998

  26. [26]

    Turkish J

    Ruan, Yongbin , TITLE =. Turkish J. Math. , FJOURNAL =. 1999 , NUMBER =

    1999

  27. [27]

    Ekholm, Tobias , TITLE =. Geom. Topol. , FJOURNAL =. 2007 , PAGES =. doi:10.2140/gt.2007.11.1083 , URL =

    work page doi:10.2140/gt.2007.11.1083 2007

  28. [28]

    Topics in symplectic

    Li, Jun and Tian, Gang , TITLE =. Topics in symplectic. 1998 , ISBN =

    1998

  29. [29]

    Peters, Stefan , TITLE =. J. Reine Angew. Math. , FJOURNAL =. 1984 , PAGES =. doi:10.1515/crll.1984.349.77 , URL =

    work page doi:10.1515/crll.1984.349.77 1984

  30. [30]

    2014 , PAGES =

    Audin, Mich\`ele and Damian, Mihai , TITLE =. 2014 , PAGES =. doi:10.1007/978-1-4471-5496-9 , URL =

    work page doi:10.1007/978-1-4471-5496-9 2014

  31. [31]

    Proceedings of the

    Wehrheim, Katrin , TITLE =. Proceedings of the. 2012 , MRCLASS =. doi:10.2140/gtm.2012.18.369 , URL =

    work page doi:10.2140/gtm.2012.18.369 2012

  32. [32]

    Smale, Stephen , TITLE =. Ann. of Math. (2) , FJOURNAL =. 1961 , PAGES =. doi:10.2307/1970311 , URL =

    work page doi:10.2307/1970311 1961

  33. [33]

    Bulletin of the London Mathematical Society , volume =

    Salamon, Dietmar , title =. Bulletin of the London Mathematical Society , volume =. doi:https://doi.org/10.1112/blms/22.2.113 , url =. https://londmathsoc.onlinelibrary.wiley.com/doi/pdf/10.1112/blms/22.2.113 , year =

    work page doi:10.1112/blms/22.2.113

  34. [34]

    Hutchings, Michael and Nelson, Jo , TITLE =. J. Topol. , FJOURNAL =. 2022 , NUMBER =

    2022

  35. [35]

    2002 , school=

    A Morse-Bott approach to contact homology , author=. 2002 , school=

    2002

  36. [36]

    and Givental, A

    Eliashberg, Y. and Givental, A. and Hofer, H. , TITLE =. Geom. Funct. Anal. , FJOURNAL =. 2000 , NUMBER =

    2000

  37. [37]

    and Wysocki, K

    Hofer, H. and Wysocki, K. and Zehnder, E. , TITLE =. J. Eur. Math. Soc. , FJOURNAL =. 2007 , NUMBER =. doi:10.4171/JEMS/99 , URL =

    work page doi:10.4171/jems/99 2007

  38. [38]

    arXiv preprint arXiv:1807.09455 , year=

    Construction of general symplectic field theory , author=. arXiv preprint arXiv:1807.09455 , year=

    Pith/arXiv arXiv

  39. [39]

    2024 , eprint=

    Morse homology and equivariance , author=. 2024 , eprint=

    2024

  40. [40]

    Pardon, John , TITLE =. J. Amer. Math. Soc. , FJOURNAL =. 2019 , NUMBER =. doi:10.1090/jams/924 , URL =

    work page doi:10.1090/jams/924 2019

  41. [41]

    2007 , PAGES =

    H\"ormander, Lars , TITLE =. 2007 , PAGES =. doi:10.1007/978-3-540-49938-1 , URL =

    work page doi:10.1007/978-3-540-49938-1 2007

  42. [42]

    Hutchings, Michael and Taubes, Clifford Henry , TITLE =. J. Symplectic Geom. , FJOURNAL =. 2009 , NUMBER =

    2009

  43. [43]

    Cohomology operations from

    Schwarz, Matthias , year=. Cohomology operations from

  44. [44]

    2023 , eprint=

    Adiabatic compactness for holomorphic curves with boundary on nearby Lagrangians , author=. 2023 , eprint=

    2023

  45. [45]

    2012 , PAGES =

    McDuff, Dusa and Salamon, Dietmar , TITLE =. 2012 , PAGES =

    2012

  46. [46]

    Bao, Erkao and Honda, Ko , TITLE =. Adv. Math. , FJOURNAL =. 2023 , PAGES =. doi:10.1016/j.aim.2023.108864 , URL =

    work page doi:10.1016/j.aim.2023.108864 2023

  47. [47]

    Bao, Erkao and Honda, Ko , TITLE =. J. Topol. , FJOURNAL =. 2018 , NUMBER =. doi:10.1112/topo.12077 , URL =

    work page doi:10.1112/topo.12077 2018

  48. [48]

    Topology , volume=

    The Maslov index for paths , author=. Topology , volume=. 1993 , publisher=

    1993

  49. [49]

    Coherent orientations in symplectic field theory , JOURNAL =

    Bourgeois, Fr\'. Coherent orientations in symplectic field theory , JOURNAL =. 2004 , NUMBER =. doi:10.1007/s00209-004-0656-x , URL =

    work page doi:10.1007/s00209-004-0656-x 2004

  50. [50]

    Erkao Bao , title =. Math. Z. , FJOURNAL =. 2023 , NUMBER =

    2023

  51. [51]

    Bao, Erkao and Honda, Ko , TITLE =. Algebr. Geom. Topol. , FJOURNAL =. 2021 , NUMBER =. doi:10.2140/agt.2021.21.1677 , URL =

    work page doi:10.2140/agt.2021.21.1677 2021

  52. [52]

    Mathematische Zeitschrift , volume=

    Introduction to contact homology , author=. Mathematische Zeitschrift , volume=. 2004 , publisher=

    2004

  53. [53]

    Floer homology, Novikov rings and clean intersections , howpublished =

  54. [54]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    Matrix factorizations from SYZ transformations. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  55. [55]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    Homological Mirror Symmetry for Calabi-Yau hypersurfaces in projective space. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  56. [56]

    Alston, Garrett , TITLE =. J. Symplectic Geom. , FJOURNAL =. 2011 , NUMBER =

    2011

  57. [57]

    Wendl, Chris , TITLE =. Comment. Math. Helv. , FJOURNAL =. 2010 , NUMBER =. doi:10.4171/CMH/199 , URL =

    work page doi:10.4171/cmh/199 2010

  58. [58]

    Alston, Garrett and Amorim, Lino , TITLE =. Int. Math. Res. Not. IMRN , FJOURNAL =. 2012 , NUMBER =. doi:10.1093/imrn/rnr125 , URL =

    work page doi:10.1093/imrn/rnr125 2012

  59. [59]

    and Walcher, Johannes , TITLE =

    Morrison, David R. and Walcher, Johannes , TITLE =. Adv. Theor. Math. Phys. , FJOURNAL =. 2009 , NUMBER =

    2009

  60. [60]

    Fukaya, Kenji and Oh, Yong-Geun , TITLE =. Asian J. Math. , FJOURNAL =. 1997 , NUMBER =. doi:10.4310/AJM.1997.v1.n1.a5 , URL =

    work page doi:10.4310/ajm.1997.v1.n1.a5 1997

  61. [61]

    ArXiv e-prints , archivePrefix = "arXiv", eprint =

    Floer cohomology of immersed Lagrangian spheres in smoothings of \ A\_N\ surfaces. ArXiv e-prints , archivePrefix = "arXiv", eprint =

  62. [62]

    Alston, Garrett , title =

  63. [63]

    Symmetries of

    Casta. Symmetries of

  64. [64]

    and Oh, Y.-G

    Fukaya, K. and Oh, Y.-G. and Ohta, H. and Ono, K. , title =

  65. [65]

    , title =

    Fukaya, K. , title =

  66. [66]

    , title =

    Solomon, J. , title =

  67. [67]

    and Oh, Y.-G

    Fukaya, K. and Oh, Y.-G. and Ohta, H. and Ono, K. , year = 2009, title =

    2009

  68. [68]

    2008 , PAGES =

    Seidel, Paul , TITLE =. 2008 , PAGES =. doi:10.4171/063 , URL =

    work page doi:10.4171/063 2008

  69. [69]

    , year = 2002, title =

    Hatcher, A. , year = 2002, title =

    2002

  70. [70]

    , year = 2000, title =

    Seidel, P. , year = 2000, title =. Bull. Soc. France , volume = 128, number = 1, pages =

    2000

  71. [71]

    , year = 1999, title =

    Oh, Y.-G. , year = 1999, title =. Comm. Anal. Geom. , volume = 7, number = 1, pages =

    1999

  72. [72]

    and Hofer, H

    Floer, A. and Hofer, H. , year = 1993, title =. Math. Z. , volume = 212, number = 1, pages =

    1993

  73. [73]

    Duke Math

    Floer, Andreas and Hofer, Helmut and Salamon, Dietmar , TITLE =. Duke Math. J. , FJOURNAL =. 1995 , NUMBER =. doi:10.1215/S0012-7094-95-08010-7 , URL =

    work page doi:10.1215/s0012-7094-95-08010-7 1995

  74. [74]

    and Salamon, D

    Robbin, J. and Salamon, D. , year = 1993, title =. Topology , volume = 32, number = 4, pages =

    1993

  75. [75]

    , year = 1986, title =

    Katz, S. , year = 1986, title =. Compositio Mathematica , volume = 60, number = 2, pages =

    1986

  76. [76]

    and de la Ossa, X

    Candelas, P. and de la Ossa, X. and Green, P. and Parkes, L. , title=. Nuclear Phys. B , volume= 359 , year=1991, number=1, pages=

    1991

  77. [77]

    , title=

    Givental, A. , title=. Intern. Math. Res. Notices , year=1996, volume=13, pages=

    1996

  78. [78]

    and Liu, K

    Lian, B. and Liu, K. and Yau, S.-T. , title=. Asian J. Math. , volume=1, year=1997, pages=

    1997

  79. [79]

    , title=

    Kontsevich, M. , title=. Proceedings of the International Congress of Mathematicians, 1994, Z\"urich , year=1995, pages=

    1994

  80. [80]

    and Zaslow, E

    Polishchuk, A. and Zaslow, E. , title=. Adv. Theor. Math. Phys. , volume=2, year=1998, number=2, pages=

    1998

Showing first 80 references.