pith. sign in

arxiv: 2302.11420 · v3 · submitted 2023-02-22 · 🧮 math-ph · math.MP· math.QA

Higher Courant-Dorfman algebras and associated higher Poisson vertex algebras

Ryo Hayami This is my paper

Pith reviewed 2026-05-24 09:30 UTC · model grok-4.3

classification 🧮 math-ph math.MPmath.QA
keywords higher Courant-Dorfman algebrashigher Poisson vertex algebrasgraded symplectic geometrydg symplectic manifoldshigher Lie conformal algebrasBFV current algebrasgraded Poisson algebras
0
0 comments X

The pith

Higher Courant-Dorfman algebras from finite-dimensional vector bundles coincide with the algebras of functions on associated dg symplectic manifolds of degree n.

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

The paper defines higher Courant-Dorfman algebras to generalize the classical link between Courant-Dorfman algebras and Poisson vertex algebras. It connects these structures to graded symplectic geometry, yielding graded Poisson algebras of degree -n when non-degenerate. For algebras arising from finite-dimensional vector bundles, the definition produces exact coincidence with the function algebras on the corresponding dg symplectic manifolds of degree n. The work further introduces higher Lie conformal algebras and higher Poisson vertex algebras, shows that the latter generate higher weak Courant-Dorfman structures, and verifies that both satisfy properties parallel to their classical versions, including an algebraic treatment of BFV current algebras.

Core claim

We define a higher Courant-Dorfman algebra and study its relationship with graded symplectic geometry. In particular, we give graded Poisson algebras of degree -n in the non-degenerate case. For higher Courant-Dorfman algebras coming from finite-dimensional vector bundles, they coincide with the algebras of functions of the associated differential-graded symplectic manifolds of degree n. We define a higher Lie conformal algebra and Poisson vertex algebra, and give a higher (weak) Courant-Dorfman algebraic structure arising from them. Moreover, we prove that the higher Lie conformal algebras and higher Poisson vertex algebras have properties like Lie conformal algebras and Poisson vertex alge

What carries the argument

higher Courant-Dorfman algebra from finite-dimensional vector bundles, shown to coincide with the function algebra on the associated dg symplectic manifold of degree n

If this is right

  • In the non-degenerate case, higher Courant-Dorfman algebras produce graded Poisson algebras of degree -n.
  • Higher Lie conformal algebras induce higher weak Courant-Dorfman algebraic structures.
  • Higher Poisson vertex algebras obey algebraic properties parallel to those of classical Poisson vertex algebras.
  • The constructions supply an algebraic description of BFV current algebras.

Where Pith is reading between the lines

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

  • The vector-bundle construction may allow direct transfer of known results on ordinary Poisson vertex algebras to the higher setting.
  • The coincidence with dg symplectic manifolds suggests the framework could apply to other geometric categories beyond finite-dimensional bundles.
  • One could check whether the higher structures admit deformations or quantizations that preserve the graded symplectic data.
  • The BFV example indicates possible use in algebraic models of gauge theories or constrained systems.

Load-bearing premise

The proposed definitions of higher Courant-Dorfman algebra and higher Lie conformal algebra are the correct and consistent extensions that preserve the intended geometric and algebraic relations.

What would settle it

A finite-dimensional vector bundle whose constructed higher Courant-Dorfman algebra fails to equal the function algebra on its associated dg symplectic manifold of degree n, or a higher Poisson vertex algebra that violates one of the stated classical-style properties.

read the original abstract

In this paper, we consider a notion of a higher version of the relation between Courant-Dorfman algebras and Poisson vertex algebras. We define a higher Courant-Dorfman algebra, and study the relationship with graded symplectic geometry. In particular, we give graded Poisson algebras of degree $-n$ in the non-degenerate case. For higher Courant-Dorfman algebras coming from finite-dimensional vector bundles, they coincide with the algebras of functions of the associated differential-graded(dg) symplectic manifolds of degree $n$. We define a higher Lie conformal algebra and Poisson vertex algebra, and give a higher (weak) Courant-Dorfman algebraic structure arising from them. Moreover, we prove that the higher Lie conformal algebras and higher Poisson vertex algebras have properties like Lie conformal algebras and Poisson vertex algebras. As an example, we obtain an algebraic description of Batalin-Fradkin-Vilkovisky(BFV) current algebras.

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

0 major / 3 minor

Summary. The paper defines higher Courant-Dorfman algebras and relates them to graded symplectic geometry, establishing that non-degenerate cases yield graded Poisson algebras of degree -n and that those arising from finite-dimensional vector bundles coincide with the algebras of functions on the associated dg symplectic manifolds of degree n. It further defines higher Lie conformal algebras and higher Poisson vertex algebras, proves that they satisfy properties analogous to their classical counterparts, derives a higher weak Courant-Dorfman structure from them, and illustrates the framework with the BFV current algebra example.

Significance. If the definitions are consistent and the stated proofs hold, the work supplies a coherent higher-degree extension of the Courant-Dorfman/Poisson-vertex correspondence together with a direct link to graded symplectic geometry. The explicit BFV example indicates immediate applicability to current algebras in gauge theory, while the finite-dimensional coincidence result furnishes a concrete geometric realization that strengthens the overall construction.

minor comments (3)
  1. [Abstract] The abstract states the coincidence result for finite-dimensional vector bundles but does not indicate whether the proof relies on any additional non-degeneracy or finite-dimensionality assumptions that should be flagged for the reader.
  2. Notation for the higher-degree brackets and anchors is introduced without an explicit side-by-side comparison to the classical (degree-1) Courant-Dorfman case; a short table or remark would improve readability.
  3. The manuscript refers to 'the expected analogous properties' for higher Lie conformal and Poisson vertex algebras; a brief sentence recalling the precise classical identities being generalized would help anchor the claims.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the clear summary of its contributions, and the recommendation of minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity; work is definitional with independent geometric relations

full rationale

The paper defines higher Courant-Dorfman algebras and higher Lie conformal/Poisson vertex algebras, then derives their relations to graded symplectic geometry and function algebras on dg manifolds for the finite-dimensional case. These are presented as theorems following from the definitions and standard graded geometry, with no equations or claims reducing by construction to fitted inputs, self-citations as load-bearing premises, or renaming of known results. The coincidence statement is a derived identification rather than a tautology, and the BFV example is an application. No load-bearing step matches the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard graded differential geometry and the classical theory of Courant-Dorfman and Poisson vertex algebras; no free parameters or invented entities are mentioned.

axioms (2)
  • standard math Standard axioms of graded symplectic geometry and dg manifolds of degree n.
    Invoked to identify the higher Courant-Dorfman algebra with the function algebra on the associated manifold.
  • domain assumption Classical properties of Lie conformal algebras and Poisson vertex algebras extend to the higher graded setting.
    Used when proving that the new higher objects satisfy analogous identities.

pith-pipeline@v0.9.0 · 5688 in / 1197 out tokens · 21230 ms · 2026-05-24T09:30:42.051390+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages · 13 internal anchors

  1. [1]

    Ted Courant.: Dirac manifolds , Transactions of the American Mathematical Society Vol. 319, No. 2 (Jun., 1990), pp. 631-661

    work page 1990

  2. [2]

    Marco Gualtieri.: Generalized complex geometry and T-duality ,arxiv:0401221

    work page

  3. [3]

    V. G. Drinfel’d.: Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of classical Yang-Baxter equations Dokl. Akad. Nauk SSSR 268.2 (1983), pp. 285–287

    work page 1983

  4. [4]

    Gil R. Cavalcanti, Marco Gualtieri.: Generalized complex geometry ,A Celebration of the Mathematical Legacy of Raoul Bott (CRM Proceed- ings and Lecture Notes), American Mathematical Society, 2010, pp. 341- 366.arXiv:1106.1747

    work page arXiv 2010

  5. [5]

    Dmitry Roytenberg.: AKSZ-BV Formalism and Courant Algebroid-induced Topological Field Theories,Lett.Math.Phys.79:143-159,2007,arxiv:0608150

    work page 2007

  6. [6]

    Supergravity as Generalised Geometry I: Type II Theories

    A. Coimbra, C. Strickland-Constable and D. Waldram,: Supergravity as Gen- eralised Geometry I: Type II Theories JHEP 1111 (2011) 091,arXiv:1107.1733

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  7. [7]

    Izu Vaisman.: Towards a double field theory on para-Hermitian manifolds ,J. Math. Phys. 54, 123507 (2013) arXiv:1209.0152

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  8. [8]

    Dmitry Roytenberg.: Courant-Dorfman algebras and their cohomology ,Lett. Math. Phys. (2009) 90:311-351, arXiv:0902.4862

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  9. [9]

    Math., Vol

    Dmitry Roytenberg.: On the structure of graded symplectic supermanifolds and Courant algebroids , Contemp. Math., Vol. 315, Amer. Math. Soc., Prov- idence, RI, 2002, arxiv:0203110

    work page 2002

  10. [10]

    Zhang-Ju Liu, Alan Weinstein, Ping Xu.: Manin Triples for Lie Bialge- broids,J. Diff. Geom., 45:547-574 arxiv:9508013

    work page

  11. [11]

    V. G. Knizhnik and A. B. Zamolodchikov.: Current algebra and Wess-Zumino model in two dimensions , Nucl.Phys.B 247 (1984) 83-103

    work page 1984

  12. [12]

    Kac; Vertex algebras for beginners, Lecture Ser., vol 10, AMS, 1996

    Victor G. Kac; Vertex algebras for beginners, Lecture Ser., vol 10, AMS, 1996. Second edition, 1998

    work page 1996

  13. [13]

    Victor Kac.: Introduction to vertex algebras, Poisson vertex algebras, and integrable Hamiltonian PDE , arXiv:1512.00821 38

    work page internal anchor Pith review Pith/arXiv arXiv

  14. [14]

    Kac: Supersymmetric vertex alge- bras,Commun.Math.Phys.271:103-178,2007 arxiv:0603633

    Reimundo Heluani, Victor G. Kac: Supersymmetric vertex alge- bras,Commun.Math.Phys.271:103-178,2007 arxiv:0603633

    work page 2007

  15. [15]

    Poisson vertex algebras in the theory of Hamiltonian equations

    Aliaa Barakat, Alberto De Sole, Victor G. Kac.: Poisson vertex algebras in the theory of Hamiltonian equations ,Jpn. J. Math. 4 (2009), no. 2, 141-252 arXiv:0907.1275

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  16. [16]

    Pedram Hekmati, Varghese Mathai.: T-duality of current algebras and their quantization,Contemporary Mathematics, 584 (2012) 17-38 arXiv:1203.1709

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  17. [17]

    Luis ´Alvarez-C´ onsul, David Fern´ andez, Reimundo Heluani.:Noncommutative Poisson vertex algebras and Courant-Dorfman algebras , arXiv:2106.00270

    work page arXiv

  18. [18]

    Giulio Bonelli, Maxim Zabzine.: From current algebras for p-branes to topo- logical M-theory,JHEP 0509 (2005) 015 arxiv:0507051

    work page 2005

  19. [19]

    Machiko Hatsuda, Tetsuji Kimura.: Canonical approach to Courant brackets for D-branes,Journal of High Energy Physics 34 (2012) arXiv:1203.5499

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  20. [20]

    Cattaneo, Florian Schaetz.: Introduction to supergeometry , arXiv:1011.3401

    Alberto S. Cattaneo, Florian Schaetz.: Introduction to supergeometry , arXiv:1011.3401

    work page arXiv

  21. [21]

    Jian Qiu, Maxim Zabzine.: Introduction to Graded Geometry, Batalin- Vilkovisky Formalism and their Applications , arXiv:1105.2680

    work page internal anchor Pith review Pith/arXiv arXiv

  22. [22]

    I. A. Batalin, G. A. Vilkovisky,: Gauge algebra and quantization , Phys. Lett., 102B:27, 1981

    work page 1981

  23. [23]

    I. A. Batalin and G. A. Vilkovisky,: Relativistic S-matrix of dynamical systems with boson and fermion constraints , Phys. Lett. B 69 (1977), 309–312

    work page 1977

  24. [24]

    I. A. Batalin and E. S. Fradkin,: A generalized canonical formalism and quan- tization of reducible gauge theories , Phys. Lett. B 122, 157–164 (1983)

    work page 1983

  25. [25]

    Anton Alekseev, Thomas Strobl.: Current Algebras and Differential Geome- try,JHEP 0503 (2005) 035 arxiv:0410183

    work page 2005

  26. [26]

    Joel Ekstrand, Maxim Zabzine.: Courant-like brackets and loop spaces ,JHEP 1103:074,2011 arXiv:0903.3215

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  27. [27]

    Frank Keller, Stefan Waldmann.: Deformation theory of Courant alge- broids via the Rothstein algebra ,Journal of Pure and Applied Algebra 219(8) arXiv:0807.0584

    work page internal anchor Pith review Pith/arXiv arXiv

  28. [28]

    On the category of Lie n-algebroids

    G. Bonavolonta, N. Poncin; On the category of Lie n-algebroids , J. Geom. Phys., 73 (2013), 70-90 arXiv:1207.3590

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  29. [29]

    Pure and Appl

    Theodore Voronov.: Higher derived brackets and homotopy algebras ,J. Pure and Appl. Algebra. 202 (2005), Issues 1-3, 1 November 2005, 133-153 arxiv:0304038 39

    work page 2005

  30. [30]

    Joel Ekstrand: Going round in circles From sigma models to vertex algebras and back , Doctoral thesis,2011

    work page 2011

  31. [31]

    Noriaki Ikeda, Kozo Koizumi.: Current Algebras and QP Mani- folds,International Journal of Geometric Methods in Modern Physics 10(6) arXiv:1108.0473

    work page internal anchor Pith review Pith/arXiv arXiv

  32. [32]

    Noriaki Ikeda, Xiaomeng Xu.: Current Algebras from DG Symplectic Pairs in Supergeometry, arXiv:1308.0100

    work page arXiv

  33. [33]

    Topological Membranes, Current Algebras and H-flux - R-flux Duality based on Courant Algebroids

    Taiki Bessho, Marc A. Heller, Noriaki Ikeda, Satoshi Watamura: Topological Membranes, Current Algebras and H-flux - R-flux Duality based on Courant Algebroids, J. High Energ. Phys. 2016, 170 (2016). arXiv:1511.03425

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  34. [34]

    Arvanitakis.: Brane current algebras and generalised geometry from QP manifolds , J

    Alex S. Arvanitakis.: Brane current algebras and generalised geometry from QP manifolds , J. High Energ. Phys. 2021, 114 (2021). arXiv:2103.08608

    work page arXiv 2021

  35. [35]

    Grigoriev, A.M

    M.A. Grigoriev, A.M. Semikhatov, I.Yu. Tipunin.: BRST Formalism and Zero Locus Reduction,J.Math.Phys. 42 (2001) 3315-3333 arxiv:0001081

    work page 2001

  36. [36]

    D52 (1995) 7146-7160 arxiv:9505012

    A.Yu.Alekseev, P.Schaller, T.Strobl.: The Topological G/G WZW Model in the Generalized Momentum Representation ,Phys.Rev. D52 (1995) 7146-7160 arxiv:9505012

    work page 1995

  37. [37]

    Noriaki Ikeda.: Lectures on AKSZ Sigma Models for Physi- cistsNoncommutative Geometry and Physics 4, Workshop on Strings, Membranes and Topological Field Theory: 79-169, 2017, World scientific, Singapore, arXiv:1204.3714 40

    work page arXiv 2017