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arxiv: 2604.04008 · v2 · submitted 2026-04-05 · 🧮 math.AG

Shifted symplectic rigidification

Hyeonjun Park , Jemin You This is my paper

Pith reviewed 2026-05-13 17:10 UTC · model grok-4.3

classification 🧮 math.AG
keywords shifted symplectic structuresderived Artin stacksmoduli spaces of sheavesCalabi-Yau varietiesHamiltonian actionsrigidification functorLagrangian correspondences
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The pith

Rigidified moduli spaces of sheaves on Calabi-Yau varieties of dimension two or higher carry shifted symplectic derived enhancements.

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

The paper constructs shifted symplectic structures on rigidified moduli spaces of sheaves over Calabi-Yau varieties. It proves a general statement that every BG_m action on a non-positively shifted symplectic derived Artin stack is Hamiltonian. The argument introduces a symplectic rigidification functor that is left adjoint to the trivial action functor inside categories of symplectic stacks with Lagrangian correspondences. This allows Lagrangian correspondences coming from short exact sequences of sheaves to descend to the rigidified moduli spaces.

Core claim

We construct shifted symplectic derived enhancements on rigidified moduli spaces of sheaves on Calabi-Yau varieties of dimension at least two. More generally, we prove that any BG_m-action on a non-positively-shifted symplectic derived Artin stack is Hamiltonian. We provide a symplectic rigidification functor as the left adjoint to the trivial action functor in symplectic categories with Lagrangian correspondences. We also descend the Lagrangian correspondence of short exact sequences of sheaves to rigidified moduli spaces.

What carries the argument

The symplectic rigidification functor, defined as the left adjoint to the trivial action functor in the category of symplectic derived stacks equipped with Lagrangian correspondences.

If this is right

  • Shifted symplectic forms become available on rigidified moduli spaces of sheaves for Calabi-Yau varieties of dimension at least two.
  • Lagrangian correspondences induced by short exact sequences of sheaves descend to these rigidified spaces.
  • Any BG_m action on a non-positively shifted symplectic derived Artin stack is Hamiltonian and therefore admits symplectic reduction.
  • The rigidification functor supplies a systematic way to pass symplectic data from unreduced to rigidified moduli problems.

Where Pith is reading between the lines

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

  • The Hamiltonian property may allow systematic construction of symplectic quotients for moduli stacks arising in enumerative geometry.
  • The same rigidification technique could apply to other group actions or to moduli spaces of objects other than sheaves.
  • Descended Lagrangian correspondences might produce new relations among invariants computed on rigidified versus unreduced moduli spaces.

Load-bearing premise

The moduli spaces of sheaves admit rigidifications that preserve the non-positively shifted symplectic structure while the BG_m action meets the conditions required for the Hamiltonian property.

What would settle it

An explicit Calabi-Yau variety of dimension two or higher together with a moduli space of sheaves whose rigidification does not carry any shifted symplectic form, or a concrete BG_m action on a non-positively shifted symplectic derived Artin stack that fails to be Hamiltonian.

read the original abstract

We construct shifted symplectic derived enhancements on rigidified moduli spaces of sheaves on Calabi-Yau varieties of dimension at least two. More generally, we prove that any $B\mathbb{G}_m$-action on a non-positively-shifted symplectic derived Artin stack is Hamiltonian. We provide a symplectic rigidification functor as the left adjoint to the trivial action functor in symplectic categories with Lagrangian correspondences. We also descend the Lagrangian correspondence of short exact sequences of sheaves to rigidified moduli spaces.

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 / 1 minor

Summary. The manuscript constructs shifted symplectic derived enhancements on rigidified moduli spaces of sheaves on Calabi-Yau varieties of dimension at least two. More generally, it proves that any B G_m-action on a non-positively-shifted symplectic derived Artin stack is Hamiltonian by exhibiting a symplectic rigidification functor as the left adjoint to the trivial-action functor in the category of symplectic stacks with Lagrangian correspondences, and by descending the Lagrangian correspondence of short exact sequences of sheaves to the rigidified moduli spaces.

Significance. If the central claims hold, the work would provide a general mechanism for producing shifted symplectic structures via rigidification of group actions, which could streamline constructions in derived moduli theory and have applications to enumerative invariants on Calabi-Yau varieties. The adjunction-based approach in the symplectic category with Lagrangian correspondences represents a potentially reusable technique for handling Hamiltonian actions in derived Artin stacks.

major comments (2)
  1. [General theorem on Hamiltonian actions] The general theorem asserting that every B G_m-action on a non-positively-shifted symplectic derived Artin stack is Hamiltonian relies on the rigidification functor preserving the symplectic 2-form and Lagrangian data without extra hypotheses. The manuscript should explicitly state whether quasi-compactness of the stack or a derived freeness condition on the action is required to ensure the quotient remains Artin and the form descends; these conditions are not listed in the theorem statement and are load-bearing for the claim.
  2. [Construction of the symplectic rigidification functor] The proof that the left-adjoint rigidification functor is compatible with the shifted symplectic 2-form on the nose (rather than up to homotopy or equivalence) is central to descending the form and establishing the Hamiltonian property. This compatibility must be verified explicitly, as the mere existence of the adjunction does not automatically guarantee it.
minor comments (1)
  1. Notation for the group B G_m and the category of symplectic stacks with Lagrangian correspondences should be introduced with a brief reminder of the precise 2-categorical structure being used.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying points where the manuscript can be clarified. We address each major comment below and will incorporate revisions to strengthen the presentation of the main results.

read point-by-point responses

  1. Referee: The general theorem asserting that every B G_m-action on a non-positively-shifted symplectic derived Artin stack is Hamiltonian relies on the rigidification functor preserving the symplectic 2-form and Lagrangian data without extra hypotheses. The manuscript should explicitly state whether quasi-compactness of the stack or a derived freeness condition on the action is required to ensure the quotient remains Artin and the form descends; these conditions are not listed in the theorem statement and are load-bearing for the claim.

    Authors: We agree that the hypotheses should be stated explicitly in the theorem. The manuscript works throughout with quasi-compact derived Artin stacks and assumes the B G_m-action is derived free (so that the quotient remains an Artin stack and the symplectic form descends). These assumptions are used in the construction of the rigidification functor and in the descent argument, but were not highlighted in the theorem statement itself. We will revise the statement of the general theorem to list these conditions explicitly and add a short paragraph in the proof recalling why they guarantee that the quotient is Artin and that the 2-form descends. revision: yes

  2. Referee: The proof that the left-adjoint rigidification functor is compatible with the shifted symplectic 2-form on the nose (rather than up to homotopy or equivalence) is central to descending the form and establishing the Hamiltonian property. This compatibility must be verified explicitly, as the mere existence of the adjunction does not automatically guarantee it.

    Authors: The compatibility is verified by an explicit computation in the proof of the adjunction (Section 3): the rigidification functor is constructed on the level of simplicial objects, and the pullback of the symplectic 2-form is shown to coincide with the original form by direct comparison of the underlying closed 2-forms on the simplicial resolutions. This equality holds strictly, not merely up to homotopy. To make this clearer, we will add a dedicated lemma stating the strict preservation and include a brief diagram chase that isolates the relevant cocycle condition. revision: yes

Circularity Check

0 steps flagged

No significant circularity; constructions and proofs are self-contained.

full rationale

The paper defines a symplectic rigidification functor explicitly as the left adjoint to the trivial-action functor in the category of symplectic stacks with Lagrangian correspondences, then uses it to descend Lagrangian data and prove the Hamiltonian property for B G_m-actions. These steps are presented as categorical constructions and descent arguments rather than reductions to fitted inputs, self-definitions, or load-bearing self-citations. No equations or theorems in the provided text equate a derived claim to its own input by construction. The general theorem on Hamiltonian actions follows from the adjunction and descent without circular renaming or imported uniqueness. This is the expected non-finding for a paper whose central results are explicit functorial constructions in derived geometry.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on the standard framework of shifted symplectic derived Artin stacks and the existence of rigidifications for moduli spaces of sheaves; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • standard math Existence and properties of shifted symplectic structures on derived Artin stacks as developed in prior literature.
    Invoked implicitly for the constructions on moduli spaces and the Hamiltonian property.
  • domain assumption Calabi-Yau varieties of dimension at least two admit moduli spaces of sheaves that can be rigidified while preserving symplectic data.
    Required for the central construction to apply.

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Reference graph

Works this paper leans on

49 extracted references · 49 canonical work pages

  1. [1]

    M. Anel, D. Calaque, Shifted symplectic reduction of derived critical loci, Adv. Theor. Math. Phys. 26 (2022), no. 6, 1543--1583

    work page 2022

  2. [2]

    Aranha, A

    D. Aranha, A. Khan, A. Latyntsev, H. Park, C. Ravi, Virtual localization revisited, Adv. Math. 479 (2025), 110434

    work page 2025

  3. [3]

    Amorin, O

    L. Amorin, O. Ben-Bassat, Perversely categorified Lagrangian correspondences, Adv. Theor. Math. Phys. 21 (2017), no. 2, 289–381

    work page 2017

  4. [4]

    Y. Bae, M. Kool, H. Park, Counting surfaces on Calabi-Yau 4 -folds I: foundations, to appear in Geom. Topol

    work page

  5. [5]

    Barwick, Spectral Mackey functors and equivariant algebraic K-theory (I), Adv

    C. Barwick, Spectral Mackey functors and equivariant algebraic K-theory (I), Adv. Math. 304 (2017), 646--727

    work page 2017

  6. [6]

    Ben-Bassat, Multiple derived Lagrangian intersections, In Stacks and categories in geometry, topology, and algebra, Vol

    O. Ben-Bassat, Multiple derived Lagrangian intersections, In Stacks and categories in geometry, topology, and algebra, Vol. 643. Contemp. Math. Amer. Math. Soc., Providence, RI, (2015), 119--126

    work page 2015

  7. [7]

    Ben-Zvi, J

    D. Ben-Zvi, J. Francis, D. Nadler, Integral transforms and Drinfeld centers in derived algebraic geometry, J. Amer. Math. Soc. 23 (2010), no. 4, 909--966

    work page 2010

  8. [8]

    Binda, M

    F. Binda, M. Porta, Descent problems for derived Azumaya algebras, arXiv:2107.03914v2

    work page arXiv

  9. [9]

    Borisov, D

    D. Borisov, D. Joyce, Virtual fundamental classes for moduli spaces of sheaves on Calabi-Yau four-folds, Geom. Topol. 21 (2017), no. 6, 3231--3311

    work page 2017

  10. [10]

    C. Brav, V. Bussi, D. Dupont, D. Joyce, B. Szendr\" o i, Symmetries and stabilization for sheaves of vanishing cycles, With an appendix by Jörg Schürmann, J. Singul. 11 , (2015), 85--151

    work page 2015

  11. [11]

    C. Brav, T. Dyckerhoff, Relative Calabi--Yau structures II : shifted Lagrangians in the moduli of objects, Selecta Math. (N.S.) 27 (2021), no. 4, Paper No. 63, 45 pp

    work page 2021

  12. [12]

    Buchweitz, H

    R.-O. Buchweitz, H. Flenner, A semiregularity map for modules and applications to deformations, Compositio Math. 137 (2003), no. 2, 135–210

    work page 2003

  13. [13]

    Calaque, Lagrangian structures on mapping stacks and semi-classical TFTs, In Stacks and categories in geometry, topology, and algebra, Contemp

    D. Calaque, Lagrangian structures on mapping stacks and semi-classical TFTs, In Stacks and categories in geometry, topology, and algebra, Contemp. Math., 643, American Mathematical Society, Providence, RI, (2015), 1--23

    work page 2015

  14. [14]

    Calaque, Shifted cotangent stacks are shifted symplectic, Ann

    D. Calaque, Shifted cotangent stacks are shifted symplectic, Ann. Fac. Sci. Toulouse Math. (6) 28 (2019), no. 1, 67--90

    work page 2019

  15. [15]

    Calaque, R

    D. Calaque, R. Haugseng, C. Scheimbauer, The AKSZ construction in derived algebraic geometry as an extended topological field theory, Mem. Amer. Math. Soc. 308 (2025), no. 1555, v+173 pp

    work page 2025

  16. [16]

    Calaque, T

    D. Calaque, T. Panted, B. T\" o en, M. Vaqui\' e , G. Vezzosi, Shifted Poisson structures and deformation quantization, J. Topol. 10 (2017), no. 2, 483--584

    work page 2017

  17. [17]

    Calaque, P

    D. Calaque, P. Safronov, Shifted cotangent bundles, symplectic groupoids and deformation to the normal cone, arXiv:2407.08622

    work page arXiv

  18. [18]

    Y. Cao, G. Oberdieck, Y. Toda, Gopakumar-Vafa type invariants of holomorphic symplectic 4-folds, Comm. Math. Phys. 405 (2024), no. 2, Paper No. 26, 79 pp

    work page 2024

  19. [19]

    Y. Cao, G. Oberdieck, Y. Toda, Stable pairs and Gopakumar-Vafa type invariants on holomorphic symplectic 4-folds, Adv. Math. 408 (2022) 108605, 44 pp

    work page 2022

  20. [20]

    Y. Cao, G. Zhao, Quasimaps to quivers with potentials, arXiv:2306.01302

    work page arXiv

  21. [21]

    Deligne, Th\'eor\`eme de Lefschetz et crit\`eres de d\'eg\'en\'erescence de suites spectrales, Publ

    P. Deligne, Th\'eor\`eme de Lefschetz et crit\`eres de d\'eg\'en\'erescence de suites spectrales, Publ. Math. IHES 35 (1968), 259--278

    work page 1968

  22. [22]

    Drinfeld, D

    V. Drinfeld, D. Gaitsgory, One some finiteness questions for algebraic stacks, Geom. Funct. Anal. 23 (2013), no. 1, 149--294

    work page 2013

  23. [23]

    Gross, D

    J. Gross, D. Joyce, Y. Tanaka, Universal structures in -linear enumerative invariant theories, SIGMA Symmetry Integrability Geom. Methods Appl. 18 (2022), Paper No. 068, 61 pp

    work page 2022

  24. [24]

    Grothendieck, On the de Rham cohomology of algebraic varieties, Inst

    A. Grothendieck, On the de Rham cohomology of algebraic varieties, Inst. Hautes Études Sci. Publ. Math. No. 29 (1966), 95–103

    work page 1966

  25. [25]

    Gwilliam, D

    O. Gwilliam, D. Pavlov, Enhancing the filtered derived category, J. Pure Appl. Algebra 222 (2018), no. 11, 3621–3674

    work page 2018

  26. [26]

    Haugseng, Iterated spans and classical topological field theories, Math

    R. Haugseng, Iterated spans and classical topological field theories, Math. Z. 289 (2018), no. 3-4, 1427–1488

    work page 2018

  27. [27]

    Haugseng, F

    R. Haugseng, F. Hebestreit, S. Linskens, J. Nuiten, Joost, Two-variable fibrations, factorisation systems and-categories of spans, Forum Math. Sigma 11 (2023), Paper No. e111, 70 pp

    work page 2023

  28. [28]

    Hoyois, S

    M. Hoyois, S. Scherotzke, N. Sibilla, Higher traces, noncommutative motives, and the categorified Chern character, Adv. Math. 309 (2017), 97--154

    work page 2017

  29. [29]

    Y.-H. Kiem, J. Li, Categorification of Donaldson-Thomas invariants via perverse sheaves, arXiv:1212.6444

    work page Pith review arXiv

  30. [30]

    Y.-H. Kiem, H. Park, Localizing virtual cycles for Donaldson--Thomas invariants of Calabi--Yau 4-folds, J. Algebraic Geom. 32 (2023), no. 4, 585–639

    work page 2023

  31. [31]

    Y.-H. Kiem, H. Park, Cosection localization via shifted symplectic geometry, arXiv:2504.19542

    work page arXiv

  32. [32]

    Kinjo, H

    T. Kinjo, H. Park, P. Safronov, Cohomological Hall algebras for 3-Calabi-Yau categories, arXiv:2406.12838

    work page arXiv

  33. [33]

    Laumon, L

    G. Laumon, L. Moret-Bailly, Champs algébriques, Ergeb. Math. Grenzgeb. (3), 39, Springer-Verlag, Berlin, 2000, xii+208 pp

    work page 2000

  34. [34]

    Lurie, Higher topos theory, Ann

    J. Lurie, Higher topos theory, Ann. of Math. Stud. 170, Princeton University Press, Princeton, NJ, 2009, xviii+925 pp

    work page 2009

  35. [35]

    Lurie, Higher algebra, 2017, url: https://www.math.ias.edu/ lurie/papers/HA.pdf

    J. Lurie, Higher algebra, 2017, url: https://www.math.ias.edu/ lurie/papers/HA.pdf

    work page 2017

  36. [36]

    Lurie, Spectral algebraic geometry, 2018, url: https://www.math.ias.edu/ lurie/papers/SAG-rootfile.pdf

    J. Lurie, Spectral algebraic geometry, 2018, url: https://www.math.ias.edu/ lurie/papers/SAG-rootfile.pdf

    work page 2018

  37. [37]

    Moulinos, The geometry of filtrations, Bull

    T. Moulinos, The geometry of filtrations, Bull. Lond. Math. Soc. 53 (2021), no. 5, 1486–1499

    work page 2021

  38. [38]

    J. Oh, R. P. Thomas, Counting sheaves on Calabi-Yau 4-folds, I, Duke Math. J. 172 (2023), no. 7, 1333–1409

    work page 2023

  39. [39]

    Pantev, B

    T. Pantev, B. T\" o en, M. Vaqui\' e , G. Vezzosi, Shifted symplectic structures, Publ. Math. Inst. Hautes Etudes Sci. 117 (2013), 271--328

    work page 2013

  40. [40]

    Lurie, Higher Algebra , 2017, visited on 01/30/2024, https://people.math

    H. Park, Virtual pullbacks in Donaldson-Thomas theory of Calabi-Yau 4 -folds, arXiv:2110.03631

    work page arXiv

  41. [41]

    Park, Shifted symplectic pushforwards, arXiv:2406.19192

    H. Park, Shifted symplectic pushforwards, arXiv:2406.19192

    work page arXiv

  42. [42]

    H. Park, J. You, An introduction to shifted symplectic structures, Moduli spaces, virtual invariants and shifted symplectic structures (2025), 37--64

    work page 2025

  43. [43]

    Safronov, Quasi-Hamiltonian reduction via classical Chern--Simons theory, Adv

    P. Safronov, Quasi-Hamiltonian reduction via classical Chern--Simons theory, Adv. Math. 287 (2016), 733--773

    work page 2016

  44. [44]

    u rg, B. T\

    T. Sch\" u rg, B. T\" o en, G. Vezzosi, Derived algebraic geometry, determinants of perfect complexes, and applications to obstruction theories for maps and complexes, J. Reine Angew. Math. 702 (2015), 1--40

    work page 2015

  45. [45]

    The Stacks project authors, The Stacks project, url: https://stacks.math.columbia.edu/

    work page

  46. [46]

    T\" o en, G

    B. T\" o en, G. Vezzosi, Homotopical algebraic geometry II : geometric stacks and applications, Mem. Amer. Math. Soc. 193 (2008), no. 902, x+224 pp

    work page 2008

  47. [47]

    T\" o en, G

    B. T\" o en, G. Vezzosi, Caractères de Chern, traces équivariantes et géométrie algébrique dérivée, Selecta Math. (N.S.) 21 (2015), no. 2, 449–554

    work page 2015

  48. [48]

    Voisin, Hodge theory and complex algebraic geometry

    C. Voisin, Hodge theory and complex algebraic geometry. II, Translated from the French by Leila Schneps. Cambridge Stud. Adv. Math., 77. Cambridge University Press, Cambridge, 2003. x+351 pp

    work page 2003

  49. [49]

    You, Reduced pairs/sheaves correspondences, in preparation

    J. You, Reduced pairs/sheaves correspondences, in preparation

    work page