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).

Key words： Nambu-Poisson structure; n-symplectic structure; (n−1, k)-Dirac structure; integrability

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