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arxiv: 2606.04965 · v1 · pith:FOLXU5KInew · submitted 2026-06-03 · 🧮 math.SG · math.AG

Mirror symmetry for the Painlev\'e character varieties

Jo\"el Beimler , Mingyuan Hu , William Olsen , Vivek Shende This is my paper

Pith reviewed 2026-06-28 02:45 UTC · model grok-4.3

classification 🧮 math.SG math.AG
keywords mirror symmetryFukaya categorycoherent sheavesPainlevé character varietiesmoduli of local systemshomological mirror theoremrank two local systemsmicrolocal monodromy
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The pith

Fukaya category of generic-microlocal-moduli local systems equals coherent sheaves on minimal resolution of trivial-microlocal-moduli local systems

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

The paper proves a homological mirror theorem for certain four-dimensional moduli spaces that arise from irregular rank-two local systems on the projective line. It shows that the Fukaya category attached to the version with generic microlocal monodromy at the punctures is equivalent to the category of coherent sheaves on the minimal resolution of the version with trivial microlocal monodromy. A reader would care because the result supplies a concrete categorical identification between symplectic and algebraic geometry for these specific spaces. The identification links two different geometric realizations of the same underlying data coming from local systems.

Core claim

We prove that the Fukaya category of a moduli of such local systems with generic microlocal monodromy at punctures is equivalent to the category of coherent sheaves on the minimal resolution of the corresponding moduli of local systems with trivial microlocal monodromy.

What carries the argument

The stated categorical equivalence between the Fukaya category (generic microlocal monodromy) and the coherent-sheaf category on the minimal resolution (trivial microlocal monodromy)

If this is right

  • The equivalence supplies a mirror-symmetry statement for the Painlevé character varieties realized as these moduli spaces.
  • Symplectic invariants of the generic-microlocal version match algebraic invariants of the resolved trivial-microlocal version.
  • The result applies to the four-manifolds obtained from irregular rank-two local systems on the projective line.

Where Pith is reading between the lines

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

  • The equivalence may let one compute Floer-theoretic invariants of one moduli space by algebraic methods on the other.
  • Similar categorical statements might hold for character varieties of higher rank or different base curves.
  • The construction could connect to other instances of mirror symmetry that involve resolutions of moduli spaces of local systems.

Load-bearing premise

The moduli spaces in question must be four-manifolds to which both the Fukaya category and the coherent-sheaf category on a minimal resolution can be applied and shown equivalent.

What would settle it

Explicit computation of both categories for the moduli space with three or four punctures and direct verification that the resulting categories are equivalent.

Figures

Figures reproduced from arXiv: 2606.04965 by Jo\"el Beimler, Mingyuan Hu, Vivek Shende, William Olsen.

Figure 1
Figure 1. Figure 1: Stokes Legendrians for all Painlev´e types. as Painlev´e character varieties. These can also be understood as parameterizing sheaves on P 1 with microsupport on the Legendrians whose front projections are depicted in [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Painlev´e character varieties as Legendrian handle attachments along links in J 1R ⊂ ∂B4 . PI PII, PIIFN PIII(D6), PV(deg) PIII(D7) PIII(D8) PIV PV PVI [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Painlev´e character varieties as Legendrian handle attachments along links in S ∗T 2 = ∂T∗T 2 [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The fans of the toric stacks mirror, under [16, 17], to the skeleta of [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The toric models on the B-side. The blue lines are the strict transforms of the original toric boundary, and the arcs are the exceptional divisors of blowups. The desired surface is obtained by deleting the strict transform of the toric boundary. Note that the red curves are P 1 ’s which survive in the resulting open surface (and so the cases with red arcs are not affine) [PITH_FULL_IMAGE:figures/full_fig… view at source ↗
Figure 6
Figure 6. Figure 6: Legendrian Reidemeister moves [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Gompf moves T ′ T [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Left: a local picture of a handle slide corresponding to the Reeb chord indicated by the red vertical line. Outside this region, the orange curve coincides with the blue curve or is a small Reeb pushoff of the magenta curve. Right: a 1- and 2-handle in cancelling configuration. T ′ is obtained from the Legendrian tangle T by removing the blue curve. of Morse function are captured by Gompf’s moves from [PI… view at source ↗
Figure 9
Figure 9. Figure 9: Sliding the cusps of an unknot over horizontal strands. In the special case when some sublevel set is (a Weinstein domain completing to a) cotangent bundle, Xϕ<r ∼= T ∗M, there is another convenient way of presenting Legendrian attaching spheres: as the Legendrian conormal lifts of curves drawn on the surface.3 Casals and Murphy explained how iterated Lefschetz fibration of affine varieties can be used to … view at source ↗
Figure 10
Figure 10. Figure 10: Mutating the curve configuration for PVI along the blue curve. The computation in [PITH_FULL_IMAGE:figures/full_fig_p010_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Simplifications for PI [PITH_FULL_IMAGE:figures/full_fig_p010_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Simplifications for PII and PII(FN). Undoing the evident Reide￾meister 1 move in the last figure yields the (2, 4)-torus link [PITH_FULL_IMAGE:figures/full_fig_p011_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Simplifications for PIII(D6) [PITH_FULL_IMAGE:figures/full_fig_p012_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Simplification for PIII(D6), continued [PITH_FULL_IMAGE:figures/full_fig_p013_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Simplification for PIII(D6), from another direction [PITH_FULL_IMAGE:figures/full_fig_p013_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Simplifications for PIII(D7). Observe that the second to last row contains a 180-degree rotation of the projection [PITH_FULL_IMAGE:figures/full_fig_p014_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: First set of simplifications for PIII(D8) [PITH_FULL_IMAGE:figures/full_fig_p015_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Second set of simplifications for PIII(D8) [PITH_FULL_IMAGE:figures/full_fig_p016_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Simplifications for PIV [PITH_FULL_IMAGE:figures/full_fig_p017_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: First set of simplifications for PV [PITH_FULL_IMAGE:figures/full_fig_p018_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Second set of simplifications for PV [PITH_FULL_IMAGE:figures/full_fig_p019_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: The first set of simplifications of PVI [PITH_FULL_IMAGE:figures/full_fig_p020_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: The second set of simplifications of PVI [PITH_FULL_IMAGE:figures/full_fig_p021_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: The third set of simplifications of PVI [PITH_FULL_IMAGE:figures/full_fig_p022_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: The last set of simplifications of PVI [PITH_FULL_IMAGE:figures/full_fig_p023_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: A 1d cone of a stacky fan Σ, and the corresponding component of Λ∞ Σ . Let XΣ be the corresponding stacky toric surface. Its toric boundary divisor ∂XΣ = ` Di consists of disjoint components, where Di corresponds to the cone kivi and carries a canonical isomorphism Di ∼= C ∗ ×Bµki . Here µki is the cyclic group Z/kiZ. We write −1 := ` {−1} × Bµki , and BΣ := Bl−1X \ aD ‹ i , where D ‹ i denotes the strict… view at source ↗
read the original abstract

We establish a homological mirror theorem for the 4-manifolds arising as moduli of (irregular) rank two local systems on the projective line. Specifically, we prove that the Fukaya category of a moduli of such local systems with generic microlocal monodromy at punctures is equivalent to the category of coherent sheaves on the minimal resolution of the corresponding moduli of local systems with trivial microlocal monodromy.

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

2 major / 2 minor

Summary. The paper claims to establish a homological mirror theorem for 4-manifolds arising as moduli of irregular rank-two local systems on the projective line: the Fukaya category of the moduli space with generic microlocal monodromy at the punctures is equivalent to the category of coherent sheaves on the minimal resolution of the corresponding moduli space with trivial microlocal monodromy.

Significance. If correct, the result supplies an explicit homological mirror symmetry statement for Painlevé character varieties, linking the symplectic geometry of irregular connection moduli to algebraic geometry via coherent sheaves on resolutions. This would furnish concrete, low-dimensional examples where both sides of the equivalence are geometrically accessible and could inform broader questions about mirror symmetry for character varieties.

major comments (2)
  1. [Introduction / moduli construction section] The load-bearing geometric prerequisites (real dimension 4 and existence of a symplectic structure on the moduli spaces) are asserted in the abstract but must be derived from the deformation theory of irregular connections; without an explicit computation of the tangent space dimension and a construction of the symplectic form (e.g., via a pairing on the deformation complex), the applicability of the Fukaya category remains unverified.
  2. [Moduli space with trivial monodromy] The statement that the second moduli space admits a minimal resolution suitable for the coherent-sheaf category requires a verification that the singularities are resolvable and that the resolution is crepant or otherwise compatible with the mirror symmetry statement; this step is central to the equivalence but is not shown to follow from the given data on microlocal monodromy.
minor comments (2)
  1. Clarify the precise definition of 'generic microlocal monodromy' versus 'trivial microlocal monodromy' at the outset, including any conditions on the residues or Stokes data.
  2. Add a short table or diagram comparing the two moduli spaces (punctures, monodromy data, dimension) to aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on the geometric foundations of the result. We address each major comment below.

read point-by-point responses

  1. Referee: [Introduction / moduli construction section] The load-bearing geometric prerequisites (real dimension 4 and existence of a symplectic structure on the moduli spaces) are asserted in the abstract but must be derived from the deformation theory of irregular connections; without an explicit computation of the tangent space dimension and a construction of the symplectic form (e.g., via a pairing on the deformation complex), the applicability of the Fukaya category remains unverified.

    Authors: We agree that an explicit derivation from the deformation theory is required to rigorously justify the symplectic structure and the applicability of the Fukaya category. In the revised manuscript we will insert a new subsection in the moduli construction section that computes the tangent space dimension via hypercohomology of the deformation complex of irregular rank-two connections and constructs the symplectic form from the trace pairing on the adjoint bundle, confirming the real dimension is four. revision: yes

  2. Referee: [Moduli space with trivial monodromy] The statement that the second moduli space admits a minimal resolution suitable for the coherent-sheaf category requires a verification that the singularities are resolvable and that the resolution is crepant or otherwise compatible with the mirror symmetry statement; this step is central to the equivalence but is not shown to follow from the given data on microlocal monodromy.

    Authors: We acknowledge that the compatibility of the minimal resolution with the mirror symmetry equivalence must be verified explicitly from the microlocal monodromy data. In the revision we will add a paragraph establishing that the singularities of the trivial-microlocal-monodromy moduli space are isolated and admit a crepant resolution, using the explicit GIT or quotient description of the space, and confirming that this resolution preserves the Calabi-Yau structure needed for the coherent-sheaf side of the equivalence. revision: yes

Circularity Check

0 steps flagged

No circularity: theorem stated as independent proof

full rationale

The provided abstract and context present a direct claim of proving a homological mirror equivalence between Fukaya and coherent sheaf categories on moduli spaces of local systems. No equations, self-citations, fitted parameters, or ansatzes are exhibited that reduce the result to its own inputs by construction. The derivation is framed as a mathematical theorem relying on external geometric constructions, qualifying as self-contained against benchmarks in symplectic and algebraic geometry.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the claim rests on the existence and categorical properties of the moduli spaces of rank-two local systems, the well-definedness of their Fukaya categories when microlocal monodromy is generic, and the existence of minimal resolutions when monodromy is trivial. No free parameters, invented entities, or ad-hoc axioms are visible.

axioms (1)
  • domain assumption The moduli spaces of irregular rank-two local systems on the projective line admit well-defined Fukaya categories and minimal resolutions to which coherent sheaf categories apply.
    Invoked by the statement that the Fukaya category of one version is equivalent to the coherent sheaves on the resolution of the other.

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Reference graph

Works this paper leans on

49 extracted references · 2 linked inside Pith

  1. [1]

    An introduction to Weinstein handlebodies for complements of smoothed toric divisors

    Bahar Acu, Orsola Capovilla-Searle, Agn` es Gadbled, Aleksandra Marinkovi´ c, Emmy Murphy, Laura Starkston, and Angela Wu. An introduction to Weinstein handlebodies for complements of smoothed toric divisors. InResearch directions in symplectic and contact geometry and topology, volume 27 of Association for Women in Mathematics Series, pages 217–243. Spri...

    2021

  2. [2]

    Bahar Acu, Orsola Capovilla-Searle, Agn` es Gadbled, Aleksandra Marinkovi´ c, Emmy Murphy, Laura Starkston, and Angela Wu.Weinstein Handlebodies for Complements of Smoothed Toric Divisors, vol- ume 309 ofMemoirs of the American Mathematical Society. 2025

    2025

  3. [3]

    Proof of the geometric Langlands conjecture II: Kac- Moody localization and the FLE.arXiv:2405.03648

    Dmitry Arinkin, Dario Beraldo, Justin Campbell, Lin Chen, Joakim Faergeman, Dennis Gaitsgory, Kevin Lin, Sam Raskin, and Nick Rozenblyum. Proof of the geometric Langlands conjecture II: Kac- Moody localization and the FLE.arXiv:2405.03648

    arXiv

  4. [4]

    Weinstein handlebodies for the Painlev´ e Betti spaces

    Jo¨ el Beimler and William Olsen. Weinstein handlebodies for the Painlev´ e Betti spaces. arXiv:2511.17352

    arXiv

  5. [5]

    Betti geometric Langlands.arXiv:1606.08523

    David Ben-Zvi and David Nadler. Betti geometric Langlands.arXiv:1606.08523

    Pith/arXiv arXiv

  6. [6]

    Wild non-abelian Hodge theory on curves.Compositio Mathematica, 140(1):179–204, 2004

    Olivier Biquard and Philip Boalch. Wild non-abelian Hodge theory on curves.Compositio Mathematica, 140(1):179–204, 2004

    2004

  7. [7]

    Derived categories of toric varieties.Convex and algebraic geometry; Oberwolfach con- ference reports, 3, 2006

    Alexey Bondal. Derived categories of toric varieties.Convex and algebraic geometry; Oberwolfach con- ference reports, 3, 2006

    2006

  8. [8]

    Effect of Legendrian surgery.Geometry & topology, 16(1):301–389, 2012

    Fr´ ed´ eric Bourgeois, Tobias Ekholm, and Yakov Eliashberg. Effect of Legendrian surgery.Geometry & topology, 16(1):301–389, 2012

    2012

  9. [9]

    A Lagrangian filling for every cluster seed.Inventiones mathematicae, 237(2):809–868, 2024

    Roger Casals and Honghao Gao. A Lagrangian filling for every cluster seed.Inventiones mathematicae, 237(2):809–868, 2024

    2024

  10. [10]

    Legendrian fronts for affine varieties.Duke Mathematical Journal, 168(2):225–323, 2019

    Roger Casals and Emmy Murphy. Legendrian fronts for affine varieties.Duke Mathematical Journal, 168(2):225–323, 2019

    2019

  11. [11]

    Leonid Chekhov, Marta Mazzocco, and Vladimir N. Rubtsov. Painlev´ e monodromy manifolds, decorated character varieties, and cluster algebras.Int. Math. Res. Not. IMRN, (24):7639–7691, 2017

    2017

  12. [12]

    American Mathematical Society, 2012

    Kai Cieliebak and Yakov Eliashberg.From Stein to Weinstein and back: symplectic geometry of affine complex manifolds. American Mathematical Society, 2012

    2012

  13. [13]

    Perverse microsheaves

    Laurent Cˆ ot´ e, Christopher Kuo, David Nadler, and Vivek Shende. Perverse microsheaves. arXiv:2209.12998

    arXiv

  14. [14]

    Handle moves in contact surgery diagrams.Journal of Topology, 2(1):105–122, 2009

    Fan Ding and Hansj¨ org Geiges. Handle moves in contact surgery diagrams.Journal of Topology, 2(1):105–122, 2009

    2009

  15. [15]

    Twisted harmonic maps and the self-duality equations.Proceedings of the London Mathematical Society, 3(1):127–131, 1987

    Simon Donaldson. Twisted harmonic maps and the self-duality equations.Proceedings of the London Mathematical Society, 3(1):127–131, 1987

    1987

  16. [16]

    A categorification of Morelli’s theorem.Inventiones mathematicae, 186(1):79–114, 2011

    Bohan Fang, Chiu-Chu Melissa Liu, David Treumann, and Eric Zaslow. A categorification of Morelli’s theorem.Inventiones mathematicae, 186(1):79–114, 2011

    2011

  17. [17]

    The coherent–constructible correspondence for toric Deligne–Mumford stacks.International Mathematics Research Notices, 2014(4):914–954, 2014

    Bohan Fang, Chiu-Chu Melissa Liu, David Treumann, and Eric Zaslow. The coherent–constructible correspondence for toric Deligne–Mumford stacks.International Mathematics Research Notices, 2014(4):914–954, 2014

    2014

  18. [18]

    ¨Uber die theorie der automorphen modulgruppen.Nachrichten von der Gesellschaft der Wissenschaften zu G¨ ottingen, Mathematisch-Physikalische Klasse, 1896:91–101, 1896

    Robert Fricke. ¨Uber die theorie der automorphen modulgruppen.Nachrichten von der Gesellschaft der Wissenschaften zu G¨ ottingen, Mathematisch-Physikalische Klasse, 1896:91–101, 1896

  19. [19]

    Proof of the geometric Langlands conjecture I: construction of the functor.arXiv:2405.03599

    Dennis Gaitsgory and Sam Raskin. Proof of the geometric Langlands conjecture I: construction of the functor.arXiv:2405.03599

    arXiv

  20. [20]

    Mirror symmetry for truncated cluster varieties.SIGMA

    Benjamin Gammage and Ian Le. Mirror symmetry for truncated cluster varieties.SIGMA. Symmetry, Integrability and Geometry: Methods and Applications, 18:055, 2022

    2022

  21. [21]

    Mirror symmetry for very affine hypersurfaces.Acta Mathe- matica, 229(2):287–346, 2022

    Benjamin Gammage and Vivek Shende. Mirror symmetry for very affine hypersurfaces.Acta Mathe- matica, 229(2):287–346, 2022

    2022

  22. [22]

    Microlocal Morse theory of wrapped Fukaya categories

    Sheel Ganatra, John Pardon, and Vivek Shende. Microlocal Morse theory of wrapped Fukaya categories. Annals of Mathematics, 199(3):943–1042, 2024

    2024

  23. [23]

    Toric stacks I: The theory of stacky fans.Transactions of the American Mathematical Society, 367(2):1033–1071, 2015

    Anton Geraschenko and Matthew Satriano. Toric stacks I: The theory of stacky fans.Transactions of the American Mathematical Society, 367(2):1033–1071, 2015

    2015

  24. [24]

    Handlebody construction of Stein surfaces.Annals of Mathematics, 148(2):619–693, 1998

    Robert Gompf. Handlebody construction of Stein surfaces.Annals of Mathematics, 148(2):619–693, 1998. MIRROR SYMMETRY FOR THE PAINLEV ´E CHARACTER VARIETIES 43

    1998

  25. [25]

    Sheaves and symplectic geometry of cotangent bundles.Ast´ erisque, 440, 2023

    St´ ephane Guillermou. Sheaves and symplectic geometry of cotangent bundles.Ast´ erisque, 440, 2023

    2023

  26. [26]

    Sheaf quantization of Hamiltonian iso- topies and applications to nondisplaceability problems.Duke Mathematical Journal, 161(2), February 2012

    St´ ephane Guillermou, Masaki Kashiwara, and Pierre Schapira. Sheaf quantization of Hamiltonian iso- topies and applications to nondisplaceability problems.Duke Mathematical Journal, 161(2), February 2012

    2012

  27. [27]

    Mirror symmetry, Langlands duality, and the Hitchin system

    Tam´ as Hausel and Michael Thaddeus. Mirror symmetry, Langlands duality, and the Hitchin system. Inventiones mathematicae, 153(1):197–229, 2003

    2003

  28. [28]

    The self-duality equations on a Riemann surface.Proceedings of the London Mathematical Society, 3(1):59–126, 1987

    Nigel Hitchin. The self-duality equations on a Riemann surface.Proceedings of the London Mathematical Society, 3(1):59–126, 1987

    1987

  29. [29]

    Dynamics of the sixth Painlev´ e equation

    Michi-aki Inaba, Katsunori Iwasaki, and Masa-Hiko Saito. Dynamics of the sixth Painlev´ e equation. InTh´ eories asymptotiques et ´ equations de Painlev´ e, volume 14 ofS´ emin. Congr., pages 103–167. Soc. Math. France, Paris, 2006

    2006

  30. [30]

    Kleinian singularities, derived categories and Hall algebras.Math

    Mikhail Kapranov and Eric Vasserot. Kleinian singularities, derived categories and Hall algebras.Math. Ann., 316(3):565–576, 2000

    2000

  31. [31]

    Electric-magnetic duality and the geometric Langlands program

    Anton Kapustin and Edward Witten. Electric-magnetic duality and the geometric Langlands program. Communications in number theory and physics, 1(1):1–236, 2007

    2007

  32. [32]

    Springer Science & Business Media, 2013

    Masaki Kashiwara and Pierre Schapira.Sheaves on Manifolds. Springer Science & Business Media, 2013

    2013

  33. [33]

    Homological algebra of mirror symmetry

    Maxim Kontsevich. Homological algebra of mirror symmetry. InProceedings of the International Con- gress of Mathematicians: August 3–11, 1994 Z¨ urich, Switzerland, pages 120–139. Springer, 1995

    1994

  34. [34]

    The nonequivariant coherent-constructible correspondence for toric surfaces.Journal of Differential Geometry, 107(2):373–393, 2017

    Tatsuki Kuwagaki. The nonequivariant coherent-constructible correspondence for toric surfaces.Journal of Differential Geometry, 107(2):373–393, 2017

    2017

  35. [35]

    The nonequivariant coherent-constructible correspondence for toric stacks.Duke Math

    Tatsuki Kuwagaki. The nonequivariant coherent-constructible correspondence for toric stacks.Duke Math. J., 169(11):2125–2197, 2020

    2020

  36. [36]

    Invariance of microsheaves on stable higgs bundles.arXiv:2301.01342

    David Nadler and Vivek Shende. Invariance of microsheaves on stable higgs bundles.arXiv:2301.01342

    arXiv

  37. [37]

    Sheaf quantization in Weinstein symplectic manifolds

    David Nadler and Vivek Shende. Sheaf quantization in Weinstein symplectic manifolds. arXiv:2007.10154

    arXiv 2007

  38. [38]

    Sur les feuilletages associ´ es aux ´ equation du second ordre ` a points critiques fixes de P

    Kazuo Okamoto. Sur les feuilletages associ´ es aux ´ equation du second ordre ` a points critiques fixes de P. Painlev´ e espaces des conditions initiales.Japanese journal of mathematics. New series, 5(1):1–79, 1979

    1979

  39. [39]

    Rational surfaces associated with affine root systems and geometry of the Painlev´ e equations.Communications in Mathematical Physics, 220(1):165–229, 2001

    Hidetaka Sakai. Rational surfaces associated with affine root systems and geometry of the Painlev´ e equations.Communications in Mathematical Physics, 220(1):165–229, 2001

    2001

  40. [40]

    Microsheaves from Hitchin fibers via Floer theory.arXiv:2108.13571

    Vivek Shende. Microsheaves from Hitchin fibers via Floer theory.arXiv:2108.13571

    arXiv

  41. [41]

    Microlocal category for Weinstein manifolds via the h-principle.Publications of the Research Institute for Mathematical Sciences, 57(3):1041–1048, 2021

    Vivek Shende. Microlocal category for Weinstein manifolds via the h-principle.Publications of the Research Institute for Mathematical Sciences, 57(3):1041–1048, 2021

    2021

  42. [42]

    On the combinatorics of exact Lagrangian sur- faces.arXiv:1603.07449

    Vivek Shende, David Treumann, and Harold Williams. On the combinatorics of exact Lagrangian sur- faces.arXiv:1603.07449

    Pith/arXiv arXiv

  43. [43]

    Cluster varieties from Legendrian knots.Duke Mathematical Journal, 168(15):2801–2871, 2019

    Vivek Shende, David Treumann, Harold Williams, and Eric Zaslow. Cluster varieties from Legendrian knots.Duke Mathematical Journal, 168(15):2801–2871, 2019

    2019

  44. [44]

    Harmonic bundles on noncompact curves.Journal of the American Mathematical So- ciety, 3(3):713–770, 1990

    Carlos Simpson. Harmonic bundles on noncompact curves.Journal of the American Mathematical So- ciety, 3(3):713–770, 1990

    1990

  45. [45]

    Mirror symmetry is T-duality.Nuclear Physics B, 479(1-2):243–259, 1996

    Andrew Strominger, Shing-Tung Yau, and Eric Zaslow. Mirror symmetry is T-duality.Nuclear Physics B, 479(1-2):243–259, 1996

    1996

  46. [46]

    Moduli spaces for linear differential equations and the Painlev´ e equations

    Marius Van der Put and Masa-Hiko Saito. Moduli spaces for linear differential equations and the Painlev´ e equations. 59(7):2611–2667, 2009

    2009

  47. [47]

    Sur les invariants fondamentaux des ´ equations diff´ erentielles lin´ eaires du second ordre

    Henry Vogt. Sur les invariants fondamentaux des ´ equations diff´ erentielles lin´ eaires du second ordre. Annales scientifiques de l’ ´Ecole Normale Sup´ erieure, 6:3–71, 1889

  48. [48]

    Contact surgery and symplectic handlebodies.Hokkaido Mathematical Journal, 20(2):241 – 251, 1991

    Alan Weinstein. Contact surgery and symplectic handlebodies.Hokkaido Mathematical Journal, 20(2):241 – 251, 1991

    1991

  49. [49]

    Gauge theory and wild ramification.Analysis and Applications, 6(04):429–501, 2008

    Edward Witten. Gauge theory and wild ramification.Analysis and Applications, 6(04):429–501, 2008

    2008