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

1 Introduction

Dirac structure is a geometric structure introduced by Courant and Weinstein [10, 11] on differential manifold M, namely the maximally isotropic subbundle of the direct sum bundle TM\oplus T^{\ast }M, covering Poisson structure, presymplectic structure, complex structures, and foliation structure. The research on Dirac structure contributed to rapid development of Courant algebroid, higher order structure and generalized complex geometry [12, 13]. The Courant algebroid introduced by Liu et al. [17] comprised of a non-degenerate symmetric pairing, a bracket operation and an anchor mapping, to study the double of Lie bialgebroid (see [17]), with its equivalent definition presented by Roytenberg [21]. One important property of Courant algebroid lies in that its sectional space is a Lie 2-algebra, which establishes relevance between Poisson geometry and higher order structure, so as to study higher order Lie theory. Especially in the recent decade, there has been research on mathematical physics and Poisson geometry from the two directions of Dirac geometry and higher order structure. For example, reference [7] proved that the Poisson homogeneous space of any Poisson Lie group could be integrated with Dirac geometry; reference [2] studied the integrability of Poisson structure and Dirac geometry on quotient manifold; references [3, 25] presented the concept of higher order Courant algebroid on TM\oplus {\wedge }^{n-1}{T}^{\ast }M; reference [5] introduced the concept of n-omni Lie algebroid; reference [16] led in the concept of omnin-Lie algebroid; references [4, 8] primarily studied higher order Dirac structure, covering Nambu-Poisson structure (see [14]) and multisymplectic geometry structure [6]; reference [22] presented the relationship between Dirac structure of higher order Courant algebroid and Lie n-algebra.

Yoichiro [20] first proposed the Nambu-Poisson bracket on a smooth manifold M. Nambu-Poisson structure, given by Takhtajan [24], was used to study the problem of Nambu-Hamilton system. Reference [3] obtained that the n-vector field π is a Nambu-Poisson structure, and its graph would be closed if and only if under the higher order Courant bracket, establishing a one-to-one correspondence between the higher order Dirac structure and Nambu-Poisson structure. References [3, 6] introduced the n-presymplectic structure, and pointed out that if the graph of the n-presymplectic structure turns out a Dirac structure, then the graph of n-symplectic form ω would be closed under higher order Courant bracket. Reference [4] introduced the concept of (p, k)-Dirac structure on higher order Courant algebroid. Reference [7] demonstrated that on the principal bundle M\to M/H, if the Dirac structure on M/H contained H-compatible presymplectic groupoid, the presymplectic groupoid could be obtained after pulling back the Dirac structure to M. Through the correspondence between the Nambu-Poisson structure, n-symplectic structure and (n−1, k)-Dirac structure, this paper further investigates the relationship between the (n−1, k)-Dirac structure and the Nambu-Poisson geometry

Section 2 gives an overview of the basic knowledge and related concepts. Section 3 presents the automorphism group ({\omega }_t,{\mathrm{\Phi }}_t)\in {\mathrm{A}\mathrm{u}\mathrm{t}}_{\mathrm{C}\mathrm{A}}\left({\mathcal{T}}^{n-1}M\right) and automorphism Lie algebra ({\gamma }_t,X_t)\in {\mathrm{A}\mathrm{u}\mathrm{t}}_{\mathrm{C}\mathrm{A}}\left({\mathcal{T}}^{n-1}M\right)({\gamma }_t,X_t)\in {\mathrm{A}\mathrm{u}\mathrm{t}}_{\mathrm{C}\mathrm{A}}\left({\mathcal{T}}^{n-1}M\right) form of Courant algebroid on (n−1, k)-Dirac structure, hence proving the integrability of infinitesimal Courant automorphism. Then (n−1, k)-Dirac morphism is introduced to study the (n−1, k)-Dirac structure which can be pulled back under diffeomorphism. Section 4 demonstrates the graph of Nambu-Poisson structure is in an (n−1, n−2)-Dirac structure, and the single parameter variety gauge transformation of Nambu-Poisson structure is related to one variety closed n-symplectic structure {\omega }_t={\int }_0^t\left(\right({\mathrm{\Phi }}_s{)}_{\ast }{\gamma }_s)\mathrm{d}s. Section 5 studies the integrability of pullback (n−1, 0)-Dirac structure based on the correspondence between (n−1, 0)-Dirac structure and Lie algebroid. Section 5 is the summary.

2 Prerequisites

In this Section, we first of all review some basic knowledge and relevant concepts mentioned in this paper.

Definition 2.1 [15]　The Nambu-Poisson structure on manifold M is an n-linear mapping

\{\cdot ,\dots ,\cdot \}:C^{\mathrm{\infty }}\left(M\right)\times \cdots \times C^{\mathrm{\infty }}\left(M\right)\to C^{\mathrm{\infty }}\left(M\right),

which satisfies the following properties.

(1) Antisymmetry. For any f_i\in C^{\mathrm{\infty }}\left(M\right)(1\le i\le n), \sigma \in S_n(S_n is an n-order symmetric group),

\{f_1,\dots ,f_n\}=(-1{)}^{\epsilon \left(\sigma \right)}\left\{f_{\sigma \left(1\right)},\dots ,f_{\sigma \left(n\right)}\right\}.

(2) Leibniz property. For any f_i,g\in C^{\mathrm{\infty }}\left(M\right)(1\le i\le n),

\left\{f_{1}g,f_2,\dots ,f_n\right\}=f_1\left\{g,f_2,\dots ,f_n\right\}+g\left\{f_1,f_2,\dots ,f_n\right\}.

(3) Fundamental identity. For any f_i,g_j\in C^{\mathrm{\infty }}\left(M\right)(1\le i\le n-1,1\le j\le n),

\left\{f_1,\dots ,f_{n-1},\left\{g_1,\dots ,g_n\right\}\right\}=\sum _{j=1}^n \left\{g_1,\dots ,\left\{f_1,\dots ,f_{n-1},g_j\right\},\dots ,g_n\right\}.

If one Nambu-Poisson bracket meets one of the above properties, there exists a multivariate vector field \pi \in \mathrm{\Gamma }\left({\wedge }^{n}TM\right) satisfying the equation

\pi (\mathrm{d}{f}_1\wedge \cdots \wedge \mathrm{d}{f}_n)=\{f_1,\dots ,f_n\}.

If {\wedge }^{n-1}{T}^{\ast }M is the (n−1)-order differential form bundle, then \pi induces a vector bundle homomorphism

{\pi }^{\mathrm{♯}}:{\wedge }^{n-1}{T}^{\ast }M\to TM.

If the multivariate vector field satisfies L_{{\pi }^{\mathrm{♯}}(\mathrm{d}{f}_1\wedge \cdots \wedge \mathrm{d}{f}_n)}\pi =0, then it is called a Nambu-Poisson vector field on manifold M. (M, \pi) refers to n-order Nambu-Poisson manifold.

Definition 2.2 [8]　The higher order Poisson structure on manifold M contains a subbundle S\subseteq {\wedge }^k{T}^{\ast }M and bundle mapping \mathrm{\Lambda }:S\to TM, such that

(a) S^{\circ }=\left\{0\right\};

(b) i_{\mathrm{\Lambda }\left(\alpha \right)}\beta =-i_{\mathrm{\Lambda }\left(\beta \right)}\alpha ,\mathrm{\forall }\alpha ,\beta \in S;

(c) [\alpha ,\beta ]:={\mathcal{L}}_{\mathrm{\Lambda }\left(\alpha \right)}\beta -i_{\mathrm{\Lambda }\left(\beta \right)}\mathrm{d}\alpha ={\mathcal{L}}_{\mathrm{\Lambda }\left(\alpha \right)}\beta -{\mathcal{L}}_{\mathrm{\Lambda }\left(\beta \right)}\alpha -\mathrm{d}\left(i_{\mathrm{\Lambda }\left(\alpha \right)}\beta \right), and \mathrm{\Gamma }(S) is closed to the bracket, where \mathrm{\Lambda }:\mathrm{\Gamma }\left(S\right)\to \mathrm{\Gamma }\left(TM\right) maintains bracket operation. Then (M, S, A) is called k-order Poisson manifold.

Definition 2.3 [9]　The multisymplectic structure on manifold M is a closed and non-degenerate differential form \omega \in {\mathrm{\Omega }}^n\left(M\right), namely i_X\omega =0 if and only if X=0, equivalently,

{\omega }^b:TM\to {\wedge }^{n-1}{T}^{\ast }M,X\mapsto i_X\omega

is injective, and we call (M, ω) a multisymplectic manifold.

As is widely known, multisymplectic manifold is a special type of Nambu-Poisson manifold. Given a multisymplectic manifold (M, ω), define

i_{X_{f_1,\dots ,f_{n-1}}}\omega =\mathrm{d}{f}_1\wedge \cdots \wedge \mathrm{d}{f}_{n-1},

the corresponding Nambu-Poisson bracket is

\{f_1,\dots ,f_{n-1},g\}=⟨X_{f_1,\dots ,f_{n-1}},\mathrm{d}g⟩.

Definition 2.4 [26]　The n-Lie algebra is the bracket operation [\cdot ,\dots ,\cdot {]}_g:{\wedge }^{n}g\to g of a vector space g and an n-antisymmetry, such that any

[x_1,x_2,\dots ,x_{n-1},[y_1,y_2,\dots ,y_n]{]}_g=\sum _{i=1}^n [y_1,y_2,\dots ,[x_1,x_2,\dots ,x_{n-1},y_i],\dots ,y_n].

Definition 2.5 [26]　Assume M is a manifold, f:E\to M is a vector bundle, one n-Lie algebroid on manifold M is a quadruple (E,\rho ,[\cdot ,\dots ,\cdot ],M), where section \mathrm{\Gamma }(E) goes with n-bracket [\cdot ,\dots ,\cdot ] satisfying generalized Jacobi identity, \rho :{\wedge }^{n-1}E\to TM is bundle mapping, which induces Li algebra homomorphism between \mathrm{\Gamma }\left({\wedge }^{n-1}E\right) and \mathfrak{X}\left(M\right), namely for any f\in C^{\mathrm{\infty }}\left(M\right),X_1,\dots ,X_{n-1},Y\in \mathrm{\Gamma }\left(E\right), there is

\left[\rho \right(X_1\wedge \cdots \wedge X_{n-1}),\rho (Y_1\wedge \cdots \wedge Y_{n-1}\left)\right]=\sum _{i=1}^{n-1} \rho (Y_1\wedge \cdots [X_1,\dots ,X_{n-1},Y_i]\wedge \cdots \wedge Y_{n-1}),

\left[X_1,\dots ,X_{n-1},fY\right]=f\left[X_1,\dots ,X_{n-1},Y\right]+\rho \left(X_1\wedge \cdots \wedge X_{n-1}\right)\left(f\right)Y,

where ρ is anchor mapping.

If \pi is a Nambu-Poisson structure, then [\pi ,\pi ]=0, the coordinates of Nambu-Possion vector field can be shown as \pi =\frac{1}{n!}{\pi }_{i_1\cdots i_n}\left(x\right)\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^{i_1}}\cdots \frac{\mathrm{\partial }}{\mathrm{\partial }{x}^{i_n}}, where {\pi }_{i_1\dots i_n} is a smooth function.

Definition 2.6 [24]　The Nambu-Poissono tensor becomes linear point by point in coordinate representation, specifically as {\pi }_{i_1\cdots i_n}\left(x\right)={\pi }_{i_1\cdots i_{i_n}}^j{x}_j, which is called linear Nambu-Poisson tensor. Meanwhile, define a linear Nambu-Poisson structure:

\{x_{i_1},\dots ,x_{i_n}\}\left(x\right)=\sum _{j=1}^n {\pi }_{i_1\cdots i_{i_n}}^j{x}_j.

The following is to prove the one-to-one correspondence between n-Lie algebroid and linear Nambu-Poisson structure vector bundle. For section X\in \mathrm{\Gamma }\left(E\right), let {\varphi }_X\in C^{\mathrm{\infty }}\left(E^{\ast }\right) be the corresponding smooth function on dual bundle E^{\ast }.

Theorem 2.1　 For any n-Lie algebroid E endowed with n-bracket [\cdot ,\dots ,\cdot ], where E\to M is the vector bundle with n as the rank, and on the space of dual bundle, there exists the only Nambu-Poisson bracket \{\cdot ,\dots ,\cdot \} such that for any section X_1,\dots ,X_{n-1},Y\in \mathrm{\Gamma }\left(E\right), there is

(2.1)\{{\varphi }_{X_1},\dots ,{\varphi }_{X_{n-1}},{\varphi }_Y\}={\varphi }_{[X_1,\dots ,X_{n-1},Y]}.

The anchor mapping under the Nambu-Poisson bracket is

(2.2)p^{\ast }\left(\rho \right(X_1\wedge \cdots \wedge X_{n-1}\left)f\right)=\{{\varphi }_{X_1},\dots ,{\varphi }_{X_{n-1}},p^{\ast }f\},

for any f\in C^{\mathrm{\infty }}\left(M\right), \{{\rho }^{\ast }{f}_1,\dots ,{\rho }^{\ast }{f}_n\}=0, then the linear Nambu-Poisson structure on E^{\ast } is defined.

Conversely, the fiber linear Nambu-Poisson structure on the vector bundle obtains the n-Lie algebroid structure on the dual bundle.

Proof　Assume E\to M is n-Lie algebroid, select locally trivial bundle E{|}_U=U\times {\mathbb{R}}^n on the open set U\subseteq M, and assume the basis on section is {\epsilon }_1,\dots ,{\epsilon }_n. Assume x^1,\dots ,x^n are local coordinates on U. Then the differential of the smooth function on dual bundle y_i={\varphi }_{{\epsilon }_i}, where function x^j{y}_i={\varphi }_{x^j{\epsilon }_i} spans T^{\ast }\left(E^{\ast }\right), the differential of linear function {\varphi }_X spans a dual space, so there exists vector field \pi \in {\mathcal{X}}^n\left(E^{\ast }\right), such that

\pi (\mathrm{d}{\varphi }_{X_1},\dots ,\mathrm{d}{\varphi }_{X_{n-1}},\mathrm{d}{\varphi }_Y)={\varphi }_{[X_1,\dots ,X_{n-1},Y]}.

To prove the existence of \pi, we use the definition of n-Lie algebroid structure constant [{\epsilon }_{i_1},\dots ,{\epsilon }_{i_n}{]}_g=\sum _{k=1}^n {\pi }_{i_1\dots i_n}^k{\epsilon }_k, and assume \rho ({\epsilon }_{i_1}\wedge \cdots \wedge {\epsilon }_{i_{j-1}}\wedge \widehat{{\epsilon }_{i_j}}\wedge {\epsilon }_{i_{j+1}}\wedge \cdots \wedge {\epsilon }_{i_n})\in \mathfrak{X}\left(U\right). y_i is the corresponding coordinates on ({\mathbb{R}}^n{)}^{\ast }, then the coordinates of the vector field can be represented as:

which is the only vector field on E^{\ast }{|}_U=U\times ({\mathbb{R}}^n{)}^{\ast }.

Conversely, assume p:V\to M is a vector bundle, \pi is the corresponding linear Nambu-Poisson structure, and assume E=V^{\ast } is a dual bundle, define n-Lie bracket (2.1) and anchor mapping (2.2) on the sectional space. We can obtain that \rho (X_1\wedge \cdots \wedge X_{n-1}) is a vector field directly from the definition of n-Lie algebroid, and the mapping (X_1\wedge \cdots \wedge X_{n-1})\mapsto \rho (X_1\wedge \cdots \wedge X_{n-1}) is a smooth linear mapping. The Jacobi identity of n-bracket can be given by the Nambu-Poisson bracket, and the derivation property of Nambu-Poisson bracket can be inferred from the Leibnitz’s rule of anchor mapping:

\begin{array}{rl}{\varphi }_{[X_1,\dots ,X_{n-1},fY]}& =\{{\varphi }_{X_1},\dots ,{\varphi }_{X_{n-1}},{\varphi }_{fY}\}\\ & =\{{\varphi }_{X_1},\dots ,{\varphi }_{X_{n-1}},(p^{\ast }f\left){\varphi }_Y\right\}\\ & =\left(p^{\ast }f\right)\{{\varphi }_{X_1},\dots ,{\varphi }_{X_{n-1}},{\varphi }_Y\}+p^{\ast }\left(\rho \right(X_1\wedge \cdots \wedge X_{n-1}\left)f\right){\varphi }_Y.\end{array}

□

Definition 2.7　Let’s call the linear mapping variety {\varphi }_E:(E_2{)}_{{\varphi }_M\left(m\right)}\to (E_1{)}_m dependent on point of m satisfying the commuting graph

\begin{array}{c}{E}_1\stackrel{\varphi E}{⇢}{E}_2\\ \downarrow \downarrow \\ M_1\underset{\varphi M}{⟶}{M}_2\end{array},

the comorphism on the vector bundle, where {\varphi }_M:M_1\to M_2 is the basis mapping.

Next research is conducted on the direct sum bundle {\mathcal{T}}^{n-1}M=TM\oplus {\wedge }^{n-1}{T}^{\ast }M, and we define the value pairing (·,·) of non-degenerate {\wedge }^{n-2}{T}^{\ast }M on the section \mathrm{\Gamma }\left({\mathcal{T}}^{n-1}M\right) as follows:

(X+\alpha ,Y+\beta {)}_{+}=\frac{1}{2}(i_X\beta +i_Y\alpha ),\mathrm{\forall }X,Y\in \mathfrak{X}(M),\alpha ,\beta \in {\mathrm{\Omega }}^{n-1}(M).

Define bracket ⟦\cdot ,\cdot ⟧ on \mathrm{\Gamma }\left({\mathcal{T}}^{n-1}M\right) as below:

⟦X+\alpha ,Y+\beta ⟧=[X,Y]+L_X\beta -i_Y\mathrm{d}\alpha ,

where anchor mapping ρ is the projection of TM\oplus {\wedge }^{n-1}{T}^{\ast }M to TM, we call (TM\oplus {\wedge }^{n-1}{T}^{\ast }M,(\cdot ,\cdot {)}_{+},⟦·,·⟧,\rho ) a higher order Courant algebroid.

Definition 2.8 [4]　Let E be the subspace of V^{n-1}, W be the projection of E on a vector space. If 0\le k\le n-2,\mathrm{\forall }{e}_1,e_2\in E,u_1,\dots ,u_k\in W, there is

i_{u_k}\cdots i_{u_1}(e_1,e_2{)}_{+}=0,

we call E the (n−1, k)-isotropic subspace. If E=E_k^{\perp }:=\{v\in {\mathcal{V}}^{n-1}\mid i_{u_k}\cdots i_{u_1}(v,E{)}_{+}=0,\mathrm{\forall }{u}_1,\dots ,u_k\in W\}, and E is (n−1, k)-maximally isotropic, we call E the linear (n−1, k)-Dirac structure.

Reference [3] proved that for any Nambu-Poisson structure \pi \in {\mathcal{X}}^n\left(M\right) on the Nambu-Poisson manifold, the graph of mapping {\pi }^{\mathrm{♯}}:{\wedge }^{n-1}{T}^{\ast }M\to TM is an (n−1)-order Nambu-Poisson structure.

Proposition 2.1 [3, 18]　 For any {\sigma }_1,{\sigma }_2,{\sigma }_3\in \mathrm{\Gamma }\left({\mathcal{T}}^{n-1}M\right),f\in C^{\mathrm{\infty }}\left(M\right), there is

(1) ⟦{\sigma }_1,f{\sigma }_2⟧=f⟦{\sigma }_1,{\sigma }_2⟧+\rho \left({\sigma }_1\right)\left(f\right){\sigma }_2;

(2) L_{\rho \left({\sigma }_1\right)}({\sigma }_2,{\sigma }_3)=(⟦{\sigma }_1,{\sigma }_2⟧,{\sigma }_3)+({\sigma }_2,⟦{\sigma }_1,{\sigma }_3⟧);

(3) [[{\sigma }_1,⟦{\sigma }_2,{\sigma }_3⟧]]=[[⟦{\sigma }_1,{\sigma }_2⟧,{\sigma }_3]]+[[{\sigma }_2,⟦{\sigma }_1,{\sigma }_3⟧]];

(4) ⟦f{\sigma }_1,{\sigma }_2⟧=f⟦{\sigma }_1,{\sigma }_2⟧-\rho \left({\sigma }_2\right)\left(f\right){\sigma }_1+2({\sigma }_1,{\sigma }_2)\text{d}f.

3 Morphism of 2(n−1, k)-Dirac structure

Similar to the description of Poisson mapping in Dirac geometry, this section presents the morphism of (n−1, k)-Dirac structure, which can be induced to pull back through transversal conditions.

Natural induction of a generalized differential mapping by diffeomorphic mapping is defined by as follows:

{\mathcal{T}}_x^{n-1}\mathrm{\Phi }:={\mathrm{d}}_x\mathrm{\Phi }+{\wedge }^{n-1}\left(\right({\mathrm{d}}_x\mathrm{\Phi }{)}^{-1}{)}^{\ast }:{\mathcal{T}}_x^{n-1}M\to {\mathcal{T}}_{\mathrm{\Phi }\left(x\right)}^{n-1}M.

Induce a pullback mapping ({\mathcal{T}}^{n-1}\mathrm{\Phi }{)}^{\ast }:\mathrm{\Gamma }({\mathcal{T}}^{n-1}M)\to \mathrm{\Gamma }({\mathcal{T}}^{n-1}M) on {\mathcal{T}}^{n-1}\mathrm{\Phi }, then it follows

\left({\mathcal{T}}^{n-1}\mathrm{\Phi }{)}^{\ast }\right(X+\alpha )={\mathrm{\Phi }}^{\ast }(X)+{\mathrm{\Phi }}^{\ast }(\alpha ),\mathrm{\forall }X\in \mathfrak{X}(M),\alpha \in {\mathrm{\Omega }}^{n-1}(M).

Here, the pullback of vector field and differential form is defined as below:

{\mathrm{\Phi }}^{\ast }\left(X\right)=(\mathrm{d}\mathrm{\Phi }{)}^{-1}\circ X\circ \mathrm{\Phi },{\mathrm{\Phi }}^{\ast }(\alpha )=(\mathrm{d}\mathrm{\Phi }{)}^{\ast }\circ \alpha \circ \mathrm{\Phi }.

Since {\mathcal{T}}^{n-1}\mathrm{\Phi } maintains bracket operation (·,·) and ⟦·,·⟧, {\mathcal{T}}^{n-1}\mathrm{\Phi } can induce (n-1,k)-Dirac structure \left({\mathcal{T}}^{n-1}\mathrm{\Phi }\right)\left(E\right) out of (n-1,k)-Dirac structure on E\in {\mathcal{T}}^{n-1}M, hence the diffeomorphic Dirac structure.

Define {\mathrm{A}\mathrm{u}\mathrm{t}}_{\mathrm{C}\mathrm{A}}\left({\mathcal{T}}^{n-1}M\right) as an automorphism group of higher order Courant algebroid on {\mathcal{T}}^{n-1}M, \mathrm{\forall }A\in {\mathrm{A}\mathrm{u}\mathrm{t}}_{\mathrm{C}\mathrm{A}}\left({\mathcal{T}}^{n-1}M\right) maintains bracket operation and anchor mapping. Assume \mathrm{\Phi }:M\to M is a basis mapping, and induce {\mathrm{\Phi }}^{\ast }:{\wedge }^{n-1}{T}^{\ast }M\to {\wedge }^{n-2}{T}^{\ast }M, then

(A_{\sigma },A_{\tau })={\mathrm{\Phi }}^{\ast }(\sigma ,\tau ),[A_{\sigma },A_{\tau }]=A[\sigma ,\tau ],\rho \circ A=\mathrm{d}\mathrm{\Phi }\circ \rho .

Any diffeomorphism \mathrm{\Phi }\in \mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}\left(M\right) on M defines standard higher order Courant algebroid automorphism {\mathcal{T}}^{n-1}\mathrm{\Phi }\in {\mathrm{A}\mathrm{u}\mathrm{t}}_{CA}\left({\mathcal{T}}^{n-1}M\right), so group homomorphism {\mathrm{A}\mathrm{u}\mathrm{t}}_{\mathrm{C}\mathrm{A}}\left({\mathcal{T}}^{n-1}M\right)\to \mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}\left(M\right) is surjective, and the standard Courant algebroid automorphism {\mathcal{T}}^{n-1}\mathrm{\Phi } induces a short exact sequence

1\to {\mathrm{\Omega }}_{\mathrm{c}\mathrm{l}}^n\left(M\right)\to {\mathrm{A}\mathrm{u}\mathrm{t}}_{\mathrm{C}\mathrm{A}}\left({\mathcal{T}}^{n-1}M\right)\to \mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}\left(M\right)\to 1,

where {\mathrm{\Omega }}_{\mathrm{c}\mathrm{l}}^n\left(M\right) is a set comprised of all closed n-forms on M.

For any closed form, induce a normal transition transformation as the vector bundle isomorphism

{\mathcal{R}}_{\omega }:{\mathcal{T}}^{n-1}M\to {\mathcal{T}}^{n-1}M,X+\alpha \to X+\alpha +i_X\omega .

Lemma 3.1 [4]　 The automorphic mapping {\mathcal{R}}_{\omega }:{\mathcal{T}}^{n-1}M\to {\mathcal{T}}^{n-1}M maintains the inner product bracket (·,·) of higher order Courant algebroid, anchor mapping ρ and higher order Dorfman bracket ⟦·,·⟧ if and only if \mathrm{d}\omega =0.

Theorem 3.1　 The automorphism group of Courant algebroid on T^{n-1}M is a semidirect product {\mathrm{A}\mathrm{u}\mathrm{t}}_{\mathrm{C}\mathrm{A}}\left({\mathcal{T}}^{n-1}M\right)={\mathrm{\Omega }}_{\mathrm{c}\mathrm{l}}^n\left(M\right)⋊\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}\left(M\right).

Proof　Assume A\in {\mathrm{A}\mathrm{u}\mathrm{t}}_{\mathrm{C}\mathrm{A}}\left({\mathcal{T}}^{n-1}M\right), \mathrm{\Phi }\in \mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}\left(M\right) is a basis mapping, then A^{\prime }=A\circ {\mathcal{T}}^{n-1}{\mathrm{\Phi }}^{-1} has an identity mapping as the basis mapping. In particular, \rho \circ A^{\prime }=\rho, and for any v_i\in \mathrm{T}\mathrm{M}\in {\mathcal{T}}^{n-1}M, A^{\prime }{v}_i-v_i\in {\wedge }^{n-1}{T}^{\ast }M, \omega (v_1,\dots ,v_n)=⟨A^{\prime }\left(v_1\right),(v_2,\dots ,v_n)⟩, because A^{\prime } maintains the bracket, hence

\begin{array}{rl}0=& \left(A^{\prime }\left(v_1\right)-v_1,A^{\prime }\left(v_2\right)-v_2\right)=\left(A^{\prime }\left(v_1\right),A^{\prime }\left(v_2\right)\right)-⟨A^{\prime }\left(v_1\right),v_2⟩-⟨v_1,A^{\prime }\left(v_2\right)⟩\\ & +⟨v_1,v_2⟩,\omega (v_1,v_2)+\omega (v_2,v_1)=0.\end{array}

Therefore, ω is antisymmetric, which proves that A^{\prime }={\mathcal{R}}_{\omega }, but {\mathcal{R}}_{\omega } maintains the Courant bracket property if and only if ω is closed. It can be seen from the Courant bracket nature that

\begin{array}{rl}& A\left(⟦f{\sigma }_1,{\sigma }_2⟧\right)=fA⟦{\sigma }_1,{\sigma }_2⟧-\rho \left({\sigma }_2\right)\left(f\right)A\left({\sigma }_1\right)+2({\sigma }_1,{\sigma }_2)A(\mathrm{d}f),\\ & ⟦A\left(f{\sigma }_1\right),A\left({\sigma }_2\right)⟧=f⟦A\left({\sigma }_1\right),A\left({\sigma }_2\right)⟧-\rho \left(A\right({\sigma }_2\left)\right)\left(f\right)A\left({\sigma }_1\right)+2\left(A\right({\sigma }_1),A({\sigma }_2\left)\right)\mathrm{d}f.\end{array}

Since mapping A maintains the bracket, it holds from comparison with the above formula

-\rho \left({\sigma }_2\right)\left(f\right)A\left({\sigma }_1\right)+2({\sigma }_1,{\sigma }_2)A\left(\text{d}f\right)=-\rho \left(A\right({\sigma }_2\left)\right)\left(f\right)A\left({\sigma }_1\right)+2\left(A\right({\sigma }_1),A({\sigma }_2\left)\right)\text{d}f.

Assign {\sigma }_1=X,{\sigma }_2=Y, then ({\sigma }_1,{\sigma }_2)=0,\rho \left({\sigma }_2\right)\left(f\right)A\left({\sigma }_1\right)=\rho \left(A\right({\sigma }_2\left)\right)\left(f\right)A\left({\sigma }_1\right)\rightrightarrows Y=\rho \left(A\right(Y\left)\right). If {\sigma }_1=\alpha ,{\sigma }_2=Y, then A\left(\mathrm{d}f\right)=\mathrm{d}f. Based on the above, A(X+\alpha )=X+\alpha +N\left(X\right), where N(X) is an (n-1)-form. Assume N={\omega }^{\mathrm{♯}},A(X+\alpha )={\mathcal{R}}_{\omega }(X+\alpha )=X+\alpha +i_X\omega, with maintained bracket if and only if ω is a closed form.□

Standard ⟦·,·⟧ bracket can be deformed by a closed 3-form [12, 23], and we assign an (n+1)-form θ for higher order ⟦·,·⟧ bracket, which can be deformed into

{⟦X+\alpha ,Y+\beta ⟧}_{\theta }=[X,Y]+L_X\beta -i_Y\mathrm{d}\alpha +i_X{i}_Y\theta .

For arbitrarily smooth function Φ, there is ⟦{\mathcal{T}}^{n-1}\mathrm{\Phi }.(X+\alpha ),{\mathcal{T}}^{n-1}\mathrm{\Phi }. (Y+\beta ){⟧}_{\theta }={\mathcal{T}}^{n-1}\mathrm{\Phi }.\left({⟦X+\alpha ,Y+\beta ⟧}_{{\mathrm{\Phi }}^{\ast }\theta }\right). Given a closed n-form ω from Lemma 3.2, {⟦{\mathcal{R}}_{\omega }(X+\alpha ),{\mathcal{R}}_{\omega }(Y+\beta )⟧}_{\theta +\mathrm{d}\omega }={\mathcal{R}}_{\omega }{⟦X+\alpha ,Y+\beta ⟧}_{\theta }.

So we can obtain that {\mathrm{A}\mathrm{u}\mathrm{t}}_{\mathrm{C}\mathrm{A}}\left({\mathcal{T}}^{n-1}M\right)={\mathrm{\Omega }}_{\mathrm{c}\mathrm{l}}^n\left(M\right)⋊\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}\left(M\right).

In the similar way, we approach the the discussion of Lie algebra {\mathrm{A}\mathrm{u}\mathrm{t}}_{\mathrm{C}\mathrm{A}}\left({\mathcal{T}}^{n-1}M\right) of infinitesimal Courant algebroid automorphism. It can be regarded as an operator on the section, that is in the linear mapping D:\mathrm{\Gamma }\left({\mathcal{T}}^{n-1}M\right)\to \mathrm{\Gamma }\left({\mathcal{T}}^{n-1}M\right), there exists a vector field X satisfying the following properties:

\begin{array}{rl}D\left(f\sigma \right)& =fD\left(\sigma \right)+X\left(f\right)\sigma ,\\ (D\sigma ,\tau )+(\sigma ,D\tau )& =X(\sigma ,\tau ),\\ ⟦D\sigma ,\tau ⟧+⟦\sigma ,D\tau ⟧& =D⟦\sigma ,\tau ⟧,\\ \rho \left(D\tau \right)& =[X,\rho (\tau \left)\right].\end{array}

Theorem 3.2　(a) The Lie algebra of infinitesimal Courant automorphism is a semidirect product

{\mathrm{A}\mathrm{u}\mathrm{t}}_{\mathrm{C}\mathrm{A}}\left({\mathcal{T}}^{n-1}M\right)={\mathrm{\Omega }}_{\mathrm{c}\mathrm{l}}^n\left(M\right)⋊\mathfrak{X}\left(M\right),

where (γ, X) functions on the section \tau =Y+\beta \in \mathrm{\Gamma }\left({\mathcal{T}}^{n}M\right) as follows:

(\gamma ,X).\tau =[X,Y]+L_X\beta -i_Y\gamma .

(b) For any section \sigma =X+\alpha \in \mathrm{\Gamma }\left({\mathcal{T}}^{n-1}M\right), the role of ⟦\cdot ,\cdot ⟧ bracket is equivalent to infinitesimal automorphism (\mathrm{d}\alpha ,X).

Proof　Assign an injection {\mathrm{\Omega }}_{\mathrm{c}\mathrm{l}}^n\left(M\right)⋊\mathfrak{X}\left(M\right)\to {\mathrm{A}\mathrm{u}\mathrm{t}}_{\mathrm{C}\mathrm{A}}\left({\mathcal{T}}^{n-1}M\right) for (a). We prove that it is also surjective. Assume the corresponding basic vector field of D\in {\mathrm{A}\mathrm{u}\mathrm{t}}_{\mathrm{C}\mathrm{A}}\left({\mathcal{T}}^{n-1}M\right) is X,L_X\in {\mathrm{A}\mathrm{u}\mathrm{t}}_{\mathrm{C}\mathrm{A}}\left({\mathcal{T}}^{n-1}M\right), then the corresponding basic vector field of D^{\prime }=D-L_X is 0. Hence D^{\prime }\left(f\sigma \right)=f{D}^{\prime }\left(\sigma \right). Furthermore, \rho \circ D^{\prime }=0, so the value of automorphism is within {\wedge }^{n-1}{T}^{\ast }M. At the same time, D^{\prime }\circ \left({\rho }^{\ast }\left({\wedge }^{n-1}{T}^{\ast }M\right)\right)=0, so bundle mapping TM\to {\wedge }^{n-1}{T}^{\ast }M can be obtained. Since D^{\prime } maintains bracket operation,

(D^{\prime }X,Y)+(X,D^{\prime }Y)=0.

Therefore, there exists n-form γ, and we obtain D^{\prime }(X+\alpha )=-i_X\gamma. Finally, D^{\prime } maintains bracket if and only if γ is closed. The demonstration of (b) can be readily obtained from (a). □

The following studies the relationship between Courant automorphism group and automorphism Lie algebra.

Theorem 3.3　 Let ({\omega }_t,{\mathrm{\Phi }}_t)\in {\mathrm{A}\mathrm{u}\mathrm{t}}_{\mathrm{C}\mathrm{A}}\left({\mathcal{T}}^{n-1}M\right) be a time-dependent automorphism variety, corresponding to infinitesimal automorphism ({\gamma }_t,X_t)\in {\mathrm{A}\mathrm{u}\mathrm{t}}_{\mathrm{C}\mathrm{A}}\left({\mathcal{T}}^{n-1}M\right), then

{\omega }_t={\int }_0^t \left(\right({\mathrm{d}}_t\mathrm{\Phi }\left){\gamma }_s\right)\mathrm{d}s,

where {\mathrm{\Phi }}_t is the flow of X_t.

Proof　The flow of the time-dependent vector field works on the function as \frac{\mathrm{d}}{\mathrm{d}t}\left({\mathrm{d}}_t\mathrm{\Phi }\right)=\left({\mathrm{d}}_t\mathrm{\Phi }\right)\circ L_{X_t}, and we assume \tau =Y+\beta \in \mathrm{\Gamma }\left({\mathcal{T}}^{n-1}M\right),({\gamma }_t,X_t)\cdot \tau =L_{X_t}\tau -i_{P_{r_1}\left(\tau \right)}{\gamma }_t, where the differential of single parameter variety ({\omega }_t,{\mathrm{\Phi }}_t) of P_{r_1}\left(\tau \right)=Y,{\mathcal{T}}^{n-1}M automorphism is ({\gamma }_t,X_t), then ({\omega }_t,{\mathrm{\Phi }}_t)\cdot \tau =\left({\mathrm{d}}_t\mathrm{\Phi }\right)\tau -i\left(\right({\mathrm{d}}_t\mathrm{\Phi }\left)Y\right){\omega }_t, and we take the derivative of single parameter t at both sides as follows:

\frac{\mathrm{d}}{\mathrm{d}t}\left(\right({\omega }_t,{\mathrm{\Phi }}_t).\tau )=({\omega }_t,{\mathrm{\Phi }}_t).({\gamma }_t,X_t).\tau .

We obtain from left side of the equation

\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}t}\left(\right({\omega }_t,{\mathrm{\Phi }}_t),\tau )& =\frac{\mathrm{d}}{\mathrm{d}t}\left(\right({\mathrm{d}}_t\mathrm{\Phi })\tau -i(\left({\mathrm{d}}_t\mathrm{\Phi }\right)Y\left){\omega }_t\right)\\ & =({\mathrm{d}}_t\mathrm{\Phi })L_{X_t}\tau -i\left(\right({\mathrm{d}}_t\mathrm{\Phi }\left)L_{X_t}Y\right){\omega }_t-i\left(\right({\mathrm{d}}_t\mathrm{\Phi }\left)Y\right)\frac{\mathrm{d}{\omega }_t}{\mathrm{d}t}.\end{array}

We obtain from right side of the equation

\begin{array}{rl}({\omega }_t,{\mathrm{\Phi }}_t).({\gamma }_t,X_t).\tau & =({\omega }_t,{\mathrm{\Phi }}_t)(L_{X_t}\tau -i(Y\left){\gamma }_t\right)\\ & =\left({\mathrm{d}}_t\mathrm{\Phi }\right)(L_{X_t}\tau -i(Y\left){\gamma }_t\right)-i\left({\mathrm{d}}_t\mathrm{\Phi }\right(P_{r_1}\left[\right(L_{X_t}\tau -i\left(Y\right){\gamma }_t\left)\right]\left)\right){\omega }_t\\ & =\left({\mathrm{d}}_t\mathrm{\Phi }\right)L_{X_t}\tau -i\left(\right({\mathrm{d}}_t\mathrm{\Phi }\left)Y\right)\left({\mathrm{d}}_t\mathrm{\Phi }\right){\gamma }_t-i\left(\right({\mathrm{d}}_t\mathrm{\Phi }\left)L_{X_t}Y\right){\omega }_t.\end{array}

We get \left({\mathrm{d}}_t\mathrm{\Phi }\right){\gamma }_t=\frac{\mathrm{d}{\omega }_t}{\mathrm{d}t} by comparing the two. □

Definition 3.1 [1]　Assume \begin{array}{ccccc}\varphi & :& N& \to & M\end{array} is a smooth mapping, vector bundle morphism is \begin{array}{ccccc}\mathrm{d}\varphi & :& TN& \to & TM\end{array}, comorphism is {\wedge }^{n-1}{T}^{\ast }\varphi :{\wedge }^{n-1}{T}^{\ast }N⇢{\wedge }^{n-1}{T}^{\ast }M,F\in {\mathcal{T}}^{n-1}N,E\in {\mathcal{T}}^{n-1}M are (n−1, k)-Dirac structure on N and M, respectively, and define the pullback morphism of (n−1, k)-Dirac structure as follows:

F:=\left\{\right(X+{\wedge }^{n-1}{T}^{\ast }\varphi \left(\alpha \right))\in {\mathcal{T}}^{n-1}N:\text{d}\varphi X+\alpha \in {\mathcal{T}}^{n-1}M\}.

Define the push-forward morphism of (n−1, k)-Dirac structure as follows:

E:=\{\text{d}\varphi X+\alpha \in {\mathcal{T}}^{n-1}M:X+{\wedge }^{n-1}{T}^{\ast }\varphi (\alpha )\in {\mathcal{T}}^{n-1}N\}.

Its graph is the sum of tangent bundle and higher order cotangent bundle mapping, with the correlation defined as below.

(1) For x=v+\mu \in {\mathcal{T}}_{\varphi \left(p\right)}^{n-1}M,y=w+\nu \in {\mathcal{T}}_p^{n-1}N, if v=\left({\mathrm{d}}_p\varphi \right)w,\nu =\left({\wedge }^{n-1}{T}_p^{\ast }\varphi \right)\mu. We call y and x \varphi relevant, written as y{\sim }_{\varphi }x\iff (x,y)\in \mathrm{G}\mathrm{r}\left({\mathcal{T}}^{n-1}\varphi \right).

(2) For \sigma =X+\alpha \in \mathrm{\Gamma }\left({\mathcal{T}}^{n-1}M\right),\tau =Y+\beta \in \mathrm{\Gamma }\left({\mathcal{T}}^{n-1}N\right), if Y{\sim }_{\varphi }X,\beta ={\varphi }^{\ast }\alpha, then there is \tau {\sim }_{\varphi }\sigma on \mathrm{\Gamma }\left(\mathrm{G}\mathrm{r}\right({\mathcal{T}}^{n-1}\varphi \left)\right).

{\mathcal{T}}^{n-1}\varphi maintains ⟦·,·⟧ bracket, and there is the following equivalence in deduction:

\begin{array}{rl}{\tau }_1{\sim }_{\varphi }{\sigma }_1,{\tau }_2{\sim }_{\varphi }{\sigma }_2& ⟹({\tau }_1,{\tau }_2)={\varphi }^{\ast }({\sigma }_1,{\sigma }_2),\\ {\tau }_1{\sim }_{\varphi }{\sigma }_1,{\tau }_2{\sim }_{\varphi }{\sigma }_2& ⟹{\tau }_1,{\tau }_2]{\sim }_{\varphi }[{\sigma }_1,{\sigma }_2],\\ \tau {\sim }_{\varphi }\sigma & ⟹\rho \left(\tau \right){\sim }_{\varphi }\rho \left(\sigma \right).\end{array}

Definition 3.2　LetF\in {\mathcal{T}}^{n-1}N, E\in {\mathcal{T}}^{n-1}M be the (n−1, k)-Dirac structure on N and M, \varphi :N\to M be smooth mapping. If for any p\in N,x\in E_{\varphi \left(p\right)}, there exists the only y\in F_p, such that y{\sim }_{\varphi }x, then define an (n−1, k)-Dirac comorphism {\mathcal{T}}^{n-1}\varphi :({\mathcal{T}}^{n-1}N,F)\to ({\mathcal{T}}^{n-1}M,E).

Definition 3.3　Let (F,{\rho }_1) and (E,{\rho }_2) be the n-Lie algebroid on manifold. If the pullback mapping of \mathrm{\Gamma }\left(E\right)\to \mathrm{\Gamma }\left(F\right) maintains bracket and anchor mapping, then n-Lie algebroid comorphism F⇢E is defined.

Assume {\mathcal{T}}^{n-1}\varphi :({\mathcal{T}}^{n-1}N,F)\to ({\mathcal{T}}^{n-1}M,E) is an (n−1, k)-Dirac comorphism. From the uniqueness of the definition, we can see that in the linear mappingE_{\varphi \left(p\right)}\to F_p, the only y\in F_p is obtained for x\in E_{\varphi \left(p\right)}, then y{\sim }_{\varphi }x. Meanwhile, it is smooth at point p, whereby the comorphism of higher order vector bundle can be defined.

Lemma 3.2　 Any (n-1,n-2)-Dirac morphism {\mathcal{T}}^{n-1}\varphi :({\mathcal{T}}^{n-1}N,F)⇢({\mathcal{T}}^{n-1}M,E) defines an n-Lie Algebroid comorphism F⇢E.

Proof　We need to prove that the pullback mapping {\varphi }^{\ast }:\mathrm{\Gamma }\left(E\right)\to \mathrm{\Gamma }\left(F\right) maintains the bracket, and the anchor mapping satisfies \mathrm{d}\varphi \left(\rho \right(y\left)\right)=\rho \left(x\right) under the condition of y{\sim }_{\varphi }x, which can be proved from the maintained bracket of {\varphi }^{\ast }:\mathrm{\Gamma }\left(E\right)\to \mathrm{\Gamma }\left(F\right).

Theorem 3.4　 Let E\subset {\mathcal{T}}^{n-1}M be an (n−1, k)-Dirac structure, \varphi :N\to M transverses across the anchor mapping of E, namely

{\mathrm{d}}_x\varphi \left(T_{x}N\right)+\rho \left(E_{\varphi \left(x\right)}\right)=T_{\varphi \left(x\right)}M,x\in M.

Then {\varphi }^{!}E=\{y\in {\mathcal{T}}^{n-1}N|\mathrm{\exists }x\in E:y{\sim }_{\varphi }x\} is an (n−1,k)-Dirac structure.

Proof　We first consider \varphi as an embedded submanifold \varphi :N\hookrightarrow M. The transversal conditions guarantee that {\varphi }^{!}E is a subbundle of {\mathcal{T}}^{n-1}N. For any y\in {\varphi }^{!}E, there exists the only x\in E:y{\sim }_{\varphi }x; we define the corresponding graph of mapping {\varphi }^{!}E\hookrightarrow E{|}_N as {\rho }^{-1}\left(TN\right)\cap E. Any given section \tau \in \mathrm{\Gamma }\left({\varphi }^{!}E\right) can be prolonged into section \sigma \in \mathrm{\Gamma }\left(E\right) in E, such that \tau {\sim }_{\varphi }\sigma. Conversely, for any given section \sigma \in \mathrm{\Gamma }\left(E\right), there exists the only \tau {\sim }_{\varphi }\sigma, such that the anchor mapping is tangent to N.

Under normal circumstances, we consider the embedded mapping of N as the graph of mapping \varphi:

j:N\to M\times N,n\mapsto \left(\varphi \left(n\right),n\right),

we get {\varphi }^{!}E=j^{!}(E\times TN)\subseteq {\mathcal{T}}^{n-1}N.

4 Nambu-Poisson (n−1, n−2)-Gauge transformation

For any n-vector field \pi, its graph Gr(\pi) is an (n−1, n−2)-Dirac structure if and only if \pi is a Nambu-Poisson structure.

Definition 4.1 [14]　A Nambu-Poisson manifold (M, \pi) is referred to as canonical if \pi is of constant rank.

Proposition 4.1　 For a canonical Nambu-Poisson manifold containing a smooth subbundle \mathrm{I}\mathrm{m}{\pi }^{\mathrm{♯}}\subset TM, we define a canonical foliation structure {\mathcal{F}}_{\pi },T{\mathcal{F}}_{\pi }=\mathrm{I}\mathrm{m}{\pi }^{\mathrm{♯}} on manifold M, and induce a symplectic structure with foliation n-form on {\mathcal{F}}_{\pi }

{\omega }_{{\mathcal{F}}_{\pi }}\in {\mathrm{\Omega }}^n\left({\mathcal{F}}_{\pi }\right)=\mathrm{\Gamma }\left({\wedge }^n{T}^{\ast }{\mathcal{F}}_{\pi }\right),

where ⟨{\omega }_{{\mathcal{F}}_{\pi }}^b\left({\pi }^{\mathrm{♯}}\left(\alpha \right)\right),{\pi }^{\mathrm{♯}}\left(\beta \right)⟩=\frac{1}{2}\left(⟨{\pi }^{\mathrm{♯}}\left(\alpha \right),\beta ⟩-⟨{\pi }^{\mathrm{♯}}\left(\beta \right),\alpha ⟩\right),\alpha ,\beta \in {\mathrm{\Omega }}^{n-1}\left(M\right).

In the (n−1, k)-Dirac structure, if \rho \left(E\right)={\mathrm{p}\mathrm{r}}_{TM}E has a constant rank, then E is called canonical Dirac manifold. We can get the following theorem.

Theorem 4.1　 There is an one-to-one correspondence between the canonical (n−1, n−2)-Dirac structure on manifold M and foliation structure {\mathcal{F}}_{\pi } in closed n-form ω on M.

Proof　The corresponding (n−1, n−2)-Dirac structure on ({\mathcal{F}}_{\pi },\omega ) is

E({\mathcal{F}}_{\pi },\omega ):=\{X+\alpha :X\in T{\mathcal{F}}_{\pi },\alpha {|}_T{\mathcal{F}}_{\pi }=i_X\omega \},

i_{u_{n-2}}\cdots i_{u_1}(X_1+i_{X_1}\omega ,X_2+i_{X_2}\omega {)}_{+}=\frac{1}{2}(i_{u_{n-2}}\cdots i_{u_1}{i}_{X_1}{i}_{X_2}\omega +i_{u_{n-2}}\cdots i_{u_1}{i}_{X_2}{i}_{X_1}\omega )=0.

There is a closed non-degenerate foliation n-form to the foliation structure {\mathcal{F}}_{\pi } shown above [25], so the one-to-one correspondence [3] between Nambu-Poisson structure and multisymplectic structure can be proved. □

There is \left({\mathcal{T}}^{n-1}\mathrm{\Phi }\right)\left(\mathrm{G}\mathrm{r}\right(\pi \left)\right)=\mathrm{G}\mathrm{r}\left(\mathrm{d}\mathrm{\Phi }\right(\pi \left)\right) for Nambu-Poisson structure (given a multivariate vector field \pi, its graph is noted as E=\mathrm{G}\mathrm{r}\left(\pi \right)\subset {\mathcal{T}}^{n-1}M). If \pi is a Nambu-Poisson structure, and {\mathcal{R}}_{\omega }\left(\mathrm{G}\mathrm{r}\right(\pi \left)\right) transverses at TM, then {\mathcal{R}}_{\omega }\left(\mathrm{G}\mathrm{r}\right(\pi \left)\right)=\mathrm{G}\mathrm{r}\left({\pi }^{\omega }\right) induces a new Nambu-Poisson structure {\pi }^{\omega }.

Lemma 4.1　 Assume \pi is a Nambu-Poisson structure on M, and \omega \in {\mathrm{\Omega }}_{\mathrm{c}\mathrm{l}}^n\left(M\right) is in closed n-form, then \mathrm{G}\mathrm{r}\left({\pi }^{\omega }\right) transverses at TM if and only if

\mathrm{i}\mathrm{d}+{\omega }^b\circ {\pi }^{\mathrm{♯}}:{\wedge }^{n-1}{T}^{\ast }M\to {\wedge }^{n-1}{T}^{\ast }M

is invertible, where the Nambu-Poisson structure {\pi }^{\omega } defined by \mathrm{G}\mathrm{r}(\pi {)}^{\omega }=\mathrm{G}\mathrm{r}({\pi }^{\omega }) satisfies

({\pi }^{\omega }{)}^{\mathrm{♯}}={\pi }^{\mathrm{♯}}\circ (\mathrm{i}\mathrm{d}+{\omega }^b\circ {\pi }^{\mathrm{♯}}{)}^{-1}.

Proof　From the definition

\mathrm{G}\mathrm{r}\left({\pi }^{\omega }\right)=\left\{{\pi }^{\mathrm{♯}}\right(\mu )+\mu +i_{{\pi }^{\mathrm{♯}}\left(\mu \right)}\omega |\mu \in {\wedge }^{n-1}{T}^{\ast }M\}.

\mathrm{G}\mathrm{r}\left({\pi }^{\omega }\right) transverses at TM if and only if the projections on {\wedge }^{n-1}{T}^{\ast }M are isomorphic, that is if and only if for any v\in {\wedge }^{n-1}{T}^{\ast }M, there exists

v=\mu +i_{{\pi }^{\mathrm{♯}}\left(\mu \right)}\omega \equiv (\mathrm{i}\mathrm{d}+{\omega }^b\circ {\pi }^{\mathrm{♯}})\mu .

Hence it is proved with ({\pi }^{\omega }{)}^{\mathrm{♯}}(\upsilon )={\pi }^{\mathrm{♯}}(\mu ).□

We call {\pi }^{\omega } the gauge transformation of \pi induced by closed n-form ω.

Lemma 4.2　 The Nambu-Poisson structures {\pi }^{\omega } and \pi define the same n-symplectic leaf, whose n-form is related to the pullback of ω.

Proof　It follows through Lemma 4.1 that the rank of ({\pi }^{\omega }{)}^{\mathrm{♯}} equals that of {\pi }^{\omega }. Let m\in M, the form on T{\mathcal{F}}_{\pi }\left(m\right)=\mathrm{I}\mathrm{m}\left({\pi }_m^{\mathrm{♯}}\right) defined by {\pi }_m is σ.

If w_1,w_2\in T{\mathcal{F}}_{\pi }\left(m\right),w_i\in {\pi }_m^{\mathrm{♯}}\left({\mu }_i\right),{\mu }_i\in {\wedge }^{n-1}{T}^{\ast }M, then

⟨\sigma \left(w_1\right),w_2⟩=-⟨{\pi }_m\left({\mu }_1\right),{\mu }_2⟩=⟨{\mu }_1,w_2⟩.

Assume {\sigma }^{\omega } is n-form defined by {\pi }^{\omega }

⟨{\sigma }^{\omega }\left(w_1\right),w_2⟩=⟨v_1,w_2⟩,

here ({\pi }^{\omega }{)}^{\mathrm{♯}}{v}_1=w_1, then

v_1=(\mathrm{i}\mathrm{d}+{\omega }^b\circ {\pi }^{\mathrm{♯}}){\mu }_1={\mu }_1+i_{w_1}\omega .

Hence ⟨\sigma \left(w_1\right),w_2⟩=⟨v_1,w_2⟩=⟨\sigma \left(w_1\right),w_2⟩=⟨\omega \left(w_1\right),w_2⟩.

From Lemma 4.2, n-symplectic foliation structure can be described with the method of (n−1,n−2)-Dirac geometry as follows.

Theorem 4.2　 Assume {\pi }_t\in {\mathfrak{X}}^n\left(M\right) is a single parameter variety of Nambu-Poisson structure, the corresponding gauge transformation is

{\pi }_t=({\pi }_0{)}^{{\omega }_t},

where {\omega }_t\in {\mathrm{\Omega }}^n\left(M\right) is in n-form with closed variety, {\omega }_0=0.

Let

\frac{\mathrm{d}{\omega }_t}{\mathrm{d}t}=-\mathrm{d}{\alpha }_t,{\alpha }_t\in {\mathrm{\Omega }}^{n-1}\left(M\right).

Define vector field X_t={\pi }_t^{\mathrm{♯}}\left({\alpha }_t\right). Let the flow of X_t be {\mathrm{\Phi }}_t, then

\left({\mathrm{d}}_t\mathrm{\Phi }\right){\pi }_t={\pi }_0.

Proof　From the definition X_t+{\alpha }_t\in \mathrm{\Gamma }\left(\mathrm{G}\mathrm{r}\right({\pi }_t\left)\right)=\mathrm{\Gamma }\left({\mathcal{R}}_{{\omega }_t}\right(\mathrm{G}\mathrm{r}\left({\pi }_0\right)\left)\right), assume {\beta }_t={\alpha }_t-i\left(X_t\right){\omega }_t, then X_t+{\beta }_t\in {\mathcal{R}}_{-{\omega }_t}\left(\mathrm{G}\mathrm{r}\right({\pi }_t\left)\right)=\mathrm{\Gamma }\left(\mathrm{G}\mathrm{r}\right({\pi }_0\left)\right). It is the section of (n−1,n−2)-Dirac structure \mathrm{G}\mathrm{r}\left({\pi }_0\right), so the flow generated by (d{\beta }_t,X_t) maintains the graph of \mathrm{G}\mathrm{r}\left({\pi }_0\right). It can be computed with Theorem 3.3 that

\left({\mathrm{d}}_t\mathrm{\Phi }\right)d{\beta }_t=\left({\mathrm{d}}_t\mathrm{\Phi }\right)(\mathrm{d}{\alpha }_t-L_{X_t{\omega }_t})=-\left({\mathrm{d}}_t\mathrm{\Phi }\right)\left(\frac{\mathrm{d}{\omega }_t}{\mathrm{d}t}+L_{X_t}{\omega }_t\right)=-\frac{\mathrm{d}}{\mathrm{d}t}\left(\right({\mathrm{d}}_t\mathrm{\Phi }\left){\omega }_t\right).

The flow is (-({\mathrm{d}}_t\mathrm{\Phi }){\omega }_t,{\mathrm{\Phi }}_t). Then

\begin{array}{rl}Gr\left({\pi }_0\right)& =(-({\mathrm{d}}_t\mathrm{\Phi }){\omega }_t,{\mathrm{\Phi }}_t).Gr\left({\pi }_0\right)\\ & ={\mathcal{R}}_{\left({\mathrm{d}}_t\mathrm{\Phi }\right){\omega }_t}\circ {\mathcal{T}}^{n-1}{\mathrm{\Phi }}_t\left(Gr\right({\pi }_0\left)\right)\\ & ={\mathcal{T}}^{n-1}{\mathrm{\Phi }}_t\circ {\mathcal{R}}_{{\omega }_t}\left(Gr\right({\pi }_0\left)\right)\\ & =Gr\left(\right({\mathrm{d}}_t\mathrm{\Phi }\left){\pi }_t\right).\end{array}

We get {\pi }_0=\left({\mathrm{d}}_t\mathrm{\Phi }\right){\pi }_t.

Definition 4.2 [27]　For the n-vector field {\pi }_N on N and n-vector field {\pi }_M on M, if {\pi }_{M_{\varphi \left(p\right)}}=({\mathrm{d}}_p\varphi {)}^{\ast }{\pi }_{N_p}, we say {\pi }_M and {\pi }_N are \varphi correlated, and \varphi is Nambu-Poisson morphism, where ({\mathrm{d}}_p\varphi {)}^{\ast }:{\wedge }^n{T}_{p}N\to {\wedge }^n{T}_{\varphi \left(p\right)}M.

Theorem 4.3　 Let (N,{\pi }_N),(M,{\pi }_M) be a Nambu-Poisson manifold, \varphi is a Nambu-Poisson morphism if and only if {\mathcal{T}}^{n-1}\varphi :({\mathcal{T}}^{n-1}N,\mathrm{G}\mathrm{r}({\pi }_N\left)\right)⇢({\mathcal{T}}^{n-1}M,\mathrm{G}\mathrm{r}({\pi }_M\left)\right) is (n−1,n−2)-Dirac morphism.

Proof　 \varphi is a Nambu-Poisson mapping if and only if

⟨{\pi }_N^{\mathrm{♯}}\left({\varphi }^{\ast }{\mu }_1\right),{\varphi }^{\ast }{\mu }_2⟩=⟨{\pi }_M^{\mathrm{♯}}\left({\mu }_1\right),{\mu }_2⟩

holds for any {\mu }_1,{\mu }_2\in {\wedge }^{n-1}{T}^{\ast }M. This is equivalent to

\mathrm{d}\varphi ({\pi }_N^{\mathrm{♯}}({\varphi }^{\ast }\mu ))={\pi }_M^{\mathrm{♯}}\left(\mu \right).

For any \mu \in {\wedge }^{n-1}{T}^{\ast }M, there is

{\pi }_N^{\mathrm{♯}}\left({\varphi }^{\ast }\mu \right)+{\varphi }^{\ast }\mu {\sim }_{\varphi }{\pi }_M^{\mathrm{♯}}\left(\mu \right)+\mu .

Which means that for any x\in \mathrm{G}\mathrm{r}({\pi }_M{)}_{\varphi \left(n\right)}, there exists the only y\in \mathrm{G}\mathrm{r}({\pi }_N{)}_n,y{\sim }_{\varphi }x□

The previous section proves that (n−1,n−2)-Dirac structure {\varphi }^{!}\mathrm{G}\mathrm{r}\left(\pi \right)\subseteq {\mathcal{T}}^{n-1}N can be pulled back with transversal conditions of \varphi, but the pulled back \pi is not a Nambu-Poisson structure. The necessary and sufficient conditions for pulled back \pi to be Nambu-Poisson structure are presented below.

Theorem 4.4　 (M,\pi ) is a Nambu-Poisson manifold, mapping \varphi :N\to M transverses at {\pi }^{\mathrm{♯}} if and only if \varphi is an immersion mapping, satisfying

TM{|}_N=TN\oplus {\pi }^{\mathrm{♯}}({\mathrm{a}\mathrm{n}\mathrm{n}}^{n-1}(TN\left)\right),{\text{ann}}^{n-1}\left(TN\right)=\{\delta \in {\wedge }^{n-1}{T}^{\ast }N|⟨X,\delta ⟩=0\},

here {\varphi }^{!}\mathrm{G}\mathrm{r}\left(\pi \right)\subset {\mathcal{T}}^{n-1}N has defined a Nambu-Poisson structure.

Proof　It can be seen from the above theorems that {\varphi }^{!}\mathrm{G}\mathrm{r}\left(\pi \right) defines a Nambu-Poisson structure if and only if it transverses at TN,TN\oplus \left({\varphi }^{!}\text{Gr}\right(\pi \left)\right)=TM{|}_N. But {\varphi }^{!}\mathrm{G}\mathrm{r}\left(\pi \right)\cap TN contains special element y\in {\mathcal{T}}^{n-1}N,y{\sim }_{\varphi }0. Let y=w+\nu ,w\in TN, \nu \in {\wedge }^{n-1}{T}^{\ast }M, which indicates that \nu =0,w\in \mathrm{k}\mathrm{e}\mathrm{r}\left(\mathrm{d}\varphi \right). Hence \mathrm{k}\mathrm{e}\mathrm{r}\left(\mathrm{d}\varphi \right) = 0

Assume \varphi is an immersion mapping, w\in TN\subseteq {\mathcal{T}}^{n-1}N, we get

w{\sim }_{\varphi }{\pi }^{\mathrm{♯}}\left(\mu \right)+\mu \in \mathrm{G}\mathrm{r}\left(\pi \right)\iff \left(\mathrm{d}\varphi \right)\left(w\right)={\pi }^{\mathrm{♯}}\left(\mu \right),\mu \in \mathrm{k}\mathrm{e}\mathrm{r}\left({\varphi }^{\ast }\right)={\mathrm{a}\mathrm{n}\mathrm{n}}^{n-1}\left(TN\right).

The condition is {\varphi }^{!}\text{Gr}\left(\pi \right)\cap TN=0\iff {\pi }^{\mathrm{♯}}\left({\text{ann}}^{n-1}\right(TN\left)\right)\cap TN=0.

5 Integrability of pullback (n−1, 0)-Dirac structure

The definition of higher order Dirac structure given in this paper [8, Definition 4.13] is a special case. From the one-to-one correspondence between higher order Dirac structure and Lie algebroid given in [8], it can be seen that the higher order Dirac structure defined in this paper also has corresponding Lie algebroid, which similarly can be integrated into (n−1)-presymplectic groupoid. Based on [7], this paper studies the integrability of the pulled high-order Dirac structure. Through the (n−1, 0)-Dirac structure pullback under transversal conditions given in Section 3, this section investigates the integrability of the pulled back (n−1, 0)-Dirac structure.

To study the integrability of (n−1, 0)-Dirac structure, we first look at the (n−1, 0)-presymplectic groupoid. Firstly the definition of (n−1, 0)-symplectic groupoid is introduced. Given a Lie groupoid \begin{array}{c}g\end{array} for any (n−1, 0)-Dirac structure E on manifold (n−1, 0)-, the n-form ω on \begin{array}{c}g\end{array} is multiplicable, if

m^{\ast }\omega =p_1^{\ast }\omega +p_2^{\ast }\omega ,

where p_i:{\mathfrak{g}}^2\to \mathfrak{g},i=1,2 is natural projection. Then (n−1)-symplectic groupoid turns out a Lie groupoid \mathfrak{g}\rightrightarrows M combining a multiplicable n-symplectic form ω.

Let A=\mathrm{k}\mathrm{e}\mathrm{r}\left(\mathrm{d}s\right){|}_M correspond to Lie algebroid on \begin{array}{c}g\end{array}, the anchor mapping is \rho =\mathrm{d}t{|}_A:A\to TM, then any closed multiplicable form \omega \in {\mathrm{\Omega }}^n\left(\mathfrak{g}\right) defines vector bundle mapping

\begin{array}{rl}& {\mu }_{\omega }:A\to {\wedge }^{n-1}{T}^{\ast }M,{\mu }_{\omega }\left(a\right)=\left(i_a\omega \right){|}_{TM},\\ & {\delta }_{\omega }:A\to {\mathcal{T}}^{n-1}M,{\delta }_{\omega }\left(a\right)=\left(\rho \right(a),{\mu }_{\omega }(a\left)\right),\end{array}

where there is one-to-one correspondence i_{u^r}\omega =t^{\ast }\left({\mu }_{\omega }\right(u\left)\right),u\in \mathrm{\Gamma }\left(A\right) between multiplicable n-form on \mathfrak{g} and IM n-form μ on A.

Definition 5.1 [1]　One (n−1)-presymplectic groupoid is a Lie groupoid \mathfrak{g}\rightrightarrows M combining a closed multiplicable n-form ω, satisfying

(1) \mathrm{k}\mathrm{e}\mathrm{r}(\omega {)}_g\cap \mathrm{k}\mathrm{e}\mathrm{r}(\mathrm{d}s{)}_g\cap \mathrm{k}\mathrm{e}\mathrm{r}(\mathrm{d}t{)}_g=\{0\};

(2) {\mathrm{d}}_{g}t(\mathrm{k}\mathrm{e}\mathrm{r}(\omega {)}_g\cap \mathrm{k}\mathrm{e}\mathrm{r}(\mathrm{d}s{)}_g)=(\mathrm{k}\mathrm{e}\mathrm{r}(\omega )\cap TM{)}_{t\left(g\right)}.

Theorem 5.1 [5]　 Assume (\mathfrak{g},\omega ) is an any source simply connected (n−1)-presymplectic groupoid on M, where ω corresponds to infinitesimal μ. Then there exists the only (n−1, 0)-Dirac structure E\subset {\mathcal{T}}^{n-1}M, where E is isomorphic to the Lie algebroid corresponding to Lie groupoid.

Theorem 5.2 [1]　 Let E be an (n−1, 0)-Dirac structure isomorphic to integrable Lie algebroid, then there exists the only source simply connected (n−1, 0)-presymplectic groupoid (\mathfrak{g},\omega ) which can be integrated by E.

In this way, (n−1, 0)-Dirac structure can define a Lie algebroid. Let the dimension of manifold N be n, there exists Nambu-Poisson structure {\pi }_N=\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_1}\wedge \cdots \wedge \frac{\mathrm{\partial }}{\mathrm{\partial }{x}_n} on {\mathcal{T}}^{n-1}N, where the graph of {\pi }_N corresponds to the (n−1, 0)-Dirac structure on N, then the Lie algebroid defined by the (n−1, 0)-Dirac structure is integrable. If it is integrated into (n−1)-presymplectic groupoid, the integrability of (n−1)-Dirac structure can be converted into a study of the integrability of the Lie algebroid defined by itself.

Assume (\mathfrak{g},\omega ) is the (n−1)-presymplectic groupoid on M, there exists anti-Lie algebroid isomorphism in the corresponding Lie algebroid isomorphism {\delta }_{\omega }:A\to E

{\delta }_{\omega }^{\mathrm{o}\mathrm{p}}:A^{\mathrm{o}\mathrm{p}}\to E^{\mathrm{o}\mathrm{p}},a\mapsto \left({\rho }^{\mathrm{o}\mathrm{p}}\left(a\right),{\mu }_{\omega }^{\mathrm{o}\mathrm{p}}\left(a\right)\right),

where {\mu }_{\omega }^{\mathrm{o}\mathrm{p}}:A^{\mathrm{o}\mathrm{p}}\to {\wedge }^{n-1}{T}^{\ast }M. If A\cong E, then A^{\mathrm{o}\mathrm{p}}\cong E^{\mathrm{o}\mathrm{p}}, conduct mapping (X,\alpha )\mapsto (X,-\alpha ), and normally we assume it is identical.

The kernel of (n−1, 0)-Dirac structure E is defined as \mathrm{k}\mathrm{e}\mathrm{r}\left(E\right):=E\cap TM\subseteq TM.

Lemma 5.1　 There exists one Nambu-Poisson structure if and only if \mathrm{k}\mathrm{e}\mathrm{r}\left(E\right)=0 on E.

Proof　Let the kernel of (n−1, 0)-Dirac structure E be 0, then E=({\pi }^{\mathrm{♯}}\alpha ,\alpha )\cap TM=0\Rightarrow {\pi }^{\mathrm{♯}}=0, so that there exists one Nambu-Poisson structure \pi on E. On the contrary, let {\pi }^{\mathrm{♯}}\left({\alpha }_1\right)=X,{\pi }^{\mathrm{♯}}\left({\alpha }_2\right)=X, then {\pi }^{\mathrm{♯}}({\alpha }_1-{\alpha }_2)=0 contains {\alpha }_1={\alpha }_2,\mathrm{k}\mathrm{e}\mathrm{r}\left(E\right)=0. □

There is a closed n-form on ω, where \mathrm{k}\mathrm{e}\mathrm{r}\left(E\right)=\mathrm{k}\mathrm{e}\mathrm{r}\left(\omega \right).

E_2{|}_{\varphi \left(x\right)}=\left\{\right(\mathrm{d}\varphi \left(X\right),\beta \left)\right|(X,{\varphi }^{\ast }\beta )\in E_1{|}_x\},

and when \mathrm{k}\mathrm{e}\mathrm{r}\left(\mathrm{d}\varphi \right)\cap \mathrm{k}\mathrm{e}\mathrm{r}\left(E_1\right)=\left\{0\right\}, mapping \varphi :M_1\to M_2 is a push-forward (n−1, 0)-Dirac morphism, hence \varphi is called strong (n−1, 0)-Dirac mapping.

Definition 5.2　Assume (M_1,E_1),(M_2,E_2) is an (n−1, 0)-Dirac manifold, if \mathrm{\forall }x\in M_1,

E_2{|}_{\varphi \left(x\right)}=\left\{\right(\mathrm{d}\varphi \left(X\right),\beta \left)\right|(X,{\varphi }^{\ast }\beta )\in E_1{|}_x\},

the following presents the relationship between the kernel of (n−1, 0)-Dirac structure and integrable (n−1, 0)-presymplectic form.

Lemma 5.2　 Let source simply connected (n−1)-presymplectic groupoid (\mathfrak{g},\omega ) be the integral of Lie algebroid on (n−1, 0)-Dirac structure E. Then,

(1) the mapping (t,s):\mathfrak{g}\to M\times M^{\mathrm{o}\mathrm{p}}is a strong Dirac map (there exists anti-higher order Dirac structure on M^{\mathrm{o}\mathrm{p}});

(2) we can obtain that under identity mapping {\delta }_{\omega }:A\to E,{\delta }_{\omega }^{\mathrm{o}\mathrm{p}}:A^{\mathrm{o}\mathrm{p}}\to E^{\mathrm{o}\mathrm{p}}

\mathrm{k}\mathrm{e}\mathrm{r}\left(\omega \right){|}_g=\{a^r+\mathrm{i}\mathrm{n}\mathrm{v}(b{)}^l\mid a\in \mathrm{k}\mathrm{e}\mathrm{r}\left(E\right){|}_{t\left(g\right)},b\in \mathrm{k}\mathrm{e}\mathrm{r}\left(E\right){|}_{s\left(g\right)},\mathrm{i}\mathrm{n}\mathrm{v}:E\to E^{\mathrm{o}\mathrm{p}}\}.

Proof　Assume g\in \mathfrak{g},s\left(g\right)=x,t\left(g\right)=y. For a\in A{|}_y,b\in A^{\mathrm{o}\mathrm{p}}{|}_x, according to the definition of Li algebroid anchor and \mathrm{d}t\left(a^r\right){|}_g=\rho \left(a\right){|}_y,\mathrm{d}s\left(b^l\right){|}_g={\rho }^{\mathrm{o}\mathrm{p}}\left(b\right){|}_x and multiplicability of multisymplectic structure [8], it can be derived that

i_{a^r}\omega =t^{\ast }\left({\mu }_{\omega }\right(a\left)\right),i_{b^l}\omega =s^{\ast }\left({\mu }_{\omega }^{\mathrm{o}\mathrm{p}}\right(b\left)\right).

Considering mapping A\to E,A^{\mathrm{o}\mathrm{p}}\to E^{\mathrm{o}\mathrm{p}}, let A\to E,A^{\mathrm{o}\mathrm{p}}\to E^{\mathrm{o}\mathrm{p}}, for Z\in T_g\mathfrak{g}, let Z=(a^r+b^l){|}_g satisfy (\mathrm{d}t,\mathrm{d}s)\left(Z\right)=(X,Y). Then

i_Z\omega =i_{a^r}\omega +i_{b^l}\omega =t^{\ast }\alpha +s^{\ast }\beta =(t,s{)}^{\ast }(\alpha ,\beta ),

which proves that (t,s):\mathfrak{g}\to M\times M^{\mathrm{o}\mathrm{p}} is a Dirac mapping, and according to the Definition 5.2, it can be seen that the mapping is a strong (n−1, 0)-Dirac mapping.

If mapping \varphi is an (n−1, 0)-Dirac mapping, then \mathrm{k}\mathrm{e}\mathrm{r}\left(E_2\right){|}_{\varphi \left(x\right)}=\mathrm{d}\varphi (\mathrm{k}\mathrm{e}\mathrm{r}(E_1){|}_x), and the condition of strong (n−1, 0)-Dirac mapping is equivalent to \mathrm{d}\varphi :\mathrm{k}\mathrm{e}\mathrm{r}\left(E_1\right){|}_x\to \mathrm{k}\mathrm{e}\mathrm{r}\left(E_2\right){|}_{\varphi \left(x\right)}, so it is isomorphic. The strong (n−1, 0)-Dirac mapping \varphi :(M_1,E_1)\to (M_2,E_2) can induce the function of Lie algebroid on manifold M_1

{\varphi }^{\ast }:E_2\to T{M}_1,(Y,\beta ){|}_{\varphi \left(x\right)}\mapsto X,

where X is well-determined by Y=\mathrm{d}\varphi \left(X\right). This mapping is restricted to {\varphi }^{\ast }\mathrm{k}\mathrm{e}\mathrm{r}\left(E_2\right)\to \mathrm{k}\mathrm{e}\mathrm{r}\left(E_1\right) on vector bundle. Hence the Lie algebroid function of E\times E^{\mathrm{o}\mathrm{p}} can be induced on groupoid \mathfrak{g}.

t^{\ast }E\times s^{\ast }{E}^{\mathrm{o}\mathrm{p}}\to T\mathfrak{g},(a,b)\mapsto a^r+b^l,

so as to define an isomorphic mapping \mathrm{k}\mathrm{e}\mathrm{r}\left(E\right){|}_{t\left(g\right)}\times \mathrm{k}\mathrm{e}\mathrm{r}\left(E^{\mathrm{o}\mathrm{p}}\right){|}_{s\left(g\right)}\to \mathrm{k}\mathrm{e}\mathrm{r}\left(\omega \right){|}_g, so the proposition is proved.□

From Section 3, it can be known that (n−1, 0)-Dirac structure can be pulled back into (n−1,0)-Dirac structure under transversal condition. The following mainly studies the integrability of pullback (n−1,0)-Dirac structure.

The function of Lie group H on Nambu-Poisson manifold M is written as q:H\times M\to M,(h,m)\mapsto h\cdot m. If the diffeomorphism q_h:M\to M of any h\in H is a Nambu-Poisson morphism, then q is called the Nambu-Poisson function. On the n-symplectic manifold (M, ω), if q is the Nambu-Poisson function, then equation q_h^{\ast }\omega =\omega is established. We study from the principle H-bundle.

Definition 5.3 [19]　The Nambu-Poisson function of Lie group H on is written as q. If q is free, appropriate and reducible, then a principal H-bundle can be defined

\pi :M\to M/H.

Theorem 5.3 [19]　 A Lie group function is free, appropriate and reducible on Nambu-Poisson manifold M. Then there exists the Nambu-Poisson structure ({\wedge }_{\mathrm{r}\mathrm{e}\mathrm{d}}^{n-1}{T}^{\ast }M,{\pi }_{\mathrm{r}\mathrm{e}\mathrm{d}}) on M/H, and the quotient mapping Π is Nambu-Poisson morphism.

Assume H is a Lie group, its corresponding Lie algebra is \mathfrak{h}, consider the Lie algebroid function on manifold M on principal H-bundle \pi :M\to M/H, the corresponding anchor mapping is {\rho }_M:\mathfrak{h}\times M\to TM, where \mathrm{I}\mathrm{m}\left({\rho }_M\right)=\mathrm{k}\mathrm{e}\mathrm{r}(\mathrm{d}\pi :TM\to T(M/H)). Assume F is any (n−1, 0)-Dirac structure on M/H, which is still an (n−1, 0)-Dirac structure after pullback from mapping Π. The definition is as follows:

E={\pi }^{!}F=\left\{\right(X,{\pi }^{\ast }\beta )\in F\}\subset T^{n-1}M.

Let F\mapsto E=\left({\pi }^{!}F\right). Then there is one-to-one correspondence between the (n−1, 0)-Dirac structure on M/H and H-invariant (n−1, 0)-Dirac structure \mathrm{I}\mathrm{m}\left({\rho }_M\right)\subset \mathrm{k}\mathrm{e}\mathrm{r}\left(E\right) on M, and under the correspondence F is the push forward of E:

F={\pi }_{!}E:=\left\{\right(\mathrm{d}\mathrm{i}\mathrm{l}\left(X\right),\alpha )\mid (X,{\pi }^{\ast }\alpha )\in E\}\subset {\mathcal{T}}^{n-1}(M/H),

corresponding to \mathrm{k}\mathrm{e}\mathrm{r}\left(F\right)=\mathrm{d}\pi (\mathrm{k}\mathrm{e}\mathrm{r}(E\left)\right),\mathrm{k}\mathrm{e}\mathrm{r}\left(E\right)=\left(\mathrm{d}\pi {)}^{-1}\right(\mathrm{k}\mathrm{e}\mathrm{r}\left(F\right)). In particular, F is the graph of Nambu-Poisson structure on M/H if and only if the higher order Dirac structure E=\left({\pi }^{!}F\right) is H-invariant on M, and \mathrm{k}\mathrm{e}\mathrm{r}\left(E\right)=\mathrm{I}\mathrm{m}\left({\rho }_M\right),

Assume (E,\psi ) is the H-Lie algebroid on M, morphism \psi :\mathfrak{h}\times M\to E,(u,x)\mapsto (u_M{|}_x,0), \mathfrak{g} is a Lie groupoid integrated by E on manifold M.

Assume \mathrm{\Psi }:H⋉M\to \mathfrak{g} is a Lie groupoid morphism obtained by Lie algebroid morphism \psi integration.

The (n−1, 0)-Dirac structure on M/H can be defined as an H-Lie algebroid ({\pi }^{!}F,\psi ), so the integrability of ({\pi }^{!}F,\psi ) can be studied with that of (n−1, 0)-Dirac structure on M/H.

Definition 5.4　Assume \mathfrak{g} is a Lie groupoid on M. It can be obtained from (Lie algebroid) integral E that there exists the Lie algebroid morphism\psi :\mathfrak{h}\times M\to E that can be integrated into a Lie groupoid morphism \mathrm{\Psi }:H⋉M\to \mathfrak{g}. Then the Lie algebroid T_{\pi }^{\ast }(M/H) defined by Nambu-Poisson structure can be integrated into a Lie groupoid \overline{\mathfrak{g}}=\mathfrak{g}/(H\times H).

Definition 5.5　Let F be a higher order Dirac structure on M/H. If the Lie algebroid morphism can be integrated into a Lie groupoid morphism , and the H\times H function ω on \mathfrak{g} remains H\times H invariant, then the presymplectic groupoid (\mathfrak{g},\omega ) integrated by the pullback higher order Dirac structure is called H-compatible.

Define a quotient Lie groupoid \overline{\mathfrak{g}}=\mathfrak{g}/(H\times H) and a quotient mapping p:\mathfrak{g}\to \overline{\mathfrak{g}}, on manifold M/H, which is also a submerged sum groupoid morphism.

Theorem 5.4　 Assume F is an (n−1, 0)-Dirac structure on M/H, and E=\left({\pi }^{!}F\right) is the pullback (n−1, 0)-Dirac structure on M. If the H-compatible (n−1)-presymplectic groupoid (\mathfrak{g},\omega ) is obtained through E integration, there exists the only pre(n−1)-symplectic structure \overline{\omega } such that p^{\ast }\overline{\omega }=\omega on quotient Lie groupoid \overline{\mathfrak{g}}=\mathfrak{g}/(H\times H). The unique source simply connected (n−1)-presymplectic groupoid (\overline{\mathfrak{g}},\overline{\omega }) can be obtained through F integration.

Proof　Firstly we deduce the function of H\times H on \mathfrak{g} through that of H\times H on \mathfrak{g},

{\rho }_{\mathfrak{g}}:(\mathfrak{h}\times \mathfrak{h})\times \mathfrak{g}\to T\mathfrak{g},{\rho }_{\mathfrak{g}}(u,v){|}_g={\left(\psi (u,t(g\left)\right)\right)}^r+{\left(\mathrm{i}\mathrm{n}\mathrm{v}\left(\psi \right(v,s\left(g\right)\left)\right)\right)}^l.

It can be known that from \mathrm{I}\mathrm{m}\left({\rho }_M\right)\subset \mathrm{k}\mathrm{e}\mathrm{r}\left(E\right) and Lemma 5.2, {\rho }_{\mathfrak{g}}(\mathfrak{h}\times \mathfrak{h})\subset \mathrm{k}\mathrm{e}\mathrm{r}\left(\omega \right) and the H\times H function of ω remains the same, then ω has a H\times H basis, and there exists \omega =p^{\ast }\overline{\omega } corresponding to the only n-form \overline{\omega } on \overline{\mathfrak{g}}. Since \pi :\mathfrak{g}\to \overline{\mathfrak{g}} is a groupoid morphism, ω is multiplicable, so that we get \mathrm{d}\mathrm{i}\mathrm{m}(\mathfrak{g}/(H\times H\left)\right)=2\mathrm{d}\mathrm{i}\mathrm{m}(M/H). Finally, let \overline{A} is the Lie algebroid of \overline{\omega }, the corresponding mapping is {\delta }_{\overline{\omega }}:\overline{A}\to {\mathcal{T}}^{n-1}(M/H). {\delta }_{\omega } is injective, so \mathrm{k}\mathrm{e}\mathrm{r}\left({\delta }_{\overline{\omega }}\right)=0 can be deduced from \mathrm{k}\mathrm{e}\mathrm{r}\left({\delta }_{\omega }\right)=0. In fact, assume \overline{\alpha }\in \mathrm{k}\mathrm{e}\mathrm{r}\left({\delta }_{\overline{\omega }}\right), we can get \overline{\rho }\left(\overline{\alpha }\right)=\mathrm{d}\pi \left(\rho \right(a\left)\right)=0,{\pi }^{\ast }{\mu }_{\overline{\omega }}\left(\overline{\alpha }\right)=0, where for any α, there is \overline{\alpha }=\mathrm{d}\pi \left(a\right). Then in \begin{array}{c}\rho \left(a\right)=\rho \left(\psi \right(u\left)\right)\end{array}, transform α into a^{\prime }=a-\psi \left(u\right), such that \mathrm{d}p\left(a^{\prime }\right)=\overline{\alpha }, \mu \left(a^{\prime }\right)=0, \rho \left(a^{\prime }\right)=0, hence a^{\prime }=0, \overline{\alpha }=0. Therefore, (\overline{\mathfrak{g}},\overline{\omega }) obtained through (M/H,F) integration is a presymplectic groupoid. □

6 Conclusions

On the basis of previous researches, this paper studies with a higher order perspective. Based on the automorphism groups and automorphism Lie algebra on higher order Courant algebroid, the gauge changes of single parameter variety of the Nambu-Poisson structure have been obtained. We have also established the relationship between the (n−1, n−2)-Dirac structure and the Nambu-Poisson structure under smooth morphism, and proved the integrability of the pullback (n−1, 0)-Dirac structure under quotient mapping.

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