Higher order Dirac structure and Nambu-Poisson geometry

Abstract

This paper studies the properties of Nambu-Poisson geometry from the (n−1, k)-Dirac structure on a smooth manifold M. Firstly, we examine the automorphism group and infinitesimal on higher order Courant algebroid, to prove the integrability of infinitesimal Courant automorphism. Under the transversal smooth morphism \varphi :N\to M and anchor mapping of M on (n−1, k)-Dirac structure, it’s holds that the pullback (n−1, k)-Dirac structure on M turns out an (n−1, k)-Dirac structure on N. Then, given that the graph of Nambu-Poisson structure takes the form of (n−1, n−2)-Dirac structure, it follows that the single parameter variety of Nambu-Poisson structure is related to one variety closed n-symplectic form under gauge transformation. When \varphi :N\to Mis taken as the immersion mapping of (n−1)-cosymplectic submanifold, the pullback Nambu-Poisson structure on M turns out the Nambu-Poisson structure on N. Finally, we discuss the (n−1, 0)-Dirac structure on M can be integrated into a problem of (n−1)-presymplectic groupoid. Under the mapping \mathrm{\Pi }: M\to M/H, the corresponding (n−1, 0)-Dirac structure is F and E respectively. If E can be integrated into (n−1)-presymplectic groupoid (\mathfrak{g},\mathrm{\Omega }), then there exists the only \overline{\omega }, such that the corresponding integral of F is (n−1)-presymplectic groupoid \left(\overline{\mathfrak{g},}\overline{\omega }\right).

Keywords

Nambu-Poisson structure / n-symplectic structure / (n−1, k)-Dirac structure / integrability

References

[1] AlbaN M. On higher Poisson and higher Dirac structures. Ph D Thesis, Rio de Janeiro: Institute Nacional de Matematica Pura e Aplicada, 2015

[2] Álvarez D. Integrability of quotients in Poisson and Dirac geometry. Pacific J Math 2021; 311(1): 1–32

[3] Bi Y H. Sheng Y H. On higher analogues of Courant algebroids. Sci China Math 2011; 54(3): 437447

[4] Bi Y H. Sheng Y H. Dirac structures for higher analogues of Courant algebroids. Int J Geom Methods Mod Phys 2015; 12(1): 1550010

[5] Bi Y H, Vitagliano L, Zhang T. Higher omni-Lie algebroids. J Lie Theory 2019; 29(3): 881–899

[6] BursztynHCabreraAIglesias-PonteD. Multisymplectic geometry and Lie groupoids. In: Geometry, Mechanics, and Dynamics, Fields Inst Commun, Vol 73. New York: Springer, 2015, 57–73

[7] BursztynHIglesias-PonteDLuJ H. Dirac geometry and integration of Poisson homogeneous spaces. 2021, arXiv: 1905.11453

[8] Bursztyn H, Martinez Alba N, Rubio R. On higher Dirac structures. Int Math Res Not IMRN 2019; 2019(5): 1503–1542

[9] Cantrijn F, Ibort A, de Leon M. On the geometry of multisymplectic manifolds. J Austral Math Soc Ser A 1999; 66(3): 303–330

[10] Courant T J. Dirac manifolds. Trans Amer Math Soc 1990; 319(2): 631–661

[11] CourantT JWeinsteinA. Beyond Poisson structures. In: Troisième Théorème de Lie, Travaux en Cours, Vol 27. Paris: Hermann, 1988, 39–49 (in French)

[12] Gualtieri M. Generalized complex geometry. Ann of Math (2) 2011; 174(1): 75–123

[13] Hitchin N. Generalized Calabi-Yau manifolds. Q J Math 2003; 54(3): 281–308

[14] Ibáñez R, de León M, Lopez B, Marrero J C, Padrón E. Duality and modular class of a Nambu-Poisson structure. J Phys A 2001; 34(17): 3623–3650

[15] Ibáñez R, de León M, Marrero J C, Martín de Diego D. Dynamics of generalized Poisson and Nambu-Poisson brackets. J Math Phys 1997; 38(5): 2332–2344

[16] Lang H L, Sheng Y H. Linearization of the higher analogue of Courant algebroids. J Geom Meeh 2020; 12(4): 585–606

[17] Liu Z J, Weinstein A, Xu P. Manin triples for Lie bialgebroids. J Differential Geom 1997; 45(3): 547–574

[18] Meinrenken E. Poisson geometry from a Dirac perspective. Lett Math Phys 2018; 10(3): 447–498

[19] Nakanishi N. On Nambu-Poisson manifolds. Rev Math Phys 1998; 10(4): 499–510

[20] Nambu Y. Generalized Hamiltonian dynamics. Phys Rev D (3) 1973; 7: 2405–2412

[21] RoytenbergD. Courant algebroids, derived brackets and even symplectic supermanifolds. Ph D Thesis, Berkeley: University of California, 1999

[22] Sheng Y H, Zhu C C. Poisson geometry and Lie n-algebras. Sci Sin Math 2017; 47: 1717–1734

[23] Ševera P, Weinstein A. Poisson geometry with a 3-form background. Progr Theoret Phys Suppl 2001; 2001(144): 145–154

[24] Takhtajan L. On foundation of the generalized Nambu mechanics. Comm Math Phys 1994; 160(2): 295–315

[25] Vaisman I. A survey on Nambu-Poisson brackets. Acta Math Univ Comenian (N S) 1999; 68(2): 213–241

[26] Vallejo J A. Nambu-Poisson manifolds and associated n-ary Lie algebroids. J Phys A 2001; 34(13): 2867–2881

[27] Zambon M. L_∞-algebras and higher analogues of Dirac structures and Courant algebroids. J Symplectic Geom 2012; 10(4): 563–599

RIGHTS & PERMISSIONS

Higher Education Press 2024