Shifted lagrangian structures in Poisson geometry

Daniel \'Alvarez; Henrique Bursztyn; Miquel Cueca

Shifted lagrangian structures in Poisson geometry

T0 review · 0 major / 2 minor · reviewed 2026-06-29 · grok-4.3

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Pith's one-line read Lagrangian morphisms into 2-shifted symplectic groups correspond to Dirac structures in transitive Courant algebroids that are products of an exact Courant algebroid and a quadratic Lie algebra.

desk verdict The paper gives a Lie-type correspondence between Lagrangian morphisms into 2-shifted symplectic groups and Dirac structures in specific transitive Courant algebroids, then uses it to define multiplicative D-valued moment maps for integrating quasi-Poisson manifolds. read the letter →

arxiv 2605.29117 v1 pith:OO3SDUOO submitted 2026-05-27 math.SG math.DG

Daniel Álvarez , Henrique Bursztyn , Miquel Cueca This is my paper · ORCID

classification math.SGmath.DG

keywords shiftedsymplecticgeometryPoissonDiracstructuresCourantalgebroidsquasi-PoissonmanifoldsLagrangianmorphismsmomentmapsgroupoids

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes a correspondence between Lagrangian morphisms into 2-shifted symplectic groups and Dirac structures in transitive Courant algebroids formed as the product of an exact Courant algebroid and a quadratic Lie algebra. This correspondence identifies multiplicative D-valued moment maps as the global objects integrating quasi-Poisson manifolds. It extends the classical integration of Poisson manifolds by symplectic groupoids and the lifting of Poisson actions to multiplicative Hamiltonian actions. The work also gives constructions of quasi-symplectic groupoids as fibred products of 2-shifted Lagrangians, placing known examples such as Poisson homogeneous spaces into a single framework.

What carries the argument

Lagrangian morphisms into 2-shifted symplectic groups, which correspond to Dirac structures in product Courant algebroids and enable multiplicative D-valued moment maps for quasi-Poisson manifolds.

What would settle it

A 2-shifted Lagrangian morphism with no corresponding Dirac structure in any such product Courant algebroid, or a quasi-Poisson manifold that admits no multiplicative D-valued moment map.

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Extended reading notes

Core claim

The paper establishes a Lie-type correspondence between Lagrangian morphisms into 2-shifted symplectic groups and Dirac structures in transitive Courant algebroids that arise as the product of an exact Courant algebroid and a quadratic Lie algebra. This yields multiplicative D-valued moment maps as the global objects integrating quasi-Poisson manifolds, extending the integration of Poisson manifolds via symplectic groupoids and the passage from Poisson actions to multiplicative Hamiltonian actions.

Load-bearing premise

The relevant Courant algebroids must be transitive and arise exactly as the product of an exact Courant algebroid and a quadratic Lie algebra.

Editorial extensions

If this is right

Multiplicative D-valued moment maps integrate quasi-Poisson manifolds globally.
Quasi-symplectic groupoids arise systematically from fibred products of 2-shifted Lagrangians.
The integration of Poisson manifolds by symplectic groupoids extends to the quasi-Poisson setting.
Poisson actions lift to multiplicative Hamiltonian actions on the integrating objects.
Integrations of Poisson homogeneous spaces and Poisson quotients fit inside the same fibred-product construction.

Reading between the lines

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

The fibred-product method may generate previously unknown quasi-symplectic groupoids from simple building blocks.
Viewing the correspondence categorically could link it to higher symplectic structures in other degrees.
The framework suggests a uniform way to reduce quasi-Poisson data by taking quotients inside the shifted setting.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

The main new pieces are the explicit correspondence and the definition of those multiplicative D-valued moment maps as the global integrators. This extends the classical symplectic groupoid integration of Poisson manifolds and the lifting of actions to the quasi-Poisson setting. The paper also shows how fibred products of 2-shifted Lagrangians produce quasi-symplectic groupoids, which puts known constructions for Poisson homogeneous spaces and quotients into one framework and produces some new examples.

It stays within a clear scope: the Courant algebroids have to be transitive and exactly the product of an exact Courant algebroid with a quadratic Lie algebra. That assumption is stated, so the claims are not overreached, but it does mean the results do not immediately apply to every quasi-Poisson situation. No circularity or fitting issues show up in the abstract or stress-test note.

The work is aimed at people already comfortable with shifted symplectic geometry and Poisson integration. A reader who knows the standard results on symplectic groupoids and moment maps will see the extensions clearly. It is worth sending to referees because the new objects and the fibred-product constructions are concrete enough to be checked and potentially used by others in the subfield.

Referee Report

0 major / 2 minor

Summary. The paper develops the interplay between shifted symplectic geometry and classical Poisson geometry by focusing on Lagrangian morphisms into 2-shifted symplectic groups. It establishes a Lie-type correspondence between such morphisms and Dirac structures in transitive Courant algebroids arising as the product of an exact Courant algebroid and a quadratic Lie algebra. As a key application, it identifies multiplicative D-valued moment maps as the global objects integrating quasi-Poisson manifolds, extending the integration of Poisson manifolds via symplectic groupoids and the lifting of Poisson actions to multiplicative Hamiltonian actions. The paper also provides systematic constructions of quasi-symplectic groupoids via fibred products of 2-shifted Lagrangians, placing known constructions such as integrations of Poisson homogeneous spaces and Poisson quotients into a broader framework while yielding new examples.

Significance. If the correspondence and applications hold, the work provides a unified conceptual framework linking shifted symplectic structures to Poisson geometry, extending classical results on integration and reduction to the quasi-Poisson setting. The explicit scoping to transitive Courant algebroids of the specified product form strengthens the claims. The identification of multiplicative D-valued moment maps and the fibred-product constructions for quasi-symplectic groupoids represent concrete advances that could impact the study of moment maps, groupoid integrations, and reduction procedures in higher geometric contexts.

minor comments (2)

The abstract and introduction would benefit from a brief explicit statement of the precise definition of '2-shifted symplectic group' used throughout, to aid readers unfamiliar with the shifted symplectic literature.
Notation for the quadratic Lie algebra and the exact Courant algebroid in the product construction could be standardized earlier in the text for consistency across sections.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, assessment of significance, and recommendation to accept the manuscript.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation self-contained from standard geometric definitions

full rationale

The paper establishes a Lie-type correspondence between Lagrangian morphisms into 2-shifted symplectic groups and Dirac structures in transitive Courant algebroids (products of exact Courant algebroids and quadratic Lie algebras), then applies this to identify multiplicative D-valued moment maps for quasi-Poisson manifolds. This is presented as a direct consequence of the interplay between shifted symplectic geometry and classical Poisson/Courant structures, with constructions via fibred products extending classical reduction. No self-definitional loops, fitted inputs renamed as predictions, load-bearing self-citations, uniqueness theorems imported from the authors' prior work, ansatzes smuggled via citation, or renamings of known results appear in the provided abstract, claims, or scoped statements. The central correspondence is scoped explicitly to the given structural premises and derives from standard definitions without reducing to its inputs by construction. The result is therefore self-contained against external geometric benchmarks.

Assumptions & free parameters 0 free parameters · 1 assumptions · 1 invented entities

The paper relies on background structures from Poisson geometry and shifted symplectic geometry without introducing fitted numerical parameters; new objects are introduced by definition rather than as unexplained entities.

assumptions (1)

domain assumption Standard axioms and properties of Courant algebroids, Dirac structures, and 2-shifted symplectic groups hold as established in the prior literature.
The correspondence and applications are built directly on these background geometric structures.

invented entities (1)

multiplicative D-valued moment maps
purpose: Global objects that integrate quasi-Poisson manifolds, extending classical moment maps and symplectic groupoid integrations.
Newly defined in the paper as the key application of the correspondence.

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Cite this review

Pith. "Pith review of Shifted lagrangian structures in Poisson geometry." pith.science (2026). https://pith.science/paper/OO3SDUOO

@misc{pith2026260529117,
  author       = {Pith},
  title        = {Pith review of: Shifted lagrangian structures in Poisson geometry},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OO3SDUOO}},
  note         = {Machine review of arXiv:2605.29117}
}

read the original abstract

This paper develops new aspects of the interplay between shifted symplectic geometry and classical Poisson geometry, focusing on lagrangian morphisms into 2-shifted symplectic groups. We establish a Lie-type correspondence between such morphisms and Dirac structures in transitive Courant algebroids given by the product of an exact Courant algebroid and a quadratic Lie algebra. As a key application, we identify the global objects integrating quasi-Poisson manifolds, which we call multiplicative D-valued moment maps; this extends the integration of Poisson manifolds to symplectic groupoids and the lifting of Poisson actions to multiplicative hamiltonian actions. We devise systematic constructions of quasi-symplectic groupoids via fibred products of 2-shifted lagrangians, extending classical reduction procedures. This places known constructions, such as the integrations of Poisson homogeneous spaces and Poisson quotients, into a broader, conceptual framework, while yielding new examples.

Figures

Figures reproduced from arXiv: 2605.29117 by the authors.

**Figure 1.** Lie theory for 2-shifted isotropic structures direction, following [39], 2-shifted symplectic groups admit a Morita-equivalent infinite-dimensional model (parallel to the Morita equivalence between quasi-hamiltonian spaces and hamiltonian loopgroup spaces in [4, 97]); the infinite-dimensional counterparts of the lagrangian morphisms considered here deserve further study. We plan to pursue these directions in future… view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

The shifted symplectic geometry of derived higher groupoids
math.SG 2026-07 conditional novelty 7.0 of 10

Derived Lie n-groupoids carry shifted symplectic and lagrangian structures whose composition is well defined under transversality, yielding a unified derived symplectic reduction at critical values.
Yang-Mills theory for multiplicative Ehresmann connections
math.DG 2026-07 conditional novelty 7.0 of 10

Generalized Yang–Mills theory on Lie algebroids is defined via connection triples with 2- and 3-form curvature, with bundle-gerbe connections appearing as a special case.

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